In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules:
Note the image of g is a generating set of M. In particular, if J is finite, then M is a finitely generated module. If I and J are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it admits a finite presentation.
Since f is a module homomorphism between free modules, it can be visualized as an (infinite) matrix with entries in R and M as its cokernel.
A free presentation always exists: any module is a quotient of a free module:
A presentation is useful for computaion. For example, since tensoring is right-exact, tensoring the above presentation with a module, say, N gives:
This says that
For left-exact functors, there is for example
Proof: Applying F to a finite presentation
and the same for G. Now apply the snake lemma.