An elementary proof of Nagel-Schenzel formula

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_۱, \ldots, a_n an \mathfrak{a}--filter regular M--sequence. The formula\begin{align*}\operatorname{H}^i_\mathfrak{a}(M)\cong\left\{\begin{array}{lll}\operatorname{H}^i_{(a_۱, \ldots, a_n)}(M) & \text{for all}\ \mathrm{i< n},\\\operatorname{H}^{i- n}_\mathfrak{a}(\operatorname{H}^n_{(a_۱, \ldots, a_n)}(M)) & \text{for all}\ \mathrm{i\geq n},\end{array}\right.\end{align*}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.

کلیدواژه ها:

Filter regular sequences ، local cohomology modules ، Nagel-Schenzel formula