The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

1 Preliminaries

  • Chapter 1: Introduction
    • Section 1.1: Overview
    • Section 1.2: Attribution
  • Chapter 2: Conventions
    • Section 2.1: Comments
    • Section 2.2: Set theory
    • Section 2.3: Categories
    • Section 2.4: Algebra
    • Section 2.5: Notation
  • Chapter 3: Set Theory
    • Section 3.1: Introduction
    • Section 3.2: Everything is a set
    • Section 3.3: Classes
    • Section 3.4: Ordinals
    • Section 3.5: The hierarchy of sets
    • Section 3.6: Cardinality
    • Section 3.7: Cofinality
    • Section 3.8: Reflection principle
    • Section 3.9: Constructing categories of schemes
    • Section 3.10: Sets with group action
    • Section 3.11: Coverings of a site
    • Section 3.12: Abelian categories and injectives
  • Chapter 4: Categories
    • Section 4.1: Introduction
    • Section 4.2: Definitions
    • Section 4.3: Opposite Categories and the Yoneda Lemma
    • Section 4.4: Products of pairs
    • Section 4.5: Coproducts of pairs
    • Section 4.6: Fibre products
    • Section 4.7: Examples of fibre products
    • Section 4.8: Fibre products and representability
    • Section 4.9: Pushouts
    • Section 4.10: Equalizers
    • Section 4.11: Coequalizers
    • Section 4.12: Initial and final objects
    • Section 4.13: Monomorphisms and Epimorphisms
    • Section 4.14: Limits and colimits
    • Section 4.15: Limits and colimits in the category of sets
    • Section 4.16: Connected limits
    • Section 4.17: Cofinal and initial categories
    • Section 4.18: Finite limits and colimits
    • Section 4.19: Filtered colimits
    • Section 4.20: Cofiltered limits
    • Section 4.21: Limits and colimits over preordered sets
    • Section 4.22: Essentially constant systems
    • Section 4.23: Exact functors
    • Section 4.24: Adjoint functors
    • Section 4.25: A criterion for representability
    • Section 4.26: Localization in categories
    • Section 4.27: Formal properties
    • Section 4.28: 2-categories
    • Section 4.29: (2, 1)-categories
    • Section 4.30: 2-fibre products
    • Section 4.31: Categories over categories
    • Section 4.32: Fibred categories
    • Section 4.33: Inertia
    • Section 4.34: Categories fibred in groupoids
    • Section 4.35: Presheaves of categories
    • Section 4.36: Presheaves of groupoids
    • Section 4.37: Categories fibred in sets
    • Section 4.38: Categories fibred in setoids
    • Section 4.39: Representable categories fibred in groupoids
    • Section 4.40: Representable 1-morphisms
  • Chapter 5: Topology
    • Section 5.1: Introduction
    • Section 5.2: Basic notions
    • Section 5.3: Hausdorff spaces
    • Section 5.4: Separated maps
    • Section 5.5: Bases
    • Section 5.6: Submersive maps
    • Section 5.7: Connected components
    • Section 5.8: Irreducible components
    • Section 5.9: Noetherian topological spaces
    • Section 5.10: Krull dimension
    • Section 5.11: Codimension and catenary spaces
    • Section 5.12: Quasi-compact spaces and maps
    • Section 5.13: Locally quasi-compact spaces
    • Section 5.14: Limits of spaces
    • Section 5.15: Constructible sets
    • Section 5.16: Constructible sets and Noetherian spaces
    • Section 5.17: Characterizing proper maps
    • Section 5.18: Jacobson spaces
    • Section 5.19: Specialization
    • Section 5.20: Dimension functions
    • Section 5.21: Nowhere dense sets
    • Section 5.22: Profinite spaces
    • Section 5.23: Spectral spaces
    • Section 5.24: Limits of spectral spaces
    • Section 5.25: Stone-Čech compactification
    • Section 5.26: Extremally disconnected spaces
    • Section 5.27: Miscellany
    • Section 5.28: Partitions and stratifications
    • Section 5.29: Colimits of spaces
    • Section 5.30: Topological groups, rings, modules
  • Chapter 6: Sheaves on Spaces
    • Section 6.1: Introduction
    • Section 6.2: Basic notions
    • Section 6.3: Presheaves
    • Section 6.4: Abelian presheaves
    • Section 6.5: Presheaves of algebraic structures
    • Section 6.6: Presheaves of modules
    • Section 6.7: Sheaves
    • Section 6.8: Abelian sheaves
    • Section 6.9: Sheaves of algebraic structures
    • Section 6.10: Sheaves of modules
    • Section 6.11: Stalks
    • Section 6.12: Stalks of abelian presheaves
    • Section 6.13: Stalks of presheaves of algebraic structures
    • Section 6.14: Stalks of presheaves of modules
    • Section 6.15: Algebraic structures
    • Section 6.16: Exactness and points
    • Section 6.17: Sheafification
    • Section 6.18: Sheafification of abelian presheaves
    • Section 6.19: Sheafification of presheaves of algebraic structures
    • Section 6.20: Sheafification of presheaves of modules
    • Section 6.21: Continuous maps and sheaves
    • Section 6.22: Continuous maps and abelian sheaves
    • Section 6.23: Continuous maps and sheaves of algebraic structures
    • Section 6.24: Continuous maps and sheaves of modules
    • Section 6.25: Ringed spaces
    • Section 6.26: Morphisms of ringed spaces and modules
    • Section 6.27: Skyscraper sheaves and stalks
    • Section 6.28: Limits and colimits of presheaves
    • Section 6.29: Limits and colimits of sheaves
    • Section 6.30: Bases and sheaves
    • Section 6.31: Open immersions and (pre)sheaves
    • Section 6.32: Closed immersions and (pre)sheaves
    • Section 6.33: Glueing sheaves
  • Chapter 7: Sites and Sheaves
    • Section 7.1: Introduction
    • Section 7.2: Presheaves
    • Section 7.3: Injective and surjective maps of presheaves
    • Section 7.4: Limits and colimits of presheaves
    • Section 7.5: Functoriality of categories of presheaves
    • Section 7.6: Sites
    • Section 7.7: Sheaves
    • Section 7.8: Families of morphisms with fixed target
    • Section 7.9: The example of G-sets
    • Section 7.10: Sheafification
    • Section 7.11: Injective and surjective maps of sheaves
    • Section 7.12: Representable sheaves
    • Section 7.13: Continuous functors
    • Section 7.14: Morphisms of sites
    • Section 7.15: Topoi
    • Section 7.16: G-sets and morphisms
    • Section 7.17: Quasi-compact objects and colimits
    • Section 7.18: Colimits of sites
    • Section 7.19: More functoriality of presheaves
    • Section 7.20: Cocontinuous functors
    • Section 7.21: Cocontinuous functors and morphisms of topoi
    • Section 7.22: Cocontinuous functors which have a right adjoint
    • Section 7.23: Cocontinuous functors which have a left adjoint
    • Section 7.24: Existence of lower shriek
    • Section 7.25: Localization
    • Section 7.26: Glueing sheaves
    • Section 7.27: More localization
    • Section 7.28: Localization and morphisms
    • Section 7.29: Morphisms of topoi
    • Section 7.30: Localization of topoi
    • Section 7.31: Localization and morphisms of topoi
    • Section 7.32: Points
    • Section 7.33: Constructing points
    • Section 7.34: Points and morphisms of topoi
    • Section 7.35: Localization and points
    • Section 7.36: 2-morphisms of topoi
    • Section 7.37: Morphisms between points
    • Section 7.38: Sites with enough points
    • Section 7.39: Criterion for existence of points
    • Section 7.40: Weakly contractible objects
    • Section 7.41: Exactness properties of pushforward
    • Section 7.42: Almost cocontinuous functors
    • Section 7.43: Subtopoi
    • Section 7.44: Sheaves of algebraic structures
    • Section 7.45: Pullback maps
    • Section 7.46: Comparison with SGA4
    • Section 7.47: Topologies
    • Section 7.48: The topology defined by a site
    • Section 7.49: Sheafification in a topology
    • Section 7.50: Topologies and sheaves
    • Section 7.51: Topologies and continuous functors
    • Section 7.52: Points and topologies
  • Chapter 8: Stacks
    • Section 8.1: Introduction
    • Section 8.2: Presheaves of morphisms associated to fibred categories
    • Section 8.3: Descent data in fibred categories
    • Section 8.4: Stacks
    • Section 8.5: Stacks in groupoids
    • Section 8.6: Stacks in setoids
    • Section 8.7: The inertia stack
    • Section 8.8: Stackification of fibred categories
    • Section 8.9: Stackification of categories fibred in groupoids
    • Section 8.10: Inherited topologies
    • Section 8.11: Gerbes
    • Section 8.12: Functoriality for stacks
    • Section 8.13: Stacks and localization
  • Chapter 9: Fields
    • Section 9.1: Introduction
    • Section 9.2: Basic definitions
    • Section 9.3: Examples of fields
    • Section 9.4: Vector spaces
    • Section 9.5: The characteristic of a field
    • Section 9.6: Field extensions
    • Section 9.7: Finite extensions
    • Section 9.8: Algebraic extensions
    • Section 9.9: Minimal polynomials
    • Section 9.10: Algebraic closure
    • Section 9.11: Relatively prime polynomials
    • Section 9.12: Separable extensions
    • Section 9.13: Linear independence of characters
    • Section 9.14: Purely inseparable extensions
    • Section 9.15: Normal extensions
    • Section 9.16: Splitting fields
    • Section 9.17: Roots of unity
    • Section 9.18: Finite fields
    • Section 9.19: Primitive elements
    • Section 9.20: Trace and norm
    • Section 9.21: Galois theory
    • Section 9.22: Infinite Galois theory
    • Section 9.23: The complex numbers
    • Section 9.24: Kummer extensions
    • Section 9.25: Artin-Schreier extensions
    • Section 9.26: Transcendence
    • Section 9.27: Linearly disjoint extensions
    • Section 9.28: Review
  • Chapter 10: Commutative Algebra
    • Section 10.1: Introduction
    • Section 10.2: Conventions
    • Section 10.3: Basic notions
    • Section 10.4: Snake lemma
    • Section 10.5: Finite modules and finitely presented modules
    • Section 10.6: Ring maps of finite type and of finite presentation
    • Section 10.7: Finite ring maps
    • Section 10.8: Colimits
    • Section 10.9: Localization
    • Section 10.10: Internal Hom
    • Section 10.11: Tensor products
    • Section 10.12: Tensor algebra
    • Section 10.13: Base change
    • Section 10.14: Miscellany
    • Section 10.15: Cayley-Hamilton
    • Section 10.16: The spectrum of a ring
    • Section 10.17: Local rings
    • Section 10.18: The Jacobson radical of a ring
    • Section 10.19: Nakayama's lemma
    • Section 10.20: Open and closed subsets of spectra
    • Section 10.21: Connected components of spectra
    • Section 10.22: Glueing properties
    • Section 10.23: Glueing functions
    • Section 10.24: Zerodivisors and total rings of fractions
    • Section 10.25: Irreducible components of spectra
    • Section 10.26: Examples of spectra of rings
    • Section 10.27: A meta-observation about prime ideals
    • Section 10.28: Images of ring maps of finite presentation
    • Section 10.29: More on images
    • Section 10.30: Noetherian rings
    • Section 10.31: Locally nilpotent ideals
    • Section 10.32: Curiosity
    • Section 10.33: Hilbert Nullstellensatz
    • Section 10.34: Jacobson rings
    • Section 10.35: Finite and integral ring extensions
    • Section 10.36: Normal rings
    • Section 10.37: Going down for integral over normal
    • Section 10.38: Flat modules and flat ring maps
    • Section 10.39: Supports and annihilators
    • Section 10.40: Going up and going down
    • Section 10.41: Separable extensions
    • Section 10.42: Geometrically reduced algebras
    • Section 10.43: Separable extensions, continued
    • Section 10.44: Perfect fields
    • Section 10.45: Universal homeomorphisms
    • Section 10.46: Geometrically irreducible algebras
    • Section 10.47: Geometrically connected algebras
    • Section 10.48: Geometrically integral algebras
    • Section 10.49: Valuation rings
    • Section 10.50: More Noetherian rings
    • Section 10.51: Length
    • Section 10.52: Artinian rings
    • Section 10.53: Homomorphisms essentially of finite type
    • Section 10.54: K-groups
    • Section 10.55: Graded rings
    • Section 10.56: Proj of a graded ring
    • Section 10.57: Noetherian graded rings
    • Section 10.58: Noetherian local rings
    • Section 10.59: Dimension
    • Section 10.60: Applications of dimension theory
    • Section 10.61: Support and dimension of modules
    • Section 10.62: Associated primes
    • Section 10.63: Symbolic powers
    • Section 10.64: Relative assassin
    • Section 10.65: Weakly associated primes
    • Section 10.66: Embedded primes
    • Section 10.67: Regular sequences
    • Section 10.68: Quasi-regular sequences
    • Section 10.69: Blow up algebras
    • Section 10.70: Ext groups
    • Section 10.71: Depth
    • Section 10.72: Functorialities for Ext
    • Section 10.73: An application of Ext groups
    • Section 10.74: Tor groups and flatness
    • Section 10.75: Functorialities for Tor
    • Section 10.76: Projective modules
    • Section 10.77: Finite projective modules
    • Section 10.78: Open loci defined by module maps
    • Section 10.79: Faithfully flat descent for projectivity of modules
    • Section 10.80: Characterizing flatness
    • Section 10.81: Universally injective module maps
    • Section 10.82: Descent for finite projective modules
    • Section 10.83: Transfinite dévissage of modules
    • Section 10.84: Projective modules over a local ring
    • Section 10.85: Mittag-Leffler systems
    • Section 10.86: Inverse systems
    • Section 10.87: Mittag-Leffler modules
    • Section 10.88: Interchanging direct products with tensor
    • Section 10.89: Coherent rings
    • Section 10.90: Examples and non-examples of Mittag-Leffler modules
    • Section 10.91: Countably generated Mittag-Leffler modules
    • Section 10.92: Characterizing projective modules
    • Section 10.93: Ascending properties of modules
    • Section 10.94: Descending properties of modules
    • Section 10.95: Completion
    • Section 10.96: Completion for Noetherian rings
    • Section 10.97: Taking limits of modules
    • Section 10.98: Criteria for flatness
    • Section 10.99: Base change and flatness
    • Section 10.100: Flatness criteria over Artinian rings
    • Section 10.101: What makes a complex exact?
    • Section 10.102: Cohen-Macaulay modules
    • Section 10.103: Cohen-Macaulay rings
    • Section 10.104: Catenary rings
    • Section 10.105: Regular local rings
    • Section 10.106: Epimorphisms of rings
    • Section 10.107: Pure ideals
    • Section 10.108: Rings of finite global dimension
    • Section 10.109: Regular rings and global dimension
    • Section 10.110: Auslander-Buchsbaum
    • Section 10.111: Homomorphisms and dimension
    • Section 10.112: The dimension formula
    • Section 10.113: Dimension of finite type algebras over fields
    • Section 10.114: Noether normalization
    • Section 10.115: Dimension of finite type algebras over fields, reprise
    • Section 10.116: Dimension of graded algebras over a field
    • Section 10.117: Generic flatness
    • Section 10.118: Around Krull-Akizuki
    • Section 10.119: Factorization
    • Section 10.120: Orders of vanishing
    • Section 10.121: Quasi-finite maps
    • Section 10.122: Zariski's Main Theorem
    • Section 10.123: Applications of Zariski's Main Theorem
    • Section 10.124: Dimension of fibres
    • Section 10.125: Algebras and modules of finite presentation
    • Section 10.126: Colimits and maps of finite presentation
    • Section 10.127: More flatness criteria
    • Section 10.128: Openness of the flat locus
    • Section 10.129: Openness of Cohen-Macaulay loci
    • Section 10.130: Differentials
    • Section 10.131: Finite order differential operators
    • Section 10.132: The naive cotangent complex
    • Section 10.133: Local complete intersections
    • Section 10.134: Syntomic morphisms
    • Section 10.135: Smooth ring maps
    • Section 10.136: Formally smooth maps
    • Section 10.137: Smoothness and differentials
    • Section 10.138: Smooth algebras over fields
    • Section 10.139: Smooth ring maps in the Noetherian case
    • Section 10.140: Overview of results on smooth ring maps
    • Section 10.141: Étale ring maps
    • Section 10.142: Local homomorphisms
    • Section 10.143: Integral closure and smooth base change
    • Section 10.144: Formally unramified maps
    • Section 10.145: Conormal modules and universal thickenings
    • Section 10.146: Formally étale maps
    • Section 10.147: Unramified ring maps
    • Section 10.148: Henselian local rings
    • Section 10.149: Filtered colimits of étale ring maps
    • Section 10.150: Henselization and strict henselization
    • Section 10.151: Serre's criterion for normality
    • Section 10.152: Formal smoothness of fields
    • Section 10.153: Constructing flat ring maps
    • Section 10.154: The Cohen structure theorem
    • Section 10.155: Japanese rings
    • Section 10.156: Nagata rings
    • Section 10.157: Ascending properties
    • Section 10.158: Descending properties
    • Section 10.159: Geometrically normal algebras
    • Section 10.160: Geometrically regular algebras
    • Section 10.161: Geometrically Cohen-Macaulay algebras
    • Section 10.162: Colimits and maps of finite presentation, II
  • Chapter 11: Brauer groups
    • Section 11.1: Introduction
    • Section 11.2: Noncommutative algebras
    • Section 11.3: Wedderburn's theorem
    • Section 11.4: Lemmas on algebras
    • Section 11.5: The Brauer group of a field
    • Section 11.6: Skolem-Noether
    • Section 11.7: The centralizer theorem
    • Section 11.8: Splitting fields
  • Chapter 12: Homological Algebra
    • Section 12.1: Introduction
    • Section 12.2: Basic notions
    • Section 12.3: Preadditive and additive categories
    • Section 12.4: Karoubian categories
    • Section 12.5: Abelian categories
    • Section 12.6: Extensions
    • Section 12.7: Additive functors
    • Section 12.8: Localization
    • Section 12.9: Serre subcategories
    • Section 12.10: K-groups
    • Section 12.11: Cohomological delta-functors
    • Section 12.12: Complexes
    • Section 12.13: Homotopy and the shift functor
    • Section 12.14: Truncation of complexes
    • Section 12.15: Graded objects
    • Section 12.16: Filtrations
    • Section 12.17: Spectral sequences
    • Section 12.18: Spectral sequences: exact couples
    • Section 12.19: Spectral sequences: differential objects
    • Section 12.20: Spectral sequences: filtered differential objects
    • Section 12.21: Spectral sequences: filtered complexes
    • Section 12.22: Spectral sequences: double complexes
    • Section 12.23: Double complexes of abelian groups
    • Section 12.24: Injectives
    • Section 12.25: Projectives
    • Section 12.26: Injectives and adjoint functors
    • Section 12.27: Essentially constant systems
    • Section 12.28: Inverse systems
    • Section 12.29: Exactness of products
  • Chapter 13: Derived Categories
    • Section 13.1: Introduction
    • Section 13.2: Triangulated categories
    • Section 13.3: The definition of a triangulated category
    • Section 13.4: Elementary results on triangulated categories
    • Section 13.5: Localization of triangulated categories
    • Section 13.6: Quotients of triangulated categories
    • Section 13.7: Adjoints for exact functors
    • Section 13.8: The homotopy category
    • Section 13.9: Cones and termwise split sequences
    • Section 13.10: Distinguished triangles in the homotopy category
    • Section 13.11: Derived categories
    • Section 13.12: The canonical delta-functor
    • Section 13.13: Triangulated subcategories of the derived category
    • Section 13.14: Filtered derived categories
    • Section 13.15: Derived functors in general
    • Section 13.16: Derived functors on derived categories
    • Section 13.17: Higher derived functors
    • Section 13.18: Injective resolutions
    • Section 13.19: Projective resolutions
    • Section 13.20: Right derived functors and injective resolutions
    • Section 13.21: Cartan-Eilenberg resolutions
    • Section 13.22: Composition of right derived functors
    • Section 13.23: Resolution functors
    • Section 13.24: Functorial injective embeddings and resolution functors
    • Section 13.25: Right derived functors via resolution functors
    • Section 13.26: Filtered derived category and injective resolutions
    • Section 13.27: Ext groups
    • Section 13.28: Unbounded complexes
    • Section 13.29: K-injective complexes
    • Section 13.30: Bounded cohomological dimension
    • Section 13.31: Derived colimits
    • Section 13.32: Derived limits
    • Section 13.33: Generators of triangulated categories
    • Section 13.34: Compact objects
    • Section 13.35: Brown representability
    • Section 13.36: Admissible subcategories
    • Section 13.37: Postnikov systems
  • Chapter 14: Simplicial Methods
    • Section 14.1: Introduction
    • Section 14.2: The category of finite ordered sets
    • Section 14.3: Simplicial objects
    • Section 14.4: Simplicial objects as presheaves
    • Section 14.5: Cosimplicial objects
    • Section 14.6: Products of simplicial objects
    • Section 14.7: Fibre products of simplicial objects
    • Section 14.8: Pushouts of simplicial objects
    • Section 14.9: Products of cosimplicial objects
    • Section 14.10: Fibre products of cosimplicial objects
    • Section 14.11: Simplicial sets
    • Section 14.12: Truncated simplicial objects and skeleton functors
    • Section 14.13: Products with simplicial sets
    • Section 14.14: Hom from simplicial sets into cosimplicial objects
    • Section 14.15: Hom from cosimplicial sets into simplicial objects
    • Section 14.16: Internal Hom
    • Section 14.17: Hom from simplicial sets into simplicial objects
    • Section 14.18: Splitting simplicial objects
    • Section 14.19: Coskeleton functors
    • Section 14.20: Augmentations
    • Section 14.21: Left adjoints to the skeleton functors
    • Section 14.22: Simplicial objects in abelian categories
    • Section 14.23: Simplicial objects and chain complexes
    • Section 14.24: Dold-Kan
    • Section 14.25: Dold-Kan for cosimplicial objects
    • Section 14.26: Homotopies
    • Section 14.27: Homotopies in abelian categories
    • Section 14.28: Homotopies and cosimplicial objects
    • Section 14.29: More homotopies in abelian categories
    • Section 14.30: Trivial Kan fibrations
    • Section 14.31: Kan fibrations
    • Section 14.32: A homotopy equivalence
    • Section 14.33: Standard resolutions
  • Chapter 15: More on Algebra
    • Section 15.1: Introduction
    • Section 15.2: Advice for the reader
    • Section 15.3: Stably free modules
    • Section 15.4: A comment on the Artin-Rees property
    • Section 15.5: Fibre products of rings, I
    • Section 15.6: Fibre products of rings, II
    • Section 15.7: Fibre products of rings, III
    • Section 15.8: Fitting ideals
    • Section 15.9: Lifting
    • Section 15.10: Zariski pairs
    • Section 15.11: Henselian pairs
    • Section 15.12: Henselization of pairs
    • Section 15.13: Lifting and henselian pairs
    • Section 15.14: Absolute integral closure
    • Section 15.15: Auto-associated rings
    • Section 15.16: Flattening stratification
    • Section 15.17: Flattening over an Artinian ring
    • Section 15.18: Flattening over a closed subset of the base
    • Section 15.19: Flattening over a closed subsets of source and base
    • Section 15.20: Flattening over a Noetherian complete local ring
    • Section 15.21: Descent of flatness along integral maps
    • Section 15.22: Torsion free modules
    • Section 15.23: Reflexive modules
    • Section 15.24: Content ideals
    • Section 15.25: Flatness and finiteness conditions
    • Section 15.26: Blowing up and flatness
    • Section 15.27: Completion and flatness
    • Section 15.28: The Koszul complex
    • Section 15.29: Koszul regular sequences
    • Section 15.30: More on Koszul regular sequences
    • Section 15.31: Regular ideals
    • Section 15.32: Local complete intersection maps
    • Section 15.33: Cartier's equality and geometric regularity
    • Section 15.34: Geometric regularity
    • Section 15.35: Topological rings and modules
    • Section 15.36: Formally smooth maps of topological rings
    • Section 15.37: Formally smooth maps of local rings
    • Section 15.38: Some results on power series rings
    • Section 15.39: Geometric regularity and formal smoothness
    • Section 15.40: Regular ring maps
    • Section 15.41: Ascending properties along regular ring maps
    • Section 15.42: Permanence of properties under completion
    • Section 15.43: Permanence of properties under étale maps
    • Section 15.44: Permanence of properties under henselization
    • Section 15.45: Field extensions, revisited
    • Section 15.46: The singular locus
    • Section 15.47: Regularity and derivations
    • Section 15.48: Formal smoothness and regularity
    • Section 15.49: G-rings
    • Section 15.50: Properties of formal fibres
    • Section 15.51: Excellent rings
    • Section 15.52: Abelian categories of modules
    • Section 15.53: Injective abelian groups
    • Section 15.54: Injective modules
    • Section 15.55: Derived categories of modules
    • Section 15.56: Computing Tor
    • Section 15.57: Derived tensor product
    • Section 15.58: Derived change of rings
    • Section 15.59: Tor independence
    • Section 15.60: Spectral sequences for Tor
    • Section 15.61: Products and Tor
    • Section 15.62: Pseudo-coherent modules
    • Section 15.63: Tor dimension
    • Section 15.64: Spectral sequences for Ext
    • Section 15.65: Projective dimension
    • Section 15.66: Injective dimension
    • Section 15.67: Hom complexes
    • Section 15.68: Derived hom
    • Section 15.69: Perfect complexes
    • Section 15.70: Lifting complexes
    • Section 15.71: Splitting complexes
    • Section 15.72: Characterizing perfect complexes
    • Section 15.73: Relatively finitely presented modules
    • Section 15.74: Relatively pseudo-coherent modules
    • Section 15.75: Pseudo-coherent and perfect ring maps
    • Section 15.76: Relatively perfect modules
    • Section 15.77: Rlim of abelian groups
    • Section 15.78: Rlim of modules
    • Section 15.79: Torsion modules
    • Section 15.80: Formal glueing of module categories
    • Section 15.81: The Beauville-Laszlo theorem
    • Section 15.82: Derived Completion
    • Section 15.83: Derived completion for a principal ideal
    • Section 15.84: Derived completion for Noetherian rings
    • Section 15.85: Taking limits of complexes
    • Section 15.86: Some evaluation maps
    • Section 15.87: Base change for derived hom
    • Section 15.88: Systems of modules
    • Section 15.89: Miscellany
    • Section 15.90: Tricks with double complexes
    • Section 15.91: Weakly étale ring maps
    • Section 15.92: Weakly étale algebras over fields
    • Section 15.93: Local irreducibility
    • Section 15.94: Branches of the completion
    • Section 15.95: Formally catenary rings
    • Section 15.96: Group actions and integral closure
    • Section 15.97: Extensions of discrete valuation rings
    • Section 15.98: Galois extensions and ramification
    • Section 15.99: Krasner's lemma
    • Section 15.100: Abhyankar's lemma and tame ramification
    • Section 15.101: Eliminating ramification
    • Section 15.102: Picard groups of rings
    • Section 15.103: Extensions of valuation rings
    • Section 15.104: Structure of modules over a PID
    • Section 15.105: Principal radical ideals
  • Chapter 16: Smoothing Ring Maps
    • Section 16.1: Introduction
    • Section 16.2: Singular ideals
    • Section 16.3: Presentations of algebras
    • Section 16.4: Intermezzo: Néron desingularization
    • Section 16.5: The lifting problem
    • Section 16.6: The lifting lemma
    • Section 16.7: The desingularization lemma
    • Section 16.8: Warmup: reduction to a base field
    • Section 16.9: Local tricks
    • Section 16.10: Separable residue fields
    • Section 16.11: Inseparable residue fields
    • Section 16.12: The main theorem
    • Section 16.13: The approximation property for G-rings
    • Section 16.14: Approximation for henselian pairs
  • Chapter 17: Sheaves of Modules
    • Section 17.1: Introduction
    • Section 17.2: Pathology
    • Section 17.3: The abelian category of sheaves of modules
    • Section 17.4: Sections of sheaves of modules
    • Section 17.5: Supports of modules and sections
    • Section 17.6: Closed immersions and abelian sheaves
    • Section 17.7: A canonical exact sequence
    • Section 17.8: Modules locally generated by sections
    • Section 17.9: Modules of finite type
    • Section 17.10: Quasi-coherent modules
    • Section 17.11: Modules of finite presentation
    • Section 17.12: Coherent modules
    • Section 17.13: Closed immersions of ringed spaces
    • Section 17.14: Locally free sheaves
    • Section 17.15: Tensor product
    • Section 17.16: Flat modules
    • Section 17.17: Constructible sheaves of sets
    • Section 17.18: Flat morphisms of ringed spaces
    • Section 17.19: Symmetric and exterior powers
    • Section 17.20: Internal Hom
    • Section 17.21: Koszul complexes
    • Section 17.22: Invertible modules
    • Section 17.23: Rank and determinant
    • Section 17.24: Localizing sheaves of rings
    • Section 17.25: Modules of differentials
    • Section 17.26: The naive cotangent complex
  • Chapter 18: Modules on Sites
    • Section 18.1: Introduction
    • Section 18.2: Abelian presheaves
    • Section 18.3: Abelian sheaves
    • Section 18.4: Free abelian presheaves
    • Section 18.5: Free abelian sheaves
    • Section 18.6: Ringed sites
    • Section 18.7: Ringed topoi
    • Section 18.8: 2-morphisms of ringed topoi
    • Section 18.9: Presheaves of modules
    • Section 18.10: Sheaves of modules
    • Section 18.11: Sheafification of presheaves of modules
    • Section 18.12: Morphisms of topoi and sheaves of modules
    • Section 18.13: Morphisms of ringed topoi and modules
    • Section 18.14: The abelian category of sheaves of modules
    • Section 18.15: Exactness of pushforward
    • Section 18.16: Exactness of lower shriek
    • Section 18.17: Global types of modules
    • Section 18.18: Intrinsic properties of modules
    • Section 18.19: Localization of ringed sites
    • Section 18.20: Localization of morphisms of ringed sites
    • Section 18.21: Localization of ringed topoi
    • Section 18.22: Localization of morphisms of ringed topoi
    • Section 18.23: Local types of modules
    • Section 18.24: Basic results on local types of modules
    • Section 18.25: Closed immersions of ringed topoi
    • Section 18.26: Tensor product
    • Section 18.27: Internal Hom
    • Section 18.28: Flat modules
    • Section 18.29: Towards constructible modules
    • Section 18.30: Flat morphisms
    • Section 18.31: Invertible modules
    • Section 18.32: Modules of differentials
    • Section 18.33: Finite order differential operators
    • Section 18.34: The naive cotangent complex
    • Section 18.35: Stalks of modules
    • Section 18.36: Skyscraper sheaves
    • Section 18.37: Localization and points
    • Section 18.38: Pullbacks of flat modules
    • Section 18.39: Locally ringed topoi
    • Section 18.40: Lower shriek for modules
    • Section 18.41: Constant sheaves
    • Section 18.42: Locally constant sheaves
    • Section 18.43: Localizing sheaves of rings
  • Chapter 19: Injectives
    • Section 19.1: Introduction
    • Section 19.2: Baer's argument for modules
    • Section 19.3: G-modules
    • Section 19.4: Abelian sheaves on a space
    • Section 19.5: Sheaves of modules on a ringed space
    • Section 19.6: Abelian presheaves on a category
    • Section 19.7: Abelian Sheaves on a site
    • Section 19.8: Modules on a ringed site
    • Section 19.9: Embedding abelian categories
    • Section 19.10: Grothendieck's AB conditions
    • Section 19.11: Injectives in Grothendieck categories
    • Section 19.12: K-injectives in Grothendieck categories
    • Section 19.13: Additional remarks on Grothendieck abelian categories
  • Chapter 20: Cohomology of Sheaves
    • Section 20.1: Introduction
    • Section 20.2: Topics
    • Section 20.3: Cohomology of sheaves
    • Section 20.4: Derived functors
    • Section 20.5: First cohomology and torsors
    • Section 20.6: First cohomology and extensions
    • Section 20.7: First cohomology and invertible sheaves
    • Section 20.8: Locality of cohomology
    • Section 20.9: Mayer-Vietoris
    • Section 20.10: The Čech complex and Čech cohomology
    • Section 20.11: Čech cohomology as a functor on presheaves
    • Section 20.12: Čech cohomology and cohomology
    • Section 20.13: Flasque sheaves
    • Section 20.14: The Leray spectral sequence
    • Section 20.15: Functoriality of cohomology
    • Section 20.16: Refinements and Čech cohomology
    • Section 20.17: Cohomology on Hausdorff quasi-compact spaces
    • Section 20.18: The base change map
    • Section 20.19: Proper base change in topology
    • Section 20.20: Cohomology and colimits
    • Section 20.21: Vanishing on Noetherian topological spaces
    • Section 20.22: Cohomology with support in a closed
    • Section 20.23: Cohomology on spectral spaces
    • Section 20.24: The alternating Čech complex
    • Section 20.25: Alternative view of the Čech complex
    • Section 20.26: Čech cohomology of complexes
    • Section 20.27: Flat resolutions
    • Section 20.28: Derived pullback
    • Section 20.29: Cohomology of unbounded complexes
    • Section 20.30: Some properties of K-injective complexes
    • Section 20.31: Unbounded Mayer-Vietoris
    • Section 20.32: Derived limits
    • Section 20.33: Producing K-injective resolutions
    • Section 20.34: Čech cohomology of unbounded complexes
    • Section 20.35: Hom complexes
    • Section 20.36: Internal hom in the derived category
    • Section 20.37: Ext sheaves
    • Section 20.38: Global derived hom
    • Section 20.39: Glueing complexes
    • Section 20.40: Strictly perfect complexes
    • Section 20.41: Pseudo-coherent modules
    • Section 20.42: Tor dimension
    • Section 20.43: Perfect complexes
    • Section 20.44: Compact objects
    • Section 20.45: Projection formula
  • Chapter 21: Cohomology on Sites
    • Section 21.1: Introduction
    • Section 21.2: Topics
    • Section 21.3: Cohomology of sheaves
    • Section 21.4: Derived functors
    • Section 21.5: First cohomology and torsors
    • Section 21.6: First cohomology and extensions
    • Section 21.7: First cohomology and invertible sheaves
    • Section 21.8: Locality of cohomology
    • Section 21.9: The Čech complex and Čech cohomology
    • Section 21.10: Čech cohomology as a functor on presheaves
    • Section 21.11: Čech cohomology and cohomology
    • Section 21.12: Second cohomology and gerbes
    • Section 21.13: Cohomology of modules
    • Section 21.14: Limp sheaves
    • Section 21.15: The Leray spectral sequence
    • Section 21.16: The base change map
    • Section 21.17: Cohomology and colimits
    • Section 21.18: Flat resolutions
    • Section 21.19: Derived pullback
    • Section 21.20: Cohomology of unbounded complexes
    • Section 21.21: Some properties of K-injective complexes
    • Section 21.22: Localization and cohomology
    • Section 21.23: Derived and homotopy limits
    • Section 21.24: Producing K-injective resolutions
    • Section 21.25: Bounded cohomological dimension
    • Section 21.26: Mayer-Vietoris
    • Section 21.27: Comparing two topologies
    • Section 21.28: Formalities on cohomological descent
    • Section 21.29: Comparing two topologies, II
    • Section 21.30: Comparing cohomology
    • Section 21.31: Cohomology on Hausdorff and locally quasi-compact spaces
    • Section 21.32: Spectral sequences for Ext
    • Section 21.33: Hom complexes
    • Section 21.34: Internal hom in the derived category
    • Section 21.35: Global derived hom
    • Section 21.36: Derived lower shriek
    • Section 21.37: Derived lower shriek for fibred categories
    • Section 21.38: Homology on a category
    • Section 21.39: Calculating derived lower shriek
    • Section 21.40: Simplicial modules
    • Section 21.41: Cohomology on a category
    • Section 21.42: Strictly perfect complexes
    • Section 21.43: Pseudo-coherent modules
    • Section 21.44: Tor dimension
    • Section 21.45: Perfect complexes
    • Section 21.46: Projection formula
    • Section 21.47: Weakly contractible objects
    • Section 21.48: Compact objects
    • Section 21.49: Complexes with locally constant cohomology sheaves
  • Chapter 22: Differential Graded Algebra
    • Section 22.1: Introduction
    • Section 22.2: Conventions
    • Section 22.3: Differential graded algebras
    • Section 22.4: Differential graded modules
    • Section 22.5: The homotopy category
    • Section 22.6: Cones
    • Section 22.7: Admissible short exact sequences
    • Section 22.8: Distinguished triangles
    • Section 22.9: Cones and distinguished triangles
    • Section 22.10: The homotopy category is triangulated
    • Section 22.11: Projective modules over algebras
    • Section 22.12: Injective modules over algebras
    • Section 22.13: P-resolutions
    • Section 22.14: I-resolutions
    • Section 22.15: The derived category
    • Section 22.16: The canonical delta-functor
    • Section 22.17: Linear categories
    • Section 22.18: Graded categories
    • Section 22.19: Differential graded categories
    • Section 22.20: Obtaining triangulated categories
    • Section 22.21: Derived Hom
    • Section 22.22: Variant of derived Hom
    • Section 22.23: Tensor product
    • Section 22.24: Derived tensor product
    • Section 22.25: Composition of derived tensor products
    • Section 22.26: Variant of derived tensor product
    • Section 22.27: Characterizing compact objects
    • Section 22.28: Equivalences of derived categories
    • Section 22.29: Resolutions of differential graded algebras
  • Chapter 23: Divided Power Algebra
    • Section 23.1: Introduction
    • Section 23.2: Divided powers
    • Section 23.3: Divided power rings
    • Section 23.4: Extending divided powers
    • Section 23.5: Divided power polynomial algebras
    • Section 23.6: Tate resolutions
    • Section 23.7: Application to complete intersections
    • Section 23.8: Local complete intersection rings
    • Section 23.9: Local complete intersection maps
  • Chapter 24: Hypercoverings
    • Section 24.1: Introduction
    • Section 24.2: Semi-representable objects
    • Section 24.3: Hypercoverings
    • Section 24.4: Acyclicity
    • Section 24.5: Čech cohomology and hypercoverings
    • Section 24.6: Hypercoverings a la Verdier
    • Section 24.7: Covering hypercoverings
    • Section 24.8: Adding simplices
    • Section 24.9: Homotopies
    • Section 24.10: Cohomology and hypercoverings
    • Section 24.11: Hypercoverings of spaces
    • Section 24.12: Constructing hypercoverings