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; thus, the cardinarity of J is the cardinarity of the generating set of M. 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. One can obviously keep continuing "resolving" the kernels; 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:

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

That is to say, $M\otimes _{R}N$ is the cokernel of $f\otimes 1$.

References

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