DOCSLIB.ORG

The Differential Brauer Group

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Differential Brauer Group The Differential Brauer Group Raymond Hoobler City College of New York and Graduate Center, CUNY Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 1 / 13 Outline 1 Review of Brauer groups of fields, rings, and D-rings 2 Cohomology 3 Cohomological interpretation of D Brauer groups (with connections to Hodge theory) Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 2 / 13 Brauer Groups of Fields Finite dimensional division algebras L over a field K are classified by Br (K ) , the Brauer group of K. If L is a central, simple algebra over K, then it is isomorphic to Mn (D) for some division algebra D. Given two such algebras L and G, L K G is again a central simple K algebra. They are said to be Brauer equivalent if there are vector spaces V , W and a K algebra isomorphism L K End (V ) u G K End (W ). This is an equivalence relation, and Br (K ) is then defined to be the group formed from the equivalence classes with as product. Br (K ) classifies division algebras over K. For any such algebra L over K, there is a Galois extension L/K such that L K L u EndL (V ). Galois cohomology is then used to classify all such equivalence classes 2 using an isomorphism Br (K ) u H GK /K , K . Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 3 / 13 Brauer Groups of Rings Azumaya algebras over a commutative ring R A finitely generated central R algebra L is an Azumaya algebra algebra if L R L is a central simple algebra over L for any homomorphism from R to a field L. An Azumaya algebra L is a central, finitely generated R algebra which is a projective L Lop algebra. R Two such Azumaya algebras L and G are Brauer equivalent if there are faithful, projective R modules P, Q and an R algebra isomorphism L R End (P) u G R End (Q) . If R is a local ring, there is an etale extension S/R such that L R S u EndS (P) for some projective S module P. Etale cohomology is then used to classify all such equivalence classes 2 using an isomorphism ¶ : Br (R) H (Ret , Gm ) . !u Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 4 / 13 Brauer Groups of D-rings Let D = d1, ..., dn be a set of n commuting derivations on R, a ring containingf Q. g A D Azumaya algebra over R is an Azumaya algebra L over R equipped with derivations extending the action of D on R. Two such D Azumaya algebras L and G are D Brauer equivalent if there are faithful, projective D R modules P, Q and a D R algebra isomorphism L R End (P) u G R End (Q). This is an equivalence relation, and BrD(R) is the resulting group on the set of equivalence classes with R as the product. If R is local, there is an etale extension S and a D S isomorphism L R S u EndS (P) for some D S projective module P. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 5 / 13 Cohomology Let C be a category with fibred products. A pretopology on C consists of specifying for all X ob(C ), a set Cov(X ) whose members are collections 2 f : U X a A Cov(X ) satisfying f a a ! j 2 g 2 1 If f : X X is an isomorphism, f Cov (X ) . ! f ga 2 a 2 If f : U X Cov (X ) and g : V U Cov (Ui ) for all f a a ! g 2 f i i ! ag 2 i , then f g a : V a X Cov (X ) . f a i i ! g 2 3 If f : U X Cov (X ) and Y X , then f a a ! g 2 ! 2 C f X Y : U X Y Y Cov (Y ) . f a a ! g 2 A presheaf F : op ((Sets)) is a sheaf if for all X and U X CovC (!X ) , 2 C f a ! g 2 F (X ) , F (U ) F U X U ! ∏ a ∏ a b is exact. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 6 / 13 Cohomology Example 1 Xet has fa : Va U fa is an etale map and U = fa (Va) = Cov (Uff) . ! j gg et S 2 XD fl has g : Va U g is a flat D map of finite type and U = g (Va) = ff a ! j a a gg CovD fl (U) . S If G is a scheme, then its functor of points defines a sheaf in either of these topologies. Moreover there is a map of sites t : XD fl Xet since 1 ! any etale map is a flat D map. Thus t ( fa ) CovD fl (U) . Moreover f g 2 H (Xet , G ) H (XD fl, G ) for sheaves G defined by smooth, !u quasi-projective group schemes over X like the sheaf of units, Gm. Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 7 / 13 Cohomological Interpretation In particular on XD fl we have the exact sequence D d ln 1 0 G Gm Z 0 ! m ! ! X ! whose cohomology sequence contains 0 1 1 D c1 1 1 H XD fl, ZX H XD fl, Gm Pic (X ) H XD fl, ZX ! ! ! 2 D H XD fl, Gm Br (X ) 0 ! ! ! 2 if X is smooth since then H (Xet , Gm ) is torsion unlike the vector space 2 1 H XD fl, ZX ! How do we interpret this?? Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 8 / 13 Cohomological Interpretation Theorem Let X be a quasi-projective variety of finite type over a field K of 2 characteristic 0. If x H (X , mN ) , then there is an Azumaya algebra L equipped with an integrable2 connection constructed from x such that 2 ¶ ([L]) = iN (x) H (Xet , Gm ) where iN : m Gm is inclusion. 2 N ! For simplicity, let’sconsider the case where X = Spec (R) is a local ring and R contains a primitive Nth root of unity. Then there is an etale extension R S Cov (Spec (R)) , i.e. U = Spec (S) Spec (R) , and ! 2 3 ˇ 2 ! a Cech 2 cocycle z mN S such that [z] = x H ((R S) , mN ) . Now by refining S we2 may assume that it is in the2 standard! form S = (R [T ] / (p (T )))g (t) where p (T ) is a monic polynomial of degree D. So we approximate S by R [t] := R [T ] / (p (T )) = D R. 1 Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 9 / 13 Cohomological Interpretation Now our cocycle z mN (S R S R S) is constant on each connected 3 2 component of S but may vary from one component to another. So we must use an index set that accounts for this. We let J = connected components of S3 D Of course : = ∏ R [t]a = a J 1 R is not usually connected F a J 2 2 3 but for each connected component a of S there is an R [t]a which admits multiplication by the value za of z on that connected component and, as an R module, is free of rank M = D (# (J)) . Then we define an R module isomorphismF · ` = z : R R R S R S z a F S S ! F MJ th by multiplying the a factor in by za. Note that this amounts to a F th diagonal block matrix where the a block is zaID . Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 10 / 13 Cohomological Interpretation We let c ( J z ) : EndR ( ) R S R S EndR ( ) R S R S be the a F ! F algebra isomorphism given by conjugation by `z. Then we get descent data from the diagram End ( ) R R F S S e2 % L End ( ) R c ( J z ) ! F S# a e1 & End ( ) R R F S S th where ei means insert 1S into the i copy of S. Note that End ( ) is the F algebra of M M matrices with d (eij ) = 0 for all d D. Here 2 c ( J z ) = c ` is the patching data used to define L and it preserves a z the action of D since c ` is given by conjugation by an Nth root of z unity on each block in End ( ) . F Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 11 / 13 Cohomological Interpretation It satisfies the cocycle condition 3 End ( ) R F S e3 c ` z % 3 End ( ) R e1 c ` F S # z e2 c ` z & 3 End ( ) R F S This commutes because 1 (e2 (c ( J z ))) (e1 (c ( J z ))) (e3 (c ( J z ))) is conjugation on a a a End ( ) R R by 1 z since z is a 2 cocycle. But this is F S S R S 1End ( ) S 3 which is the cocycle condition for descent. F Raymond Hoobler (City College of New York and GraduateThe Differential Center, CUNY) Brauer Group 12 / 13 Cohomological Interpretation Thus the Cech cocycle provides the needed descent data and we immediately see that 2 ¶ ([c ( J z )]) = [z] Hˇ (X , m ) a 2 n where L = [c ( J z )] is the desired D Azumaya algebra.
Load more

Recommended publications
  • AZUMAYA, SEMISIMPLE and IDEAL ALGEBRAS* Introduction. If a Is An
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972 AZUMAYA, SEMISIMPLE AND IDEAL ALGEBRAS* BY M. L. RANGA RAO Communicated by Dock S. Rim, January 31, 1972 Introduction. If A is an algebra over a commutative ring, then for any ideal I of R the image of the canonical map / ® R A -» A is an ideal of A, denoted by IA and called a scalar ideal. The algebra A is called an ideal algebra provided that / -> IA defines a bijection from the ideals of R to the ideals of A. The contraction map denned by 21 -> 91 n JR, being the inverse mapping, each ideal of A is a scalar ideal and every ideal of R is a contraction of an ideal of A. It is known that any Azumaya algebra is an ideal algebra (cf. [2], [4]). In this paper we initiate the study of ideal algebras and obtain a new characterization of Azumaya algebras. We call an algebra finitely genera­ ted (or projective) if the R-module A is, and show that a finitely generated algebra A is Azumaya iff its enveloping algebra is ideal (1.7). Hattori introduced in [9] the notion of semisimple algebras over a commutative ring as those algebras whose relative global dimension [10] is zero. If R is a Noetherian ring, central, finitely generated semisimple algebras have the property that their maximal ideals are all scalar ideals [6, Theorem 1.6]. It is also known that for R, a Noetherian integrally closed domain, finitely generated, projective, central semisimple algebras are maximal orders in central simple algebras [9, Theorem 4.6].
    [Show full text]
  • No Transcendental Brauer-Manin Obstructions on Abelian Varieties
    THERE ARE NO TRANSCENDENTAL BRAUER-MANIN OBSTRUCTIONS ON ABELIAN VARIETIES BRENDAN CREUTZ Abstract. Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer-Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate-Shafarevich group. Our results are unconditional. 1. Introduction Let X be a smooth projective and geometrically integral variety over a number field k. In order that X possesses a k-rational point it is necessary that X has points everywhere locally, i.e., that the set X(Ak) of adelic points on X is nonempty. The converse to this statement is called the Hasse principle, and it is known that this can fail. When X(k) is nonempty one can ask if weak approximation holds, i.e., if X(k) is dense in X(Ak) in the adelic topology. Manin [Man71] showed that the failure of the Hasse principle or weak approximation can, in many cases, be explained by a reciprocity law on X(Ak) imposed by the Brauer group, 2 Br X := Hét(X, Gm). Specifically, each element α ∈ Br X determines a continuous map, α∗ : X(Ak) → Q/Z, between the adelic and discrete topologies with the property that the subset X(k) ⊂ X(Ak) of rational points is mapped to 0.
    [Show full text]
  • Brauer Groups of Abelian Schemes
    ANNALES SCIENTIFIQUES DE L’É.N.S. RAYMOND T. HOOBLER Brauer groups of abelian schemes Annales scientifiques de l’É.N.S. 4e série, tome 5, no 1 (1972), p. 45-70 <http://www.numdam.org/item?id=ASENS_1972_4_5_1_45_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1972, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scienL EC. Norm. Sup., 4® serie, t. 5, 1972, p. 45 ^ 70. BRAUER GROUPS OF ABELIAN SCHEMES BY RAYMOND T. HOOBLER 0 Let A be an abelian variety over a field /c. Mumford has given a very beautiful construction of the dual abelian variety in the spirit of Grothen- dieck style algebraic geometry by using the theorem of the square, its corollaries, and cohomology theory. Since the /c-points of Pic^n is H1 (A, G^), it is natural to ask how much of this work carries over to higher cohomology groups where the computations must be made in the etale topology to render them non-trivial. Since H2 (A, Gm) is essentially a torsion group, the representability of the corresponding functor does not have as much geometric interest as for H1 (A, G^).
    [Show full text]
  • Uniform Distribution in Subgroups of the Brauer Group of an Algebraic Number Field
    Pacific Journal of Mathematics UNIFORM DISTRIBUTION IN SUBGROUPS OF THE BRAUER GROUP OF AN ALGEBRAIC NUMBER FIELD GARY R. GREENFIELD Vol. 107, No. 2 February 1983 PACIFIC JOURNAL OF MATHEMATICS Vol. 107, No. 2, 1983 UNIFORM DISTRIBUTION IN SUBGROUPS OF THE BRAUER GROUP OF AN ALGEBRAIC NUMBER FIELD GARY R. GREENFIELD We construct subgroups of the Brauer group of an algebraic number field whose member classes have Hasse invariants satisfying a rigid arithmetic structure — that of (relative) uniform distribution. After ob- taining existence and structure theorems for these subgroups, we focus on the problem of describing algebraic properties satisfied by the central simple algebras in these subgroups. Key results are that splitting fields are determined up to isomorphism, and there exists a distinguished subgroup of central automorphisms which can be extended. 1. Introduction. Let K be an algebraic number field, and let denote the class of the finite dimensional central simple X-algebra A in the Brauer group B(K) of K. The class [A] is determined arithmetically by its Hasse invariants at the primes of K. Algebraic properties of A often impose severe but interesting arithmetic properties on its invariants. As evidence we cite the important work of M. Benard and M. Schacher [2] concerning the invariants when [A] is in S(K) the Schur subgroup of K, and the surprising result of G. Janusz [4] obtained in considering the problem of when an automorphism of K extends to A. In this paper we offer a construction which gives rise to subgroups of B(K) whose member classes have invariants which possess a rigid arith- metic structure — that of uniform distribution — then search for corre- sponding algebraic properties.
    [Show full text]
  • Non-Additive Ring and Module Theory IV the Brauer Group of a Symmetric
    Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 549 Brauer Groups Proceedings of the Conference Held at Evanston, October 11-15,1975 * 416 109 546 200 16 Edited by D. Zelinsky Springer-Verlag Berlin · Heidelberg · NewYork1976 Editor Daniel Zelinsky Northwestern University Department of Mathematics Evanston, II. 60091/USA Library of Congress Cataloging in Publication Data Main entry under title: Brauer groups. (Lecture notes in mathematics ; 5^9) "Sponsored by Northwestern University." Bi bli ography: ρ. Includes index. 1. Brauer group--Congresses. 2. Separable algebras—Congresses. I. Zelinsky, Daniel. II. Northwestern University, Evanston, 111. III. Series: Lecture notes in mathematics (Berlin) ; 5^9. QA3.L28 no. 5^9 [QA251.3] 510'.8s [512».2*0 76-kekie AMS Subject Classifications (1970): 13A20, 16A16, 18H20, 14C20, 14H99, 14L15, 18D10 ISBN 3-540-07989-0 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07989-0 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, re­ printing, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. CONTENTS Lindsay N, Childs On Brauer groups of some normal local rings..... 1 P.
    [Show full text]
  • The Brauer Group of the Moduli Stack of Elliptic Curves Over Algebraically Closed fields of Characteristic 2, J
    The Brauer group of the moduli stack of elliptic curves Benjamin Antieau∗ and Lennart Meier† Abstract We compute the Brauer group of M1,1, the moduli stack of elliptic curves, over Spec Z, its localizations, finite fields of odd characteristic, and algebraically closed fields of characteristic not 2. The methods involved include the use of the parameter space of Legendre curves and the moduli stack M(2) of curves with full (naive) level 2 structure, the study of the Leray–Serre spectral sequence in ´etale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of S3 in a certain integral representation, the classification of cubic Galois extensions of Q, the computation of Hilbert symbols in the ramified case for the primes 2 and 3, and finding p-adic elliptic curves with specified properties. Key Words. Brauer groups, moduli of elliptic curves, level structures, Hilbert sym- bols. Mathematics Subject Classification 2010. Primary: 14F22, 14H52, 14K10. Secondary: 11G05, 11G07. Contents 1 Introduction 2 2 Brauer groups, cyclic algebras, and ramification 4 3 The low-dimensional Gm-cohomology of BCm 10 4 A presentation of the moduli stack of elliptic curves 13 arXiv:1608.00851v3 [math.AG] 5 May 2020 5 Beginning of the computation 17 6 The p-primary torsion in Br(MZ[ 1 ]) for primes p 5 21 2 ≥ 7 The 3-primary torsion in Br(M ) 22 Z 1 [ 6 ] 8 The ramification of the 3-torsion 24 9 The 2-primary torsion in Br(M 1 ) 26 Z[ 2 ] 10 The Brauer group of M 30 11 The Brauer group of M over Fq with q odd 31 ∗Benjamin Antieau was supported by NSF Grants DMS-1461847 and DMS-1552766.
    [Show full text]
  • Arxiv:1711.06456V4 [Math.AG] 1 Dec 2018
    PURITY FOR THE BRAUER GROUP KĘSTUTIS ČESNAVIČIUS Abstract. A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension ě 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic p0,pq. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover. 1. The purity conjecture of Grothendieck and Auslander–Goldman .............. 1 Acknowledgements ................................... ................................ 3 2. Passage to a finite flat cover ................................................... .... 3 3. Passage to the completion ................................................... ....... 6 4. The p-primary Brauer group in the perfectoid case .............................. 7 5. Passage to perfect or perfectoid towers ........................................... 11 6. Global conclusions ................................................... ............... 13 Appendix A. Fields of dimension ď 1 ................................................ 15 References ................................................... ........................... 16 1. The purity conjecture of Grothendieck and Auslander–Goldman Grothendieck predicted in [Gro68b, §6] that the cohomological Brauer group of a regular scheme X is insensitive to removing a closed subscheme Z Ă X of codimension ě 2. This purity conjecture is known in many cases (as we discuss in detail below), for instance, for cohomology classes of order 2 invertible on X, and its codimension requirement is necessary: the Brauer group of AC does not agree with that of the complement of the coordinate axes (see [DF84, Rem. 3]). In this paper, we finish the remaining cases, that is, we complete the proof of the following theorem.
    [Show full text]
  • Brauer Groups and Galois Cohomology of Function Fields Of
    Brauer groups and Galois cohomology of function fields of varieties Jason Michael Starr Department of Mathematics, Stony Brook University, Stony Brook, NY 11794 E-mail address: [email protected] Contents 1. Acknowledgments 5 2. Introduction 7 Chapter 1. Brauer groups and Galois cohomology 9 1. Abelian Galois cohomology 9 2. Non-Abelian Galois cohomology and the long exact sequence 13 3. Galois cohomology of smooth group schemes 22 4. The Brauer group 29 5. The universal cover sequence 34 Chapter 2. The Chevalley-Warning and Tsen-Lang theorems 37 1. The Chevalley-Warning Theorem 37 2. The Tsen-Lang Theorem 39 3. Applications to Brauer groups 43 Chapter 3. Rationally connected fibrations over curves 47 1. Rationally connected varieties 47 2. Outline of the proof 51 3. Hilbert schemes and smoothing combs 54 4. Ramification issues 63 5. Existence of log deformations 68 6. Completion of the proof 70 7. Corollaries 72 Chapter 4. The Period-Index theorem of de Jong 75 1. Statement of the theorem 75 2. Abel maps over curves and sections over surfaces 78 3. Rational simple connectedness hypotheses 79 4. Rational connectedness of the Abel map 81 5. Rational simply connected fibrations over a surface 82 6. Discriminant avoidance 84 7. Proof of the main theorem for Grassmann bundles 86 Chapter 5. Rational simple connectedness and Serre’s “Conjecture II” 89 1. Generalized Grassmannians are rationally simply connected 89 2. Statement of the theorem 90 3. Reductions of structure group 90 Bibliography 93 3 4 1. Acknowledgments Chapters 2 and 3 notes are largely adapted from notes for a similar lecture series presented at the Clay Mathematics Institute Summer School in G¨ottingen, Germany in Summer 2006.
    [Show full text]
  • Unramified Division Algebras Do Not Always Contain Azumaya Maximal
    Unramified division algebras do not always contain Azumaya maximal orders BENJAMIN ANTIEAU AND BEN WILLIAMS ABSTRACT. We show that, in general, over a regular integral noetherian affine scheme X of dimension at least 6, there exist Brauer classes on X for which the associated division algebras over the generic point have no Azumaya maximal orders over X. Despite the algebraic nature of the result, our proof relies on the topology of classifying spaces of algebraic groups. 1. INTRODUCTION Let K be a field. The Artin–Wedderburn Theorem implies that every central simple K- algebra A is isomorphic to an algebra Mn(D) of n × n matrices over a finite dimensional ′ central K-division algebra D. One says Mn(D) and Mn′ (D ) are Brauer-equivalent when D and D′ are isomorphic over K. The set of Brauer-equivalence classes forms a group under tensor product, Br(K), the Brauer group of K. The index of an equivalence class α = cl(Mn(D)) ∈ Br(K) is the degree of the minimal representative, D itself. Let X be a connected scheme. The notion of a central simple algebra over a field was generalized by Auslander–Goldman [3] and by Grothendieck [11] to the concept of an Azu- maya algebra over X. An Azumaya algebra A is a locally-free sheaf of algebras which ´etale-locally takes the form of a matrix algebra. That is, there is an ´etale cover π : U → X ∗ ∼ such that π A = Mn(OU). In this case, the degree of A is n. Brauer equivalence and a con- travariant Brauer group functor may be defined in this context, generalizing the definition of the Brauer group over a field.
    [Show full text]
  • The Geometry of Representations of 3- Dimensional Sklyanin Algebras
    The Geometry of Representations of 3- Dimensional Sklyanin Algebras Kevin De Laet & Lieven Le Bruyn Algebras and Representation Theory ISSN 1386-923X Volume 18 Number 3 Algebr Represent Theor (2015) 18:761-776 DOI 10.1007/s10468-014-9515-6 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Algebr Represent Theor (2015) 18:761–776 DOI 10.1007/s10468-014-9515-6 The Geometry of Representations of 3-Dimensional Sklyanin Algebras Kevin De Laet Lieven Le Bruyn · Received: 9 May 2014 / Accepted: 17 December 2014 / Published online: 30 January 2015 ©SpringerScience+BusinessMediaDordrecht2015 Abstract The representation scheme repn A of the 3-dimensional Sklyanin algebra A associated to a plane elliptic curve and n-torsion point contains singularities over the aug- mentation ideal m.Weinvestigatethesemi-stablerepresentationsofthenoncommutative blow-up algebra B A mt m2t2 ... to obtain a partial resolution of the central singularity = ⊕ ⊕ ⊕ such that the remaining singularities in the exceptional fiber determine an elliptic curve and 2 are all of type C C /Zn.
    [Show full text]
  • Brauer Groups of Linear Algebraic Groups
    Volume 71, Number 2, September 1978 BRAUERGROUPS OF LINEARALGEBRAIC GROUPS WITH CHARACTERS ANDY R. MAGID Abstract. Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d - l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, characterless, linear algebraic group over an algebraically closed field of characteristic zero: the Brauer group is finite-in fact, it is the Schur multiplier of the fundamental group of the algebraic group [6, Theorem 4.1, p. 299]. In this note, we extend these calculations to an arbitrary connected linear group in characteristic zero. The main result is the determination of the Brauer group of a ¿/-dimen- sional affine algebraic torus, which is shown to be (Q/Z)(n) where n = d(d — l)/2. (This result is noted in [6, 4.8, p. 301] when d = 2.) We then show that if G is a connected linear algebraic group whose character group has Z-rank d, then the Brauer group of G is C X (Q/Z)<n) where n is as above and C is finite. We adopt the following notational conventions: 7ms an algebraically closed field of characteristic zero, and T = (F*)id) is a ^-dimensional affine torus over F.
    [Show full text]
  • There Are Enough Azumaya Algebras on Surfaces
    THERE ARE ENOUGH AZUMAYA ALGEBRAS ON SURFACES STEFAN SCHROER¨ Final version, 26 February 2001 Abstract. Using Maruyama’s theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for sep- arated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces. Introduction Generalizing the classical theory of central simple algebras over fields, Grothen- dieck [12] introduced the Brauer group Br(X) and the cohomological Brauer group Br0(X) for schemes. Let me recall the definitions. The Brauer group Br(X) comprises equivalence classes of Azumaya algebras. Two Azumaya algebras A, B are called equivalent if there are everywhere nonzero vector bundles E, F with A ⊗ End(E) ' B ⊗ End(F). Let us define the cohomological Brauer group Br0(X) as the torsion part of the 2 ´etalecohomology group H (X, Gm). Nonabelian cohomology gives an inclusion Br(X) ⊂ Br0(X), and Grothendieck asked whether this is bijective. It would be nice to know this for the following reason: The cohomological Brauer group is related to various other cohomology groups via exact sequences, and this is useful for computations. In contrast, it is almost impossible to calculate the Brauer group of a scheme directly from the definition. Here is a list of schemes with Br(X) = Br0(X): (1) Schemes of dimension ≤ 1 and regular surfaces (Grothendieck [12]). (2) Abelian varieties (Hoobler [16]). (3) The union of two affine schemes with affine intersection (Gabber [6]). (4) Smooth toric varieties (DeMeyer and Ford [4]). On the other hand, a nonseparated normal surface with Br(X) 6= Br0(X) recently appeared in [5].
    [Show full text]