Free presentation

In algebra, a module over a ring

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

⨁ i ∈ I R   → f   ⨁ j ∈ J R   → g   M → 0. {\displaystyle \bigoplus _{i\in I}R\ {\overset {f}{\to }}\ \bigoplus _{j\in 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   → g   M → 0 {\displaystyle F\ {\overset {g}{\to }}\ M\to 0} , but then the kernel of g is again a quotient of a free module: F ′   → f   ker ⁡ g → 0 {\displaystyle 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:

⨁ i ∈ I N   → f ⊗ 1   ⨁ j ∈ J N → M ⊗ R N → 0. {\displaystyle \bigoplus _{i\in I}N\ {\overset {f\otimes 1}{\to }}\ \bigoplus _{j\in J}N\to M\otimes _{R}N\to 0.}

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

For left-exact functors, there is for example

Proposition — Let F, G be left-exact contravariant functors from the category of modules over a commutative ring R to abelian groups and θ a natural transformation from F to G. If θ : F ( R ⊕ n ) → G ( R ⊕ n ) {\displaystyle \theta :F(R^{\oplus n})\to G(R^{\oplus n})} is an isomorphism for each natural number n, then θ : F ( M ) → G ( M ) {\displaystyle \theta :F(M)\to G(M)} is an isomorphism for any finitely-presented module M.

Proof: Applying F to a finite presentation R ⊕ n → R ⊕ m → M → 0 {\displaystyle R^{\oplus n}\to R^{\oplus m}\to M\to 0} results in

0 → F ( M ) → F ( R ⊕ m ) → F ( R ⊕ n ) . {\displaystyle 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}

This can be trivially extended to

0 → 0 → F ( M ) → F ( R ⊕ m ) → F ( R ⊕ n ) . {\displaystyle 0\to 0\to F(M)\to F(R^{\oplus m})\to F(R^{\oplus n}).}

The same thing holds for G {\displaystyle G} . Now apply the five lemma. ◻ {\displaystyle \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, ISBN 0-387-94268-8.


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