Support of a module with extended scalars I have a question which should be pretty basic commutative algebra, but I can't find a reference and I'm stuck on proving a result myself, so here it goes:
Let $\varphi: S \to R$ be a morphism of commutative rings and let $M$ be a finitely generated $S$-module. Then, we can "extend the scalars of $M$" by considering $M \otimes_S R$, which is naturally an $R$-module. I would like to get a description of $\text{Supp}(M \otimes_S R)$ (as an $R$-module) involving $\text{Supp}(M)$ and $\varphi$.
Maybe this helps: by Eisenbud's Commutative algebra with a view toward algebraic geometry, Corollary 2.7, the problem can probably be solved by finding a description of $\text{Ann}(M \otimes_S R)$ and it is not difficult to see that $\varphi(\text{Ann}(M))R \subset \text{Ann}(M \otimes_S R)$, but I fail to prove the other inclusion.
 A: The support of $M$ is the set $\{ \mathfrak q \in \text{ Spec } S \, | \, \kappa(q)\otimes_S M
\neq 0\}$, where $\kappa(\mathfrak q)$ denotes the fraction field of the domain $S/\mathfrak q$.  (Since $M$ is finitely generated, one sees by Nakayama's lemma that the stalk $M_{\mathfrak q}$ is non-zero if and only if $\kappa(\mathfrak q)\otimes_S M$ is non-zero).  
The support of $R\otimes_S M$ is defined analogously.
Now if $\mathfrak p$ is an element of Spec $R$, then
$$\kappa(\mathfrak p)\otimes_R (R\otimes _S M) = \kappa(\mathfrak p)\otimes_S M
= \kappa(\mathfrak p)\otimes_{\kappa(\mathfrak q)} (\kappa(\mathfrak q)\otimes_S M),$$
where $\mathfrak q$ is the preimage of $\mathfrak p$ in $S$.
Since $\kappa(\mathfrak q) \to \kappa(\mathfrak p)$ is just an inclusion of fields,
we see that $\kappa(\mathfrak p)\otimes_R (R\otimes_S M) \neq 0$ if and only if
$\kappa(q)\otimes_S M \neq 0$.
In conclusion, the support of $R\otimes_S M$ is the preimage in Spec $R$ of the support
of $M$ in Spec $S$.
All of this has a geometric interpretation:
The first description of the support says that, when $M$ is finitely generated, we can
check that the stalks of $M$ are non-zero by instead checking if the fibres are non-zero.
Then the computation relating the fibre of $R\otimes_S M$ at $\mathfrak p$ to
the fibre of $M$ at $\mathfrak q$ just says that the fibre of the pull-back is the pull-back of the fibre.
Finally, note that if you think geometrically, the result makes intuitive sense:
we are pulling back a sheaf (i.e. applying $\varphi^*$), and we can think of the sheaf as being a subset of
Spec $R$ (namely its support) with extra decoration (i.e. each point has a certain fibre
of the sheaf lying over it).  When we pull back the sheaf, we pull back the subset 
(i.e. we pull back the support), and then at each point of the pull back we also pull back
the extra decoration.  Thinking about supports just involves forgetting the extra decoration.
A: Since you say that you can't find a reference, look at Atiyah Macdonald Chapter 3, Exercise 19(viii). 
