# 5 Topics in Geometry

• Chapter 82: Chow Groups of Spaces
• Section 82.1: Introduction
• Section 82.2: Setup
• Section 82.3: Cycles
• Section 82.4: Multiplicities
• Section 82.5: Cycle associated to a closed subspace
• Section 82.6: Cycle associated to a coherent sheaf
• Section 82.7: Preparation for proper pushforward
• Section 82.8: Proper pushforward
• Section 82.9: Preparation for flat pullback
• Section 82.10: Flat pullback
• Section 82.11: Push and pull
• Section 82.12: Preparation for principal divisors
• Section 82.13: Principal divisors
• Section 82.14: Principal divisors and pushforward
• Section 82.15: Rational equivalence
• Section 82.16: Rational equivalence and push and pull
• Section 82.17: The divisor associated to an invertible sheaf
• Section 82.18: Intersecting with an invertible sheaf
• Section 82.19: Intersecting with an invertible sheaf and push and pull
• Section 82.20: The key formula
• Section 82.21: Intersecting with an invertible sheaf and rational equivalence
• Section 82.22: Intersecting with effective Cartier divisors
• Section 82.23: Gysin homomorphisms
• Section 82.24: Relative effective Cartier divisors
• Section 82.25: Affine bundles
• Section 82.26: Bivariant intersection theory
• Section 82.27: Projective space bundle formula
• Section 82.28: The Chern classes of a vector bundle
• Section 82.29: Polynomial relations among Chern classes
• Section 82.30: Additivity of Chern classes
• Section 82.31: The splitting principle
• Section 82.32: Degrees of zero cycles
• Chapter 83: Quotients of Groupoids
• Section 83.1: Introduction
• Section 83.2: Conventions and notation
• Section 83.3: Invariant morphisms
• Section 83.4: Categorical quotients
• Section 83.5: Quotients as orbit spaces
• Section 83.6: Coarse quotients
• Section 83.7: Topological properties
• Section 83.8: Invariant functions
• Section 83.9: Good quotients
• Section 83.10: Geometric quotients
• Chapter 84: More on Cohomology of Spaces
• Section 84.1: Introduction
• Section 84.2: Conventions
• Section 84.3: Transporting results from schemes
• Section 84.4: Proper base change
• Section 84.5: Comparing big and small topoi
• Section 84.6: Comparing fppf and étale topologies
• Section 84.7: Comparing fppf and étale topologies: modules
• Section 84.8: Comparing ph and étale topologies
• Chapter 85: Simplicial Spaces
• Section 85.1: Introduction
• Section 85.2: Simplicial topological spaces
• Section 85.3: Simplicial sites and topoi
• Section 85.4: Augmentations of simplicial sites
• Section 85.5: Morphisms of simplicial sites
• Section 85.6: Ringed simplicial sites
• Section 85.7: Morphisms of ringed simplicial sites
• Section 85.8: Cohomology on simplicial sites
• Section 85.9: Cohomology and augmentations of simplicial sites
• Section 85.10: Cohomology on ringed simplicial sites
• Section 85.11: Cohomology and augmentations of ringed simplicial sites
• Section 85.12: Cartesian sheaves and modules
• Section 85.13: Simplicial systems of the derived category
• Section 85.14: Simplicial systems of the derived category: modules
• Section 85.15: The site associated to a semi-representable object
• Section 85.16: The site associate to a simplicial semi-representable object
• Section 85.17: Cohomological descent for hypercoverings
• Section 85.18: Cohomological descent for hypercoverings: modules
• Section 85.19: Cohomological descent for hypercoverings of an object
• Section 85.20: Cohomological descent for hypercoverings of an object: modules
• Section 85.21: Hypercovering by a simplicial object of the site
• Section 85.22: Hypercovering by a simplicial object of the site: modules
• Section 85.23: Unbounded cohomological descent for hypercoverings
• Section 85.24: Glueing complexes
• Section 85.25: Proper hypercoverings in topology
• Section 85.26: Simplicial schemes
• Section 85.27: Descent in terms of simplicial schemes
• Section 85.28: Quasi-coherent modules on simplicial schemes
• Section 85.29: Groupoids and simplicial schemes
• Section 85.30: Descent data give equivalence relations
• Section 85.31: An example case
• Section 85.32: Simplicial algebraic spaces
• Section 85.33: Fppf hypercoverings of algebraic spaces
• Section 85.34: Fppf hypercoverings of algebraic spaces: modules
• Section 85.35: Fppf descent of complexes
• Section 85.36: Proper hypercoverings of algebraic spaces
• Chapter 86: Duality for Spaces
• Section 86.1: Introduction
• Section 86.2: Dualizing complexes on algebraic spaces
• Section 86.3: Right adjoint of pushforward
• Section 86.4: Right adjoint of pushforward and base change, I
• Section 86.5: Right adjoint of pushforward and base change, II
• Section 86.6: Right adjoint of pushforward and trace maps
• Section 86.7: Right adjoint of pushforward and pullback
• Section 86.8: Right adjoint of pushforward for proper flat morphisms
• Section 86.9: Relative dualizing complexes for proper flat morphisms
• Section 86.10: Comparison with the case of schemes
• Chapter 87: Formal Algebraic Spaces
• Section 87.1: Introduction
• Section 87.2: Formal schemes à la EGA
• Section 87.3: Conventions and notation
• Section 87.4: Topological rings and modules
• Section 87.5: Taut ring maps
• Section 87.6: Adic ring maps
• Section 87.7: Weakly adic rings
• Section 87.8: Descending properties
• Section 87.9: Affine formal algebraic spaces
• Section 87.10: Countably indexed affine formal algebraic spaces
• Section 87.11: Formal algebraic spaces
• Section 87.12: The reduction
• Section 87.13: Colimits of algebraic spaces along thickenings
• Section 87.14: Completion along a closed subset
• Section 87.15: Fibre products
• Section 87.16: Separation axioms for formal algebraic spaces
• Section 87.17: Quasi-compact formal algebraic spaces
• Section 87.18: Quasi-compact and quasi-separated formal algebraic spaces
• Section 87.19: Morphisms representable by algebraic spaces
• Section 87.20: Types of formal algebraic spaces
• Section 87.21: Morphisms and continuous ring maps
• Section 87.22: Taut ring maps and representability by algebraic spaces
• Section 87.24: Morphisms of finite type
• Section 87.25: Surjective morphisms
• Section 87.26: Monomorphisms
• Section 87.27: Closed immersions
• Section 87.28: Restricted power series
• Section 87.29: Algebras topologically of finite type
• Section 87.30: Separation axioms for morphisms
• Section 87.31: Proper morphisms
• Section 87.32: Formal algebraic spaces and fpqc coverings
• Section 87.33: Maps out of affine formal schemes
• Section 87.34: The small étale site of a formal algebraic space
• Section 87.35: The structure sheaf
• Section 87.36: Colimits of formal algebraic spaces
• Section 87.37: Recompletion
• Section 87.38: Completion along a closed subspace
• Chapter 88: Algebraization of Formal Spaces
• Section 88.1: Introduction
• Section 88.2: Two categories
• Section 88.3: A naive cotangent complex
• Section 88.4: Rig-smooth algebras
• Section 88.5: Deformations of ring homomorphisms
• Section 88.6: Algebraization of rig-smooth algebras over G-rings
• Section 88.7: Algebraization of rig-smooth algebras
• Section 88.8: Rig-étale algebras
• Section 88.9: A pushout argument
• Section 88.10: Algebraization of rig-étale algebras
• Section 88.11: Finite type morphisms
• Section 88.12: Finite type on reductions
• Section 88.13: Flat morphisms
• Section 88.14: Rig-closed points
• Section 88.15: Rig-flat homomorphisms
• Section 88.16: Rig-flat morphisms
• Section 88.17: Rig-smooth homomorphisms
• Section 88.18: Rig-smooth morphisms
• Section 88.19: Rig-étale homomorphisms
• Section 88.20: Rig-étale morphisms
• Section 88.21: Rig-surjective morphisms
• Section 88.22: Formal algebraic spaces over cdvrs
• Section 88.23: The completion functor
• Section 88.24: Formal modifications
• Section 88.25: Completions and morphisms, I
• Section 88.26: Rig glueing of morphisms
• Section 88.27: Algebraization of rig-étale morphisms
• Section 88.28: Completions and morphisms, II
• Section 88.29: Artin's theorem on dilatations
• Section 88.30: Application to modifications
• Chapter 89: Resolution of Surfaces Revisited
• Section 89.1: Introduction
• Section 89.2: Modifications
• Section 89.3: Strategy
• Section 89.4: Dominating by quadratic transformations
• Section 89.5: Dominating by normalized blowups
• Section 89.6: Base change to the completion
• Section 89.7: Implied properties
• Section 89.8: Resolution
• Section 89.9: Examples