11.2 Grothendieck topologies

Grothendieck topologies generalize classical topological spaces to abstract categories. They provide a framework for defining sheaves on categories, crucial for studying algebraic and arithmetic properties of schemes. This approach enables the development of cohomology theories in arithmetic geometry, extending beyond traditional topological settings.

Introduced by Alexander Grothendieck in the 1960s, these topologies arose from the need to study étale cohomology for schemes lacking a suitable classical topology. They allow for a more flexible notion of "covering" in categories, extending beyond open subsets and enabling the study of geometric properties in non-classical settings.

Definition of Grothendieck topologies

• Grothendieck topologies generalize classical topological spaces to abstract categories
• Provide a framework for defining sheaves on categories, crucial for studying algebraic and arithmetic properties of schemes
• Enable the development of cohomology theories in arithmetic geometry, extending beyond traditional topological settings

Motivation and historical context

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• Introduced by Alexander Grothendieck in the 1960s to unify various cohomology theories
• Arose from the need to study étale cohomology for schemes lacking a suitable classical topology
• Allowed for a more flexible notion of "covering" in categories, extending beyond open subsets

Relation to site theory

• Sites consist of a category equipped with a Grothendieck topology
• Provide a foundation for defining sheaves on categories
• Enable the study of geometric properties in non-classical settings (schemes, algebraic stacks)
• Generalize the concept of topological spaces to abstract categories

Comparison with classical topologies

• Classical topologies defined by open sets, Grothendieck topologies use coverings in categories
• Grothendieck topologies allow for "local" properties in categories without a notion of points
• Provide a more general framework for studying sheaves and cohomology
• Include classical topologies as special cases (small site of a topological space)

Categories and sieves

• Categories form the underlying structure for Grothendieck topologies
• Sieves generalize the notion of open coverings in classical topology
• Understanding categories and sieves essential for grasping Grothendieck topologies in arithmetic geometry

Category theory prerequisites

• Categories consist of objects and morphisms between them
• Functors map between categories, preserving structure
• Natural transformations compare functors
• Limits and colimits generalize products and coproducts (pullbacks, pushouts)

Sieve definition and properties

• Sieve on an object X consists of morphisms with codomain X
• Closed under composition with arbitrary morphisms
• Represent potential "coverings" of X in a Grothendieck topology
• Form a complete lattice under inclusion for each object X

Pullback of sieves

• Pullback of a sieve S along a morphism f: Y → X defined as $f^*(S) = \{g: Z → Y | f \circ g \in S\}$
• Preserves sieve structure and inclusion relations
• Essential for defining the stability axiom in Grothendieck topologies
• Allows for "localizing" coverings to different objects in the category

Grothendieck topology axioms

• Axioms formalize the properties expected of a "covering" in a category
• Generalize the behavior of open coverings in classical topology
• Provide a consistent framework for defining sheaves and cohomology theories

Identity axiom

• The maximal sieve on any object X (all morphisms with codomain X) is a covering
• Ensures every object is "covered by itself"
• Analogous to the entire space being an open cover in classical topology
• Formally stated as: For all X, the sieve $\{f: Y → X | f \text{ is any morphism}\}$ is a covering sieve

Stability axiom

• Coverings are stable under pullbacks
• If S is a covering sieve on X and f: Y → X is any morphism, then f*(S) is a covering sieve on Y
• Ensures "local" nature of coverings persists under change of base object
• Allows for consistent definition of sheaf properties across different objects

Transitivity axiom

• Composition of coverings yields a covering
• If S is a covering sieve on X and R is a sieve on X such that f*(R) is a covering sieve for all f in S, then R is a covering sieve
• Ensures coverings can be "refined" consistently
• Analogous to the transitivity of open coverings in classical topology

Examples of Grothendieck topologies

• Grothendieck topologies provide various "geometric" perspectives on schemes and other algebraic objects
• Different topologies capture distinct aspects of arithmetic and algebraic geometry
• Understanding these examples crucial for applying Grothendieck topologies in arithmetic geometry

Zariski topology

• Defined on the category of schemes or rings
• Coverings given by open immersions that jointly surject onto the target
• Corresponds to the classical Zariski topology on affine schemes
• Useful for studying local properties of schemes (regular, normal, Cohen-Macaulay)

Étale topology

• Defined on the category of schemes
• Coverings given by étale morphisms that jointly surject onto the target
• Finer than the Zariski topology, allows for studying "infinitesimal" properties
• Crucial for defining étale cohomology, with applications to the Weil conjectures

Flat topology

• Defined on the category of schemes
• Coverings given by flat morphisms that jointly surject onto the target
• Includes both Zariski and étale topologies as special cases
• Useful for studying descent problems and moduli spaces

Crystalline topology

• Defined on the category of schemes in characteristic p
• Coverings given by certain divided power thickenings
• Used to define crystalline cohomology, a p-adic cohomology theory
• Important for studying deformations and p-adic periods

Sheaves on Grothendieck topologies

• Sheaves generalize the notion of locally defined functions to arbitrary categories
• Provide a framework for studying global objects from local data
• Essential for defining cohomology theories in arithmetic geometry

Sheaf definition

• Presheaf F on a site (C, J) assigns objects F(X) to objects X in C and morphisms F(f): F(Y) → F(X) to morphisms f: X → Y in C
• Sheaf condition: For any covering sieve S on X, the diagram $F(X) \to \prod_{f \in S} F(\text{dom}(f)) \rightrightarrows \prod_{f,g \in S} F(\text{dom}(f) \times_X \text{dom}(g))$ is an equalizer
• Ensures local data can be uniquely glued to form global sections
• Generalizes the classical sheaf condition for topological spaces

Sheafification process

• Transforms a presheaf into a sheaf while preserving its local properties
• Involves two steps: separation (making the presheaf separated) and sheafification
• Separated presheaf satisfies the uniqueness part of the sheaf condition
• Sheafification adds missing "local" sections to satisfy the existence part of the sheaf condition

Sheaf cohomology

• Measures obstructions to extending local sections to global ones
• Defined using injective resolutions or Čech cohomology
• Generalizes classical cohomology theories (singular, de Rham) to arbitrary Grothendieck topologies
• Applications include studying Galois cohomology, class field theory, and motivic cohomology

Grothendieck topoi

• Generalize the category of sheaves on a topological space
• Provide a categorical framework for studying "generalized spaces"
• Essential for understanding the geometric aspects of Grothendieck topologies

Definition and properties

• Grothendieck topos defined as a category equivalent to the category of sheaves on a site
• Satisfies axioms including: having all small limits and colimits, being locally small, and having a small generating set
• Possesses an internal logic, allowing for intuitionistic reasoning within the topos
• Examples include the category of sets, sheaves on a topological space, and étale sheaves on a scheme

Comparison with classical topoi

• Classical topoi based on the category of sheaves on a topological space
• Grothendieck topoi generalize this concept to arbitrary sites
• Both types of topoi share many properties (existence of subobject classifier, internal logic)
• Grothendieck topoi allow for studying "spaces" without underlying point-set topology

Geometric morphisms

• Functors between topoi that preserve the geometric structure
• Consist of an adjoint pair of functors (f*, f_) with f preserving finite limits
• Generalize continuous maps between topological spaces
• Examples include morphisms of schemes inducing geometric morphisms between their étale topoi

Applications in arithmetic geometry

• Grothendieck topologies provide powerful tools for studying arithmetic properties of schemes and varieties
• Enable the development of cohomology theories beyond classical settings
• Crucial for understanding deep connections between number theory and algebraic geometry

Étale cohomology

• Cohomology theory for schemes based on the étale topology
• Provides a good cohomology theory for varieties over finite fields
• Key to the proof of the Weil conjectures by Deligne
• Applications in studying zeta functions, L-functions, and Galois representations

Motivic cohomology

• Universal cohomology theory for algebraic varieties
• Based on algebraic cycles and motivic complexes
• Conjectured to unify various cohomology theories (Betti, de Rham, l-adic)
• Applications in studying special values of L-functions and algebraic K-theory

Derived categories

• Categorical framework for studying complexes of sheaves up to quasi-isomorphism
• Allows for a unified treatment of various cohomology theories
• Applications in perverse sheaves, intersection cohomology, and the geometric Langlands program
• Provides a setting for studying derived algebraic geometry and higher categorical structures

Generalizations and variations

• Extend the ideas of Grothendieck topologies to more abstract or higher-dimensional settings
• Provide new frameworks for studying arithmetic and algebraic geometry
• Explore connections between different areas of mathematics (topology, algebra, category theory)

Higher Grothendieck topologies

• Generalize Grothendieck topologies to higher categories
• Allow for studying higher stacks and derived algebraic geometry
• Provide a framework for defining higher topoi and ∞-topoi
• Applications in homotopy theory and higher categorical structures

Condensed mathematics

• Developed by Clausen and Scholze as a foundation for p-adic geometry
• Based on sheaves on a category of profinite sets with the "condensed" topology
• Unifies various approaches to topological vector spaces and adic spaces
• Applications in p-adic Hodge theory and perfectoid spaces

∞-topoi

• Generalize Grothendieck topoi to the setting of ∞-categories
• Provide a framework for homotopy-coherent sheaf theory
• Allow for studying higher stacks and derived algebraic geometry
• Applications in homotopy theory, higher category theory, and derived algebraic geometry

Computational aspects

• Develop algorithms and software tools for working with Grothendieck topologies
• Enable practical applications of Grothendieck topologies in arithmetic geometry
• Facilitate exploration and verification of conjectures involving Grothendieck topologies

Algorithms for sieve calculations

• Implement efficient methods for computing and manipulating sieves in various categories
• Develop algorithms for checking covering properties in specific Grothendieck topologies
• Optimize calculations of pullbacks and compositions of sieves
• Implement methods for computing cohomology groups using Čech or derived functor approaches

Software tools for Grothendieck topologies

• Computer algebra systems with support for category theory and Grothendieck topologies (SageMath, Macaulay2)
• Specialized libraries for working with sheaves and topoi (PARI/GP, GAP)
• Visualization tools for exploring the structure of Grothendieck topologies and sheaves
• Interfaces for defining custom Grothendieck topologies and performing calculations

Open problems and current research

• Active areas of investigation in arithmetic geometry involving Grothendieck topologies
• Ongoing efforts to extend and apply Grothendieck topologies to new domains
• Connections between Grothendieck topologies and other areas of mathematics

Conjectures involving Grothendieck topologies

• Grothendieck's standard conjectures on algebraic cycles and motives
• Tate conjecture relating algebraic cycles to Galois representations
• Hodge conjecture and its variants in different cohomology theories
• Conjectures on the behavior of motivic cohomology and its relation to other cohomology theories

Recent developments in the field

• Advances in derived algebraic geometry using higher Grothendieck topologies
• Applications of condensed mathematics to p-adic Hodge theory and perfectoid spaces
• Progress on the Langlands program using geometric methods and Grothendieck topologies
• Developments in motivic homotopy theory and its applications to algebraic K-theory