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 .