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Slices of co-operations for KGL

Slices of co-operations for KGL We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic $$K$$-theory. Introduction Fix a finite-dimensional Noetherian separated base scheme $$S$$, and consider the motivic stable homotopy category $${\cal {{SH}}} (S)$$ as defined in [18]. We write $$KGL$$ for the motivic spectrum representing homotopy invariant $$K$$-theory in $${\cal {{SH}}} (S)$$. In this paper, we study Voevodsky's conjecture on slices of co-operations for $$KGL$$ (see [19, Conjecture 8]), which describes the motivic slices of the motivic spectra $$KGL\wedge \ldots \wedge KGL$$. We verify the conjecture when $$S$$ is smooth over a perfect field (see Theorem 0.1). To describe the conjecture, recall that for any motivic spectrum $$E$$, Voevodsky introduced a natural ‘slice’ tower $$\cdots \to f_{q+1}E\to f_qE\to \cdots$$ of motivic spectra, and defined triangulated ‘slice’ functors $$s_q$$ fitting into cofibration sequences $$f_{q+1}E\to f_qE\to s_qE$$. Roughly, $${\{ } f_qE{\} }$$ is the analogue of the Postnikov tower in topology; $$s_q$$ is the analogue of the functor $$X\mapsto K(\pi _qX,q)$$. If $$E$$ is a topological ring spectrum, the ring $$E_*E=\pi _* (E\wedge E)$$ is called the ring of co-operations for $$E$$; the name comes from [3], where $$\pi _* (KU\wedge KU)$$ is worked out (see Section 1). The title of this paper comes from viewing the motivic spectrum $$KGL\wedge KGL$$ as giving rise to co-operations for $$K$$-theory. The easy part of Voevodsky's Conjecture 8 says that there is an isomorphism   \[s_q(KGL\wedge KGL) \cong (T^q\wedge H {\Bbb {Z}}) \otimes \pi _{2q}(KU\wedge KU).\] Here, $$H {\Bbb {Z}}$$ is the motivic Eilenberg–Mac  Lane spectrum in $${\cal {{SH}}} (S)$$, $$T$$ is the motivic space represented by the pointed projective line, and the tensor product of a spectrum with an abelian group has its usual meaning (see Notation 0.5). Since a ring spectrum $$E$$ is a monoid object in spectra, co-operations fit into a cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} E = E^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. This is a general construction: in any category with product $$\wedge$$, a monoid $$E$$ determines a triple $$\top X=X\wedge E$$ and an augmented cosimplicial object $$n\mapsto X\wedge E^{\wedge n+1}$$ for any object $$X$$; the cofaces are given by the unit $${{\bf{1}}} \to E$$ and the codegeneracies are given by the product $$E\wedge E\to E$$; see [23, 8.6.4]. We write $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} E$$ when $$X= {{\bf{1}}}$$, so $$N^{n}E = E^{\wedge n+1}$$. Voevodsky's Conjecture 8 (Slices of co-operations for $$KGL$$) As cosimplicial motivic spectra, the $$q$$th slice satisfies   \[s_q N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL) \cong (T^q\wedge H {\Bbb {Z}}) \otimes \pi _{2q} N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU).\] One reformulation of this conjecture is to take the direct sum over $$q$$ and use the fact that $$s_*E=\bigoplus s_q(E)$$ is a graded motivic ring spectrum (see [11, 3.6.13]). It is convenient to adopt the notation that $$E \otimes A_* $$ denotes $$\bigoplus _q (T^q\wedge E) \otimes A_q$$ for a motivic spectrum $$E$$ and a graded ring $$A_* $$. In this notation, we prove: Theorem 0.1 Assume that$$S$$is smooth over a perfect field. Then there is an isomorphism of cosimplicial ring spectra  \[s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL) \cong H {\Bbb {Z}} \otimes \pi _{2* }N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU).\] Remark If $$S$$ is over a field of characteristic 0, it seems likely that Theorem 0.1 would follow from the work of Spitzweck [16] on Landweber exact spectra. Such an approach would depend heavily on the Hopkins–Morel–Hoyois theorem [7]. Conjecture 8 is intertwined with Voevodsky's Conjectures 1, 7 and 10 in [19], that $$H {\Bbb {Z}} \leftarrow {{\bf{1}}} \to KGL$$ induces isomorphisms   \[H {\Bbb {Z}} \cong s_0(H {\Bbb {Z}}) \,{\mathop {\longleftarrow }\limits ^{{\cong } }}\, s_0({{\bf{1}}}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_0(KGL),\] and thus that $${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))\cong H^0(S, {\Bbb {Z}})$$. These are known to hold when the base $$S$$ is smooth over a perfect field by the work of Voevodsky and Levine (see [9, 10.5.1 and 11.3.6; 21]), or singular over a field of characteristic 0 (see [13]). Here is our main result, which evidently implies Theorem 0.1. Theorem 0.2 Let$$S$$be a finite-dimensional separated Noetherian scheme. Then we have the following. There are isomorphisms for all$$n\ge 0$$:  \[s_0(KGL) \otimes \pi _{2* }(KU^{\wedge n}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n}).\]These isomorphisms commute with all of the coface and codegeneracy operators except possibly$$\partial ^0$$and$$\sigma ^0$$. Assume in addition that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism in$${\cal {{SH}}} (S),$$and that$${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))$$is torsion-free. Then the maps in (a) are the components of an isomorphism of graded cosimplicial motivic ring spectra$${:}$$  \[s_0(KGL) \otimes \pi _{2* }N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL).\] The case $$n=0$$ of Theorem 0.2(a), that $$s_0(KGL) \otimes \pi _{2* }KU\cong s_*KGL$$, is immediate from the periodicity isomorphism $$T\wedge KGL\cong KGL$$ defining the motivic spectrum $$KGL$$ and the formula $$\pi _{2* }KU\cong {\Bbb {Z}} [u,u^{-1}]$$. The need to pass to slices is clear at this stage, because $$\pi _{2n,n}KGL\cong K_n(S)$$ for $$S$$ smooth over a perfect field. The left side of Theorem 0.2 is algebraic in nature, involving only the cosimplicial ring $$\pi _{2* }(KU^{\wedge n+1})$$ and $$H=KU_0(KU)$$. In fact, $$\pi _{2* }(KU^{\wedge n+1})$$ is the cobar construction $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_\Gamma (R,R)\cong KU_* \otimes H^{ \otimes n}$$ for the Hopf algebroid $$(R,\Gamma )=(KU_* ,KU_*KU)$$. We devote the first four sections to an analysis of this algebra, focussing on the rings $$F=KU_0({\Bbb {C}} {\mathbb P}^{\infty })$$ and $$H$$. Much of this material is well known, and due to Frank Adams [1]. If $$E$$ is an oriented motivic spectrum, the projective bundle theorem says that $$E\wedge {\mathbb P}^{\infty } \cong E \otimes F$$ and hence that $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n}\cong E \otimes F^{ \otimes n}$$. Using this, we establish a toy version of Theorem 0.2 in Propositions 5.8 and 6.4 that, as cosimplicial spectra, $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KGL \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ (the cobar construction on $$F$$) and   \[s_* (KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_* (KGL) \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_0(KGL) \otimes KU_* ({ {\Bbb {C}} {\mathbb P}^{\infty } }^{\wedge n}). \] (0.3) Using a theorem of Snaith [5, 17], we use the toy model to show that $$KGL\wedge KGL\cong KGL \otimes H$$ and more generally (in Corollary 7.6) that   \[KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2} = KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \] (0.4) as cosimplicial spectra. Taking slices in (0.4) gives the isomorphisms in Theorem 0.2(a), and (with a little decoding) also proves compatibility with every coface and codegeneracy map except for $$\partial ^0$$ and $$\sigma ^0$$. This proves Theorem 0.2(a). Compatibility with the coface maps $$\partial ^0$$ is established in Proposition 8.5, using the extra hypothesis that $$s_0({{\bf{1}}})\cong s_0(KGL)$$, and compatibility with the codegeneracy map $$\sigma ^0$$ is established in Lemma 8.7, using the torsion-free hypothesis. This proves Theorem 0.2(b). The paper is organized as follows. In Sections 1 and 2, we introduce the binomial rings $$F$$ and $$H$$. As noted in Remark 1.7, $$KU_* ({\Bbb {C}} {\mathbb P}^{\infty })=KU_* \otimes F$$ and $$KU_* (KU)=KU_* \otimes H$$. In Section 3, we quickly review Hopf algebroids and the algebroid structure on $$(KU_* ,KU_*KU)$$. In Section 4, we recall how $$\pi _*N^{ {{{\bf{\scriptscriptstyle \bullet }}}} }KU$$ is the cobar complex for this algebroid, by showing that $$KU\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KU \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$, where $$C^{ {{{\bf{\scriptscriptstyle \bullet }}}} }(F,F)$$ is the cobar complex of the Hopf algebra $$F$$. Most of this material is due (at least in spirit) to Adams. In Section 5, we show that $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$, by mimicking the development of Section 4. In Section 6, we show that the slice functors commute with direct sums, and deduce (0.3). Section 7 establishes (0.4), and Theorem 0.2 is established in Section 8. Notation 0.5 We will write $$S$$ for a Noetherian separated base scheme of finite Krull dimension, and $${\cal {{SH}}} (S)$$ or simply $${\cal {{SH}}}$$ for the Morel–Voevodsky motivic stable homotopy category of $$T$$-spectra. For any graded abelian group $$A$$ and motivic spectrum $$E$$, we can form a motivic spectrum $$E \otimes A$$, as follows. If $$A$$ is free with a basis of elements $$a_i$$ in degrees $$d_i$$, then $$E \otimes A$$ is the wedge of the $$E\wedge T^{d_i}$$, and we may regard $$a_i$$ as a map from $$E\wedge T^{d_i}$$ to $$E \otimes A$$. For general $$A$$, choose a free graded resolution $$0\to P_1\to P_0\to A\to 0$$ and define $$E \otimes A$$ to be the cofibre of $$E \otimes P_1\to E \otimes P_0$$; Shanuel's Lemma implies that this is independent of the choice of resolution, up to isomorphism. The choice of a lift of a homomorphism $$A\to B$$ is unique up to chain homotopy, so it yields a map $$E \otimes A\to E \otimes B$$ unique up to homotopy. Given homomorphisms $$A\to B\to C$$, this yields a homotopy between the composition $$E \otimes A\to E \otimes B\to E \otimes C$$ and $$E \otimes A\to E \otimes C$$. That is, this construction gives a lax functor from abelian groups to strict motivic spectra over $$E$$, and a functor to motivic spectra over $$E$$. There is a natural associative map $$({{\bf{1}}} \otimes A)\wedge ({{\bf{1}}} \otimes B)\to {{\bf{1}}} \otimes A \otimes B$$, at least if $$A$$ and $$B$$ have no summands $${\Bbb {Z}} /2$$, $${\Bbb {Z}} /3$$ or $${\Bbb {Z}} /4$$ [24, IV.2.8]; it is an isomorphism if $$A$$ and $$B$$ are free abelian groups. Thus, if $$E$$ is a ring spectrum and $$A$$ is a ring, the composition $$(E \otimes A)\wedge (E \otimes A)\to E \otimes A$$ makes $$E \otimes A$$ into a ring spectrum. 1. Universal binomial rings Recall that a binomial ring is a subring $$R$$ of a $${\Bbb {Q}}$$-algebra which is closed under the operations $$r\mapsto \binom {r}{n}$$. It is a $$\lambda$$-ring with operations $$\lambda ^n(r) = {\binom rn}$$. For example, consider the subring $$F$$ of $${\Bbb {Q}} [t]$$ consisting of numerical polynomials, polynomials $$f(t)$$ with $$f(n)\in {\Bbb {Z}}$$ for all integers $$n\gg 0$$. It is well known that $$F$$ is free as an abelian group, and that the $$\alpha _n=\binom tn$$ form a basis. It is not hard to verify the formula that   \[t{\binom tn} = n{\binom tn} + (n+1){\binom t{n+1}}. \] (1.1) The general ring structure of $$F$$ is determined by the combinatorial identity:   \[\alpha _i * \alpha _j = \sum _{k\le i+j}\binom {k}{k-i,k-j,i+j-k}\alpha _k. \] (1.2) Here, $$\binom {k}{a,b,c}$$ denotes $$k!/a!b!c!$$. (To derive (1.2), note that the left-hand side counts pairs of subsets of a set with $$t$$ elements. If the union of an $$i$$-element set and a $$j$$-element set has $$k$$ elements, the sets intersect in $$i+j-k$$ elements.) The universal polynomials for $$\lambda ^m(\lambda ^n(r))$$ show that the numerical polynomials form a binomial ring. In fact, $$F$$ is the free binomial ring on one generator $$t$$: if $$R$$ is binomial and $$r\in R$$, the canonical extension of the universal ring map $${\Bbb {Z}} [t]\to R$$ to $$F\to R\otimes {\Bbb {Q}}$$ factors uniquely through a map $$F\to R$$. Definition 1.3 Let $$H$$ denote the localization $$F[1/t]$$ of the ring of numerical polynomials; it is a subring of $${\Bbb {Q}} [t,1/t]$$. Here is a useful criterion for membership in $$H$$. Lemma 1.4 $$H=F[1/t]$$is the ring of all$$f(t)\in {\Bbb {Q}} [t,1/t]$$such that for any positive integer$$a$$we have$$f(a)\in {\Bbb {Z}} [1/a]$$. Proof (Cf. [3, 5.3]) Multiplying $$f$$ by a suitable power of $$t$$, we may assume that $$f(t)\in t^\nu {\Bbb {Q}} [t]$$, where $$\nu$$ is the highest exponent of any prime occurring in the denominators of the coefficients of $$f$$. It suffices to show that $$f$$ is a numerical polynomial. Fix $$a>0$$ and let $$p$$ be a prime. If $$p|a,$$ then $$p$$ does not appear in the denominator by construction; if $$p\nmid a$$ then $$p$$ does not appear in the denominator of $$f(a)$$ by hypothesis. Hence, $$f(a)\in {\Bbb {Z}}$$, as desired. Recall that $${\Bbb {Q}} [t,1/t]$$ is a Hopf algebra with $$\Delta (t)=t_1t_2$$. The usual proof [6, I.7.3] that the functions $$\binom tn$$ form a basis of the ring $$F$$ of numerical polynomials $$f(t)$$ is easily modified to show that the functions $$\binom {t_1}{m}\binom {t_2}{n}$$ form a $${\Bbb {Z}}$$-basis of the ring of numerical polynomials $$f(t_1,t_2)$$ in $${\Bbb {Q}} [t_1,t_2]$$. Identifying $${\Bbb {Q}} [t_1,t_2]$$ with $${\Bbb {Q}} [t] \otimes {\Bbb {Q}} [t]$$, we obtain a canonical isomorphism between the subring $$F\otimes F$$ of $${\Bbb {Q}} [t] \otimes {\Bbb {Q}} [t]$$ and the ring of numerical polynomials in $${\Bbb {Q}} [t_1,t_2]$$. Theorem 1.5 $$H$$is a Hopf subalgebra of$${\Bbb {Q}} [t,1/t]$$. $$H$$is a binomial ring; it is the free binomial ring on a unit. Proof For (a), it suffices to show that $$\Delta : {\Bbb {Q}} [t]\to {\Bbb {Q}} [t_1,t_2]$$ sends $$F[1/t]\subset {\Bbb {Q}} [1/t]$$ into the subring $$F[1/t]\otimes F[1/t]$$. But $$\Delta$$ sends $$\binom tn$$ to $$f(t_1,t_2)=\binom {t_1t_2}{n}$$, which is a numerical polynomial and hence belongs to $$F\otimes F$$. Thus, $$\Delta$$ sends $$t^{-k}\binom {t}{n}$$ to a $${\Bbb {Z}}$$-linear combination of the functions $$(t_1t_2)^{-k}\binom {t_1}{m}\binom {t_2}{n}$$, which is in $$H \otimes H$$. For (b), fix $$f(t)/t^m$$ in $$F[1/t]$$, and set $$g_k(t)=\lambda ^k(f/t^m)$$. For each non-zero $$a\in {\Bbb {Z}}$$, $${\Bbb {Z}} [1/a]$$ is a binomial ring, so $$g_k(a)= \lambda ^k(f(a)/a^m)$$ is in $${\Bbb {Z}} [1/a]$$. Theorem 1.6 As an abelian group,$$H$$is free. Proof For $$m\le n$$, let $$F(m,n)$$ denote the intersection of $$H$$ with the $${\Bbb {Q}}$$-span of $$t^m,\ldots ,t^n$$ in $${\Bbb {Q}} [t,1/t]$$. Then the proof of [2, 2.2] goes through to prove that $$F(m,n)\cong R^{1+n-m}$$ and that $$H$$ is free abelian, given the following remark: For each $$k,m,n$$, $$\binom {kt}n$$ is a numerical polynomial, so $$(kt)^{-m}{\binom {kt}n}$$ is certainly in $$F[1/t][1/k]$$. Remark 1.7 The ring $$KU_{\ast } \otimes F$$ is $$KU_* ({\Bbb {C}} {\mathbb P}^{\infty })$$, and $$KU_{\ast } \otimes H$$ is isomorphic to the ring $$KU_0(KU)$$. These observations follow from [1, II.3; 3, 2.3, 4.1, 5.3]. A $${\Bbb {Z}}$$-basis of $$KU_* (KU)$$ was given in [8, Corollary 13]. Since we will be interested in the algebras $$R\otimes F[1/t]$$ over different base rings $$R$$, we now give a slightly different presentation of $$F_R=R\otimes F$$ and $$F_R[1/t]$$. As an $$R$$-module, $$F_R$$ is free with countable basis $${\{ } \alpha _{0}, \alpha _{1}, \ldots {\} }$$, and we are given an $$R$$-module map $$T:F_R\to F_R$$ (multiplication by $$t$$), which by (1.1) is defined as   \[T(\alpha _n)= n\alpha _n + (n+1)\alpha _{n+1}. \] (1.8) Note that $$T(\alpha _0)=\alpha _1$$. The localization $$F_R[1/t]$$ is the colimit of the system   \[F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, \cdots . \] (1.9) To describe it, we introduce the bookkeeping index $$t^{-j}$$ to indicate the $$j$$th term in this sequence. By [23, 2.6.8], there is a short exact sequence of $$R$$-modules:   \[0\longrightarrow \bigoplus _{j=0}^{\infty }F_R t^{-j} \, {\mathop {\longrightarrow }\limits ^{{\Phi } }}\, \bigoplus _{j=0}^{\infty }F_R t^{-j} \longrightarrow F_R[1/t] \longrightarrow 0,\] where $$\Phi (\alpha _{n}t^{-j})=T(\alpha _n)t^{-j-1}-\alpha _n t^{-j}$$. Since $$\Phi$$ is $$t^{-1}$$-linear, we may regard $$\Phi$$ as an endomorphism of the free $$R[t^{-1}]$$-module $$F_R[t^{-1}]=\bigoplus F_R t^{-n}$$ with basis $${\{ }\alpha _m{\} }$$, with   \[\Phi (\alpha _n)=T(\alpha _n)t^{-1}-\alpha _n = (n+1)t^{-1}\alpha _{n+1}+ (nt^{-1}-1)\alpha _n.\] By abuse of notation, we write $$\alpha _n$$ for the image in $$F_R[1/t]$$ of the basis element $$\alpha _n$$ of $$F_R$$. Thus, $$F_R[1/t]$$ may be presented as the $$R[1/t]$$-module with generators $$\alpha _n$$ and relations   \[(1-nt^{-1})\alpha _n=(n+1)t^{-1}\alpha _{n+1}, \quad n\ge 0. \] (1.10) Note that $$\alpha _{0}=t^{-1}\alpha _{1}$$. It is not hard to verify directly, beginning with (1.9), that multiplication by $$t^{-1}$$ is an isomorphism on $$F_R[1/t]$$. It also follows directly from the ring structure on $$F$$ and the identification of the colimit with $$F_R[1/t]$$. If $$R$$ contains no $${\Bbb {Z}}$$-torsion, so that $$R\subseteq R\otimes {\Bbb {Q}}$$, it is easy to see that $$F_R[1/t]$$ embeds in $$F_{R\otimes {\Bbb {Q}} }[1/t]=R\otimes {\Bbb {Q}} [t,t^{-1}]$$, as we saw at the beginning of this section. The presentation of $$R \otimes H$$ as an $$R$$-module looks different when $$R$$ has $${\Bbb {Z}}$$-torsion, as we shall see in Section 2. Example 1.11 (See [3, 2.2]) If $$R$$ contains $${\Bbb {Q}}$$, the presentation (1.10) shows that $$F_R[1/t]$$ is the $$R[t^{-1}]$$-module with generators $${\{ }\alpha _{0},\alpha _{1},\ldots {\} }$$ modulo the relations   \[t^{-n}\alpha _{n+1} =\frac {1}{(n+1)!}(1-t^{-1})(1-2t^{-1})\ldots (1-nt^{-1})\alpha _1.\] That is, $$F_R[1/t]\cong R[t,t^{-1}]$$ on generator $$\alpha _0$$ with the relations   \[\alpha _{n} = \binom {t}{n}\alpha _0.\] Variant 1.12 Adams uses a variant of the above construction. Fix a unit $$u$$ of $$R$$ and set $$v=ut$$, so $$t=v/u$$. We consider $$B=uT$$ to be multiplication by $$v$$ with (1.8) replaced by $$B(\beta _n)=nu\beta _n + (n+1)\beta _{n+1}$$, where $$\beta _n=u^n\alpha _n$$. Replacing the bookkeeping index $$t^{-1}$$ by $$v^{-1}$$, (1.10) becomes $$(1-nu/v)\beta _n=(n+1)v^{-1}\beta _{n+1}$$, or $$(n+1)\beta _{n+1}=(t-n)u\beta _n$$, and we recover   \[\beta _{n} = u^n{\binom {v/u}n}\beta _0 = u^n{\binom tn}\beta _0.\] These $$\beta _n$$ are the elements described by Adams in [1, II.13.7] as the generators of $$KU_* (KU)$$, regarded as a submodule of $$KU_* (KU) \otimes {\Bbb {Q}} = {\Bbb {Q}} [u,1/u,v,1/v]$$. 2. The $$\ell$$-primary decomposition In this section, we suppose that $$R$$ is an algebra over $${\Bbb {Z}} /\ell ^\nu$$ and give a basis for the colimit $$H_R=F_R[1/t]$$ of the sequence $$F_R \, {\mathop {\to }\limits ^{{T} }}\, F_R \, {\mathop {\to }\limits ^{{T} }}\, F_R \, {\mathop {\to }\limits ^{{T} }}\, \cdots$$ of (1.9). Recall from Section 1 that the elements of $$F$$ are polynomial functions $${\Bbb {N}}\to {\Bbb {Z}}$$, and that the $$\alpha _n=\binom tn$$ form a basis of $$F$$. Lemma 2.1 For each$$q\in {\Bbb {Z}}$$,$$F \otimes {\Bbb {Z}} /q$$embeds in the ring of all functions$${\Bbb {N}}\to {\Bbb {Z}} /q$$. Proof If $$f\in F$$ satisfies $$f(a)\equiv 0\pmod {q}$$ for every $$a>0,$$ then $$h(t)=f(t)/q$$ is a numerical polynomial, and $$f(t)=q\,h(t)$$ is in $$qF$$. Example 2.2 Suppose that $${\Bbb {Z}} /2\subseteq R$$. Then the relations $$\alpha _{2k}=t^{-1}\alpha _{2k+1}$$ and $$(1-t^{-1})\alpha _{2k+1}=0$$ imply that $$H_R=F_R[1/t]$$ is the free $$R$$-module with basis $${\{ }\alpha _{2k+1}, k\ge 0{\} }$$, with $$\alpha _{2k}=\alpha _{2k+1}$$. In fact, $$F/2F$$ and $$H/2H$$ are Boolean rings by Lemma 2.1. In particular, each $$\alpha _n$$ is idempotent in the ring $$F/2F$$, including $$t=\alpha _1$$, and $$H/2H=t\,F/2F$$ is a factor ring of $$F/2F$$. In addition, if $$m\le 2^r-1,$$ then (1.2) implies that $$\alpha _m\alpha _{2^r-1}=\alpha _{2^r-1}$$. When $$R$$ is a $${\Bbb {Z}} /\ell ^\nu$$-algebra, $$F_R$$ has a similar block decomposition. To prepare for it, let $$L'$$ denote the free $$R$$-module on basis $$\alpha _0, \alpha _1,\ldots ,\alpha _{\ell -1}$$, and let $$L$$ denote the submodule of $$L'$$ on $$\alpha _1,\ldots ,\alpha _{\ell -1}$$. Following (1.8), we define maps $$b_k:L'\to L'$$ by   \[b_{k}(\alpha _i)=(k\ell +i)\alpha _i + (k\ell +i+1)\alpha _{i+1}, \quad i=0,\ldots ,\ell -2 \] (2.3) and $$b_k(\alpha _{\ell -1})=(k\ell + \ell -1)\alpha _{\ell -1}$$. Note that $$b_k(L)\subseteq L$$. Lemma 2.4 The maps$$L\to L[1/b_k]\to L'[1/b_k]$$are isomorphisms for all$$k$$. Proof The restriction of $$b_k$$ to $$L$$ is represented by a lower triangular matrix, whose determinant $$\prod _{i=1}^{\ell -1}(k\ell +i)$$ is a unit in $$R$$. Thus, each $$b_k$$ restricts to an automorphism of $$L$$. Since $$\ell ^\nu =0$$ in $$R$$, $$(b_k)^{\nu }$$ maps $$\alpha _0$$ into $$L$$. The result is now straightforward. We can now describe the $$R$$-module $$H_R=R \otimes F_R[1/t]$$ when $$R= {\Bbb {Z}} /\ell$$; the case $$\ell =2$$ was given in Example 2.2. The maps $$\phi _k:L'\to F_R$$, $$\phi _k(\alpha _i)=\alpha _{k\ell +i}$$, induce an isomorphism $$\bigoplus \phi _k:\bigoplus _{k=0}^\infty L' \to F_R$$ under which the map $$\bigoplus b_k$$ is identified with the map $$T$$ of (1.8). Corollary 2.5 If$${\Bbb {Z}} /\ell \subseteq R$$with$$\ell$$prime, the map$$\bigoplus \phi _k: \bigoplus _{k=0}^\infty L \to H_R$$is an isomorphism. Thus, the elements$$\alpha _n$$,$$n\not\equiv 0\pmod {\ell }$$form a basis of$$H_R$$. Proof The maps $$\phi _k:L'\to F$$ satisfy $$\phi _k\circ \bigoplus b_k = T\circ \phi$$. Hence, the map $$\bigoplus \phi _k$$ induces an isomorphism between $$\bigoplus L\cong \bigoplus L'[1/b_k]$$ and $$H_R=F_R[1/t]$$. Theorem 2.6 If$$R$$is a$${\Bbb {Z}} /\ell ^\nu$$-algebra, the$$R$$-module map$$\bigoplus \phi _k:\bigoplus L \, {\mathop {\to }\limits ^{{\cong } }}\, H_R$$is an isomorphism. In particular, the elements$$\alpha _n$$,$$n\not\equiv 0\pmod {\ell }$$,$$n>0$$, form a basis. Proof Let $$F_n$$ denote the free $$R$$-submodule of $$F_R$$ on basis $${\{ }\alpha _{nq+i}: 0\le i<q{\} }$$, where $$q=\ell ^\nu$$. Then $$F_R$$ is the direct sum of the $$F_n$$. Since $$T(\alpha _{nq-1})=(nq-1)\alpha _{nq-1}$$, $$T$$ sends $$F_n$$ into itself, and there are isomorphisms $$F_0\to F_n$$, $$\alpha _i\mapsto \alpha _{nq+i}$$ commuting with $$T$$. Therefore, $$H_R=F_R[1/t]$$ is isomorphic to $$\bigoplus F_n[1/t]$$, and it suffices to show that $$F_0[1/t]$$ is free on the $$\alpha _n$$ with $$0<n<q$$ and $$n\not\equiv 0\pmod {\ell }$$. For $$k=0,\ldots ,\ell ^{\nu -1}$$, let $$F_0^{\ge k\ell }$$ denote the $$R$$-submodule of $$F_0$$ generated by the $$\alpha _i$$ with $$k\ell \le i<q$$. These form a filtration of $$F_0$$, and the maps $$\phi _k:L'\to F_0^{\ge k\ell }$$ induce $$R$$-module isomorphisms with the filtration quotients   \[\bar \phi _k: L' \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, F_0^{\ge k\ell }/F_0^{\ge (k+1)\ell }\] such that $$T\circ \bar \phi _k = b_k\circ \phi _k$$. By Lemma 2.4, it follows that $$\phi _k$$ induces an isomorphism   \[L \cong L'[1/b_k] \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, F_0^{\ge k\ell }/F_0^{\ge k\ell + \ell }[1/t].\] By induction on $$k$$, it follows that $$\bigoplus \phi _k$$ induces an isomorphism $$\bigoplus L \cong F_0[1/t]$$. 3. Hopf algebroids Recall from [14, A1.1.1] that a Hopf algebroid $$(R,\Gamma )$$ is a pair of commutative rings, with maps $$\eta _L,\eta _R:R\to \Gamma$$, $$\varepsilon :\Gamma \to R$$, $$c:\Gamma \to \Gamma$$ and $$\Delta :\Gamma \to \Gamma \otimes _R\Gamma$$ satisfying certain axioms, listed in [14]. Let $$M$$ be a $$\Gamma$$-comodule with structure map $$M \, {\mathop {\to }\limits ^{{\psi } }}\, M \otimes _R\Gamma$$. Recall [14, A1.2.11] that the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,R)$$ is the cosimplicial comodule with $$C^0=M$$ and $$C^n=M \otimes _R\Gamma ^{ \otimes _Rn}$$, with cofaces given by $$\psi$$, $$\eta _L$$ and $$\Delta :\Gamma \to \Gamma \otimes _R\Gamma$$. In particular, when $$M=R$$ (with $$\psi =\eta _R$$), the cobar complex is a cosimplicial ring. Example 3.1 For any commutative algebra $$R$$, $$(R,R \otimes R)$$ is a Hopf algebroid with $$\eta _L(r)=r \otimes 1$$, $$\eta _R(r)=1 \otimes r$$, $$c(r \otimes s)=s \otimes r$$ and $$\Delta (r \otimes s)=r \otimes 1 \otimes s$$. The cobar complex $$C_{R \otimes R}^ {{{\bf{\scriptscriptstyle \bullet }}}} (R,R)$$ is the standard cosimplicial module $$n\mapsto R^{ \otimes n+1}$$. If $$\Gamma$$ is a Hopf algebra over $$R$$, then $$(R,\Gamma )$$ is a Hopf algebroid with $$\eta _L=\eta _R$$ the unit, $$\varepsilon$$ the counit, $$c$$ the antipode and $$\Delta$$ the coproduct. In this case, the cobar complex is classical. Example 3.2 The pair $$(R_ {\Bbb {Q}},\Gamma _ {\Bbb {Q}})=({\Bbb {Q}} [u,1/u], {\Bbb {Q}} [u,1/u,t,1/t])$$ is a Hopf algebroid with $$\eta _L(u)=u \otimes 1$$, $$\eta _R(u)=c(u)=tu$$, $$c(t)=1/t$$, $$\varepsilon (t)=1$$ and $$\Delta (t)=t \otimes t$$. Recall from Definition 1.3 that $$H$$ is the subalgebra of $${\Bbb {Q}} [t,1/t]$$ generated by the $$\binom tn$$. If $$R= {\Bbb {Z}} [u,1/u]$$ and $$\Gamma =R \otimes H$$, then $$(R,\Gamma )$$ is a sub-Hopf algebroid. If we set $$v=tu,$$ then we have $$\eta _R(u)=v$$, $$c(u)=v$$ and   \[\Delta (u)=u \otimes 1,\ \Delta (t)=t \otimes t,\ \Delta (v)=ut \otimes t = 1 \otimes v.\] Of course, $$KU_* =R$$ and $$\pi _* (KU\wedge KU)=\Gamma$$, and the formulas given by Adams in [1, II.13.4] show that $$(R,\Gamma )$$ is the original Hopf algebroid $$(KU_* ,KU_*KU)$$. Lemma 3.3 The Hopf algebroid$$(R,R \otimes H)$$of Example 3.2 is split. Proof The natural inclusion of the Hopf algebra $$({\Bbb {Z}},H)$$ into $$(R,R \otimes H)$$ is a map of algebroids [14, A1.1.9], and the identity map on $$R \otimes H$$ is an algebra map. Fix $$R$$ and $$\Gamma \cong R \otimes H$$ as in Example 3.2. Because the algebroid $$(R,\Gamma )$$ is split, we can say more: $$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,R)=C_H^ {{{\bf{\scriptscriptstyle \bullet }}}} (M, {\Bbb {Z}})$$ (see [14, A1.2.17]), and the latter is the usual cobar complex of $$M$$ as a comodule over the Hopf algebra $$H$$. For $$M=R$$, we have the following description. Corollary 3.4 The cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (R,R)$$has$$C^n=R \otimes H^{ \otimes n}$$. The coface maps$$C^n\to C^{n+1}$$are given by the units$$\eta _L,\eta _R:R\to R \otimes H=\Gamma$$of$$\Gamma$$and$$\Delta _H$$. 4. Oriented topological spectra This section is intended to serve as a template for the motivic constructions which follow, and is restricted to topological spectra. No originality is claimed for these results. To emphasize our analogy, we will write $${\mathbb P}^{\infty }$$ for $${\Bbb {C}} {\mathbb P}^{\infty }_+ $$ in this section. If $$E$$ is a topological ring spectrum, then combining the ring structure of $$E$$ with the $$H$$-space structure of $${\mathbb P}^{\infty }$$ we obtain a natural ring spectrum $$E\wedge {\mathbb P}^{\infty }$$. Following Adams [1] and Snaith [15], the motivation for studying $$F[1/t]$$ comes from the computation of $$E\wedge KU$$, where $$E$$ is an oriented spectrum. Recall that a commutative topological ring spectrum $$E$$ with unit $${{\bf{1}}} \, {\mathop {\to }\limits ^{{\eta _E} }}\, E$$ is said to be oriented if there is a map $${\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\to }\limits ^{{x} }}\, S^2\wedge E$$ whose restriction to $${\Bbb {C}} {\mathbb P} ^1_+ \wedge {{\bf{1}}}$$ is $$1\wedge \eta _E$$ (using $$S^2= {\Bbb {C}} {\mathbb P} ^1$$). As in [1, II.2], these data yield a formal group law on $$E_* [\![x]\!]\cong E^* ({\mathbb P}^{\infty })$$, and dual elements $$\beta _n$$ in $$E_{2n}({\mathbb P}^{\infty })=\pi _{2n}(E\wedge {\mathbb P}^{\infty })$$. When $$E$$ is orientable, the $$\beta _n$$ induce maps   \[ E\wedge S^{2n} {\mathop {\longrightarrow}\limits^{1\wedge \beta _n}} E\wedge E\wedge {\mathbb P}^{\infty } \longrightarrow E\wedge {\mathbb P}^{\infty },\] and these maps induce an isomorphism from $$\bigoplus _{n=0}^{\infty }E\wedge S^{2n}$$ to $$E\wedge {\mathbb P}^{\infty }$$ (by [1, p. 42]). Consequently, we have   \[E_* ({\mathbb P}^{\infty })=\pi _* (E\wedge {\mathbb P}^{\infty }) \cong \bigoplus _{n=0}^{\infty }\pi _* (E\wedge S^{2n}).\] Thus, $$E_* ({\mathbb P}^{\infty })$$ is a free graded $$E_* $$-module on generators $$\beta _n$$ in degree $$2n$$, $$n\ge 0$$. Let $$\xi :S^{2}_+ \cong {\Bbb {C}} \mathbb P^{1}_+ \to {\mathbb P}^{\infty }$$ be the map that classifies the tautological line bundle on $${\Bbb {C}} \mathbb P^{1}$$, and let $$b: {\mathbb P}^{\infty } \wedge {{\bf{1}}} \to S^{-2}\wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}}$$ denote the adjoint of the stable map   \[ S^2 \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{\mathrm {Hopf}}} S^2_+ \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{\xi \wedge 1}} {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{m}} {\mathbb P}^{\infty } \wedge {{\bf{1}}}.\] Let $$j=\xi -1$$ denote the map $${\mathbb P}^{\infty } \to BU_+ \to KU$$ classifying the virtual tautological line bundle of rank $$0$$. By Snaith [15, 2.12], the map $$j$$ induces an equivalence from the homotopy colimit of the sequence   \[{\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\longrightarrow }\limits ^{{b} }}\, S^{-2} {\mathbb P}^{\infty } \wedge {{\bf{1}}} \; \, {\mathop {\longrightarrow }\limits ^{{S^{-2}b\;} }}\, S^{-4} {\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\longrightarrow }\limits ^{{} }}\, \cdots\] to the spectrum $$KU$$. We write $$u=u_{KU}$$ for the map $$S^2\to {\mathbb P}^{\infty } \, {\mathop {\to }\limits ^{{j} }}\, KU$$. The map   \[S^2\wedge KU \, {\mathop {\longrightarrow }\limits ^{{u\wedge 1} }}\, KU\wedge KU \, {\mathop {\longrightarrow }\limits ^{{m} }}\, KU\] is the periodicity isomorphism for $$KU$$ (cf. (7.1)). Composing $$j$$ with the inverse of the periodicity isomorphism, we obtain an orientation for $$KU$$ whose formal group law is $$x+y+uxy$$. Smashing with $$E$$ yields maps $$b_E:E\wedge {\mathbb P}^{\infty } \to S^{-2}\wedge E\wedge {\mathbb P}^{\infty }$$ and an equivalence between $$E\wedge KU$$ and the homotopy colimit of the sequence of spectra:   \[E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b_E}} S^{-2}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{S^{-2} b_E}} S^{-4}\wedge E\wedge {\mathbb P}^{\infty } \longrightarrow \cdots . \] (4.1) We now make the connection with the algebraic considerations of Section 1. Lemma 4.2 The coefficient of$$x^iy^j$$in$$(x+y+uxy)^k$$is$$\binom {k}{k-i,k-j,i+j-k}u^{i+j-k}$$. Proof The product has $$3^k$$ terms. The only terms producing the monomial $$x^iy^j$$ are those with exactly $$k-j$$ of the factors must be $$x$$, exactly $$k-i$$ must be $$y$$, and the remaining $$i+j-k$$ must be $$uxy$$. Corollary 4.3 Suppose that$$E$$is an oriented spectrum with a multiplicative group law$$\mu (x,y)=x+y+uxy$$such that$$u$$is a unit of$$E_* $$. Then the homotopy groups of the cosimplicial spectrum$$E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$form a cosimplicial group$$E_* \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1},$$isomorphic to the cobar complex$$E_* \otimes C_{F \otimes F}^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$of Example 3.1. Proof By the above remarks, $$\pi _* (E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1})$$ is a free $$E_* $$-module with basis $$\beta _{i_0}\wedge \ldots \beta _{i_n}$$, which we identify with $$E_* \otimes F^{ \otimes n+1}$$ by sending $$\beta _i$$ to $$u^i\alpha _i$$, as in 1.12. The coface maps are easily seen to correspond, and for the codegeneracies it suffices to consider $$\sigma ^0:\pi _* (E\wedge {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty })\to \pi _* (E\wedge {\mathbb P}^{\infty }).$$ Adams showed in [1, II(3.4–5)] that the coefficient $$a_{ij}^k$$ of $$\sigma ^0(\beta _i \otimes \beta _j)$$ is the coefficient of $$x^iy^j$$ in $$(x+y+uxy)^k$$, which is given by Lemma 4.2, and agrees with $$(u^i\alpha _i)\ast (u^j\alpha _j)$$ by (1.2). Lemma 4.4 Suppose that$$E$$is an oriented spectrum with a multiplicative group law$$\mu (x,y)=x+y+uxy$$such that$$u$$is a unit of$$E_* $$. Then  \[\pi _* (E\wedge KU)\cong \varinjlim \pi _* (S^{-2n}\wedge E\wedge {\mathbb P}^{\infty }) \cong E_* \otimes \varinjlim F \cong E_* \otimes H.\] Proof Identifying $$R=E_* $$ and $$F_R=E_* ({\mathbb P}^{\infty })=\pi _* (E\wedge {\mathbb P}^{\infty })$$, Adams shows in [1, II(3.6)] that multiplication by $$\beta _1$$ is given by the formula (1.8), with $$t$$ replaced by $$v=ut$$ as in Variant 1.12. Comparing with (1.9), we see that the ring $$E_* ({\mathbb P}^{\infty })[1/\beta _1]$$ is isomorphic to $$E_* \otimes H$$. From (4.1), we obtain the result. Remark 4.5 Lemma 4.4 fails dramatically if $$u=0$$, for example when $$E=H {\Bbb {Z}}$$. Replacing $$E$$ by $$E\wedge KU$$, which we orient by $$x\wedge \eta _{KU}$$, Lemma 4.4 yields $$\pi _* (E\wedge KU\wedge KU)\cong E_* \otimes H \otimes H$$. The two maps $$E_* \otimes H\to E_* \otimes H \otimes H$$, induced by $$E\wedge KU \to E\wedge KU^{\wedge 2}$$, are given by $$\eta _E \otimes 1$$ and $$1 \otimes \Delta _H$$, and $$E_* \otimes H \otimes H \, {\mathop {\to }\limits ^{{\sigma ^0} }}\, E_* \otimes H$$ is given by Corollary 4.3. An inductive argument establishes: Corollary 4.6 The homotopy groups of the cosimplicial spectrum$$n\mapsto E\wedge KU^{\wedge n+1}$$form the cosimplicial group:  \[\pi _* (E\wedge KU^{\wedge n+1}) = E_* \otimes H^{ \otimes n+1}.\]This is the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,E_* )$$of Example 3.2 with$$M=E_* \otimes H$$. When $$E=KU$$, we can also form the cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU)$$, as in the Introduction, and we have: Proposition 4.7 The cosimplicial ring$$\pi _* (N^ {{{\bf{\scriptscriptstyle \bullet }}}} KU)$$is$$KU_* \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} },$$the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU_* ,KU_* )$$of 3.4. In particular, for every$$n$$we have an isomorphism  \[\pi _* (KU^{\wedge n+1}) \cong KU_* \otimes H^{ \otimes n}.\] $$KU^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} }$$with finite coefficients Fix a prime $$\ell$$ and a power $$q=\ell ^\nu$$, and let $$P$$ denote the Moore spectrum for $${\Bbb {Z}} /q$$. We will assume that $$q$$ is chosen so that $$P$$ is a ring spectrum with unit. For example, if $$\ell \neq 2,3$$, then this is the case for any $$\nu \ge 1$$; see [24, IV.2.8]. Smashing $$P$$ with $$KU$$ yields the standard cosimplicial spectrum $$P\wedge KU^{\wedge n+1}$$ associated to the triple $$-\wedge KU$$. Set $$R=\pi _* (P\wedge KU)$$; the Universal Coefficient Theorem yields $$R= {\Bbb {Z}} /q[u,1/u]$$. Since $$P\wedge KU$$ is oriented by the orientation on $$KU$$, we see from Lemma 4.4 and Theorem 2.6 that $$\pi _* (P\wedge KU\wedge KU)=R \otimes H$$ is a free $$R$$-module with basis $${\{ }\beta _n:n\not\equiv 0\pmod \ell {\} }$$, and from ( 4.6) that $$\pi _* (P\wedge KU^{\wedge n+1})=R \otimes H^{ \otimes n}$$. In fact, $$(R,R \otimes H)$$ is the mod $$q$$ reduction of the Hopf algebroid of Lemma 3.3. Set $$E=P\wedge KU$$. Then each $$\beta _n$$ determines a map $$S^{2n}\wedge E\to E\wedge {\mathbb P}^{\infty } \to E\wedge KU$$. By Lemma 4.4, the direct sum of these maps is a homotopy equivalence of spectra   \[\bigoplus E\wedge S^{2n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge {\mathbb P}^{\infty } {. }\] There is a sequence (4.1) for $$E\wedge {\mathbb P}^{\infty }$$, with colimit $$E\wedge KU$$. By Example 3.2, $$\pi _* (E\wedge KU)\cong E_* \otimes H$$. Since $$R=\pi _* (E)$$ is a $${\Bbb {Z}} /q$$-module, Theorem 2.6 applies: the $$\beta _n$$ with $$n\not\equiv 0\pmod \ell$$ form a basis of $$E_* \otimes H$$. Proposition 4.8 For$$E=P\wedge KU$$, there is a homotopy equivalence of spectra  \[\bigoplus _{n\not\equiv 0 \,{\mathrm {mod}}\,\ell } E\wedge S^{2n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge KU.\] Proof The argument we gave in Corollary 2.5 shows that the left side is the colimit of (4.1). That is, $$E\wedge KU$$. In effect, we just note that the components $$E\wedge S^{2n}\to E\wedge S^{2n-2}$$ of $$b$$ are multiplication by $$n$$ and are thus null-homotopic when $$\ell |n$$, while the maps $$E\wedge S^{2n}\to E\wedge S^{2n}$$ of $$b$$ are multiplication by $$(n+1)$$ and are thus null-homotopic when $$\ell |n+1$$. 5. The motivic $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })$$ We now turn to motivic spectra. The goal of this section is to describe the cosimplicial spectrum $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })= E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ associated to an oriented motivic spectrum $$E$$. Here, the motivic space $${\mathbb P}^{\infty }$$ is defined as the union of the spaces represented by the schemes $${\mathbb P}^n$$. The Segre maps $${\mathbb P} ^m\times {\mathbb P} ^n\to {\mathbb P}^{\infty }$$ induce a map $$m: {\mathbb P}^{\infty } \times {\mathbb P}^{\infty } \to {\mathbb P}^{\infty }$$ making $${\mathbb P}^{\infty } \wedge {{\bf{1}}}$$ into a ring $$T$$-spectrum. Then $$n\mapsto E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$ is the standard cosimplicial spectrum associated to a spectrum $$E$$ and the triple $$-\wedge {\mathbb P}^{\infty }$$ [23, 8.6.4]. If $$E$$ is a ring $$T$$-spectrum, then this is a cosimplicial ring spectrum. A commutative ring $$T$$-spectrum $$E$$ with unit $${{\bf{1}}} \, {\mathop {\to }\limits ^{{\eta } }}\, E$$ is said to be oriented if there is a map $${\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\to }\limits ^{{t} }}\, T\wedge E$$ whose restriction to $${\mathbb P} ^1\wedge {{\bf{1}}}$$ is $$T\wedge \eta$$. (Compare with Section 4.) As pointed out in Example 2.6.4 of [10], $$KGL$$ is an oriented ring spectrum in $${\cal {{SH}}}$$ with a multiplicative formal group law [10, 1.3.1 and 1.3.3]. When $$E$$ is oriented, the projective bundle theorem in [4, 3.2] yields an isomorphism with inverse $$\bigoplus c^n$$:   \[\bigoplus \beta _n: \bigoplus _{n=0}^{\infty }E\wedge T^{n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge {\mathbb P}^{\infty }. \] (5.1) If we let $$F_ {\mathrm {gr}}$$ denote the free graded abelian group on generators $$\beta _n$$ in degree $$n\ge 0$$, then $$E\wedge {\mathbb P}^{\infty }$$ is the free graded $$E$$-module spectrum $$E \otimes F_ {\mathrm {gr}}$$. By (5.1), the terms $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$ are isomorphic to the direct sum of terms $$E\wedge T^{i_0}\wedge \ldots \wedge T^{i_n}$$, $$i_j\ge 0$$. Lemma 5.2 The coface map$$\partial ^j:E\wedge { {\mathbb P}^{\infty } }^{\wedge n}\to E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$sends the summand$$E\wedge T^{i_1}\wedge \ldots \wedge T^{i_n}$$to$$E\wedge T^{i_1}\wedge \ldots T^{i_j}\wedge T^0\wedge T^{i_{j+1}}\ldots \wedge T^{i_n}.$$ Proof Since the coface maps are determined by the unit map $${{\bf{1}}} \to {\mathbb P}^{\infty }$$, it suffices to observe that the map   \[E=E\wedge T^0\longrightarrow E\wedge {\mathbb P}^{\infty } \cong \bigoplus E\wedge T^n\] is just the inclusion of $$T^0$$ into $$\bigoplus T^n$$, by the projective bundle theorem. At this point, we may interpret the coface maps in $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })$$ in terms of $$F_ {\mathrm {gr}}$$. Recall (from [23, 8.1.9]) that a semi-cosimplicial object in a category is a sequence of objects $$K^n$$ with coface operators $$\partial ^i:K^{n-1}\to K^n$$ ($$0\le i \le n$$) satisfying the cosimplicial identities $$\partial ^j\partial ^i=\partial ^i\partial ^{j-1}$$ if $$i<j$$. For example, $$n\mapsto F_ {\mathrm {gr}} ^{ \otimes n+1}$$ is a semi-cosimplicial graded abelian group, where the codegeneracies insert $$1=\beta _0$$. (The generators $$\beta _n$$ of $$F_ {\mathrm {gr}}$$ lie in degree $$n$$.) We may also form the semi-cosimplicial spectrum $$n\mapsto E \otimes F_ {\mathrm {gr}} ^{ \otimes n+1}$$. Note that the codegeneracy maps $$F_ {\mathrm {gr}} ^{ \otimes n+1}\to F_ {\mathrm {gr}} ^{ \otimes n}$$ associated to the product $$F \otimes F\to F$$ are not graded, as (1.2) shows, so we do not get a cosimplicial spectrum in this way. Corollary 5.3 Let$$E$$be an oriented spectrum$$E$$. Then there is an isomorphism of semi-cosimplicial ring spectra,$$E \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \, {\mathop {\to }\limits ^{{\cong } }}\, E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. Lemma 5.4 For any oriented$$E,$$the component maps$$E\wedge T^n\to E\wedge T^{i}\wedge T^j$$of the diagonal$$E\wedge {\mathbb P}^{\infty } \to E\wedge {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty }$$are the canonical associativity isomorphisms when$$i+j=n,$$and zero otherwise. Proof The maps $$\beta _n:E\wedge T^n\to E\wedge {\mathbb P}^{\infty }$$ in (5.1) satisfy $$c^k\circ \beta _n =\delta _{kn}$$, where $$c^n$$ is the projection $$E\wedge {\mathbb P}^{\infty } \to E\wedge T^n$$. We have a commutative diagram:   The top horizontal composite is $$c^{i+j}\circ \beta _n=\delta _{n,i+j}$$, and the entire composition is the component map in question. The result follows. We now turn to the codegeneracies $$\sigma ^j: E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}\to E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$. These are all induced from the product $${\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } \to {\mathbb P}^{\infty }$$. After taking the smash product with $$E$$, and using (5.1), we may rewrite the product as an $$E$$-module map   \[\bigoplus _{i,j\ge 0}E\wedge T^i\wedge T^j = (E\wedge {\mathbb P}^{\infty })\wedge (E\wedge {\mathbb P}^{\infty }) \longrightarrow E\wedge {\mathbb P}^{\infty } = \bigoplus _{k\ge 0} E\wedge T^k.\] Since each term $$E\wedge T^i\wedge T^j$$ is a compact $$E$$-module, its image lies in a finite sum. Thus, the product is given by a matrix whose entries $$a_{ij}^k$$ are the component maps $$E\wedge T^i\wedge T^j\to E\wedge T^k$$. In terms of motivic homotopy groups, $$E_{* ,* }({\mathbb P}^{\infty })\cong \bigoplus _{n=0}^{\infty }E_{* ,* }(T^{n})$$ is a free graded $$E_{* ,* }$$-module with generators $$\beta _n\in E_{2n,n}({\mathbb P}^{\infty })$$, $$n\ge 0$$, corresponding to the unit of $$E_{2n,n}(T^n)\cong E_{0,0}$$. Dually, $$E^{* ,* }({\mathbb P}^{\infty })\cong E^{* ,* }[\![x]\!]$$ and (as in topology) the orientation yields a formal group law of the form $$\mu =x+y+ \sum _{i,j\ge 1}a_{i,j}x^{i}y^{j}$$ in $$E_* [\![x,y]\!]\cong E^* ({\mathbb P}^{\infty } \times {\mathbb P}^{\infty })$$, where   \[a_{i,j}\in E_{2i+2j-2,i+j-1} = {\mathrm {Hom}}_{ {\cal {{SH}}} }(T^{i+j},E\wedge T). \] (5.5) We can now extend a result of Adams to the motivic setting. Theorem 5.6 If$$E$$is an oriented motivic ring spectrum, then$$a_{1j}^k = k a_{1,1+j-k},$$with$$a_{1,1+j-k}$$as in (5.5). Proof This follows directly from Lemma 5.4, together with Adams’ calculation in [1, II(3.6), p. 46]. Remark For fixed $$i,j$$ the maps $$a_{ij}^k$$ are zero unless $$k\le i+j$$. Therefore, the image of $$E\wedge T^i\wedge T^j$$ under the product lands in the finitely many terms $$E\wedge T^k$$ with $$k\le i+j$$. Our next task is to replace $$E \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ by $$E \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ to get a cosimplicial spectrum. This is possible when $$E$$ is periodic in the following sense. If $$E_* $$ has a unit in $$E_{2,1}$$, represented by a map $$T \, {\mathop {\to }\limits ^{{u} }}\, E$$, then $$u\wedge E$$ induces an isomorphism $$T\wedge E\to E$$ with inverse $$u^{-1}\wedge E$$. We call it a periodicity map for $$E$$, and say that $$E$$ is periodic. If $$E$$ is oriented and periodic, we can use the periodicity map for $$E$$ to define maps $$\alpha _n:E\to E\wedge {\mathbb P}^{\infty }$$ such that $$\beta _n=u^n\alpha _n$$, resulting in a rewriting of the projective bundle formula (5.1) as   \[\bigoplus \alpha _n: \bigoplus _{n=0}^\infty E \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, E\wedge {\mathbb P}^{\infty }. \] (5.7) Using this new basis, each $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$ is isomorphic to $$E \otimes F^{ \otimes n}$$. Now suppose that $$E$$ is an oriented spectrum with a multiplicative formal group law $$\mu =x+y+uxy$$, with $$u$$ a unit in $$E_{2,1}$$. In this case, it is convenient to change our orientation to eliminate $$u$$, as suggested in [1, II(2.1)]. This produces the new formal group law $$x+y+xy$$. To see this, let $$t'$$ denote the element $$ut$$ of $$E^0({\mathbb P}^{\infty })$$; then the formal group law implies that   \[\mu (t')=(ux)+ (uy)+ (ux)(uy)=x'+y'+x'y'.\] Proposition 5.8 Let$$E$$be an oriented ring spectrum$$E$$with multiplicative group law$$x+y+uxy,$$$$u$$a unit. Then the cosimplicial ring spectrum$$E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$is$$E \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. In particular,$$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$is isomorphic to$$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. Proof After the change in orientation indicated above, the projection bundle formula yields isomorphisms $$E \otimes F^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$. The respective coface maps are insertion of $$1$$ and $$T^0$$ (by Lemma 5.2), so they agree. The codegeneracies of the left side are given by the product $$F \otimes F\to F$$, whose coefficients are given by (1.2). These agree with the matrix entries $$a_{ij}^k$$ on the right by Lemma 4.2 and Corollary 4.3, as the proof of 4.3 shows. 6. The slice filtration Recall [19, 2.2] that $$s_q(E)$$ denotes the $$q$$th slice of a motivic spectrum $$E$$, and that $$s_q(E\wedge T^i)=s_{q-i}(E)\wedge T^i$$. By [12], $$s_0(E)$$ has an additive formal group law. The following result is well known to experts. Proposition 6.1 The slice functors$$s_q$$commute with direct sums. Proof It suffices to prove that $$f_{q}$$ commutes with direct sums, since there is a distinguished triangle of the form $$f_{q+1}\rightarrow f_{q} \rightarrow s_{q}$$. For simplicity, we will restrict ourselves to the case $$q=0$$; the argument for arbitrary $$q$$ is exactly the same. Consider a direct sum $$E=\bigoplus _{\alpha }E_{\alpha }$$ in $${\cal {{SH}}}$$. Since $${\cal {{SH}}} ^{ {\mathrm {eff}} }$$ is closed under direct sums, $$\bigoplus f_0(E_\alpha )$$ is effective. By [11, 3.1.14], there is a family of compact objects $$K$$ such that, for every $$X\to Y$$, $$f_0(X)\to f_0(Y)$$ is an isomorphism if and only if each $$[K,X]\to [K,Y]$$ is an isomorphism. In the case at hand, for every such $$K$$ we have   \[\left [K,\bigoplus f_0(E_\alpha )\right ]\cong \bigoplus [K,f_0(E_\alpha )]\cong \bigoplus [K,E_\alpha ]\cong \left [K,\bigoplus E_\alpha \right ].\] Therefore $$\bigoplus f_0(E_\alpha )\to f_0(\bigoplus E_\alpha )$$ is an isomorphism. Example 6.2 Suppose that $$E$$ is oriented. Applying $$s_q$$ to (5.1) yields the formula $$\bigoplus s_{q-n}(E)\wedge T^n \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(E\wedge {\mathbb P}^{\infty })$$ with component maps $$s_q(\beta _n)$$. More generally, the slice $$s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge n})$$ is isomorphic to   \[\bigoplus s_{r}(E)\wedge T^{i_1}\wedge \ldots \wedge T^{i_n},\] where $$q=r+i_1+ \ldots +i_n$$, and the coface maps in $$s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} })$$ are given by Lemma 5.2, with $$E$$ replaced by $$s_q(E)$$. Thus, except for the codegeneracy maps, the cosimplicial spectrum $$\bigoplus _q s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1})$$ is isomorphic to $$\bigoplus s_q(E) \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. We will need the following observation: for any motivic ring spectrum $$E$$, $$s_*E=\bigoplus s_q(E)$$ is a graded motivic ring spectrum; see [11, 3.6.13]). Since $${\Bbb {Z}} [u,u^{-1}]$$ is a graded ring (with $$u$$ in degree 1), we can form the graded motivic ring spectrum $$s_0(E) \otimes {\Bbb {Z}} [u,u^{-1}] = \bigoplus T^q\wedge s_0(E)$$. Lemma 6.3 As ring spectra,$$s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \, {\mathop {\to }\limits ^{{\cong } }}\, \bigoplus s_q(KGL)$$. More generally,$$s_0 E \otimes {\Bbb {Z}} [u,u^{-1}] \, {\mathop {\to }\limits ^{{\cong } }}\, s_*E$$for any oriented ring spectrum$$E$$with a unit$$u$$in$$E_{2,1}$$. Proof The periodicity isomorphisms $$u^q:KGL\wedge T^q\to KGL$$, and more generally $$E\wedge T^q\to E$$, induce isomorphisms $$s_0 E\wedge T^q\cong s_q E$$ compatible with multiplication. These assemble to give the result. (Cf. [16, 6.2].) Proposition 6.4 Suppose that$$E$$is oriented, and has a multiplicative group law$$x+y+uxy$$. If$$u$$is a unit, we have isomorphisms of cosimplicial ring spectra:  \[s_* (E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_* (E) \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_0(E) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F).\]In particular,$$s_* (KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. Proof Applying $$s_q$$ to 5.7 yields the simple formula, with component maps $$s_q(\alpha _n)$$:   \[s_q(E) \otimes F = \bigoplus _{n=0}^\infty s_q(E) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_q(E\wedge {\mathbb P}^{\infty }).\] The rest is immediate from Proposition 5.8 and Example 6.2. When $$E=KGL$$, Propositions 5.8 and 6.4 give the formulas for $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ and its slices which were mentioned in the Introduction. 7. $$E\wedge KGL$$ In this section, we describe the augmented cosimplicial spectrum $$E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ associated to an oriented spectrum $$E$$ and the triple $$-\wedge KGL$$.   \[E {\mathop {\longrightarrow }\limits ^{\eta _L}} E\wedge KGL \substack{{\longrightarrow}\\ {\longrightarrow}} E\wedge KGL\wedge KGL \ \substack{{\longrightarrow}\\ {\longrightarrow} \\ \longrightarrow} \ E\wedge KGL^{\wedge 3} \cdots\] Replacing $$E$$ by $$E\wedge KGL^{\wedge n}$$, we are largely reduced to the description of $$E\wedge KGL$$, $$\eta _L$$ and the map $$\sigma ^0:E\wedge KGL\wedge KGL\to E\wedge KGL$$. We begin with a few generalities. The $$T$$-spectrum $$KGL$$ comes with a periodicity isomorphism $$T\wedge KGL\to KGL$$; see [18, 6.8; 19, 3.3]; our description of it is taken from [10, 1.3]. Let $$\xi :T \to {\mathbb P}^{\infty }$$ be the map that classifies the tautological line bundle on $$\mathbb P^{1}$$, and let $$b: {{\bf{1}}} \wedge {\mathbb P}^{\infty } \to T^{-1}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty }$$ denote the adjoint of the map   \[T\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{\xi \wedge 1}} {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{m}} {\mathbb P}^{\infty }.\] As observed by Gepner–Snaith [5, 4.17] and Spitzweck–Østvæ r [17], $$KGL$$ is the homotopy colimit of the resulting sequence,   \[ {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b}} T^{-1}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-1}b}} T^{-2}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-2}b}} \cdots ,\] and the colimit map $${{\bf{1}}} \wedge {\mathbb P}^{\infty } \to KGL$$ is the map classifying the virtual tautological line bundle $$\xi -1$$ of degree 0. We write $$u_K$$ for the map $$T \, {\mathop {\to }\limits ^{{} }}\, {{\bf{1}}} \wedge {\mathbb P}^{\infty } \, {\mathop {\to }\limits ^{{\xi -1} }}\, KGL$$. Then multiplication by $$u_K$$ is the periodicity isomorphism   \[ T\wedge KGL {\mathop {\longrightarrow}\limits^{u_K\wedge 1}} KGL\wedge KGL {\mathop {\longrightarrow}\limits^{m}}, KGL \] (7.1) for the $$T$$-spectrum $$KGL$$; see [10, 1.3.3]. Smashing $$b$$ with any spectrum $$E$$ yields maps $$b_E:E\wedge {\mathbb P}^{\infty } \to T^{-1}\wedge E\wedge {\mathbb P}^{\infty }$$, and yields a sequence of $$T$$-spectra   \[ E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b_E}} T^{-1}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-1}b_E}} T^{-2}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-2}b_E}} \cdots , \] (7.2) with homotopy colimit $$E\wedge KGL$$, parallel to the sequence (4.1) in topology. When $$E$$ is oriented, the adjoint $$E\wedge T\wedge {\mathbb P}^{\infty } \to E\wedge {\mathbb P}^{\infty }$$ of $$b_E$$ is the composition of the map $$\beta _1: E\wedge T\to E\wedge {\mathbb P}^{\infty }$$ of (5.1), smashed with $${\mathbb P}^{\infty }$$, with the product on $${\mathbb P}^{\infty }$$. Therefore, $$b_E$$ corresponds to multiplication by $$\beta _1$$, and the following observation follows from Theorem 5.6. Lemma 7.3 If$$E$$is oriented, the components$$E\wedge T^j\to T^{-1}\wedge E\wedge T^k$$of the map$$b_E$$of (7.2) are multiplication by$$k\ a_{1,1+j-k}$$. We now assume that $$E$$ has a multiplicative formal group law $$\mu =x+y+uxy$$ with $$u\in E_{2,1}$$ a unit of $$E_* $$. If $$\alpha _n:E\to E \otimes F\cong E\wedge {\mathbb P}^{\infty }$$ is the map defined by (5.7), Lemma 7.3 says that   \[b_E(\beta _n) = n\, u\, \beta _n + (n+1)\beta _{n+1}, \quad {\mathrm {or}}\quad b_E(\alpha _n) = n\,\alpha _n + (n+1)\alpha _{n+1}. \] (7.4) Comparing with (1.8), this shows that the map $$b_E: E\wedge {\mathbb P}^{\infty } \to T^{-1}\wedge E\wedge {\mathbb P}^{\infty }$$ is induced from the homomorphism $$T:F\to F$$, $$T(f)=tf$$. Recall from Definition 1.3 that the Hopf algebra $$H$$ is defined by $$H=F[1/t]$$ and that $$C_H^n({\Bbb {Z}}, {\Bbb {Z}})=H^{ \otimes n}$$ in the classical cobar complex (see Example 3.1). Theorem 7.5 If$$E$$has a multiplicative group law$$x+y+uxy$$with$$u$$a unit, we have$$E\wedge KGL \cong E \otimes H$$, and isomorphisms of cosimplicial spectra for all$$q\in {\Bbb {Z}} {:}$$  \[\begin {array}{rl} E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1} & \cong E \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong E \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{H}({\Bbb {Z}}, {\Bbb {Z}}) {, } \\ s_* (E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) & \cong s_* (E) \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_* (E) \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{H}({\Bbb {Z}}, {\Bbb {Z}}). \end {array}\]In addition,$$E\to E\wedge KGL$$corresponds to$$E \otimes {\Bbb {Z}} \to E \otimes H$$. Proof By (7.4), multiplication by $$t$$ on $$E \otimes F$$ is given by the same formula as multiplication by $$b_E$$ on $$E\wedge {\mathbb P}^{\infty }$$. It follows that the homotopy colimit $$E\wedge KGL$$ of the sequence (7.2) is the same as $$E \otimes {\mathrm {colim}}(F \, {\mathop {\to }\limits ^{{t} }}\, F\to \cdots ) = E \otimes H$$. Replacing $$E$$ by $$E\wedge KGL^{\wedge n}$$ shows that $$E\wedge KGL^{\wedge n+1} \cong E \otimes H^{ \otimes n+1}$$ and hence that $$s_* (E\wedge KGL^{\wedge n+1}) \cong s_* (E) \otimes H^{ \otimes n+1}$$. The coface and codegeneracies of these cosimplicial spectra are identified by Propositions 5.8 and 6.4. When $$E=KGL,$$ we get the formula mentioned in the Introduction: Corollary 7.6 $$KGL\wedge KGL \cong KGL \otimes H,$$and$$KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2} \cong KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. In addition,$$\eta _L:KGL\to KGL\wedge KGL$$corresponds to$$KGL \otimes {\Bbb {Z}} \to KGL \otimes H$$. Example 7.7 When $$E=E \otimes {\Bbb {Q}}$$, we have $$E\wedge KGL^{\wedge n}\cong E \otimes {\Bbb {Q}} [t_1,t_1^{-1},\ldots ,t_n,t_n^{-1}]$$, because $${\Bbb {Q}} \otimes H= {\Bbb {Q}} [t,t^{-1}]$$. Set $$q=\ell ^\nu$$ and write $$E/q$$ for the cofibre $$E \otimes {\Bbb {Z}} /q$$ of $$E \, {\mathop {\to }\limits ^{{q} }}\, E$$. An elementary calculation shows that $$(E/q) \otimes H\cong E \otimes H/qH$$. Corollary 7.8 Suppose that$$E$$has a multiplicative formal group law$$x+y+uxy,$$such that$$u$$is a unit of$$E_{* ,* }$$. Then the family of maps$$\beta _n:E\wedge T^{n}\to E\wedge {\mathbb P}^{\infty } \to E\wedge KGL$$with$$n\not\equiv 0\pmod {\ell }$$induces an isomorphism  \[E \otimes H/\ell ^\nu H \cong \bigoplus _{n\not\equiv 0} (E/\ell ^\nu )\wedge T^{n} \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, (E/\ell ^\nu )\wedge KGL.\]Moreover, the slices$$s_m(E/\ell ^\nu \wedge KGL)$$are isomorphic to$$\bigoplus _{n\not\equiv 0} s_{m-n}(E/\ell ^\nu )\wedge T^n$$. Proof Immediate from Theorems 2.6 and 7.5. 8. The cosimplicial $$KGL$$ spectrum In this section, we complete the information in the previous section to determine the slices of the cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$, with $$N^nKGL=KGL^{\wedge n+1}$$.   \[KGL \substack{\longrightarrow\\ {\longrightarrow}} KGL\wedge KGL \ \substack{\longrightarrow\\ {\longrightarrow}\\ \longrightarrow} \ KGL^{\wedge 3} \ \substack{\longrightarrow\\ {\longrightarrow}\\ {\longrightarrow} \\{\longrightarrow}} \ \cdots\] Its codegeneracies are given by the product on $$KGL$$, and its coface maps are given by insertion of the unit $$\eta _K: {{\bf{1}}} \to KGL$$; in particular, $$\partial ^0,\partial ^1:KGL\to KGL\wedge KGL$$ are the canonical maps $$\eta _R=\eta _K\wedge 1$$ and $$\eta _L=1\wedge \eta _K$$, respectively. There is also an involution $$c$$ of $$KGL\wedge KGL$$ swapping the two factors, and we have $$\eta _R=c\circ \eta _L$$. By Corollary 7.6, there are isomorphisms $$KGL \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, KGL^{\wedge n+1}$$ such that the diagram   commutes. As $$\partial ^1=\eta _L$$, this establishes the initial case of the following lemma. Lemma 8.1 The isomorphisms$$KGL \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, KGL^{\wedge n+1}$$of Corollary 7.6 are compatible with all the coface and codegeneracy operators of$$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL,$$except possibly for$$\partial ^0$$and$$\sigma ^0$$. Proof Recall the standard construction (the dual path space of [23, 8.3.14]) which takes a cosimplicial object $$X^ {{{\bf{\scriptscriptstyle \bullet }}}}$$ and produces a new cosimplicial object $$Y^ {{{\bf{\scriptscriptstyle \bullet }}}}$$ with $$Y^n=X^{n+1},$$ coface maps $$\partial _Y^i=\partial _X^{i+1}$$ and codegeneracy maps $$\sigma _Y^i=\sigma _X^{i+1}$$. Applying this construction to $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$ yields the cosimplicial spectrum $$KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2}$$ of Corollary 7.6. Since this is isomorphic to the cosimplicial spectrum $$KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} }$$ by Theorem 7.5, the result follows. Proof of Theorem 0.2(a) Recall that for a motivic spectrum $$E$$ and a graded ring $$A_* $$, we are using the notation $$E \otimes A_* $$ to denote $$\bigoplus _q (T^q\wedge E) \otimes A_q$$. The reason for the index $$2* $$ in the formula that we want to prove:   \[s_0(KGL) \otimes \pi _{2* }(KU^{\wedge n}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n})\] is that the coefficients for complex $$K$$-theory $$\pi _{\ast }(KU)$$ are concentrated in degree $$2$$, that is, as graded rings $$\pi _{\ast }(KU) \cong \mathbb Z[t,t^{-1}],$$ where $$t$$ has degree $$2$$. Now, we proceed to prove the formula. Taking slices in Lemma 8.1, we see that the isomorphisms $$s_* (KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n+1})$$ are compatible with all of the coface and codegeneracy operators, except possibly for $$\partial ^0$$ and $$\sigma ^0$$. Remark 8.2 It is tempting to consider the cosimplicial spectrum $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_H({\Bbb {Z}}, {\Bbb {Z}})$$, where $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_H({\Bbb {Z}}, {\Bbb {Z}})$$ is the cobar complex over $$H$$. However, its coface maps $$\partial ^0$$ and codegeneracy maps $$\sigma ^0$$ are not the same as in $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$. To finish the proof of Theorem 0.2(b), we need to show that the isomorphisms in Lemma 8.1 are indeed compatible with $$\partial ^0$$ and $$\sigma ^0$$, under the two additional assumptions. We will do so in Propositions 8.5 and 8.11. Definition 8.3 The map $$v_K:T\wedge {{\bf{1}}} \to KGL\wedge KGL$$ is defined to be $$\eta _R(u_K)$$, where $$u_K:T\to KGL$$ is the map of 7.1. We define $$v: T\wedge KGL\wedge KGL \to KGL\wedge KGL$$ to be multiplication by $$v_K$$. Lemma 8.4 We have commutative diagrams:  Proof For reasons of space, we write $$K$$ for $$KGL$$. In the diagram below, the top rectangle commutes by definition of $$v_K$$, and the bottom rectangle commutes because $$\eta _R$$ is a morphism of ring spectra.   The left and right verticals are the maps $$u$$ and $$v$$, respectively. This establishes commutativity of the first square in Lemma 8.4. As the second square is the composition of the natural isomorphism $$T\wedge s_q(E) \, {\mathop {\to }\limits ^{{\cong } }}\, s_{q+1}(T\wedge E)$$ and the $$q$$th slice of the first square, it also commutes. Recall that $$s_0({{\bf{1}}})\to s_0(KGL)$$ is an isomorphism if the base $$S$$ is smooth over a perfect field $$k$$, or any scheme over a field of characteristic 0. This is so because it is true over any perfect field (see [9, 21]), and $$s_0$$ commutes with the pullback $${\cal {{SH}}} (k)\to {\cal {{SH}}} (S)$$ in these cases; see [13, 2.16 and 3.7]. Proposition 8.5 Assume that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism. Then the isomorphisms$$s_q(KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL^{\wedge n+1})$$are compatible with$$\partial ^0$$. Proof Since $${{\bf{1}}}$$ is an initial object, there is a unique map from $${{\bf{1}}}$$ to $$KGL\wedge KGL$$; the assumption implies that $$\eta _L=\eta _R$$ as maps $$s_0(KGL) \to s_0(KGL\wedge KGL)$$. Now for any $$q\in {\Bbb {Z}}$$, Lemma 6.3 says that $$u^q:T^q\wedge s_0(KGL) \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL)$$. By Lemma 8.4, we have a diagram   This determines the maps $$\eta _R:s_{q}(KGL)\to s_{q}(KGL \wedge KGL)$$. Summing over $$q$$, and invoking Lemma 6.3 twice, we see that $$\eta _R$$ is the map   \[s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \longrightarrow s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1},v,v^{-1}]\] sending the copy of $$s_0(KGL)$$ indexed by $$u^q$$ to the copy of $$s_0(KGL)$$ indexed by $$v^q$$. Since we saw in Example 3.2 that this is the same as the map induced from $$\eta _R: {\Bbb {Z}} [u,u^{-1}]\to {\Bbb {Z}} [u,u^{-1}] \otimes H$$, this shows that we have a commutative diagram:   The result follows for $$n>0$$, since $$\partial ^0:KGL^{\wedge n}\to KGL^{\wedge n+1}$$ is $$\eta _R\wedge KGL^{\wedge n-1}$$. To conclude the proof of Theorem 0.2, we have to compare the slices of the codegeneracy $$\sigma ^0:KGL\wedge KGL\to KGL$$ (the product) with the map of Example 3.2,   \[s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes H \, {\mathop {\longrightarrow }\limits ^{{\varepsilon } }}\, s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}], \quad \varepsilon (u)=\varepsilon (v)=u. \] (8.6) Recall from 1.3 that $${\Bbb {Z}} [t,t^{-1}]$$ is a subring of $$H$$, and that $$\varepsilon :H\to {\Bbb {Z}}$$ sends $$t$$ to $$1$$. Lemma 8.7 The restriction of the product$$s_* (KGL\wedge KGL)\to s_* (KGL)$$to  \[s_* (KGL) \otimes {\Bbb {Z}} [t,t^{-1}] \longrightarrow s_* (KGL) \otimes H \cong s_* (KGL\wedge KGL)\]equals the composition  \[s_* (KGL) \otimes {\Bbb {Z}} [t,t^{-1}] \, {\mathop {\longrightarrow }\limits ^{{\varepsilon } }}\, s_* (KGL) \otimes {\Bbb {Z}} \cong s_* (KGL).\] Proof Since $$\sigma ^0:KGL\wedge KGL\to KGL$$ is a left inverse to both $$\eta _L$$ and $$\eta _R$$, and $$u=\eta _L(u)$$, $$v=\eta _R(u)$$, we see that $$\sigma ^0$$ sends the copies of $$s_0(KGL)$$ indexed by the monomials $$u^q$$ and $$v^q$$ (in $${\Bbb {Z}} [u,u^{-1}] \otimes H$$) to the copies indexed by $$u^q$$. Since $$\sigma ^0$$ is a map of ring spectra, the copies indexed by $$u^i \otimes t^j=u^{i-j}v^j$$ map to the copies indexed by $$u^{i}$$. Recall from Example 7.7 that $$KGL^{\wedge n+1} \otimes {\Bbb {Q}} \cong KGL \otimes {\Bbb {Q}} [t_0,t_0^{-1},\ldots , t_n,t_n^{-1}]$$. This is $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{ {\Bbb {Q}} [t,t^{-1}]}({\Bbb {Z}}, {\Bbb {Z}})$$, where $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{ {\Bbb {Q}} [t,t^{-1}]}({\Bbb {Z}}, {\Bbb {Z}})$$ is the $$n$$th term of the cobar complex of Example 3.2 for the Hopf algebra $${\Bbb {Q}} [u,u^{-1}]$$. Now consider $$(R,\Gamma )$$ with $$R= {\Bbb {Q}} [u,u^{-1}]$$ and $$\Gamma =R[t,t^{-1}]$$, as in Example 3.2. Combining Lemma 8.1, Proposition 8.5 and Lemma 8.7, we obtain the rational version of the KGL slice conjecture 8: Corollary 8.8 Let$$(R,\Gamma )$$be as above, and assume that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism. As cosimplicial motivic ring spectra,  \[s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL \otimes {\Bbb {Q}}) \cong s_0(KGL) \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{\Gamma }(R,R).\] As in [3, 6.1], it is easy to verify that the diagram   (8.9) commutes, where $$b_K$$ is $$KGL\wedge b$$, $$b$$ is the map in (7.2) and the right side is multiplication by $$v_K$$ (smashed with $$T^{-1}$$). Lemma 8.10 If$$S$$is smooth over a perfect field, then$$s_0({{\bf{1}}})\to H {\Bbb {Z}}$$is an isomorphism in$${\cal {{SH}}} (S)$$. Proof This is true over the ground field $$F$$, by [9, 11.3.6]. Now the slice functors $$s_q$$ commute with the pullback $$\pi ^* $$ over $$\pi :S\to \mathrm {Spec} (F)$$; see [13, 2.16]. Because $$S$$ is smooth, $$H {\Bbb {Z}}$$ also pulls back over $$\pi ^* $$ (see [18, 6.1; 22, 3.18]), and we have   \[s_0({{\bf{1}}}_S)=\pi ^*s_0({{\bf{1}}}_F) = \pi ^* (H {\Bbb {Z}}_F) = H {\Bbb {Z}}_S.\] Now the group $${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(H {\Bbb {Z}},H {\Bbb {Z}})\cong H^0(S, {\Bbb {Z}})$$ is torsion-free, and equal to $${\Bbb {Z}}$$ if $$S$$ is connected; cf. [20, 3.7]. It follows that if $$S$$ is smooth over a perfect field, so that $$s_0({{\bf{1}}})\cong H {\Bbb {Z}}$$, the hypothesis of the following proposition is satisfied. Proposition 8.11 Assume that$${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))$$is torsion-free. Then the isomorphisms$$s_q(KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL^{\wedge n+1})$$are compatible with$$\sigma ^0$$. Proof Every element of $$H$$ is a power of $$t$$ times a numerical polynomial, so it suffices to check the test maps $$\alpha _n:s_0(K)\to s_0(KGL) \otimes H\cong s_0(KGL\wedge KGL)$$ corresponding to the elements $$\binom {t}{n}$$ of $$H$$. If $$n=0,1,$$ we are done by Lemma 8.7. For $$n\ge 2,$$ the composition   \[s_0(KGL) \, {\mathop {\longrightarrow }\limits ^{{n!} }}\, s_0(KGL) \, {\mathop {\longrightarrow }\limits ^{{\alpha _n} }}\, s_0(KGL\wedge KGL) \, {\mathop {\longrightarrow }\limits ^{{\sigma ^0} }}\, s_0(KGL)\] is given by $$\sigma ^0(t(t-1)\ldots (t-n+1))=0$$ in the ring $$[s_0(KGL),s_0(KGL)]$$, which is $${\Bbb {Z}}$$ by Lemma 8.10. Since $$\sigma ^0\alpha _n$$ corresponds to an integer, and is killed by $$n!$$, we must have $$\sigma ^0\alpha _n=0$$. Acknowledgements The authors are grateful to Oliver Röndigs and Paul Østvæ r for pointing out the calculations in Adams’ book [1], to Markus Spitzweck for conversations about [16], and to all three for useful discussions about the slice conjectures. References 1 Adams J. F., Stable homotopy and generalised homology , Chicago Lectures in Mathematics ( University of Chicago Press, Chicago, IL, 1974). 2 Adams J. F. Clarke F., ‘ Stable operations on complex $$K$$-theory’, Ill. J. Math.  21 ( 1977) 826– 829. 3 Adams J. F. Harris A. Switzer B., ‘ Hopf algebras of cooperations for real and complex K-theory’, Proc. London Math. Soc.  23 ( 1971) 385– 408. Google Scholar CrossRef Search ADS   4 Déglise F., ‘ Around the Gysin triangle II’, Doc. Math.  13 ( 2008) 613– 675. 5 Gepner D. Snaith V., ‘ On the motivic spectra representing algebraic cobordism and algebraic $$K$$-theory’, Doc. Math.  14 ( 2009) 359– 396. 6 Hartshorne R., Algebraic geometry  ( Springer, Berlin, 1977). 7 Hoyois M., ‘ From algebraic cobordism to motivic cohomology’, J. Reine u. Ang. Math , to appear; doi:10.1515/crelle-2013-0038, arXiv:1210.7182. 8 Johnson K., ‘ The action of the stable operations of complex $$K$$-theory on coefficient groups’, Ill. J. Math.  28 ( 1984) 57– 63. 9 Levine M., ‘ The homotopy coniveau tower’, J. Top.  1 ( 2008) 217– 267. Google Scholar CrossRef Search ADS   10 Panin I. Pimenov K. Röndigs O., ‘ On Voevodsky's algebraic $$K$$-theory spectrum’, Algebraic Topology, Abel Symp.  4, 279– 330. 11 Pelaez P., ‘ Multiplicative properties of the slice filtration’, Astérisque  ( 2011). 12 Pelaez P., ‘ On the orientability of the slice filtration’, Homology Homotopy Appl.  13 ( 2011) 293– 300. Google Scholar CrossRef Search ADS   13 Pelaez P., ‘ On the functoriality of the slice filtration’, J. K-theory  11 ( 2013) 55– 71. Google Scholar CrossRef Search ADS   14 Ravenel D., Complex cobordism and stable homotopy groups of spheres , Pure and Applied Mathematics 121 ( Academic Press, New York, NY, 1986). 15 Snaith V., ‘ Localized stable homotopy of some classifying spaces’, Math. Proc. Camb. Phil. Soc.  89 ( 1981) 325– 30. Google Scholar CrossRef Search ADS   16 Spitzweck M., ‘ Slices of motivic Landweber spectra’, J. K-theory  9 ( 2012) 103– 117. Google Scholar CrossRef Search ADS   17 Spitzweck M. Østvaer P. A., ‘ The Bott inverted infinite projective space is homotopy algebraic $$K$$-theory’, Bull. Lond. Math. Soc.  41 ( 2009) 281– 292. Google Scholar CrossRef Search ADS   18 Voevodsky V., ‘ $${\Bbb {A}}^1$$-homotopy theory’, Proceedings of the International Congress of Mathematicians , Vol. I (Berlin, 1998), number Extra Vol. I (electronic), 1998, 579– 604. 19 Voevodsky V., ‘ Open problems in the motivic stable homotopy theory, I’, Motives, polylogarithms and Hodge theory , Part I (Irvine, CA, 1998), International Press Lecture Series 3 ( International Press, Somerville, MA, 2002) 3– 34. 20 Voevodsky V., ‘ Reduced power operations in motivic cohomology’, Publ. Math. Inst. Hautes Études Sci.  ( 2003) 1– 57. 21 Voevodsky V., ‘ On the zero slice of the sphere spectrum’, Tr. Mat. Inst. Steklova  246 ( 2004) 106– 115. 22 Voevodsky V., ‘ Motivic Eilenberg-MacLane spaces’, Publ. Math. Inst. Hautes Études Sci.  ( 2010) 1– 99. 23 Weibel C., An introduction to homological algebra  ( Cambridge University Press, Cambridge, 1994). 24 Weibel C., The K-book , Graduate Studies in Mathematics 145 ( American Mathematical Society, Providence, RI, 2013). © 2014 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Slices of co-operations for KGL

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Oxford University Press
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© 2014 London Mathematical Society
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0024-6093
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1469-2120
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10.1112/blms/bdu015
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Abstract

We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic $$K$$-theory. Introduction Fix a finite-dimensional Noetherian separated base scheme $$S$$, and consider the motivic stable homotopy category $${\cal {{SH}}} (S)$$ as defined in [18]. We write $$KGL$$ for the motivic spectrum representing homotopy invariant $$K$$-theory in $${\cal {{SH}}} (S)$$. In this paper, we study Voevodsky's conjecture on slices of co-operations for $$KGL$$ (see [19, Conjecture 8]), which describes the motivic slices of the motivic spectra $$KGL\wedge \ldots \wedge KGL$$. We verify the conjecture when $$S$$ is smooth over a perfect field (see Theorem 0.1). To describe the conjecture, recall that for any motivic spectrum $$E$$, Voevodsky introduced a natural ‘slice’ tower $$\cdots \to f_{q+1}E\to f_qE\to \cdots$$ of motivic spectra, and defined triangulated ‘slice’ functors $$s_q$$ fitting into cofibration sequences $$f_{q+1}E\to f_qE\to s_qE$$. Roughly, $${\{ } f_qE{\} }$$ is the analogue of the Postnikov tower in topology; $$s_q$$ is the analogue of the functor $$X\mapsto K(\pi _qX,q)$$. If $$E$$ is a topological ring spectrum, the ring $$E_*E=\pi _* (E\wedge E)$$ is called the ring of co-operations for $$E$$; the name comes from [3], where $$\pi _* (KU\wedge KU)$$ is worked out (see Section 1). The title of this paper comes from viewing the motivic spectrum $$KGL\wedge KGL$$ as giving rise to co-operations for $$K$$-theory. The easy part of Voevodsky's Conjecture 8 says that there is an isomorphism   \[s_q(KGL\wedge KGL) \cong (T^q\wedge H {\Bbb {Z}}) \otimes \pi _{2q}(KU\wedge KU).\] Here, $$H {\Bbb {Z}}$$ is the motivic Eilenberg–Mac  Lane spectrum in $${\cal {{SH}}} (S)$$, $$T$$ is the motivic space represented by the pointed projective line, and the tensor product of a spectrum with an abelian group has its usual meaning (see Notation 0.5). Since a ring spectrum $$E$$ is a monoid object in spectra, co-operations fit into a cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} E = E^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. This is a general construction: in any category with product $$\wedge$$, a monoid $$E$$ determines a triple $$\top X=X\wedge E$$ and an augmented cosimplicial object $$n\mapsto X\wedge E^{\wedge n+1}$$ for any object $$X$$; the cofaces are given by the unit $${{\bf{1}}} \to E$$ and the codegeneracies are given by the product $$E\wedge E\to E$$; see [23, 8.6.4]. We write $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} E$$ when $$X= {{\bf{1}}}$$, so $$N^{n}E = E^{\wedge n+1}$$. Voevodsky's Conjecture 8 (Slices of co-operations for $$KGL$$) As cosimplicial motivic spectra, the $$q$$th slice satisfies   \[s_q N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL) \cong (T^q\wedge H {\Bbb {Z}}) \otimes \pi _{2q} N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU).\] One reformulation of this conjecture is to take the direct sum over $$q$$ and use the fact that $$s_*E=\bigoplus s_q(E)$$ is a graded motivic ring spectrum (see [11, 3.6.13]). It is convenient to adopt the notation that $$E \otimes A_* $$ denotes $$\bigoplus _q (T^q\wedge E) \otimes A_q$$ for a motivic spectrum $$E$$ and a graded ring $$A_* $$. In this notation, we prove: Theorem 0.1 Assume that$$S$$is smooth over a perfect field. Then there is an isomorphism of cosimplicial ring spectra  \[s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL) \cong H {\Bbb {Z}} \otimes \pi _{2* }N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU).\] Remark If $$S$$ is over a field of characteristic 0, it seems likely that Theorem 0.1 would follow from the work of Spitzweck [16] on Landweber exact spectra. Such an approach would depend heavily on the Hopkins–Morel–Hoyois theorem [7]. Conjecture 8 is intertwined with Voevodsky's Conjectures 1, 7 and 10 in [19], that $$H {\Bbb {Z}} \leftarrow {{\bf{1}}} \to KGL$$ induces isomorphisms   \[H {\Bbb {Z}} \cong s_0(H {\Bbb {Z}}) \,{\mathop {\longleftarrow }\limits ^{{\cong } }}\, s_0({{\bf{1}}}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_0(KGL),\] and thus that $${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))\cong H^0(S, {\Bbb {Z}})$$. These are known to hold when the base $$S$$ is smooth over a perfect field by the work of Voevodsky and Levine (see [9, 10.5.1 and 11.3.6; 21]), or singular over a field of characteristic 0 (see [13]). Here is our main result, which evidently implies Theorem 0.1. Theorem 0.2 Let$$S$$be a finite-dimensional separated Noetherian scheme. Then we have the following. There are isomorphisms for all$$n\ge 0$$:  \[s_0(KGL) \otimes \pi _{2* }(KU^{\wedge n}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n}).\]These isomorphisms commute with all of the coface and codegeneracy operators except possibly$$\partial ^0$$and$$\sigma ^0$$. Assume in addition that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism in$${\cal {{SH}}} (S),$$and that$${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))$$is torsion-free. Then the maps in (a) are the components of an isomorphism of graded cosimplicial motivic ring spectra$${:}$$  \[s_0(KGL) \otimes \pi _{2* }N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL).\] The case $$n=0$$ of Theorem 0.2(a), that $$s_0(KGL) \otimes \pi _{2* }KU\cong s_*KGL$$, is immediate from the periodicity isomorphism $$T\wedge KGL\cong KGL$$ defining the motivic spectrum $$KGL$$ and the formula $$\pi _{2* }KU\cong {\Bbb {Z}} [u,u^{-1}]$$. The need to pass to slices is clear at this stage, because $$\pi _{2n,n}KGL\cong K_n(S)$$ for $$S$$ smooth over a perfect field. The left side of Theorem 0.2 is algebraic in nature, involving only the cosimplicial ring $$\pi _{2* }(KU^{\wedge n+1})$$ and $$H=KU_0(KU)$$. In fact, $$\pi _{2* }(KU^{\wedge n+1})$$ is the cobar construction $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_\Gamma (R,R)\cong KU_* \otimes H^{ \otimes n}$$ for the Hopf algebroid $$(R,\Gamma )=(KU_* ,KU_*KU)$$. We devote the first four sections to an analysis of this algebra, focussing on the rings $$F=KU_0({\Bbb {C}} {\mathbb P}^{\infty })$$ and $$H$$. Much of this material is well known, and due to Frank Adams [1]. If $$E$$ is an oriented motivic spectrum, the projective bundle theorem says that $$E\wedge {\mathbb P}^{\infty } \cong E \otimes F$$ and hence that $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n}\cong E \otimes F^{ \otimes n}$$. Using this, we establish a toy version of Theorem 0.2 in Propositions 5.8 and 6.4 that, as cosimplicial spectra, $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KGL \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ (the cobar construction on $$F$$) and   \[s_* (KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_* (KGL) \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_0(KGL) \otimes KU_* ({ {\Bbb {C}} {\mathbb P}^{\infty } }^{\wedge n}). \] (0.3) Using a theorem of Snaith [5, 17], we use the toy model to show that $$KGL\wedge KGL\cong KGL \otimes H$$ and more generally (in Corollary 7.6) that   \[KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2} = KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \] (0.4) as cosimplicial spectra. Taking slices in (0.4) gives the isomorphisms in Theorem 0.2(a), and (with a little decoding) also proves compatibility with every coface and codegeneracy map except for $$\partial ^0$$ and $$\sigma ^0$$. This proves Theorem 0.2(a). Compatibility with the coface maps $$\partial ^0$$ is established in Proposition 8.5, using the extra hypothesis that $$s_0({{\bf{1}}})\cong s_0(KGL)$$, and compatibility with the codegeneracy map $$\sigma ^0$$ is established in Lemma 8.7, using the torsion-free hypothesis. This proves Theorem 0.2(b). The paper is organized as follows. In Sections 1 and 2, we introduce the binomial rings $$F$$ and $$H$$. As noted in Remark 1.7, $$KU_* ({\Bbb {C}} {\mathbb P}^{\infty })=KU_* \otimes F$$ and $$KU_* (KU)=KU_* \otimes H$$. In Section 3, we quickly review Hopf algebroids and the algebroid structure on $$(KU_* ,KU_*KU)$$. In Section 4, we recall how $$\pi _*N^{ {{{\bf{\scriptscriptstyle \bullet }}}} }KU$$ is the cobar complex for this algebroid, by showing that $$KU\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KU \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$, where $$C^{ {{{\bf{\scriptscriptstyle \bullet }}}} }(F,F)$$ is the cobar complex of the Hopf algebra $$F$$. Most of this material is due (at least in spirit) to Adams. In Section 5, we show that $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ is $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$, by mimicking the development of Section 4. In Section 6, we show that the slice functors commute with direct sums, and deduce (0.3). Section 7 establishes (0.4), and Theorem 0.2 is established in Section 8. Notation 0.5 We will write $$S$$ for a Noetherian separated base scheme of finite Krull dimension, and $${\cal {{SH}}} (S)$$ or simply $${\cal {{SH}}}$$ for the Morel–Voevodsky motivic stable homotopy category of $$T$$-spectra. For any graded abelian group $$A$$ and motivic spectrum $$E$$, we can form a motivic spectrum $$E \otimes A$$, as follows. If $$A$$ is free with a basis of elements $$a_i$$ in degrees $$d_i$$, then $$E \otimes A$$ is the wedge of the $$E\wedge T^{d_i}$$, and we may regard $$a_i$$ as a map from $$E\wedge T^{d_i}$$ to $$E \otimes A$$. For general $$A$$, choose a free graded resolution $$0\to P_1\to P_0\to A\to 0$$ and define $$E \otimes A$$ to be the cofibre of $$E \otimes P_1\to E \otimes P_0$$; Shanuel's Lemma implies that this is independent of the choice of resolution, up to isomorphism. The choice of a lift of a homomorphism $$A\to B$$ is unique up to chain homotopy, so it yields a map $$E \otimes A\to E \otimes B$$ unique up to homotopy. Given homomorphisms $$A\to B\to C$$, this yields a homotopy between the composition $$E \otimes A\to E \otimes B\to E \otimes C$$ and $$E \otimes A\to E \otimes C$$. That is, this construction gives a lax functor from abelian groups to strict motivic spectra over $$E$$, and a functor to motivic spectra over $$E$$. There is a natural associative map $$({{\bf{1}}} \otimes A)\wedge ({{\bf{1}}} \otimes B)\to {{\bf{1}}} \otimes A \otimes B$$, at least if $$A$$ and $$B$$ have no summands $${\Bbb {Z}} /2$$, $${\Bbb {Z}} /3$$ or $${\Bbb {Z}} /4$$ [24, IV.2.8]; it is an isomorphism if $$A$$ and $$B$$ are free abelian groups. Thus, if $$E$$ is a ring spectrum and $$A$$ is a ring, the composition $$(E \otimes A)\wedge (E \otimes A)\to E \otimes A$$ makes $$E \otimes A$$ into a ring spectrum. 1. Universal binomial rings Recall that a binomial ring is a subring $$R$$ of a $${\Bbb {Q}}$$-algebra which is closed under the operations $$r\mapsto \binom {r}{n}$$. It is a $$\lambda$$-ring with operations $$\lambda ^n(r) = {\binom rn}$$. For example, consider the subring $$F$$ of $${\Bbb {Q}} [t]$$ consisting of numerical polynomials, polynomials $$f(t)$$ with $$f(n)\in {\Bbb {Z}}$$ for all integers $$n\gg 0$$. It is well known that $$F$$ is free as an abelian group, and that the $$\alpha _n=\binom tn$$ form a basis. It is not hard to verify the formula that   \[t{\binom tn} = n{\binom tn} + (n+1){\binom t{n+1}}. \] (1.1) The general ring structure of $$F$$ is determined by the combinatorial identity:   \[\alpha _i * \alpha _j = \sum _{k\le i+j}\binom {k}{k-i,k-j,i+j-k}\alpha _k. \] (1.2) Here, $$\binom {k}{a,b,c}$$ denotes $$k!/a!b!c!$$. (To derive (1.2), note that the left-hand side counts pairs of subsets of a set with $$t$$ elements. If the union of an $$i$$-element set and a $$j$$-element set has $$k$$ elements, the sets intersect in $$i+j-k$$ elements.) The universal polynomials for $$\lambda ^m(\lambda ^n(r))$$ show that the numerical polynomials form a binomial ring. In fact, $$F$$ is the free binomial ring on one generator $$t$$: if $$R$$ is binomial and $$r\in R$$, the canonical extension of the universal ring map $${\Bbb {Z}} [t]\to R$$ to $$F\to R\otimes {\Bbb {Q}}$$ factors uniquely through a map $$F\to R$$. Definition 1.3 Let $$H$$ denote the localization $$F[1/t]$$ of the ring of numerical polynomials; it is a subring of $${\Bbb {Q}} [t,1/t]$$. Here is a useful criterion for membership in $$H$$. Lemma 1.4 $$H=F[1/t]$$is the ring of all$$f(t)\in {\Bbb {Q}} [t,1/t]$$such that for any positive integer$$a$$we have$$f(a)\in {\Bbb {Z}} [1/a]$$. Proof (Cf. [3, 5.3]) Multiplying $$f$$ by a suitable power of $$t$$, we may assume that $$f(t)\in t^\nu {\Bbb {Q}} [t]$$, where $$\nu$$ is the highest exponent of any prime occurring in the denominators of the coefficients of $$f$$. It suffices to show that $$f$$ is a numerical polynomial. Fix $$a>0$$ and let $$p$$ be a prime. If $$p|a,$$ then $$p$$ does not appear in the denominator by construction; if $$p\nmid a$$ then $$p$$ does not appear in the denominator of $$f(a)$$ by hypothesis. Hence, $$f(a)\in {\Bbb {Z}}$$, as desired. Recall that $${\Bbb {Q}} [t,1/t]$$ is a Hopf algebra with $$\Delta (t)=t_1t_2$$. The usual proof [6, I.7.3] that the functions $$\binom tn$$ form a basis of the ring $$F$$ of numerical polynomials $$f(t)$$ is easily modified to show that the functions $$\binom {t_1}{m}\binom {t_2}{n}$$ form a $${\Bbb {Z}}$$-basis of the ring of numerical polynomials $$f(t_1,t_2)$$ in $${\Bbb {Q}} [t_1,t_2]$$. Identifying $${\Bbb {Q}} [t_1,t_2]$$ with $${\Bbb {Q}} [t] \otimes {\Bbb {Q}} [t]$$, we obtain a canonical isomorphism between the subring $$F\otimes F$$ of $${\Bbb {Q}} [t] \otimes {\Bbb {Q}} [t]$$ and the ring of numerical polynomials in $${\Bbb {Q}} [t_1,t_2]$$. Theorem 1.5 $$H$$is a Hopf subalgebra of$${\Bbb {Q}} [t,1/t]$$. $$H$$is a binomial ring; it is the free binomial ring on a unit. Proof For (a), it suffices to show that $$\Delta : {\Bbb {Q}} [t]\to {\Bbb {Q}} [t_1,t_2]$$ sends $$F[1/t]\subset {\Bbb {Q}} [1/t]$$ into the subring $$F[1/t]\otimes F[1/t]$$. But $$\Delta$$ sends $$\binom tn$$ to $$f(t_1,t_2)=\binom {t_1t_2}{n}$$, which is a numerical polynomial and hence belongs to $$F\otimes F$$. Thus, $$\Delta$$ sends $$t^{-k}\binom {t}{n}$$ to a $${\Bbb {Z}}$$-linear combination of the functions $$(t_1t_2)^{-k}\binom {t_1}{m}\binom {t_2}{n}$$, which is in $$H \otimes H$$. For (b), fix $$f(t)/t^m$$ in $$F[1/t]$$, and set $$g_k(t)=\lambda ^k(f/t^m)$$. For each non-zero $$a\in {\Bbb {Z}}$$, $${\Bbb {Z}} [1/a]$$ is a binomial ring, so $$g_k(a)= \lambda ^k(f(a)/a^m)$$ is in $${\Bbb {Z}} [1/a]$$. Theorem 1.6 As an abelian group,$$H$$is free. Proof For $$m\le n$$, let $$F(m,n)$$ denote the intersection of $$H$$ with the $${\Bbb {Q}}$$-span of $$t^m,\ldots ,t^n$$ in $${\Bbb {Q}} [t,1/t]$$. Then the proof of [2, 2.2] goes through to prove that $$F(m,n)\cong R^{1+n-m}$$ and that $$H$$ is free abelian, given the following remark: For each $$k,m,n$$, $$\binom {kt}n$$ is a numerical polynomial, so $$(kt)^{-m}{\binom {kt}n}$$ is certainly in $$F[1/t][1/k]$$. Remark 1.7 The ring $$KU_{\ast } \otimes F$$ is $$KU_* ({\Bbb {C}} {\mathbb P}^{\infty })$$, and $$KU_{\ast } \otimes H$$ is isomorphic to the ring $$KU_0(KU)$$. These observations follow from [1, II.3; 3, 2.3, 4.1, 5.3]. A $${\Bbb {Z}}$$-basis of $$KU_* (KU)$$ was given in [8, Corollary 13]. Since we will be interested in the algebras $$R\otimes F[1/t]$$ over different base rings $$R$$, we now give a slightly different presentation of $$F_R=R\otimes F$$ and $$F_R[1/t]$$. As an $$R$$-module, $$F_R$$ is free with countable basis $${\{ } \alpha _{0}, \alpha _{1}, \ldots {\} }$$, and we are given an $$R$$-module map $$T:F_R\to F_R$$ (multiplication by $$t$$), which by (1.1) is defined as   \[T(\alpha _n)= n\alpha _n + (n+1)\alpha _{n+1}. \] (1.8) Note that $$T(\alpha _0)=\alpha _1$$. The localization $$F_R[1/t]$$ is the colimit of the system   \[F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, F_R \, {\mathop {\longrightarrow }\limits ^{{T} }}\, \cdots . \] (1.9) To describe it, we introduce the bookkeeping index $$t^{-j}$$ to indicate the $$j$$th term in this sequence. By [23, 2.6.8], there is a short exact sequence of $$R$$-modules:   \[0\longrightarrow \bigoplus _{j=0}^{\infty }F_R t^{-j} \, {\mathop {\longrightarrow }\limits ^{{\Phi } }}\, \bigoplus _{j=0}^{\infty }F_R t^{-j} \longrightarrow F_R[1/t] \longrightarrow 0,\] where $$\Phi (\alpha _{n}t^{-j})=T(\alpha _n)t^{-j-1}-\alpha _n t^{-j}$$. Since $$\Phi$$ is $$t^{-1}$$-linear, we may regard $$\Phi$$ as an endomorphism of the free $$R[t^{-1}]$$-module $$F_R[t^{-1}]=\bigoplus F_R t^{-n}$$ with basis $${\{ }\alpha _m{\} }$$, with   \[\Phi (\alpha _n)=T(\alpha _n)t^{-1}-\alpha _n = (n+1)t^{-1}\alpha _{n+1}+ (nt^{-1}-1)\alpha _n.\] By abuse of notation, we write $$\alpha _n$$ for the image in $$F_R[1/t]$$ of the basis element $$\alpha _n$$ of $$F_R$$. Thus, $$F_R[1/t]$$ may be presented as the $$R[1/t]$$-module with generators $$\alpha _n$$ and relations   \[(1-nt^{-1})\alpha _n=(n+1)t^{-1}\alpha _{n+1}, \quad n\ge 0. \] (1.10) Note that $$\alpha _{0}=t^{-1}\alpha _{1}$$. It is not hard to verify directly, beginning with (1.9), that multiplication by $$t^{-1}$$ is an isomorphism on $$F_R[1/t]$$. It also follows directly from the ring structure on $$F$$ and the identification of the colimit with $$F_R[1/t]$$. If $$R$$ contains no $${\Bbb {Z}}$$-torsion, so that $$R\subseteq R\otimes {\Bbb {Q}}$$, it is easy to see that $$F_R[1/t]$$ embeds in $$F_{R\otimes {\Bbb {Q}} }[1/t]=R\otimes {\Bbb {Q}} [t,t^{-1}]$$, as we saw at the beginning of this section. The presentation of $$R \otimes H$$ as an $$R$$-module looks different when $$R$$ has $${\Bbb {Z}}$$-torsion, as we shall see in Section 2. Example 1.11 (See [3, 2.2]) If $$R$$ contains $${\Bbb {Q}}$$, the presentation (1.10) shows that $$F_R[1/t]$$ is the $$R[t^{-1}]$$-module with generators $${\{ }\alpha _{0},\alpha _{1},\ldots {\} }$$ modulo the relations   \[t^{-n}\alpha _{n+1} =\frac {1}{(n+1)!}(1-t^{-1})(1-2t^{-1})\ldots (1-nt^{-1})\alpha _1.\] That is, $$F_R[1/t]\cong R[t,t^{-1}]$$ on generator $$\alpha _0$$ with the relations   \[\alpha _{n} = \binom {t}{n}\alpha _0.\] Variant 1.12 Adams uses a variant of the above construction. Fix a unit $$u$$ of $$R$$ and set $$v=ut$$, so $$t=v/u$$. We consider $$B=uT$$ to be multiplication by $$v$$ with (1.8) replaced by $$B(\beta _n)=nu\beta _n + (n+1)\beta _{n+1}$$, where $$\beta _n=u^n\alpha _n$$. Replacing the bookkeeping index $$t^{-1}$$ by $$v^{-1}$$, (1.10) becomes $$(1-nu/v)\beta _n=(n+1)v^{-1}\beta _{n+1}$$, or $$(n+1)\beta _{n+1}=(t-n)u\beta _n$$, and we recover   \[\beta _{n} = u^n{\binom {v/u}n}\beta _0 = u^n{\binom tn}\beta _0.\] These $$\beta _n$$ are the elements described by Adams in [1, II.13.7] as the generators of $$KU_* (KU)$$, regarded as a submodule of $$KU_* (KU) \otimes {\Bbb {Q}} = {\Bbb {Q}} [u,1/u,v,1/v]$$. 2. The $$\ell$$-primary decomposition In this section, we suppose that $$R$$ is an algebra over $${\Bbb {Z}} /\ell ^\nu$$ and give a basis for the colimit $$H_R=F_R[1/t]$$ of the sequence $$F_R \, {\mathop {\to }\limits ^{{T} }}\, F_R \, {\mathop {\to }\limits ^{{T} }}\, F_R \, {\mathop {\to }\limits ^{{T} }}\, \cdots$$ of (1.9). Recall from Section 1 that the elements of $$F$$ are polynomial functions $${\Bbb {N}}\to {\Bbb {Z}}$$, and that the $$\alpha _n=\binom tn$$ form a basis of $$F$$. Lemma 2.1 For each$$q\in {\Bbb {Z}}$$,$$F \otimes {\Bbb {Z}} /q$$embeds in the ring of all functions$${\Bbb {N}}\to {\Bbb {Z}} /q$$. Proof If $$f\in F$$ satisfies $$f(a)\equiv 0\pmod {q}$$ for every $$a>0,$$ then $$h(t)=f(t)/q$$ is a numerical polynomial, and $$f(t)=q\,h(t)$$ is in $$qF$$. Example 2.2 Suppose that $${\Bbb {Z}} /2\subseteq R$$. Then the relations $$\alpha _{2k}=t^{-1}\alpha _{2k+1}$$ and $$(1-t^{-1})\alpha _{2k+1}=0$$ imply that $$H_R=F_R[1/t]$$ is the free $$R$$-module with basis $${\{ }\alpha _{2k+1}, k\ge 0{\} }$$, with $$\alpha _{2k}=\alpha _{2k+1}$$. In fact, $$F/2F$$ and $$H/2H$$ are Boolean rings by Lemma 2.1. In particular, each $$\alpha _n$$ is idempotent in the ring $$F/2F$$, including $$t=\alpha _1$$, and $$H/2H=t\,F/2F$$ is a factor ring of $$F/2F$$. In addition, if $$m\le 2^r-1,$$ then (1.2) implies that $$\alpha _m\alpha _{2^r-1}=\alpha _{2^r-1}$$. When $$R$$ is a $${\Bbb {Z}} /\ell ^\nu$$-algebra, $$F_R$$ has a similar block decomposition. To prepare for it, let $$L'$$ denote the free $$R$$-module on basis $$\alpha _0, \alpha _1,\ldots ,\alpha _{\ell -1}$$, and let $$L$$ denote the submodule of $$L'$$ on $$\alpha _1,\ldots ,\alpha _{\ell -1}$$. Following (1.8), we define maps $$b_k:L'\to L'$$ by   \[b_{k}(\alpha _i)=(k\ell +i)\alpha _i + (k\ell +i+1)\alpha _{i+1}, \quad i=0,\ldots ,\ell -2 \] (2.3) and $$b_k(\alpha _{\ell -1})=(k\ell + \ell -1)\alpha _{\ell -1}$$. Note that $$b_k(L)\subseteq L$$. Lemma 2.4 The maps$$L\to L[1/b_k]\to L'[1/b_k]$$are isomorphisms for all$$k$$. Proof The restriction of $$b_k$$ to $$L$$ is represented by a lower triangular matrix, whose determinant $$\prod _{i=1}^{\ell -1}(k\ell +i)$$ is a unit in $$R$$. Thus, each $$b_k$$ restricts to an automorphism of $$L$$. Since $$\ell ^\nu =0$$ in $$R$$, $$(b_k)^{\nu }$$ maps $$\alpha _0$$ into $$L$$. The result is now straightforward. We can now describe the $$R$$-module $$H_R=R \otimes F_R[1/t]$$ when $$R= {\Bbb {Z}} /\ell$$; the case $$\ell =2$$ was given in Example 2.2. The maps $$\phi _k:L'\to F_R$$, $$\phi _k(\alpha _i)=\alpha _{k\ell +i}$$, induce an isomorphism $$\bigoplus \phi _k:\bigoplus _{k=0}^\infty L' \to F_R$$ under which the map $$\bigoplus b_k$$ is identified with the map $$T$$ of (1.8). Corollary 2.5 If$${\Bbb {Z}} /\ell \subseteq R$$with$$\ell$$prime, the map$$\bigoplus \phi _k: \bigoplus _{k=0}^\infty L \to H_R$$is an isomorphism. Thus, the elements$$\alpha _n$$,$$n\not\equiv 0\pmod {\ell }$$form a basis of$$H_R$$. Proof The maps $$\phi _k:L'\to F$$ satisfy $$\phi _k\circ \bigoplus b_k = T\circ \phi$$. Hence, the map $$\bigoplus \phi _k$$ induces an isomorphism between $$\bigoplus L\cong \bigoplus L'[1/b_k]$$ and $$H_R=F_R[1/t]$$. Theorem 2.6 If$$R$$is a$${\Bbb {Z}} /\ell ^\nu$$-algebra, the$$R$$-module map$$\bigoplus \phi _k:\bigoplus L \, {\mathop {\to }\limits ^{{\cong } }}\, H_R$$is an isomorphism. In particular, the elements$$\alpha _n$$,$$n\not\equiv 0\pmod {\ell }$$,$$n>0$$, form a basis. Proof Let $$F_n$$ denote the free $$R$$-submodule of $$F_R$$ on basis $${\{ }\alpha _{nq+i}: 0\le i<q{\} }$$, where $$q=\ell ^\nu$$. Then $$F_R$$ is the direct sum of the $$F_n$$. Since $$T(\alpha _{nq-1})=(nq-1)\alpha _{nq-1}$$, $$T$$ sends $$F_n$$ into itself, and there are isomorphisms $$F_0\to F_n$$, $$\alpha _i\mapsto \alpha _{nq+i}$$ commuting with $$T$$. Therefore, $$H_R=F_R[1/t]$$ is isomorphic to $$\bigoplus F_n[1/t]$$, and it suffices to show that $$F_0[1/t]$$ is free on the $$\alpha _n$$ with $$0<n<q$$ and $$n\not\equiv 0\pmod {\ell }$$. For $$k=0,\ldots ,\ell ^{\nu -1}$$, let $$F_0^{\ge k\ell }$$ denote the $$R$$-submodule of $$F_0$$ generated by the $$\alpha _i$$ with $$k\ell \le i<q$$. These form a filtration of $$F_0$$, and the maps $$\phi _k:L'\to F_0^{\ge k\ell }$$ induce $$R$$-module isomorphisms with the filtration quotients   \[\bar \phi _k: L' \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, F_0^{\ge k\ell }/F_0^{\ge (k+1)\ell }\] such that $$T\circ \bar \phi _k = b_k\circ \phi _k$$. By Lemma 2.4, it follows that $$\phi _k$$ induces an isomorphism   \[L \cong L'[1/b_k] \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, F_0^{\ge k\ell }/F_0^{\ge k\ell + \ell }[1/t].\] By induction on $$k$$, it follows that $$\bigoplus \phi _k$$ induces an isomorphism $$\bigoplus L \cong F_0[1/t]$$. 3. Hopf algebroids Recall from [14, A1.1.1] that a Hopf algebroid $$(R,\Gamma )$$ is a pair of commutative rings, with maps $$\eta _L,\eta _R:R\to \Gamma$$, $$\varepsilon :\Gamma \to R$$, $$c:\Gamma \to \Gamma$$ and $$\Delta :\Gamma \to \Gamma \otimes _R\Gamma$$ satisfying certain axioms, listed in [14]. Let $$M$$ be a $$\Gamma$$-comodule with structure map $$M \, {\mathop {\to }\limits ^{{\psi } }}\, M \otimes _R\Gamma$$. Recall [14, A1.2.11] that the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,R)$$ is the cosimplicial comodule with $$C^0=M$$ and $$C^n=M \otimes _R\Gamma ^{ \otimes _Rn}$$, with cofaces given by $$\psi$$, $$\eta _L$$ and $$\Delta :\Gamma \to \Gamma \otimes _R\Gamma$$. In particular, when $$M=R$$ (with $$\psi =\eta _R$$), the cobar complex is a cosimplicial ring. Example 3.1 For any commutative algebra $$R$$, $$(R,R \otimes R)$$ is a Hopf algebroid with $$\eta _L(r)=r \otimes 1$$, $$\eta _R(r)=1 \otimes r$$, $$c(r \otimes s)=s \otimes r$$ and $$\Delta (r \otimes s)=r \otimes 1 \otimes s$$. The cobar complex $$C_{R \otimes R}^ {{{\bf{\scriptscriptstyle \bullet }}}} (R,R)$$ is the standard cosimplicial module $$n\mapsto R^{ \otimes n+1}$$. If $$\Gamma$$ is a Hopf algebra over $$R$$, then $$(R,\Gamma )$$ is a Hopf algebroid with $$\eta _L=\eta _R$$ the unit, $$\varepsilon$$ the counit, $$c$$ the antipode and $$\Delta$$ the coproduct. In this case, the cobar complex is classical. Example 3.2 The pair $$(R_ {\Bbb {Q}},\Gamma _ {\Bbb {Q}})=({\Bbb {Q}} [u,1/u], {\Bbb {Q}} [u,1/u,t,1/t])$$ is a Hopf algebroid with $$\eta _L(u)=u \otimes 1$$, $$\eta _R(u)=c(u)=tu$$, $$c(t)=1/t$$, $$\varepsilon (t)=1$$ and $$\Delta (t)=t \otimes t$$. Recall from Definition 1.3 that $$H$$ is the subalgebra of $${\Bbb {Q}} [t,1/t]$$ generated by the $$\binom tn$$. If $$R= {\Bbb {Z}} [u,1/u]$$ and $$\Gamma =R \otimes H$$, then $$(R,\Gamma )$$ is a sub-Hopf algebroid. If we set $$v=tu,$$ then we have $$\eta _R(u)=v$$, $$c(u)=v$$ and   \[\Delta (u)=u \otimes 1,\ \Delta (t)=t \otimes t,\ \Delta (v)=ut \otimes t = 1 \otimes v.\] Of course, $$KU_* =R$$ and $$\pi _* (KU\wedge KU)=\Gamma$$, and the formulas given by Adams in [1, II.13.4] show that $$(R,\Gamma )$$ is the original Hopf algebroid $$(KU_* ,KU_*KU)$$. Lemma 3.3 The Hopf algebroid$$(R,R \otimes H)$$of Example 3.2 is split. Proof The natural inclusion of the Hopf algebra $$({\Bbb {Z}},H)$$ into $$(R,R \otimes H)$$ is a map of algebroids [14, A1.1.9], and the identity map on $$R \otimes H$$ is an algebra map. Fix $$R$$ and $$\Gamma \cong R \otimes H$$ as in Example 3.2. Because the algebroid $$(R,\Gamma )$$ is split, we can say more: $$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,R)=C_H^ {{{\bf{\scriptscriptstyle \bullet }}}} (M, {\Bbb {Z}})$$ (see [14, A1.2.17]), and the latter is the usual cobar complex of $$M$$ as a comodule over the Hopf algebra $$H$$. For $$M=R$$, we have the following description. Corollary 3.4 The cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (R,R)$$has$$C^n=R \otimes H^{ \otimes n}$$. The coface maps$$C^n\to C^{n+1}$$are given by the units$$\eta _L,\eta _R:R\to R \otimes H=\Gamma$$of$$\Gamma$$and$$\Delta _H$$. 4. Oriented topological spectra This section is intended to serve as a template for the motivic constructions which follow, and is restricted to topological spectra. No originality is claimed for these results. To emphasize our analogy, we will write $${\mathbb P}^{\infty }$$ for $${\Bbb {C}} {\mathbb P}^{\infty }_+ $$ in this section. If $$E$$ is a topological ring spectrum, then combining the ring structure of $$E$$ with the $$H$$-space structure of $${\mathbb P}^{\infty }$$ we obtain a natural ring spectrum $$E\wedge {\mathbb P}^{\infty }$$. Following Adams [1] and Snaith [15], the motivation for studying $$F[1/t]$$ comes from the computation of $$E\wedge KU$$, where $$E$$ is an oriented spectrum. Recall that a commutative topological ring spectrum $$E$$ with unit $${{\bf{1}}} \, {\mathop {\to }\limits ^{{\eta _E} }}\, E$$ is said to be oriented if there is a map $${\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\to }\limits ^{{x} }}\, S^2\wedge E$$ whose restriction to $${\Bbb {C}} {\mathbb P} ^1_+ \wedge {{\bf{1}}}$$ is $$1\wedge \eta _E$$ (using $$S^2= {\Bbb {C}} {\mathbb P} ^1$$). As in [1, II.2], these data yield a formal group law on $$E_* [\![x]\!]\cong E^* ({\mathbb P}^{\infty })$$, and dual elements $$\beta _n$$ in $$E_{2n}({\mathbb P}^{\infty })=\pi _{2n}(E\wedge {\mathbb P}^{\infty })$$. When $$E$$ is orientable, the $$\beta _n$$ induce maps   \[ E\wedge S^{2n} {\mathop {\longrightarrow}\limits^{1\wedge \beta _n}} E\wedge E\wedge {\mathbb P}^{\infty } \longrightarrow E\wedge {\mathbb P}^{\infty },\] and these maps induce an isomorphism from $$\bigoplus _{n=0}^{\infty }E\wedge S^{2n}$$ to $$E\wedge {\mathbb P}^{\infty }$$ (by [1, p. 42]). Consequently, we have   \[E_* ({\mathbb P}^{\infty })=\pi _* (E\wedge {\mathbb P}^{\infty }) \cong \bigoplus _{n=0}^{\infty }\pi _* (E\wedge S^{2n}).\] Thus, $$E_* ({\mathbb P}^{\infty })$$ is a free graded $$E_* $$-module on generators $$\beta _n$$ in degree $$2n$$, $$n\ge 0$$. Let $$\xi :S^{2}_+ \cong {\Bbb {C}} \mathbb P^{1}_+ \to {\mathbb P}^{\infty }$$ be the map that classifies the tautological line bundle on $${\Bbb {C}} \mathbb P^{1}$$, and let $$b: {\mathbb P}^{\infty } \wedge {{\bf{1}}} \to S^{-2}\wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}}$$ denote the adjoint of the stable map   \[ S^2 \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{\mathrm {Hopf}}} S^2_+ \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{\xi \wedge 1}} {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } \wedge {{\bf{1}}} {\mathop {\longrightarrow}\limits^{m}} {\mathbb P}^{\infty } \wedge {{\bf{1}}}.\] Let $$j=\xi -1$$ denote the map $${\mathbb P}^{\infty } \to BU_+ \to KU$$ classifying the virtual tautological line bundle of rank $$0$$. By Snaith [15, 2.12], the map $$j$$ induces an equivalence from the homotopy colimit of the sequence   \[{\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\longrightarrow }\limits ^{{b} }}\, S^{-2} {\mathbb P}^{\infty } \wedge {{\bf{1}}} \; \, {\mathop {\longrightarrow }\limits ^{{S^{-2}b\;} }}\, S^{-4} {\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\longrightarrow }\limits ^{{} }}\, \cdots\] to the spectrum $$KU$$. We write $$u=u_{KU}$$ for the map $$S^2\to {\mathbb P}^{\infty } \, {\mathop {\to }\limits ^{{j} }}\, KU$$. The map   \[S^2\wedge KU \, {\mathop {\longrightarrow }\limits ^{{u\wedge 1} }}\, KU\wedge KU \, {\mathop {\longrightarrow }\limits ^{{m} }}\, KU\] is the periodicity isomorphism for $$KU$$ (cf. (7.1)). Composing $$j$$ with the inverse of the periodicity isomorphism, we obtain an orientation for $$KU$$ whose formal group law is $$x+y+uxy$$. Smashing with $$E$$ yields maps $$b_E:E\wedge {\mathbb P}^{\infty } \to S^{-2}\wedge E\wedge {\mathbb P}^{\infty }$$ and an equivalence between $$E\wedge KU$$ and the homotopy colimit of the sequence of spectra:   \[E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b_E}} S^{-2}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{S^{-2} b_E}} S^{-4}\wedge E\wedge {\mathbb P}^{\infty } \longrightarrow \cdots . \] (4.1) We now make the connection with the algebraic considerations of Section 1. Lemma 4.2 The coefficient of$$x^iy^j$$in$$(x+y+uxy)^k$$is$$\binom {k}{k-i,k-j,i+j-k}u^{i+j-k}$$. Proof The product has $$3^k$$ terms. The only terms producing the monomial $$x^iy^j$$ are those with exactly $$k-j$$ of the factors must be $$x$$, exactly $$k-i$$ must be $$y$$, and the remaining $$i+j-k$$ must be $$uxy$$. Corollary 4.3 Suppose that$$E$$is an oriented spectrum with a multiplicative group law$$\mu (x,y)=x+y+uxy$$such that$$u$$is a unit of$$E_* $$. Then the homotopy groups of the cosimplicial spectrum$$E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$form a cosimplicial group$$E_* \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1},$$isomorphic to the cobar complex$$E_* \otimes C_{F \otimes F}^ {{{\bf{\scriptscriptstyle \bullet }}}} (F,F)$$of Example 3.1. Proof By the above remarks, $$\pi _* (E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1})$$ is a free $$E_* $$-module with basis $$\beta _{i_0}\wedge \ldots \beta _{i_n}$$, which we identify with $$E_* \otimes F^{ \otimes n+1}$$ by sending $$\beta _i$$ to $$u^i\alpha _i$$, as in 1.12. The coface maps are easily seen to correspond, and for the codegeneracies it suffices to consider $$\sigma ^0:\pi _* (E\wedge {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty })\to \pi _* (E\wedge {\mathbb P}^{\infty }).$$ Adams showed in [1, II(3.4–5)] that the coefficient $$a_{ij}^k$$ of $$\sigma ^0(\beta _i \otimes \beta _j)$$ is the coefficient of $$x^iy^j$$ in $$(x+y+uxy)^k$$, which is given by Lemma 4.2, and agrees with $$(u^i\alpha _i)\ast (u^j\alpha _j)$$ by (1.2). Lemma 4.4 Suppose that$$E$$is an oriented spectrum with a multiplicative group law$$\mu (x,y)=x+y+uxy$$such that$$u$$is a unit of$$E_* $$. Then  \[\pi _* (E\wedge KU)\cong \varinjlim \pi _* (S^{-2n}\wedge E\wedge {\mathbb P}^{\infty }) \cong E_* \otimes \varinjlim F \cong E_* \otimes H.\] Proof Identifying $$R=E_* $$ and $$F_R=E_* ({\mathbb P}^{\infty })=\pi _* (E\wedge {\mathbb P}^{\infty })$$, Adams shows in [1, II(3.6)] that multiplication by $$\beta _1$$ is given by the formula (1.8), with $$t$$ replaced by $$v=ut$$ as in Variant 1.12. Comparing with (1.9), we see that the ring $$E_* ({\mathbb P}^{\infty })[1/\beta _1]$$ is isomorphic to $$E_* \otimes H$$. From (4.1), we obtain the result. Remark 4.5 Lemma 4.4 fails dramatically if $$u=0$$, for example when $$E=H {\Bbb {Z}}$$. Replacing $$E$$ by $$E\wedge KU$$, which we orient by $$x\wedge \eta _{KU}$$, Lemma 4.4 yields $$\pi _* (E\wedge KU\wedge KU)\cong E_* \otimes H \otimes H$$. The two maps $$E_* \otimes H\to E_* \otimes H \otimes H$$, induced by $$E\wedge KU \to E\wedge KU^{\wedge 2}$$, are given by $$\eta _E \otimes 1$$ and $$1 \otimes \Delta _H$$, and $$E_* \otimes H \otimes H \, {\mathop {\to }\limits ^{{\sigma ^0} }}\, E_* \otimes H$$ is given by Corollary 4.3. An inductive argument establishes: Corollary 4.6 The homotopy groups of the cosimplicial spectrum$$n\mapsto E\wedge KU^{\wedge n+1}$$form the cosimplicial group:  \[\pi _* (E\wedge KU^{\wedge n+1}) = E_* \otimes H^{ \otimes n+1}.\]This is the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (M,E_* )$$of Example 3.2 with$$M=E_* \otimes H$$. When $$E=KU$$, we can also form the cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU)$$, as in the Introduction, and we have: Proposition 4.7 The cosimplicial ring$$\pi _* (N^ {{{\bf{\scriptscriptstyle \bullet }}}} KU)$$is$$KU_* \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} },$$the cobar complex$$C_\Gamma ^ {{{\bf{\scriptscriptstyle \bullet }}}} (KU_* ,KU_* )$$of 3.4. In particular, for every$$n$$we have an isomorphism  \[\pi _* (KU^{\wedge n+1}) \cong KU_* \otimes H^{ \otimes n}.\] $$KU^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} }$$with finite coefficients Fix a prime $$\ell$$ and a power $$q=\ell ^\nu$$, and let $$P$$ denote the Moore spectrum for $${\Bbb {Z}} /q$$. We will assume that $$q$$ is chosen so that $$P$$ is a ring spectrum with unit. For example, if $$\ell \neq 2,3$$, then this is the case for any $$\nu \ge 1$$; see [24, IV.2.8]. Smashing $$P$$ with $$KU$$ yields the standard cosimplicial spectrum $$P\wedge KU^{\wedge n+1}$$ associated to the triple $$-\wedge KU$$. Set $$R=\pi _* (P\wedge KU)$$; the Universal Coefficient Theorem yields $$R= {\Bbb {Z}} /q[u,1/u]$$. Since $$P\wedge KU$$ is oriented by the orientation on $$KU$$, we see from Lemma 4.4 and Theorem 2.6 that $$\pi _* (P\wedge KU\wedge KU)=R \otimes H$$ is a free $$R$$-module with basis $${\{ }\beta _n:n\not\equiv 0\pmod \ell {\} }$$, and from ( 4.6) that $$\pi _* (P\wedge KU^{\wedge n+1})=R \otimes H^{ \otimes n}$$. In fact, $$(R,R \otimes H)$$ is the mod $$q$$ reduction of the Hopf algebroid of Lemma 3.3. Set $$E=P\wedge KU$$. Then each $$\beta _n$$ determines a map $$S^{2n}\wedge E\to E\wedge {\mathbb P}^{\infty } \to E\wedge KU$$. By Lemma 4.4, the direct sum of these maps is a homotopy equivalence of spectra   \[\bigoplus E\wedge S^{2n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge {\mathbb P}^{\infty } {. }\] There is a sequence (4.1) for $$E\wedge {\mathbb P}^{\infty }$$, with colimit $$E\wedge KU$$. By Example 3.2, $$\pi _* (E\wedge KU)\cong E_* \otimes H$$. Since $$R=\pi _* (E)$$ is a $${\Bbb {Z}} /q$$-module, Theorem 2.6 applies: the $$\beta _n$$ with $$n\not\equiv 0\pmod \ell$$ form a basis of $$E_* \otimes H$$. Proposition 4.8 For$$E=P\wedge KU$$, there is a homotopy equivalence of spectra  \[\bigoplus _{n\not\equiv 0 \,{\mathrm {mod}}\,\ell } E\wedge S^{2n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge KU.\] Proof The argument we gave in Corollary 2.5 shows that the left side is the colimit of (4.1). That is, $$E\wedge KU$$. In effect, we just note that the components $$E\wedge S^{2n}\to E\wedge S^{2n-2}$$ of $$b$$ are multiplication by $$n$$ and are thus null-homotopic when $$\ell |n$$, while the maps $$E\wedge S^{2n}\to E\wedge S^{2n}$$ of $$b$$ are multiplication by $$(n+1)$$ and are thus null-homotopic when $$\ell |n+1$$. 5. The motivic $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })$$ We now turn to motivic spectra. The goal of this section is to describe the cosimplicial spectrum $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })= E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ associated to an oriented motivic spectrum $$E$$. Here, the motivic space $${\mathbb P}^{\infty }$$ is defined as the union of the spaces represented by the schemes $${\mathbb P}^n$$. The Segre maps $${\mathbb P} ^m\times {\mathbb P} ^n\to {\mathbb P}^{\infty }$$ induce a map $$m: {\mathbb P}^{\infty } \times {\mathbb P}^{\infty } \to {\mathbb P}^{\infty }$$ making $${\mathbb P}^{\infty } \wedge {{\bf{1}}}$$ into a ring $$T$$-spectrum. Then $$n\mapsto E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$ is the standard cosimplicial spectrum associated to a spectrum $$E$$ and the triple $$-\wedge {\mathbb P}^{\infty }$$ [23, 8.6.4]. If $$E$$ is a ring $$T$$-spectrum, then this is a cosimplicial ring spectrum. A commutative ring $$T$$-spectrum $$E$$ with unit $${{\bf{1}}} \, {\mathop {\to }\limits ^{{\eta } }}\, E$$ is said to be oriented if there is a map $${\mathbb P}^{\infty } \wedge {{\bf{1}}} \, {\mathop {\to }\limits ^{{t} }}\, T\wedge E$$ whose restriction to $${\mathbb P} ^1\wedge {{\bf{1}}}$$ is $$T\wedge \eta$$. (Compare with Section 4.) As pointed out in Example 2.6.4 of [10], $$KGL$$ is an oriented ring spectrum in $${\cal {{SH}}}$$ with a multiplicative formal group law [10, 1.3.1 and 1.3.3]. When $$E$$ is oriented, the projective bundle theorem in [4, 3.2] yields an isomorphism with inverse $$\bigoplus c^n$$:   \[\bigoplus \beta _n: \bigoplus _{n=0}^{\infty }E\wedge T^{n} \, {\mathop {\longrightarrow }\limits ^{{\simeq } }}\, E\wedge {\mathbb P}^{\infty }. \] (5.1) If we let $$F_ {\mathrm {gr}}$$ denote the free graded abelian group on generators $$\beta _n$$ in degree $$n\ge 0$$, then $$E\wedge {\mathbb P}^{\infty }$$ is the free graded $$E$$-module spectrum $$E \otimes F_ {\mathrm {gr}}$$. By (5.1), the terms $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$ are isomorphic to the direct sum of terms $$E\wedge T^{i_0}\wedge \ldots \wedge T^{i_n}$$, $$i_j\ge 0$$. Lemma 5.2 The coface map$$\partial ^j:E\wedge { {\mathbb P}^{\infty } }^{\wedge n}\to E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}$$sends the summand$$E\wedge T^{i_1}\wedge \ldots \wedge T^{i_n}$$to$$E\wedge T^{i_1}\wedge \ldots T^{i_j}\wedge T^0\wedge T^{i_{j+1}}\ldots \wedge T^{i_n}.$$ Proof Since the coface maps are determined by the unit map $${{\bf{1}}} \to {\mathbb P}^{\infty }$$, it suffices to observe that the map   \[E=E\wedge T^0\longrightarrow E\wedge {\mathbb P}^{\infty } \cong \bigoplus E\wedge T^n\] is just the inclusion of $$T^0$$ into $$\bigoplus T^n$$, by the projective bundle theorem. At this point, we may interpret the coface maps in $$E\wedge N^ {{{\bf{\scriptscriptstyle \bullet }}}} ({\mathbb P}^{\infty })$$ in terms of $$F_ {\mathrm {gr}}$$. Recall (from [23, 8.1.9]) that a semi-cosimplicial object in a category is a sequence of objects $$K^n$$ with coface operators $$\partial ^i:K^{n-1}\to K^n$$ ($$0\le i \le n$$) satisfying the cosimplicial identities $$\partial ^j\partial ^i=\partial ^i\partial ^{j-1}$$ if $$i<j$$. For example, $$n\mapsto F_ {\mathrm {gr}} ^{ \otimes n+1}$$ is a semi-cosimplicial graded abelian group, where the codegeneracies insert $$1=\beta _0$$. (The generators $$\beta _n$$ of $$F_ {\mathrm {gr}}$$ lie in degree $$n$$.) We may also form the semi-cosimplicial spectrum $$n\mapsto E \otimes F_ {\mathrm {gr}} ^{ \otimes n+1}$$. Note that the codegeneracy maps $$F_ {\mathrm {gr}} ^{ \otimes n+1}\to F_ {\mathrm {gr}} ^{ \otimes n}$$ associated to the product $$F \otimes F\to F$$ are not graded, as (1.2) shows, so we do not get a cosimplicial spectrum in this way. Corollary 5.3 Let$$E$$be an oriented spectrum$$E$$. Then there is an isomorphism of semi-cosimplicial ring spectra,$$E \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \, {\mathop {\to }\limits ^{{\cong } }}\, E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. Lemma 5.4 For any oriented$$E,$$the component maps$$E\wedge T^n\to E\wedge T^{i}\wedge T^j$$of the diagonal$$E\wedge {\mathbb P}^{\infty } \to E\wedge {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty }$$are the canonical associativity isomorphisms when$$i+j=n,$$and zero otherwise. Proof The maps $$\beta _n:E\wedge T^n\to E\wedge {\mathbb P}^{\infty }$$ in (5.1) satisfy $$c^k\circ \beta _n =\delta _{kn}$$, where $$c^n$$ is the projection $$E\wedge {\mathbb P}^{\infty } \to E\wedge T^n$$. We have a commutative diagram:   The top horizontal composite is $$c^{i+j}\circ \beta _n=\delta _{n,i+j}$$, and the entire composition is the component map in question. The result follows. We now turn to the codegeneracies $$\sigma ^j: E\wedge { {\mathbb P}^{\infty } }^{\wedge n+1}\to E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$. These are all induced from the product $${\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } \to {\mathbb P}^{\infty }$$. After taking the smash product with $$E$$, and using (5.1), we may rewrite the product as an $$E$$-module map   \[\bigoplus _{i,j\ge 0}E\wedge T^i\wedge T^j = (E\wedge {\mathbb P}^{\infty })\wedge (E\wedge {\mathbb P}^{\infty }) \longrightarrow E\wedge {\mathbb P}^{\infty } = \bigoplus _{k\ge 0} E\wedge T^k.\] Since each term $$E\wedge T^i\wedge T^j$$ is a compact $$E$$-module, its image lies in a finite sum. Thus, the product is given by a matrix whose entries $$a_{ij}^k$$ are the component maps $$E\wedge T^i\wedge T^j\to E\wedge T^k$$. In terms of motivic homotopy groups, $$E_{* ,* }({\mathbb P}^{\infty })\cong \bigoplus _{n=0}^{\infty }E_{* ,* }(T^{n})$$ is a free graded $$E_{* ,* }$$-module with generators $$\beta _n\in E_{2n,n}({\mathbb P}^{\infty })$$, $$n\ge 0$$, corresponding to the unit of $$E_{2n,n}(T^n)\cong E_{0,0}$$. Dually, $$E^{* ,* }({\mathbb P}^{\infty })\cong E^{* ,* }[\![x]\!]$$ and (as in topology) the orientation yields a formal group law of the form $$\mu =x+y+ \sum _{i,j\ge 1}a_{i,j}x^{i}y^{j}$$ in $$E_* [\![x,y]\!]\cong E^* ({\mathbb P}^{\infty } \times {\mathbb P}^{\infty })$$, where   \[a_{i,j}\in E_{2i+2j-2,i+j-1} = {\mathrm {Hom}}_{ {\cal {{SH}}} }(T^{i+j},E\wedge T). \] (5.5) We can now extend a result of Adams to the motivic setting. Theorem 5.6 If$$E$$is an oriented motivic ring spectrum, then$$a_{1j}^k = k a_{1,1+j-k},$$with$$a_{1,1+j-k}$$as in (5.5). Proof This follows directly from Lemma 5.4, together with Adams’ calculation in [1, II(3.6), p. 46]. Remark For fixed $$i,j$$ the maps $$a_{ij}^k$$ are zero unless $$k\le i+j$$. Therefore, the image of $$E\wedge T^i\wedge T^j$$ under the product lands in the finitely many terms $$E\wedge T^k$$ with $$k\le i+j$$. Our next task is to replace $$E \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ by $$E \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ to get a cosimplicial spectrum. This is possible when $$E$$ is periodic in the following sense. If $$E_* $$ has a unit in $$E_{2,1}$$, represented by a map $$T \, {\mathop {\to }\limits ^{{u} }}\, E$$, then $$u\wedge E$$ induces an isomorphism $$T\wedge E\to E$$ with inverse $$u^{-1}\wedge E$$. We call it a periodicity map for $$E$$, and say that $$E$$ is periodic. If $$E$$ is oriented and periodic, we can use the periodicity map for $$E$$ to define maps $$\alpha _n:E\to E\wedge {\mathbb P}^{\infty }$$ such that $$\beta _n=u^n\alpha _n$$, resulting in a rewriting of the projective bundle formula (5.1) as   \[\bigoplus \alpha _n: \bigoplus _{n=0}^\infty E \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, E\wedge {\mathbb P}^{\infty }. \] (5.7) Using this new basis, each $$E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$ is isomorphic to $$E \otimes F^{ \otimes n}$$. Now suppose that $$E$$ is an oriented spectrum with a multiplicative formal group law $$\mu =x+y+uxy$$, with $$u$$ a unit in $$E_{2,1}$$. In this case, it is convenient to change our orientation to eliminate $$u$$, as suggested in [1, II(2.1)]. This produces the new formal group law $$x+y+xy$$. To see this, let $$t'$$ denote the element $$ut$$ of $$E^0({\mathbb P}^{\infty })$$; then the formal group law implies that   \[\mu (t')=(ux)+ (uy)+ (ux)(uy)=x'+y'+x'y'.\] Proposition 5.8 Let$$E$$be an oriented ring spectrum$$E$$with multiplicative group law$$x+y+uxy,$$$$u$$a unit. Then the cosimplicial ring spectrum$$E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$is$$E \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. In particular,$$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$is isomorphic to$$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. Proof After the change in orientation indicated above, the projection bundle formula yields isomorphisms $$E \otimes F^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, E\wedge { {\mathbb P}^{\infty } }^{\wedge n}$$. The respective coface maps are insertion of $$1$$ and $$T^0$$ (by Lemma 5.2), so they agree. The codegeneracies of the left side are given by the product $$F \otimes F\to F$$, whose coefficients are given by (1.2). These agree with the matrix entries $$a_{ij}^k$$ on the right by Lemma 4.2 and Corollary 4.3, as the proof of 4.3 shows. 6. The slice filtration Recall [19, 2.2] that $$s_q(E)$$ denotes the $$q$$th slice of a motivic spectrum $$E$$, and that $$s_q(E\wedge T^i)=s_{q-i}(E)\wedge T^i$$. By [12], $$s_0(E)$$ has an additive formal group law. The following result is well known to experts. Proposition 6.1 The slice functors$$s_q$$commute with direct sums. Proof It suffices to prove that $$f_{q}$$ commutes with direct sums, since there is a distinguished triangle of the form $$f_{q+1}\rightarrow f_{q} \rightarrow s_{q}$$. For simplicity, we will restrict ourselves to the case $$q=0$$; the argument for arbitrary $$q$$ is exactly the same. Consider a direct sum $$E=\bigoplus _{\alpha }E_{\alpha }$$ in $${\cal {{SH}}}$$. Since $${\cal {{SH}}} ^{ {\mathrm {eff}} }$$ is closed under direct sums, $$\bigoplus f_0(E_\alpha )$$ is effective. By [11, 3.1.14], there is a family of compact objects $$K$$ such that, for every $$X\to Y$$, $$f_0(X)\to f_0(Y)$$ is an isomorphism if and only if each $$[K,X]\to [K,Y]$$ is an isomorphism. In the case at hand, for every such $$K$$ we have   \[\left [K,\bigoplus f_0(E_\alpha )\right ]\cong \bigoplus [K,f_0(E_\alpha )]\cong \bigoplus [K,E_\alpha ]\cong \left [K,\bigoplus E_\alpha \right ].\] Therefore $$\bigoplus f_0(E_\alpha )\to f_0(\bigoplus E_\alpha )$$ is an isomorphism. Example 6.2 Suppose that $$E$$ is oriented. Applying $$s_q$$ to (5.1) yields the formula $$\bigoplus s_{q-n}(E)\wedge T^n \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(E\wedge {\mathbb P}^{\infty })$$ with component maps $$s_q(\beta _n)$$. More generally, the slice $$s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge n})$$ is isomorphic to   \[\bigoplus s_{r}(E)\wedge T^{i_1}\wedge \ldots \wedge T^{i_n},\] where $$q=r+i_1+ \ldots +i_n$$, and the coface maps in $$s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} })$$ are given by Lemma 5.2, with $$E$$ replaced by $$s_q(E)$$. Thus, except for the codegeneracy maps, the cosimplicial spectrum $$\bigoplus _q s_q(E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1})$$ is isomorphic to $$\bigoplus s_q(E) \otimes F_ {\mathrm {gr}} ^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. We will need the following observation: for any motivic ring spectrum $$E$$, $$s_*E=\bigoplus s_q(E)$$ is a graded motivic ring spectrum; see [11, 3.6.13]). Since $${\Bbb {Z}} [u,u^{-1}]$$ is a graded ring (with $$u$$ in degree 1), we can form the graded motivic ring spectrum $$s_0(E) \otimes {\Bbb {Z}} [u,u^{-1}] = \bigoplus T^q\wedge s_0(E)$$. Lemma 6.3 As ring spectra,$$s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \, {\mathop {\to }\limits ^{{\cong } }}\, \bigoplus s_q(KGL)$$. More generally,$$s_0 E \otimes {\Bbb {Z}} [u,u^{-1}] \, {\mathop {\to }\limits ^{{\cong } }}\, s_*E$$for any oriented ring spectrum$$E$$with a unit$$u$$in$$E_{2,1}$$. Proof The periodicity isomorphisms $$u^q:KGL\wedge T^q\to KGL$$, and more generally $$E\wedge T^q\to E$$, induce isomorphisms $$s_0 E\wedge T^q\cong s_q E$$ compatible with multiplication. These assemble to give the result. (Cf. [16, 6.2].) Proposition 6.4 Suppose that$$E$$is oriented, and has a multiplicative group law$$x+y+uxy$$. If$$u$$is a unit, we have isomorphisms of cosimplicial ring spectra:  \[s_* (E\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_* (E) \otimes F^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_0(E) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F).\]In particular,$$s_* (KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) \cong s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{F \otimes F}(F,F)$$. Proof Applying $$s_q$$ to 5.7 yields the simple formula, with component maps $$s_q(\alpha _n)$$:   \[s_q(E) \otimes F = \bigoplus _{n=0}^\infty s_q(E) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_q(E\wedge {\mathbb P}^{\infty }).\] The rest is immediate from Proposition 5.8 and Example 6.2. When $$E=KGL$$, Propositions 5.8 and 6.4 give the formulas for $$KGL\wedge { {\mathbb P}^{\infty } }^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ and its slices which were mentioned in the Introduction. 7. $$E\wedge KGL$$ In this section, we describe the augmented cosimplicial spectrum $$E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$ associated to an oriented spectrum $$E$$ and the triple $$-\wedge KGL$$.   \[E {\mathop {\longrightarrow }\limits ^{\eta _L}} E\wedge KGL \substack{{\longrightarrow}\\ {\longrightarrow}} E\wedge KGL\wedge KGL \ \substack{{\longrightarrow}\\ {\longrightarrow} \\ \longrightarrow} \ E\wedge KGL^{\wedge 3} \cdots\] Replacing $$E$$ by $$E\wedge KGL^{\wedge n}$$, we are largely reduced to the description of $$E\wedge KGL$$, $$\eta _L$$ and the map $$\sigma ^0:E\wedge KGL\wedge KGL\to E\wedge KGL$$. We begin with a few generalities. The $$T$$-spectrum $$KGL$$ comes with a periodicity isomorphism $$T\wedge KGL\to KGL$$; see [18, 6.8; 19, 3.3]; our description of it is taken from [10, 1.3]. Let $$\xi :T \to {\mathbb P}^{\infty }$$ be the map that classifies the tautological line bundle on $$\mathbb P^{1}$$, and let $$b: {{\bf{1}}} \wedge {\mathbb P}^{\infty } \to T^{-1}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty }$$ denote the adjoint of the map   \[T\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{\xi \wedge 1}} {\mathbb P}^{\infty } \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{m}} {\mathbb P}^{\infty }.\] As observed by Gepner–Snaith [5, 4.17] and Spitzweck–Østvæ r [17], $$KGL$$ is the homotopy colimit of the resulting sequence,   \[ {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b}} T^{-1}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-1}b}} T^{-2}\wedge {{\bf{1}}} \wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-2}b}} \cdots ,\] and the colimit map $${{\bf{1}}} \wedge {\mathbb P}^{\infty } \to KGL$$ is the map classifying the virtual tautological line bundle $$\xi -1$$ of degree 0. We write $$u_K$$ for the map $$T \, {\mathop {\to }\limits ^{{} }}\, {{\bf{1}}} \wedge {\mathbb P}^{\infty } \, {\mathop {\to }\limits ^{{\xi -1} }}\, KGL$$. Then multiplication by $$u_K$$ is the periodicity isomorphism   \[ T\wedge KGL {\mathop {\longrightarrow}\limits^{u_K\wedge 1}} KGL\wedge KGL {\mathop {\longrightarrow}\limits^{m}}, KGL \] (7.1) for the $$T$$-spectrum $$KGL$$; see [10, 1.3.3]. Smashing $$b$$ with any spectrum $$E$$ yields maps $$b_E:E\wedge {\mathbb P}^{\infty } \to T^{-1}\wedge E\wedge {\mathbb P}^{\infty }$$, and yields a sequence of $$T$$-spectra   \[ E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{b_E}} T^{-1}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-1}b_E}} T^{-2}\wedge E\wedge {\mathbb P}^{\infty } {\mathop {\longrightarrow}\limits^{T^{-2}b_E}} \cdots , \] (7.2) with homotopy colimit $$E\wedge KGL$$, parallel to the sequence (4.1) in topology. When $$E$$ is oriented, the adjoint $$E\wedge T\wedge {\mathbb P}^{\infty } \to E\wedge {\mathbb P}^{\infty }$$ of $$b_E$$ is the composition of the map $$\beta _1: E\wedge T\to E\wedge {\mathbb P}^{\infty }$$ of (5.1), smashed with $${\mathbb P}^{\infty }$$, with the product on $${\mathbb P}^{\infty }$$. Therefore, $$b_E$$ corresponds to multiplication by $$\beta _1$$, and the following observation follows from Theorem 5.6. Lemma 7.3 If$$E$$is oriented, the components$$E\wedge T^j\to T^{-1}\wedge E\wedge T^k$$of the map$$b_E$$of (7.2) are multiplication by$$k\ a_{1,1+j-k}$$. We now assume that $$E$$ has a multiplicative formal group law $$\mu =x+y+uxy$$ with $$u\in E_{2,1}$$ a unit of $$E_* $$. If $$\alpha _n:E\to E \otimes F\cong E\wedge {\mathbb P}^{\infty }$$ is the map defined by (5.7), Lemma 7.3 says that   \[b_E(\beta _n) = n\, u\, \beta _n + (n+1)\beta _{n+1}, \quad {\mathrm {or}}\quad b_E(\alpha _n) = n\,\alpha _n + (n+1)\alpha _{n+1}. \] (7.4) Comparing with (1.8), this shows that the map $$b_E: E\wedge {\mathbb P}^{\infty } \to T^{-1}\wedge E\wedge {\mathbb P}^{\infty }$$ is induced from the homomorphism $$T:F\to F$$, $$T(f)=tf$$. Recall from Definition 1.3 that the Hopf algebra $$H$$ is defined by $$H=F[1/t]$$ and that $$C_H^n({\Bbb {Z}}, {\Bbb {Z}})=H^{ \otimes n}$$ in the classical cobar complex (see Example 3.1). Theorem 7.5 If$$E$$has a multiplicative group law$$x+y+uxy$$with$$u$$a unit, we have$$E\wedge KGL \cong E \otimes H$$, and isomorphisms of cosimplicial spectra for all$$q\in {\Bbb {Z}} {:}$$  \[\begin {array}{rl} E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1} & \cong E \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong E \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{H}({\Bbb {Z}}, {\Bbb {Z}}) {, } \\ s_* (E\wedge KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +1}) & \cong s_* (E) \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1} \cong s_* (E) \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{H}({\Bbb {Z}}, {\Bbb {Z}}). \end {array}\]In addition,$$E\to E\wedge KGL$$corresponds to$$E \otimes {\Bbb {Z}} \to E \otimes H$$. Proof By (7.4), multiplication by $$t$$ on $$E \otimes F$$ is given by the same formula as multiplication by $$b_E$$ on $$E\wedge {\mathbb P}^{\infty }$$. It follows that the homotopy colimit $$E\wedge KGL$$ of the sequence (7.2) is the same as $$E \otimes {\mathrm {colim}}(F \, {\mathop {\to }\limits ^{{t} }}\, F\to \cdots ) = E \otimes H$$. Replacing $$E$$ by $$E\wedge KGL^{\wedge n}$$ shows that $$E\wedge KGL^{\wedge n+1} \cong E \otimes H^{ \otimes n+1}$$ and hence that $$s_* (E\wedge KGL^{\wedge n+1}) \cong s_* (E) \otimes H^{ \otimes n+1}$$. The coface and codegeneracies of these cosimplicial spectra are identified by Propositions 5.8 and 6.4. When $$E=KGL,$$ we get the formula mentioned in the Introduction: Corollary 7.6 $$KGL\wedge KGL \cong KGL \otimes H,$$and$$KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2} \cong KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} +1}$$. In addition,$$\eta _L:KGL\to KGL\wedge KGL$$corresponds to$$KGL \otimes {\Bbb {Z}} \to KGL \otimes H$$. Example 7.7 When $$E=E \otimes {\Bbb {Q}}$$, we have $$E\wedge KGL^{\wedge n}\cong E \otimes {\Bbb {Q}} [t_1,t_1^{-1},\ldots ,t_n,t_n^{-1}]$$, because $${\Bbb {Q}} \otimes H= {\Bbb {Q}} [t,t^{-1}]$$. Set $$q=\ell ^\nu$$ and write $$E/q$$ for the cofibre $$E \otimes {\Bbb {Z}} /q$$ of $$E \, {\mathop {\to }\limits ^{{q} }}\, E$$. An elementary calculation shows that $$(E/q) \otimes H\cong E \otimes H/qH$$. Corollary 7.8 Suppose that$$E$$has a multiplicative formal group law$$x+y+uxy,$$such that$$u$$is a unit of$$E_{* ,* }$$. Then the family of maps$$\beta _n:E\wedge T^{n}\to E\wedge {\mathbb P}^{\infty } \to E\wedge KGL$$with$$n\not\equiv 0\pmod {\ell }$$induces an isomorphism  \[E \otimes H/\ell ^\nu H \cong \bigoplus _{n\not\equiv 0} (E/\ell ^\nu )\wedge T^{n} \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, (E/\ell ^\nu )\wedge KGL.\]Moreover, the slices$$s_m(E/\ell ^\nu \wedge KGL)$$are isomorphic to$$\bigoplus _{n\not\equiv 0} s_{m-n}(E/\ell ^\nu )\wedge T^n$$. Proof Immediate from Theorems 2.6 and 7.5. 8. The cosimplicial $$KGL$$ spectrum In this section, we complete the information in the previous section to determine the slices of the cosimplicial spectrum $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$, with $$N^nKGL=KGL^{\wedge n+1}$$.   \[KGL \substack{\longrightarrow\\ {\longrightarrow}} KGL\wedge KGL \ \substack{\longrightarrow\\ {\longrightarrow}\\ \longrightarrow} \ KGL^{\wedge 3} \ \substack{\longrightarrow\\ {\longrightarrow}\\ {\longrightarrow} \\{\longrightarrow}} \ \cdots\] Its codegeneracies are given by the product on $$KGL$$, and its coface maps are given by insertion of the unit $$\eta _K: {{\bf{1}}} \to KGL$$; in particular, $$\partial ^0,\partial ^1:KGL\to KGL\wedge KGL$$ are the canonical maps $$\eta _R=\eta _K\wedge 1$$ and $$\eta _L=1\wedge \eta _K$$, respectively. There is also an involution $$c$$ of $$KGL\wedge KGL$$ swapping the two factors, and we have $$\eta _R=c\circ \eta _L$$. By Corollary 7.6, there are isomorphisms $$KGL \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, KGL^{\wedge n+1}$$ such that the diagram   commutes. As $$\partial ^1=\eta _L$$, this establishes the initial case of the following lemma. Lemma 8.1 The isomorphisms$$KGL \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, KGL^{\wedge n+1}$$of Corollary 7.6 are compatible with all the coface and codegeneracy operators of$$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL,$$except possibly for$$\partial ^0$$and$$\sigma ^0$$. Proof Recall the standard construction (the dual path space of [23, 8.3.14]) which takes a cosimplicial object $$X^ {{{\bf{\scriptscriptstyle \bullet }}}}$$ and produces a new cosimplicial object $$Y^ {{{\bf{\scriptscriptstyle \bullet }}}}$$ with $$Y^n=X^{n+1},$$ coface maps $$\partial _Y^i=\partial _X^{i+1}$$ and codegeneracy maps $$\sigma _Y^i=\sigma _X^{i+1}$$. Applying this construction to $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$ yields the cosimplicial spectrum $$KGL^{\wedge {{{\bf{\scriptscriptstyle \bullet }}}} +2}$$ of Corollary 7.6. Since this is isomorphic to the cosimplicial spectrum $$KGL \otimes H^{ \otimes {{{\bf{\scriptscriptstyle \bullet }}}} }$$ by Theorem 7.5, the result follows. Proof of Theorem 0.2(a) Recall that for a motivic spectrum $$E$$ and a graded ring $$A_* $$, we are using the notation $$E \otimes A_* $$ to denote $$\bigoplus _q (T^q\wedge E) \otimes A_q$$. The reason for the index $$2* $$ in the formula that we want to prove:   \[s_0(KGL) \otimes \pi _{2* }(KU^{\wedge n}) \, {\mathop {\longrightarrow }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n})\] is that the coefficients for complex $$K$$-theory $$\pi _{\ast }(KU)$$ are concentrated in degree $$2$$, that is, as graded rings $$\pi _{\ast }(KU) \cong \mathbb Z[t,t^{-1}],$$ where $$t$$ has degree $$2$$. Now, we proceed to prove the formula. Taking slices in Lemma 8.1, we see that the isomorphisms $$s_* (KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_* (KGL^{\wedge n+1})$$ are compatible with all of the coface and codegeneracy operators, except possibly for $$\partial ^0$$ and $$\sigma ^0$$. Remark 8.2 It is tempting to consider the cosimplicial spectrum $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_H({\Bbb {Z}}, {\Bbb {Z}})$$, where $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_H({\Bbb {Z}}, {\Bbb {Z}})$$ is the cobar complex over $$H$$. However, its coface maps $$\partial ^0$$ and codegeneracy maps $$\sigma ^0$$ are not the same as in $$N^ {{{\bf{\scriptscriptstyle \bullet }}}} KGL$$. To finish the proof of Theorem 0.2(b), we need to show that the isomorphisms in Lemma 8.1 are indeed compatible with $$\partial ^0$$ and $$\sigma ^0$$, under the two additional assumptions. We will do so in Propositions 8.5 and 8.11. Definition 8.3 The map $$v_K:T\wedge {{\bf{1}}} \to KGL\wedge KGL$$ is defined to be $$\eta _R(u_K)$$, where $$u_K:T\to KGL$$ is the map of 7.1. We define $$v: T\wedge KGL\wedge KGL \to KGL\wedge KGL$$ to be multiplication by $$v_K$$. Lemma 8.4 We have commutative diagrams:  Proof For reasons of space, we write $$K$$ for $$KGL$$. In the diagram below, the top rectangle commutes by definition of $$v_K$$, and the bottom rectangle commutes because $$\eta _R$$ is a morphism of ring spectra.   The left and right verticals are the maps $$u$$ and $$v$$, respectively. This establishes commutativity of the first square in Lemma 8.4. As the second square is the composition of the natural isomorphism $$T\wedge s_q(E) \, {\mathop {\to }\limits ^{{\cong } }}\, s_{q+1}(T\wedge E)$$ and the $$q$$th slice of the first square, it also commutes. Recall that $$s_0({{\bf{1}}})\to s_0(KGL)$$ is an isomorphism if the base $$S$$ is smooth over a perfect field $$k$$, or any scheme over a field of characteristic 0. This is so because it is true over any perfect field (see [9, 21]), and $$s_0$$ commutes with the pullback $${\cal {{SH}}} (k)\to {\cal {{SH}}} (S)$$ in these cases; see [13, 2.16 and 3.7]. Proposition 8.5 Assume that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism. Then the isomorphisms$$s_q(KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL^{\wedge n+1})$$are compatible with$$\partial ^0$$. Proof Since $${{\bf{1}}}$$ is an initial object, there is a unique map from $${{\bf{1}}}$$ to $$KGL\wedge KGL$$; the assumption implies that $$\eta _L=\eta _R$$ as maps $$s_0(KGL) \to s_0(KGL\wedge KGL)$$. Now for any $$q\in {\Bbb {Z}}$$, Lemma 6.3 says that $$u^q:T^q\wedge s_0(KGL) \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL)$$. By Lemma 8.4, we have a diagram   This determines the maps $$\eta _R:s_{q}(KGL)\to s_{q}(KGL \wedge KGL)$$. Summing over $$q$$, and invoking Lemma 6.3 twice, we see that $$\eta _R$$ is the map   \[s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \longrightarrow s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1},v,v^{-1}]\] sending the copy of $$s_0(KGL)$$ indexed by $$u^q$$ to the copy of $$s_0(KGL)$$ indexed by $$v^q$$. Since we saw in Example 3.2 that this is the same as the map induced from $$\eta _R: {\Bbb {Z}} [u,u^{-1}]\to {\Bbb {Z}} [u,u^{-1}] \otimes H$$, this shows that we have a commutative diagram:   The result follows for $$n>0$$, since $$\partial ^0:KGL^{\wedge n}\to KGL^{\wedge n+1}$$ is $$\eta _R\wedge KGL^{\wedge n-1}$$. To conclude the proof of Theorem 0.2, we have to compare the slices of the codegeneracy $$\sigma ^0:KGL\wedge KGL\to KGL$$ (the product) with the map of Example 3.2,   \[s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}] \otimes H \, {\mathop {\longrightarrow }\limits ^{{\varepsilon } }}\, s_0(KGL) \otimes {\Bbb {Z}} [u,u^{-1}], \quad \varepsilon (u)=\varepsilon (v)=u. \] (8.6) Recall from 1.3 that $${\Bbb {Z}} [t,t^{-1}]$$ is a subring of $$H$$, and that $$\varepsilon :H\to {\Bbb {Z}}$$ sends $$t$$ to $$1$$. Lemma 8.7 The restriction of the product$$s_* (KGL\wedge KGL)\to s_* (KGL)$$to  \[s_* (KGL) \otimes {\Bbb {Z}} [t,t^{-1}] \longrightarrow s_* (KGL) \otimes H \cong s_* (KGL\wedge KGL)\]equals the composition  \[s_* (KGL) \otimes {\Bbb {Z}} [t,t^{-1}] \, {\mathop {\longrightarrow }\limits ^{{\varepsilon } }}\, s_* (KGL) \otimes {\Bbb {Z}} \cong s_* (KGL).\] Proof Since $$\sigma ^0:KGL\wedge KGL\to KGL$$ is a left inverse to both $$\eta _L$$ and $$\eta _R$$, and $$u=\eta _L(u)$$, $$v=\eta _R(u)$$, we see that $$\sigma ^0$$ sends the copies of $$s_0(KGL)$$ indexed by the monomials $$u^q$$ and $$v^q$$ (in $${\Bbb {Z}} [u,u^{-1}] \otimes H$$) to the copies indexed by $$u^q$$. Since $$\sigma ^0$$ is a map of ring spectra, the copies indexed by $$u^i \otimes t^j=u^{i-j}v^j$$ map to the copies indexed by $$u^{i}$$. Recall from Example 7.7 that $$KGL^{\wedge n+1} \otimes {\Bbb {Q}} \cong KGL \otimes {\Bbb {Q}} [t_0,t_0^{-1},\ldots , t_n,t_n^{-1}]$$. This is $$KGL \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{ {\Bbb {Q}} [t,t^{-1}]}({\Bbb {Z}}, {\Bbb {Z}})$$, where $$C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{ {\Bbb {Q}} [t,t^{-1}]}({\Bbb {Z}}, {\Bbb {Z}})$$ is the $$n$$th term of the cobar complex of Example 3.2 for the Hopf algebra $${\Bbb {Q}} [u,u^{-1}]$$. Now consider $$(R,\Gamma )$$ with $$R= {\Bbb {Q}} [u,u^{-1}]$$ and $$\Gamma =R[t,t^{-1}]$$, as in Example 3.2. Combining Lemma 8.1, Proposition 8.5 and Lemma 8.7, we obtain the rational version of the KGL slice conjecture 8: Corollary 8.8 Let$$(R,\Gamma )$$be as above, and assume that$$s_0({{\bf{1}}})\to s_0(KGL)$$is an isomorphism. As cosimplicial motivic ring spectra,  \[s_*N^ {{{\bf{\scriptscriptstyle \bullet }}}} (KGL \otimes {\Bbb {Q}}) \cong s_0(KGL) \otimes C^ {{{\bf{\scriptscriptstyle \bullet }}}}_{\Gamma }(R,R).\] As in [3, 6.1], it is easy to verify that the diagram   (8.9) commutes, where $$b_K$$ is $$KGL\wedge b$$, $$b$$ is the map in (7.2) and the right side is multiplication by $$v_K$$ (smashed with $$T^{-1}$$). Lemma 8.10 If$$S$$is smooth over a perfect field, then$$s_0({{\bf{1}}})\to H {\Bbb {Z}}$$is an isomorphism in$${\cal {{SH}}} (S)$$. Proof This is true over the ground field $$F$$, by [9, 11.3.6]. Now the slice functors $$s_q$$ commute with the pullback $$\pi ^* $$ over $$\pi :S\to \mathrm {Spec} (F)$$; see [13, 2.16]. Because $$S$$ is smooth, $$H {\Bbb {Z}}$$ also pulls back over $$\pi ^* $$ (see [18, 6.1; 22, 3.18]), and we have   \[s_0({{\bf{1}}}_S)=\pi ^*s_0({{\bf{1}}}_F) = \pi ^* (H {\Bbb {Z}}_F) = H {\Bbb {Z}}_S.\] Now the group $${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(H {\Bbb {Z}},H {\Bbb {Z}})\cong H^0(S, {\Bbb {Z}})$$ is torsion-free, and equal to $${\Bbb {Z}}$$ if $$S$$ is connected; cf. [20, 3.7]. It follows that if $$S$$ is smooth over a perfect field, so that $$s_0({{\bf{1}}})\cong H {\Bbb {Z}}$$, the hypothesis of the following proposition is satisfied. Proposition 8.11 Assume that$${\mathrm {Hom}}_{ {\cal {{SH}}} (S)}(s_0({{\bf{1}}}),s_0({{\bf{1}}}))$$is torsion-free. Then the isomorphisms$$s_q(KGL) \otimes H^{ \otimes n} \, {\mathop {\to }\limits ^{{\cong } }}\, s_q(KGL^{\wedge n+1})$$are compatible with$$\sigma ^0$$. Proof Every element of $$H$$ is a power of $$t$$ times a numerical polynomial, so it suffices to check the test maps $$\alpha _n:s_0(K)\to s_0(KGL) \otimes H\cong s_0(KGL\wedge KGL)$$ corresponding to the elements $$\binom {t}{n}$$ of $$H$$. If $$n=0,1,$$ we are done by Lemma 8.7. For $$n\ge 2,$$ the composition   \[s_0(KGL) \, {\mathop {\longrightarrow }\limits ^{{n!} }}\, s_0(KGL) \, {\mathop {\longrightarrow }\limits ^{{\alpha _n} }}\, s_0(KGL\wedge KGL) \, {\mathop {\longrightarrow }\limits ^{{\sigma ^0} }}\, s_0(KGL)\] is given by $$\sigma ^0(t(t-1)\ldots (t-n+1))=0$$ in the ring $$[s_0(KGL),s_0(KGL)]$$, which is $${\Bbb {Z}}$$ by Lemma 8.10. Since $$\sigma ^0\alpha _n$$ corresponds to an integer, and is killed by $$n!$$, we must have $$\sigma ^0\alpha _n=0$$. 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Published: Apr 3, 2014

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