Multiplicities and a Dimension Inequality for Unmixed Modules

We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serreʹs positivity conjecture. Assume that (R, ) is a local ring with finitely generated module M such that R/Ann(M) is quasi-unmixed and contains a field, and that and are prime ideals in the support of M such that is analytically unramified, and e(M ) = e(M). Then We also prove a similar theorem in a special case of mixed characteristic. Finally, we provide several examples to explain our assumptions as well as an example of a noncatenary local domain R with prime ideal such that e(Rp) > e(R) = 1.