DOCSLIB.ORG

INTRODUCTION to REDUCTIVE GROUP SCHEMES OVER RINGS in Construction P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
INTRODUCTION to REDUCTIVE GROUP SCHEMES OVER RINGS in Construction P INTRODUCTION TO REDUCTIVE GROUP SCHEMES OVER RINGS In construction P. GILLE Contents 1. Introduction 3 2. Sorites 5 2.1. R-Functors 5 2.2. Zariski sheaves 5 2.3. Functors in groups 6 2.4. Definition 6 3. Examples 8 3.1. Constant group schemes 8 3.2. Vector groups 8 3.3. Group of invertible elements, linear groups 9 3.4. Diagonalizable group schemes 9 4. Sequences of group functors 11 4.1. Exactness 11 4.2. Semi-direct product 13 4.3. Monomorphisms of group schemes 13 5. Flatness 15 5.1. The DVR case 15 5.2. A necessary condition 15 5.3. Examples 16 6. Representations 16 6.1. Representations of diagonalizable group schemes 18 6.2. Existence of faithful finite dimensional representations 19 6.3. Hochschild cohomology 20 6.4. First Hochschild cohomology group 23 6.5. H2 and group extensions 23 6.6. Linearly reductive algebraic groups 24 7. Weil restriction 25 8. Tangent spaces and Lie algebras 28 8.1. Tangent spaces 28 8.2. Lie algebras 29 8.3. More infinitesimal properties 33 9. Fixed points of diagonalizable groups 34 9.1. Representatibility 34 9.2. Smoothness of the fixed point locus 35 10. Lifting homomorphisms 35 10.1. Rigidity principle 35 Date: May 12, 2021. 1 2 10.2. Formal smoothness 37 10.3. Algebraization 38 10.4. Rank in family 40 11. Reductive group schemes 41 12. Limit groups 42 12.1. Limit functors 43 12.2. The group case 43 12.3. Parabolic and Borel subgroup schemes 44 13. Root data, type of reductive group schemes 45 13.1. Definition 45 13.2. Geometric case 46 13.3. Root datum Ψ(G; T) 46 13.4. Center 48 14. Flat sheaves 50 14.1. Covers 50 14.2. Definition 50 14.3. Monomorphisms and covering morphisms 51 14.4. Sheafification 52 14.5. Group actions, quotients sheaves and contracted products 54 14.6. Contracted products 56 14.7. Sheaf torsors 56 14.8. Quotient by a finite constant group 58 14.9. Quotient by a normalizer 58 15. Non-abelian cohomology, I 58 15.1. Definition 58 15.2. Twisting 59 15.3. Weil restriction II 59 16. Quotients by diagonalizable groups 60 16.1. Torsors 60 16.2. Quotients 61 16.3. Homomorphisms to a group scheme 62 17. Groups of multiplicative type 62 17.1. Definitions 62 17.2. Splitting results 64 17.3. Back to the recognition statement 67 18. Splitting reductive group schemes 68 18.1. Local splitting 68 18.2. Weyl groups 69 18.3. Center of reductive groups 70 18.4. Isotriviality issues 70 18.5. Killing pairs 72 19. Towards the classification of semisimple group schemes 73 19.1. Kernel of the adjoint representation 73 19.2. Pinnings 74 19.3. Automorphism group 75 19.4. Unicity and existence theorems 75 20. Appendix: smoothness 77 21. Appendix: universal cover of a connected ring 79 3 21.1. Embedding ´etalecovers into Galois covers 79 21.2. Construction of the universal cover 82 21.3. Examples 84 21.4. Local rings and henselizations 87 21.5. Artin's approximation theorem 89 References 89 1. Introduction The theory of reductive group schemes is due to Demazure and Grothendieck and was achieved fifty years ago in the seminar SGA 3, see also Demazure's thesis [De]. Roughly speaking it is the theory of reductive groups in family focusing to subgroups and classification issues. It occurs in several areas: representation theory, model theory, automorphic forms, arithmetic groups and buildings, infinite dimensional Lie theory, . The story started as follows. Demazure asked Serre whether there is a good reason for the map SLn(Z) ! SLn(Z=dZ) to be surjective for all d > 0. Serre answered it is a question for Grothendieck ... Grothendieck answered it is not the right question ! The right question was the development of a theory of reductive groups over schemes and especially the classification of the \split" ones. The general underlying statement is now that the specialization map G(Z) ! G(Z=dZ) is onto for each semisimple group split (or Chevalley) simply connected scheme G=Z. It is a special case of strong approximation. Demazure-Grothendieck's theory assume known the theory of reductive groups over an algebraically closed field due mainly to C. Chevalley ([Ch], see also [Bo], [Sp]) and we will do the same. In the meantime, Borel-Tits achieved the theory of reductive groups over an arbitrary field [BT65] and Tits classified the semisimple groups [Ti1]. In the general setting, Borel-Tits theory extends to the case of a local base. Let us warn the reader by pointing out that we do not plan to prove all hard theorems of the theory, for example the unicity and existence theorem of split reductive groups. Our purpose is more to take the user viewpoint by explaining how such results permit to analyse and classify algebraic struc- tures. It is not possible to enter into that theory without some background on affine group schemes and strong technical tools of algebraic geometry (descent, Grothendieck topologies,...). Up to improve afterwards certain results, half of the lectures avoid descent theory and general schemes. 4 The aim of the notes is to try to help people attending the lectures. It is very far to be self-contained and quotes a lot in several references starting with [SGA3], Demazure-Gabriel's book [DG], and also the material of the Luminy's summer school provided by Brochard [Br], Conrad [C] and Oesterl´e[O]. Acknowledgments: Several people helped me by comments and ques- tions. I would like to thank N. Bhaskhar, Z. Chang, V. Egorov, M. Jeannin, T.-Y. Lee, N. Reyssaire and A. Stavrova. 5 Affine group schemes I We shall work over a base ring R (commutative and unital). 2. Sorites 2.1. R-Functors. We denote by Aff R the category of affine R-schemes. We are interested in R{functors, i.e. covariant functors from Aff R to the category of sets. If X an R-scheme, it defines a covariant functor hX : Aff R ! Sets; S 7! X(S): Given a map f : Y ! X of R-schemes, there is a natural morphism of functors f∗ : hY ! hX of R-functors. We recall now Yoneda's lemma. Let F be an R-functor. If X = Spec(R[X]) is an affine R{scheme and ζ 2 F (R[X]), we define a morphism of R-functors ζe : hX ! F by ζe(S): hX(S) = HomR(R[X];S) ! F (S), f 7! F (f)(ζ). Each morphism ' : hX ! F is of this shape for a unique ζ 2 F (R[X]): ζ is the image of IdR[X] by the mapping ' : hX (R[X]) ! F (R[X]). ∼ In particular, each morphism of functors hY −! hX is of the shape hv for a unique R{morphism v : Y ! X. An R-functor F is representable by an affine scheme if there exists an affine scheme X and an isomorphism of functors hX ! F . We say that X represents F . The isomorphism hX ! F comes from an element ζ 2 F (R[X]) which is called the universal element of F (R[X]). The pair (X; ζ) satisfies the following universal property: For each affine R-scheme T and for each η 2 F (R[T]), there exists a unique morphism u : T ! X such that F (u∗)(ζ) = η. Given a morphism of ring j : R ! R0, an R{functor F defines by restric- 0 tion an R {functor denoted by j∗F or FR0 . If F = hX for an affine R-scheme 0 0 X, we have FR = hX×RR . 2.2. Zariski sheaves. We say that an R{functor F is a Zariski sheaf if it satisfies the following requirements: for each R{ring S and each decomposition 1 = f1 + ··· + fn in S, then ∼ n Y o F (S) −! (α ) 2 F (S ) j (α ) = (α ) for i; j = 1; :::; n : i fi i Sfifj j Sfifj i=1;::;n 2.2.1. Lemma. Let F be an R{functor F which a Zariski sheaf. (1) If F is not the empty functor, then F (0) = {•}. (2) F is additive, i.e. the map F (S1 × S2) ! F (S1) × F (S2) is bijective for each pair (S1;S2) of R{algebras. 6 Proof. We prove both statements in the same time assuming that F is not the empty R{functor. It means there exists an R{ring S such that F (S) 6= ;. By taking the map S ! 0, it follows that F (0) 6= ;. We are given an R{ring S = S1 × S2; we write it S = S1 × S2 = Se1 + Se2 where e1; e2 are idempotents satisfying e1 + e2 = 1, we have S1 = Se1 , S2 = Se2 and Se1e2 = 0 [Sta, C. Algebra, §20]. Then ∼ F (S) −! (α1; α2) 2 F (S1) × F (S2) j α1;0 = α2;0 2 F (0) : Applying that to S1 = S2 = 0, we see that F (O) = {•}. Coming back to the general case, we conclude that F (S) = F (S1) × F (S2). Representable R-functors are Zariski sheaves. 2.2.2. Lemma. Let 1 = f1 + ··· + fn. Let F be an R{functor which is a Zariski sheaf and such that F is representable by an affine R -scheme for Rfi fi i = 1; :::; n. Then F is representable by an affine R{scheme. ∼ Proof. Let Xi be an Rfi -scheme together with an isomorphism ζi : hXi −! F of R {functors for i = 1; ::; n. Then for i 6= j, F is represented by Rfi fi Rfifj −1 Xi ×Rf Rfifj and Xj ×Rf Rfifj .
Load more

Recommended publications
  • The Tate–Oort Group Scheme Top
    The Tate{Oort group scheme TOp Miles Reid In memory of Igor Rostislavovich Shafarevich, from whom we have all learned so much Abstract Over an algebraically closed field of characteristic p, there are 3 group schemes of order p, namely the ordinary cyclic group Z=p, the multiplicative group µp ⊂ Gm and the additive group αp ⊂ Ga. The Tate{Oort group scheme TOp of [TO] puts these into one happy fam- ily, together with the cyclic group of order p in characteristic zero. This paper studies a simplified form of TOp, focusing on its repre- sentation theory and basic applications in geometry. A final section describes more substantial applications to varieties having p-torsion in Picτ , notably the 5-torsion Godeaux surfaces and Calabi{Yau 3-folds obtained from TO5-invariant quintics. Contents 1 Introduction 2 1.1 Background . 3 1.2 Philosophical principle . 4 2 Hybrid additive-multiplicative group 5 2.1 The algebraic group G ...................... 5 2.2 The given representation (B⊕2)_ of G .............. 6 d ⊕2 _ 2.3 Symmetric power Ud = Sym ((B ) ).............. 6 1 3 Construction of TOp 9 3.1 Group TOp in characteristic p .................. 9 3.2 Group TOp in mixed characteristic . 10 3.3 Representation theory of TOp . 12 _ 4 The Cartier dual (TOp) 12 4.1 Cartier duality . 12 4.2 Notation . 14 4.3 The algebra structure β : A_ ⊗ A_ ! A_ . 14 4.4 The Hopf algebra structure δ : A_ ! A_ ⊗ A_ . 16 5 Geometric applications 19 5.1 Background . 20 2 5.2 Plane cubics C3 ⊂ P with free TO3 action .
    [Show full text]
  • Automorphism Groups of Locally Compact Reductive Groups
    PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 2, June 1989 AUTOMORPHISM GROUPS OF LOCALLY COMPACT REDUCTIVE GROUPS T. S. WU (Communicated by David G Ebin) Dedicated to Mr. Chu. Ming-Lun on his seventieth birthday Abstract. A topological group G is reductive if every continuous finite di- mensional G-module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let G be a locally compact reductive group such that G/Gq is compact. Then /(Go) is dense in Aq(G) . Let G be a locally compact topological group. Let A(G) be the group of all bi-continuous automorphisms of G. Then A(G) has the natural topology (the so-called Birkhoff topology or ^-topology [1, 3, 4]), so that it becomes a topo- logical group. We shall always adopt such topology in the following discussion. When G is compact, it is well known that the identity component A0(G) of A(G) is the group of all inner automorphisms induced by elements from the identity component G0 of G, i.e., A0(G) = I(G0). This fact is very useful in the study of the structure of locally compact groups. On the other hand, it is also well known that when G is a semi-simple Lie group with finitely many connected components A0(G) - I(G0). The latter fact had been generalized to more general groups ([3]).
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]
  • 1:34 Pm April 11, 2013 Red.Tex
    1:34 p.m. April 11, 2013 Red.tex The structure of reductive groups Bill Casselman University of British Columbia, Vancouver [email protected] An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices A 0 0 a with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for 1 example a = det− (A). The formulas for matrix multiplication are certainly algebraic, and the inverse of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic. Formally, this means that we are given (a) an F •rational multiplication map G × G −→ G; (b) an F •rational inverse map G −→ G; (c) an identity element—i.e. an F •rational point of G. I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties). In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are respectively equivalent to the specification of (a’) an F •homomorphism AF [G] −→ AF [G] ⊗F AF [G]; (b’) an F •involution AF [G] −→ AF [G]; (c’) a distinguished homomorphism AF [G] −→ F . The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example, in the case of GLn it takes xik −→ xij ⊗ xjk . j In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of such groups, but I should say that in practice the specification of an algebraic group is often indirect—as a subgroup or quotient, say, of another simpler one.
    [Show full text]
  • GROUPOID SCHEMES 022L Contents 1. Introduction 1 2
    GROUPOID SCHEMES 022L Contents 1. Introduction 1 2. Notation 1 3. Equivalence relations 2 4. Group schemes 4 5. Examples of group schemes 5 6. Properties of group schemes 7 7. Properties of group schemes over a field 8 8. Properties of algebraic group schemes 14 9. Abelian varieties 18 10. Actions of group schemes 21 11. Principal homogeneous spaces 22 12. Equivariant quasi-coherent sheaves 23 13. Groupoids 25 14. Quasi-coherent sheaves on groupoids 27 15. Colimits of quasi-coherent modules 29 16. Groupoids and group schemes 34 17. The stabilizer group scheme 34 18. Restricting groupoids 35 19. Invariant subschemes 36 20. Quotient sheaves 38 21. Descent in terms of groupoids 41 22. Separation conditions 42 23. Finite flat groupoids, affine case 43 24. Finite flat groupoids 50 25. Descending quasi-projective schemes 51 26. Other chapters 52 References 54 1. Introduction 022M This chapter is devoted to generalities concerning groupoid schemes. See for exam- ple the beautiful paper [KM97] by Keel and Mori. 2. Notation 022N Let S be a scheme. If U, T are schemes over S we denote U(T ) for the set of T -valued points of U over S. In a formula: U(T ) = MorS(T,U). We try to reserve This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 GROUPOID SCHEMES 2 the letter T to denote a “test scheme” over S, as in the discussion that follows. Suppose we are given schemes X, Y over S and a morphism of schemes f : X → Y over S.
    [Show full text]
  • Linear Algebraic Groups
    Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G.
    [Show full text]
  • COHOMOLOGY of LIE GROUPS MADE DISCRETE Abstract
    Publicacions Matemátiques, Vol 34 (1990), 151-174 . COHOMOLOGY OF LIE GROUPS MADE DISCRETE PERE PASCUAL GAINZA Abstract We give a survey of the work of Milnor, Friedlander, Mislin, Suslin, and other authors on the Friedlander-Milnor conjecture on the homology of Lie groups made discrete and its relation to the algebraic K-theory of fields. Let G be a Lie group and let G6 denote the same group with the discrete topology. The natural homomorphism G6 -+ G induces a continuous map between classifying spaces rt : BG6 -----> BG . E.Friedlander and J. Milnor have conjectured that rl induces isomorphisms of homology and cohomology with finite coefficients. The homology of BG6 is the Eilenberg-McLane homology of the group G6 , hard to compute, and one of the interests of the conjecture is that it permits the computation of these groups with finite coefficients through the computation (much better understood) of the homology and cohomology of BG. The Eilenberg-McLane homology groups of a topological group G are of interest in a variety of contexts such as the theory of foliations [3], [11], the scissors congruence [6] and algebraic K-theory [26] . For example, the Haefliger classifying space of the theory of foliations is closely related to the group of homeomorphisms of a topological manifold, for which one can prove analogous results of the Friedlander-Milnor conjecture with entire coefFicients, cf. [20], [32] . These notes are an exposition of the context of the conjecture, some known results mainly due to Milnor, Friedlander, Mislin and Suslin, and its application to the study of the groups K;(C), i > 0.
    [Show full text]
  • Coalgebras from Formulas
    Coalgebras from Formulas Serban Raianu California State University Dominguez Hills Department of Mathematics 1000 E Victoria St Carson, CA 90747 e-mail:[email protected] Abstract Nichols and Sweedler showed in [5] that generic formulas for sums may be used for producing examples of coalgebras. We adopt a slightly different point of view, and show that the reason why all these constructions work is the presence of certain representative functions on some (semi)group. In particular, the indeterminate in a polynomial ring is a primitive element because the identity function is representative. Introduction The title of this note is borrowed from the title of the second section of [5]. There it is explained how each generic addition formula naturally gives a formula for the action of the comultiplication in a coalgebra. Among the examples chosen in [5], this situation is probably best illus- trated by the following two: Let C be a k-space with basis {s, c}. We define ∆ : C −→ C ⊗ C and ε : C −→ k by ∆(s) = s ⊗ c + c ⊗ s ∆(c) = c ⊗ c − s ⊗ s ε(s) = 0 ε(c) = 1. 1 Then (C, ∆, ε) is a coalgebra called the trigonometric coalgebra. Now let H be a k-vector space with basis {cm | m ∈ N}. Then H is a coalgebra with comultiplication ∆ and counit ε defined by X ∆(cm) = ci ⊗ cm−i, ε(cm) = δ0,m. i=0,m This coalgebra is called the divided power coalgebra. Identifying the “formulas” in the above examples is not hard: the for- mulas for sin and cos applied to a sum in the first example, and the binomial formula in the second one.
    [Show full text]
  • Real Group Orbits on Flag Manifolds
    Real group orbits on flag manifolds Dmitri Akhiezer Abstract In this survey, we gather together various results on the action of a real form G0 of a complex semisimple group G on its flag manifolds. We start with the finiteness theorem of J. Wolf implying that at least one of the G0-orbits is open. We give a new proof of the converse statement for real forms of inner type, essentially due to F.M. Malyshev. Namely, if a real form of inner type G0 ⊂ G has an open orbit on a complex algebraic homogeneous space G/H then H is parabolic. In order to prove this, we recall, partly with proofs, some results of A.L. Onishchik on the factorizations of reductive groups. Finally, we discuss the cycle spaces of open G0-orbits and define the crown of a symmetric space of non- compact type. With some exceptions, the cycle space agrees with the crown. We sketch a complex analytic proof of this result, due to G. Fels, A. Huckleberry and J. Wolf. 1 Introduction The first systematic treatment of the orbit structure of a complex flag manifold X = G/P under the action of a real form G0 ⊂ G is due to J. Wolf [38]. Forty years after his paper, these real group orbits and their cycle spaces are still an object of intensive research. We present here some results in this area, together with other related results on transitive and locally transitive actions of Lie groups on complex manifolds. The paper is organized as follows. In Section 2 we prove the celebrated finiteness theorem for G0-orbits on X (Theorem 2.3).
    [Show full text]
  • NON-SPLIT REDUCTIVE GROUPS OVER Z Brian Conrad
    NON-SPLIT REDUCTIVE GROUPS OVER Z Brian Conrad Abstract. | We study the following phenomenon: some non-split connected semisimple Q-groups G admit flat affine Z-group models G with \everywhere good reduction" (i.e., GFp is a connected semisimple Fp-group for every prime p). Moreover, considering such G up to Z-group isomorphism, there can be more than one such G for a given G. This is seen classically for types B and D by using positive-definite quadratic lattices. The study of such Z-groups provides concrete applications of many facets of the theory of reductive groups over rings (scheme of Borel subgroups, auto- morphism scheme, relative non-abelian cohomology, etc.), and it highlights the role of number theory (class field theory, mass formulas, strong approximation, point-counting over finite fields, etc.) in analyzing the possibilities. In part, this is an expository account of [G96]. R´esum´e. | Nous ´etudions le ph´enom`ene suivant : certains Q-groupes G semi-simples connexes non d´eploy´es admettent comme mod`eles des Z-groupes G affines et plats avec \partout bonne r´eduction" (c'est `adire, GFp est un Fp- groupe Q-groupes G pour chaque premier p). En outre, consid´erant de tels G `a Z-groupe isomorphisme pr`es, il y a au plus un tel G pour un G donn´e.Ceci est vu classiquement pour les types B et D en utilisant des r´eseaux quadratiques d´efinis positifs. L'´etude de ces Z-groupes donne lieu `ades applications concr`etes d'aspects multiples, de la th´eorie des groupes r´eductifs sur des anneaux (sch´emas de sous-groupes de Borel, sch´emas d'automorphismes, cohomologie relative non ab´elienne, etc.), et met en ´evidence le r^ole de la th´eorie des nombres (th´eorie du corps de classes, formules de masse, approximation forte, comptage de points sur les corps finis, etc.) dans l'analyse des possibilit´es.En partie, ceci est un article d'exposition sur [G96].
    [Show full text]
  • Some Structure Theorems for Algebraic Groups
    Proceedings of Symposia in Pure Mathematics Some structure theorems for algebraic groups Michel Brion Abstract. These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures. Their aim is to present structure results for group schemes of finite type over a field, with applications to Picard varieties and automorphism groups. Contents 1. Introduction 2 2. Basic notions and results 4 2.1. Group schemes 4 2.2. Actions of group schemes 7 2.3. Linear representations 10 2.4. The neutral component 13 2.5. Reduced subschemes 15 2.6. Torsors 16 2.7. Homogeneous spaces and quotients 19 2.8. Exact sequences, isomorphism theorems 21 2.9. The relative Frobenius morphism 24 3. Proof of Theorem 1 27 3.1. Affine algebraic groups 27 3.2. The affinization theorem 29 3.3. Anti-affine algebraic groups 31 4. Proof of Theorem 2 33 4.1. The Albanese morphism 33 4.2. Abelian torsors 36 4.3. Completion of the proof of Theorem 2 38 5. Some further developments 41 5.1. The Rosenlicht decomposition 41 5.2. Equivariant compactification of homogeneous spaces 43 5.3. Commutative algebraic groups 45 5.4. Semi-abelian varieties 48 5.5. Structure of anti-affine groups 52 1991 Mathematics Subject Classification. Primary 14L15, 14L30, 14M17; Secondary 14K05, 14K30, 14M27, 20G15. c 0000 (copyright holder) 1 2 MICHEL BRION 5.6. Commutative algebraic groups (continued) 54 6. The Picard scheme 58 6.1. Definitions and basic properties 58 6.2. Structure of Picard varieties 59 7. The automorphism group scheme 62 7.1.
    [Show full text]
  • The Picard Scheme of an Abelian Variety
    Chapter VI. The Picard scheme of an abelian variety. 1. Relative Picard functors. § To place the notion of a dual abelian variety in its context, we start with a short discussion of relative Picard functors. Our goal is to sketch some general facts, without much discussion of proofs. Given a scheme X we write Pic(X)=H1(X, O∗ )= isomorphism classes of line bundles on X , X { } which has a natural group structure. (If τ is either the Zariski, or the ´etale, or the fppf topology 1 on Sch/X then we can also write Pic(X)=Hτ (X, Gm), viewing the group scheme Gm = Gm,X as a τ-sheaf on Sch/X ;seeExercise??.) If C is a complete non-singular curve over an algebraically closed field k then its Jacobian Jac(C) is an abelian variety parametrizing the degree zero divisor classes on C or, what is the same, the degree zero line bundles on C. (We refer to Chapter 14 for further discussion of Jacobians.) Thus, for every k K the degree map gives a homomorphism Pic(C ) Z, and ⊂ K → we have an exact sequence 0 Jac(C)(K) Pic(C ) Z 0 . −→ −→ K −→ −→ In view of the importance of the Jacobian in the theory of curves one may ask if, more generally, the line bundles on a variety X are parametrized by a scheme which is an extension of a discrete part by a connected group variety. If we want to study this in the general setting of a scheme f: X S over some basis S,we → are led to consider the contravariant functor P :(Sch )0 Ab given by X/S /S → P : T Pic(X )=H1(X T,G ) .
    [Show full text]