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: Categorically compact objects

Section 4.27: Localization in categories

Section 4.28: Formal properties

Section 4.29: 2categories

Section 4.30: (2, 1)categories

Section 4.31: 2fibre 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 2Yoneda lemma

Section 4.42: Representable 1morphisms

Section 4.43: Monoidal categories

Section 4.44: Categories of dotted arrows

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: Quasicompact spaces and maps

Section 5.13: Locally quasicompact 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: StoneČech 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 Gsets

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: Gsets and morphisms

Section 7.17: Quasicompact 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: 2morphisms 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: ArtinSchreier 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: CayleyHamilton

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 metaobservation 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: Kgroups

Section 10.56: Graded rings

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: Quasiregular 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: MittagLeffler systems

Section 10.87: Inverse systems

Section 10.88: MittagLeffler modules

Section 10.89: Interchanging direct products with tensor

Section 10.90: Coherent rings

Section 10.91: Examples and nonexamples of MittagLeffler modules

Section 10.92: Countably generated MittagLeffler 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: CohenMacaulay modules

Section 10.104: CohenMacaulay 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: AuslanderBuchsbaum

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 KrullAkizuki

Section 10.120: Factorization

Section 10.121: Orders of vanishing

Section 10.122: Quasifinite 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 CohenMacaulay 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 quasifinite 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 quasifinite 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 CohenMacaulay 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: SkolemNoether

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: JordanHölder

Section 12.10: Serre subcategories

Section 12.11: Kgroups

Section 12.12: Cohomological deltafunctors

Section 12.13: Complexes

Section 12.14: Homotopy and the shift functor

Section 12.15: Truncation of complexes

Section 12.16: Graded objects

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 deltafunctor

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: CartanEilenberg 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: Kgroups

Section 13.29: Unbounded complexes

Section 13.30: Deriving adjoints

Section 13.31: Kinjective 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 13.40: Admissible subcategories

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: DoldKan

Section 14.25: DoldKan 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 15.2: Advice for the reader

Section 15.3: Stably free modules

Section 15.4: A comment on the ArtinRees 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: Autoassociated 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: Grings

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: Pseudocoherent modules, I

Section 15.65: Pseudocoherent 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 pseudocoherent modules

Section 15.82: Pseudocoherent 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 BeauvilleLaszlo 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 Kgroups

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 Grings

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: Quasicoherent 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: The annihilator of a sheaf of modules

Section 17.24: Koszul complexes

Section 17.25: Invertible modules

Section 17.26: Rank and determinant

Section 17.27: Localizing sheaves of rings

Section 17.28: Modules of differentials

Section 17.29: Finite order differential operators

Section 17.30: The de Rham complex

Section 17.31: 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: 2morphisms 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: Gmodules

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: Kinjectives in Grothendieck categories

Section 19.13: Additional remarks on Grothendieck abelian categories

Section 19.14: The GabrielPopescu 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: MayerVietoris

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 quasicompact 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 subset

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 Kinjective complexes

Section 20.33: Unbounded MayerVietoris

Section 20.34: Cohomology with support in a closed subset, II

Section 20.35: Inverse systems and cohomology, I

Section 20.36: Inverse systems and cohomology, II

Section 20.37: Derived limits

Section 20.38: Producing Kinjective resolutions

Section 20.39: Inverse systems and cohomology, III

Section 20.40: Čech cohomology of unbounded complexes

Section 20.41: Hom complexes

Section 20.42: Internal hom in the derived category

Section 20.43: Ext sheaves

Section 20.44: Global derived hom

Section 20.45: Glueing complexes

Section 20.46: Strictly perfect complexes

Section 20.47: Pseudocoherent modules

Section 20.48: Tor dimension

Section 20.49: Perfect complexes

Section 20.50: Duals

Section 20.51: Miscellany

Section 20.52: Invertible objects in the derived category

Section 20.53: Compact objects

Section 20.54: Projection formula

Section 20.55: 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 Kinjective 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 Kinjective resolutions

Section 21.25: Bounded cohomological dimension

Section 21.26: MayerVietoris

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 quasicompact 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: Pseudocoherent 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: Presolutions

Section 22.21: Iresolutions

Section 22.22: The derived category

Section 22.23: The canonical deltafunctor

Section 22.24: Linear categories

Section 22.25: Graded 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 24.5: The graded category of 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 24.8: Sheaves of graded bimodules and tensorhom adjunction

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 24.17: Sheaves of differential graded bimodules and tensorhom adjunction

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 24.24: The differential graded hull of a graded module

Section 24.25: Kinjective differential graded modules

Section 24.26: The derived category

Section 24.27: The canonical deltafunctor

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: Semirepresentable 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

Section 25.8: Adding simplices

Section 25.9: Homotopies

Section 25.10: Cohomology and hypercoverings

Section 25.11: Hypercoverings of spaces

Section 25.12: Constructing hypercoverings