Let $M$ be a positively graded finitely generated module over a positively graded commutative ring $R$. Assume that $R\_0$ is a local ring with maximal ideal $m\_0$. Let $d$ be the Krull dimension of $M$. In the book "Local cohomology: An algebraic introduction with geometric application" by M.P.Brodmann and R.Y.Sharp, the authors claim that if $d=0$ then the set of associated prime ideals of $M$ is $\lbrace m\_0 \oplus R\_+ \rbrace$, and therefore there exists a $t\in \mathbb{N}$ such that $R\_{+}^{t}M=0$.


My question is : 


1. Why do we have Ass(M)= $\lbrace m\_0 \oplus R\_+ \rbrace$?
2. Why does this imply $R\_{+}^{t}M=0$ for some $t\in \mathbb{N}$?



If $M$ is a positively graded finitely generated module of dim 0, then why $R_{+}^{t}M=0$ for some $t\in \mathbb{N}$?
First, claim 1 as stated is wrong - consider the zero module.


Second, I guess that you talk about a step in the proof of Theorem 15.3.1 in Brodmann-Sharp. If so, then you have more hypotheses than you mentioned. Beside others, $R$ is noetherian and - most important - $M$ is $0$-dimensional. (And $M$ is not "positively graded", a notion that seems not reasonable for graded modules.)


So, what you want to show is that under these hypotheses, $M$ is $R\_+$-torsion. For this it suffices to show that $M$ is $\mathfrak{m}\_0+R\_+$-torsion. More general, it suffices to show that a $0$-dimensional finitely generated graded module $M$ over a \*local graded ring with \*maximal ideal $\mathfrak{m}$ is $\mathfrak{m}$-torsion. And this is indeed the case. Namely, $0$-dimensionality means that the graded ring $R/(0:\_RM)$ is $0$-dimensional. Now, $\sqrt{(0:\_RM)}$ is the intersection of the graded primes containing $(0:\_RM)$. But as $R/(0:\_RM)$ is $0$-dimensional, $\mathfrak{m}$ is the only such prime, implying $\sqrt{(0:\_RM)}=\mathfrak{m}$. Since $\mathfrak{m}$ is finitely generated (as $R$ is supposed to be noetherian), there exists $t\in\mathbb{N}$ with $\mathfrak{m}^t\subseteq(0:\_RM)$, and this implies that $M$ is an $\mathfrak{m}$-torsion module as desired.


