nLab projective presentation

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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.

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

0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0 \,,

hence exhibiting $X$ as the cokernel

X \simeq coker(N \hookrightarrow P)

such that $P$ is a projective object.

Last revised on September 4, 2012 at 20:29:40. See the history of this page for a list of all contributions to it.