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 $\mathfrak{ b}$
Categories: math.AC