Free presentation
In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules:
- [math]\displaystyle{ \bigoplus_{i \in I} R \ \overset{f} \to\ \bigoplus_{j \in J} R \ \overset{g}\to\ M \to 0. }[/math]
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: [math]\displaystyle{ F \ \overset{g}\to\ M \to 0 }[/math], but then the kernel of g is again a quotient of a free module: [math]\displaystyle{ F' \ \overset{f} \to\ \ker g \to 0 }[/math]. 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:
- [math]\displaystyle{ \bigoplus_{i \in I} N \ \overset{f \otimes 1} \to\ \bigoplus_{j \in J} N \to M \otimes_R N \to 0. }[/math]
This says that [math]\displaystyle{ M \otimes_R N }[/math] is the cokernel of [math]\displaystyle{ f \otimes 1 }[/math]. If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module [math]\displaystyle{ M \otimes_R N }[/math]; 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 [math]\displaystyle{ \theta: F(R^{\oplus n}) \to G(R^{\oplus n}) }[/math] is an isomorphism for each natural number n, then [math]\displaystyle{ \theta: F(M) \to G(M) }[/math] is an isomorphism for any finitely-presented module M.
Proof: Applying F to a finite presentation [math]\displaystyle{ R^{\oplus n} \to R^{\oplus m} \to M \to 0 }[/math] results in
- [math]\displaystyle{ 0 \to F(M) \to F(R^{\oplus m}) \to F(R^{\oplus n}). }[/math]
This can be trivially extended to
- [math]\displaystyle{ 0 \to 0 \to F(M) \to F(R^{\oplus m}) \to F(R^{\oplus n}). }[/math]
The same thing holds for [math]\displaystyle{ G }[/math]. Now apply the five lemma. [math]\displaystyle{ \square }[/math]
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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