Cominimaxness properties of formal local cohomology modules
aa r X i v : . [ m a t h . A C ] N ov COMINIMAXNESS PROPERTIES OF FORMAL LOCAL COHOMOLOGY MODULES
BEHRUZ SADEQIA
BSTRACT . Let a be an ideal of local ring ( R , m ) and M a finitely generated R -module and n ∈ N . Itis shown that some results concerning cominimaxness of formal local cohomology modules.
1. I
NTRODUCTION
Throughout this paper, R is a commutative Noetherian ring with identity, a is an ideal of R and M is an R -module. Recall that the i -th local cohomology module of M with respect to a is denotedby H i a ( M ) . For basic facts local cohomology refer to [2]. Let a be an ideal of a local ring ( R , m ) and M a ?nitely generated R -module. For each i ≥ F i a ( M ) = lim ←− n H i m ( M / a n M ) is called the i -thformal local cohomology of with respect to a . The basic properties of formal local cohomologymodules are found in [6], [1]. Recall that a module M is a minimax module if there is a finitelygenerated submodule N of M such that the quotient module M / N is Artinian. An R -module M is an a -cominimax module if Supp R ( M ) ⊆ V ( a ) and Ext iR ( R / a , M ) is a minimax module for all i ≥
0. The class of cominimax modules includes all cofinite and all Artinian modules. The notionsof weakly Laskerian modules were introduced by Divaani-Aazar and Mafi in [3]. An R module M is said to be weakly Laskerian if the set of associated primes of any quotient module of M isfinite. Moreover it is closed under taking submodules, quotients and extensions, i.e., it is a Serresubcategory of the category of R -modules.In this paper we investigate some cominimaxness properties of formal local cohomology mod-ules. 2. C OMINIMAXNESS OF FORMAL LOCAL COHOMOLOGY MODULES
We begin with an example show that the class of cofinite modules with respect to an ideal isstrictly contained in the class of cominimax modules with respect to the same ideal.
Example . Let ( R , m ) be a local ring and p a prime ideal of R such that dimR / p =
1. Then it iseasy to see that the R -module E ( R / p ) is p -cominimax but not p -cofinite.The following lemma is used in the sequel. Lemma 2.2.
Let a be an ideal of a Noetherian ring R and M an minimax R-module such that Supp R ( M ) ⊆ V ( a ) . Then the following statements are equivalent: (a) M is a -cominimax. (b) The R-module Hom R ( R / I , M ) is minimax. Mathematics Subject Classification.
Key words and phrases. formal local cohomology, local cohomology, cominimax.
Proof.
We know by definitions that (b) follows from (a). Let N be a finite submodule of M suchthat M / N is Artinian and suppose the R -module Hom R ( R / a , M ) is minimax. The exactness of0 → Hom R ( R / a , N ) → Hom R ( R / a , M ) → Hom R ( R / a , M / N ) → Ext R ( R / a , N ) implies that Hom R ( R / a , M / N ) is minimax.Since M / N is Artinian, it is easy to see that Hom R ( R / a , M / N ) is an Artinian R -module. As M / N is a -torsion, it follows by Melkersson’s theorem that M / N is Artinian. Thus M is minimax. The a -torsionness of M imples that it is a -cominimax. (cid:3) Theorem 2.3.
Let a be an ideal of a Noetherian ring R and M an R-module such that dimM ≤ andSuppM ⊆ V ( a ) . Then the following statements are equivalent: (a) M is a -cominimax. (b) The R-modules Hom R ( R / a , M ) and Ext R ( R / a , M ) are minimax.Proof. The conclusion (b) follows from (a) is obvious. In order to prove ( b ) ⇒ ( a ) using lemma(2.2), we may assume dimM =
1. Now use Lemma (2.2) instead of [5], Lemma 2.1, and the a -cominimaxness instead of a -cofiniteness in the proof of [5], Theorem 2.3. (cid:3) Theorem 2.4.
Let a be an ideal of a Noetherian ring R and M an weakly Laskerian R-module such thatSuppM ⊆ V ( a ) . Then the following statements are equivalent: (a) M is a -cominimax. (b) The R-modules Hom R ( R / a , M ) and Ext R ( R / a , M ) are minimax.Proof. The conclusion (b) follows from (a) is obvious. In order to prove (a) follows from (b),by definition there is a finitely generated submodule N of M such that dim ( M / N ) ≤ SuppM / N ⊆ V ( a ) . Also, the exact sequence0 −→ N −→ M −→ M / N −→ ( ⋆ ) induces the exact sequence0 −→ Hom R ( R / a , N ) −→ Hom R ( R / a , M ) −→ Hom R ( R / a , M / N ) −→ Ext R ( R / a , N ) −→ Ext R ( R / a , M ) −→ Ext R ( R / a , M / N ) −→ Ext R ( R / a , N ) Hence, it follows that the R -modules Hom R ( R / a , M / N ) and Ext R ( R / a , M / N ) are finitely gener-ated. Therefore, in view of lemma (2.2), the R -module M / N is a -cominimax. Now it follows fromthe exact sequence ( ⋆ ) that M is a -cominimax. (cid:3) Theorem 2.5.
Let a be an ideal of a local ring ( R , m ) and M a is nonzero ?nitely generated R-module.Let t ∈ N . Suppose that the R-module F i a ( M ) is a -cominimax for all i < t, and the R-modulesExt tR ( R / a , M ) and Ext t + R ( R / a , M ) are minimax. Then the R-modules Hom R ( R / a , F t a ( M ) andExt R ( R / a , F t a ( M ) are minimax.Proof. We use induction on t .The exact sequence0 −→ Γ a ( M ) −→ M −→ M / Γ a ( M ) −→ ( ⋆ ) OMINIMAXNESS PROPERTIES 3 induces the exact sequence:0 −→ Hom R ( R / a , Γ a ( M )) −→ Hom R ( R / a , M ) −→ Hom R ( R / a , M / Γ a ( M )) −→ Ext R ( R / a , Γ a ( M )) −→ Ext R ( R / a , M ) Since
Hom R ( R / a , M / Γ a ( M )) = Hom R ( R / a , Γ a ( M )) and Ext R ( R / a , Γ a ( M )) are minimax.Assume inductively that t > t . By applying the functor Hom R ( R / a , − ) to the exact sequence ( ⋆ ) , we can deducethat Ext jR ( R / a , M / Γ a ( M )) is minimax for j = t , t +
1. On the other hand, F a ( M / Γ a ( M )) = F j a ( M / Γ a ( M )) ≃ F j a ( M ) for all j >
0. Therefore we may assume that Γ a ( M ) =
0. Let E be aninjective hull of M and put N = E / M . Then Hom R ( R / a , E ) = = Γ a ( E ) . Hence Ext jR ( R / a , N ) ≃ Ext j + R ( R / a , M ) and F j a ( N ) ≃ F j + a ( M ) for all j ≥
0. Now, the induction hypothsis yields that
Hom R ( R / a , F t a ( M )) and Ext R ( R / a , F t a ( M )) are minimax, as required. (cid:3) We are now ready to state and prove the main tteorem.
Theorem 2.6.
Let a be an ideal of a local ring ( R , m ) and M a ?nitely generated R-module, and t ∈ N such that Ext iR ( R / a , M ) are minimax for all i ≤ t + . Let the R-modules F i a ( M ) be weakly laskerianR-modules for all i < t. Then the following assertions hold: (a) The R-modules F i a ( M ) are a -cominimax for all i < t. (b) For all minimax submodules N of F i a ( M ) , the R-modulesHom R ( R / a , F t a ( M ) / N ) , Ext R ( R / a , F t a ( M ) / N ) are minimax. In particular, the set Ass R ( F t a ( M ) / N ) is finite.Proof. (a) We proceed by induction on t . In the case t = t > t . By the inductive assump-tion, F i a ( M ) is a -cominimax for i =
0, 1, · · · , t −
2. Hence by theorem 2.5 and the assumption,
Hom R ( R / a , F t − a ( M )) and Ext R ( R / a , F t − a ( M )) are minimax. Therefore by Theorem 2.4, F i a ( M ) is a -cominimax for all i < t . This completes the inductive step.(b) In view of (a) and theorem 2.5, Hom R ( R / a , F t a ( M )) and Ext R ( R / a , F t a ( M )) are minimax. Onthe other hand, according to Lemma 2.2, N is a -cominimax. Now, the exact sequence0 −→ N −→ F t a ( M ) −→ F t a ( M ) / N −→ −→ Hom R ( R / a , F t a ( M )) −→ Hom R ( R / a , F t a ( M ) / N ) −→ Ext R ( R / a , N ) −→ Ext R ( R / a , F t a ( M )) −→ Ext R ( R / a , F t a ( M ) / N ) −→ Ext R ( R / a , N ) Consequently,
Hom R ( R / a , F t a ( M ) / N ) , Ext R ( R / a , F t a ( M ) / N ) are minimax, as required. (cid:3) B. SADEQI
Corollary 2.7.
Let a be an ideal of a local ring ( R , m ) and M a ?nitely generated R-module such thatExt iR ( R / a , M ) are minimax for all i and the R-modules F i a ( M ) are weakly laskerian R-modules for all i.Then: (a) The R-modules F i a ( M ) are a -cominimax for all i. (b) For any i ≥ and for any minimax submodule N of F i a ( M ) , the R-module F i a ( M ) / N is a -cominimax.Proof. ( a ) Clear. ( b ) In view of ( a ) the R -module F i a ( M ) is a -cominimax for all i . Hence the R -module Hom R ( R / a , N ) is minimax, and so it follows from Lemma 2.2 that N is a -cominimax.Now, the exact sequence 0 −→ N −→ F i a ( M ) −→ F i a ( M ) / N −→ R -module F i a ( M ) / N is a -cominimax. (cid:3) Corollary 2.8.
Let a be an ideal of a local ring ( R , m ) and M a ?nitely generated R-module such that theR-modules F i a ( M ) are weakly laskerian R-modules for all i. Then the following conditions are equivalent: (a) The R-modules Ext iR ( R / a , M ) are minimax for all i. (b) The R-modules F i a ( M ) are a -cominimax for all i.Proof. ( a ) ⇒ ( b ) follows by Corollary 2.7. ( a ) ⇒ ( b ) follows by [4], Proposition 3.9. (cid:3) Acknowledgement .
My thanks are due to my phd. adviser, Drs. KH. Ahmadi Amoli, for herguidance to prepare this paper and useful hints and to the reviewer for suggesting several im-provements. R
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