Free presentation explained

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

o plus _{i} R \overset{f} \to o plus _{j} R \overset{g}\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{g}\to M \to 0 , but then the kernel of g is again a quotient of a free module: F' \overset{f} \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:

o plus _{i} N \overset{f ⊗ 1 } \to o plus _{j} N \to M ⊗ _{R} N \to 0 .

This says that

M ⊗ _{R} N 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 ⊗ _{R} N ; that is, the presentation extends under base extension.

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

R ^{⊕} \to R ^{⊕} \to M \to 0 results in0 \to F(M) \to F(R ^{⊕} ) \to F(R ^{⊕} ). This can be trivially extended to0 \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, .