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Computations of the Cohomological Brauer Group of Some Algebraic Stacks

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Computations of the cohomological Brauer group of some algebraic stacks by Minseon Shin A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Martin C. Olsson, Chair Professor Sug Woo Shin Professor Daniel Tataru Professor Surjeet Rajendran Spring 2019 Computations of the cohomological Brauer group of some algebraic stacks Copyright 2019 by Minseon Shin 1 Abstract Computations of the cohomological Brauer group of some algebraic stacks by Minseon Shin Doctor of Philosophy in Mathematics University of California, Berkeley Professor Martin C. Olsson, Chair The theme of this dissertation is the Brauer group of algebraic stacks. Antieau and Meier showed that if k is an algebraically closed field of char k 6= 2, then Br(M1;1;k) = 0, where M1;1 is the moduli stack of elliptic curves. We show that if char k = 2 then Br(M1;1;k) = Z=(2). In another direction, we compute the cohomological Brauer group of Gm-gerbes; this is an analogue of a result of Gabber which computes the cohomological Brauer group of Brauer- Severi schemes. We also discuss two kinds of algebraic stacks X for which not all torsion 2 0 classes in H´et(X; Gm) are represented by Azumaya algebras on X (i.e. Br 6= Br ). i Contents Acknowledgements iii Introduction iv 1. Generalities 1 1.1. Skolem-Noether for locally ringed sites 1 1.2. Azumaya algebras and the Brauer group 3 1.3. Gerbes and twisted sheaves 7 1.4. Brauer map 11 1.5. The cup product and cyclic algebras 13 2. Brauer groups of algebraic stacks: generalities and examples 17 2.1. Surjectivity of Brauer map 17 2.2. Low-dimensional stacks 19 2.3. Quotient stacks by discrete groups 20 2.4. Classifying stack of an elliptic curve 21 2.5. Classifying stack of diagonalizable groups 22 2.6. Classifying stack of GLn 26 3. The Brauer group of the moduli stack of elliptic curves 30 3.1. Introduction 30 3.2. Preliminary observations 30 3.3. The case char k is not 2 32 3.4. The case char k is 2 37 4. The cohomological Brauer group of Gm-gerbes 49 4.1. Main theorem and introductory remarks 49 4.2. Gerbes and the transgression map 50 4.3. Picard groups of (Laurent) polynomial rings 52 4.4. Unit groups of Laurent polynomial rings 59 4.5. Proof of the main theorem 61 5. Variants 66 5.1. The Azumaya Brauer group of a Brauer-Severi scheme 66 5.2. The Azumaya Brauer group of a Gm-gerbe 67 5.3. A1-homotopy invariance of the Brauer group 68 Appendix A. Torsors under torsion-free abelian groups 71 Appendix B. Cohomology and spectral sequences 71 B.1. Cohomological descent spectral sequence 72 B.2. Higher direct images of sheaves on classifying stacks of discrete groups 72 Appendix C. Inverse image of gerbes 73 Appendix D. The Weierstrass and Hesse presentations of M1;1 77 D.1. Full level N structures 77 D.2. Comparing the Weierstrass and Hesse presentations 78 ii Appendix E. Computation using Magma 84 References 85 iii Acknowledgements I would like to thank my advisor Martin Olsson. He suggested this research topic, gen- erously shared his ideas, and patiently answered my questions. I greatly admire his mathe- matical style and taste and hope to continue learning from him. I had many helpful conversations with friends, for whom I am very grateful. They taught me many things, mathematical and otherwise, and they continue to humble me. I thank my family for their support in general. I received support from NSF grant DMS-1646385 (the Berkeley RTG in Arithmetic Geom- etry and Number Theory) during Fall 2017 and Fall 2018, and the Raymond H. Sciobereti Fellowship (the Berkeley mathematics department spring fellowship) during Spring 2017. iv Introduction The Brauer group of a field k is a classical invariant which classifies central simple k- algebras, and the Brauer group of an algebraic variety X classifies Azumaya OX -algebras, which are \twists" of the matrix algebra Matn×n(OX ) over the structure sheaf OX . In complex geometry, the fact that the Brauer group is an invariant for birational maps can be used to determine whether a variety is rational. In number theory, the Tate conjecture for divisors for a smooth projective surface X over a finite field is known to be equivalent to the finiteness of Br X. In this dissertation, we are interested in Brauer groups of algebraic stacks. An Azumaya algebra on a moduli stack corresponds to a family of Azumaya algebras on the objects of the moduli stack compatible with all morphisms between the objects. The Brauer group of a quotient stack [X=G] corresponds to Azumaya algebras on X that are equivariant with respect to the G-action. The Brauer group of stacks may sometimes be used to answer questions about algebraic varieties; for example, Lieblich [61] considered the Brauer group of classifying stacks Bµn to prove new cases of the period-index conjecture for function fields of curves over local fields. For the moduli stack of elliptic curves M1;1;k over an algebraically closed field k, Antieau and Meier had shown that Br(M1;1;k) = 0 if char k 6= 2. In Section 3 we compute Br(M1;1;k) in the characteristic 2 case: Theorem A1 ([4, 11.2] in char k 6= 2). Let k be an algebraically closed field. Then Br M1;1;k is 0 unless char k = 2, in which case Br M1;1;k = Z=(2). Theorem A2. Let k be a finite field of characteristic 2. Then ( Z=(12) ⊕ Z=(2) if x2 + x + 1 has a root in k Br M1;1;k = Z=(24) otherwise. The methods of [4] do not apply to the characteristic 2 case since they rely on the finite Galois cover of M1;1;k obtained by fixing a full level 2 structure. We study the char k = 2 case by considering full level 3 structures instead, which comes at the cost of increasing the size of the group (by which M1;1;k is a quotient stack) from j GL2(F2)j = 6 to j GL2(F3)j = 48. Gabber proved that, given a Brauer-Severi scheme π : X ! S, the cohomological Brauer group Br0(X) is the quotient of Br0(S) by the class [X]. In Section 4 we prove an analogous result on the cohomological Brauer group of Gm-gerbes: 0 Theorem B. Let S be a scheme and let πG : G! S be a Gm;S-gerbe with [G] 2 Br S. Then the sequence π∗ 0 0 G 0 H´et(S; Z) ! Br S ! Br G! 0 is exact, where the first map sends 1 7! [G]. This is one of the first computations of the Brauer group of an algebraic stack that is not a Deligne-Mumford stack. It may be viewed as a generalization of Gabber's result since the 1 2 image of a Brauer-Severi scheme X under the coboundary H´et(S; PGLn) ! H´et(S; Gm) is a torsion Gm-gerbe. Assuming additional hypotheses on S (i.e. that it is regular and its fraction field has characteristic 0), we give another proof of Gabber's result. v We discuss two kinds examples of algebraic stacks for which Br 6= Br0, namely the classi- fying stack of Z ⊕ Z over a regular local ring (see Example 2.3.5), and the classifying stack of an elliptic curve (see Corollary 2.4.6). In Section 5 we explore some questions related to the Brauer group of algebraic stacks. Whereas in Gabber's result and our Theorem B we are concerned with the cohomological Brauer group, it would also be interesting to ask whether the same statements hold with \Br0" replaced by \Br" (the Azumaya Brauer group). Partial positive results are discussed in Section 5.1 and Section 5.2. In Section 5.3 we discuss whether the Brauer group functor is an \A1-homotopy invariant", 1 namely whether Br S ! Br AS is an isomorphism. This is a question that arises naturally when trying to compute the Brauer group via the descent spectral sequence associated to a smooth covering; it may be viewed as being part of a collection of questions asking when i the ´etalecohomology functor H´et(−; Gm) for i ≥ 0 is invariant with respect to polynomial extensions of the ground ring. It is known that the unit group (i.e. the i = 0 case) is invariant exactly when the ring is reduced, and the Picard group (i.e. the i = 1 case) is invariant exactly when the ring is seminormal; it would be nice to find a similar, purely ring-theoretic criterion which corresponds exactly to those cases in which the Brauer group (roughly the i = 2 case) is invariant. The only unknown part is concerning the `-torsion for primes ` that are not invertible in the base. We extend a result of Knus and Ojanguren to show that the Brauer group is A1-homotopy invariant if the base is a monoid algebra over a small class of regular rings. 1 1. Generalities In this section we discuss Azumaya algebras on locally ringed sites. Some standard refer- ences are Giraud [40], Lieblich [59], and the Stacks Project [88]. 1.1. Skolem-Noether for locally ringed sites. Definition 1.1.1 (Locally ringed site, locally ringed topos). 1 A locally ringed site is a ringed site (C; O) such that, for every object U 2 C and f 2 Γ(U; O), there exists a covering fUi ! Ugi2I of U such that for each i 2 I, either fjUi or (1 − f)jUi is a unit of Γ(Ui; O).
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