Proposition 4

Suppose that $H$ is noetherian and $M$ finitely generated. Then $\dim_H(M) = \dim_{S^{-1}H}(S^{-1}M)$.

Let a be the annihilator of the $H$-module $M$; it is a graded ideal of $H$. Since $M$ is a finitely generated $H$-module, the annihilator of the $S^{-1}H$-module $S^{-1}M$ is the ideal $S^{-1}\alpha$ of $S^{-1}H$. One has $\dim_H(M) = \dim(H/\alpha)$ and $\dim_{S^{-1}H}(S^{-1}M) = \dim(S^{-1}H/S^{-1}\alpha)$. Proposition 4 then follows from Theorem 1, c) of No. 2 applied to the graded ring $H/\alpha$.