If i take $M$ a positively graded finitely generated $A$-module, with A a positively graded finite generated $A_0$ algebra, with $A_0$ artinian ring, is true that holder length $$l(M_n)=l(R_0)l_{R_0}(M_n)$$
I've tried in this way: Since $R_0$ is artinian, it has finite Holder length, so i can take: $R_{0,1} \subset \dots \subset R_{0,r}$. Then, if i take a set of minimal generator for M as $R_0$ module, i can write down $x_1 R_{0,1}\subset x_1 R_{0,2}\subset \dots \subset x_1 R_0+x_2 R_{0,1}\subset \dots \subset M_n$
So i can say that $l(M_n)\leq (R_0)l_{R_0}(M_n)$. Right?