# 1 Preliminaries

• Chapter 1: Introduction
• Section 1.1: Overview
• Chapter 2: Conventions
• 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 4.25: A criterion for representability
• Section 4.26: Categorically compact objects
• Section 4.27: Localization in categories
• Section 4.28: Formal properties
• Section 4.29: 2-categories
• Section 4.30: (2, 1)-categories
• Section 4.31: 2-fibre products
• Section 4.32: Categories over categories
• Section 4.33: Fibred categories
• Section 4.34: Inertia
• Section 4.35: Categories fibred in groupoids
• Section 4.36: Presheaves of categories
• Section 4.37: Presheaves of groupoids
• Section 4.38: Categories fibred in sets
• Section 4.39: Categories fibred in setoids
• Section 4.40: Representable categories fibred in groupoids
• Section 4.41: The 2-Yoneda lemma
• Section 4.42: Representable 1-morphisms
• Section 4.43: Monoidal categories
• 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: Characterizing finite and finitely presented modules
• Section 10.12: Tensor products
• Section 10.13: Tensor algebra
• Section 10.14: Base change
• Section 10.15: Miscellany
• Section 10.16: Cayley-Hamilton
• Section 10.17: The spectrum of a ring
• Section 10.18: Local rings
• Section 10.19: The Jacobson radical of a ring
• Section 10.20: Nakayama's lemma
• Section 10.21: Open and closed subsets of spectra
• Section 10.22: Connected components of spectra
• Section 10.23: Glueing properties
• Section 10.24: Glueing functions
• Section 10.25: Zerodivisors and total rings of fractions
• Section 10.26: Irreducible components of spectra
• Section 10.27: Examples of spectra of rings
• Section 10.28: A meta-observation about prime ideals
• Section 10.29: Images of ring maps of finite presentation
• Section 10.30: More on images
• Section 10.31: Noetherian rings
• Section 10.32: Locally nilpotent ideals
• Section 10.33: Curiosity
• Section 10.34: Hilbert Nullstellensatz
• Section 10.35: Jacobson rings
• Section 10.36: Finite and integral ring extensions
• Section 10.37: Normal rings
• Section 10.38: Going down for integral over normal
• Section 10.39: Flat modules and flat ring maps
• Section 10.40: Supports and annihilators
• Section 10.41: Going up and going down
• Section 10.42: Separable extensions
• Section 10.43: Geometrically reduced algebras
• Section 10.44: Separable extensions, continued
• Section 10.45: Perfect fields
• Section 10.46: Universal homeomorphisms
• Section 10.47: Geometrically irreducible algebras
• Section 10.48: Geometrically connected algebras
• Section 10.49: Geometrically integral algebras
• Section 10.50: Valuation rings
• Section 10.51: More Noetherian rings
• Section 10.52: Length
• Section 10.53: Artinian rings
• Section 10.54: Homomorphisms essentially of finite type
• Section 10.55: K-groups
• Section 10.57: Proj of a graded ring
• Section 10.58: Noetherian graded rings
• Section 10.59: Noetherian local rings
• Section 10.60: Dimension
• Section 10.61: Applications of dimension theory
• Section 10.62: Support and dimension of modules
• Section 10.63: Associated primes
• Section 10.64: Symbolic powers
• Section 10.65: Relative assassin
• Section 10.66: Weakly associated primes
• Section 10.67: Embedded primes
• Section 10.68: Regular sequences
• Section 10.69: Quasi-regular sequences
• Section 10.70: Blow up algebras
• Section 10.71: Ext groups
• Section 10.72: Depth
• Section 10.73: Functorialities for Ext
• Section 10.74: An application of Ext groups
• Section 10.75: Tor groups and flatness
• Section 10.76: Functorialities for Tor
• Section 10.77: Projective modules
• Section 10.78: Finite projective modules
• Section 10.79: Open loci defined by module maps
• Section 10.80: Faithfully flat descent for projectivity of modules
• Section 10.81: Characterizing flatness
• Section 10.82: Universally injective module maps
• Section 10.83: Descent for finite projective modules
• Section 10.84: Transfinite dévissage of modules
• Section 10.85: Projective modules over a local ring
• Section 10.86: Mittag-Leffler systems
• Section 10.87: Inverse systems
• Section 10.88: Mittag-Leffler modules
• Section 10.89: Interchanging direct products with tensor
• Section 10.90: Coherent rings
• Section 10.91: Examples and non-examples of Mittag-Leffler modules
• Section 10.92: Countably generated Mittag-Leffler modules
• Section 10.93: Characterizing projective modules
• Section 10.94: Ascending properties of modules
• Section 10.95: Descending properties of modules
• Section 10.96: Completion
• Section 10.97: Completion for Noetherian rings
• Section 10.98: Taking limits of modules
• Section 10.99: Criteria for flatness
• Section 10.100: Base change and flatness
• Section 10.101: Flatness criteria over Artinian rings
• Section 10.102: What makes a complex exact?
• Section 10.103: Cohen-Macaulay modules
• Section 10.104: Cohen-Macaulay rings
• Section 10.105: Catenary rings
• Section 10.106: Regular local rings
• Section 10.107: Epimorphisms of rings
• Section 10.108: Pure ideals
• Section 10.109: Rings of finite global dimension
• Section 10.110: Regular rings and global dimension
• Section 10.111: Auslander-Buchsbaum
• Section 10.112: Homomorphisms and dimension
• Section 10.113: The dimension formula
• Section 10.114: Dimension of finite type algebras over fields
• Section 10.115: Noether normalization
• Section 10.116: Dimension of finite type algebras over fields, reprise
• Section 10.117: Dimension of graded algebras over a field
• Section 10.118: Generic flatness
• Section 10.119: Around Krull-Akizuki
• Section 10.120: Factorization
• Section 10.121: Orders of vanishing
• Section 10.122: Quasi-finite maps
• Section 10.123: Zariski's Main Theorem
• Section 10.124: Applications of Zariski's Main Theorem
• Section 10.125: Dimension of fibres
• Section 10.126: Algebras and modules of finite presentation
• Section 10.127: Colimits and maps of finite presentation
• Section 10.128: More flatness criteria
• Section 10.129: Openness of the flat locus
• Section 10.130: Openness of Cohen-Macaulay loci
• Section 10.131: Differentials
• Section 10.132: The de Rham complex
• Section 10.133: Finite order differential operators
• Section 10.134: The naive cotangent complex
• Section 10.135: Local complete intersections
• Section 10.136: Syntomic morphisms
• Section 10.137: Smooth ring maps
• Section 10.138: Formally smooth maps
• Section 10.139: Smoothness and differentials
• Section 10.140: Smooth algebras over fields
• Section 10.141: Smooth ring maps in the Noetherian case
• Section 10.142: Overview of results on smooth ring maps
• Section 10.143: Étale ring maps
• Section 10.144: Local structure of étale ring maps
• Section 10.145: Étale local structure of quasi-finite ring maps
• Section 10.146: Local homomorphisms
• Section 10.147: Integral closure and smooth base change
• Section 10.148: Formally unramified maps
• Section 10.149: Conormal modules and universal thickenings
• Section 10.150: Formally étale maps
• Section 10.151: Unramified ring maps
• Section 10.152: Local structure of unramified ring maps
• Section 10.153: Henselian local rings
• Section 10.154: Filtered colimits of étale ring maps
• Section 10.155: Henselization and strict henselization
• Section 10.156: Henselization and quasi-finite ring maps
• Section 10.157: Serre's criterion for normality
• Section 10.158: Formal smoothness of fields
• Section 10.159: Constructing flat ring maps
• Section 10.160: The Cohen structure theorem
• Section 10.161: Japanese rings
• Section 10.162: Nagata rings
• Section 10.163: Ascending properties
• Section 10.164: Descending properties
• Section 10.165: Geometrically normal algebras
• Section 10.166: Geometrically regular algebras
• Section 10.167: Geometrically Cohen-Macaulay algebras
• Section 10.168: 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 12.4: Karoubian categories
• Section 12.5: Abelian categories
• Section 12.6: Extensions
• Section 12.8: Localization
• Section 12.9: Jordan-Hölder
• Section 12.10: Serre subcategories
• Section 12.11: K-groups
• Section 12.12: Cohomological delta-functors
• Section 12.13: Complexes
• Section 12.14: Homotopy and the shift functor
• Section 12.15: Truncation of complexes
• Section 12.17: Additive monoidal categories
• Section 12.18: Double complexes and associated total complexes
• Section 12.19: Filtrations
• Section 12.20: Spectral sequences
• Section 12.21: Spectral sequences: exact couples
• Section 12.22: Spectral sequences: differential objects
• Section 12.23: Spectral sequences: filtered differential objects
• Section 12.24: Spectral sequences: filtered complexes
• Section 12.25: Spectral sequences: double complexes
• Section 12.26: Double complexes of abelian groups
• Section 12.27: Injectives
• Section 12.28: Projectives
• Section 12.29: Injectives and adjoint functors
• Section 12.30: Essentially constant systems
• Section 12.31: Inverse systems
• Section 12.32: 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: Filtered derived categories
• Section 13.14: Derived functors in general
• Section 13.15: Derived functors on derived categories
• Section 13.16: Higher derived functors
• Section 13.17: Triangulated subcategories of the derived category
• 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: K-groups
• Section 13.29: Unbounded complexes
• Section 13.31: K-injective complexes
• Section 13.32: Bounded cohomological dimension
• Section 13.33: Derived colimits
• Section 13.34: Derived limits
• Section 13.35: Operations on full subcategories
• Section 13.36: Generators of triangulated categories
• Section 13.37: Compact objects
• Section 13.38: Brown representability
• Section 13.39: Brown representability, bis
• Section 13.41: Postnikov systems
• Section 13.42: Essentially constant 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: Preparation for standard resolutions
• Section 14.34: Standard resolutions
• Chapter 15: More on Algebra
• Section 15.1: Introduction
• 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: The extended alternating Čech complex
• Section 15.30: Koszul regular sequences
• Section 15.31: More on Koszul regular sequences
• Section 15.32: Regular ideals
• Section 15.33: Local complete intersection maps
• Section 15.34: Cartier's equality and geometric regularity
• Section 15.35: Geometric regularity
• Section 15.36: Topological rings and modules
• Section 15.37: Formally smooth maps of topological rings
• Section 15.38: Formally smooth maps of local rings
• Section 15.39: Some results on power series rings
• Section 15.40: Geometric regularity and formal smoothness
• Section 15.41: Regular ring maps
• Section 15.42: Ascending properties along regular ring maps
• Section 15.43: Permanence of properties under completion
• Section 15.44: Permanence of properties under étale maps
• Section 15.45: Permanence of properties under henselization
• Section 15.46: Field extensions, revisited
• Section 15.47: The singular locus
• Section 15.48: Regularity and derivations
• Section 15.49: Formal smoothness and regularity
• Section 15.50: G-rings
• Section 15.51: Properties of formal fibres
• Section 15.52: Excellent rings
• Section 15.53: Abelian categories of modules
• Section 15.54: Injective abelian groups
• Section 15.55: Injective modules
• Section 15.56: Derived categories of modules
• Section 15.57: Computing Tor
• Section 15.58: Tensor products of complexes
• Section 15.59: Derived tensor product
• Section 15.60: Derived change of rings
• Section 15.61: Tor independence
• Section 15.62: Spectral sequences for Tor
• Section 15.63: Products and Tor
• Section 15.64: Pseudo-coherent modules, I
• Section 15.65: Pseudo-coherent modules, II
• Section 15.66: Tor dimension
• Section 15.67: Spectral sequences for Ext
• Section 15.68: Projective dimension
• Section 15.69: Injective dimension
• Section 15.70: Modules which are close to being projective
• Section 15.71: Hom complexes
• Section 15.72: Sign rules
• Section 15.73: Derived hom
• Section 15.74: Perfect complexes
• Section 15.75: Lifting complexes
• Section 15.76: Splitting complexes
• Section 15.77: Recognizing perfect complexes
• Section 15.78: Characterizing perfect complexes
• Section 15.79: Strong generators and regular rings
• Section 15.80: Relatively finitely presented modules
• Section 15.81: Relatively pseudo-coherent modules
• Section 15.82: Pseudo-coherent and perfect ring maps
• Section 15.83: Relatively perfect modules
• Section 15.84: Two term complexes
• Section 15.85: The naive cotangent complex
• Section 15.86: Rlim of abelian groups
• Section 15.87: Rlim of modules
• Section 15.88: Torsion modules
• Section 15.89: Formal glueing of module categories
• Section 15.90: The Beauville-Laszlo theorem
• Section 15.91: Derived Completion
• Section 15.92: The category of derived complete modules
• Section 15.93: Derived completion for a principal ideal
• Section 15.94: Derived completion for Noetherian rings
• Section 15.95: An operator introduced by Berthelot and Ogus
• Section 15.96: Perfect complexes and the eta operator
• Section 15.97: Taking limits of complexes
• Section 15.98: Some evaluation maps
• Section 15.99: Base change for derived hom
• Section 15.100: Systems of modules
• Section 15.101: Systems of modules, bis
• Section 15.102: Miscellany
• Section 15.103: Tricks with double complexes
• Section 15.104: Weakly étale ring maps
• Section 15.105: Weakly étale algebras over fields
• Section 15.106: Local irreducibility
• Section 15.107: Miscellaneous on branches
• Section 15.108: Branches of the completion
• Section 15.109: Formally catenary rings
• Section 15.110: Group actions and integral closure
• Section 15.111: Extensions of discrete valuation rings
• Section 15.112: Galois extensions and ramification
• Section 15.113: Krasner's lemma
• Section 15.114: Abhyankar's lemma and tame ramification
• Section 15.115: Eliminating ramification
• Section 15.116: Eliminating ramification, II
• Section 15.117: Picard groups of rings
• Section 15.118: Determinants
• Section 15.119: Perfect complexes and K-groups
• Section 15.120: Determinants of endomorphisms of finite length modules
• Section 15.121: A regular local ring is a UFD
• Section 15.122: Determinants of complexes
• Section 15.123: Extensions of valuation rings
• Section 15.124: Structure of modules over a PID
• Section 15.125: Principal radical ideals
• Section 15.126: Invertible objects in the derived category
• Section 15.127: Splitting off a free module
• Section 15.128: Big projective modules are free
• 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: Bilinear maps
• Section 17.16: Tensor product
• Section 17.17: Flat modules
• Section 17.18: Duals
• Section 17.19: Constructible sheaves of sets
• Section 17.20: Flat morphisms of ringed spaces
• Section 17.21: Symmetric and exterior powers
• Section 17.22: Internal Hom
• Section 17.23: Koszul complexes
• Section 17.24: Invertible modules
• Section 17.25: Rank and determinant
• Section 17.26: Localizing sheaves of rings
• Section 17.27: Modules of differentials
• Section 17.28: Finite order differential operators
• Section 17.29: The de Rham complex
• Section 17.30: 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: Duals
• Section 18.30: Towards constructible modules
• Section 18.31: Flat morphisms
• Section 18.32: Invertible modules
• Section 18.33: Modules of differentials
• Section 18.34: Finite order differential operators
• Section 18.35: The naive cotangent complex
• Section 18.36: Stalks of modules
• Section 18.37: Skyscraper sheaves
• Section 18.38: Localization and points
• Section 18.39: Pullbacks of flat modules
• Section 18.40: Locally ringed topoi
• Section 18.41: Lower shriek for modules
• Section 18.42: Constant sheaves
• Section 18.43: Locally constant sheaves
• Section 18.44: Localizing sheaves of rings
• Section 18.45: Sheaves of pointed sets
• 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
• Section 19.14: The Gabriel-Popescu theorem
• Section 19.15: Brown representability and Grothendieck abelian categories
• Chapter 20: Cohomology of Sheaves
• Section 20.1: Introduction
• Section 20.2: Cohomology of sheaves
• Section 20.3: Derived functors
• Section 20.4: First cohomology and torsors
• Section 20.5: First cohomology and extensions
• Section 20.6: First cohomology and invertible sheaves
• Section 20.7: Locality of cohomology
• Section 20.8: Mayer-Vietoris
• Section 20.9: The Čech complex and Čech cohomology
• Section 20.10: Čech cohomology as a functor on presheaves
• Section 20.11: Čech cohomology and cohomology
• Section 20.12: Flasque sheaves
• Section 20.13: The Leray spectral sequence
• Section 20.14: Functoriality of cohomology
• Section 20.15: Refinements and Čech cohomology
• Section 20.16: Cohomology on Hausdorff quasi-compact spaces
• Section 20.17: The base change map
• Section 20.18: Proper base change in topology
• Section 20.19: Cohomology and colimits
• Section 20.20: Vanishing on Noetherian topological spaces
• Section 20.21: Cohomology with support in a closed
• Section 20.22: Cohomology on spectral spaces
• Section 20.23: The alternating Čech complex
• Section 20.24: Alternative view of the Čech complex
• Section 20.25: Čech cohomology of complexes
• Section 20.26: Flat resolutions
• Section 20.27: Derived pullback
• Section 20.28: Cohomology of unbounded complexes
• Section 20.29: Cohomology of filtered complexes
• Section 20.30: Godement resolution
• Section 20.31: Cup product
• Section 20.32: Some properties of K-injective complexes
• Section 20.33: Unbounded Mayer-Vietoris
• Section 20.34: Cohomology with support in a closed, II
• Section 20.35: Inverse systems and cohomology
• Section 20.36: Derived limits
• Section 20.37: Producing K-injective resolutions
• Section 20.38: Čech cohomology of unbounded complexes
• Section 20.39: Hom complexes
• Section 20.40: Internal hom in the derived category
• Section 20.41: Ext sheaves
• Section 20.42: Global derived hom
• Section 20.43: Glueing complexes
• Section 20.44: Strictly perfect complexes
• Section 20.45: Pseudo-coherent modules
• Section 20.46: Tor dimension
• Section 20.47: Perfect complexes
• Section 20.48: Duals
• Section 20.49: Miscellany
• Section 20.50: Invertible objects in the derived category
• Section 20.51: Compact objects
• Section 20.52: Projection formula
• Section 20.53: An operator introduced by Berthelot and Ogus
• Chapter 21: Cohomology on Sites
• Section 21.1: Introduction
• Section 21.2: Cohomology of sheaves
• Section 21.3: Derived functors
• Section 21.4: First cohomology and torsors
• Section 21.5: First cohomology and extensions
• Section 21.6: First cohomology and invertible sheaves
• Section 21.7: Locality of cohomology
• Section 21.8: The Čech complex and Čech cohomology
• Section 21.9: Čech cohomology as a functor on presheaves
• Section 21.10: Čech cohomology and cohomology
• Section 21.11: Second cohomology and gerbes
• Section 21.12: Cohomology of modules
• Section 21.13: Totally acyclic sheaves
• Section 21.14: The Leray spectral sequence
• Section 21.15: The base change map
• Section 21.16: Cohomology and colimits
• Section 21.17: Flat resolutions
• Section 21.18: Derived pullback
• Section 21.19: Cohomology of unbounded complexes
• Section 21.20: Some properties of K-injective complexes
• Section 21.21: Localization and cohomology
• Section 21.22: Inverse systems 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: Cup product
• Section 21.34: Hom complexes
• Section 21.35: Internal hom in the derived category
• Section 21.36: Global derived hom
• Section 21.37: Derived lower shriek
• Section 21.38: Derived lower shriek for fibred categories
• Section 21.39: Homology on a category
• Section 21.40: Calculating derived lower shriek
• Section 21.41: Simplicial modules
• Section 21.42: Cohomology on a category
• Section 21.43: Modules on a category
• Section 21.44: Strictly perfect complexes
• Section 21.45: Pseudo-coherent modules
• Section 21.46: Tor dimension
• Section 21.47: Perfect complexes
• Section 21.48: Duals
• Section 21.49: Invertible objects in the derived category
• Section 21.50: Projection formula
• Section 21.51: Weakly contractible objects
• Section 21.52: Compact objects
• Section 21.53: 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: Left modules
• Section 22.12: Tensor product
• Section 22.13: Hom complexes and differential graded modules
• Section 22.14: Projective modules over algebras
• Section 22.15: Projective modules over graded algebras
• Section 22.16: Projective modules and differential graded algebras
• Section 22.17: Injective modules over algebras
• Section 22.18: Injective modules over graded algebras
• Section 22.19: Injective modules and differential graded algebras
• Section 22.20: P-resolutions
• Section 22.21: I-resolutions
• Section 22.22: The derived category
• Section 22.23: The canonical delta-functor
• Section 22.24: Linear categories
• Section 22.26: Differential graded categories
• Section 22.27: Obtaining triangulated categories
• Section 22.28: Bimodules
• Section 22.29: Bimodules and tensor product
• Section 22.30: Bimodules and internal hom
• Section 22.31: Derived Hom
• Section 22.32: Variant of derived Hom
• Section 22.33: Derived tensor product
• Section 22.34: Composition of derived tensor products
• Section 22.35: Variant of derived tensor product
• Section 22.36: Characterizing compact objects
• Section 22.37: Equivalences of derived categories
• Section 22.38: 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
• Section 23.10: Smooth ring maps and diagonals
• Section 23.11: Freeness of the conormal module
• Section 23.12: Koszul complexes and Tate resolutions
• Chapter 24: Differential Graded Sheaves
• Section 24.1: Introduction
• Section 24.2: Conventions
• Section 24.3: Sheaves of graded algebras
• Section 24.4: Sheaves of graded modules
• Section 24.6: Tensor product for sheaves of graded modules
• Section 24.7: Internal hom for sheaves of graded modules
• Section 24.9: Pull and push for sheaves of graded modules
• Section 24.10: Localization and sheaves of graded modules
• Section 24.11: Shift functors on sheaves of graded modules
• Section 24.12: Sheaves of differential graded algebras
• Section 24.13: Sheaves of differential graded modules
• Section 24.14: The differential graded category of modules
• Section 24.15: Tensor product for sheaves of differential graded modules
• Section 24.16: Internal hom for sheaves of differential graded modules
• Section 24.18: Pull and push for sheaves of differential graded modules
• Section 24.19: Localization and sheaves of differential graded modules
• Section 24.20: Shift functors on sheaves of differential graded modules
• Section 24.21: The homotopy category
• Section 24.22: Cones and triangles
• Section 24.23: Flat resolutions
• Section 24.25: K-injective differential graded modules
• Section 24.26: The derived category
• Section 24.27: The canonical delta-functor
• Section 24.28: Derived pullback
• Section 24.29: Derived pushforward
• Section 24.30: Equivalences of derived categories
• Section 24.31: Resolutions of differential graded algebras
• Section 24.32: Miscellany
• Section 24.33: Differential graded modules on a category
• Section 24.34: Differential graded modules on a category, bis
• Section 24.35: Inverse systems of differential graded algebras
• Chapter 25: Hypercoverings
• Section 25.1: Introduction
• Section 25.2: Semi-representable objects
• Section 25.3: Hypercoverings
• Section 25.4: Acyclicity
• Section 25.5: Čech cohomology and hypercoverings
• Section 25.6: Hypercoverings a la Verdier
• Section 25.7: Covering hypercoverings