Free presentation

{{Short description|In algebra, a module over a ring}} {{About|describing a module over a ring|specifying generators and relations of a group|presentation of a group}} {{one source |date=May 2024}} In algebra, a '''free presentation''' of a module ''M'' over a commutative ring ''R'' is an exact sequence of ''R''-modules:

:<math>\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>F \ \overset{g}\to\ M \to 0</math>, but then the kernel of ''g'' is again a quotient of a free module: <math>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>\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>M \otimes_R N</math> is the cokernel of <math>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>M \otimes_R N</math>; that is, the presentation extends under base extension.

For left-exact functors, there is for example {{math_theorem|name=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>\theta: F(R^{\oplus n}) \to G(R^{\oplus n})</math> is an isomorphism for each natural number ''n'', then <math>\theta: F(M) \to G(M)</math> is an isomorphism for any finitely-presented module ''M''.}} Proof: Applying ''F'' to a finite presentation <math>R^{\oplus n} \to R^{\oplus m} \to M \to 0</math> results in :<math>0 \to F(M) \to F(R^{\oplus m}) \to F(R^{\oplus n}).</math> This can be trivially extended to :<math>0 \to 0 \to F(M) \to F(R^{\oplus m}) \to F(R^{\oplus n}).</math> The same thing holds for <math>G</math>. Now apply the five lemma. <math>\square</math>

== References == * Eisenbud, David, ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, {{ISBN|0-387-94268-8}}.

Category:Abstract algebra