5 Topics in Geometry

Chapter 80: Chow Groups of Spaces
 Section 80.1: Introduction
 Section 80.2: Setup
 Section 80.3: Cycles
 Section 80.4: Multiplicities
 Section 80.5: Cycle associated to a closed subspace
 Section 80.6: Cycle associated to a coherent sheaf
 Section 80.7: Preparation for proper pushforward
 Section 80.8: Proper pushforward
 Section 80.9: Preparation for flat pullback
 Section 80.10: Flat pullback
 Section 80.11: Push and pull
 Section 80.12: Preparation for principal divisors
 Section 80.13: Principal divisors
 Section 80.14: Principal divisors and pushforward
 Section 80.15: Rational equivalence
 Section 80.16: Rational equivalence and push and pull
 Section 80.17: The divisor associated to an invertible sheaf
 Section 80.18: Intersecting with an invertible sheaf
 Section 80.19: Intersecting with an invertible sheaf and push and pull
 Section 80.20: The key formula
 Section 80.21: Intersecting with an invertible sheaf and rational equivalence
 Section 80.22: Intersecting with effective Cartier divisors
 Section 80.23: Gysin homomorphisms
 Section 80.24: Relative effective Cartier divisors
 Section 80.25: Affine bundles
 Section 80.26: Bivariant intersection theory
 Section 80.27: Projective space bundle formula
 Section 80.28: The Chern classes of a vector bundle
 Section 80.29: Polynomial relations among chern classes
 Section 80.30: Additivity of chern classes
 Section 80.31: The splitting principle
 Section 80.32: Degrees of zero cycles

Chapter 81: Quotients of Groupoids
 Section 81.1: Introduction
 Section 81.2: Conventions and notation
 Section 81.3: Invariant morphisms
 Section 81.4: Categorical quotients
 Section 81.5: Quotients as orbit spaces
 Section 81.6: Coarse quotients
 Section 81.7: Topological properties
 Section 81.8: Invariant functions
 Section 81.9: Good quotients
 Section 81.10: Geometric quotients

Chapter 82: More on Cohomology of Spaces
 Section 82.1: Introduction
 Section 82.2: Conventions
 Section 82.3: Transporting results from schemes
 Section 82.4: Proper base change
 Section 82.5: Comparing big and small topoi
 Section 82.6: Comparing fppf and étale topologies
 Section 82.7: Comparing fppf and étale topologies: modules
 Section 82.8: Comparing ph and étale topologies

Chapter 83: Simplicial Spaces
 Section 83.1: Introduction
 Section 83.2: Simplicial topological spaces
 Section 83.3: Simplicial sites and topoi
 Section 83.4: Augmentations of simplicial sites
 Section 83.5: Morphisms of simplicial sites
 Section 83.6: Ringed simplicial sites
 Section 83.7: Morphisms of ringed simplicial sites
 Section 83.8: Cohomology on simplicial sites
 Section 83.9: Cohomology and augmentations of simplicial sites
 Section 83.10: Cohomology on ringed simplicial sites
 Section 83.11: Cohomology and augmentations of ringed simplicial sites
 Section 83.12: Cartesian sheaves and modules
 Section 83.13: Simplicial systems of the derived category
 Section 83.14: Simplicial systems of the derived category: modules
 Section 83.15: The site associated to a semirepresentable object
 Section 83.16: The site associate to a simplicial semirepresentable object
 Section 83.17: Cohomological descent for hypercoverings
 Section 83.18: Cohomological descent for hypercoverings: modules
 Section 83.19: Cohomological descent for hypercoverings of an object
 Section 83.20: Cohomological descent for hypercoverings of an object: modules
 Section 83.21: Hypercovering by a simplicial object of the site
 Section 83.22: Hypercovering by a simplicial object of the site: modules
 Section 83.23: Unbounded cohomological descent for hypercoverings
 Section 83.24: Glueing complexes
 Section 83.25: Proper hypercoverings in topology
 Section 83.26: Simplicial schemes
 Section 83.27: Descent in terms of simplicial schemes
 Section 83.28: Quasicoherent modules on simplicial schemes
 Section 83.29: Groupoids and simplicial schemes
 Section 83.30: Descent data give equivalence relations
 Section 83.31: An example case
 Section 83.32: Simplicial algebraic spaces
 Section 83.33: Fppf hypercoverings of algebraic spaces
 Section 83.34: Fppf hypercoverings of algebraic spaces: modules
 Section 83.35: Fppf descent of complexes
 Section 83.36: Proper hypercoverings of algebraic spaces

Chapter 84: Duality for Spaces
 Section 84.1: Introduction
 Section 84.2: Dualizing complexes on algebraic spaces
 Section 84.3: Right adjoint of pushforward
 Section 84.4: Right adjoint of pushforward and base change, I
 Section 84.5: Right adjoint of pushforward and base change, II
 Section 84.6: Right adjoint of pushforward and trace maps
 Section 84.7: Right adjoint of pushforward and pullback
 Section 84.8: Right adjoint of pushforward for proper flat morphisms
 Section 84.9: Relative dualizing complexes for proper flat morphisms
 Section 84.10: Comparison with the case of schemes

Chapter 85: Formal Algebraic Spaces
 Section 85.1: Introduction
 Section 85.2: Formal schemes à la EGA
 Section 85.3: Conventions and notation
 Section 85.4: Topological rings and modules
 Section 85.5: Affine formal algebraic spaces
 Section 85.6: Countably indexed affine formal algebraic spaces
 Section 85.7: Formal algebraic spaces
 Section 85.8: Colimits of algebraic spaces along thickenings
 Section 85.9: Completion along a closed subset
 Section 85.10: Fibre products
 Section 85.11: Separation axioms for formal algebraic spaces
 Section 85.12: Quasicompact formal algebraic spaces
 Section 85.13: Quasicompact and quasiseparated formal algebraic spaces
 Section 85.14: Morphisms representable by algebraic spaces
 Section 85.15: Types of formal algebraic spaces
 Section 85.16: Morphisms and continuous ring maps
 Section 85.17: Adic morphisms
 Section 85.18: Morphisms of finite type
 Section 85.19: Monomorphisms
 Section 85.20: Closed immersions
 Section 85.21: Restricted power series
 Section 85.22: Algebras topologically of finite type
 Section 85.23: Separation axioms for morphisms
 Section 85.24: Proper morphisms
 Section 85.25: Formal algebraic spaces and fpqc coverings
 Section 85.26: Maps out of affine formal schemes
 Section 85.27: The small étale site of a formal algebraic space
 Section 85.28: The structure sheaf

Chapter 86: Restricted Power Series
 Section 86.1: Introduction
 Section 86.2: Two categories
 Section 86.3: A naive cotangent complex
 Section 86.4: Rigétale homomorphisms
 Section 86.5: Rigétale morphisms
 Section 86.6: Glueing rings along a principal ideal
 Section 86.7: Glueing rings along an ideal
 Section 86.8: In case the base ring is a Gring
 Section 86.9: Rigsurjective morphisms
 Section 86.10: Algebraization
 Section 86.11: Application to modifications

Chapter 87: Resolution of Surfaces Revisited
 Section 87.1: Introduction
 Section 87.2: Modifications
 Section 87.3: Strategy
 Section 87.4: Dominating by quadratic transformations
 Section 87.5: Dominating by normalized blowups
 Section 87.6: Base change to the completion
 Section 87.7: Implied properties
 Section 87.8: Resolution
 Section 87.9: Examples