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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 1.1. 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 1.1 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 1.1 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@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4249 87 . D.T. Huong / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 87-90 88 Definition 2.1. 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 ideals containing . This condition is equivalent to for all and for all . Remark 2.2. It should be noted that for any we always can choose a -filter regular sequence on M . Indeed, by the prime avoidance lemma we can choose and for all . For assume that we have , then we choose and for all by the prime avoidance lemma again. For more details, see [1, Section 2]. The -filter regular sequence can be seen as a generalization of the well-known notion of regular sequence (cf. [4, Proposition 2.2]). Lemma 2.3. A sequence is an -filter regular sequence on M if and only if for all , and for all such that we have is an -sequence. Corollary 2.4. Let is -torsion for all an be an -filter regular sequence on Proof. For each we have either -regular sequence by Lemma 2.3. For the first case we have ( for all . Then . ) ( or is ) . For the second case we have ( for all have ( torsion for all ) by the Grothendieck vanishing theorem [2, Theorem 6.2.7]. Therefore we for all and for all . So ) is  . It is well-known that local cohomology agrees with the -th cohomology of the Čech complex with respect to the sequence → → → → The following simple fact is the crucial key for our proof. Lemma 2.5. Let be any element of Proof. Obiviously the multiplication map . Then is an isomorphism. It induces isomorphism maps for all . But is - torsion, so it Therefore for all . We are ready to prove the theorem of Nagel and Schenzel. -torsion since .  D.T. Huong / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 87-90 Proof of Theorem 1.1. We set complex and set short exact sequences the ideal generated by and . Let for all 89 the -th chain of Čech . We split the Čech complex into ( ) ( ) … ( ) ( ) ( ) By Lemma 2.3 we have of for all we have torsion for all ( ) for all ( ) ( ) ( by Corollary 2.4, so and for all and all for all ) ( ) Since and are submodules . We also note that ( ) and ( ( ) ( ) ) is for all . Now applying the functor observations we have to the short exact sequence and using the above ( ) and for all For each sequence we have , applying the local cohomology functor 0 and the isomorphism ( ) for all . Furthermore, if we apply exact sequence ( ) ( ) and the isomorphism ( ) for all . Note that ( ) for the short exact sequence ( ) as above, so ( ) to the short exact ( ) ), then we get the short D.T. Huong / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 87-90 90 We next show that and for all ( ) . Indeed, using isomorphisms consecutively, we have ( D1 ) (1)  ( D2 ) (C1 )  (Ci1 )   ( 2)   ( ) Therefore, we have showed the first case of Nagel-Schenzel's isomorphism for all . Finally, for (1)  by similar arguments we have ( D1 )  (C1 )  On the other hand, by applying the functor ( ( ) for all ( ) ( D2 )  (C t 1 )  to the short exact sequence we have ) . Thus ( ) for all , and we finish the proof.  References [1] J. Asadollahi and P. Schenzel, Some results on associated primes of local cohomology modules, Japanese J. Mathematics 29 (2003), 285–296. [2] M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998. [3] H. Dao and P.H. Quy, On the associated primes of local cohomology, Nagoya Math. J., to appear. [4] U. Nagel and P. Schenzel, Cohomological annihilators and Castelnuovo-Mumford regularity, in Commutative algebra: Syzygies, multiplicities, and birational algebra, Contemp. Math. 159 (1994), Amer. Math. Soc. Providence, R.I., 307–328. [5] P.H. Quy and K. Shimomoto, F -injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0, Adv. Math. 313 (2017), 127–166.