Nagel-Schenzel’s isomorphism that has many applications was proved by using spectral sequence theory. In this short note, we present a simple proof for the theorem of Nagel and Schenzel. | VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 87-90 A Simple Proof for a Theorem of Nagel and Schenzel Duong Thi Huong* Department of Mathematics, Thang Long University, Hanoi, Vietnam Received 12 December 2017 Revised 25 December 2017; Accepted 28 December 2017 Abstract: Nagel-Schenzel’s isomorphism that has many applications was proved by using spectral sequence theory. In this short note, we present a simple proof for the theorem of Nagel and Schenzel. Keywords: Local cohomology, filter regular sequence. 1. Introduction Throughout this paper, let be a commutative Noetherian ring, a finitely genrated -module and an ideal of . Local cohomology introduced by Grothendieck, is an important tool in both algebraic geometry and commutative algebra (cf. [2]). Moreover, the notion of -filter regular sequences on is an useful technique in study local cohomology. In [4] Nagel and Schenzel proved the following theorem (see also [1]). Theorem . Let be an ideal of a Noetherian ring and a finitely generated -module. Let an -filter regular sequence of . Then we have { The most important case of Theorem is , and is a submodule of . Recently, many applications of this fact have been found [3,5]. It should be noted that Nagel-Schenzel’s theorem was proved by using spectral sequence theory. The aim of this short note is to give a simple proof for Theorem based on standard argument on local cohomology [2]. 2. Proofs Firstly, we recall the notion of -filter regular sequence on _ Corresponding author. Tel.: 84-983602625. Email: duonghuongtlu@ https// 87 . . Huong / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 87-90 88 Definition . Let be a finitely generated module over a local ring , k) and let be a sequence of elements of . Then we say that is a -filter regular sequence on if the following conditions hold: Supp for all where denotes the set of prime
