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 Ext-functor for examples.

The dual notion is that of an injective presentation.

Definition

In abelian categories

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.

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