An English translation of the Séminaire de Géométrie Algébrique du Bois Marie

- Introduction and foreword
*(@thosgood)*

- Basics of differential calculus
- Quasi-finite morphisms
- Unramified morphisms
- Étale morphisms. Étale covers
- Fundamental property of étale morphisms
- Application to étale extensions of complete local rings
- Local construction of unramified and étale morphisms
- Infinitesimal lifting of étale schemes. Applications to formal schemes
- Invariance properties
- Étale covers of a normal scheme
- Various addenda

- Generalities
*(2)* - Some smoothness criteria for morphisms
*(3)* - Invariance properties
*(1)* - Differential properties of smooth morphisms
*(17)* - The case of a base field
*(6)*

- Formally smooth homomorphisms
*(4)* - Characteristic lifting property of formally smooth homomorphisms
*(5)* - Local infinitesimal extension of morphisms in a smooth 𝑆-scheme
*(2)* - Local infinitesimal extension of smooth 𝑆-schemes
*(1)* - Global infinitesimal extension of morphisms
*(7)* - Global infinitesimal extension of smooth 𝑆-schemes
*(5)* - Application to the construction of smooth formal schemes and of smooth ordinary schemes over a complete local ring 𝐴
*(5)*

- Syllogisms on flat modules
*(3)* - Faithfully flat modules
*(3)* - Relations to completion
*(1)* - Relations to free modules
*(2)* - Local flatness criteria
*(5)* - Flat morphisms and open sets
*(5)*

- Introduction
*(1)* - Preschemes with finite operator groups. Quotient preschemes
*(5)* - Decomposition groups and inertia groups. Étale case
*(6)* - Automorphisms and morphisms of étale coverings
*(2)* - Axiomatic conditions for a Galois theory
*(9)* - Galois categories
*(7)* - Exact functors from one Galois category to another
*(6)* - Case of preschemes
*(3)* - Case of a normal base prescheme
*(1)* - Case of non-connected preschemes: multi-Galois categories
*(1)*

- Introduction
*(1)* - Universes, categories, equivalence of categories
*(2)* - Categories over one another
*(4)* - Base change for categories over ℰ
*(6)* - Fibred categories. Equivalence of ℰ-categories
*(3)* - Cartesian morphisms, inverse images, cartesian functors
*(3)* - Fibred categories and pre-fibred categories
*(6)* - Cloven categories over ℰ
*(5)* - Cloven category defined by a pseudofunctor ℰ°→𝖢𝖺𝗍
*(4)* - Example: cloven category defined by a functor ℰ°→𝖢𝖺𝗍. Categories split over ℰ
*(2)* - Co-fibred categories, bi-fibred categories
*(1)* - Various examples
*(7)* - Functors on a cloven category
*(5)* - Bibliography
*(1)*

- Descent for quasi-coherent modules
*(7)* - Descent for affine preschemes over one another
*(1)* - Descent of set-theoretic properties and finiteness properties of morphisms
*(2)* - Descent of topological properties
*(5)* - Descent of morphisms of preschemes
*(6)* - Applications to finite and quasi-finite morphisms
*(3)* - Effectiveness criteria for a descent data
*(8)* - Bibliography
*(1)*

- Reminders on étale morphisms
*(3)* - Submersive and universally submersive morphisms
*(2)* - Descent of étale morphisms of preschemes
*(2)* - Descent of étale preschemes: effectiveness criteria
*(7)* - Translation in terms of the fundamental group
*(11)* - A fundamental exact sequence. Descent by morphisms with relatively connected fibres
*(7)* - Bibliography
*(1)*

- Exact sequence of homotopy for a proper and separable morphism
*(7)* - Application to the existence theorem of sheaves: semi-continuity theorem for fundamental groups of fibres of a proper and separable morphism
*(7)* - Application to the purity theorem: continuity theorem for fundamental groups of fibres of a proper and simple morphism
*(9)* - Bibliography
*(1)*

- Projective spaces, unirational varieties
*(1)* - Abelian varieties
*(4)* - Projecting cones. Zariski’s example
*(2)* - Exact sequence of cohomology
*(7)* - Particular cases of principal bundles
*(3)* - Applications to principal coverings: Kummer and Artinschreier theories
*(8)* - Bibliography
*(1)*

- Analytic space associated to a scheme
*(4)* - Comparison of properties of a scheme and the associated analytic space
*(4)* - Comparison of properties of morphisms
*(6)* - Cohomological comparison theorems and existence theorems
*(6)* - Comparison theorems for étale coverings
*(11)* - Bibliography
*(1)*

- Reminders on the theory of stacks
*(1)* - Cohomological properness
*(3)* - Particular case of cohomological properness: relative normal crossing divisors
*(26)* - Cohomological properness and generic local acyclicity
*(19)* - Exact sequences of homotopy
*(15)* - Appendix I: Variations on Abhyankar’s lemma
*(8)* - Appendix II: Finiteness theorem for direct images of stacks
*(4)* - Bibliography
*(1)*

- The functors 𝛤𝑍 and 𝛤𝑍
*(7)* - The functors 𝐻𝑍•(𝑋,𝐹) and 𝐻𝑍•(𝑋,𝐹)
*(6)*

- Applications to quasi-coherent sheaves on preschemes
*(8)*

- Reminders
*(1)* - Depth
*(6)* - Depth and topological properties
*(9)*

- Generalities on functors of modules
*(4)* - Characterisation of exact functors
*(1)* - Study of the case where 𝑇 is left exact and 𝑇(𝑀) is of finite type for all 𝑀
*(3)* - Dualising module. Dualising functor
*(5)* - Consequences of the theory of dualising modules
*(5)*

- Complexes of homomorphisms
*(3)* - The local duality theorem for a local regular ring
*(1)* - Application to the structure of the 𝐻ⁱ(𝑀)
*(7)*

`\operatorname{Ext}(X;F,G)`

and `\underline{\operatorname{Ext}}(F,G)`

- Generalities
*(3)* - Application to quasi-coherent sheaves on preschemes
*(2)*

`\underline{\operatorname{Ext}}(F,G)`

- Study of 𝑖<𝑛
*(5)* - Study of 𝑖>𝑛
*(2)*

- Bi-duality spectral sequence
*(5)* - Finiteness theorem
*(7)* - Applications
*(3)*

- Comparison theorem
*(8)* - Existence theorem
*(4)*

- Comparison of
`\mathsf{Et}(\widehat{X})`

with`\mathsf{Et}(Y)`

*(1)* - Comparison of
`\mathsf{Et}(Y)`

with`\mathsf{Et}(U)`

, for varying 𝑈*(5)* - Comparison of 𝜋₁(𝑋) with 𝜋₁(𝑈)
*(7)*

- Comparison of
`\operatorname{Pic}(\widehat{X})`

with`\operatorname{Pic}(Y)`

*(1)* - Comparison of
`\operatorname{Pic}(Y)`

with`\operatorname{Pic}(U)`

, for varying`U`

*(5)* - Comparison of
`\mathsf{P}(X)`

with`\mathsf{P}(U)`

*(7)*

- Projective duality theorem and finiteness theorem
*(7)* - Lefschetz theory for a projective morphism: Grauert’s comparison theorem
*(4)* - Lefschetz theory for a projective morphism: existence theorem
*(7)* - Formal completion and normal flatness
*(10)* - Universal finiteness conditions for a non-proper morphism
*(8)*

- Links between local and global results. Affine problems relating to duality
*(4)* - Problems relating to 𝜋₀: local Bertini theorems
*(5)* - Problems relating to 𝜋₁
*(2)* - Problems relating to higher 𝜋ᵢ: local and global Lefschetz theorems for complex analytic spaces
*(6)* - Problems relating to local Picard groups
*(5)* - Comments
*(7)*

- Cohomological and homotopic depth
*(30)* - Technical lemmas
*(7)* - Converse of the affine Lefschetz theorem
*(12)* - Main theorem and variations
*(21)* - Geometric depth
*(6)* - Open questions
*(5)*

- I. Algebraic structures. Group cohomology

- Generalities
*(12)* - Algebraic structures
*(7)* - Category of
`\mathcal{O}`

-modules, category of`G`

-`\mathcal{O}`

-modules*(2)* - Algebraic structures in the category of preschemes
*(15)* - Group cohomology
*(6)*

- II. Tangent bundles. Lie algebras

`\underline{\operatorname{Hom}}_{Z/S}(X,Y)`

functors*(2)*- The preschemes
`I_S(M)`

*(3)* - The tangent bundle, the (E) condition
*(11)* - Tangent space of a group. Lie algebras
*(15)* - Calculation of some Lie algebras
*(6)* - Various remarks
*(3)*

- III. Infinitesimal extensions

- Reminders from SGA 1 III. Various remarks
*(15)* - Extensions and cohomology
*(12)* - Infinitesimal extensions of a morphism of group preschemes
*(9)* - Infinitesimal extensions of a group prescheme
*(6)* - Infinitesimal extensions of closed subgroups
*(32)*

- IV. Topologies and sheaves

- Universal effective epimorphisms
*(6)* - Descent morphisms
*(5)* - Universal effective equivalence relations
*(14)* - Topologies and sheaves
*(43)* - Passage to the quotient and algebraic structures
*(10)* - Topologies in the category of schemes
*(12)*

- V. Construction of quotient preschemes

`\mathcal{C}`

-groupoids*(4)*- Examples of
`\mathcal{C}`

-groupoids*(2)* - Some syllogisms for
`\mathcal{C}`

-groupoids*(5)* - Passage to the quotient by a finite and flat equivalence prerelation
*(5)* - Passage to the quotient by a finite and flat equivalence relation
*(4)* - Passage to the quotient when there exists a quasi-section
*(5)* - Passage to the quotient by a proper and flat equivalence prerelation
*(5)* - Passage to the quotient by a flat and non-necessarily proper equivalence prerelation
*(3)* - Elimination of Noetherian hypotheses
*(3)*

- VIa. Generalities on algebraic groups

- Preliminary remarks
*(4)* - Local properties of an 𝐴-group of locally finite type
*(4)* - Connected components of an 𝐴-group of locally finite type
*(5)* - Construction of quotient groups: case of groups of finite type
*(6)* - Construction of quotient groups: general case
*(6)* - Addenda
*(5)*

- VIb. Generalities on group preschemes

- Morphisms of groups of locally finite type over a field
*(9)* - “Open properties” of groups and morphisms of groups of locally finite presentation
*(12)* - Identity component of a group of locally finite presentation
*(7)* - Dimension of fibres of groups of locally finite presentation
*(4)* - Separation of groups and homogeneous spaces
*(6)* - Sub-functors and group sub-preschemes
*(5)* - Generated subgroups; commutator group
*(12)* - Solvable and nilpotent group preschemes
*(5)* - Quotient sheaves
*(6)* - Passage to the projective limit for group preschemes and operator group preschemes
*(11)* - Affine group preschemes
*(16)*

- VIIa. Infinitesimal study of group schemes: differential operators and Lie 𝑝-algebras

- Differential operators
*(5)* - Invariant differential operators on group preschemes
*(7)* - Coalgebras and Cartier duality
*(10)* - “Frobeniuseries”
*(11)* - Lie 𝑝-algebras
*(9)* - Lie 𝑝-algebras of a group 𝑆-prescheme
*(7)* - Radicial groups of height 1
*(8)* - Case of a base field
*(6)*

- VIIb. Infinitesimal study of group schemes: formal groups

- Reminders on pseudocompact rings and modules
*(15)* - Formal varieties over a pseudocompact ring
*(20)* - Generalities on formal groups
*(19)* - Phenomena particular to characteristic 0
*(10)* - Phenomena particular to characteristic 𝑝>0
*(10)* - Homogeneous spaces of infinitesimal formal groups over a field
*(13)*

- Biduality
*(5)* - Scheme-theoretic properties of diagonalisable groups
*(1)* - Exactness properties of the functor 𝐷ₛ
*(4)* - Torsors under a diagonalisable group
*(4)* - Quotient of an affine scheme by a diagonalisable group acting freely
*(5)* - Essentially free morphisms, and representability of certain functors of the form
`\prod_{Y/S}Z/Y`

*(5)* - Appendix: Monomorphisms of group preschemes
*(25)*

- Definitions
*(3)* - Extension of certain properties of diagonalisable groups to groups of multiplicative type
*(6)* - Infinitesimal properties: lifting and conjugation theorem
*(4)* - Density theorem
*(8)* - Central homomorphisms of groups of multiplicative type
*(5)* - Monomorphisms of groups of multiplicative type and canonical factorisation of a homomorphism of such a group
*(5)* - Algebraicity of formal homomorphisms to an affine group
*(6)* - Subgroups, quotient groups, and extensions of groups of multiplicative type over a field
*(3)*

- Classification of isotrivial groups: case of a base field
*(4)* - Infinitesimal variations of structure
*(5)* - Infinitesimal finite variations of structure: case of a complete base ring
*(6)* - Case of an arbitrary base. Quasi-isotriviality theorem
*(6)* - Scheme of homomorphisms from one multiplicative type group to another. Twisted constant groups and groups of multiplicative type
*(8)* - Infinite principal Galois coverings and the enlarged fundamental group
*(6)* - Classification of twisted constant preschemes and finite groups of multiplicative type in terms of the enlarged fundamental group
*(4)* - Appendix: Elimination of certain affine hypotheses
*(11)*

- Introduction
*(1)* - Reminders on smooth, étale, and unramified morphisms
*(9)* - Examples of formally smooth functors extracted from the theory of groups of multiplicative type
*(4)* - Auxiliary results on representability
*(16)* - Scheme of subgroups of multiplicative type of an affine smooth group
*(7)* - First corollaries of the representability theorem
*(7)* - On a rigidity property for homomorphisms of certain group schemes, and the representability of certain transporters
*(9)*

- Maximal toruses
*(1)* - The Weyl group
*(10)* - Cartan subgroups
*(5)* - The reductive centre
*(2)* - Application to the scheme of subgroups of multiplicative type
*(12)* - Maximal toruses and Cartan subgroups of not-necessarily affine algebraic groups (over an algebraically closed base field)
*(6)* - Application to non-necessarily affine smooth group preschemes
*(23)* - Semi-simple elements, union and intersection of maximal toruses in non-necessarily affine group schemes
*(2)*

- An auxiliary lemma on varieties with operators
*(4)* - Density theorem and the theory of regular points of 𝐺
*(16)* - Case of a prescheme over an arbitrary base
*(7)* - Lie algebras over a field: rank, regular elements, Cartan sub-algebras
*(7)* - Case of the Lie algebra of a smooth algebraic group: density theorem
*(8)* - Cartan sub-algebras and subgroups of type (C), with respect to a smooth algebraic group
*(5)*

- Construction of Cartan subgroups and maximal toruses for a smooth algebraic group
*(3)* - Lie algebras on an arbitrary prescheme: regular sections and Cartan sub-algebras
*(13)* - Subgroups of type (C) of group preschemes over an arbitrary prescheme
*(11)* - Digression on Borel subgroups
*(7)* - Relations between Cartan subgroups and Cartan sub-algebras
*(4)* - Applications to the structure of algebraic groups
*(8)* - Appendix: Existence of regular elements over finite fields
*(7)*

- Introduction
*(1)* - Lifting finite subgroups
*(7)* - Infinitesimal lifting of sub-toruses
*(17)* - Characterisation of a sub-torus by its underlying set
*(24)* - Characterisation of a sub-torus 𝑇 by the subgroups ₙ𝑇
*(11)* - Representability of the functor: smooth subgroups identical to their connected normaliser
*(13)* - Functor of Cartan subgroups and functor of parabolic subgroups
*(13)* - Cartan subgroups of a smooth group
*(14)* - Representability criteria of the functor of sub-toruses of a smooth group
*(25)*

- An immersion criterion
*(19)* - Representability theorem for quotients
*(7)* - Groups with flat centre
*(10)* - Groups with affine fibres, of unitpotent rank zero
*(4)* - Application to reductive and semi-simple groups
*(3)* - Applications: extension of certain rigidity properties of toruses of groups of unipotent rank zero
*(5)*

- Some notation
*(2)* - Definition of unipotent algebraic groups
*(4)* - First properties of unipotent groups
*(9)* - Unipotent groups acting on a vector space
*(10)* - Characterisation of unipotent groups
*(16)* - Extension of a group of multiplicative type by a unipotent group
*(29)* - Extension of a unipotent group by a group of multiplicative type
*(9)* - Nilpotent affine algebraic groups
*(7)* - Appendix I: Hochschild cohomology and extensions of algebraic groups
*(5)* - Appendix II: Reminders and addenda on radicial groups
*(4)* - Appendix III: Remarks and addenda for chapters XV, XVI, and XVII
*(5)*

- Introduction
*(1)* - “Reminders” on rational maps
*(2)* - Local determination of a morphism of groups
*(4)* - Construction of a group from a rational law
*(15)*

- Reminders on groups over an algebraically closed field
*(9)* - Reductive group schemes: definitions and first properties
*(5)* - Roots and root systems of reductive group schemes
*(4)* - Roots and vector group schemes
*(6)* - An instructive example
- Local existence of maximal toruses. The Weyl group
*(4)*

- Elementary systems. The groups 𝑃ᵣ and 𝑃₋ᵣ
*(12)* - Structure of elementary systems
*(13)* - The Weyl group
*(11)* - The isomorphism theorem
*(2)* - Examples of elementary systems, applications
*(7)* - Generators and relations for an elementary system
*(5)*

- Generalities
*(7)* - Relations between two roots
*(5)* - Simple roots, positive roots
*(20)* - Reduced raditical data of semi-simple rank 2
*(4)* - The Weyl group: generators and relations
*(6)* - Morphisms of radicial data
*(17)* - Structure
*(12)*

- Roots and coroots. Split groups and radicial data
*(9)* - Existence of split groups. Type of a reductive group
*(3)* - The Weyl group
*(3)* - Homomorphisms of split groups
*(16)* - Subgroups of type (R)
*(64)* - The derived group
*(12)*

- Pinnings
*(8)* - Generators and relations for a pinned group
*(14)* - Groups of semi-simple rank 2
*(20)* - Uniqueness of pinned groups: fundamental theorem
*(8)* - Corollaries of the fundamental theorem
*(5)* - Chevalley systems
*(5)*