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Free presentation ：ウィキペディア英語版

Free presentation
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 ''g'' then maps each basis element to each generator of ''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.
A free presentation always exists: any module is a quotient of 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 computaion. 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 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

results in
:

0 \to F(M) \to F(R^) \to F(R^)

and the same for ''G''. Now apply the snake lemma.

\square

== See also ==

*coherent module
*quasi-coherent sheaf
*Fitting ideal
*finitely-related module

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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