nonabelian homological algebra
additive and abelian categories
category of chain complexes
chain map, quasi-isomorphism
chain homology and cohomology
injective object, projective object
injective resolution, projective resolution
homotopy limit, homotopy colimit
abelian sheaf cohomology
triangulated category, enhanced triangulated category
stable model category
(∞,1)-category of chain complexes
Koszul-Tate resolution, BRST-BV complex
spectral sequence of a filtered complex
spectral sequence of a double complex
Grothendieck spectral sequence
Leray spectral sequence
Serre spectral sequence
Hochschild-Serre spectral sequence
four lemma, five lemma
snake lemma, connecting homomorphism
Dold-Kan correspondence / monoidal, operadic
universal coefficient theorem
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A chain complex M • of modules over a commutative ring A is called perfect if it quasi-isomorphic to
a bounded chain complex;
and finitely generated modules.
Let A be a commutative ring and let D(A) denote the derived category of A-modules. A chain complex M • of A-modules is perfect if and only if it is a compact object of D(A).
See (Stacks Project, 07LT).
Let (X,𝒪 X) be a ringed space. A chain complex of 𝒪 X-modules is called perfect if it is locally quasi-isomorphic to a bounded complex of free? 𝒪 X-modules of finite type?.
Let D(Mod(𝒪 X)) be the derived category of 𝒪 X-modules. Let Pf(X)⊂D(Mod(𝒪 X)) denote the full subcategory of perfect complexes. This is a triangulated subcategory, see triangulated categories of sheaves.
For perfect complexes of sheaves see the references at triangulated categories of sheaves.