# Free presentation

In algebra, a **free presentation** of a module *M* over a commutative ring *R* is an exact sequence of *R*-modules:

Note the image under *g* of the standard basis generates *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: , but then the kernel of *g* is again a quotient of a free module: . The combination of *f* and *g* is a free presentation of *M*. Now, one can obviously keep "resolving" the kernels in this fashion; the result is called a free resolution. Thus, a free presentation is the early part of the free resolution.

A presentation is useful for computation. For example, since tensoring is right-exact, tensoring the above presentation with a module, say, *N* gives:

This says that is the cokernel of . If *N* is an *R*-algebra, then this is the presentation of the *N*-module ; that is, the presentation extends under base extension.

For left-exact functors, there is for example

**Proposition** — Let *F*, *G* be left-exact contravariant functors from the category of modules over a commutative ring *R* to abelian groups and θ a natural transformation from *F* to *G*. If is an isomorphism for each natural number *n*, then is an isomorphism for any finitely-presented module *M*.

Proof: Applying *F* to a finite presentation results in

and the same for *G*. Now apply the snake lemma.

## References

- Eisenbud, David,
*Commutative Algebra with a View Toward Algebraic Geometry*, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.