aa r X i v : . [ m a t h . A C ] J a n ASSOCIATED PRIMES OF FORMAL LOCAL COHOMOLOGYMODULES
BEHRUZ SADEQI
Abstract.
Let a be an ideal of a commutative Noetherian ring R and, M , afinitely generated R -module. In this paper, we proved that if Supp F i a ( M ) is finitefor all i < t , then Ass( F t a ( M )) is finite. Introduction
In this paper, ( R, m ) is commutative Noterian local ring with nonzero identityand all modules are finitely generated. Recall that the i -th formal local cohomologymodule of M with respect to a is denoted by F i a ( M ) for all i ∈ N (See[2]). In thispaper, we proved that if Supp F i a ( M ) is finite for all i < t , then Ass( F t a ( M )) is finite.Our terminology and notation of formal local cohomology modules come from [5].2. The results
Lemma 2.1.
Let M be an a -torsion R -module: M = S ∞ n =1 (0 : M a n ) . Then Ass M =Ass(0 : M a ) .Proof. By (0 : M a ) ⊆ M then Ass(0 : M a ) ⊆ Ass M . But p ∈ Ass M then thereis (0 = x ) ∈ M such that p = (0 : M x ), but M is an a -torsion R -module,then(0 : M a ) ⊆ S ∞ n =1 (0 : M a n ), and then Ass M ⊆ Ass(0 : M a ). (cid:3) Lemma 2.2.
Let R be a ring, and M , an R -module. If N is submodule of M then Ass(
M/N ) ⊆ Ass M ∪ Supp N . In particular, if the set Supp( N ) is finite, then Ass(
M/N ) is finite if and only if Ass M is finite.Proof. let p ∈ Ass(
M/N ) \ Supp N . So (0 = x ) ∈ M such that p = ( N : R x );we have p x ⊆ N . Set Rad (Ann( p x )) = T ni =1 q i . Then there exists a posotiveinteger t such that ( q · · · q n ) t p x = 0. Set q := ( q · · · q n ) t , then qp x = 0, therefore Mathematics Subject Classification.
Key words and phrases. formal local cohomology, associated prime ideals, cofinitness, weaklyLaskerian modules. p ⊆ Ann( q x ) ⊆ ( N : R q x ). Let a ∈ ( N : R q x ),then a q x ⊆ N and a q ⊆ p .Then a ∈ p and p = Ann( q x ); therefore p ∈ Ass( q x ) ,and hence p ∈ Ass( M ). (cid:3) Theorem 2.3.
Let ( R, m ) be a local ring. M is a finitely generated R -module.Suppose that there is an integer t ∈ N such that for all i < t the set Supp( F i a ( M )) is finite. Then Ass( F t a ( M )) is finite.Proof. We proceed by induction on t . If t = 0, then F a ( M ), by [3, lemma 2.1], isartinian, and hence Ass( F a ( M )) is finite. So, suppose that t >
0. Let Supp( F i a ( M ))is finite for all i < t . We prove(by induction) that Ass( F t a ( M )) is finite. By [5,theorem 3.11] there is the exact sequence:0 −→ Γ a ( M ) −→ M −→ M/ Γ a ( M ) −→ · · · −→ F i a ( M ) −→ F i a ( M/ Γ a ( M )) −→ F i +1 a (Γ a ( M )) −→ · · · is exact sequence. ButAss( F i a ( M/ Γ a ( M ))) ⊆ Ass( F i a ( M )) ∪ Ass( F i +1 a (Γ a ( M )))Let i = t −
1, and hence Ass( F t − a ( M/ Γ a ( M ))) is finite if and only if Ass( F t a (Γ a ( M )))isfinite and Ass( F t − a ( M/ Γ a ( M ))) = Ass( F t − a ( M )), Thus there is an M -reqular ele-ment x ∈ a .The exact sequence0 −→ M .x −→ M −→ M/xM −→ · · · −→ F t − a ( M ) .x −→ F t − a ( M ) g −→ F t − a ( M/xM ) f −→ F t a ( M ) −→ · · · It can be seen that Supp( F i a ( M/xM )) is finite set for all i < t . From inductionhypothesis, we deduce Ass( F t − a ( M/xM )) is finite. By applying lemma (2.2) to theexact sequence 0 −→ Im g −→ F t − a ( M/xM ) −→ Im f −→ Im f ) is finite. By noting that
Im f = (0 : F t a ( M ) x ) and usinglemma (2.2) the result now follows. (cid:3) Corollary 2.4.
Let
Supp( F i a ( M )) be finite for all i < t , and N , a submodule of F i a ( M ) such that Ass(Tor R ( R/ a , N )) is finite, then Ass( F t a ( M ) /N ) is finite. SSOCIATED PRIMES OF FORMAL LOCAL COHOMOLOGY MODULES 3
Proof.
The exact sequence0 −→ N −→ F t a ( M ) −→ F t a ( M ) /N −→ · · · −→ Tor Ri ( R/ a , F t a ( M ) /N ) −→ Tor Ri +1 ( R/ a , N ) −→ Tor Ri +1 ( R/ a , F t a ( M )) · · · . The Ass( R/ a ⊗ F t a ( M )) is finite by lemma(2.1) and Ass(Tor R ( R/ a , N )) is finite byhypothesis, hence Ass( F n a ( M ) /N ) is finite. (cid:3) Corollary 2.5.
Let ( R, m ) be a local ring. M is a finitely generated R -module. Sup-pose that there is n N such that for all i < n , F i a ( M ) is Artinian. Then Ass( F n a ( M )) is finite.Proof. Since Artinian modules have finite support and by (2.3), corollary is an im-mediate consequence. (cid:3)
References [1] M. Aghapournahr and L. Melkersson,
Local cohomology and Serre subcategories. , Journal ofAlgebra, , no. 3, (2008), pp.1275-1287.[2] M. Asgharzadeh and K. Divaani-Aazar, finiteness properties of formal local cohomology mod-ules and Cohen-Macaulayness , Commun. Alg. , no. 3, (2011), 1082-1103(22).[3] Bijan-Zadeh, Mohammad Hasan and Rezaei, Shahram, Artinianness and attached primes offormal local cohomology modules , Algebra Colloq. (2014), no. 2, 307–316.[4] M. Brodmann and R.Y. Sharp, Local cohomology: an algebraic introduction with geometricapplications , Cambridge Univ. Press, , Cambridge, (1998).[5] P. Schenzel, On formal local cohomology and connectedness , J. Algebra, (2), (2007), 894-923.
Departement of Sciences, Marand Branch, Islamic Azad university, Marand, Iran
Email address : [email protected] Email address ::