Remark 43.19.5. Let $(A, \mathfrak m, \kappa )$ be a regular local ring. Let $M$ and $N$ be nonzero finite $A$-modules such that $M \otimes _ A N$ is supported in $\{ \mathfrak m\}$. Then

$\chi (M, N) = \sum (-1)^ i \text{length}_ A \text{Tor}_ i^ A(M, N)$

is finite. Let $r = \dim (\text{Supp}(M))$ and $s = \dim (\text{Supp}(N))$. In it is shown that $r + s \leq \dim (A)$ and the following conjectures are made:

1. if $r + s < \dim (A)$, then $\chi (M, N) = 0$, and

2. if $r + s = \dim (A)$, then $\chi (M, N) > 0$.

The arguments that prove Lemma 43.19.4 and Proposition 43.19.3 can be leveraged (as is done in Serre's text) to show that (1) and (2) are true if $A$ contains a field. Currently, conjecture (1) is known in general and it is known that $\chi (M, N) \geq 0$ in general (Gabber). Positivity is, as far as we know, still an open problem.

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