Anton: Reverted edits by 120.203.1.250 (talk) to last revision by Anton

2011-11-22T19:14:30Z

Reverted edits by 120.203.1.250 (talk) to last revision by Anton

← Older revision		Revision as of 11:14, 22 November 2011
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	:5. Group cohomology		:5. Group cohomology

	~~I have been so bweliedred in the past but now it all makes sense!~~		=====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=====		=====III. Infinitesimal extensions, by M. Demazure=====

120.203.1.250: /* II. Tangent bundles. Lie algebras, by M. Demazure */

2011-11-22T17:27:24Z

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

← Older revision		Revision as of 09:27, 22 November 2011
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	:5. Group cohomology		:5. Group cohomology

	~~=====II. Tangent bundles. Lie algebras, by M. Demazure=====~~		I have been so bweliedred in the past but now it all makes sense!
	~~: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=====		=====III. Infinitesimal extensions, by M. Demazure=====

Anton at 21:20, 2 November 2011

2011-11-02T21:20:41Z

← Older revision		Revision as of 13:20, 2 November 2011
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			__NOTOCNUM__

	=SGA 1=		=SGA 1=
	=SGA 2=		=SGA 2=

Anton at 21:15, 2 November 2011

2011-11-02T21:15:16Z

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 =SGA 3: Group Schemes=
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 :6. Parabolic subgroups and trivial tori
 :7. Relative radicial datum

Anton: Created page with "=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 cat..."

2011-11-02T21:12:11Z

Created page with "=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 cat..."

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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, Ã©le, 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

SGA contents - Revision history

Anton: Reverted edits by 120.203.1.250 (talk) to last revision by Anton

120.203.1.250: /* II. Tangent bundles. Lie algebras, by M. Demazure */

Anton at 21:20, 2 November 2011

Anton at 21:15, 2 November 2011

Anton: Created page with "=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 cat..."