# EGA contents

From stacky wiki

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Here is a table of contents for EGA, taken from Mark Haiman's handout. The full texts are available from numdam:

I, II, III(1), III(2), IV(1), IV(2), IV(3), IV(4)

## Contents

- 1 Chapter 0. Preliminary material
- 1.1 §1. Rings of fractions
- 1.2 §2. Irreducible and Noetherian spaces
- 1.3 §3. Supplement on sheaves
- 1.4 §4. Ringed spaces
- 1.5 §5. Quasi-coherent and coherent sheaves
- 1.6 §6. Flatness
- 1.7 §7. $I$-adic rings
- 1.8 §8. Representable functors
- 1.9 §9. Constructible sets
- 1.10 §10. Supplement on flat modules
- 1.11 §11. Supplement on homological algebra
- 1.12 §12. Supplement on sheaf cohomology
- 1.13 §13. Projective limits in homological algebra
- 1.14 §14. Combinatorial dimension of a topological space
- 1.15 §15. $M$-regular and $\mathcal F$-regular sequences
- 1.16 §16. Dimension and depth of Noetherian local rings
- 1.17 §17. Regular rings
- 1.18 §18. Supplement on extension of algebras
- 1.19 §19. Formally smooth algebras and Cohen rings
- 1.20 §20. Derivations and differentials
- 1.21 §21. Differentials in rings of characteristic $p$
- 1.22 §22. Differential criteria for smoothness and regularity
- 1.23 §23. Japanese rings

- 2 Volume I. The language of schemes
- 2.1 §1. Affine schemes
- 2.2 §2. Preschemes and their morphisms
- 2.3 §3. Product of preschemes
- 2.4 §4. Sub-preschemes and immersions
- 2.5 §5. Reduced preschemes; separatedness
- 2.6 §6. Finiteness conditions
- 2.7 §7. Rational maps
- 2.8 §8. Chevalley schemes
- 2.9 §9. Supplement on quasi-coherent sheaves
- 2.10 §10. Formal schemes

- 3 Volume II. Basic global properties of some classes of morphisms.
- 3.1 §1. Affine morphisms
- 3.2 §2. Homogeneous prime spectra
- 3.3 §3. Homogeneous prime spectrum of a sheaf of graded algebras
- 3.4 §4. Projective bundles; ample sheaves.
- 3.5 §5. Quasi-affine, quasi-projective, proper and projective morphisms
- 3.6 §6. Integral and finite morphisms
- 3.7 §7. Valuative criteria
- 3.8 §8. Blowup schemes; projective cones; projective closure

- 4 Volume III. Cohomological study of coherent sheaves
- 4.1 §1. Cohomology of affine schemes
- 4.2 §2. Cohomological study of projective morphisms
- 4.3 §3. Finiteness theorem for proper morphisms
- 4.4 §4. Fundamental theorem on proper morphisms, and applications
- 4.5 §5. An existence theorem for coherent sheaves
- 4.6 §6. Local and global Tor, Künneth formula
- 4.7 §7. Base change for homological functors on sheaves of modules

- 5 Volume IV. Local study of schemes and morphisms
- 5.1 §1. Relative finiteness conditions; constructible sets in preschemes
- 5.2 §2. Base change and flatness
- 5.3 §3. Associated prime cycles and primary decomposition
- 5.4 §4. Change of ground field for algebraic preschemes
- 5.5 §5. Dimension, depth, and regularity for locally Noetherian preschemes
- 5.6 §6. Flat morphisms of locally Noetherian preschemes
- 5.7 §7. Noetherian local rings and their completions; excellent rings
- 5.8 §8. Projective limits of preschemes
- 5.9 §9. Constructible properties
- 5.10 §10. Jacobson preschemes
- 5.11 §11. Topological properties of finitely presented flat morphisms; flatness criteria.
- 5.12 §12. Fibers of finitely presented flat morphisms
- 5.13 §13. Equidimensional morphisms
- 5.14 §14. Universally open morphisms
- 5.15 §15. Fibers of a universally open morphism
- 5.16 §16. Differential invariants; differentially smooth morphisms
- 5.17 §17. Smooth, unramified and étale morphisms
- 5.18 §18. Supplement on étale morphisms; Henselian local rings and strictly local rings
- 5.19 §19. Regular immersions and normal flatness
- 5.20 §20. Meromorphic functions and pseudo-morphisms
- 5.21 §21. Divisors

## Chapter 0. Preliminary material

(Volume I)

##### §1. Rings of fractions

- 1.0 Rings and algebras
- 1.1 Radical of an ideal; nilradical and radical of a ring
- 1.2 Modules and rings of fractions
- 1.3 Functorial properties
- 1.4 Change of multiplicative sets
- 1.5 Change of rings
- 1.6 $M_f$ as a direct limit
- 1.7 Support of a module

##### §2. Irreducible and Noetherian spaces

- 2.1 Irreducible spaces
- 2.2 Noetherian spaces

##### §3. Supplement on sheaves

- 3.1 Sheaves with values in a category
- 3.2 Presheaves on a base of open sets
- 3.3 Gluing sheaves
- 3.4 Direct images of presheaves
- 3.5 Inverse images of presheaves
- 3.6 Constant and locally constant sheaves
- 3.7 Inverse images of presheaves of groups or rings
- 3.8 Pseudo-discrete sheaves of spaces

##### §4. Ringed spaces

- 4.1 Ringed spaces, sheaves of $\mathcal A$-modules, $\mathcal A$-algebras
- 4.2 Direct image of an $\mathcal A$-module
- 4.3 Inverse image of an $\mathcal A$-module
- 4.4 Relation between direct and inverse images

##### §5. Quasi-coherent and coherent sheaves

- 5.1 Quasi-coherent sheaves
- 5.2 Sheaves of finite type
- 5.3 Coherent sheaves
- 5.4 Locally free sheaves
- 5.5 Sheaves on a locally ringed space

##### §6. Flatness

- 6.1 Flat modules
- 6.2 Change of rings
- 6.3 Local nature of flatness
- 6.4 Faithfully flat modules
- 6.5 Restriction of scalars
- 6.6 Faithfully flat rings
- 6.7 Flat morphisms of ringed spaces

##### §7. $I$-adic rings

- 7.1 Admissisble rings
- 7.2 $I$-adic rings and projective limits
- 7.3 Pre-$I$-adic Noetherian rings
- 7.4 Quasi-finite modules over local rings
- 7.5 Restricted formal series rings
- 7.6 Completed rings of fractions
- 7.7 Completed tensor products
- 7.8 Topologies on Hom modules

(Volume III)

##### §8. Representable functors

- 8.1 Representable functors
- 8.2 Algebraic structures in categories

##### §9. Constructible sets

- 9.1 Constructible sets
- 9.2 Constructible subsets of Noetherian spaces
- 9.3 Constructible functions

##### §10. Supplement on flat modules

- 10.1 Relations between free and flat modules
- 10.2 Local flatness criteria
- 10.3 Existence of flat extensions of local rings

##### §11. Supplement on homological algebra

- 11.1 Reminder on spectral sequences
- 11.2 Spectral sequence of a filtered complex
- 11.3 Spectral sequences of a double complex
- 11.4 Hypercohomology of a functor on a complex $K^\bullet$
- 11.5 Inductive limits in hypercohomology
- 11.6 Hypercohomology of a functor on a complex $K_\bullet$
- 11.7 Hypercohomology of a functor on a double complex $K_{\bullet\bullet}$
- 11.8 Supplement on cohomology of simplicial complexes
- 11.9 A lemma on complexes of finite type
- 11.10 Euler-Poincaré characteristic of a complex of finite-length modules

##### §12. Supplement on sheaf cohomology

- 12.1 Cohomology of sheaves of modules on a ringed space
- 12.2 Higher direct images
- 12.3 Supplement on Ext of sheaves
- 12.4 Hypercohomology of the direct image functor

##### §13. Projective limits in homological algebra

- 13.1 Mittag-Leffler condition
- 13.2 Mittag-Leffler condition for abelian groups
- 13.3 Application to cohomology of a projective limit of sheaves
- 13.4 Mittag-Leffler condition and graded objects associated to projective systems
- 13.5 Projective limits of spectral sequences of filtered complexes
- 13.6 Spectral sequence of a functor relative to a finitely filtered object
- 13.7 Derived functors on projective limits

(Volume IV)

##### §14. Combinatorial dimension of a topological space

- 14.1 Combinatorial dimension of a topological space
- 14.2 Codimension of a closed subset
- 14.3 Chain condition

##### §15. $M$-regular and $\mathcal F$-regular sequences

- 15.1 $M$-regular and $M$-quasi-regular sequences
- 15.2 $\mathcal F$-regular sequences

##### §16. Dimension and depth of Noetherian local rings

- 16.1 Dimension of a ring
- 16.2 Dimension of a semi-local Noetherian ring
- 16.3 Systems of parameters in a Noetherian local ring
- 16.4 Depth and co-depth
- 16.5 Cohen-Macaulay modules

##### §17. Regular rings

- 17.1 Definition of regular ring
- 17.2 Reminder on projective and injective dimension
- 17.3 Cohomological theory of regular rings

##### §18. Supplement on extension of algebras

- 18.1 Inverse images of augmented rings
- 18.2 Extension of ring by a bi-module
- 18.3 The group of $A$-extension classes
- 18.4 Extensions of algebras
- 18.5 Case of topological rings

##### §19. Formally smooth algebras and Cohen rings

- 19.0 Introduction
- 19.1 Formal epi- and monomorphisms
- 19.2 Formally projective modules
- 19.3 Formally smooth algebras
- 19.4 First criteria for formal smoothness
- 19.5 Formal smoothness and associated graded rings
- 19.6 Case of algebras over a field
- 19.7 Case of local homomorphisms; existence and uniqueness theorems
- 19.8 Cohen algebras and $p$-rings; structure of complete local rings
- 19.9 Relatively formally smooth algebras
- 19.10 Formally unramified and formally étale algebras

##### §20. Derivations and differentials

- 20.1 Derivations and extensions of algebras
- 20.2 Functorial properties of derivations
- 20.3 Continuous derivations of topological rings
- 20.4 Principal and differential subsets
- 20.5 Basic functorial properties of $\Omega^1_{B/A}$
- 20.6 Imperfection modules and characteristic homomorphisms
- 20.7 Generalizations to topological rings

##### §21. Differentials in rings of characteristic $p$

- 21.1 Systems of $p$-generators and $p$-bases
- 21.2 $p$-bases and formal smoothness
- 21.3 $p$-bases and imperfection modules
- 21.4 Case of a field extension
- 21.5 Application: separability criteria
- 21.6 Admissible fields for an extension
- 21.7 Cartier's identity
- 21.8 Admissibility criteria
- 21.9 Completed modules of differentials for formal power series rings

##### §22. Differential criteria for smoothness and regularity

- 22.1 Lifting of formal smoothness
- 22.2 Differential characterization of formally smooth local algebras over a field
- 22.3 Application to relations between certain local rins and their completions
- 22.4 Preliminary results on finite extensions of local rings in which $\mathfrak m^2 = 0$.
- 22.5 Geometrically regular and formally smooth algebras
- 22.6 Zariski's Jacobian criterion
- 22.7 Nagata's Jacobian criterion

##### §23. Japanese rings

- 23.1 Japanese rings
- 23.2 Integral closure of a Noetherian local domain

## Volume I. The language of schemes

##### §1. Affine schemes

- 1.1 Prime spectrum (Spec) of a ring
- 1.2 Functorial properties of Spec
- 1.3 Sheaf associated to a module
- 1.4 Quasi-coherent sheaves on Spec
- 1.5 Coherent sheaves on Spec
- 1.6 Functorial properties of quasi-coherent sheaves on Spec
- 1.7 Characterization of morphisms of affine schemes

##### §2. Preschemes and their morphisms

- 2.1 Definition of prescheme
- 2.2 Morphisms of preschemes
- 2.3 Gluing preschemes
- 2.4 Local schemes
- 2.5 Preschemes over a prescheme

##### §3. Product of preschemes

- 3.1 Disjoint union of preschemes
- 3.2 Product of preschemes
- 3.3 Formal properties of the product; change of base prescheme
- 3.4 Points of a prescheme with values in a prescheme; geometric points
- 3.5 Surjections and injections
- 3.6 Fibers
- 3.7 Application: reduction of a prescheme mod $\mathcal I$

##### §4. Sub-preschemes and immersions

- 4.1 Sub-preschemes
- 4.2 Immersions
- 4.3 Product of immersions
- 4.4 Inverse image of a prescheme
- 4.5 Local immersions and local isomorphisms

##### §5. Reduced preschemes; separatedness

- 5.1 Reduced preschemes
- 5.2 Existence of sub-prescheme with a given underlying space
- 5.3 Diagonal; graph of a morphism
- 5.4 Separated morphisms and preschemes
- 5.5 Criteria for separatedness

##### §6. Finiteness conditions

- 6.1 Noetherian and locally Noetherian preschemes
- 6.2 Artinian preschemes
- 6.3 Morphisms of finite type
- 6.4 Algebraic preschemes
- 6.5 Local determination of a morphism
- 6.6 Quasi-compact morphisms and morphisms locally of finite type

##### §7. Rational maps

- 7.1 Rational maps and rational functions
- 7.2 Domain of definition of a rational map
- 7.3 Sheaf of rational functions
- 7.4 Torsion sheaves and torsion-free sheaves

##### §8. Chevalley schemes

- 8.1 Allied local rings
- 8.2 Local rings of an integral scheme
- 8.3 Chevalley schemes

##### §9. Supplement on quasi-coherent sheaves

- 9.1 Tensor product of quasi-coherent sheaves
- 9.2 Direct image of a quasi-coherent sheaf
- 9.3 Extension of sections of quasi-coherent sheaves
- 9.4 Extension of quasi-coherent sheaves
- 9.5 Closed image of a prescheme; closure of a sub-prescheme
- 9.6 Quasi-coherent sheaves of algebras; change of structure sheaf

##### §10. Formal schemes

- 10.1 Affine formal schemes
- 10.2 Morphisms of affine formal schemes
- 10.3 Ideals of definition of a formal affine scheme
- 10.4 Formal preschemes and their morphisms
- 10.5 Ideals of definition of formal preschemes
- 10.6 Formal preschemes as inductive limits of schemes
- 10.7 Product of formal schemes
- 10.8 Formal completion of a prescheme along a closed subset
- 10.9 Extension of morphisms to completions
- 10.10 Application to coherent sheaves on formal schemes

## Volume II. Basic global properties of some classes of morphisms.

##### §1. Affine morphisms

- 1.1 $S$-preschemes and $\mathcal O_S$-algebras
- 1.2 Preschemes affine over a prescheme
- 1.3 Affine prescheme over $S$ associated to an $O_S$-algebra
- 1.4 Quasi-coherent sheaves on a prescheme affine over $S$
- 1.5 Change of base prescheme
- 1.6 Affine morphisms
- 1.7 Vector bundle associated a sheaf of modules

##### §2. Homogeneous prime spectra

- 2.1 Generalities on graded rings and modules
- 2.2 Rings of fractions of a graded ring
- 2.3 Homogeneous prime spectrum of a graded ring
- 2.4 The scheme structure of $Proj(S)$
- 2.5 Sheaf associated to a graded module
- 2.6 Graded S-module associatedto a sheaf on $Proj(S)$
- 2.7 Finiteness conditions
- 2.8 Functorial behavior
- 2.9 Closed sub-preschemes of $Proj(S)$

##### §3. Homogeneous prime spectrum of a sheaf of graded algebras

- 3.1 Homogeneous prime spectrum of a graded, quasi-coherent $\mathcal O_Y$-algebra
- 3.2 Sheaf on $Proj(S)$ associated to a sheaf of graded $S$-modules
- 3.3 Sheaf of graded $S$-modules associated to a sheaf on $Proj(S)$
- 3.4 Finiteness conditions
- 3.5 Functorial behavior
- 3.6 Closed sub-preschemes of $Proj(S)$
- 3.7 Morphisms from a prescheme to a Proj
- 3.8 Criteria for immersion into a Proj

##### §4. Projective bundles; ample sheaves.

- 4.1 Definition of projective bundles
- 4.2 Morphisms from a prescheme to a projective bundle
- 4.3 The Segre morphism
- 4.4 Immersions in projective bundles; very ample sheaves
- 4.5 Ample sheaves
- 4.6 Relative ample sheaves

##### §5. Quasi-affine, quasi-projective, proper and projective morphisms

- 5.1 Quasi-affine morphisms
- 5.2 Serre's criterion
- 5.3 Quasi-projective morphisms
- 5.4 Universally closed and proper morphisms
- 5.5 Projective morphisms
- 5.6 Chow's lemma

##### §6. Integral and finite morphisms

- 6.1 Preschemes integral over another
- 6.2 Quasi-finite morphisms
- 6.3 Integral closure of a prescheme
- 6.4 Determinant of an endomorphism of a sheaf of $\mathcal O_X$-modules
- 6.5 Norm of an invertible sheaf
- 6.6 Application: criteria for ampleness
- 6.7 Chevalley's theorem

##### §7. Valuative criteria

- 7.1 Reminder on valuation rings
- 7.2 Valuative criterion for separatedness
- 7.3 Valuative criterion for properness
- 7.4 Algebraic curves and function fields of dimension 1

##### §8. Blowup schemes; projective cones; projective closure

- 8.1 Blowup preschemes
- 8.2 Preliminary results on localization of graded rings
- 8.3 Projective cones
- 8.4 Projective closure of a vector bundle
- 8.5 Functorial behavior
- 8.6 A canonical isormorphism for pointed cones
- 8.7 Blowing up projective cones
- 8.8 Ample sheaves and contractions
- 8.9 Grauert's ampleness criterion: statement
- 8.10 Grauert's ampleness criterion: proof
- 8.11 Uniqueness of contractions
- 8.12 Quasi-coherent sheaves on projective cones
- 8.13 Projective closure of sub-sheaves and closed subschemes
- 8.14 Supplement on sheaves associated to graded $\mathcal S$-modules

## Volume III. Cohomological study of coherent sheaves

(Part 1)

##### §1. Cohomology of affine schemes

- 1.1 Reminder on the exterior algebra complex
- 1.2 Čech cohomology of an open cover
- 1.3 Cohomology of an affine scheme
- 1.4 Application to cohomology of general preschemes

##### §2. Cohomological study of projective morphisms

- 2.1 Explicit calculation of some cohomology groups
- 2.2 Fundamental theorem on projective morphisms
- 2.3 Application to sheaves of graded algebras and modules
- 2.4 Generalization of the fundamental theorem
- 2.5 Euler-Poincaré characteristic and Hilbert polynomial
- 2.6 Application: criteria for ampleness

##### §3. Finiteness theorem for proper morphisms

- 3.1 "Dévissage" lemma
- 3.2 Finiteness theorem for ordinary schemes
- 3.3 Generalization of the finiteness theorem
- 3.4 Finiteness theorem for formal schemes

##### §4. Fundamental theorem on proper morphisms, and applications

- 4.1 The fundamental theorem
- 4.2 Special cases and variations
- 4.3 Zariski's connectedness theorem
- 4.4 Zariski's "main theorem"
- 4.5 Completion of Hom modules
- 4.6 Relations between ordinary and formal morphisms
- 4.7 An ampleness criterion
- 4.8 Finite morphisms of formal preschemes

##### §5. An existence theorem for coherent sheaves

- 5.1 Statement of the theorem
- 5.2 Proof in the projective & quasi-projective case
- 5.3 Proof in the general case
- 5.4 Application: comparison between morphism of ordinary and formal schemes; algebraisable formal schemes
- 5.5 Decomposition of certain schemes

(Part 2)

##### §6. Local and global Tor, Künneth formula

- 6.1 Introduction
- 6.2 Hypercohomology of complexes of sheaves of modules on a prescheme
- 6.3 Hypertor of two complexes
- 6.4 Local hypertor for quasi-coherent complexes, affine case
- 6.5 Local hypertor for quasi-coherent complexes, general case
- 6.6 Global hypertor for quasi-coherent complexes and Künneth spectral sequence, case of an affine base
- 6.7 Global hypertor for quasi-coherent complexes and Künneth spectral sequence, general case
- 6.8 Associativity spectral sequence for global hypertor
- 6.9 Base-change spectral sequence for global hypertor
- 6.10 Local nature of certain cohomological functors

##### §7. Base change for homological functors on sheaves of modules

- 7.1 Functors on $A$-modules
- 7.2 Characterization of the tensor product functor
- 7.3 Exactness criteria for homological functors on modules
- 7.4 Exactness criteria for the functors $H_\bullet(P_\bullet\otimes_A M)$
- 7.5 Case of Noetherian local rings
- 7.6 Descent of exactness properties; semi-continuity theorem and Grauert's exactness criterion
- 7.7 Application to proper morphisms: I. Exchange property
- 7.8 Application to proper morphisms: II. Cohomological flatness criteria
- 7.9 Application to proper morphisms: III. Invariance of Euler-Poincaré characteristic and Hilbert polynomial

## Volume IV. Local study of schemes and morphisms

(Part 1)

##### §1. Relative finiteness conditions; constructible sets in preschemes

- 1.1 Quasi-compact morphisms
- 1.2 Quasi-separted morphisms
- 1.3 Morphisms locally of finite type
- 1.4 Locally finitely presented morphisms
- 1.5 Morphisms of finite type
- 1.6 Finitely presented morphisms
- 1.7 Improvements of preceding results
- 1.8 Finitely presented morphisms and constructible sets
- 1.9 Pro- and ind-constructible morphisms
- 1.10 Application to open morphisms

(Part 2)

##### §2. Base change and flatness

- 2.1 Flat sheaves of modules on preschemes
- 2.2 Faithfully flat sheaves on preschemes
- 2.3 Topological properties of flat morphisms
- 2.4 Universally open morphisms and flat morphisms
- 2.5 Persistence of properties of sheaves under faithfully flat descent
- 2.6 Persistence of set-theoretic and topological properties under faithfully flat descent
- 2.7 Persistence of various properties of morphisms under faithfully flat descent
- 2.8 Preschemes over a regular base scheme of dimension 1; closed subschemes in the closure of the generic fiber

##### §3. Associated prime cycles and primary decomposition

- 3.1 Associated prime cycles of a sheaf of modules
- 3.2 Irredundant decompositions
- 3.3 Relations with flatness
- 3.4 Properties of sheaves of the form $\mathcal F/t\mathcal F$

##### §4. Change of ground field for algebraic preschemes

- 4.1 Dimension of algebraic preschemes
- 4.2 Associated prime cycles on algebraic preschemes
- 4.3 Reminder on tensor products of fields
- 4.4 Irreducible and connected preschemes over an algebraically closed field
- 4.5 Geometrically irreducible and connected preschemes
- 4.6 Geometrically reduced preschemes
- 4.7 Multiplicities in primary decomposition on an algebraic prescheme
- 4.8 Fields of definition
- 4.9 Subsets defined over a field

##### §5. Dimension, depth, and regularity for locally Noetherian preschemes

- 5.1 Dimension of preschemes
- 5.2 Dimension of algebraic preschemes
- 5.3 Dimension of the support of a sheaf; Hilbert polynomial
- 5.4 Dimension of the image of a morphism
- 5.5 Dimension formula for a morphism of finite type
- 5.6 Dimension formula and universally catenary rings
- 5.7 Depth and property $(S_k)$
- 5.8 Regular preschemes and property $(R_k)$; Serre's criterion for normality
- 5.9 $Z$-pure and $Z$-closed sheaves of modules
- 5.10 Property $(S_2)$ and $Z$-closure
- 5.11 Coherence criteria for sheaves $\mathcal H^0_{X/Z}(\mathcal F)$
- 5.12 Relations between the properties of a Noetherian local ring $A$ and a quotient $A/tA$.
- 5.13 Properties that persist under inductive limits

##### §6. Flat morphisms of locally Noetherian preschemes

- 6.1 Flatness and dimension
- 6.2 Flatness and projective dimension
- 6.3 Flatness and depth
- 6.4 Flatness and property $(S_k)$
- 6.5 Flatness and property $(R_k)$
- 6.6 Transitiviy properties
- 6.7 Application to change of base for algebraic preschemes
- 6.8 Regular, normal, reduced and smooth morphisms
- 6.9 Theorem on generic flatness
- 6.10 Dimension and depth of a sheaf normally flat along a closed sub-prescheme
- 6.11 Criteria for $U_{S_n}(\mathcal F)$ and $U_{C_n}(\mathcal F)$ to be open
- 6.12 Nagata's criteria for Reg($X$) to be open
- 6.13 Criteria for Nor($X$) to be open
- 6.14 Base change and integral closure
- 6.15 Geometrically unibranched preschemes

##### §7. Noetherian local rings and their completions; excellent rings

- 7.1 Formal equidensionality and formally catenary rings
- 7.2 Strictly formally catenary rings
- 7.3 Formal fibers of Noetherian local rings
- 7.4 Persistence of properties of formal fibers
- 7.5 A criterion for $P$-morphisms
- 7.6 Application I: Locally Japanese rings
- 7.7 Application II: Universally Japanese rings
- 7.8 Excellent rings
- 7.9 Excellent rings and resolution of singularities

(Part 3)

##### §8. Projective limits of preschemes

- 8.1 Introduction
- 8.2 Projective limits of preschemes
- 8.3 Constructible subsets of a projective limit of preschemes
- 8.4 Irreducibility and connectedness criteria for projective limits of preschemes
- 8.5 Finitely presented sheaves of modules on a projective limit of preschemes
- 8.6 Finitely presented subschemes of a projective limit of preschemes
- 8.7 Criteria for a projective limits of preschemes to be a reduced (resp. integral) prescheme
- 8.8 Preschemes finitely presented over a projective limit of preschemes
- 8.9 Initial applications to elimination of Noetherian hypotheses
- 8.10 Properties of morphisms persistent under projective limits
- 8.11 Application to quasi-finite morphisms
- 8.12 Another proof and generalization of Zariski's "main theorem"
- 8.13 Translation into the language of pro-objects
- 8.14 Characterization of a prescheme locally finitely presented over another, in terms of the functor it represents

##### §9. Constructible properties

- 9.1 Principle of finite extension
- 9.2 Constructible and ind-constructible properties
- 9.3 Constructible properties of morphisms of algebraic preschemes
- 9.4 Constructibility of certain properties of sheaves of modules
- 9.5 Constructibility of topological properties
- 9.6 Constructibility of certain properties of morphisms
- 9.7 Constructibility of the properties of separability, and geometric irreducibility and connectedness
- 9.8 Primary decomposition in the neighborhood of a generic fiber
- 9.9 Constructibility of local properties of fibers

##### §10. Jacobson preschemes

- 10.1 Very dense subsets of a topological space
- 10.2 Quasi-homeomorphisms
- 10.3 Jacobson spaces
- 10.4 Jacobson preschemes and rings
- 10.5 Noetherian Jacobson preschemes
- 10.6 Dimension of Jacobson preschemes
- 10.7 Examples and counterexamples
- 10.8 Rectified depth
- 10.9 Maximal spectra and ultra-preschemes
- 10.10 Algebraic spaces in the sense of Serre

##### §11. Topological properties of finitely presented flat morphisms; flatness criteria.

- 11.1 Flatness loci (Noetherian case)
- 11.2 Flatness of a projective limit of preschemes
- 11.3 Application to elimination of Noetherian hypotheses
- 11.4 Descent of flatness by arbitrary morphisms: case of a prescheme over an Artinian base
- 11.5 Descent of flatness by arbitrary morphisms: general case
- 11.6 Descent of flatness by arbitrary morphisms: case of a prescheme over a unibranched base
- 11.7 Counterexamples
- 11.8 Valuative criterion for flatness
- 11.9 Separated and universally separtated families of homomorphisms of sheaves of modules
- 11.10 Schematically dominant families of morphisms and schematically dense families of sub-preschemes

##### §12. Fibers of finitely presented flat morphisms

- 12.0 Introduction
- 12.1 Local properties of the fibers of a locally finitely presented flat morphism
- 12.2 Local and global properties of the fibers of a proper, flat, finitely presented morphism
- 12.3 Local cohomological properties of the fibers of a locally finitely presented flat morphism

##### §13. Equidimensional morphisms

- 13.1 Chevalley's semi-continuity theorem
- 13.2 Equidimensional morphisms: case of domeinant morphisms of irreducible preschemes
- 13.3 Equidimensional morphisms: general case

##### §14. Universally open morphisms

- 14.1 Open morphisms
- 14.2 Open morphisms and dimension formula
- 14.3 Universally open morphisms
- 14.4 Chevalley's criterion for universally open morphisms
- 14.5 Universally open morphisms and quasi-sections

##### §15. Fibers of a universally open morphism

- 15.1 Multiplicities of fibers of a universally open morphism
- 15.2 Flatness of universally open morphisms with geometrically reduced fibers
- 15.3 Application: criteria for reducedness and irreducibility
- 15.4 Supplement on Cohen-Macaulay morphisms
- 15.5 Separable rank of the fibers of a quasi-finite and universally open morphism; application to geometrically connected components of the fibers of a proper morphism
- 15.6 Connected components of fibers along a section
- 15.7 Appendix: local valuative criteria for properness

(Part 4)

##### §16. Differential invariants; differentially smooth morphisms

- 16.1 Normal invariants of an immersion
- 16.2 Functorial properties of normal invariants
- 16.3 Basic differential invariants of a morphism of preschemes
- 16.4 Functorial properties of differential invariants
- 16.5 Relative tangent sheaves and bundles; derivations
- 16.6 Sheaves of $p$-differentials and exterior differentials
- 16.7 The sheaves $P^n_{X/S}(\mathcal F)$
- 16.8 Differential operators
- 16.9 Regular and quasi-regular immersions
- 16.10 Differentially smooth morphisms
- 16.11 Differential operators on a differentially smooth $S$-prescheme
- 16.12 Characteristic 0 case: Jacobian criterion for differentially smooth morphisms

##### §17. Smooth, unramified and étale morphisms

- 17.1 Formally smooth, unramified and étale morphisms
- 17.2 General differential properties
- 17.3 Smooth, unramified and étale morphisms
- 17.4 Characterization of unramified morphisms
- 17.5 Characterization of smooth morphisms
- 17.6 Characterization of étale morphisms
- 17.7 Properties of descent and passage to the limit
- 17.8 Criteria for smoothness and unramification in terms of fibers
- 17.9 Etale morphisms and open immersions
- 17.10 Relative dimension of a prescheme smooth over another
- 17.11 Smooth morphisms of smooth preschemes
- 17.12 Smooth subschemes of a smooth prescheme; smooth and differentially smooth morphisms
- 17.13 Transverse morphisms
- 17.14 Local and infinitesimal characterizations of smooth, unramified and étale morphisms
- 17.15 Case of preschemes over a field
- 17.16 Quasi-sections of flat and smooth morphisms

##### §18. Supplement on étale morphisms; Henselian local rings and strictly local rings

- 18.1 A remarkable equivalence of categories
- 18.2 étale covers
- 18.3 Finite étale algebras
- 18.4 Local structure of unramified and étale morphisms
- 18.5 Henselian local rings
- 18.6 Henselization
- 18.7 Henselizatoin and excellent rings
- 18.8 Strictly local rings and strict Henselization
- 18.9 Formal fibers of Noetherian Henselian rings
- 18.10 Preschemes étale over a geometrically unibranched or normal prescheme
- 18.11 Application to complete Noetherian local algebras over a field
- 18.12 Applications of étale localization to quasi-finite morphisms (generalizations of preceding results)

##### §19. Regular immersions and normal flatness

- 19.1 Properties of regular immersions
- 19.2 Transversally regular immersions
- 19.3 Relative complete intersections (flat case)
- 19.4 Application: criteria for regularity and smoothness of blowups
- 19.5 Criteria for $M$-regularity
- 19.6 Regular sequences relative to a filtered quotient module
- 19.7 Hironaka's criterion for normal flatness
- 19.8 Properties of projective limits
- 19.9 $\mathcal F$-regular sequences and depth

##### §20. Meromorphic functions and pseudo-morphisms

- 20.0 Introduction
- 20.1 Meromorphic functions
- 20.2 Pseudo-morphisms and pseudo-functions
- 20.3 Composition of pseudo-morphisms
- 20.4 Properties of domains of definition of rational functions
- 20.5 Relative pseudo-morphisms
- 20.6 Relative meromorphic functions

##### §21. Divisors

- 21.1 Divisors on a ringed space
- 21.2 Divisors and invertible fractional ideal sheaves
- 21.3 Linear equivalence of divisors
- 21.4 Inverse images of divisors
- 21.5 Direct images of divisors
- 21.6 1-codimensional cycle associated to a divisor
- 21.7 Interpretation of positive 1-codimensional cycles in terms of sub-preschemes
- 21.8 Divisors and normalization
- 21.9 Divisors on preschemes of dimension 1
- 21.10 Inverse and direct images of 1-codimensional cycles
- 21.11 Factoriality of regular rings
- 21.12 Van der Waerden's purity theorem for the ramification locus of a birational morphism
- 21.13 Parafactorial pairs; parafactorial local rings
- 21.14 The Ramanujan-Samuel theorem
- 21.15 Relative divisors