SGA contents

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SGA 3: Group Schemes

I. Algebraic structures. Group cohomology, by M. Demazure

1. Generalities

2. Algebraic structures

3. The category of \cal{O}-modules, the category of G-\cal{O}-modules

4. Algebraic structures in the category of preschemes

5. Group cohomology

II. Tangent bundles. Lie algebras, by M. Demazure

1. The functors \Hom_{Z/S}(X,Y)

2. The preschemes I_S(M)

3. The tangent bundle, condition (E)

4. Tangent space of a group. Lie algebras

5. Calculation of certain Lie algebras

6. Various remarks

III. Infinitesimal extensions, by M. Demazure

0. Recollection of SGA I, III. Various remarks

1. Extensions and cohomology

2. Infinitesimal extensions of a morphism of group preschemes

3. Infinitesimal extensions of group prescheme

4. Infinitesimal extensions closed subgroups

IV. Topologies and sheaves, by M. Demazure

1. Universal effective epimorphisms

2. Descent morphisms

3. Universal effective equivalence relations

4. Topologies and sheaves

5. Passage to the quotient and algebraic structures

6. Topologies on the category of schemes

V. Construction of quotient preschemes, by P. Gabriel

1. \cal{C}-groupoids

2. Examples of \cal{C}-groupoids

3. Some arguments on \cal{C}-groupoids

4. Passage to the quotient by a finite and flat equivalence prerelation

5. Passage to the quotient by a finite and flat equivalence relation

6. Passage to the quotient given the existence of a quasi-section

7. Quotient by a proper and flat equivalence prerelation

8. Quotient by a flat equivalence prerelation which is not necessarily proper

9. Elimination of noetherian hypotheses

VIA. Generalities on algebraic groups, by P. Gabriel

0. Preliminary remarks

1. Local properties of an A-group locally of finite type

2. Connected components of an A-group locally of finite type

3. Construction of quotient groups (case of groups of finite type)

4. Construction of quotient groups (general case)

5. Complements

VIB. Generalities on group preschemes, by P. Gabriel

1. Morphisms of groups of finite type over a field

2. "Open properties" of groups and of morphisms of groups locally of finite presentation

3. Connected component of the identity of a group locally of finite presentation

4. Dimension of fibres of groups locally of finite presentation

5. Separation of groups and homogeneous spaces

6. Sub-functors and group sub-schemes

7. Generated sub-groups; group of commutators

8. Solvable and nilpotent group schemes

9. Quotient sheaves

10. Passage to the projective limit in the category of group preschemes and the category of operator-group preschemes

11. Affine group schemes

VIIA. Infinitesimal study of group schemes. Differential operators and Lie p-algebras, by P. Gabriel

1. Differential operators

2. Invariant differential operators on group preschemes

3. Coalgebras and Cartier duality

4. "Frobeniisms"

5. Lie p-algebras

6. Lie p-algebras on an S-group scheme

5. Radical groups of height 1

5. Case of base field

VIIB. Infinitesimal study of group schemes. Formal groups, by P. Gabriel

0. Recollection on pseudocompact rings and modules

1. Formal varieties over a pseudocompact ring

2. Generalities on formal groups

3. Phenomena particular to the characteristic-0 case

4. Phenomena particular to the characteristic-p case

5. Homogeneous spaces for infinitesimal formal groups over a field

VIII. Diagonalisable groups, by A. Grothendieck

1. Biduality

2. Schematic properties of diagonalisable groups

3. Exactness properties of the functor D_S

4. Torsors under a diagonalisable group

5. Quotient of an affine scheme by a diagonalisable group operating freely

6. Essentially free morphisms, and representability of certain functors of the form \prod_{Y/S}Z/Y

7. Appendix: one monomorphisms of group schemes

IX. Groups of multiplicative type: Homomorphisms into a group scheme, by A. Grothendieck

1. Definitions

2. Extensions of certain properties of diagonalisable groups to groups of multiplicative type

3. Infinitesimal properties: lifting and conjugation theorem

4. The density theorem

5. Central homomorphisms of groups of multiplicative type

6. Monomorphisms of groups of multiplicative type and the canonical factorisation of a homomorphism of such a group

7. Algebraicity of formal homomorphisms into an affine group

8. Subgroups, quotient groups, and extensions of groups of multiplicative type over a field

X. Characterisation and classification of groups of multiplicative type, by A. Grothendieck

1. Classification of isotrivial groups---case of a base field

2. Infinitesimal variations of structure

3. Finite variations of structure: complete base ring

4. Case of arbitrary base. Quasi-isotriviality theorem

5. Scheme of homomorphisms from one group of multiplicative type to another. Constant torsion groups and groups of multiplicative type

6. Infinite principal Galois covers and enlarged fundamental group

7. Classification of constant torsion preschemes and of groups of multiplicative type and of finite type in terms of the enlarged fundamental group

8. Appendix: elimination of certain affineness hypotheses

XI. Criteria of representability. Applications to subgroups of multiplicative type of affine group schemes, by A. Grothendieck

0. Introduction

1. Recollections on smooth, éle, unramified morphisms

2. Examples of formally smooth functors drawn from the theory of groups of multiplicative type

3. Auxiliary representability results

4. The scheme of subgroups of multiplicative type of a smooth affine group

6. First corollaries of the representability theorem

7. On a rigidity property for the homomorphisms of certain group schemes, and the representability of certain transporters

XII. Maximal tori, Weyl group, Cartan subgroup, reductive centre of smooth and affine group schemes, by A. Grothendieck

1. Maximal tori

2. The Weyl subgroup

3. Cartan subgroups

4. The reductive centre

5. Application to the scheme of subgroups of multiplicative type

6. Maximal tori and Cartan subgroups of algebraic groups not necessarily affine (with algebraically closed base field)

7. Application to smooth group preschemes not necessarily affine

8. Semisimple elements, union and intersection of maximal tori in group schemes not necessarily affine

XIII. Regular elements of algebraic groups and Lie groups, by A. Grothendieck

1. An auxiliary lemma on varieties with operators

2. Density theorem and theory of regular points of G

3. Cas of a prescheme of arbitrary base

4. Lie algebras over a field: rank, regular elements, Cartan sub-algebras

5. Case of the Lie algebra of a smooth algebraic group: density theorem

6. Cartan subalgebras and subgroups of type (C) relative to a smooth algebraic group

XIV. Regular elements: continuation, Applications to algebraic groups, by A. Grothendieck

1. Construction of Cartan subgroups and maximal tori for a smooth algebraic group

2. Lie algebras over an arbitrary prescheme: regular sections and Cartan subalgebras

3. Subgroups of type (C) of group preschemes over an arbitrary prescheme

4. A digression on Borel subgroups

5. Relations between Cartan subgroups and Cartan subalgebras

6. Applications to the structure of algebraic groups

7. Appendix: Existence of regular elements over the finite fields

XV. Complements on the subtori of a group prescheme. Application to smooth groups, by M. Raynaud

0. Introduction

1. Lifting of finite subgroups

2. Infinitesimal lifting of subtori

3. Characterisation of a subtorus by its underlying set

4. Characterisation of a subtorus T by the subgroups T_n

5. Representability of the functor of smooth subgroups identical to the connected normaliser

6. Functor of Cartan subgroups and functor of parabolic subgroups

7. Cartan subgroups of a smooth group

8. Criterion for representability of the functor of subtori of a smooth group

XVI. Groups of unipotent rank 0, by M. Raynaud

1. A criterion for immersion

2. A theorem for the representability of quotients

3. Groups of flat centre

4. Groups of affine fibres, of unipotent rank 0

5. Application to reductive and semisimple groups

6. Applications: Extension of certain rigidity properties of tori to groups of unipotent rank 0

XVII. Unipotent algebraic groups. Extensions between unipotent groups and groups of multiplicative type, by M. Raynaud

0. Some notations

1. Definition of algebraic unipotent groups

2. First properties of unipotent groups

3. Unipotent groups operating on a vector space

4. A characterisation of unipotent groups

5. Extension of a group of multiplicative type by a unipotent group

6. Extension of a unipotent group by a group of multiplicative type

7. Affine nilpotent algebraic groups

8. Appendix I: Hochschild cohomology and extensions of algebraic groups

9. Appendix II: Recollections and complements on radicial groups

10. Appendix III: Remarks and complements for the exposes XV, XVI, XVII

XVIII. Weil's theorem on the construction of a group beginning with a rational law, by M. Artin

0. Introduction

1. "Recollections" on rational morphisms

2. Local determination of a group morphism

3. Construction of a group beginning from a rational law

XIX. Reductive groups. Generalities, by M. Demazure

1. Recollections on groups over an algebraically closed field

2. Reductive group schemes. Definition and first properties

3. Roots and root systems of reductive group schemes

4. Roots and vector groups

5. An instructive example

6. Local existence of maximal tori. The Weyl group

XX. Reductive groups of semisimple rank 1, by M. Demazure

1. Elementary systems. The groups P_r and P_{-r}

1. Structure of elementary systems

2. The Weyl group

3. The isomorphism theorem

4. Examples of elementary systems, applications

5. Generators and relations for an elementary system

XXI. Radicial data, by M. Demazure

1. Generalities

2. Relations between two roots

3. Simple roots, positive roots

5. Reduced radicial data of semisimple rank 2

6. The Weyl group: generators and relations

7. Radicial morphisms and data

8. Structure

XXII. Reductive groups: Deployment, subgroups, quotient groups, by M. Demazure

1. Roots and coroots. Deployed groups and radicial data

1. Existence of a deployment. Type of a reductive group

2. The Weyl group

3. Homomorphisms of deployed groups

4. Subgroups of type (R)

XXIII. Reductive groups: unicity of pinned (aka split) groups, by M. Demazure

1. Pinning (aka splitting)

2. Generators and relations for a pinned (aka split) group

3. Groups of semisimple rank 2

4. Unicity of pinned (aka split) groups: fundamental theorem

5. Corollaries of the fundamental theorem

6. Chevalley systems

XXIV. Automorphisms of reductive groups, by M. Demazure

1. Scheme of automorphisms of a reductive group

2. Automorphisms and subgroups

3. Dynkin scheme of a reductive group. Quasi-deployed groups

4. Isotriviality of reductive groups and of principal fibres with respect to reductive groups

5. Canonical decomposition of a an adjoint or simply connected group

6. Automorphisms of the Borel groups of reductive groups

7. Representability of the functors \Hom_{S\groups}(G,H), with G reductive

8. Appendix: cohomology of a smooth group over a henselian ring, cohomology and the functor \prod

XXV. The existence theorem, by M. Demazure

1. Announcement of the theorem

2. Existence theorem: construction of a piece of the group

3. Existence theorem: end of the proof

4. Appendix

XXVI. Parabolic subgroups of reductive groups, by M. Demazure

1. Recollections, Levi subgroups

2. Structure of the nilpotent radical of a parabolic subgroup

3. Scheme of parabolic subgroups of a reductive group

4. Relative position of two parabolic groups

5. Conjugation theorem

6. Parabolic subgroups and trivial tori

7. Relative radicial datum