14.2 Cohomology theories and topos theory

Topos theory unifies cohomology, providing a general setting to define and compare different theories. It connects étale cohomology to Galois cohomology and crystalline cohomology to de Rham cohomology, using topoi to bridge these concepts.

Sheaf cohomology in topoi calculates cohomology of sheaves, using derived categories and Grothendieck topologies. This framework extends to algebraic geometry, defining schemes and exploring various topologies, while also connecting to motivic cohomology and non-commutative geometry.

Topos Theory and Cohomology

Topos theory and cohomology relationships

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• Topos theory unifies cohomology theories providing general setting for defining and studying cohomology allowing comparison between different cohomology theories
• Étale cohomology defined in terms of étale topology on schemes connects to Galois cohomology for fields applied in arithmetic geometry (Weil conjectures)
• Crystalline cohomology defined for schemes in characteristic p relates to de Rham cohomology in characteristic 0 uses crystalline site as a topos
• Topos theory compares cohomology theories through functorial properties and spectral sequences relating different theories (Leray spectral sequence)

Topoi in sheaf cohomology

• Sheaf cohomology in topoi defines sheaves on a site calculates cohomology of sheaves in a topos
• Derived categories construct derived category of sheaves define derived functors and their universal properties (derived pushforward)
• Grothendieck topologies and sites define and exemplify sheaves on a site (Zariski, étale, fppf)
• Cohomological descent relates Čech cohomology to sheaf cohomology uses hypercoverings in descent theory

Topos theory in algebraic geometry

• Grothendieck's work on schemes and topoi defines schemes as locally ringed spaces explores Zariski topology and étale topology
• Geometric morphisms between topoi relate to morphisms of schemes define pullback and pushforward functors
• Points of a topos correspond to geometric points of schemes classify points in various topoi (classifying topos)
• Coherent topoi define properties relate to coherent schemes (noetherian schemes)

Topoi for motivic cohomology

• Motivic cohomology defined using category of mixed motives relates to algebraic K-theory (Bloch-Lichtenbaum spectral sequence)
• Voevodsky's approach uses simplicial presheaves develops A1-homotopy theory
• l-adic cohomology compares with étale cohomology applies to Weil conjectures (Riemann hypothesis for varieties over finite fields)
• De Rham-Witt cohomology relates to crystalline cohomology applies to p-adic Hodge theory (Fontaine's period rings)
• Topoi in non-commutative geometry follows Connes' approach to non-commutative spaces interprets cyclic cohomology topos-theoretically