# Support sets along a ring homomorphism.

Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-module, then what is the relation between $Supp_s(M\otimes S)$ and $Supp_R(M)$? Thanks in advance!

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Please try harder to figure it out for yourself. I will give you two hints: pass to quotient by annihilator of $M$ to focus on the case when the natural map of $R$-modules $R \rightarrow {\rm{End}}_R(M)$ is injective. Now make use of flatness (locality is irrelevant), including how flat scalar extension interacts with the formation of Hom-modules among finitely generated noetherian rings. (There are alternative approaches if you know how to think geometrically about modules.) –  BCnrd Aug 11 '10 at 3:16
In the 2nd hint above, please replace "finitely generated noetherian rings" with "finitely generated modules over noetherian rings". –  BCnrd Aug 11 '10 at 5:11