nLab
perfect chain complex

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

A chain complex M M_\bullet of modules over a commutative ring AA is called perfect if it quasi-isomorphic to

Properties

Relation to compact objects

Proposition

Let AA be a commutative ring and let D(A)D(A) denote the derived category of AA-modules. A chain complex M M_\bullet of AA-modules is perfect if and only if it is a compact object of D(A)D(A).

For instance (Stacks Project, 07LT).

Perfect complexes on a ringed space

Let (X,𝒪 X)(X, \mathcal{O}_X) be a ringed space. A chain complex of 𝒪 X\mathcal{O}_X-modules is called perfect if it is locally quasi-isomorphic to a bounded complex of free? 𝒪 X\mathcal{O}_X-modules of finite type?.

Let D(Mod(𝒪 X))D(Mod(\mathcal{O}_X)) be the derived category of 𝒪 X\mathcal{O}_X-modules. Let Pf(X)D(Mod(𝒪 X))Pf(X) \subset D(Mod(\mathcal{O}_X)) denote the full subcategory of perfect complexes. This is a triangulated subcategory, see triangulated categories of sheaves.

finite objects:

geometrymonoidal category theorycategory theory
perfect module(fully-)dualizable objectcompact object

References

For perfect complexes of sheaves see the references at triangulated categories of sheaves.

Revised on February 10, 2014 06:07:48 by Urs Schreiber (89.204.135.153)