An elementary proof of Nagel-Schenzel formula

Abstract

Let $R$ be a commutative Noetherian ring with non-zero identity, $mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $a_1, ldots, a_n$ an $mathfrak{a}$--filter regular $M$--sequence. The formula<br />begin{align*}<br />operatorname{H}^i_mathfrak{a}(M)cong<br />left{begin{array}{lll}<br />operatorname{H}^i_{(a_1, ldots, a_n)}(M) & text{for all} mathrm{i< n},<br />operatorname{H}^{i- n}_mathfrak{a}(operatorname{H}^n_{(a_1, ldots, a_n)}(M)) & text{for all} mathrm{igeq n},<br />end{array}right.<br />end{align*}<br />is known as Nagel-Schenzel formula and is a useful result to express the local cohomology modules in terms of filter regular sequences. In this paper, we provide an elementary proof to this formula.