An elementary proof of Nagel-Schenzel formula

Document Type : Research Paper

Author

Department of Mathematics, Payame Noor University (PNU), P.O.BOX, 19395-4697, Tehran, Iran

10.29252/as.2019.1359

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
\begin{align*}
\operatorname{H}^i_\mathfrak{a}(M)\cong
\left\{\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{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.

Keywords


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