Primary decomposition of zero-dimensional modules

Question

(I removed my motivation because it may be misleading :) )

Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) = 0$) $A$-module. Then the submodule $0 < M$ has primary decomposition $0 = \bigcap_{\mathfrak{p} \in \mathrm{Supp}(M)} M(\mathfrak{p})$, where $M(\mathfrak{p})$ is the $\mathfrak{p}$-primary component of $M$, i.e. the kernel of the canonical morphism $M \rightarrow M_{\mathfrak{p}}$. I have proven (it's hopefully correct) that the canonical morphism $\mathbf{j}:M \rightarrow \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M/M(\mathfrak{p})$ ist an isomorphism and that $\mathbf{j}(M \lbrack \mathfrak{q}^\infty \rbrack) \leq M/M(\mathfrak{q}) \leq \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M/M(\mathfrak{p})$, where $M \lbrack \mathfrak{q}^\infty \rbrack$ is the $\mathfrak{q}$-torsion part of $M$ ($0$-th local cohomology with support $\mathfrak{q}$). Now, my question is if in general $\mathbf{j}(M \lbrack \mathfrak{q}^\infty \rbrack) = M/M(\mathfrak{q})$ so that $M = \bigoplus_{\mathfrak{p} \in \mathrm{Supp}(M)} M \lbrack \mathfrak{p}^\infty \rbrack$, or do I need to assume that $A$ is a Dedekind ring (and perhaps also that $M$ is a torsion module) and if so, how can I prove this?

Answer 1

If $M$ is finitely generated and has $0$-dimensional support, then $M$ is in fact supported at finitely many maximal ideals (a $0$-dimensional closed subset of the Spec of a Noetherian ring is just a finite union of maximal ideals), and one has the isomorphism $M = \oplus_{\mathfrak p} M_{\mathfrak p} = \oplus_{\mathfrak p} M[\mathfrak p^{\infty}].$

The first equality just specifies the fact that, since the quasi-coherent sheaf on Spec $A$ attached to $M$ is supported at finitely many closed points, it is a sky-scraper sheaf at these points, and its global sections are just the sum of its stalks at those finitely many points. (In particular, at all ${\mathfrak p}$ not in the support of $M$, the localization $M_{\mathfrak p}$ vanishes, and so does not contribute to the direct sum, so really the direct sum is just over the finitely many points in the support.) To see the second equality (which is the crux of the question as far as I can tell), note that we reduce to the local case: $M_{\mathfrak p}$ is finitely generated over $A_{\mathfrak p}$, and has support equal to the closed point ${\mathfrak p}$ of Spec $A_{\mathfrak p}$. A consideration of the very definition of support will now show that each element of $M_{\mathfrak p}$ is annihilated by some power of ${\mathfrak p}$, and hence that $M_{\mathfrak p} = M[\mathfrak p^{\infty}]$.

From this decomposition everything else is easy to work out: for example, $M(\mathfrak p)$ (which if I understand correctly is defined to be the kernel of the map to $M_{\mathfrak p}$) is precisely $\oplus_{\mathfrak q \neq \mathfrak p} M_{\mathfrak q} = \oplus_{\mathfrak q \neq \mathfrak p} M[\mathfrak q^{\infty}].$

In particular, if we want to isolate $M[\mathfrak q^{\infty}],$ we just intersect the $M(\mathfrak p)$ for all $\mathfrak p \neq \mathfrak q$. This explains the formula in Arminius's answer.

In short: rather than having to memorize or quote technical results from Eisenbud (or elsewhere), one can use geometric reasoning on Spec $A$ to answer such questions. (All the technical reasoning has been embedded in one step: the proof of the correspondence between $A$-modules and quasi-coherent sheaves on Spec $A$.)

Answer 2

I could answer this now myself (so, sorry for asking!): From a general result (presented e.g. in Eisenbud's commutative algebra book) it follows that $M \lbrack \mathfrak{q}^\infty \rbrack = \bigcap_{\mathfrak{p} \in \mathrm{Supp}(M), \mathfrak{p} \neq \mathfrak{q}} M(\mathfrak{p})$ and then it follows that $\mathbf{j}(M \lbrack \mathfrak{q}^\infty \rbrack) = M/M(\mathfrak{q})$.