$M$ is finitely generated as an $A$-module iff $M/A_{>0}M$ is finitely generated as an $A$-module?

Question

Let $A$ be a nonnegative graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. How do I see that $M$ is finitely generated as an $A$-module if and only if $M/A_{>0}M$ is finitely generated as an $A$-module?

Answer 1

If $M$ is finitely generated, then clearly $M/A_+M$ is finitely generated, since it is a quotient of $M$.

Conversely suppose that $M/A_+M$ is finitely generated, let $B=\{b_1,\dots,b_n\}$ be a finite set of homogeneous generators of cardinal $n$, and let $\phi:A^n\to M$ be the unique $A$-linear map such that $\phi(e_i)=b_i$ for each $i\in\{1,\dots,n\}$. The construction implies at once that $$M=\phi(A^n)+A_+M.$$ Now you need some form of Nakayama's lemma to conclude that this implies that $\phi(A^n)=M$ and, therefore, that $\phi$ is surjective and $M$ finitely generated.