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: formula_2, but then the kernel of "g" is again a quotient of a free module: formula_3. 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 formula_5 is the cokernel of formula_6. If "N" is an "R"-algebra, then this is the presentation of the "N"-module formula_5; that is, the presentation extends under base extension.
For left-exact functors, there is for example
Proof: Applying "F" to a finite presentation formula_8 results in
and the same for "G". Now apply the snake lemma. formula_10