# sga

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

View the Project on GitHub thosgood/sga

# Séminaire de Géométrie Algébrique du Bois Marie

## 1. Étale covers and the fundamental group

• Introduction and foreword (@thosgood)
• I. Étale morphisms (@thosgood)
• II. Smooth morphisms: generalities, differential properties
1. Generalities
2. Some smoothness criteria for morphisms (3)
3. Invariance properties (1)
4. Differential properties of smooth morphisms (17)
5. The case of a base field (6)
• III. Smooth morphisms: extension properties
1. Formally smooth homomorphisms (4)
2. Characteristic lifting property of formally smooth homomorphisms (5)
3. Local infinitesimal extension of morphisms in a smooth $S$-scheme (2)
4. Local infinitesimal extension of smooth $S$-schemes (1)
5. Global infinitesimal extension of morphisms (7)
6. Global infinitesimal extension of smooth $S$-schemes (5)
7. Application to the construction of smooth formal schemes and of smooth ordinary schemes over a complete local ring $A$ (5)
• IV. Flat morphisms
1. Syllogisms on flat modules (3)
2. Faithfully flat modules (3)
3. Relations to completion (1)
4. Relations to free modules (2)
5. Local flatness criteria (5)
6. Flat morphisms and open sets (5)
• V. The fundamental group: generalities
1. Introduction (1)
2. Preschemes with finite operator groups. Quotient preschemes (5)
3. Decomposition groups and inertia groups. Étale case (6)
4. Automorphisms and morphisms of étale coverings (2)
5. Axiomatic conditions for a Galois theory (9)
6. Galois categories (7)
7. Exact functors from one Galois category to another (6)
8. Case of preschemes (3)
9. Case of a normal base prescheme (1)
10. Case of non-connected preschemes: multi-Galois categories (1)
• VI. Fibred categories and descent
1. Introduction (1)
2. Universes, categories, equivalence of categories (2)
3. Categories over one another (4)
4. Base change for categories over ℰ (6)
5. Fibred categories. Equivalence of ℰ-categories (3)
6. Cartesian morphisms, inverse images, cartesian functors (3)
7. Fibred categories and pre-fibred categories (6)
8. Cloven categories over ℰ (5)
9. Cloven category defined by a pseudofunctor ℰ°→𝖢𝖺𝗍 (4)
10. Example: cloven category defined by a functor ℰ°→𝖢𝖺𝗍. Categories split over ℰ (2)
11. Co-fibred categories, bi-fibred categories (1)
12. Various examples (7)
13. Functors on a cloven category (5)
14. Bibliography (1)
• VII. (Does not exist)
• VIII. Faithfully flat descent
1. Descent for quasi-coherent modules (7)
2. Descent for affine preschemes over one another (1)
3. Descent of set-theoretic properties and finiteness properties of morphisms (2)
4. Descent of topological properties (5)
5. Descent of morphisms of preschemes (6)
6. Applications to finite and quasi-finite morphisms (3)
7. Effectiveness criteria for a descent data (8)
8. Bibliography (1)
• IX. Descent of étale morphisms. Applications to the fundamental group
1. Reminders on étale morphisms (3)
2. Submersive and universally submersive morphisms (2)
3. Descent of étale morphisms of preschemes (2)
4. Descent of étale preschemes: effectiveness criteria (7)
5. Translation in terms of the fundamental group (11)
6. A fundamental exact sequence. Descent by morphisms with relatively connected fibres (7)
7. Bibliography (1)
• X. Specialisation theory of the fundamental group
1. Exact sequence of homotopy for a proper and separable morphism (7)
2. Application to the existence theorem of sheaves: semi-continuity theorem for fundamental groups of fibres of a proper and separable morphism (7)
3. Application to the purity theorem: continuity theorem for fundamental groups of fibres of a proper and simple morphism (9)
4. Bibliography (1)
1. Projective spaces, unirational varieties (1)
2. Abelian varieties (4)
3. Projecting cones. Zariski’s example (2)
4. Exact sequence of cohomology (7)
5. Particular cases of principal bundles (3)
6. Applications to principal coverings: Kummer and Artinschreier theories (8)
7. Bibliography (1)
• XII. Algebraic geometry and analytic geometry
1. Analytic space associated to a scheme (4)
2. Comparison of properties of a scheme and the associated analytic space (4)
3. Comparison of properties of morphisms (6)
4. Cohomological comparison theorems and existence theorems (6)
5. Comparison theorems for étale coverings (11)
6. Bibliography (1)
• XIII. Cohomological properties of sheaves of sets and of sheaves of non-commutative groups
1. Reminders on the theory of stacks (1)
2. Cohomological properness (3)
3. Particular case of cohomological properness: relative normal crossing divisors (26)
4. Cohomological properness and generic local acyclicity (19)
5. Exact sequences of homotopy (15)
6. Appendix I: Variations on Abhyankar’s lemma (8)
7. Appendix II: Finiteness theorem for direct images of stacks (4)
8. Bibliography (1)

## 2. Local cohomology of coherent sheaves and local and global Lefschetz theoremshttps://thosgood.com/sga/sga-2/index.ht

• Introduction
• I. Global and local cohomological invariants with respect to a closed subspace
1. The functors $\Gamma_Z$ and $\underline{\Gamma}_Z$ (7)
2. The functors $\operatorname{H}_Z^\bullet(X,F)$ and $\underline{\operatorname{H}}_Z^\bullet(F)$ (6)
• II. Applications to quasi-coherent sheaves on preschemes (8)
• III. Cohomological invariants and depth
1. Reminders (1)
2. Depth (6)
3. Depth and topological properties (9)
• IV. Dualising modules and dualising functors
1. Generalities on functors of modules (4)
2. Characterisation of exact functors (1)
3. Study of the case where $T$ is left exact and $T(M)$ is of finite type for all $M$ (3)
4. Dualising module. Dualising functor (5)
5. Consequences of the theory of dualising modules (5)
• V. Local duality and structure of the $\operatorname{H}^i(M)$
1. Complexes of homomorphisms (3)
2. The local duality theorem for a local regular ring (1)
3. Application to the structure of the $\operatorname{H}^i(M)$ (7)
• VI. The functors $\operatorname{Ext}(X;F,G)$ and $\underline{\operatorname{Ext}}(F,G)$
1. Generalities (3)
2. Application to quasi-coherent sheaves on preschemes (2)
• VII. Nullity criteria. Coherence conditions for the sheaves $\underline{\operatorname{Ext}}(F,G)$
1. Study of $i<n$ (5)
2. Study of $i>n$ (2)
• VIII. Finiteness theorem
1. Bi-duality spectral sequence (5)
2. Finiteness theorem (7)
3. Applications (3)
• IX. Algebraic geometry and formal geometry
1. Comparison theorem (8)
2. Existence theorem (4)
• X. Applications to the fundamental group
1. Comparison of $\mathsf{Et}(\widehat{X})$ with $\mathsf{Et}(Y)$ (1)
2. Comparison of $\mathsf{Et}(Y)$ with $\mathsf{Et}(U)$, for varying 𝑈 (5)
3. Comparison of $\pi_1(X)$ with $\pi_1(U)$ (7)
• XI. Applications to the Picard group
1. Comparison of $\operatorname{Pic}(\widehat{X})$ with $\operatorname{Pic}(Y)$ (1)
2. Comparison of $\operatorname{Pic}(Y)$ with $\operatorname{Pic}(U)$, for varying $U$ (5)
3. Comparison of $\mathsf{P}(X)$ with $\mathsf{P}(U)$ (7)
• XII. Applications to projective algebraic schemes
1. Projective duality theorem and finiteness theorem (7)
2. Lefschetz theory for a projective morphism: Grauert’s comparison theorem (4)
3. Lefschetz theory for a projective morphism: existence theorem (7)
4. Formal completion and normal flatness (10)
5. Universal finiteness conditions for a non-proper morphism (8)
• XIII. Problems and conjectures
1. Links between local and global results. Affine problems relating to duality (4)
2. Problems relating to $\pi_0$: local Bertini theorems (5)
3. Problems relating to $\pi_1$ (2)
4. Problems relating to higher $\pi_i$: local and global Lefschetz theorems for complex analytic spaces (6)
5. Problems relating to local Picard groups (5)
• XIV. Depth and Lefschetz theorems in étale cohomology
1. Cohomological and homotopic depth (30)
2. Technical lemmas (7)
3. Converse of the affine Lefschetz theorem (12)
4. Main theorem and variations (21)
5. Geometric depth (6)
6. Open questions (5)

## 3-I. Group schemes I: General properties of group schemes

• I. Algebraic structures. Group cohomology
1. Generalities (12)
2. Algebraic structures (7)
3. Category of $\mathcal{O}$-modules, category of $G$-$\mathcal{O}$-modules (2)
4. Algebraic structures in the category of preschemes (15)
5. Group cohomology (6)
• II. Tangent bundles. Lie algebras
1. $\underline{\operatorname{Hom}}_{Z/S}(X,Y)$ functors (2)
2. The preschemes $I_S(M)$ (3)
3. The tangent bundle, the (E) condition (11)
4. Tangent space of a group. Lie algebras (15)
5. Calculation of some Lie algebras (6)
6. Various remarks (3)
• III. Infinitesimal extensions
1. Reminders from SGA 1 III. Various remarks (15)
2. Extensions and cohomology (12)
3. Infinitesimal extensions of a morphism of group preschemes (9)
4. Infinitesimal extensions of a group prescheme (6)
5. Infinitesimal extensions of closed subgroups (32)
• IV. Topologies and sheaves
1. Universal effective epimorphisms (6)
2. Descent morphisms (5)
3. Universal effective equivalence relations (14)
4. Topologies and sheaves (43)
5. Passage to the quotient and algebraic structures (10)
6. Topologies in the category of schemes (12)
• V. Construction of quotient preschemes
1. $\mathcal{C}$-groupoids (4)
2. Examples of $\mathcal{C}$-groupoids (2)
3. Some syllogisms for $\mathcal{C}$-groupoids (5)
4. Passage to the quotient by a finite and flat equivalence prerelation (5)
5. Passage to the quotient by a finite and flat equivalence relation (4)
6. Passage to the quotient when there exists a quasi-section (5)
7. Passage to the quotient by a proper and flat equivalence prerelation (5)
8. Passage to the quotient by a flat and non-necessarily proper equivalence prerelation (3)
9. Elimination of Noetherian hypotheses (3)
• VIa. Generalities on algebraic groups
1. Preliminary remarks (4)
2. Local properties of an $A$-group of locally finite type (4)
3. Connected components of an $A$-group of locally finite type (5)
4. Construction of quotient groups: case of groups of finite type (6)
5. Construction of quotient groups: general case (6)
• VIb. Generalities on group preschemes
1. Morphisms of groups of locally finite type over a field (9)
2. “Open properties” of groups and morphisms of groups of locally finite presentation (12)
3. Identity component of a group of locally finite presentation (7)
4. Dimension of fibres of groups of locally finite presentation (4)
5. Separation of groups and homogeneous spaces (6)
6. Sub-functors and group sub-preschemes (5)
7. Generated subgroups; commutator group (12)
8. Solvable and nilpotent group preschemes (5)
9. Quotient sheaves (6)
10. Passage to the projective limit for group preschemes and operator group preschemes (11)
11. Affine group preschemes (16)
• VIIa. Infinitesimal study of group schemes: differential operators and Lie $p$-algebras
1. Differential operators (5)
2. Invariant differential operators on group preschemes (7)
3. Coalgebras and Cartier duality (10)
4. “Frobeniuseries” (11)
5. Lie $p$-algebras (9)
6. Lie $p$-algebras of a group $S$-prescheme (7)
7. Radicial groups of height 1 (8)
8. Case of a base field (6)
• VIIb. Infinitesimal study of group schemes: formal groups
1. Reminders on pseudocompact rings and modules (15)
2. Formal varieties over a pseudocompact ring (20)
3. Generalities on formal groups (19)
4. Phenomena particular to characteristic 0 (10)
5. Phenomena particular to characteristic $p>0$ (10)
6. Homogeneous spaces of infinitesimal formal groups over a field (13)

## 3-II. Group schemes II: Groups of multiplicative type, and structure of general group schemes

• VIII. Diagonalisable groups
1. Biduality (5)
2. Scheme-theoretic properties of diagonalisable groups (1)
3. Exactness properties of the functor $D_S$ (4)
4. Torsors under a diagonalisable group (4)
5. Quotient of an affine scheme by a diagonalisable group acting freely (5)
6. Essentially free morphisms, and representability of certain functors of the form $\prod_{Y/S}Z/Y$ (5)
7. Appendix: Monomorphisms of group preschemes (25)
• IX. Groups of multiplicative type: homomorphisms to a group scheme
1. Definitions (3)
2. Extension of certain properties of diagonalisable groups to groups of multiplicative type (6)
3. Infinitesimal properties: lifting and conjugation theorem (4)
4. Density theorem (8)
5. Central homomorphisms of groups of multiplicative type (5)
6. Monomorphisms of groups of multiplicative type and canonical factorisation of a homomorphism of such a group (5)
7. Algebraicity of formal homomorphisms to an affine group (6)
8. Subgroups, quotient groups, and extensions of groups of multiplicative type over a field (3)
• X. Characterisation and classification of group of multiplicative types
1. Classification of isotrivial groups: case of a base field (4)
2. Infinitesimal variations of structure (5)
3. Infinitesimal finite variations of structure: case of a complete base ring (6)
4. Case of an arbitrary base. Quasi-isotriviality theorem (6)
5. Scheme of homomorphisms from one multiplicative type group to another. Twisted constant groups and groups of multiplicative type (8)
6. Infinite principal Galois coverings and the enlarged fundamental group (6)
7. Classification of twisted constant preschemes and finite groups of multiplicative type in terms of the enlarged fundamental group (4)
8. Appendix: Elimination of certain affine hypotheses (11)
• XI. Representability criteria. Applications to multiplicative subgroups of affine group schemes
1. Introduction (1)
2. Reminders on smooth, étale, and unramified morphisms (9)
3. Examples of formally smooth functors extracted from the theory of groups of multiplicative type (4)
4. Auxiliary results on representability (16)
5. Scheme of subgroups of multiplicative type of an affine smooth group (7)
6. First corollaries of the representability theorem (7)
7. On a rigidity property for homomorphisms of certain group schemes, and the representability of certain transporters (9)
• XII. Maximal toruses, Weyl group, Cartan subgroup, reductive centre of smooth and affine group schemes
1. Maximal toruses (1)
2. The Weyl group (10)
3. Cartan subgroups (5)
4. The reductive centre (2)
5. Application to the scheme of subgroups of multiplicative type (12)
6. Maximal toruses and Cartan subgroups of not-necessarily affine algebraic groups (over an algebraically closed base field) (6)
7. Application to non-necessarily affine smooth group preschemes (23)
8. Semi-simple elements, union and intersection of maximal toruses in non-necessarily affine group schemes (2)
• XIII. Regular elements of algebraic groups and of Lie algebras
1. An auxiliary lemma on varieties with operators (4)
2. Density theorem and the theory of regular points of 𝐺 (16)
3. Case of a prescheme over an arbitrary base (7)
4. Lie algebras over a field: rank, regular elements, Cartan sub-algebras (7)
5. Case of the Lie algebra of a smooth algebraic group: density theorem (8)
6. Cartan sub-algebras and subgroups of type (C), with respect to a smooth algebraic group (5)
• XIV. Regular elements (continued). Applications to algebraic groups
1. Construction of Cartan subgroups and maximal toruses for a smooth algebraic group (3)
2. Lie algebras on an arbitrary prescheme: regular sections and Cartan sub-algebras (13)
3. Subgroups of type (C) of group preschemes over an arbitrary prescheme (11)
4. Digression on Borel subgroups (7)
5. Relations between Cartan subgroups and Cartan sub-algebras (4)
6. Applications to the structure of algebraic groups (8)
7. Appendix: Existence of regular elements over finite fields (7)
• XV. Addenda on sub-toruses of group preschemes. Application to smooth groups
1. Introduction (1)
2. Lifting finite subgroups (7)
3. Infinitesimal lifting of sub-toruses (17)
4. Characterisation of a sub-torus by its underlying set (24)
5. Characterisation of a sub-torus $T$ by the subgroups ${}_nT$ (11)
6. Representability of the functor: smooth subgroups identical to their connected normaliser (13)
7. Functor of Cartan subgroups and functor of parabolic subgroups (13)
8. Cartan subgroups of a smooth group (14)
9. Representability criteria of the functor of sub-toruses of a smooth group (25)
• XVI. Groups of unipotent rank zero
1. An immersion criterion (19)
2. Representability theorem for quotients (7)
3. Groups with flat centre (10)
4. Groups with affine fibres, of unitpotent rank zero (4)
5. Application to reductive and semi-simple groups (3)
6. Applications: extension of certain rigidity properties of toruses of groups of unipotent rank zero (5)
• XVII. Unipotent algebraic groups. Extensions between unipotent groups and group of multiplicative types
1. Some notation (2)
2. Definition of unipotent algebraic groups (4)
3. First properties of unipotent groups (9)
4. Unipotent groups acting on a vector space (10)
5. Characterisation of unipotent groups (16)
6. Extension of a group of multiplicative type by a unipotent group (29)
7. Extension of a unipotent group by a group of multiplicative type (9)
8. Nilpotent affine algebraic groups (7)
9. Appendix I: Hochschild cohomology and extensions of algebraic groups (5)
11. Appendix III: Remarks and addenda for chapters XV, XVI, and XVII (5)
• XVIII. Weil’s theorem on the construction of a group from a rational law
1. Introduction (1)
2. “Reminders” on rational maps (2)
3. Local determination of a morphism of groups (4)
4. Construction of a group from a rational law (15)

## 3-III. Group schemes III: Structure of reductive group schemes

• XIX. Reductive groups: generalities
1. Reminders on groups over an algebraically closed field (9)
2. Reductive group schemes: definitions and first properties (5)
3. Roots and root systems of reductive group schemes (4)
4. Roots and vector group schemes (6)
5. An instructive example
6. Local existence of maximal toruses. The Weyl group (4)
• XX. Reductive groups of semi-simple rank 1
1. Elementary systems. The groups 𝑃ᵣ and 𝑃₋ᵣ (12)
2. Structure of elementary systems (13)
3. The Weyl group (11)
4. The isomorphism theorem (2)
5. Examples of elementary systems, applications (7)
6. Generators and relations for an elementary system (5)
1. Generalities (7)
2. Relations between two roots (5)
3. Simple roots, positive roots (20)
4. Reduced raditical data of semi-simple rank 2 (4)
5. The Weyl group: generators and relations (6)
6. Morphisms of radicial data (17)
7. Structure (12)
• XXII. Reductive groups: split groups, subgroups, quotient groups
1. Roots and coroots. Split groups and radicial data (9)
2. Existence of split groups. Type of a reductive group (3)
3. The Weyl group (3)
4. Homomorphisms of split groups (16)
5. Subgroups of type (R) (64)
6. The derived group (12)
• XXIII. Reductive groups: uniticity of pinned groups
1. Pinnings (8)
2. Generators and relations for a pinned group (14)
3. Groups of semi-simple rank 2 (20)
4. Uniqueness of pinned groups: fundamental theorem (8)
5. Corollaries of the fundamental theorem (5)
6. Chevalley systems (5)
• XXIV. Automorphisms of reductive groups
• XXV. Existence theorem
• XXVI Parabolic subgroups of reductive groups

## 4-I. Topos theory and étale cohomology of schemes I

• I. Presheaves
• II. Topologies and sheaves
• III. Functoriality of categories of sheaves
• IV. Toposes

## 4-II. Topos theory and étale cohomology of schemes II

• V. Cohomology in toposes
• Vb. Techniques for cohomological descent
• VI. Finiteness conditions. Fibred toposes and sites. Applications to questions of passing to the limit
• VII. Étale site and topos of a scheme
• VIII. Fibred functors, supports, cohomological study of finite morphisms

## 4-II. Topos theory and étale cohomology of schemes III

• IX. Constructible sheaves. Cohomology of an algebraic curve
• X. Cohomological dimension: first results
• XI. Comparison with classical cohomology: the case of a smooth prescheme
• XII. Base change theorem for a proper morphism
• XIII. Base change theorem for a proper morphism: end of proof
• XIV. Finiteness theorem for a proper morphism; cohomological dimension of affine algebraic schemes
• XV. Acyclic morphisms
• XVI. Base change theorem for a smooth morphism, and applications
• XVII. Cohomology with proper support
• XVIII. The global duality formula
• XIX. Cohomology of excellent preschemes of equal characteristic

## 4½. Étale cohomology

• 0. An Ariadne’s thread for SGA 4, SGA 4½, and SGA 5
• 1. Étale cohomology: starting points
• 2. Relation to the trace formula
• 3. $L$-functions modulo $\ell^n$ and modulo $p$
• 4. Cohomology class associated to a cycle
• 5. Duality
• 6. Applications of the trace formula to trigonometric sums
• 7. Finiteness theorems in $\ell$-adic cohomology
• 8. Derived categories

## 5. $\ell$-adic cohomology and $L$-functions

• I. Dualising complexes
• II. (Does not exist)
• III. The Lefschetz formula
• IIIb. Calculations of local terms
• IV. (Does not exist)
• VI. $\ell$-adic cohomology
• VII. Cohomology of some classical schemes; cohomological theory of Chern classes
• VIII. Groups of classes of abelian and triangulated categories, perfect complexes
• IX. (Does not exist)
• X. The Euler–Poincaré formula in étale cohomology
• XI. (Does not exist)
• XII. The Nielsen–Wecken and Lefschetz formulas in algebraic geometry
• XIII. (Does not exist)
• XIV. The Frobenius morphism, and rationality of $L$-functions

## 7-I. Monodromy groups in algebraic geometry I

• I. Summary of the first talks by A. Grothendieck
• II. Finiteness properties of the fundamental group
• III. (Does not exist)
• IV. (Does not exist)
• V. (Does not exist)
• VI. Formal deformation theory
• VII. Bi-extension of sheaves of groups
• VIII. Addenda on bi-extensions. General properties of bi-extensions of group schemes
• IX. Néron models and monodromy

## 7-II. Monodromy groups in algebraic geometry II

• X. Intersections on regular surfaces
• XI. Cohomology of complete intersections
• XIII. Formalism of vanishing cycles
• XIV. Comparison with transcendental theory
• XV. The Picard–Lefschetz formula
• XVI. The Milnor formula
• XVII. Lefschetz pencils: existence theorem
• XVIII. Cohomological study of Lefschetz pencils
• XIX. Noether’s theorem
• XX Griffiths’s theorem
• XXI. Level of cohomology of complete intersections
• XXII. Congruence formula for the 𝜻-function