projective presentation



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.


In abelian categories

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

0NiPpX0, 0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0 \,,

hence exhibiting XX as the cokernel

Xcoker(NP) X \simeq coker(N \hookrightarrow P)

such that PP is a projective object.