Associated Primes of Tensor Product

Let $R$ be a Noetherian ring, and let $M$ and $N$ be finitely generated $R$ module. Do we know any formulas for $\operatorname{Ass}(M\otimes_R N)$ in terms of $\operatorname{Ass}(M)$, $\operatorname{Ass}(N)$ or in terms of $\operatorname{Supp}(M)$ or $\operatorname{Supp}(N)$?

Recall that we have such a formula for the support, i.e., $\operatorname{Supp}(M\otimes_R N)=\operatorname{Supp}M\cap \operatorname{Supp}N$. We also have a formula for $\operatorname{Ass}(\operatorname{Hom}(M,N))$.

I have not seen any formula for Ass of tensor products, it would be nice to have such a formula in at least a few special cases.

• I think that in general almost nothing can be said. See for example $M =A/I$ and $N = A/J$. Apr 10, 2013 at 12:42

The following is THM 23.2 in Matsumura's "Commutative Ring Theory":

Let $$\phi:A \rightarrow B$$ be a homomorphism of Noetherian rings, and let $$E$$ be an $$A$$-module and $$G$$ be a $$B$$-module. Suppose $$G$$ is flat over $$A$$; then:

(i) If $$\mathfrak{p} \in \text{Spec}(A)$$ and $$G/\mathfrak{p}G \ne 0$$, then, letting $$\phi^a: \text{Spec}(B) \rightarrow \text{Spec}(A)$$ be the induced map (this is Matsumura's notation) we have $$\phi^a(\text{Ass}_B(G/\mathfrak{p}G))=\text{Ass}_A(G/\mathfrak{p}G)=\{\mathfrak{p}$$}

(ii) $$\text{Ass}_B(E \otimes_A G) = \bigcup_{\mathfrak{p} \in \text{Ass}_A(E)}\text{Ass}_B(G/\mathfrak{p}G)$$

So, this answers the question in the case where $$M$$ is a flat $$R$$-module; take $$A=R=B$$, $$E=N$$, and $$G=M$$, and apply (ii).

So this isn't very interesting when $$R$$ is local, for instance...but it is the best I can do.

• I know this result, but as you mention, it is not as useful once the ring is local. Apr 11, 2013 at 10:18