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Cohomology descends to the derived category
PDF p. 4: Definition 2.2
Statement
For each integer , cohomology defines a functor . Here is the derived category of cochain complexes of abelian groups, obtained by inverting quasi-isomorphisms; a quasi-isomorphism is a cochain map inducing an isomorphism on every cohomology group.
Facts & Assumptions
The derived category is the localization of the homotopy category of complexes at quasi-isomorphisms and has the corresponding universal property (Derived category of an abelian category).
Cochain maps induce functorial cohomology maps, and homotopic maps induce equal maps (Cohomology maps induced by cochain maps, Homotopic maps act equally on cohomology).
Proof
Given: An integer and the derived category described in the statement.
Goal: Construct the stated functor.
By [F2], taking th cohomology is a functor on the homotopy category. Every quasi-isomorphism becomes an isomorphism under this functor by the definition of quasi-isomorphism.
The localization universal property in [F1] therefore gives a functor on whose composite with localization is the functor of step 1.1. Thus cohomology and its maps depend only on the derived-category object and morphism.
Depends on
Used by · 10
- Motivic complexes and cohomology
- Motivic dimension vanishing
- Functorial finite coefficients for derived complexes
- Finite derived products and motivic coefficient operations
- Prime-field evaluation and compatibility with the standing theory
- Gluing upper bounds through an actual fiber sequence
- External prime-field cdh comparison and cdh dimension bound
- External dimension criterion for a cdh-acyclic Nisnevich sheaf
- Cohomology of equivalent derived complexes
- Cofiber bounds imply low-degree cohomology isomorphisms
Dependency tree · 3 levels
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