Serre's positivity conjecture for intersection multiplicities

Theorem 1.1: if $(R,\mathfrak m)$ is a commutative Noetherian regular local ring of dimension $d$ and $M,N$ are nonzero finitely generated modules with $\ell(M\otimes_R N)<\infty$ and $\dim M+\dim N=d$, then $\chi^R(M,N)>0$. This covers ramified mixed characteristic and needs no lift to an unramified ring. Consequences recorded: the Kurano-Roberts symbolic-power containment $P\cap Q^{(r)}\subseteq\mathfrak m^{r+1}$ in ramified mixed characteristic, and a strict-transform inequality $\chi^A(A/P,A/Q)\ge e(A/P)e(A/Q)$ with an equality criterion. It does not address Serre's other questions beyond positivity (vanishing and nonnegativity were already known), and gives no explicit lower bound beyond $\chi\ge1$.

Theorem 1.1: if $(R,\mathfrak m)$ is a commutative Noetherian regular local ring of dimension $d$ and $M,N$ are nonzero finitely generated modules with $\ell(M\otimes_R N)<\infty$ and $\dim M+\dim N=d$, then $\chi^R(M,N)>0$. This covers ramified mixed characteristic and needs no lift to an unramified ring. Consequences recorded: the Kurano-Roberts symbolic-power containment $P\cap Q^{(r)}\subseteq\mathfrak m^{r+1}$ in ramified mixed characteristic, and a strict-transform inequality $\chi^A(A/P,A/Q)\ge e(A/P)e(A/Q)$ with an equality criterion. It does not address Serre's other questions beyond positivity (vanishing and nonnegativity were already known), and gives no explicit lower bound beyond $\chi\ge1$.