nLab
projective presentation

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

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

hence exhibiting $X$ as the cokernel

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

such that $P$ is a projective object.

Related concepts

• projective object, projective presentation, projective cover, projective resolution

  • projective module

• injective object, injective presentation, injective envelope, injective resolution

  • injective module

• free object, free resolution

  • free module

• flat object, flat resolution

  • flat module