Free Presentation Of A Module

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

:$\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0.$

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: $F \ \overset\to\ M \to 0$, but then the kernel of ''g'' is again a quotient of a free module: $F' \ \overset \to\ \ker g \to 0$. 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:

: $\bigoplus_ N \ \overset \to\ \bigoplus_ N \to M \otimes_R N \to 0.$

This says that $M \otimes_R N$ is the cokernel of $f \otimes 1$. If ''N'' is also a ring (and hence an ''R''-algebra), then this is the presentation of the ''N''-module $M \otimes_R N$; that is, the presentation extends under base extension.

For left-exact functors, there is for example

Proof: Applying ''F'' to a finite presentation $R^ \to R^ \to M \to 0$ results in
:$0 \to F(M) \to F(R^) \to F(R^).$
This can be trivially extended to
:$0 \to 0 \to F(M) \to F(R^) \to F(R^).$
The same thing holds for $G$. Now apply the five lemma. $\square$

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, .

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Algebra