Cohomological and projective dimensions

TL;DR

This paper establishes an upper bound for the cohomological dimension of certain graded ideals in polynomial rings, providing new examples of prime ideals that are not set-theoretically Cohen-Macaulay, applicable in all characteristics.

Contribution

It introduces a new upper bound for cohomological dimension in characteristic 0 and extends the result to positive characteristic, with applications to Cohen-Macaulay properties.

Findings

Upper bound for cohomological dimension in characteristic 0

Extension of bound to positive characteristic via Peskine and Szpiro's theorem

New examples of prime ideals not set-theoretically Cohen-Macaulay

Abstract

In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a well-known theorem of Peskine and Szpiro. As a corollary, we give new examples of prime ideals that are not set-theoretically Cohen-Macaulay.