Let $R$ be a Noetherian commutative ring and $M$ a finitely generated $R$-module.



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> What is known about the following subset of $ \mathrm {Spec}(R)$:
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> $$\mathrm{supp}\_{fl}(M)=\{P\in \mathrm {Spec}(R):\ \mathrm{Tor}^R\_1(M, R/P)= 0\}?$$
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> 


In particular,



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> Is $\mathrm{supp}\_{fl}(M)$ open in $\mathrm {Spec}(R)$?
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The flat support of a module
I suspect that Graham Denham's argument only shows that the weaker condition $\mathrm{Tor}\_1(M,\kappa(P))=0$ is open, where $\kappa(P)$ is the residue field of $P$. With the problem as stated, here is a counterexample: for a field $k$, take $R=k[x,y]$ and $M=R/(x,y)$. Then it is easy to check that $\mathrm{supp}\_{fl}(M)$ consists of:  

$\bullet$ the generic point of $\mathrm{Spec}(R)$,  

$\bullet$ all closed points except the origin $(x,y)$, and  

$\bullet$ all principal primes $(f)$ where $f$ is irreducible and not in $(x,y)$, i.e. all generic points of curves not containing the origin.


This is not open in $\mathrm{Spec}(R)$.


