category theory

# Contents

## Idea

A projective presentation of an object is a realization of that object as a suitable quotient of a projective object.

In homological algebra projective presentations can sometimes be used in place of genuine projective resolutions in the computation of derived functors. See for instance at Ext-functor for examples.

The dual notion is that of injective presentation.

## Definition

### In abelian categories

Let $𝒜$ be an abelian category. For $X\in 𝒜$ any object, a projective presentation of $X$ is a short exact sequence of the form

$0\to N\stackrel{i}{↪}P\stackrel{p}{\to }X\to 0\phantom{\rule{thinmathspace}{0ex}},$0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0 \,,

hence exhibiting $X$ as the cokernel

$X\simeq \mathrm{coker}\left(N↪P\right)$X \simeq coker(N \hookrightarrow P)

such that $P$ is a projective object.

Revised on September 4, 2012 20:29:40 by Urs Schreiber (131.174.190.104)