# 5 Topics in Geometry

• Chapter 79: Chow Groups of Spaces
• Section 79.1: Introduction
• Section 79.2: Setup
• Section 79.3: Cycles
• Section 79.4: Multiplicities
• Section 79.5: Cycle associated to a closed subspace
• Section 79.6: Cycle associated to a coherent sheaf
• Section 79.7: Preparation for proper pushforward
• Section 79.8: Proper pushforward
• Section 79.9: Preparation for flat pullback
• Section 79.10: Flat pullback
• Section 79.11: Push and pull
• Section 79.12: Preparation for principal divisors
• Section 79.13: Principal divisors
• Section 79.14: Principal divisors and pushforward
• Section 79.15: Rational equivalence
• Section 79.16: Rational equivalence and push and pull
• Section 79.17: The divisor associated to an invertible sheaf
• Section 79.18: Intersecting with an invertible sheaf
• Section 79.19: Intersecting with an invertible sheaf and push and pull
• Section 79.20: The key formula
• Section 79.21: Intersecting with an invertible sheaf and rational equivalence
• Section 79.22: Intersecting with effective Cartier divisors
• Section 79.23: Gysin homomorphisms
• Section 79.24: Relative effective Cartier divisors
• Section 79.25: Affine bundles
• Section 79.26: Bivariant intersection theory
• Section 79.27: Projective space bundle formula
• Section 79.28: The Chern classes of a vector bundle
• Section 79.29: Polynomial relations among chern classes
• Section 79.30: Additivity of chern classes
• Section 79.31: The splitting principle
• Section 79.32: Degrees of zero cycles
• Chapter 80: Quotients of Groupoids
• Section 80.1: Introduction
• Section 80.2: Conventions and notation
• Section 80.3: Invariant morphisms
• Section 80.4: Categorical quotients
• Section 80.5: Quotients as orbit spaces
• Section 80.6: Coarse quotients
• Section 80.7: Topological properties
• Section 80.8: Invariant functions
• Section 80.9: Good quotients
• Section 80.10: Geometric quotients
• Chapter 81: More on Cohomology of Spaces
• Section 81.1: Introduction
• Section 81.2: Conventions
• Section 81.3: Transporting results from schemes
• Section 81.4: Proper base change
• Section 81.5: Comparing big and small topoi
• Section 81.6: Comparing fppf and étale topologies
• Section 81.7: Comparing fppf and étale topologies: modules
• Section 81.8: Comparing ph and étale topologies
• Chapter 82: Simplicial Spaces
• Section 82.1: Introduction
• Section 82.2: Simplicial topological spaces
• Section 82.3: Simplicial sites and topoi
• Section 82.4: Augmentations of simplicial sites
• Section 82.5: Morphisms of simplicial sites
• Section 82.6: Ringed simplicial sites
• Section 82.7: Morphisms of ringed simplicial sites
• Section 82.8: Cohomology on simplicial sites
• Section 82.9: Cohomology and augmentations of simplicial sites
• Section 82.10: Cohomology on ringed simplicial sites
• Section 82.11: Cohomology and augmentations of ringed simplicial sites
• Section 82.12: Cartesian sheaves and modules
• Section 82.13: Simplicial systems of the derived category
• Section 82.14: Simplicial systems of the derived category: modules
• Section 82.15: The site associated to a semi-representable object
• Section 82.16: The site associate to a simplicial semi-representable object
• Section 82.17: Cohomological descent for hypercoverings
• Section 82.18: Cohomological descent for hypercoverings: modules
• Section 82.19: Cohomological descent for hypercoverings of an object
• Section 82.20: Cohomological descent for hypercoverings of an object: modules
• Section 82.21: Hypercovering by a simplicial object of the site
• Section 82.22: Hypercovering by a simplicial object of the site: modules
• Section 82.23: Unbounded cohomological descent for hypercoverings
• Section 82.24: Glueing complexes
• Section 82.25: Proper hypercoverings in topology
• Section 82.26: Simplicial schemes
• Section 82.27: Descent in terms of simplicial schemes
• Section 82.28: Quasi-coherent modules on simplicial schemes
• Section 82.29: Groupoids and simplicial schemes
• Section 82.30: Descent data give equivalence relations
• Section 82.31: An example case
• Section 82.32: Simplicial algebraic spaces
• Section 82.33: Fppf hypercoverings of algebraic spaces
• Section 82.34: Fppf hypercoverings of algebraic spaces: modules
• Section 82.35: Fppf descent of complexes
• Section 82.36: Proper hypercoverings of algebraic spaces
• Chapter 83: Duality for Spaces
• Section 83.1: Introduction
• Section 83.2: Dualizing complexes on algebraic spaces
• Section 83.3: Right adjoint of pushforward
• Section 83.4: Right adjoint of pushforward and base change, I
• Section 83.5: Right adjoint of pushforward and base change, II
• Section 83.6: Right adjoint of pushforward and trace maps
• Section 83.7: Right adjoint of pushforward and pullback
• Section 83.8: Right adjoint of pushforward for proper flat morphisms
• Section 83.9: Relative dualizing complexes for proper flat morphisms
• Section 83.10: Comparison with the case of schemes
• Chapter 84: Formal Algebraic Spaces
• Section 84.1: Introduction
• Section 84.2: Formal schemes à la EGA
• Section 84.3: Conventions and notation
• Section 84.4: Topological rings and modules
• Section 84.5: Affine formal algebraic spaces
• Section 84.6: Countably indexed affine formal algebraic spaces
• Section 84.7: Formal algebraic spaces
• Section 84.8: Colimits of algebraic spaces along thickenings
• Section 84.9: Completion along a closed subset
• Section 84.10: Fibre products
• Section 84.11: Separation axioms for formal algebraic spaces
• Section 84.12: Quasi-compact formal algebraic spaces
• Section 84.13: Quasi-compact and quasi-separated formal algebraic spaces
• Section 84.14: Morphisms representable by algebraic spaces
• Section 84.15: Types of formal algebraic spaces
• Section 84.16: Morphisms and continuous ring maps
• Section 84.18: Morphisms of finite type
• Section 84.19: Monomorphisms
• Section 84.20: Closed immersions
• Section 84.21: Restricted power series
• Section 84.22: Algebras topologically of finite type
• Section 84.23: Separation axioms for morphisms
• Section 84.24: Proper morphisms
• Section 84.25: Formal algebraic spaces and fpqc coverings
• Section 84.26: Maps out of affine formal schemes
• Section 84.27: The small étale site of a formal algebraic space
• Section 84.28: The structure sheaf
• Chapter 85: Restricted Power Series
• Section 85.1: Introduction
• Section 85.2: Two categories
• Section 85.3: A naive cotangent complex
• Section 85.4: Rig-étale homomorphisms
• Section 85.5: Rig-étale morphisms
• Section 85.6: Glueing rings along a principal ideal
• Section 85.7: Glueing rings along an ideal
• Section 85.8: In case the base ring is a G-ring
• Section 85.9: Rig-surjective morphisms
• Section 85.10: Algebraization
• Section 85.11: Application to modifications
• Chapter 86: Resolution of Surfaces Revisited
• Section 86.1: Introduction
• Section 86.2: Modifications
• Section 86.3: Strategy
• Section 86.4: Dominating by quadratic transformations
• Section 86.5: Dominating by normalized blowups
• Section 86.6: Base change to the completion
• Section 86.7: Implied properties
• Section 86.8: Resolution
• Section 86.9: Examples