An elementary proof of Nagel-Schenzel formula

Document Type : Research Paper


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


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
\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},
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.


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