A dimension inequality for excellent, Cohen-Macaulay rings related to the positivity of Serre's intersection multiplicity

Assume that (R, m ) is a Noetherian local ring. Kurano and Roberts have made the following conjecture related to the positivity of Serre's intersection multiplicity. Assume that R is regular and contains prime ideals p and q such that p+q=m and dim(R/ p ) + dim(R/ q ) = dim(R ); then pn ∩q⊆m n+1for alln≥1. We consider this conjecture and the following question, which is a generalization of the conjecture. Assume that R is quasi-unmixed with prime ideals p and q such that p+q=m and eRp =eR . Does the inequality dimR/p +dimR/q ≤dim R hold? We answer this question in the affirmative in the following cases: (1) R is excellent and contains a field. (2) ht ( p ) = 0. (3) R is Nagata and ht ( q ) = 0. (4) dim(R/ q ) = 1. (5) R is Nagata and dim(R/ p ) = 1. (6) R is Nagata and R/ p is regular. We also verify the original conjecture of Kurano and Roberts in a number of cases (with no excellence restriction), most notably when (1) R contains a field. (2) p is generated by a regular sequence. (3) q is generated by part of a regular system of parameters. We also present a number of examples that demonstrate the necessity of each of the assumptions of the conjectures as well as the limitations of some of our results.