An Elementary Proof of Nagel-Schenzel Formula

Let $r$ be a commutative noetherian ring with nonzero 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 formulabegin{align*}operatorname{h}^i_mathfrak{a}(m)congleft{begin{array}{lll}operatorname{h}^i_{(a_1, ldots, a_n)}(m) & text{for all} mathrm{i< n},operatorname{h}^{i n}_mathfrak{a}(operatorname{h}^n_{(a_1, ldots, a_n)}(m)) & text{for all} mathrm{igeq n},end{array}right.end{align*}is known as nagelschenzel 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.