Proposition 2

Suppose that $H_0$ is a field, and let $M$ be a finitely generated graded $H$-module. Let $K$ be the field of fractions of $H$. Then $c_M$ is equal to the rank of the $H$-module $M$, that is to say to the dimension of the vector space $M \otimes_H K$ over $K$.

This is clear if $M = H$, since $c_H = 1$. On the other hand, let $x \in H$ be homogeneous of degree $d$, and nonzero; one has $(H/xH) \otimes_H K = 0$; from the exact sequence

and remarks 2 and 3, one deduces that $c_{H/xH} = 0$. The proposition is therefore verified when $M$ is generated by a homogeneous element. The general case follows from this, since every finitely generated graded $H$-module has a composition series whose quotients are of the preceding form.