Serre's positivity conjecture

Let $R$ be a commutative Noetherian regular local ring, let $\dim$ denote Krull dimension, and for an $R$-module $M$ let $\ell_R(M)$ denote its length. Let $P$ and $Q$ be prime ideals of $R$ such that

$$\ell_R\big((R/P)\otimes_R(R/Q)\big)<\infty .$$

Then every module $\operatorname{Tor}_i^R(R/P,R/Q)$ has finite length, and, $R$ being regular, $\operatorname{Tor}_i^R(R/P,R/Q)=0$ for $i>\dim(R)$; hence the intersection multiplicity

$$\chi(R/P,R/Q):=\sum_{i=0}^{\infty}(-1)^{i}\,\ell_R\big(\operatorname{Tor}_i^R(R/P,R/Q)\big)$$

is a well-defined integer, and $\dim(R/P)+\dim(R/Q)\le \dim(R)$.

Under these hypotheses, if

$$\dim(R/P)+\dim(R/Q)=\dim(R),$$

then

$$\chi(R/P,R/Q)>0 .$$