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

What is known about the following subset of $ \mathrm {Spec}(R)$:

$$\mathrm{supp}_{fl}(M)=\{P\in \mathrm {Spec}(R):\ \mathrm{Tor}^R_1(M, R/P)= 0\}?$$

In particular,

Is $\mathrm{supp}_{fl}(M)$ open in $\mathrm {Spec}(R)$?


2 Answers 2


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 $\mathrm{Spec}(R)$.

  • $\begingroup$ Yes, you're right. I'll edit my answer accordingly. $\endgroup$ Oct 21, 2016 at 11:40

At least $\mathrm{supp}_{fl}(M)$ is open in $\mathrm{mSpec}(R)$. To see this, let $(C_\cdot,d)$ be a resolution of $M$ by finitely-generated free modules. By looking at matrices over $R$, one sees $\mathrm{Tor}^R_1(M,R/P)\neq0$ if and only if a maximal prime $P$ contains the (maximal) Fitting ideal of $d_1\oplus d_0$.

These closed sets are the homology jump loci of the complex $C_\cdot$, and the argument I sketch appears in

Nero Budur, MR 2875860 Complements and higher resonance varieties of hyperplane arrangements, Math. Res. Lett. 18 (2011), no. 5, 859--873.

though it goes back further.


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