On Minimax Modules and Generalized Local Cohomology with Respect to a Pair of Ideals
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On Minimax Modules and Generalized Local Cohomology with Respect to a Pair of Ideals Nguyen Thanh Nam1 · Tran Tuan Nam2 · Nguyen Minh Tri3 Received: 8 December 2019 / Revised: 22 February 2020 / Accepted: 3 March 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We study the minimax properties and the artinianness of the generalized local cohomology i (M, N ) with respect to a pair of ideals (I, J ). We also show some results on modules HI,J top generalized local cohomology modules. Keywords Attached primes · Generalized local cohomology · Minimax Mathematics Subject Classification (2010) 13D45
1 Introduction Throughout this paper, R is a noetherian commutative (with non-zero identity) ring. It is well-known that the local cohomology theory of Grothendieck is an important tool in commutative algebra and algebraic geometry. There are some generalizations of this theory, one of them was introduced by Herzog [9]. He defined the generalized local cohomology modules HIi (M, N ) of M and N with respect to I by HIi (M, N ) = lim ExtiR (M/I n M, N ). −→ n
Tran Tuan Nam
[email protected]; [email protected] Nguyen Thanh Nam [email protected] Nguyen Minh Tri [email protected]; [email protected] 1
Faculty of Mathematics & Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam
2
Department of Mathematics-Informatics, Ho Chi Minh University of Pedagogy, Ho Chi Minh City, Vietnam
3
Department of Natural Science Education, Dong Nai University, Dong Nai, Vietnam
N.T. Nam et al.
In 2009, Takahashi, Yoshino, and Yoshizawa introduced local cohomology modules with respect to a pair of ideals (I, J ) [16]. For an R-module M, the (I, J )-torsion submodule of i M is ΓI,J (M) = {x ∈ M | I n x ⊆ J x for some positive integer n}. They denoted by HI,J the ith right derived functor of the functor ΓI,J . It is clear that when J = 0, the functor i coincides with the usual local cohomology functor H i . In [1], Aghapournahr studied HI,0 I the cofiniteness of local cohomology modules for a pair of ideals for small dimensions. In [13], for two R-modules M and N, we defined ΓI,J (M, N ) to be the (I, J )torsion submodule of HomR (M, N ). For each R-module M, there is a covariant functor ΓI,J (M, −) from the category of R-modules to itself. The ith generalized local cohomoli (M, −) with respect to a pair of ideals (I, J ) is the ith right derived functor ogy functor HI,J of the functor ΓI,J (M, −). This definition is really a generalization of the local cohomoli . On the other hand, H i (M, N ) is also an extension of the generalized ogy functor HI,J I,J local cohomology module HIi (M, N ). Also, in [13], we study some basic properties of i (M, N ). HI,J The organization of the paper is as follows. In Section 2, we study the minimax propi (M, N ). It should be mentioned that the minimax modules were introduced erties of HI,J by H. Z¨oschinger [18]. Let M be a finitely generated R-module. The fi
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