# nLab perfect chain complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

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

## Properties

### Relation to compact objects

###### Proposition

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

For instance (Stacks Project, 07LT).

### Perfect complexes on a ringed space

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

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

finite objects:

## 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)