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

Idea

A perfect complex over a commutative ring AA is a perfect module over the Eilenberg-Mac Lane spectrum H(A)H(A). Under the stable Dold-Kan correspondence, perfect complexes correspond to bounded chain complexes of finitely generated projective modules.

Viewing commutative rings as affine schemes, this definition generalizes to arbitrary stacks. In this generality, perfect modules still coincide with the dualizable objects, but not always with the compact objects. The latter does hold for quasi-compact quasi-separated schemes by work of Thomason, Neeman, Bondal-Van den Bergh.

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

Revised on February 6, 2015 14:20:40 by Ingo Blechschmidt (137.250.162.16)