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.


[1] M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applica-
tions, Cambridge University Press, Cambridge, 1998.
[2] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1998.
[3] K. Khashyarmanesh and Sh. Salarian, Filter regular sequences and the niteness of local cohomology mod-
ules, Comm. Algebra 26 (1998) 2483-2490.
[4] U. Nagel and P. Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, Commutative
algebra: Syzygies, multiplicities, and birational algebra, Contemp. Math. 159 (1994) 307-328.
[5] J. J. Rotman, An Introduction to Homological Algebra, Academic Press, San Diego, 1979.
[6] P. Schenzel, N. V. Trung, and N. T. Cuong, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85
(1978) 57-73.
[7] J. Stuckrad and W. Vogel, Buchsbaum Rings and Applications, VEB Deutscher Verlag der Wissenschaften,
Berlin, 1986.