Editing Finitely generated module (section)

== Generic rank ==
Let ''M'' be a finitely generated module over an integral domain ''A'' with the field of fractions ''K''. Then the dimension <math>\operatorname{dim}_K (M \otimes_A K)</math> is called the '''generic rank''' of ''M'' over ''A''. This number is the same as the number of maximal ''A''-linearly independent vectors in ''M'' or equivalently the rank of a maximal free submodule of ''M'' (''cf. [[Rank of an abelian group]]''). Since <math>(M/F)_{(0)} = M_{(0)}/F_{(0)} = 0</math>, <math>M/F</math> is a [[torsion module]]. When ''A'' is Noetherian, by [[generic freeness]], there is an element ''f'' (depending on ''M'') such that <math>M[f^{-1}]</math> is a free <math>A[f^{-1}]</math>-module. Then the rank of this free module is the generic rank of ''M''.

Now suppose the integral domain ''A'' is an <math>\mathbb{N}</math>-[[graded algebra]] over a field ''k'' generated by finitely many homogeneous elements of degrees <math>d_i</math>. Suppose ''M'' is graded as well and let <math>P_M(t) = \sum (\operatorname{dim}_k M_n) t^n</math> be the [[Poincaré series (modular form)|Poincaré series]] of ''M''.
By the [[Hilbert–Serre theorem]], there is a polynomial ''F'' such that <math>P_M(t) = F(t) \prod (1-t^{d_i})^{-1}</math>. Then <math>F(1)</math> is the generic rank of ''M''.<ref>{{harvnb|Springer|1977|loc=Theorem 2.5.6.}}</ref>

A finitely generated module over a [[principal ideal domain]] is [[torsion-free module|torsion-free]] if and only if it is free. This is a consequence of the [[structure theorem for finitely generated modules over a principal ideal domain]], the basic form of which says a finitely generated module over a PID is a direct sum of a torsion module and a free module. But it can also be shown directly as follows: let ''M'' be a torsion-free finitely generated module over a PID ''A'' and ''F'' a maximal free submodule. Let ''f'' be in ''A'' such that <math>f M \subset F</math>. Then <math>fM</math> is free since it is a submodule of a free module and ''A'' is a PID. But now <math>f: M \to fM</math> is an isomorphism since ''M'' is torsion-free.

By the same argument as above, a finitely generated module over a [[Dedekind domain]] ''A'' (or more generally a [[semi-hereditary ring]]) is torsion-free if and only if it is [[projective module|projective]]; consequently, a finitely generated module over ''A'' is a direct sum of a torsion module and a projective module. A finitely generated projective module over a Noetherian integral domain has constant rank and so the generic rank of a finitely generated module over ''A'' is the rank of its projective part.