In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules:
⨁ i ∈ I R → f ⨁ j ∈ J R → g M → 0. {\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: F → g M → 0 {\displaystyle F\ {\overset {g}{\to }}\ M\to 0} , but then the kernel of g is again a quotient of a free module: F ′ → f ker g → 0 {\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:
⨁ i ∈ I N → f ⊗ 1 ⨁ j ∈ J N → M ⊗ R N → 0. {\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 M ⊗ R N {\displaystyle M\otimes _{R}N} is the cokernel of f ⊗ 1 {\displaystyle f\otimes 1} . If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module M ⊗ R N {\displaystyle M\otimes _{R}N} ; that is, the presentation extends under base extension. For left-exact functors, there is for example
Proof: Applying F to a finite presentation R ⊕ n → R ⊕ m → M → 0 {\displaystyle R^{\oplus n}\to R^{\oplus m}\to M\to 0} results in
0 → F ( M ) → F ( R ⊕ m ) → F ( R ⊕ n ) . {\displaystyle 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}
This can be trivially extended to
0 → 0 → F ( M ) → F ( R ⊕ m ) → F ( R ⊕ n ) . {\displaystyle 0\to 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}
The same thing holds for G {\displaystyle G} . Now apply the five lemma. ◻ {\displaystyle \square }
See also Coherent module Finitely related module Fitting ideal Quasi-coherent sheaf
References Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.