Cohomological dimension with respect to the linked ideals

TL;DR

This paper explores the cohomological dimension of ideals in Noetherian rings using a new linkage concept, revealing relationships with local cohomology and invariance under geometric linkage.

Contribution

It introduces a novel linkage framework for ideals over modules and establishes formulas connecting cohomological dimension with local cohomology and ideal linkage.

Findings

Cohomological dimension relates to the grade of ideals and local cohomology.

Cohomological dimension remains invariant under geometric linkage.

Provides explicit formulas for cohomological dimension in linked ideals.

Abstract

Let $R$ be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if $\mathfrak{a}$ is an ideal of $R$ which is linked by the ideal $I$, then $cd(\mathfrak{a},R) \in \{ grad \mathfrak{a}, cd(\fa, H^{grad \mathfrak{a}}_ {\mathfrak{c}} (R)) + grad \mathfrak{a}\}, $ where $\mathfrak{c} : = \bigcap_{\mathfrak{p} \in Ass \frac{R}{I}- V(\mathfrak{a})}\mathfrak{p}$. Also, it is shown that for every ideal $\mathfrak{b}$ which is geometrically linked with $\mathfrak{a},$ $cd(\mathfrak{a}, H^{grad \mathfrak{b}}_ {\mathfrak{b}} (R))$ does not depend on $\mathfrak{b}$