# Free presentation

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

${\displaystyle \bigoplus _{i\in I}R{\overset {f}{\to }}\bigoplus _{j\in J}R{\overset {g}{\to }}M\to 0.}$

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: ${\displaystyle F{\overset {g}{\to }}M\to 0}$, but then the kernel of g is again a quotient of a free module: ${\displaystyle F'{\overset {f}{\to }}\ker g\to 0}$. 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:

${\displaystyle \bigoplus _{i\in I}N{\overset {f\otimes 1}{\to }}\bigoplus _{j\in J}N\to M\otimes _{R}N\to 0.}$

This says that ${\displaystyle M\otimes _{R}N}$ is the cokernel of ${\displaystyle f\otimes 1}$. If N is an R-algebra, then this is the presentation of the N-module ${\displaystyle M\otimes _{R}N}$; 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 ${\displaystyle \theta :F(R^{\oplus n})\to G(R^{\oplus n})}$ is an isomorphism for each natural number n, then ${\displaystyle \theta :F(M)\to G(M)}$ is an isomorphism for any finitely-presented module M.

Proof: Applying F to a finite presentation ${\displaystyle R^{\oplus n}\to R^{\oplus m}\to M}$ results in

${\displaystyle 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n})}$

and the same for G. Now apply the snake lemma. ${\displaystyle \square }$