Free presentation

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

${\displaystyle \bigoplus _{i\in I}R\ {\overset {f}{\to }}\ \bigoplus _{j\in J}R\ {\overset {g}{\to }}\ M\to 0.}${\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: ${\displaystyle F\ {\overset {g}{\to }}\ M\to 0}${\displaystyle F\ {\overset {g}{\to }}\ M\to 0}, but then the kernel of g is again a quotient of a free module: ${\displaystyle F'\ {\overset {f}{\to }}\ \ker g\to 0}${\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:

${\displaystyle \bigoplus _{i\in I}N\ {\overset {f\otimes 1}{\to }}\ \bigoplus _{j\in J}N\to M\otimes _{R}N\to 0.}${\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 ${\displaystyle M\otimes _{R}N}${\displaystyle M\otimes _{R}N} is the cokernel of ${\displaystyle f\otimes 1}${\displaystyle f\otimes 1}. If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module ${\displaystyle M\otimes _{R}N}${\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 ${\displaystyle R^{\oplus n}\to R^{\oplus m}\to M\to 0}${\displaystyle R^{\oplus n}\to R^{\oplus m}\to M\to 0} results in

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

This can be trivially extended to

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

The same thing holds for ${\displaystyle G}${\displaystyle G}. Now apply the five lemma. ${\displaystyle \square }${\displaystyle \square }

• Coherent module
• Finitely related module
• Fitting ideal
• Quasi-coherent sheaf

• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

• Abstract algebra
• Algebra stubs

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from May 2024
• All articles needing additional references
• All stub articles