On set theoretically and cohomologically complete intersection ideals

TL;DR

This paper investigates conditions under which classical inequalities relating height, cohomological dimension, arithmetical rank, and minimal number of generators of ideals in local rings become equalities, using formal grade.

Contribution

It provides new conditions involving formal grade that characterize when these ideal invariants are equal, advancing understanding of set-theoretically and cohomologically complete intersection ideals.

Findings

Identifies conditions for equality of ideal invariants using formal grade.

Clarifies the relationship between height, cohomological dimension, and arithmetical rank.

Enhances the theoretical framework for complete intersection ideals.

Abstract

Let $(R,\fm)$ be a local ring and $\fa$ be an ideal of $R$. The inequalities $$\begin{array}{ll} \ \Ht(\fa) \leq \cd(\fa,R) \leq \ara(\fa) \leq l(\fa) \leq \mu(\fa) \end{array}$$ are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.