A special case of positivity (II)

Abstract:

In this note we prove the following special case of Serre’s conjecture on Intersection Multiplicity: Let $(R,m)$ be a regular local ring and let $P$, $Q$ be two prime ideals such that $\ell (R/(P+Q))<\infty$, $\dim R/P +\dim R/Q=\dim R$ and dimension of $G_{m}(R/P)\otimes _{G_{m}(R)}G_{m}(R/Q)<2$. Then $\chi (R/P,R/Q)\geq e_{m}(R/P) e_{m}(R/Q)$; here $e_{m}(T)$ denotes the Hilbert-Samuel multiplicity for any finitely generated module $T$ with respect to $m$.