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.
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.
If N is also a ring (and hence 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 This can be trivially extended to The same thing holds for