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Let $k$ be a field and $S=k[x_1,\dots,x_n]$ the polynomial ring with the usual grading. Let $M$ be a finitely generated graded $S$-module.

Question 1: How can we see that $\dim M = 0$ implies that the length of $M$ is finite, i.e. $l(M)< \infty$?

Remark 1: If $R$ was local Noetherian, then the result would follow immediately from the Fundamental Theorem of Dimension Theory. I suspect that the analogue of this theorem holds for *local rings as well, even though i can't find a source that verifies that.

Question 2: Does the fundamental theorem of dimension theory for local Noetherian rings admit an analogue for *local rings? Any reference in the literature?

Remark 2: By Remark 1 we have that $\dim M_P =0$ for every $P \in \operatorname{Supp}M$ and so $l(M_P) < \infty$. This motivates

Question 3: Is it true that for a finitely generated module $M$ over a Noetherian ring $R$ (could also be local or *local) we have that $l(M) = \sup \left\{(l(M_P) : P \in \operatorname{Supp} M \right\}$?

Manos
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  • I don't understand what this has to do with local or *local rings: $\dim M=0$ implies $R/Ann(M)$ artinian and since $M$ is f.g. we get $M$ artinian, that is, $l(M)<\infty$. 2. There is a dimension theory for graded modules over graded rings closely related to the non-graded case, but what do you call by the "Fundamental Theorem of Dimension Theory"?
    • – user26857 Apr 25 '14 at 20:09
  • I see. I was trying to prove it via a dimension-theoretic argument. 2. I am referring to theorem 13.4 in Matsumura. Where can i find this dimension theory for graded modules? I am a lot interested in that. B&H have some theorems but not anything analogous to 13.4 in Matsumura.
    • – Manos Apr 25 '14 at 20:18