Extension of scalars and support of a non-finitely generated module

Question

This is a question about support of modules under extension of scalars.

Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module. Recall that the support of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localization $M_{\mathfrak{p}}$ is nonzero. Then $\mathrm{Supp} _B(B \otimes_A M) = f^{*-1}(\mathrm{Supp}_A(M))$, the set of prime ideals of $B$ whose contractions are in the support of $M$.

The $\subseteq$ containment is true for any $M$. What's an obvious example of a non-finitely generated module where the other containment doesn't hold?

Answer 1

Take $A = \mathbb Z$, $M = \mathbb Q$, $B = \mathbb Z/p\mathbb Z$, were $p$ is a prime.

Answer 2

Okay, a further question. Despite its simplicity, it is not homework.

Atiyah and Macdonald (Ch. 2, Ex. 19.iv)) claim that if $M$ is a sum of submodules $M_i$, then $\text{Supp}(M) = \bigcup \text{Supp}(M_i)$, not specifying what size of set the indices $i$ are to run over. I assumed initially the index set was to be assumed arbitrary, and came up with an argument, which it now occurs to me may be wrong, as your example makes me wonder if the sum should really be finite. (Or I'm possibly just (continuing to be) stupid.)

Continue with your example, so $M = \mathbb Q$ is a $A =\mathbb Z$-module, and $B = \mathbb Z / p \mathbb Z$. Now $\mathbb Q$ is generated over $\mathbb Z$ by the elements $\frac 1 n \in \mathbb Z$, so it is a sum of the submodules $\frac 1 n \mathbb Z$, Thus the elements $z_n = 1 \otimes \frac 1 n$ generate $B \otimes_A M = \mathbb Z / p \mathbb Z \ \otimes_{\mathbb Z} \mathbb Q$; in fact all are zero, since the generator $z_n = p \otimes \frac 1 {pn}$ is zero already in $\mathbb Z / p \mathbb Z \ \otimes_{\mathbb Z} \frac{1}{pn}\mathbb Z$.

So $\mathbb Z / p \mathbb Z \ \otimes_{\mathbb Z} \mathbb Q = 0$ has empty support, even though the $\mathbb Z / p \mathbb Z$-submodules $\mathbb Z / p \mathbb Z \ \otimes_{\mathbb Z} \frac 1{n} \mathbb Z \cong \frac 1{n} \mathbb Z / \frac p{n} \mathbb Z \cong \mathbb Z / p \mathbb Z$ all have $(0)$ in their support.

Is this right, or am I making an obvious mistake? Should the sum of modules be assumed finite for the proposition to hold?

Answer 3

The question is amount to the statement that if $f:(A,\mathfrak{m})\to (B,\mathfrak{n})$ is a local homomorphism and $M$ is a non-zero $A$-module then $M\otimes _AB$ is also non-zero. Notice that this statement is true if $M$ as an $A$-module possesses a minimal generating set. A proof would be as follows: In a presentation of $M$ over $A$ the entries of the representing matrix belong to $\mathfrak{m}$. After tensoring into $B$ over $A$, the entries go to $\mathfrak{n}$; so that the cokernel of the new matrix (which is isomorphic to $M\otimes _AB$) won't be zero.

In the example, by Angelo, $\mathbb{Q}$ does not have a minimal generating set as a $\mathbb{Z}$-module.