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Galois structure of S-units

Galois structure of S-units Abstract Let $$K/k$$ be a finite Galois extension of number fields with Galois group $$G,$$$$S$$ a large$$G$$-stable set of primes of $$K,$$ and $$E$$ (respectively, $$\boldsymbol {\mu } )$$ the $$G$$-module of $$S$$-units of $$K$$ (respectively, roots of unity). Previous work using the Tate sequence of $$E$$ and the Chinburg class $$\Omega _m$$ has shown that the stable isomorphism class of $$E$$ is determined by the data$$\Delta S,\boldsymbol {\mu }, \Omega _m,$$ and a special character $$\varepsilon $$ of $$H^2(G,{\rm Hom}(\Delta S,\boldsymbol {\mu } )).$$ This paper explains how to build a $$G$$-module $$M$$ from this data that is stably isomorphic to $$E\oplus {\mathbb Z} G^n,$$ for some integer $$n.$$ 1. Introduction Let $$K/k$$ be a finite Galois extension of number fields with Galois group $$G$$ and let $$S$$ be a finite $$G$$-stable set of primes of $$K$$ containing all archimedean primes. Assume that $$S$$ is large in the sense that it contains all ramified primes of $$K/k$$ and that the $$S$$-class group of $$K$$ is trivial. Let $$E$$ denote the $$G$$-module of $$S$$-units of $$K$$ and $$\boldsymbol {\mu } $$ the roots of unity in $$K.$$ The purpose of this paper is to specify the stable isomorphism class of the $$G$$-module $$E$$ in a much more explicit way than in [7, Theorem B]. More precisely, and continuing in the notation of [7], we recall that [11, 12] defines a canonical 2-extension class of $$G$$-modules, represented by Tate sequences   \[ 0\longrightarrow E\longrightarrow A\longrightarrow B\longrightarrow\Delta S\longrightarrow 0, \] with $$A$$ a finitely generated cohomologically trivial $${\mathbb Z} G$$-module, $$B$$ a finitely generated projective $${\mathbb Z} G$$-module and $$\Delta S$$ the kernel of the $$G$$-map $${\mathbb Z} S\to {\mathbb Z}$$ which sends every element of $$S$$ to $$1.$$ From this, [2] obtains the Chinburg $$\Omega (3)$$-class   \[ \Omega_m:= [A] - [B] \] in the locally free class group $${\rm Cl} ({\mathbb Z} G)\subseteq K_0 ({\mathbb Z} G),$$ which is an invariant of $$K/k$$ that is independent of $$S,$$ and conjectures that $$\Omega _m$$ equals the root number class in $${\rm Cl}({\mathbb Z} G).$$ The method of [7] analyzes the $$G$$-module $$E$$ in terms of a fixed envelope of $$\boldsymbol {\mu }.$$ This is an exact sequence   \begin{equation} 0\longrightarrow \boldsymbol{\mu} \longrightarrow \boldsymbol\omega \longrightarrow \overline{\boldsymbol\omega} \longrightarrow 0, \end{equation} (1.1) with $$\boldsymbol \omega $$ cohomologically trivial and $$\overline {\boldsymbol \omega } $$ the $${\mathbb Z} G$$-lattice obtained from $$\boldsymbol \omega $$ by factoring by its $${\mathbb Z}$$-torsion. By Theorem B, the $$G$$-module $$E$$ is determined, up to stable isomorphism, by knowledge of the $$G$$-set $$S,$$ the $$G$$-module $$\boldsymbol {\mu },$$ the Chinburg class $$\Omega _m(K/k)\in {\rm Cl}({\mathbb Z} [G]),$$ and an arithmetically defined character   \[ \varepsilon \in H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu} ))^*, \] where $$\underline {\,\,}^*$$ means $${\rm Hom} (\,\underline {\,\,},{\mathbb Q}/{\mathbb Z}).$$ Let $$L_1:= \Delta G\otimes \Delta S$$ and $$L_2:= \Delta G\otimes L_1$$ with $$\otimes = \otimes _{{\mathbb Z}}$$ and diagonal action by $$G.$$ Choose the envelope $$\boldsymbol \omega $$ to be related to the Chinburg class by the condition   \begin{equation} [\boldsymbol\omega ] - w[{\mathbb Z} G] = \Omega_m(K/k)\quad \text{in}\ {\rm Cl}({\mathbb Z} G), \end{equation} (1.2) with $$\vert G\vert w$$ equal to the $${\mathbb Q}$$-dimension of $${\mathbb Q} \otimes \boldsymbol \omega .$$ We will construct a canonical isomorphism $$H^2(G,{\rm Hom}(\Delta S,\boldsymbol {\mu } ))^* \to H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2))$$ so that our main result is the following theorem. Theorem. Let $$M=M(\varepsilon )$$ denote the $$G$$-module in a $${\mathbb Z}$$-split 1-extension   \[ 0\longrightarrow L_2\longrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \] with extension class equal to the image $$\varepsilon ^{(1)}$$ of $$-\varepsilon $$ in $$H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2)).$$ Then $$E\oplus ({\mathbb Z} G)^n$$ is stably isomorphic to $$M(\varepsilon ),$$ with $$n:= (\vert G\vert -2)(\vert S\vert -1) + w$$ when $$G\ne 1.$$ This improves Theorem B by explaining how its data determines $$M,$$ a model for the stable isomorphism class of $$E.$$ The remaining problem becomes not only to understand the ingredients $$\Delta S,\boldsymbol \omega ,\Omega _m,\varepsilon ,n$$ of the Theorem, but to do so in a way that improves $$M$$ into a better approximation of $$E.$$ As a first example of this, we show how to get a smaller $$n,$$ and an $$M^{\prime },$$ in Corollary 5.1. There is also a continuing discussion on the relation of the Theorem with [7], including Proposition 3.2, and especially on the role of the distinguished character $$\varepsilon , $$ in Remark 2 and Lemma 5.2. Our proof of the Theorem, based on [7], is presented in three sections: the first recalling relevant results, the second reformulating the Theorem in their terms, and the third containing a proof. The last section discusses some basic aspects of the many new problems that arise. 2. Review of [7] Applying $$\underline {\,\,}\,\otimes \Delta S$$ to the $$({\mathbb Z}$$-split) augmentation sequence $$0\to \Delta G\to {\mathbb Z} G\to {\mathbb Z} \to 0$$ gives the $$({\mathbb Z}$$-split) $$G$$-module sequence   \begin{equation} 0\longrightarrow L_1\longrightarrow {\mathbb Z} G\otimes \Delta S\longrightarrow \Delta S\longrightarrow 0, \end{equation} (2.1) with $${\mathbb Z} G\otimes \Delta S$$ a free $${\mathbb Z} G$$-module, and $$L_1:= \Delta G\otimes \Delta S.$$ Applying $${\rm Hom}(\,\underline {\,\,},\boldsymbol {\mu } )$$ to this gives the exact $$G$$-module sequence   \[ 0\longrightarrow\,{\rm Hom}(\Delta S,\boldsymbol{\mu} ) \longrightarrow\,{\rm Hom}({\mathbb Z} G\otimes \Delta S,\boldsymbol{\mu})\longrightarrow\,{\rm Hom}(L_1,\boldsymbol{\mu} ) \longrightarrow 0, \] inducing the connecting isomorphism in Tate cohomology   \begin{equation} \partial_1: H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu})) \longrightarrow H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu})) \end{equation} (2.2) and defining $$\varepsilon _1:=\varepsilon \circ \partial _1\in H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*.$$ Similarly, applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ to our fixed envelope (1.1) of $$\boldsymbol {\mu } $$ and then $$G$$-cohomology gives the isomorphism, still in Tate cohomology,   \begin{equation} \partial^{\prime}_0: \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega})) \longrightarrow H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu})), \end{equation} (2.3) and defines $$\varepsilon _0:=\varepsilon _1\circ \partial ^{\prime }_0\in \widehat H^0(G,{\rm Hom}(L_1,\overline {\boldsymbol \omega }))^*.$$ We now use the isomorphism   \begin{equation} \widehat H^0(G,{\rm Hom}(\overline{\boldsymbol\omega},L_1))\longrightarrow \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega}))^*, \end{equation} (2.4) from (1.2) of [7], that sends $$[f]$$ to $$[f]^*$$ with $$[f]^*$$ represented by the element $$g\mapsto (1/\vert G\vert )$$ trace $$(f\circ g) + {\mathbb Z}$$ of $${\rm Hom}_G(L_1,\overline {\boldsymbol \omega } )^*.$$ Here we are using the identification $$[L,N] = \widehat H\,^0(G,{\rm Hom}(L,N))$$ from (5.1) of [6]; this will reappear near our (3.8) and (4.4). It follows that   \begin{equation} \varepsilon_0 = [f]^*\quad \text{for some}\ G\text{-homomorphism} \ f:\overline {\boldsymbol\omega} \longrightarrow L_1. \end{equation} (2.5) Extension classes in Tate cohomology are as in [6, §11] (cf. Remark after 11.1): a $${\mathbb Z}$$-split 1-extension $$(M):0\to X\to M\to Y\to 0$$ of $$G$$-modules remains exact on applying $${\rm Hom} (Y,\underline {\,\,}\,),$$ and the connecting homomorphism   \begin{equation} \partial_{(M)}: \widehat H^0(G,{\rm Hom}(Y,Y))\longrightarrow H^1(G,{\rm Hom}(Y,X)) \end{equation} (2.6) on its $$G$$-cohomology allows the definition $$\xi _{(M)} := \partial _{(M)}({\rm id}_Y) \in H^1(G,{\rm Hom}(Y,X))$$ of the extension class of $$(M).$$ Note that $$(M)\mapsto \xi _{(M)}$$ induces a bijection between the set of equivalence classes of $${\mathbb Z}$$-split 1-extensions $$(M)$$ and $$H^1(G,{\rm Hom}(Y,X)).$$ This induces a monomorphism $$H^1(G,{\rm Hom}(Y,X))\to \text {Ext}^1_{{\mathbb Z} G}(Y,X),$$ which is an isomorphism when $$Y$$ is a lattice. The basic idea of [7, §5] is to use the homotopy class $$[f_0]\in [\boldsymbol {\omega },L_1]$$ to ‘reconstruct’ $$E.$$ However, the formation of the ‘homotopy’ kernel $$M^{\prime }:=\ker (f)$$ of $$f_0$$ does not provide a description of $$M^{\prime }.$$ This obstacle to improving Theorem B to our present main result will be removed by using extension classes. This occurs in the middle column of our present diagram (4.1) with $$h,\eta $$ playing the roles of $$f_0,f.$$ 3. Reformulation The notational deviation $$L_1,\varepsilon _1$$ from the $$L,\varepsilon $$ of [7] in (2.2) is intended to separate the role of $$\varepsilon _1,$$ which is at the center of the envelope focus of [7], from that of the more fundamental $$\varepsilon .$$ So every $$\varepsilon $$ after the first two pages of [7] should now be read as $$\varepsilon _1$$ for our present purposes. First, applying $$\underline {\,\,}\,\otimes L_1$$ to the augmentation sequence, as in (2.1), gives a ($${\mathbb Z}$$-split) $$G$$-module sequence   \begin{equation} 0\longrightarrow L_2\longrightarrow {\mathbb Z} G\otimes L_1\overset{p_{1}}\longrightarrow L_1\longrightarrow 0, \end{equation} (3.1) with $${\mathbb Z} G\otimes L_1$$  $${\mathbb Z} G$$-free and $$L_2:=\Delta G\otimes L_1.$$ Thus applying $${\rm Hom}(\boldsymbol \omega ,\,\underline {\,\,}\,),$$ as in §1, and then $$G$$-cohomology gives the connecting isomorphism   \begin{equation} \delta_0: \widehat H^0(G,{\rm Hom}(\boldsymbol \omega ,L_1)\longrightarrow H^1(G,{\rm Hom}(\boldsymbol\omega ,L_2)). \end{equation} (3.2) Our reformulation starts from the trivial observation that the $$G$$-map $$\boldsymbol \omega \to \overline {\boldsymbol \omega }$$ of (2.1) induces an equality of the functors $${\rm Hom}(\overline {\boldsymbol \omega },\,\underline {\,\,}\,) \to {\rm Hom}(\boldsymbol \omega ,\,\underline {\,\,}\,)$$ on $${\mathbb Z} G$$-lattices $$X.$$ Then   \begin{equation} \widehat H\,^0 (G,{\rm Hom}(\overline{\boldsymbol\omega}, L_1)) =\widehat H^0(G,{\rm Hom}(\boldsymbol \omega ,L_1)) \end{equation} (3.3) allows us to rewrite (2.4) as an isomorphism   \begin{equation} \widehat H^0(G,{\rm Hom}(\boldsymbol\omega ,L_1)) \longrightarrow \widehat H^0(G,{\rm Hom}(L_1, \overline{\boldsymbol\omega}))^* \end{equation} (3.4) that sends $$[h]$$ to $$[h]^*$$ with $$[h]^*$$ represented by the element $$g\mapsto (1/\vert G\vert )\,{\rm trace}\,(\overline h \circ g) + {\mathbb Z}$$ of $${\rm Hom}_G(L_1, \overline {\boldsymbol \omega })^*.$$ It follows that   \begin{equation} \varepsilon_0 = [h]^* \quad \text{for some}\ h\in {\rm Hom}(\boldsymbol\omega, L_1)^G. \end{equation} (3.5) We now define the isomorphism before the Theorem of the introduction to be the composition of the isomorphisms   \begin{align} H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu} ))^* &\longrightarrow H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu}))^* \longrightarrow \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega}))^*\notag\\ &\longleftarrow \widehat H^0 (G,{\rm Hom}(\boldsymbol\omega ,L_1)) \longrightarrow H^1(G,{\rm Hom}(\boldsymbol\omega , L_2)) \end{align} (3.6) of (2.2)$$^*$$, (2.3)$$^*$$, (3.4), (3.2), and observe that it takes $$-\varepsilon $$ to $$-\delta _0([h]).$$ It follows that $$\varepsilon ^{(1)} = -\delta _0([h])$$ in the statement of the Theorem of the introduction, which is therefore equivalent to the following reformulation. Theorem 3.1. Let$$[h]\in \widehat H^0 (G,{\rm Hom}(\boldsymbol \omega ,L_1))$$be the image of$$\varepsilon $$under the composite of the first three maps in (3.6), and let$$\delta _0$$be the last map of that composite, as in (3.2). Let$$M$$be the$$G$$-module in a$${\mathbb Z}$$-split 1-extension  \[ 0 \longrightarrow L_2 \longrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \]with extension class equal to$$-\delta _0([h])$$in$$H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2)).$$Then$$E\oplus ({\mathbb Z} G)^n$$is stably isomorphic to$$M,$$with$$n:= (\vert G\vert -2)(\vert S\vert -1) + w$$when$$G\ne 1.$$ In particular, the character$$\varepsilon $$and the extension class of$$M(\varepsilon )$$determine each other uniquely. The connection between $$\varepsilon $$ and $$\varepsilon _1$$ is a consequence of the relationship between Tate sequences and Tate envelopes, or, more precisely, between the Tate canonical class $$\alpha _3 \in H^2(G,{\rm Hom}(\Delta S,E))$$ and Tate envelopes. We now adapt this connection, sketched on [7, p. 972], to our needs, with proof. Thus, following the last four paragraphs of Tate's proof of Theorem 5.1 of [12, Chapter 2], we select a special Tate sequence representing $$\alpha _3$$ and define the Tate envelope to be the left half of this special Tate sequence. Proposition 3.2. A Tate envelope$$0\to E\to A\to L_1 \to 0$$has  \[ \Omega_m = A-(\vert S\vert -1 )[{\mathbb Z} G] \quad \text{in}\ {\rm Cl}({\mathbb Z} G). \] Proof. We specialize Tate's initial exact sequence by selecting the one   \begin{equation} 0 \longrightarrow L_2 \longrightarrow B^{\prime} \longrightarrow B \longrightarrow \Delta S \longrightarrow 0, \end{equation} (3.7) obtained by splicing (2.1) and (3.1); Tate's first paragraph ends with isomorphisms   \[ \widehat H^r(G,{\rm Hom}(L_2,E)) \simeq \widehat H^{r+2}(G,{\rm Hom}(\Delta S,E)), \] for all $$r\in {\mathbb Z},$$ in our notation. The second paragraph chooses $$\alpha \in {\rm Hom}_G(L_2,E)$$ corresponding to $$\alpha _3\in H^2(G,{\rm Hom}(\Delta S,E))$$ and deduces, from his (5.2), that $$\alpha $$ induces isomorphisms $$\widehat H^r(G,L_2) \to \widehat H^r(G,E),$$ for all $$r;$$ the third paragraph extends $$\alpha $$ to a surjective $$\alpha :L_2\oplus F\to E,$$ with $$F$$ free, and replaces $$L_2\to B^{\prime }$$ in (3.7) by $$L_2\oplus F\to B^{\prime } \oplus F$$ to obtain a new (3.7) and the exact sequence $$0\to \,\ker (\alpha )\to L_2\oplus F\to E\to 0.$$ The fourth paragraph deduces that $$\ker (\alpha ),$$ and thus $$A:= (B^{\prime } \oplus F)/\ker (\alpha ),$$ is cohomologically trivial. Combining with the new (3.7) gives the Tate sequence $$0\to E\to A\to B\to \Delta S\to 0,$$ the left half $$0\to E\to A\to L_1\to 0$$ of which is our Tate envelope. Now $$B={\mathbb Z} G\otimes \Delta S\simeq ({\mathbb Z} G)^ {\vert S\vert -1}$$ implies that $$\Omega _m =[A]-[B]= [A] - ( \vert S\vert -1)[{\mathbb Z} G].$$ Preparing for §4: if $$(C): 0\to M\to C\to L_1\to 0$$ is an envelope with $${\mathbb Z}$$-torsion $$j: \boldsymbol {\mu }\hookrightarrow M ,$$ applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ and $$G$$-cohomology gives an isomorphism   \begin{equation} \partial_{(C)}:[L_1,L_1]\longrightarrow H^1\big(G,{\rm Hom}(L_1,M)\big), \end{equation} (3.8) of right $$[L_1,L_1]$$-modules. Then $$\tau _1\partial _{(C)}^{-1}j_*$$ is in $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*$$ and we say, following (1.6) of [7], that $$(C)$$ is linked to its $${\rm Aut}_G(\boldsymbol {\mu } )$$-orbit. Here $$\tau _1:[L_1,L_1]\to {\mathbb Q}/{\mathbb Z}$$ is as in (5.6) of [6]. This orbit is here insensitive to the choice of $$j,$$ because $${\rm Aut}_G(\boldsymbol {\mu } ) = {\rm Aut}(\boldsymbol \mu )$$ since $$\boldsymbol {\mu } $$ cyclic implies that $${\rm Aut}(\boldsymbol {\mu })$$ is abelian. 4. Proof of the reformulated theorem The proof is now straightforward. We assume that $$G\ne 1$$ (since the $$G=1$$ case, while true with the obvious interpretation, is trivial), and start by fixing an envelope   \[ 0\longrightarrow \boldsymbol{\mu} \longrightarrow \boldsymbol \omega \longrightarrow \overline{\boldsymbol\omega } \longrightarrow 0, \] satisfying (1.1) and (1.2). The existence of such an $$\boldsymbol \omega $$ follows from (2.1) in [6] and (39.12), (32.13) in [4]: start with any envelope $$0\to \boldsymbol {\mu } \to C\to \overline C \to 0,$$ define $$c$$ by $$\vert G\vert c ={\rm dim} \, {\mathbb Q} \otimes C,$$ and observe that $$\Omega _m - ([C] - c[{\mathbb Z} G]) = [P] - [{\mathbb Z} G]$$ in $${\rm Cl}({\mathbb Z} G),$$ for some projective $${\mathbb Z} G$$-module $$P$$ with $${\rm dim}\,{\mathbb Q} \otimes P = \vert G\vert ,$$ hence $$C^{\prime } := C\oplus P$$ gives an envelope $$0\to \boldsymbol {\mu } \to C^{\prime } \to \overline {(C^{\prime })} \to 0$$ with $$\Omega _m = [C^{\prime }] -c^{\prime } [{\mathbb Z} G],$$ as required. Letting $$[h],$$ with $$h\in \,{\rm Hom}(\boldsymbol \omega ,L_1)^G,$$ be as in the assertion of Theorem 3.1, define $$\eta :({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega \to L_1$$ by $$\eta ((x,y)) = p_1(x) + h(y),$$ and form the big diagram as follows: start from the commutative square containing $$p_1$$ and $$\eta ,$$ use it to form the bottom two rows with the additional map sending $$(x,y)$$ to $$y,$$ and then get the top row by taking kernels, and using (3.1) as the first column. We put $$M:=\ker (\eta )$$ and focus first on the column and then on the row containing $$M.$$ Now let $$0\to E\to A\to L_1\to 0$$ be a fixed Tate envelope, and form the envelope   \begin{equation} 0 \longrightarrow ({\mathbb Z} G)^n \oplus E \longrightarrow ({\mathbb Z} G)^n\oplus A\longrightarrow L_1\longrightarrow 0, \end{equation} (4.2) from it by adding $$({\mathbb Z} G)^n = ({\mathbb Z} G)^n.$$ This is an envelope with $${\mathbb Z}$$-torsion $$\boldsymbol {\mu } $$ and lattice $$L_1,$$ as is the middle column   \begin{equation} 0\longrightarrow M\longrightarrow ({\mathbb Z} G\otimes L_1)\oplus \boldsymbol\omega \longrightarrow L_1\longrightarrow 0, \end{equation} (4.3) of (4.1). We now apply [7, Theorem 4.7] to show that the left ends of these envelopes are stably isomorphic. This requires two conditions to be verified. The quicker condition to check is that $$[({\mathbb Z} G\otimes L_1) \oplus \boldsymbol \omega ]$$ is equal to $$[({\mathbb Z} G)^n\oplus A]$$ in $$K_0({\mathbb Z} G).$$ Now $${\mathbb Z} G\otimes L_1\simeq ({\mathbb Z} G)^{(\vert G\vert -1)(\vert S\vert -1)},$$ because it is $${\mathbb Z} G$$-free; and (1.2) applies to $$[\boldsymbol \omega ],$$ hence $$[({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega ] = (\vert G\vert -1)(\vert S\vert -1)[{\mathbb Z} G] + w[{\mathbb Z} G] +\Omega _m.$$ Similarly, the second expression equals $$n[{\mathbb Z} G] + (\vert S\vert -1)[{\mathbb Z} G] + \Omega _m,$$ by Proposition 3.2. These agree by the choice of $$n$$. The other condition is that both of these envelopes are linked to the same$${\rm Aut}_G(\boldsymbol \mu )$$-orbit on $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*,$$ which we will show is $$\varepsilon _1\,{\rm Aut}_G(\boldsymbol \mu ).$$ First, by definition, the Tate envelope is linked to $$\tau _1\partial _{(A)}^{-1}j_*,$$ with $$j:\boldsymbol {\mu } \hookrightarrow E$$ the inclusion, which is $$t_Ej_*$$ by definition of the trace character $$t_E$$ in §7, that is, the ‘restriction’ $$\varepsilon _1$$ of $$t_E$$ to $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } )).$$ To get the same conclusion for the envelope (4.2), consider the commutative diagram defined by inclusion of the Tate envelope into (4.2), and apply $${\rm Hom}\,(L_1,\,\underline {\,\,}\,)$$ and $$G$$-cohomology to obtain the commutative square, with all maps isomorphisms, inside the commutative diagram with left triangle from composing the inclusions $$\boldsymbol {\mu } \hookrightarrow E$$ and $$E\hookrightarrow ({\mathbb Z} G)^n \oplus E.$$ The top composite from $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))$$ to $${\mathbb Q}/{\mathbb Z}$$ is equal to $$\varepsilon _1,$$ by the first sentence of this paragraph, hence so is the bottom one. Next, to see that the envelope (4.3) is linked to $$\varepsilon _1,$$ consider the commutative diagram with top row the envelope $$(\boldsymbol \omega )$$ of (1.1), (1.2), bottom row the vertical envelope $$(C)$$ of (4.1) with $$C= ({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega ,$$ and $$k(y) = (0,y)$$ for all $$y\in \boldsymbol \omega .$$ Here, forming the right square first defines $$j^{\prime }.$$ Applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ and $$G$$-cohomology gives the commutative square with horizontal isomorphisms and $$(C)$$ linked to $$\tau _1\partial _{(C)}^{-1}(j^{\prime })_* \in H^1({\rm Hom}\,(L_1,\boldsymbol {\mu } ))^*,$$ by the definition (3.8), with $$\tau _1:[L_1,L_1] \to {\mathbb Q}/{\mathbb Z}.$$ Our hypothesis on $$[h]$$ implies the $$[\overline h]^* =\varepsilon _1\partial ^{\prime }_0$$ by (3.5), (2.5), and (2.3), with $$\partial ^ {\prime }_0 = \partial _{(\boldsymbol \omega )},$$ that is, $$[\overline h]^* =\varepsilon _1\partial _{(\boldsymbol \omega )}.$$ Now, quoting [7], $$\varepsilon _1\in H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*$$ implies that $$\varepsilon _1 = \tau _1\theta $$ for some right $$[L_1,L_1]$$-homomorphism $$\theta :H^1({\rm Hom}(L_1,\boldsymbol {\mu } ))\to [L_1,L_1],$$ by (2.3). Then $$\theta \partial _{(\boldsymbol \omega )}$$ is a right $$[L_1,L_1]$$-homomorphism: $$[L_1,\overline {\boldsymbol \omega }]\to [L_1,L_1]$$ so that $$[\overline h]\in [\overline {\boldsymbol \omega }, L_1]$$ having $$[\overline h]^*= \varepsilon _1\partial _{(\boldsymbol \omega )} = \tau _1\theta \partial _{(\boldsymbol \omega )},$$ by the previous paragraph, implies that $$\theta \partial _{(\boldsymbol \omega )} = [{\rm id}_{L_1},\overline h],$$ by (2.4). Combining with (4.4) above gives $$\tau _1\partial _{(C)}^{-1}(j^{\prime })_* =\tau _1[{\rm id}_{L_1}, \overline h]\partial _{(\boldsymbol \omega )}^{-1} = \tau _1\theta =\varepsilon _1,$$ as required. Finally, we must show that the top row   \[ (M) : 0\longrightarrow L_2 \hookrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \] of the big diagram (4.1) has extension class $$-\delta _0([h]),$$ in the notation of (2.6). To obtain a 1-cocycle representing $$-\delta _0([h]),$$ one applies $${\rm Hom}(\boldsymbol \omega ,\underline {\,\,}\,)$$ to (3.1), obtaining the exact sequence $$0\to {\rm Hom}(\boldsymbol \omega ,L_2)\to {\rm Hom}(\boldsymbol \omega ,{\mathbb Z} G\otimes L_1)\to {\rm Hom}(\boldsymbol \omega ,L_1)\to 0,$$ chooses a pre-image of $$h$$ in $${\rm Hom} (\boldsymbol \omega ,{\mathbb Z} G\otimes L_1),$$ say the map $$1\otimes h$$ taking every $$y\in \boldsymbol \omega $$ to $$1\otimes h(y),$$ and then forms the 1-cocycle $$g\mapsto (1\otimes h) - g(1\otimes h)$$ (with $$g\in G)$$ taking values in $${\rm Hom}(\boldsymbol \omega ,L_2),$$ namely $${}[(1\otimes h)-g\!\cdot (1\otimes h)](y) = (1\otimes h)(y) -g\!\cdot (1\otimes h)(g^{-1}y) \,{=}\,1\otimes h(y)\,{-}\,g\!\cdot (1\otimes h(g^{-1}y)) \,{=}\,1\otimes h(y)\,{-}\,g\otimes g\!\cdot \! h(g^{-1}y) \,{=}\, 1\otimes h(y) \,{-}\,g\otimes h(y) \,{=}\,(1-g)\otimes h(y)\in \Delta G\otimes L_1= L_2.$$ On the other hand, the extension class $$\xi _{(M)} $$ of $$(M)$$ is, by definition, obtained from $$(M) $$ by applying $${\rm Hom}(\boldsymbol \omega ,\underline {\,\,}\,)$$ to $$(M),$$ obtaining $$0\to {\rm Hom}\,(\boldsymbol \omega ,L_2)\to {\rm Hom}\,(\boldsymbol \omega ,M)\to {\rm Hom}\,(\boldsymbol \omega ,\boldsymbol \omega )\to 0,$$ lifting $${\rm id}_{\boldsymbol \omega }$$ to some $$s\in {\rm Hom}\,(\boldsymbol \omega ,M),$$ and forming the class of the 1-cocycle $$g\mapsto gs-s$$ with values in $${\rm Hom}\,(\boldsymbol \omega ,L_2).$$ Setting $$s(y) = (-1\otimes h(y),y)$$ works, since $$\eta (s(y)) = p_1(-1\otimes h(y)) + h(y) =0$$ and $$p_0(s(y)) = y.$$ Now $$(gs-s)(y) = g(-1\otimes h(g^{-1}y), g^{-1}y) - (-1\otimes h(y),y) = (-g\otimes h(y),y) + (1\otimes h(y),-y) = ((1-g)\otimes h(y),0),$$ which is the image of $$(1-g)\otimes h(y)\in L_2.$$ This agrees with the 1-cocycle of the previous paragraph. 5. Discussion We begin with a consequence of the Theorem, for which we prepare with a naturality property of the Gruenberg resolution. We start with a subset, of $$d$$ elements $$g_i$$ of $$G\backslash \{1\},$$ which generates$$G,$$ form the free group $$F$$ on $$x_i,\ 1\le i\le d,$$ and define the relation module $$R_d$$ by the exact sequence   \begin{equation} 0\longrightarrow R_d\longrightarrow {\mathbb Z} G\otimes_{{\mathbb Z} F}\Delta F\longrightarrow \Delta G\longrightarrow 0 \end{equation} (5.1) (cf. [8, p. 199 and 218]). Here the middle term is $$\simeq ({\mathbb Z} G)^d$$ since $$\Delta F$$ is $${\mathbb Z} F$$-free on the elements $$(x_i-1)$$, and the right map sends the $${\mathbb Z} G$$-basis $$1\otimes _F (x_i-1)$$ to $$g_i-1.$$ In the special case $$d=\vert G\vert -1,$$ write $${\mathcal R}, {\mathcal F}$$ for $$R_d,\,F,$$ respectively. For general $$d,$$ the inclusion $$F\to {\mathcal F}$$ induces a map from the relation sequence for $$R_d$$ to $${\mathcal R},$$ which on middle terms is an inclusion of the respective $${\mathbb Z} G$$-bases, so has cokernel $$\simeq ({\mathbb Z} G)^{\vert G\vert -1-d},$$ yielding the exact sequence $$0\to R_d \to {\mathcal R} \to ({\mathbb Z} G)^{\vert G\vert -1-d}\to 0$$ on the left terms. Similarly, the relation module sequence for $${\mathcal R}$$ maps to the exact sequence obtained by applying $$\underline {\,\,}\otimes \Delta G$$ to the augmentation sequence, with middle map matching $${\mathbb Z} G$$-bases by $$1\otimes _{\mathcal F} (x_i-1)\mapsto 1\otimes (g_i-1),$$ inducing an isomorphism $${\mathcal R}\to \Delta G\otimes \Delta G.$$ This implies that   \begin{equation} 0\longrightarrow R_d \overset\beta \longrightarrow \Delta G\otimes \Delta G\longrightarrow ({\mathbb Z} G)^m \longrightarrow 0 \end{equation} (5.2) is exact with an explicit map $$\beta $$ and $$m=\vert G\vert -1-d,$$ when $$G\ne 1.$$ Let $$d(G)$$ be the minimum number of generators of $$G,$$ and set $$R:=R_{d(G)},$$ to state the following corollary. Corollary 5.1. There is an explicit$$G$$-homomorphism$$\beta ^{\prime } :R\otimes \Delta S\to L_2$$so that the induced isomorphism$$\beta ^{\prime }_*: H^1(G,{\rm Hom}(\boldsymbol \omega ,R\otimes \Delta S))\to H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2))$$has the following property: let$$M^{\prime }$$be the$$G$$-module in a$${\mathbb Z}$$-split 1-extension  \[ 0\longrightarrow R\otimes \Delta S\longrightarrow M^{\prime} \longrightarrow \boldsymbol\omega \longrightarrow 0 \]with extension class mapping to$$\varepsilon ^{(1)}$$under$$\beta ^{\prime }_*.$$Then$$E\oplus ({\mathbb Z} G)^{n^{\prime }}$$is stably isomorphic to$$M^{\prime }$$with$$n^{\prime } = (d(G) -1)(\vert S\vert -1) + w$$when$$G\ne 1.$$ Proof. By $$L_2 = \Delta G\otimes L_1 =\Delta G\otimes (\Delta G\otimes \Delta S)\simeq (\Delta G\otimes \Delta G)\otimes \Delta S,$$ applying $$\underline {\,\,} \otimes \Delta S$$ to (5.2) gives the exact sequence   \begin{equation} 0\longrightarrow R\otimes \Delta S\overset{\beta^{\prime}}\longrightarrow L_2 \longrightarrow ({\mathbb Z} G)^{n-n^{\prime}} \longrightarrow 0, \end{equation} (5.3) defining $$\beta ^{\prime }.$$ This follows from $$({\mathbb Z} G)^m \otimes \Delta S\simeq ({\mathbb Z} G)^{m(\vert S\vert -1)}$$ with $$m(\vert S\vert -1)=n-n^{\prime }.$$ Now the extension class of the 1-extension $$(M^{\prime })$$ has the property that its pushout along $$\beta ^{\prime }$$ has extension class $$\varepsilon ^{(1)}$$, so there is a commutative diagram Since $$\beta ^{\prime }$$ has cokernel $$({\mathbb Z} G)^{n-n^{\prime }}$$, so does the middle arrow, hence there is an exact sequence $$0\to M^{\prime } \to M\to ({\mathbb Z} G)^{n-n^{\prime }} \to 0.$$ Thus, by the Theorem, $$E\oplus ({\mathbb Z} G)^n\approx M\approx M^{\prime } \oplus ({\mathbb Z} G)^{n-n^{\prime }},$$ which implies that $$E\oplus ({\mathbb Z} G)^{n^{\prime }} \approx M^{\prime }.$$ Remark 1. The relation module $$R$$ has no non-zero projective summand if $$G$$ is solvable or, more generally, when $$G$$ has generation gap $$=0$$ (cf. (24) in [5]), in which case we cannot expect bigger $${\mathbb Z} G$$-free summands from the above approach. Note that $$R_d$$ is determined up to stable isomorphism by $$d,$$ as follows from (5.1) by Schanuel's lemma. Corollary 5.1 is a first step toward the important goal of excising as many $${\mathbb Z} G$$-free summands of $$M$$ as explicitly as possible. There are many aspects of this problem but still no systematic approach. There has been considerable work on Chinburg's conjecture as a special case of the Equivariant Tamagawa Number Conjecture; a recent reference is [1] (cf. Corollary 2.8 and Remark 2.9). Since Chinburg's conjecture predicts that $$\Omega _m=0$$ whenever $$G$$ has no irreducible symplectic representation (cf. [3, §3]), an envelope $$\boldsymbol \omega $$ of $$\boldsymbol \mu $$ with $$[ \boldsymbol \omega ] - w[{\mathbb Z} G]=0$$ and $$w =d(G) $$ (cf. Chapter 3 of [9]) is a useful ingredient for examples. On the other hand, the condition (1.2) on $$\boldsymbol \omega $$ could be replaced in the Theorem by   \[ [\boldsymbol\omega ] - w[{\mathbb Z} G] \equiv \Omega_m\mod B[\varepsilon_1], \] as the appeal to [7, Theorem 4.7] in its proof shows. This shows that the full strength of Chinburg's conjecture may not be needed. Remark 2. The emphasis on $$\varepsilon _1$$ in [7] comes from the envelope focus. In particular, Theorem A for $$\varepsilon _1$$ is proved by this method, but its statement depends on the local and global invariant maps on $$H^2,$$ the natural habitat of $$\varepsilon .$$ Theorem A can be translated from $$\varepsilon _1$$ to $$\varepsilon $$ by using the formalism of [11], in the direction of the last paragraph of the Remark on p. 971 of [7]. The $$\varepsilon \text {-analogue}$$ of Theorem A is strikingly simpler than it was for $$\varepsilon _1.$$ More precisely, let $${\rm Hom}(({\mathbb Z} S),(J))$$ be the $$G$$-module consisting of all triples $$(f_1,f_2,f_3)$$ of $${\mathbb Z}$$-homomorphisms so that the diagram commutes. This leads to an exact sequence   \begin{align*} 0&\longrightarrow H^2(G,{\rm Hom}(({\mathbb Z} S),(J)))\longrightarrow H^2 (G,{\rm Hom}(\Delta S,E))\oplus H^2(G,{\rm Hom}({\mathbb Z} S,J))\\ &\longrightarrow H^2(G,{\rm Hom}(\Delta S,J))\longrightarrow 0 \end{align*} allowing us to study the trace character $$T_E:H^2(G,{\rm Hom}(\Delta S,E))\to {\mathbb Q}/{\mathbb Z}$$ defined by dimension shifting $$t_E$$ using the exact sequence (2.1). This implies that $$\varepsilon =T_E\circ j_*,$$ with $$j:\boldsymbol \mu \hookrightarrow E$$ the inclusion, but now the point is that $$T_E$$ can be described in terms of the $$H^2$$-sequence above without further dimension shifting. More precisely, given $$x\in H^2(G,{\rm Hom}(\Delta S,E)),$$ there exists $$y\in H^2(G,{\rm Hom}({\mathbb Z} S,J))$$ so $$(x,y)$$ maps to 0 in $$H^2(G,{\rm Hom}(\Delta S,E)),$$ hence there is a unique $$T\in H^2(G,{\rm Hom}(({\mathbb Z} S),(J)))$$ mapping to $$(x,y).$$ Taking a 2-cocycle of triples representing $$T$$ and projecting on the third component gives a 2-cocycle defining $$z\in H^2(G,{\rm Hom}({\mathbb Z},C_K)).$$ Then (cf. Chapter 4 of [9])   \begin{equation} T_E(x) =\text{inv}(z) -\sum_{{{\mathfrak p}}\in S_*}\,\text{inv}_{{\mathfrak p}}(y_{{\mathfrak p}}), \end{equation} (5.4) where $$S_*$$ is a transversal to the $$G$$-orbits on $$S,y_{{\mathfrak p}} = k_{{\mathfrak p}}(\text {res}\,y)i_{{\mathfrak p}}$$ with $$k_{{\mathfrak p}}:J\to K^\times _{{\mathfrak p}}$$ the projection and $$i_{{\mathfrak p}}:{\mathbb Z} \to {\mathbb Z} S$$ with $$i_{{\mathfrak p}}(1)={{\mathfrak p}}.$$ This description has the weakness that the existence of $$y$$ apparently depends on the vanishing of $$H^3(G,J).$$ This situation is improved by the following lemma. Lemma 5.2. The map$$H^2(G,{\rm Hom}({\mathbb Z} S,J))\to H^2(G,{\rm Hom}(\Delta S,J))$$has a special splitting. Proof. The $$S$$-idele group $$J$$ is a finite product, over $${{\mathfrak p}}\in S_*,$$ of components $$V_{{\mathfrak p}}:= \prod _{\mathfrak q}K^\times _{\mathfrak q},$$ with $${\mathfrak q}$$ running through the $$G$$-orbit of $${{\mathfrak p}},$$ up to a large cohomologically trivial component of unit ideles. So it suffices to show that $$H^2(G,{\rm Hom}({\mathbb Z} S,V_{{\mathfrak p}}))\to H^2(G,{\rm Hom} (\Delta S,V_{{\mathfrak p}}))$$ is split for each $${{\mathfrak p}}.$$ If $$H$$ is a subgroup of $$G,$$ and $$B$$ any $$H$$-module, define the coinduced $$G$$-module $$\text {coind}\,(B),$$ from $$H$$ to $$G,$$ to be $${\rm Hom}_{{\mathbb Z} H}({\mathbb Z} G,B)$$ with $$g\in G$$ acting on elements $$\varphi $$ by $$(g\varphi )(z)=\varphi (zg)$$ for all $$z\in {\mathbb Z} G$$ (cf VII, [10, §5]). If $$D$$ is any $${\mathbb Z} G$$-lattice, viewing $${\rm Hom}(D,\text {coind}\,(B))$$ as $$G$$-module and $${\rm Hom}(\text {res}\, D,B)$$ as $$H$$-module, both by diagonal action, then there are natural Shapiro isomorphisms   \[ H^n(G,{\rm Hom}(D,\text{coind}\, B)) \longrightarrow H^n(H,{\rm Hom}(\text{res}\,D,B)). \] Take $$H=G_{{\mathfrak p}},$$$$B=K^\times _{{\mathfrak p}}$$ and identify $$\text {coind}\,K^\times _{{\mathfrak p}}$$ with $$V_{{\mathfrak p}},$$ via $$\varphi \mapsto \prod _t(t\!\cdot \varphi (t^{-1}),$$ with $$t$$ a choice of representatives of $$G/G_{{\mathfrak p}}.$$ This choice does not matter, since $$(th)\!\cdot \varphi ((th)^{-1}) = t\!\cdot (h\!\cdot \varphi (h^{-1}t^{-1})) = t\!\cdot \varphi (t^{-1})$$ for $$h\in G_{{\mathfrak p}}.$$ The map is bijective, since the components $$tK^\times _{{\mathfrak p}}$$ of $$V_{{\mathfrak p}}$$ are disjoint, and is a $$G$$-homomorphism because $$g(\prod _t(t\!\cdot \varphi (t^{-1})) = \prod _t((gt)\!\cdot \varphi ((gt)^{-1}g)) = \prod _t((gt)\!\cdot (g\varphi )((gt)^{-1})) =\prod _t(t\!\cdot (g\varphi )(t^{-1})).$$ This identifies our map of the first paragraph with the top row of the commutative square with vertical isomorphisms, and horizontal maps from $$0\to \Delta S\overset {a}\hookrightarrow {\mathbb Z} S\overset {a^{\prime }}\to {\mathbb Z}\to 0.$$ This exact sequence is $$G_{{\mathfrak p}}$$-split, by the $$G_{{\mathfrak p}}$$-map $$\lambda _{{\mathfrak p}}: d\mapsto d-a^{\prime }(d){{\mathfrak p}}$$ having $$\lambda _{{{\mathfrak p}}}\circ a =\text {id}_{\Delta S}.$$ Thus $$\lambda _{{\mathfrak p}}$$ induces $$H^2({\rm Hom}(\text {res}\, \Delta S,K^\times _{{\mathfrak p}}))\to H^2(G_{{\mathfrak p}},{\rm Hom}(\text {res}\, {\mathbb Z} S,K^\times _{{\mathfrak p}}))$$ splitting the bottom $$a^*$$ of the commutative square, which then completes the argument. References 1 Burns D., ‘ On main conjectures in non-commutative Iwasawa theory and related conjectures’, J. reine angew. Math.  698 ( 2015) 105– 159. 2 Chinburg T., ‘ On the Galois module structure of algebraic integers and $$S$$-units’, Invent. Math.  74 ( 1983) 321– 349. Google Scholar CrossRef Search ADS   3 Chinburg T., ‘ Exact sequences and Galois module structure’, Ann. of Math.  121 ( 1985) 351– 378. Google Scholar CrossRef Search ADS   4 Curtis C. W. Reiner I., Methods of representation theory  ( J. Wiley, New York, 1981/1987). 5 Gruenberg K. W., ‘Free abelianised extensions of finite groups’, Homological group theory (ed. C. T. C. Wall), London Mathematical Society Lecture Note Series 36 (Cambridge University Press, Cambridge, 1979), 71–104. 6 Gruenberg K. W. Weiss A., ‘ Galois invariants for units’, Proc. London Math. Soc.  70 ( 1995) 264– 283. Google Scholar CrossRef Search ADS   7 Gruenberg K. W. Weiss A., ‘ Galois invariants for $$S$$-units’, Amer. J. Math.  119 ( 1997) 953– 983. Google Scholar CrossRef Search ADS   8 Hilton P. Stammbach U., A course in homological algebra  ( Springer, Berlin, 1971). 9 Riveros Pacheco D. R., ‘On the Galois structure of the $$S$$-units for cyclotomic extensions over $$\mathbb {Q}$$’, PhD Thesis, University of Alberta, 2015. 10 Serre J.-P., Local fields  ( Springer, Berlin, 1979). 11 Tate J., ‘ The cohomology groups of tori in finite Galois extensions of number fields’, Nagoya Math. J.  27 ( 1966) 709– 719. Google Scholar CrossRef Search ADS   12 Tate J., Le conjectures de Stark sur les fonctions L d'Artin en s = 0 , Progress in Mathematics 47 ( Birkhauser, Boston, 1984). © 2016 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Galois structure of S-units

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Oxford University Press
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© 2016 London Mathematical Society
ISSN
0024-6093
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1469-2120
DOI
10.1112/blms/bdv100
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Abstract

Abstract Let $$K/k$$ be a finite Galois extension of number fields with Galois group $$G,$$$$S$$ a large$$G$$-stable set of primes of $$K,$$ and $$E$$ (respectively, $$\boldsymbol {\mu } )$$ the $$G$$-module of $$S$$-units of $$K$$ (respectively, roots of unity). Previous work using the Tate sequence of $$E$$ and the Chinburg class $$\Omega _m$$ has shown that the stable isomorphism class of $$E$$ is determined by the data$$\Delta S,\boldsymbol {\mu }, \Omega _m,$$ and a special character $$\varepsilon $$ of $$H^2(G,{\rm Hom}(\Delta S,\boldsymbol {\mu } )).$$ This paper explains how to build a $$G$$-module $$M$$ from this data that is stably isomorphic to $$E\oplus {\mathbb Z} G^n,$$ for some integer $$n.$$ 1. Introduction Let $$K/k$$ be a finite Galois extension of number fields with Galois group $$G$$ and let $$S$$ be a finite $$G$$-stable set of primes of $$K$$ containing all archimedean primes. Assume that $$S$$ is large in the sense that it contains all ramified primes of $$K/k$$ and that the $$S$$-class group of $$K$$ is trivial. Let $$E$$ denote the $$G$$-module of $$S$$-units of $$K$$ and $$\boldsymbol {\mu } $$ the roots of unity in $$K.$$ The purpose of this paper is to specify the stable isomorphism class of the $$G$$-module $$E$$ in a much more explicit way than in [7, Theorem B]. More precisely, and continuing in the notation of [7], we recall that [11, 12] defines a canonical 2-extension class of $$G$$-modules, represented by Tate sequences   \[ 0\longrightarrow E\longrightarrow A\longrightarrow B\longrightarrow\Delta S\longrightarrow 0, \] with $$A$$ a finitely generated cohomologically trivial $${\mathbb Z} G$$-module, $$B$$ a finitely generated projective $${\mathbb Z} G$$-module and $$\Delta S$$ the kernel of the $$G$$-map $${\mathbb Z} S\to {\mathbb Z}$$ which sends every element of $$S$$ to $$1.$$ From this, [2] obtains the Chinburg $$\Omega (3)$$-class   \[ \Omega_m:= [A] - [B] \] in the locally free class group $${\rm Cl} ({\mathbb Z} G)\subseteq K_0 ({\mathbb Z} G),$$ which is an invariant of $$K/k$$ that is independent of $$S,$$ and conjectures that $$\Omega _m$$ equals the root number class in $${\rm Cl}({\mathbb Z} G).$$ The method of [7] analyzes the $$G$$-module $$E$$ in terms of a fixed envelope of $$\boldsymbol {\mu }.$$ This is an exact sequence   \begin{equation} 0\longrightarrow \boldsymbol{\mu} \longrightarrow \boldsymbol\omega \longrightarrow \overline{\boldsymbol\omega} \longrightarrow 0, \end{equation} (1.1) with $$\boldsymbol \omega $$ cohomologically trivial and $$\overline {\boldsymbol \omega } $$ the $${\mathbb Z} G$$-lattice obtained from $$\boldsymbol \omega $$ by factoring by its $${\mathbb Z}$$-torsion. By Theorem B, the $$G$$-module $$E$$ is determined, up to stable isomorphism, by knowledge of the $$G$$-set $$S,$$ the $$G$$-module $$\boldsymbol {\mu },$$ the Chinburg class $$\Omega _m(K/k)\in {\rm Cl}({\mathbb Z} [G]),$$ and an arithmetically defined character   \[ \varepsilon \in H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu} ))^*, \] where $$\underline {\,\,}^*$$ means $${\rm Hom} (\,\underline {\,\,},{\mathbb Q}/{\mathbb Z}).$$ Let $$L_1:= \Delta G\otimes \Delta S$$ and $$L_2:= \Delta G\otimes L_1$$ with $$\otimes = \otimes _{{\mathbb Z}}$$ and diagonal action by $$G.$$ Choose the envelope $$\boldsymbol \omega $$ to be related to the Chinburg class by the condition   \begin{equation} [\boldsymbol\omega ] - w[{\mathbb Z} G] = \Omega_m(K/k)\quad \text{in}\ {\rm Cl}({\mathbb Z} G), \end{equation} (1.2) with $$\vert G\vert w$$ equal to the $${\mathbb Q}$$-dimension of $${\mathbb Q} \otimes \boldsymbol \omega .$$ We will construct a canonical isomorphism $$H^2(G,{\rm Hom}(\Delta S,\boldsymbol {\mu } ))^* \to H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2))$$ so that our main result is the following theorem. Theorem. Let $$M=M(\varepsilon )$$ denote the $$G$$-module in a $${\mathbb Z}$$-split 1-extension   \[ 0\longrightarrow L_2\longrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \] with extension class equal to the image $$\varepsilon ^{(1)}$$ of $$-\varepsilon $$ in $$H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2)).$$ Then $$E\oplus ({\mathbb Z} G)^n$$ is stably isomorphic to $$M(\varepsilon ),$$ with $$n:= (\vert G\vert -2)(\vert S\vert -1) + w$$ when $$G\ne 1.$$ This improves Theorem B by explaining how its data determines $$M,$$ a model for the stable isomorphism class of $$E.$$ The remaining problem becomes not only to understand the ingredients $$\Delta S,\boldsymbol \omega ,\Omega _m,\varepsilon ,n$$ of the Theorem, but to do so in a way that improves $$M$$ into a better approximation of $$E.$$ As a first example of this, we show how to get a smaller $$n,$$ and an $$M^{\prime },$$ in Corollary 5.1. There is also a continuing discussion on the relation of the Theorem with [7], including Proposition 3.2, and especially on the role of the distinguished character $$\varepsilon , $$ in Remark 2 and Lemma 5.2. Our proof of the Theorem, based on [7], is presented in three sections: the first recalling relevant results, the second reformulating the Theorem in their terms, and the third containing a proof. The last section discusses some basic aspects of the many new problems that arise. 2. Review of [7] Applying $$\underline {\,\,}\,\otimes \Delta S$$ to the $$({\mathbb Z}$$-split) augmentation sequence $$0\to \Delta G\to {\mathbb Z} G\to {\mathbb Z} \to 0$$ gives the $$({\mathbb Z}$$-split) $$G$$-module sequence   \begin{equation} 0\longrightarrow L_1\longrightarrow {\mathbb Z} G\otimes \Delta S\longrightarrow \Delta S\longrightarrow 0, \end{equation} (2.1) with $${\mathbb Z} G\otimes \Delta S$$ a free $${\mathbb Z} G$$-module, and $$L_1:= \Delta G\otimes \Delta S.$$ Applying $${\rm Hom}(\,\underline {\,\,},\boldsymbol {\mu } )$$ to this gives the exact $$G$$-module sequence   \[ 0\longrightarrow\,{\rm Hom}(\Delta S,\boldsymbol{\mu} ) \longrightarrow\,{\rm Hom}({\mathbb Z} G\otimes \Delta S,\boldsymbol{\mu})\longrightarrow\,{\rm Hom}(L_1,\boldsymbol{\mu} ) \longrightarrow 0, \] inducing the connecting isomorphism in Tate cohomology   \begin{equation} \partial_1: H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu})) \longrightarrow H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu})) \end{equation} (2.2) and defining $$\varepsilon _1:=\varepsilon \circ \partial _1\in H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*.$$ Similarly, applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ to our fixed envelope (1.1) of $$\boldsymbol {\mu } $$ and then $$G$$-cohomology gives the isomorphism, still in Tate cohomology,   \begin{equation} \partial^{\prime}_0: \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega})) \longrightarrow H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu})), \end{equation} (2.3) and defines $$\varepsilon _0:=\varepsilon _1\circ \partial ^{\prime }_0\in \widehat H^0(G,{\rm Hom}(L_1,\overline {\boldsymbol \omega }))^*.$$ We now use the isomorphism   \begin{equation} \widehat H^0(G,{\rm Hom}(\overline{\boldsymbol\omega},L_1))\longrightarrow \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega}))^*, \end{equation} (2.4) from (1.2) of [7], that sends $$[f]$$ to $$[f]^*$$ with $$[f]^*$$ represented by the element $$g\mapsto (1/\vert G\vert )$$ trace $$(f\circ g) + {\mathbb Z}$$ of $${\rm Hom}_G(L_1,\overline {\boldsymbol \omega } )^*.$$ Here we are using the identification $$[L,N] = \widehat H\,^0(G,{\rm Hom}(L,N))$$ from (5.1) of [6]; this will reappear near our (3.8) and (4.4). It follows that   \begin{equation} \varepsilon_0 = [f]^*\quad \text{for some}\ G\text{-homomorphism} \ f:\overline {\boldsymbol\omega} \longrightarrow L_1. \end{equation} (2.5) Extension classes in Tate cohomology are as in [6, §11] (cf. Remark after 11.1): a $${\mathbb Z}$$-split 1-extension $$(M):0\to X\to M\to Y\to 0$$ of $$G$$-modules remains exact on applying $${\rm Hom} (Y,\underline {\,\,}\,),$$ and the connecting homomorphism   \begin{equation} \partial_{(M)}: \widehat H^0(G,{\rm Hom}(Y,Y))\longrightarrow H^1(G,{\rm Hom}(Y,X)) \end{equation} (2.6) on its $$G$$-cohomology allows the definition $$\xi _{(M)} := \partial _{(M)}({\rm id}_Y) \in H^1(G,{\rm Hom}(Y,X))$$ of the extension class of $$(M).$$ Note that $$(M)\mapsto \xi _{(M)}$$ induces a bijection between the set of equivalence classes of $${\mathbb Z}$$-split 1-extensions $$(M)$$ and $$H^1(G,{\rm Hom}(Y,X)).$$ This induces a monomorphism $$H^1(G,{\rm Hom}(Y,X))\to \text {Ext}^1_{{\mathbb Z} G}(Y,X),$$ which is an isomorphism when $$Y$$ is a lattice. The basic idea of [7, §5] is to use the homotopy class $$[f_0]\in [\boldsymbol {\omega },L_1]$$ to ‘reconstruct’ $$E.$$ However, the formation of the ‘homotopy’ kernel $$M^{\prime }:=\ker (f)$$ of $$f_0$$ does not provide a description of $$M^{\prime }.$$ This obstacle to improving Theorem B to our present main result will be removed by using extension classes. This occurs in the middle column of our present diagram (4.1) with $$h,\eta $$ playing the roles of $$f_0,f.$$ 3. Reformulation The notational deviation $$L_1,\varepsilon _1$$ from the $$L,\varepsilon $$ of [7] in (2.2) is intended to separate the role of $$\varepsilon _1,$$ which is at the center of the envelope focus of [7], from that of the more fundamental $$\varepsilon .$$ So every $$\varepsilon $$ after the first two pages of [7] should now be read as $$\varepsilon _1$$ for our present purposes. First, applying $$\underline {\,\,}\,\otimes L_1$$ to the augmentation sequence, as in (2.1), gives a ($${\mathbb Z}$$-split) $$G$$-module sequence   \begin{equation} 0\longrightarrow L_2\longrightarrow {\mathbb Z} G\otimes L_1\overset{p_{1}}\longrightarrow L_1\longrightarrow 0, \end{equation} (3.1) with $${\mathbb Z} G\otimes L_1$$  $${\mathbb Z} G$$-free and $$L_2:=\Delta G\otimes L_1.$$ Thus applying $${\rm Hom}(\boldsymbol \omega ,\,\underline {\,\,}\,),$$ as in §1, and then $$G$$-cohomology gives the connecting isomorphism   \begin{equation} \delta_0: \widehat H^0(G,{\rm Hom}(\boldsymbol \omega ,L_1)\longrightarrow H^1(G,{\rm Hom}(\boldsymbol\omega ,L_2)). \end{equation} (3.2) Our reformulation starts from the trivial observation that the $$G$$-map $$\boldsymbol \omega \to \overline {\boldsymbol \omega }$$ of (2.1) induces an equality of the functors $${\rm Hom}(\overline {\boldsymbol \omega },\,\underline {\,\,}\,) \to {\rm Hom}(\boldsymbol \omega ,\,\underline {\,\,}\,)$$ on $${\mathbb Z} G$$-lattices $$X.$$ Then   \begin{equation} \widehat H\,^0 (G,{\rm Hom}(\overline{\boldsymbol\omega}, L_1)) =\widehat H^0(G,{\rm Hom}(\boldsymbol \omega ,L_1)) \end{equation} (3.3) allows us to rewrite (2.4) as an isomorphism   \begin{equation} \widehat H^0(G,{\rm Hom}(\boldsymbol\omega ,L_1)) \longrightarrow \widehat H^0(G,{\rm Hom}(L_1, \overline{\boldsymbol\omega}))^* \end{equation} (3.4) that sends $$[h]$$ to $$[h]^*$$ with $$[h]^*$$ represented by the element $$g\mapsto (1/\vert G\vert )\,{\rm trace}\,(\overline h \circ g) + {\mathbb Z}$$ of $${\rm Hom}_G(L_1, \overline {\boldsymbol \omega })^*.$$ It follows that   \begin{equation} \varepsilon_0 = [h]^* \quad \text{for some}\ h\in {\rm Hom}(\boldsymbol\omega, L_1)^G. \end{equation} (3.5) We now define the isomorphism before the Theorem of the introduction to be the composition of the isomorphisms   \begin{align} H^2(G,{\rm Hom}(\Delta S,\boldsymbol{\mu} ))^* &\longrightarrow H^1(G,{\rm Hom}(L_1,\boldsymbol{\mu}))^* \longrightarrow \widehat H^0(G,{\rm Hom}(L_1,\overline{\boldsymbol\omega}))^*\notag\\ &\longleftarrow \widehat H^0 (G,{\rm Hom}(\boldsymbol\omega ,L_1)) \longrightarrow H^1(G,{\rm Hom}(\boldsymbol\omega , L_2)) \end{align} (3.6) of (2.2)$$^*$$, (2.3)$$^*$$, (3.4), (3.2), and observe that it takes $$-\varepsilon $$ to $$-\delta _0([h]).$$ It follows that $$\varepsilon ^{(1)} = -\delta _0([h])$$ in the statement of the Theorem of the introduction, which is therefore equivalent to the following reformulation. Theorem 3.1. Let$$[h]\in \widehat H^0 (G,{\rm Hom}(\boldsymbol \omega ,L_1))$$be the image of$$\varepsilon $$under the composite of the first three maps in (3.6), and let$$\delta _0$$be the last map of that composite, as in (3.2). Let$$M$$be the$$G$$-module in a$${\mathbb Z}$$-split 1-extension  \[ 0 \longrightarrow L_2 \longrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \]with extension class equal to$$-\delta _0([h])$$in$$H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2)).$$Then$$E\oplus ({\mathbb Z} G)^n$$is stably isomorphic to$$M,$$with$$n:= (\vert G\vert -2)(\vert S\vert -1) + w$$when$$G\ne 1.$$ In particular, the character$$\varepsilon $$and the extension class of$$M(\varepsilon )$$determine each other uniquely. The connection between $$\varepsilon $$ and $$\varepsilon _1$$ is a consequence of the relationship between Tate sequences and Tate envelopes, or, more precisely, between the Tate canonical class $$\alpha _3 \in H^2(G,{\rm Hom}(\Delta S,E))$$ and Tate envelopes. We now adapt this connection, sketched on [7, p. 972], to our needs, with proof. Thus, following the last four paragraphs of Tate's proof of Theorem 5.1 of [12, Chapter 2], we select a special Tate sequence representing $$\alpha _3$$ and define the Tate envelope to be the left half of this special Tate sequence. Proposition 3.2. A Tate envelope$$0\to E\to A\to L_1 \to 0$$has  \[ \Omega_m = A-(\vert S\vert -1 )[{\mathbb Z} G] \quad \text{in}\ {\rm Cl}({\mathbb Z} G). \] Proof. We specialize Tate's initial exact sequence by selecting the one   \begin{equation} 0 \longrightarrow L_2 \longrightarrow B^{\prime} \longrightarrow B \longrightarrow \Delta S \longrightarrow 0, \end{equation} (3.7) obtained by splicing (2.1) and (3.1); Tate's first paragraph ends with isomorphisms   \[ \widehat H^r(G,{\rm Hom}(L_2,E)) \simeq \widehat H^{r+2}(G,{\rm Hom}(\Delta S,E)), \] for all $$r\in {\mathbb Z},$$ in our notation. The second paragraph chooses $$\alpha \in {\rm Hom}_G(L_2,E)$$ corresponding to $$\alpha _3\in H^2(G,{\rm Hom}(\Delta S,E))$$ and deduces, from his (5.2), that $$\alpha $$ induces isomorphisms $$\widehat H^r(G,L_2) \to \widehat H^r(G,E),$$ for all $$r;$$ the third paragraph extends $$\alpha $$ to a surjective $$\alpha :L_2\oplus F\to E,$$ with $$F$$ free, and replaces $$L_2\to B^{\prime }$$ in (3.7) by $$L_2\oplus F\to B^{\prime } \oplus F$$ to obtain a new (3.7) and the exact sequence $$0\to \,\ker (\alpha )\to L_2\oplus F\to E\to 0.$$ The fourth paragraph deduces that $$\ker (\alpha ),$$ and thus $$A:= (B^{\prime } \oplus F)/\ker (\alpha ),$$ is cohomologically trivial. Combining with the new (3.7) gives the Tate sequence $$0\to E\to A\to B\to \Delta S\to 0,$$ the left half $$0\to E\to A\to L_1\to 0$$ of which is our Tate envelope. Now $$B={\mathbb Z} G\otimes \Delta S\simeq ({\mathbb Z} G)^ {\vert S\vert -1}$$ implies that $$\Omega _m =[A]-[B]= [A] - ( \vert S\vert -1)[{\mathbb Z} G].$$ Preparing for §4: if $$(C): 0\to M\to C\to L_1\to 0$$ is an envelope with $${\mathbb Z}$$-torsion $$j: \boldsymbol {\mu }\hookrightarrow M ,$$ applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ and $$G$$-cohomology gives an isomorphism   \begin{equation} \partial_{(C)}:[L_1,L_1]\longrightarrow H^1\big(G,{\rm Hom}(L_1,M)\big), \end{equation} (3.8) of right $$[L_1,L_1]$$-modules. Then $$\tau _1\partial _{(C)}^{-1}j_*$$ is in $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*$$ and we say, following (1.6) of [7], that $$(C)$$ is linked to its $${\rm Aut}_G(\boldsymbol {\mu } )$$-orbit. Here $$\tau _1:[L_1,L_1]\to {\mathbb Q}/{\mathbb Z}$$ is as in (5.6) of [6]. This orbit is here insensitive to the choice of $$j,$$ because $${\rm Aut}_G(\boldsymbol {\mu } ) = {\rm Aut}(\boldsymbol \mu )$$ since $$\boldsymbol {\mu } $$ cyclic implies that $${\rm Aut}(\boldsymbol {\mu })$$ is abelian. 4. Proof of the reformulated theorem The proof is now straightforward. We assume that $$G\ne 1$$ (since the $$G=1$$ case, while true with the obvious interpretation, is trivial), and start by fixing an envelope   \[ 0\longrightarrow \boldsymbol{\mu} \longrightarrow \boldsymbol \omega \longrightarrow \overline{\boldsymbol\omega } \longrightarrow 0, \] satisfying (1.1) and (1.2). The existence of such an $$\boldsymbol \omega $$ follows from (2.1) in [6] and (39.12), (32.13) in [4]: start with any envelope $$0\to \boldsymbol {\mu } \to C\to \overline C \to 0,$$ define $$c$$ by $$\vert G\vert c ={\rm dim} \, {\mathbb Q} \otimes C,$$ and observe that $$\Omega _m - ([C] - c[{\mathbb Z} G]) = [P] - [{\mathbb Z} G]$$ in $${\rm Cl}({\mathbb Z} G),$$ for some projective $${\mathbb Z} G$$-module $$P$$ with $${\rm dim}\,{\mathbb Q} \otimes P = \vert G\vert ,$$ hence $$C^{\prime } := C\oplus P$$ gives an envelope $$0\to \boldsymbol {\mu } \to C^{\prime } \to \overline {(C^{\prime })} \to 0$$ with $$\Omega _m = [C^{\prime }] -c^{\prime } [{\mathbb Z} G],$$ as required. Letting $$[h],$$ with $$h\in \,{\rm Hom}(\boldsymbol \omega ,L_1)^G,$$ be as in the assertion of Theorem 3.1, define $$\eta :({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega \to L_1$$ by $$\eta ((x,y)) = p_1(x) + h(y),$$ and form the big diagram as follows: start from the commutative square containing $$p_1$$ and $$\eta ,$$ use it to form the bottom two rows with the additional map sending $$(x,y)$$ to $$y,$$ and then get the top row by taking kernels, and using (3.1) as the first column. We put $$M:=\ker (\eta )$$ and focus first on the column and then on the row containing $$M.$$ Now let $$0\to E\to A\to L_1\to 0$$ be a fixed Tate envelope, and form the envelope   \begin{equation} 0 \longrightarrow ({\mathbb Z} G)^n \oplus E \longrightarrow ({\mathbb Z} G)^n\oplus A\longrightarrow L_1\longrightarrow 0, \end{equation} (4.2) from it by adding $$({\mathbb Z} G)^n = ({\mathbb Z} G)^n.$$ This is an envelope with $${\mathbb Z}$$-torsion $$\boldsymbol {\mu } $$ and lattice $$L_1,$$ as is the middle column   \begin{equation} 0\longrightarrow M\longrightarrow ({\mathbb Z} G\otimes L_1)\oplus \boldsymbol\omega \longrightarrow L_1\longrightarrow 0, \end{equation} (4.3) of (4.1). We now apply [7, Theorem 4.7] to show that the left ends of these envelopes are stably isomorphic. This requires two conditions to be verified. The quicker condition to check is that $$[({\mathbb Z} G\otimes L_1) \oplus \boldsymbol \omega ]$$ is equal to $$[({\mathbb Z} G)^n\oplus A]$$ in $$K_0({\mathbb Z} G).$$ Now $${\mathbb Z} G\otimes L_1\simeq ({\mathbb Z} G)^{(\vert G\vert -1)(\vert S\vert -1)},$$ because it is $${\mathbb Z} G$$-free; and (1.2) applies to $$[\boldsymbol \omega ],$$ hence $$[({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega ] = (\vert G\vert -1)(\vert S\vert -1)[{\mathbb Z} G] + w[{\mathbb Z} G] +\Omega _m.$$ Similarly, the second expression equals $$n[{\mathbb Z} G] + (\vert S\vert -1)[{\mathbb Z} G] + \Omega _m,$$ by Proposition 3.2. These agree by the choice of $$n$$. The other condition is that both of these envelopes are linked to the same$${\rm Aut}_G(\boldsymbol \mu )$$-orbit on $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*,$$ which we will show is $$\varepsilon _1\,{\rm Aut}_G(\boldsymbol \mu ).$$ First, by definition, the Tate envelope is linked to $$\tau _1\partial _{(A)}^{-1}j_*,$$ with $$j:\boldsymbol {\mu } \hookrightarrow E$$ the inclusion, which is $$t_Ej_*$$ by definition of the trace character $$t_E$$ in §7, that is, the ‘restriction’ $$\varepsilon _1$$ of $$t_E$$ to $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } )).$$ To get the same conclusion for the envelope (4.2), consider the commutative diagram defined by inclusion of the Tate envelope into (4.2), and apply $${\rm Hom}\,(L_1,\,\underline {\,\,}\,)$$ and $$G$$-cohomology to obtain the commutative square, with all maps isomorphisms, inside the commutative diagram with left triangle from composing the inclusions $$\boldsymbol {\mu } \hookrightarrow E$$ and $$E\hookrightarrow ({\mathbb Z} G)^n \oplus E.$$ The top composite from $$H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))$$ to $${\mathbb Q}/{\mathbb Z}$$ is equal to $$\varepsilon _1,$$ by the first sentence of this paragraph, hence so is the bottom one. Next, to see that the envelope (4.3) is linked to $$\varepsilon _1,$$ consider the commutative diagram with top row the envelope $$(\boldsymbol \omega )$$ of (1.1), (1.2), bottom row the vertical envelope $$(C)$$ of (4.1) with $$C= ({\mathbb Z} G\otimes L_1)\oplus \boldsymbol \omega ,$$ and $$k(y) = (0,y)$$ for all $$y\in \boldsymbol \omega .$$ Here, forming the right square first defines $$j^{\prime }.$$ Applying $${\rm Hom}(L_1,\underline {\,\,}\,)$$ and $$G$$-cohomology gives the commutative square with horizontal isomorphisms and $$(C)$$ linked to $$\tau _1\partial _{(C)}^{-1}(j^{\prime })_* \in H^1({\rm Hom}\,(L_1,\boldsymbol {\mu } ))^*,$$ by the definition (3.8), with $$\tau _1:[L_1,L_1] \to {\mathbb Q}/{\mathbb Z}.$$ Our hypothesis on $$[h]$$ implies the $$[\overline h]^* =\varepsilon _1\partial ^{\prime }_0$$ by (3.5), (2.5), and (2.3), with $$\partial ^ {\prime }_0 = \partial _{(\boldsymbol \omega )},$$ that is, $$[\overline h]^* =\varepsilon _1\partial _{(\boldsymbol \omega )}.$$ Now, quoting [7], $$\varepsilon _1\in H^1(G,{\rm Hom}(L_1,\boldsymbol {\mu } ))^*$$ implies that $$\varepsilon _1 = \tau _1\theta $$ for some right $$[L_1,L_1]$$-homomorphism $$\theta :H^1({\rm Hom}(L_1,\boldsymbol {\mu } ))\to [L_1,L_1],$$ by (2.3). Then $$\theta \partial _{(\boldsymbol \omega )}$$ is a right $$[L_1,L_1]$$-homomorphism: $$[L_1,\overline {\boldsymbol \omega }]\to [L_1,L_1]$$ so that $$[\overline h]\in [\overline {\boldsymbol \omega }, L_1]$$ having $$[\overline h]^*= \varepsilon _1\partial _{(\boldsymbol \omega )} = \tau _1\theta \partial _{(\boldsymbol \omega )},$$ by the previous paragraph, implies that $$\theta \partial _{(\boldsymbol \omega )} = [{\rm id}_{L_1},\overline h],$$ by (2.4). Combining with (4.4) above gives $$\tau _1\partial _{(C)}^{-1}(j^{\prime })_* =\tau _1[{\rm id}_{L_1}, \overline h]\partial _{(\boldsymbol \omega )}^{-1} = \tau _1\theta =\varepsilon _1,$$ as required. Finally, we must show that the top row   \[ (M) : 0\longrightarrow L_2 \hookrightarrow M\longrightarrow \boldsymbol\omega \longrightarrow 0 \] of the big diagram (4.1) has extension class $$-\delta _0([h]),$$ in the notation of (2.6). To obtain a 1-cocycle representing $$-\delta _0([h]),$$ one applies $${\rm Hom}(\boldsymbol \omega ,\underline {\,\,}\,)$$ to (3.1), obtaining the exact sequence $$0\to {\rm Hom}(\boldsymbol \omega ,L_2)\to {\rm Hom}(\boldsymbol \omega ,{\mathbb Z} G\otimes L_1)\to {\rm Hom}(\boldsymbol \omega ,L_1)\to 0,$$ chooses a pre-image of $$h$$ in $${\rm Hom} (\boldsymbol \omega ,{\mathbb Z} G\otimes L_1),$$ say the map $$1\otimes h$$ taking every $$y\in \boldsymbol \omega $$ to $$1\otimes h(y),$$ and then forms the 1-cocycle $$g\mapsto (1\otimes h) - g(1\otimes h)$$ (with $$g\in G)$$ taking values in $${\rm Hom}(\boldsymbol \omega ,L_2),$$ namely $${}[(1\otimes h)-g\!\cdot (1\otimes h)](y) = (1\otimes h)(y) -g\!\cdot (1\otimes h)(g^{-1}y) \,{=}\,1\otimes h(y)\,{-}\,g\!\cdot (1\otimes h(g^{-1}y)) \,{=}\,1\otimes h(y)\,{-}\,g\otimes g\!\cdot \! h(g^{-1}y) \,{=}\, 1\otimes h(y) \,{-}\,g\otimes h(y) \,{=}\,(1-g)\otimes h(y)\in \Delta G\otimes L_1= L_2.$$ On the other hand, the extension class $$\xi _{(M)} $$ of $$(M)$$ is, by definition, obtained from $$(M) $$ by applying $${\rm Hom}(\boldsymbol \omega ,\underline {\,\,}\,)$$ to $$(M),$$ obtaining $$0\to {\rm Hom}\,(\boldsymbol \omega ,L_2)\to {\rm Hom}\,(\boldsymbol \omega ,M)\to {\rm Hom}\,(\boldsymbol \omega ,\boldsymbol \omega )\to 0,$$ lifting $${\rm id}_{\boldsymbol \omega }$$ to some $$s\in {\rm Hom}\,(\boldsymbol \omega ,M),$$ and forming the class of the 1-cocycle $$g\mapsto gs-s$$ with values in $${\rm Hom}\,(\boldsymbol \omega ,L_2).$$ Setting $$s(y) = (-1\otimes h(y),y)$$ works, since $$\eta (s(y)) = p_1(-1\otimes h(y)) + h(y) =0$$ and $$p_0(s(y)) = y.$$ Now $$(gs-s)(y) = g(-1\otimes h(g^{-1}y), g^{-1}y) - (-1\otimes h(y),y) = (-g\otimes h(y),y) + (1\otimes h(y),-y) = ((1-g)\otimes h(y),0),$$ which is the image of $$(1-g)\otimes h(y)\in L_2.$$ This agrees with the 1-cocycle of the previous paragraph. 5. Discussion We begin with a consequence of the Theorem, for which we prepare with a naturality property of the Gruenberg resolution. We start with a subset, of $$d$$ elements $$g_i$$ of $$G\backslash \{1\},$$ which generates$$G,$$ form the free group $$F$$ on $$x_i,\ 1\le i\le d,$$ and define the relation module $$R_d$$ by the exact sequence   \begin{equation} 0\longrightarrow R_d\longrightarrow {\mathbb Z} G\otimes_{{\mathbb Z} F}\Delta F\longrightarrow \Delta G\longrightarrow 0 \end{equation} (5.1) (cf. [8, p. 199 and 218]). Here the middle term is $$\simeq ({\mathbb Z} G)^d$$ since $$\Delta F$$ is $${\mathbb Z} F$$-free on the elements $$(x_i-1)$$, and the right map sends the $${\mathbb Z} G$$-basis $$1\otimes _F (x_i-1)$$ to $$g_i-1.$$ In the special case $$d=\vert G\vert -1,$$ write $${\mathcal R}, {\mathcal F}$$ for $$R_d,\,F,$$ respectively. For general $$d,$$ the inclusion $$F\to {\mathcal F}$$ induces a map from the relation sequence for $$R_d$$ to $${\mathcal R},$$ which on middle terms is an inclusion of the respective $${\mathbb Z} G$$-bases, so has cokernel $$\simeq ({\mathbb Z} G)^{\vert G\vert -1-d},$$ yielding the exact sequence $$0\to R_d \to {\mathcal R} \to ({\mathbb Z} G)^{\vert G\vert -1-d}\to 0$$ on the left terms. Similarly, the relation module sequence for $${\mathcal R}$$ maps to the exact sequence obtained by applying $$\underline {\,\,}\otimes \Delta G$$ to the augmentation sequence, with middle map matching $${\mathbb Z} G$$-bases by $$1\otimes _{\mathcal F} (x_i-1)\mapsto 1\otimes (g_i-1),$$ inducing an isomorphism $${\mathcal R}\to \Delta G\otimes \Delta G.$$ This implies that   \begin{equation} 0\longrightarrow R_d \overset\beta \longrightarrow \Delta G\otimes \Delta G\longrightarrow ({\mathbb Z} G)^m \longrightarrow 0 \end{equation} (5.2) is exact with an explicit map $$\beta $$ and $$m=\vert G\vert -1-d,$$ when $$G\ne 1.$$ Let $$d(G)$$ be the minimum number of generators of $$G,$$ and set $$R:=R_{d(G)},$$ to state the following corollary. Corollary 5.1. There is an explicit$$G$$-homomorphism$$\beta ^{\prime } :R\otimes \Delta S\to L_2$$so that the induced isomorphism$$\beta ^{\prime }_*: H^1(G,{\rm Hom}(\boldsymbol \omega ,R\otimes \Delta S))\to H^1(G,{\rm Hom}(\boldsymbol \omega ,L_2))$$has the following property: let$$M^{\prime }$$be the$$G$$-module in a$${\mathbb Z}$$-split 1-extension  \[ 0\longrightarrow R\otimes \Delta S\longrightarrow M^{\prime} \longrightarrow \boldsymbol\omega \longrightarrow 0 \]with extension class mapping to$$\varepsilon ^{(1)}$$under$$\beta ^{\prime }_*.$$Then$$E\oplus ({\mathbb Z} G)^{n^{\prime }}$$is stably isomorphic to$$M^{\prime }$$with$$n^{\prime } = (d(G) -1)(\vert S\vert -1) + w$$when$$G\ne 1.$$ Proof. By $$L_2 = \Delta G\otimes L_1 =\Delta G\otimes (\Delta G\otimes \Delta S)\simeq (\Delta G\otimes \Delta G)\otimes \Delta S,$$ applying $$\underline {\,\,} \otimes \Delta S$$ to (5.2) gives the exact sequence   \begin{equation} 0\longrightarrow R\otimes \Delta S\overset{\beta^{\prime}}\longrightarrow L_2 \longrightarrow ({\mathbb Z} G)^{n-n^{\prime}} \longrightarrow 0, \end{equation} (5.3) defining $$\beta ^{\prime }.$$ This follows from $$({\mathbb Z} G)^m \otimes \Delta S\simeq ({\mathbb Z} G)^{m(\vert S\vert -1)}$$ with $$m(\vert S\vert -1)=n-n^{\prime }.$$ Now the extension class of the 1-extension $$(M^{\prime })$$ has the property that its pushout along $$\beta ^{\prime }$$ has extension class $$\varepsilon ^{(1)}$$, so there is a commutative diagram Since $$\beta ^{\prime }$$ has cokernel $$({\mathbb Z} G)^{n-n^{\prime }}$$, so does the middle arrow, hence there is an exact sequence $$0\to M^{\prime } \to M\to ({\mathbb Z} G)^{n-n^{\prime }} \to 0.$$ Thus, by the Theorem, $$E\oplus ({\mathbb Z} G)^n\approx M\approx M^{\prime } \oplus ({\mathbb Z} G)^{n-n^{\prime }},$$ which implies that $$E\oplus ({\mathbb Z} G)^{n^{\prime }} \approx M^{\prime }.$$ Remark 1. The relation module $$R$$ has no non-zero projective summand if $$G$$ is solvable or, more generally, when $$G$$ has generation gap $$=0$$ (cf. (24) in [5]), in which case we cannot expect bigger $${\mathbb Z} G$$-free summands from the above approach. Note that $$R_d$$ is determined up to stable isomorphism by $$d,$$ as follows from (5.1) by Schanuel's lemma. Corollary 5.1 is a first step toward the important goal of excising as many $${\mathbb Z} G$$-free summands of $$M$$ as explicitly as possible. There are many aspects of this problem but still no systematic approach. There has been considerable work on Chinburg's conjecture as a special case of the Equivariant Tamagawa Number Conjecture; a recent reference is [1] (cf. Corollary 2.8 and Remark 2.9). Since Chinburg's conjecture predicts that $$\Omega _m=0$$ whenever $$G$$ has no irreducible symplectic representation (cf. [3, §3]), an envelope $$\boldsymbol \omega $$ of $$\boldsymbol \mu $$ with $$[ \boldsymbol \omega ] - w[{\mathbb Z} G]=0$$ and $$w =d(G) $$ (cf. Chapter 3 of [9]) is a useful ingredient for examples. On the other hand, the condition (1.2) on $$\boldsymbol \omega $$ could be replaced in the Theorem by   \[ [\boldsymbol\omega ] - w[{\mathbb Z} G] \equiv \Omega_m\mod B[\varepsilon_1], \] as the appeal to [7, Theorem 4.7] in its proof shows. This shows that the full strength of Chinburg's conjecture may not be needed. Remark 2. The emphasis on $$\varepsilon _1$$ in [7] comes from the envelope focus. In particular, Theorem A for $$\varepsilon _1$$ is proved by this method, but its statement depends on the local and global invariant maps on $$H^2,$$ the natural habitat of $$\varepsilon .$$ Theorem A can be translated from $$\varepsilon _1$$ to $$\varepsilon $$ by using the formalism of [11], in the direction of the last paragraph of the Remark on p. 971 of [7]. The $$\varepsilon \text {-analogue}$$ of Theorem A is strikingly simpler than it was for $$\varepsilon _1.$$ More precisely, let $${\rm Hom}(({\mathbb Z} S),(J))$$ be the $$G$$-module consisting of all triples $$(f_1,f_2,f_3)$$ of $${\mathbb Z}$$-homomorphisms so that the diagram commutes. This leads to an exact sequence   \begin{align*} 0&\longrightarrow H^2(G,{\rm Hom}(({\mathbb Z} S),(J)))\longrightarrow H^2 (G,{\rm Hom}(\Delta S,E))\oplus H^2(G,{\rm Hom}({\mathbb Z} S,J))\\ &\longrightarrow H^2(G,{\rm Hom}(\Delta S,J))\longrightarrow 0 \end{align*} allowing us to study the trace character $$T_E:H^2(G,{\rm Hom}(\Delta S,E))\to {\mathbb Q}/{\mathbb Z}$$ defined by dimension shifting $$t_E$$ using the exact sequence (2.1). This implies that $$\varepsilon =T_E\circ j_*,$$ with $$j:\boldsymbol \mu \hookrightarrow E$$ the inclusion, but now the point is that $$T_E$$ can be described in terms of the $$H^2$$-sequence above without further dimension shifting. More precisely, given $$x\in H^2(G,{\rm Hom}(\Delta S,E)),$$ there exists $$y\in H^2(G,{\rm Hom}({\mathbb Z} S,J))$$ so $$(x,y)$$ maps to 0 in $$H^2(G,{\rm Hom}(\Delta S,E)),$$ hence there is a unique $$T\in H^2(G,{\rm Hom}(({\mathbb Z} S),(J)))$$ mapping to $$(x,y).$$ Taking a 2-cocycle of triples representing $$T$$ and projecting on the third component gives a 2-cocycle defining $$z\in H^2(G,{\rm Hom}({\mathbb Z},C_K)).$$ Then (cf. Chapter 4 of [9])   \begin{equation} T_E(x) =\text{inv}(z) -\sum_{{{\mathfrak p}}\in S_*}\,\text{inv}_{{\mathfrak p}}(y_{{\mathfrak p}}), \end{equation} (5.4) where $$S_*$$ is a transversal to the $$G$$-orbits on $$S,y_{{\mathfrak p}} = k_{{\mathfrak p}}(\text {res}\,y)i_{{\mathfrak p}}$$ with $$k_{{\mathfrak p}}:J\to K^\times _{{\mathfrak p}}$$ the projection and $$i_{{\mathfrak p}}:{\mathbb Z} \to {\mathbb Z} S$$ with $$i_{{\mathfrak p}}(1)={{\mathfrak p}}.$$ This description has the weakness that the existence of $$y$$ apparently depends on the vanishing of $$H^3(G,J).$$ This situation is improved by the following lemma. Lemma 5.2. The map$$H^2(G,{\rm Hom}({\mathbb Z} S,J))\to H^2(G,{\rm Hom}(\Delta S,J))$$has a special splitting. Proof. The $$S$$-idele group $$J$$ is a finite product, over $${{\mathfrak p}}\in S_*,$$ of components $$V_{{\mathfrak p}}:= \prod _{\mathfrak q}K^\times _{\mathfrak q},$$ with $${\mathfrak q}$$ running through the $$G$$-orbit of $${{\mathfrak p}},$$ up to a large cohomologically trivial component of unit ideles. So it suffices to show that $$H^2(G,{\rm Hom}({\mathbb Z} S,V_{{\mathfrak p}}))\to H^2(G,{\rm Hom} (\Delta S,V_{{\mathfrak p}}))$$ is split for each $${{\mathfrak p}}.$$ If $$H$$ is a subgroup of $$G,$$ and $$B$$ any $$H$$-module, define the coinduced $$G$$-module $$\text {coind}\,(B),$$ from $$H$$ to $$G,$$ to be $${\rm Hom}_{{\mathbb Z} H}({\mathbb Z} G,B)$$ with $$g\in G$$ acting on elements $$\varphi $$ by $$(g\varphi )(z)=\varphi (zg)$$ for all $$z\in {\mathbb Z} G$$ (cf VII, [10, §5]). If $$D$$ is any $${\mathbb Z} G$$-lattice, viewing $${\rm Hom}(D,\text {coind}\,(B))$$ as $$G$$-module and $${\rm Hom}(\text {res}\, D,B)$$ as $$H$$-module, both by diagonal action, then there are natural Shapiro isomorphisms   \[ H^n(G,{\rm Hom}(D,\text{coind}\, B)) \longrightarrow H^n(H,{\rm Hom}(\text{res}\,D,B)). \] Take $$H=G_{{\mathfrak p}},$$$$B=K^\times _{{\mathfrak p}}$$ and identify $$\text {coind}\,K^\times _{{\mathfrak p}}$$ with $$V_{{\mathfrak p}},$$ via $$\varphi \mapsto \prod _t(t\!\cdot \varphi (t^{-1}),$$ with $$t$$ a choice of representatives of $$G/G_{{\mathfrak p}}.$$ This choice does not matter, since $$(th)\!\cdot \varphi ((th)^{-1}) = t\!\cdot (h\!\cdot \varphi (h^{-1}t^{-1})) = t\!\cdot \varphi (t^{-1})$$ for $$h\in G_{{\mathfrak p}}.$$ The map is bijective, since the components $$tK^\times _{{\mathfrak p}}$$ of $$V_{{\mathfrak p}}$$ are disjoint, and is a $$G$$-homomorphism because $$g(\prod _t(t\!\cdot \varphi (t^{-1})) = \prod _t((gt)\!\cdot \varphi ((gt)^{-1}g)) = \prod _t((gt)\!\cdot (g\varphi )((gt)^{-1})) =\prod _t(t\!\cdot (g\varphi )(t^{-1})).$$ This identifies our map of the first paragraph with the top row of the commutative square with vertical isomorphisms, and horizontal maps from $$0\to \Delta S\overset {a}\hookrightarrow {\mathbb Z} S\overset {a^{\prime }}\to {\mathbb Z}\to 0.$$ This exact sequence is $$G_{{\mathfrak p}}$$-split, by the $$G_{{\mathfrak p}}$$-map $$\lambda _{{\mathfrak p}}: d\mapsto d-a^{\prime }(d){{\mathfrak p}}$$ having $$\lambda _{{{\mathfrak p}}}\circ a =\text {id}_{\Delta S}.$$ Thus $$\lambda _{{\mathfrak p}}$$ induces $$H^2({\rm Hom}(\text {res}\, \Delta S,K^\times _{{\mathfrak p}}))\to H^2(G_{{\mathfrak p}},{\rm Hom}(\text {res}\, {\mathbb Z} S,K^\times _{{\mathfrak p}}))$$ splitting the bottom $$a^*$$ of the commutative square, which then completes the argument. References 1 Burns D., ‘ On main conjectures in non-commutative Iwasawa theory and related conjectures’, J. reine angew. Math.  698 ( 2015) 105– 159. 2 Chinburg T., ‘ On the Galois module structure of algebraic integers and $$S$$-units’, Invent. Math.  74 ( 1983) 321– 349. Google Scholar CrossRef Search ADS   3 Chinburg T., ‘ Exact sequences and Galois module structure’, Ann. of Math.  121 ( 1985) 351– 378. Google Scholar CrossRef Search ADS   4 Curtis C. W. Reiner I., Methods of representation theory  ( J. Wiley, New York, 1981/1987). 5 Gruenberg K. W., ‘Free abelianised extensions of finite groups’, Homological group theory (ed. C. T. C. Wall), London Mathematical Society Lecture Note Series 36 (Cambridge University Press, Cambridge, 1979), 71–104. 6 Gruenberg K. W. Weiss A., ‘ Galois invariants for units’, Proc. London Math. Soc.  70 ( 1995) 264– 283. Google Scholar CrossRef Search ADS   7 Gruenberg K. W. Weiss A., ‘ Galois invariants for $$S$$-units’, Amer. J. Math.  119 ( 1997) 953– 983. Google Scholar CrossRef Search ADS   8 Hilton P. Stammbach U., A course in homological algebra  ( Springer, Berlin, 1971). 9 Riveros Pacheco D. R., ‘On the Galois structure of the $$S$$-units for cyclotomic extensions over $$\mathbb {Q}$$’, PhD Thesis, University of Alberta, 2015. 10 Serre J.-P., Local fields  ( Springer, Berlin, 1979). 11 Tate J., ‘ The cohomology groups of tori in finite Galois extensions of number fields’, Nagoya Math. J.  27 ( 1966) 709– 719. Google Scholar CrossRef Search ADS   12 Tate J., Le conjectures de Stark sur les fonctions L d'Artin en s = 0 , Progress in Mathematics 47 ( Birkhauser, Boston, 1984). © 2016 London Mathematical Society

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Published: Feb 4, 2016

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