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denis.levchenko@math.uzh.ch Institute of Mathematics, Using the exceptional inverse image functor for quasi-ﬁnite proper morphisms of University of Zurich, separated tame Deligne–Mumford stacks of ﬁnite type over a ﬁeld k, Serre duality is Winterthurerstrasse 190, 8057 Zurich, Switzerland obtained in varying degrees of generality for tame Deligne–Mumford stacks. The approach follows that for schemes. Keywords: Serre duality, Deligne–Mumford stacks, Tame stacks, Quotient stacks Mathematics Subject Classiﬁcation: 14A20, 14F06 1 Introduction Serre duality for Deligne–Mumford stacks is a result of foundational importance. In the preprint [13], the general theory of Grothendieck duality for Deligne–Mumford stacks is developed and used to prove several Serre duality results over an algebraically closed ﬁeld. The paper [1] cites the preprint and expands on these results, also working over an algebraically closed ﬁeld. A simple valid argument for Serre duality for tame Deligne–Mumford stacks over any ﬁeld may be obtained without using the derived machinery of Grothendieck duality, by following the argument for schemes in [11, Section 6.4]. The purpose of this article is to record this argument. We follow the conventions for stacks of [10]. The reader may consult [11, Section 6.4] for a treatment of the topics mentioned here (dualizing sheaf, canonical sheaf, etc.) in the scheme case. The argument uses an auxiliary result that is perhaps of interest in its own right. We show in Proposition 4 and Remark 3 that a separated tame Deligne–Mumford stack of ﬁnite type over a ﬁeld has an open substack that is a quotient stack with complement everywhere of codimension at least 2. 2 Exceptional inverse image functor In the derived setting for algebraic stacks, existence of the right adjoint to the direct image functor was established in [13, Theorem 1.16] and [6, Theorem 4.14]. Here, we are only exhibiting the simpler non-derived version in the generality that is suﬃcient for our application to Serre duality. We write QCoh for the category of quasi-coherent sheaves on a Deligne–Mumford stack X. © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 67 Page 2 of 5 Levchenko Res Math Sci (2022) 9:67 Proposition 1 Let X and Y separated tame Deligne–Mumford stacks of ﬁnite type over a ﬁeld k. Let f : Y → X be a quasi-ﬁnite proper morphism of stacks. Then, f : QCoh → ∗ Y QCoh has a right adjoint f . Proof By [8]and [4, Corollary 18.5.10], we get an étale-local presentation of f , i.e., a Cartesian diagram Y [Spec(C)/H] f (1) X Spec(A), where H is a ﬁnite group acting on Spec(C) of order prime to the characteristic of k and the lower map is étale and surjective. First, we exhibit an explicit construction of the exceptional inverse image functor for the map g. Given a quasi-coherent sheaf N for an A-module N on Spec(A), we deﬁne g by sending N to the quasi-coherent sheaf associated with the C-module Hom (C, N), which has an H-action induced from the H-action on C. Given a quasi-coherent sheaf on [Spec(C)/H] associated with a C-module M with a compatible H-action, its direct image under g is the sheaf associated with M ,the A- submodule of H-invariant elements of M. To show that g is indeed the right adjoint of g , we exhibit a bijection H H Hom (M ,N) = Hom (M, Hom (C, N)) . A C A Indeed, Hom (M, N) = Hom (M, Hom (C, N)); this is an H-equivariant isomorphism A C A for the induced H-actions. Since the order of H is invertible in k, the averaging map 1 H h(−): M → M is well-deﬁned and, letting M denote the kernel, gives M = |H | h M ⊕ M , and hence Hom (M, N) = Hom (M ,N) ⊕ Hom (M ,N). A A A As H acts trivially on Hom (M ,N), we get the desired isomorphism from the vanishing of Hom (M ,N) . Applying the averaging map to a morphism M → N, we see it is invariant if and only if it is zero. It is clear that the exceptional inverse image functor is compatible with étale base change. Now for a general f : Y → X, we use the étale-local presentation g as above ! ! ! to deﬁne f by descent from g .The fact that f is indeed the right adjoint to f ,and in particular independent of the choice of presentation g, follows directly from étale base change compatibility. Remark 1 The hypothesis of tameness is essential in Proposition 1. As we see in easy examples like B(Z/2Z) → Spec k with k a ﬁeld of characteristic 2, without tameness f need not be right exact and hence may not have a right adjoint. As noted by Hall [5, Section 1], non-tame algebraic stacks may have inﬁnite cohomological dimension, preventing the use of many standard scheme techniques. 3 Serre duality Proposition 2 Let X be a separated tame Deligne–Mumford stack, of pure dimension n and of ﬁnite type over a ﬁeld k with projective coarse moduli space. Then, X admits a dualizing sheaf. Levchenko Res Math Sci (2022) 9:67 Page 3 of 5 67 ¯ ¯ Proof Let π : X → X be the coarse moduli space of X.Since X is projective and of pure n n ¯ ¯ dimension n over k, we can choose a ﬁnite morphism f : X → P .Let f : X → P be the k k composition. By Proposition 2 we have the functor f .Let ω be the dualizing sheaf of P . Then, for a quasi-coherent sheaf E on X we have: ! n n n ∼ ∼ ∼ n n Hom(E,f ω ) Hom(f E, ω ) Hom(H (P ,f E),k) Hom(H (X, E),k), (2) = = = P ∗ P ∗ k k where the ﬁrst isomorphism holds by the adjunction, second by the deﬁnition of the dualizing sheaf of P and the third by tameness, using a Leray spectral sequence argument. All three are functorial, hence so is the composition, and f ω is a dualizing sheaf of X. Proposition 3 With hypothesis of Proposition 2, assume further that X is Cohen– Macaulay. Then, for locally free coherent sheaves E, the isomorphism 2 extends uniquely to functorial isomorphisms j ∨ n−j ∨ H (X, E ⊗ ω ) → H (X, E) that are compatible with short exact sequences of locally free coherent sheaves. Proof Since X is Cohen–Macaulay, the map f : X → P from the proof of Proposition 2 is ﬂat. By locally applying an averaging argument as in Sect. 2,wesee that f E is locally free on P .Sowehave j ∨ j n ∨ j n ∨ n−j ∨ ∼ ∼ ∼ H (X, E ⊗ ω ) = H (P ,f (E ⊗ ω )) = H (P , (f E) ⊗ ω ) = H (X, E) . X ∗ X ∗ P k k Proposition 4 Let X be a separated tame pure-dimensional Deligne–Mumford stack, of ﬁnite type over a ﬁeld k with projective coarse moduli space. Then, there exists open U ⊂ X, such that U is a quotient stack, with X \ U empty or of codimension ≥ 2. Remark 2 By quotient stack, we mean the stack quotient of an algebraic space by a linear algebraic group. We recall useful statements from [9]: (Remark 4.2) there is no loss of generality in supposing the linear algebraic group to be GL for some N, (Remark 4.3) the algebraic space has to be a quasi-projective scheme when the coarse moduli space is quasi-projective, (Proposition 5.1) we get a linear action on this quasi-projective scheme, (Theorem 5.3, stated with the hypothesis char(k) = 0 but applicable to arbitrary k with modiﬁed conclusion) there is a closed immersion U → W where W is a smooth separated tame Deligne–Mumford stack of ﬁnite type over k with quasi-projective coarse moduli space. Proof Let f : X → P be a quasi-ﬁnite proper morphism, where n is the dimension of X. red A dense open substack of X is normal. Shrinking, we may suppose that the compo- nents have constant geometric stabilizer groups. Further shrinking, we may take this to be −1 n the reduced substack of f (S) for aﬃne open S ⊂ P , deﬁned by the nonvanishing of a 0 n section h ∈ H (P , O (d)) (d > 0). We extend, after possibly replacing d by a multiple 0 P and h by a power, to h , h , ... , h , spanning a base-point free linear system. We denote 0 0 1 n n n by h: P → P the corresponding ﬁnite morphism, so for the appropriate coordinate n −1 n hyperplane H ⊂ P we have S = h (P \ H). The complement of another coordinate hyperplane of P is an aﬃne space, whose intersection with H is deﬁned by the vanishing of one of the coordinates. This lets us 67 Page 4 of 5 Levchenko Res Math Sci (2022) 9:67 express the aﬃne space as the aﬃne line over an aﬃne chart of H. Localizing, we obtain 1 1 A .By[12, Thm. 59], the base change of X to A is a quotient stack. This stack k(H) k(H) has, by [2, Lemma 2.12], a locally free coherent sheaf with faithful geometric stabilizer group actions. This is the restriction of such a locally free coherent sheaf E on the base change of X to the aﬃne line over a nonempty aﬃne open subset of H.Wedenoteby R the pre-image of the latter by h,so R is aﬃne open in P , and the generic point of every irreducible component of P \ S is in R. −1 red The restriction of E to (f (R ∩ S)) extends to a torsion-free coherent sheaf on −1 red (f (S)) . So there exists aﬃne open S ⊂ S, containing the generic point of every −1 red irreducible component of S \ R, such that the restriction of E to (f (R ∩ S )) extends −1 red −1 red to a locally free coherent sheaf on (f (S )) (here we use the normality of (f (S)) : torsion-free implies locally free in a neighborhood of the generic point of any divisor). Arguing as in the proof of [12, Thm. 59], this may be extended further to a locally free −1 −1 red coherent sheaf on f (S ), and an isomorphism with the restriction of E to (f (R ∩S )) −1 −1 may be extended to an isomorphism on f (R ∩ S ). So, by [2, Lemma 2.12], f (R ∪ S ) is a quotient stack. Remark 3 The hypothesis of Proposition 4 can be relaxed to quasi-projective coarse mod- uli space with no change in argument. By [3, Theorem 1.5], this can be relaxed further to coarse moduli space a scheme. But as can be seen by Chevalley’s Theorem and Chow’s Lemma for algebraic spaces, an algebraic space as coarse moduli space also leads to the desired conclusion, see also [14, Tag 0ADD]. Removing the intersection of any pair of irreducible components of unequal dimensions lets us also remove the hypothesis of pure dimensionality. Theorem 1 Let X be a separated tame pure-dimensional Deligne–Mumford stack, of ﬁnite type over a ﬁeld k with projective coarse moduli space. Suppose further that X is locally a complete intersection (i.e., the classical local complete intersection condition holds for an étale atlas of X). Then, the dualizing sheaf of X is the canonical sheaf on X. Proof We take U as in the statement of Proposition 4 and U → W as in Remark 2.Since X is locally a complete intersection, X is Cohen–Macaulay. Then, we may appeal to [4, (5.10.5)], which has several consequences: restriction to U induces a bijection on regular functions, and the same is valid upon passage to an arbitrary étale chart of X; for a pair of invertible sheaves on X an isomorphism of their restrictions to U extends uniquely to an isomorphism on X; an invertible sheaf L on U extends as an invertible sheaf to X if and only if every point of X has an étale neighborhood V , such that the restriction of L to U × V is trivial. We have a canonical sheaf ω , whose restriction to U is deﬁned X X as in the scheme case using the embedding in W , and whose unique extension to X follows from the local complete intersection hypothesis. Then, we can show that there are 0 ∨ n ∨ functorial isomorphisms H (X, E ⊗ ω ) → H (X, E) for coherent sheaves E, where n is the dimension of X, making ω a dualizing sheaf for X. The argument starts as in the scheme case, taking f : X → P to be a quasi-ﬁnite proper morphism, so that we have the ! ! dualizing sheaf f ω n. The construction of f ω n, using ﬁnite quotient presentations of the P P n ! base change of X to a suitable étale cover of P , reveals that f ω n is an invertible sheaf; indeed, in the setting of (1) and upon forgetting the H-action, we obtain the exceptional inverse image to Spec(C) which in our setting is a local complete intersection scheme. Levchenko Res Math Sci (2022) 9:67 Page 5 of 5 67 −1 n Shrinking U, we may suppose that U = f (T) for some open T ⊂ P , still with X \ U empty or of codimension ≥ 2. The description of the restriction of f ω n to U, that we obtain from the factorization U → T × W as in [7, Section I.4], leads to an isomorphism with the restriction to U of ω . This extends uniquely to an isomorphism f ω = ω . X P X Funding Open access funding provided by University of Zurich. Data Availability Statement Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. Received: 16 February 2022 Accepted: 3 November 2022 Published online: 19 November 2022 References 1. Bruzzo, U., Sala, F.: Framed sheaves on projective stacks. Adv. Math. 272, 20–95 (2015) 2. Edidin, D., Hassett, B., Kresch, A., Vistoli, A.: Brauer groups and quotient stacks. Amer. J. Math. 123, 761–777 (2001) 3. Gross, P.: The resolution property of algebraic surfaces. Compositio Math. 148, 209–226 (2012) 4. Grothendieck, A.: Eléments de géométrie algébrique IV. Publ. Math. IHES 20, 5–259 (1964) 5. 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The Stacks Project Authors, The Stacks Project, https://stacks.math.columbia.edu, (2021) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.
Research in the Mathematical Sciences – Springer Journals
Published: Dec 1, 2022
Keywords: Serre duality; Deligne–Mumford stacks; Tame stacks; Quotient stacks; 14A20; 14F06
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