Free presentation

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

$\oplus _{i\in I}R{\overset {f}{\to }}\oplus _{j\in J}R{\overset {g}{\to }}M\to 0,$

where g maps each basis element to each generator of M (in general, the image of a generating set generates the image). In particular, if J is finite, then M is a finitely generated module.

A free presentation always exists: any module is a quotient of free module: $F{\overset {g}{\to }}M\to 0$, but then the kernel of g is again a quotient of a free module: $F'{\overset {f}{\to }}\operatorname {ker} g\to 0$. A combination of f' and g is a free presentation. 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 computaion. For example, since tensoring is right-exact, tensoring the above presentation with a module, say, N gives:

$\oplus _{i\in I}N{\overset {f\otimes 1}{\to }}\oplus _{j\in J}N\to M\otimes _{R}N\to 0.$

This says that $M\otimes _{R}N$ is the cokernel of $f\otimes 1$. If N is an R-algebra, then this is the presentation of the N-module $M\otimes _{R}N$; that is, the presentation extends under base extension.

See also

• coherent module
• quasi-coherent sheaf
• finitely-presented module
• Fitting ideal

References

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