A dimension inequality for Cohen-Macaulay rings

Abstract:

The recent work of Kurano and Roberts on Serre’s positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak {p}$ and $\mathfrak {q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak {n})$ such that $e(A_{\mathfrak {p}})=e(A)$ we have $\dim (A/\mathfrak {p})+\dim (A/\mathfrak {q})\leq \dim (A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak {p}$ is regular.