Free presentation explained

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

oplus_iR \overset{f}\to oplus_jR \overset{g}\to M\to0.

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{g}\to M\to0

, but then the kernel of g is again a quotient of a free module:

F' \overset{f}\to \kerg\to0

. 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:

oplus_iN \overset{f ⊗ 1}\to oplus_jN\toM ⊗ _RN\to0.

This says that

M ⊗ _RN

is the cokernel of

f ⊗ 1

. If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module

M ⊗ _RN

; that is, the presentation extends under base extension.

For left-exact functors, there is for exampleProof: Applying F to a finite presentation

R^⊕\toR^⊕\toM\to0

results in

0\toF(M)\toF(R^⊕)\toF(R^⊕).

This can be trivially extended to

0\to0\toF(M)\toF(R^⊕)\toF(R^⊕).

The same thing holds for

. Now apply the five lemma.

\square

References

Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, .

Free presentation explained

See also

References