2002/12/09 by Sean Sather-Wagstaff
Mathematics · #math.AC
published as Journal of Algebra 238 (2001), 372-388, doi:10.1006/jabr.2000.8630
arxiv created 2002/12/09 · arxiv updated 2009/11/30
We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely-generated module M such that R/ann(M) is quasi-unmixed and contains a field, and that p and q are prime ideals in the support of M such that p is analytically unramified, p+q is m-primary and e(Mp)=e(M). Then dim(R/p)+dim(R/q)≤ dim(M). 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 p such that e(Rp)>e(R)=1.