nLab
filtered derived category

Idea

Given an abelian category, the filtered derived category is the category of chain complexes of filtered objects up to quasi-isomorphism.

Definition

Given an abelian category AA, one may consider the additive category Fil(A)Fil(A) of filtered objects in AA, whose objects are pairs (a,F *)(a, F^*) with aa an object of AA and F *F^* a filtration on aa. Let Comp(Fil(A))Comp(Fil(A)) denote the category of chain complexes in Fil(A)Fil(A). One defines a morphism in Comp(Fil(A))Comp(Fil(A)) to be a quasi-isomorphism if it induces quasi-isomorphisms on each degree of the associated graded objects. The filtered derived category of AA,

D fil(A)=Comp(Fil(A))[qis 1] D_{fil}(A) = Comp(Fil(A))[qis^{-1}]

is the localization of Comp(Fil(A))Comp(Fil(A)) at the class of quasi-isomorphisms.

See also

References

Created on October 27, 2014 at 15:39:27. See the history of this page for a list of all contributions to it.