Free presentation

In algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules: 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: , but then the kernel of g is again a quotient of a free module: . 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: This says that is the cokernel of . If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module ; that is, the presentation extends under base extension. For left-exact functors, there is for example 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 is an isomorphism for each natural number n, then is an isomorphism for any finitely-presented module M. Proof: Applying F to a finite presentation results in This can be trivially extended to The same thing holds for . Now apply the five lemma.