Cohomological dimension with respect to the 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 $\mathfrak{b}$