The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

5 Topics in Geometry

  • Chapter 74: Chow Groups of Spaces
    • Section 74.1: Introduction
    • Section 74.2: Setup
    • Section 74.3: Cycles
    • Section 74.4: Multiplicities
    • Section 74.5: Cycle associated to a closed subspace
    • Section 74.6: Cycle associated to a coherent sheaf
    • Section 74.7: Preparation for proper pushforward
    • Section 74.8: Proper pushforward
    • Section 74.9: Preparation for flat pullback
    • Section 74.10: Flat pullback
    • Section 74.11: Push and pull
    • Section 74.12: Preparation for principal divisors
    • Section 74.13: Principal divisors
    • Section 74.14: Principal divisors and pushforward
    • Section 74.15: Rational equivalence
    • Section 74.16: Rational equivalence and push and pull
    • Section 74.17: The divisor associated to an invertible sheaf
    • Section 74.18: Intersecting with an invertible sheaf
    • Section 74.19: Intersecting with an invertible sheaf and push and pull
    • Section 74.20: The key formula
    • Section 74.21: Intersecting with an invertible sheaf and rational equivalence
    • Section 74.22: Intersecting with effective Cartier divisors
    • Section 74.23: Gysin homomorphisms
    • Section 74.24: Relative effective Cartier divisors
    • Section 74.25: Affine bundles
    • Section 74.26: Bivariant intersection theory
    • Section 74.27: Projective space bundle formula
    • Section 74.28: The Chern classes of a vector bundle
    • Section 74.29: Polynomial relations among chern classes
    • Section 74.30: Additivity of chern classes
    • Section 74.31: The splitting principle
    • Section 74.32: Degrees of zero cycles
  • Chapter 75: Quotients of Groupoids
    • Section 75.1: Introduction
    • Section 75.2: Conventions and notation
    • Section 75.3: Invariant morphisms
    • Section 75.4: Categorical quotients
    • Section 75.5: Quotients as orbit spaces
    • Section 75.6: Coarse quotients
    • Section 75.7: Topological properties
    • Section 75.8: Invariant functions
    • Section 75.9: Good quotients
    • Section 75.10: Geometric quotients
  • Chapter 76: More on Cohomology of Spaces
    • Section 76.1: Introduction
    • Section 76.2: Conventions
    • Section 76.3: Transporting results from schemes
    • Section 76.4: Proper base change
    • Section 76.5: Comparing big and small topoi
    • Section 76.6: Comparing fppf and étale topologies
    • Section 76.7: Comparing fppf and étale topologies: modules
    • Section 76.8: Comparing ph and étale topologies
  • Chapter 77: Simplicial Spaces
    • Section 77.1: Introduction
    • Section 77.2: Simplicial topological spaces
    • Section 77.3: Simplicial sites and topoi
    • Section 77.4: Augmentations of simplicial sites
    • Section 77.5: Morphisms of simplicial sites
    • Section 77.6: Ringed simplicial sites
    • Section 77.7: Morphisms of ringed simplicial sites
    • Section 77.8: Cohomology on simplicial sites
    • Section 77.9: Cohomology and augmentations of simplicial sites
    • Section 77.10: Cohomology on ringed simplicial sites
    • Section 77.11: Cohomology and augmentations of ringed simplicial sites
    • Section 77.12: Cartesian sheaves and modules
    • Section 77.13: Simplicial systems of the derived category
    • Section 77.14: Simplicial systems of the derived category: modules
    • Section 77.15: The site associated to a semi-representable object
    • Section 77.16: The site associate to a simplicial semi-representable object
    • Section 77.17: Cohomological descent for hypercoverings
    • Section 77.18: Cohomological descent for hypercoverings: modules
    • Section 77.19: Cohomological descent for hypercoverings of an object
    • Section 77.20: Cohomological descent for hypercoverings of an object: modules
    • Section 77.21: Hypercovering by a simplicial object of the site
    • Section 77.22: Hypercovering by a simplicial object of the site: modules
    • Section 77.23: Unbounded cohomological descent for hypercoverings
    • Section 77.24: Glueing complexes
    • Section 77.25: Proper hypercoverings in topology
    • Section 77.26: Simplicial schemes
    • Section 77.27: Descent in terms of simplicial schemes
    • Section 77.28: Quasi-coherent modules on simplicial schemes
    • Section 77.29: Groupoids and simplicial schemes
    • Section 77.30: Descent data give equivalence relations
    • Section 77.31: An example case
    • Section 77.32: Simplicial algebraic spaces
    • Section 77.33: Fppf hypercoverings of algebraic spaces
    • Section 77.34: Fppf hypercoverings of algebraic spaces: modules
    • Section 77.35: Fppf descent of complexes
    • Section 77.36: Proper hypercoverings of algebraic spaces
  • Chapter 78: Duality for Spaces
    • Section 78.1: Introduction
    • Section 78.2: Dualizing complexes on algebraic spaces
    • Section 78.3: Right adjoint of pushforward
    • Section 78.4: Right adjoint of pushforward and base change, I
    • Section 78.5: Right adjoint of pushforward and base change, II
    • Section 78.6: Right adjoint of pushforward and trace maps
    • Section 78.7: Right adjoint of pushforward and pullback
    • Section 78.8: Right adjoint of pushforward for proper flat morphisms
    • Section 78.9: Relative dualizing complexes for proper flat morphisms
    • Section 78.10: Comparison with the case of schemes
  • Chapter 79: Formal Algebraic Spaces
    • Section 79.1: Introduction
    • Section 79.2: Formal schemes à la EGA
    • Section 79.3: Conventions and notation
    • Section 79.4: Topological rings and modules
    • Section 79.5: Affine formal algebraic spaces
    • Section 79.6: Countably indexed affine formal algebraic spaces
    • Section 79.7: Formal algebraic spaces
    • Section 79.8: Colimits of algebraic spaces along thickenings
    • Section 79.9: Completion along a closed subset
    • Section 79.10: Fibre products
    • Section 79.11: Separation axioms for formal algebraic spaces
    • Section 79.12: Quasi-compact formal algebraic spaces
    • Section 79.13: Quasi-compact and quasi-separated formal algebraic spaces
    • Section 79.14: Morphisms representable by algebraic spaces
    • Section 79.15: Types of formal algebraic spaces
    • Section 79.16: Morphisms and continuous ring maps
    • Section 79.17: Adic morphisms
    • Section 79.18: Morphisms of finite type
    • Section 79.19: Monomorphisms
    • Section 79.20: Closed immersions
    • Section 79.21: Restricted power series
    • Section 79.22: Algebras topologically of finite type
    • Section 79.23: Separation axioms for morphisms
    • Section 79.24: Proper morphisms
    • Section 79.25: Formal algebraic spaces and fpqc coverings
    • Section 79.26: Maps out of affine formal schemes
    • Section 79.27: The small étale site of a formal algebraic space
    • Section 79.28: The structure sheaf
  • Chapter 80: Restricted Power Series
    • Section 80.1: Introduction
    • Section 80.2: Two categories
    • Section 80.3: A naive cotangent complex
    • Section 80.4: Rig-étale homomorphisms
    • Section 80.5: Rig-étale morphisms
    • Section 80.6: Glueing rings along a principal ideal
    • Section 80.7: Glueing rings along an ideal
    • Section 80.8: In case the base ring is a G-ring
    • Section 80.9: Rig-surjective morphisms
    • Section 80.10: Algebraization
    • Section 80.11: Application to modifications
  • Chapter 81: Resolution of Surfaces Revisited
    • Section 81.1: Introduction
    • Section 81.2: Modifications
    • Section 81.3: Strategy
    • Section 81.4: Dominating by quadratic transformations
    • Section 81.5: Dominating by normalized blowups
    • Section 81.6: Base change to the completion
    • Section 81.7: Implied properties
    • Section 81.8: Resolution
    • Section 81.9: Examples