In algebra, a module over a ring
This article is about describing a module over a ring. For specifying generators and relations of a group, see
presentation of a group.
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ebbc1750fba6f275b4d766265b8ea0d88ec5e72)
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:
![{\displaystyle \bigoplus _{i\in I}N\ {\overset {f\otimes 1}{\to }}\ \bigoplus _{j\in J}N\to M\otimes _{R}N\to 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6645f8364d73cba4053a04b261e7a2142135dd07)
This says that
is the cokernel of
. 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
![{\displaystyle 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e1963e73f3c86a7bae1d20a604f475c91404dd)
This can be trivially extended to
![{\displaystyle 0\to 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2529cf65cfe5bf3cc95c0de180aa83edb986c8)
The same thing holds for
. Now apply the five lemma.
See also
References