aa r X i v : . [ m a t h . A C ] M a r GENERALIZED IDEAL TRANSFORMS
TRAN TUAN NAM AND NGUYEN MINH TRI
Abstract.
We study basic properties of the generalized idealtransforms D I ( M, N ) and the set of associated primes of the mod-ules R i D I ( M, N ) . Key words : (generalized) local cohomology, (generalized) ideal trans-form, associated prime. : 13D45.1.
Introduction
Throughout this paper, R is a Noetherian commutative ring withnon-zero identity and I is an ideal of R . In [6], Brodmann definedideal transform D I ( M ) of an R − module M with respect to I by D I ( M ) = lim −→ n Hom R ( I n , M ) . Ideal transforms turn out to be a powerful tool in various fields ofcommutative algebra and they are closed to local cohomology modulesof Grothendieck.In [11], Herzog introduced the definition of generalized local coho-mology modules which is an extension of local cohomology modules ofGrothendieck. The i -th generalized local cohomology module of mod-ules M and N with respect to I was given as H iI ( M, N ) = lim −→ n Ext iR ( M/I n M, N ) . A natural way, we have a generalization of the ideal transform. In [10],the generalized ideal transform functor with respect to an ideal I isdefined by D I ( M, − ) = lim −→ n Hom R ( I n M, − ) . This research is funded by Vietnam National Foundation for Science and Tech-nology Development (NAFOSTED).
Also in [10] they used the generalized ideal transforms to study thecofiniteness of generalized local cohomology modules. Let R i D I ( M, − )denote the i -th right derived functor of D I ( M, − ) . It is clear that R i D I ( M, − ) ∼ = lim −→ n Ext iR ( I n M, − )for all i ≥ . The organization of our paper is as follows. In the next sectionwe study basic properties of the generalized ideal transform functor D I ( M, − ) and its right derived functors R i D I ( M, − ) . The first result isTheorem 2.1 which says that if M is a finitely generated R -module and N is an I -torsion R -module, then R i D I ( M, N ) = 0 for all i ≥ . Next,Theorem 2.7 gives us isomorphisms D I (Hom R ( M, N )) ∼ = D I ( M, N )and D aR ( M, N ) ∼ = D aR ( M, N ) a . In Theorem 2.10 we see that the mod-ule Hom R ( R/I, R t D I ( M, N )) is a finitely generated R -module providedthe modules R i D I ( M, N ) are finitely generated for all i < t . The sec-tion is closed by Theorem 2.12 which shows the Artinianness of themodules R i D I ( M, N ) . The last section is devoted to study the set of associated primesAss( R i D I ( M, N )) . Theorem 3.3 shows that if M is a finitely generated R -module and N is a weakly Laskerian R -module, then Supp( H i m ( M, N ))and Ass( R i D m ( M, N )) are finite sets for all i ≥ . Finally, Theorem 3.5gives us two interesting consequences about the finiteness of the setsAss( R t D I ( M, N )) (Corollary 3.6) and Supp R ( R t D I ( M, N )) (Corollary3.7).2.
Some basic properties of generalized ideal transforms An R − module N is called I -torsion if Γ I ( N ) ∼ = N. We have the firstfollowing result.
Theorem 2.1.
Let M be a finitely generated R -module and N an I -torsion R -module. Then R i D I ( M, N ) = 0 for all i ≥ .Proof. We first prove D I ( M, N ) = 0.Consider the n -th injection λ n : Hom R ( I n M, N ) → L Hom R ( I n M, N )and the homomorphisms ϕ ij : Hom R ( I i M, N ) → Hom R ( I j M, N ) suchthat ϕ ij ( f i ) = f i | I j M for all i ≤ j .Let S be an R -submodule of L Hom R ( I n M, N ) which is generatedby elements λ j ϕ ij ( f i ) − λ i f i , f i ∈ Hom R ( I i M, N ) and i ≤ j. Thenlim −→ n Hom R ( I n M, N ) = (cid:0) M
Hom R ( I n M, N ) (cid:1) /S. ENERALIZED IDEAL TRANSFORMS 3
For any u ∈ D I ( M, N ) = lim −→ n Hom R ( I n M, N ) , we have u = λ t f t + S ,where f t ∈ Hom R ( I t M, N ).Since I t M is a finitely generated R -module and N is an I -torsion R -module, there exists a positive integer p such that ϕ tp + t ( f t ) = 0.It follows from [17, 2.17 (ii)] that u = 0 and then D I ( M, N ) = 0.The proof will be complete if we show R i D I ( M, N ) = 0 for all i > N is I -torsion, there is an injective resolution E • of N suchthat each term of the resolution is an I -torsion R -module. By theabove proof, we have lim −→ n Hom R ( I n M, E i ) = 0 for all i ≥ . Therefore R i D I ( M, N ) = 0 for all i ≥ (cid:3) Corollary 2.2.
Let M be a finitely generated R -module and N an R -module such that D I ( N ) = 0 . Then R i D I ( M, N ) = 0 for all i ≥ .Proof. We consider the exact sequence0 → Γ I ( N ) → N → D I ( N ) → H I ( N ) → I ( N ) ∼ = N that means N is I -torsion.From 2.1 we have the conclusion. (cid:3) The following lemmas will be used to prove the next propositions.
Lemma 2.3. ([10, 2.2])
Let
M, N be R -modules. Then, there is anexact sequence → H I ( M, N ) → Hom R ( M, N ) → D I ( M, N ) → H I ( M, N ) → · · ·· · · H iI ( M, N ) → Ext iR ( M, N ) → R i D I ( M, N ) → H i +1 I ( M, N ) → · · · Moreover, if pd( M ) < ∞ , then R i D I ( M, N ) ∼ = H i +1 I ( M, N ) for all i ≥ pd( M ) + 1 . Lemma 2.4. ([5, Theorem 1])
The following conditions on an R -module M are equivalent: (i) M admits a resolution by finitely generated projectives; (ii) The functors
Ext nR ( M, − ) preserve direct limits for all n ;(iii) The functors
Tor Rn ( − , M ) preserve products for all n. The following lemma shows some basic properties of generalized idealtransforms that we shall use.
Lemma 2.5.
Let M be a finitely generated R -module and N an R -module. Then (i) D I ( M, N ) is an I -torsion-free R -module; (ii) R i D I ( M, N ) ∼ = R i D I ( M, N/ Γ I ( N )) for all i ≥ ; TRAN TUAN NAM AND NGUYEN MINH TRI (iii) R i D I ( M, N ) ∼ = R i D I ( M, D I ( N )) for all i ≥ ; (iv) D I ( D I ( M, N )) ∼ = D I ( M, N ) ; (v) D I (Hom R ( M, N )) ∼ = Hom R ( M, D I ( N )) .Proof. (i) We have by 2.4Γ I ( D I ( M, N )) = lim −→ n Hom R ( R/I n , D I ( M, N )) ∼ = lim −→ n lim −→ t Hom R ( R/I n , Hom R ( I t M, N )) ∼ = lim −→ n lim −→ t Hom R ( R/I n ⊗ I t M, N ) ∼ = lim −→ t lim −→ n Hom R ( I t M, Hom R ( R/I n , N )) ∼ = lim −→ t Hom R ( I t M, Γ I ( N )) ∼ = D I ( M, Γ I ( N )) . Since Γ I ( N ) is an I -torsion R -module and from 2.1, we get D I ( M, Γ I ( N )) =0. Thus D I ( M, N ) is an I -torsion-free R -module.(ii) The short exact sequence0 → Γ I ( N ) → N → N/ Γ I ( N ) → → D I ( M, Γ I ( N )) → D I ( M, N ) → D I ( M, N/ Γ I ( N )) → · · ·· · · → R i D I ( M, N ) → R i D I ( M, N/ Γ I ( N )) → R i +1 D I ( M, Γ I ( N )) · · · Then R i D I ( M, N ) ∼ = R i D I ( M, N/ Γ I ( N )) for all i ≥ , as R i D I ( M, Γ I ( N )) =0. (iii) The short exact sequence0 → N/ Γ I ( N ) → D I ( N ) → H I ( N ) → → D I ( M, N/ Γ I ( N )) → D I ( M, D I ( N )) → D I ( M, H I ( N )) → · · ·→ R i D I ( M, N/ Γ I ( N )) → R i D I ( M, D I ( N )) → R i D I ( M, H I ( N )) → As R i D I ( M, H I ( N )) = 0, R i D I ( M, N/ Γ I ( N )) ∼ = R i D I ( M, D I ( N ))for all i ≥ ENERALIZED IDEAL TRANSFORMS 5 D I ( D I ( M, N )) = lim −→ n Hom R ( I n , D I ( M, N )) ∼ = lim −→ n lim −→ t Hom R ( I n , Hom R ( I t M, N )) ∼ = lim −→ n lim −→ t Hom R ( I n ⊗ I t M, N ) ∼ = lim −→ t lim −→ n Hom R ( I t M, Hom R ( I n , N )) ∼ = lim −→ t Hom R ( I t M, D I ( N )) ∼ = D I ( M, D I ( N )) ∼ = D I ( M, N ) . (v) From 2.4 we have D I (Hom R ( M, N )) = lim −→ n Hom R ( I n , Hom R ( M, N )) ∼ = lim −→ n Hom R ( M, Hom R ( I n , N )) ∼ = Hom R ( M, D I ( N ))as required. (cid:3) If f : N → N ′ is an R -module homomorphism such that Ker f andCoker f are both I -torsion R -modules, then R i D I ( N ) ∼ = R i D I ( N ′ ) forall i ≥ Proposition 2.6.
Let f : N → N ′ be an R -module homomorphismsuch that Ker f and Coker f are both I -torsion R -modules. Then R i D I ( M, N ) ∼ = R i D I ( M, N ′ ) for all non-negative integer i .Proof. Two short exact sequences0 → Ker f → N → Im f → → Im f → N ′ → Coker f → → D I ( M, Ker f ) → D I ( M, N ) → D I ( M, Im f ) → R D I ( M, Ker f ) · · · → D I ( M, Im f ) → D I ( M, N ′ ) → D I ( M, Coker f ) → R D I ( M, Im f ) · · · Since Ker f and Coker f are both I -torsion R -modules, R i D I ( M, Ker f ) =0 and R i D I ( M, Coker f ) = 0 for all i ≥
0. Hence R i D I ( M, N ) ∼ = R i D I ( M, Im f ) and R i D I ( M, Im f ) ∼ = R i D I ( M, N ′ ). Finally, we get R i D I ( M, N ) ∼ = R i D I ( M, N ′ ) for all i ≥ (cid:3) Let N a denote the localization of N respect to the multiplicativelyclosed subset S = { a i | i ∈ N } . We have the following theorem. TRAN TUAN NAM AND NGUYEN MINH TRI
Theorem 2.7.
Let M be a finitely generated R -module and N an R -module. Then (i) D I (Hom R ( M, N )) ∼ = D I ( M, N );(ii) If I = aR is a principal ideal of R, then D aR ( M, N ) ∼ = D aR ( M, N ) a . Proof. (i). The long exact sequence0 → Γ I ( M, N ) → Hom R ( M, N ) → D I ( M, N ) f → H I ( M, N ) → · · · deduces an exact sequence0 → Γ I ( M, N ) → Hom R ( M, N ) → D I ( M, N ) → Im f → . Note that Im f is an R -submodule of H I ( M, N ), then Im f is an I -torsion R -module.Since Γ I ( M, N ) and Im f are both I -torsion R -modules, there areisomorphisms D I (Hom R ( M, N )) ∼ = D I ( D I ( M, N )) ∼ = D I ( M, N ) . (ii). From 2.5 we have D I ( M, N ) ∼ = D I ( M, D I ( N )). We now considerthe module D I ( M, D I ( N )) . Since I = aR is a principal ideal, it follows D I ( N ) ∼ = N a . As I n M is finitely generated, we have by [17, 3.83]Hom R ( I n M, N ⊗ S − R ) ∼ = S − R ⊗ Hom R ( I n M, N ) . It followslim −→ n Hom R ( I n M, N ⊗ S − R ) ∼ = lim −→ n S − R ⊗ Hom R ( I n M, N ) . Hence D I ( M, D I ( N )) ∼ = S − R ⊗ D I ( M, N ) . Finally, we get D aR ( M, N ) ∼ = D aR ( M, N ) a . (cid:3) If E is an injective R -module, then Γ I ( E ) is also injective and H I ( E ) =0. Hence, the short exact sequence0 → Γ I ( E ) → E → D I ( E ) → D I ( E ) is an injective R -module. Proposition 2.8.
Let M be a finitely generated R -module, N an R -module and J • an injective resolution of N. Then R i D I ( M, N ) ∼ = H i (Hom R ( M, D I ( J • ))) . ENERALIZED IDEAL TRANSFORMS 7
Proof.
Combining 2.7 with 2.5 yields R i D I ( M, N ) = H i ( D I ( M, J • )) ∼ = H i ( D I (Hom( M, J • ))) ∼ = H i (Hom( M, D I ( J • )))as required. (cid:3) Next, we study the finiteness of generalized ideal transforms whichrelates to generalized local cohomology modules.
Proposition 2.9.
Let
M, N be two finitely generated R -modules and i a positive integer. Then H iI ( M, N ) is finitely generated if and only if R i − D I ( M, N ) is finitely generated.Proof. Since
M, N are finitely generated R -modules, Ext iR ( M, N ) isalso a finitely generated R -module for all i ≥
0. By 2.3 we have theconclusion. (cid:3)
Theorem 2.10.
Let M be a finitely generated R -module and N an R -module. If t is a non-negative integer such that R i D I ( M, N ) is finitelygenerated for all i < t, then Hom R ( R/I, R t D I ( M, N )) is a finitely gen-erated R -module.Proof. We use induction on t .Let t = 0. We have Hom R ( R/I, D I ( M, N )) = 0, since D I ( M, N ) is I -torsion-free.When t >
0, from 2.5, it follows R i D I ( M, N ) ∼ = R i D I ( M, D I ( N ))for all i ≥ R ( R/I, R t D I ( M, D I ( N ))) is finitelygenerated.Since D I ( N ) is an I -torsion-free R -module, there is a D I ( N )-regularelement x ∈ I. Now the short exact sequence0 → D I ( N ) x −→ D I ( N ) → D I ( N ) → , where D I ( N ) = D I ( N ) /xD I ( N ) gives rise a long exact sequence0 → D I ( M, D I ( N )) x −→ D I ( M, D I ( N )) → D I ( M, D I ( N )) · · ·· · · R t − D I ( M, D I ( N )) h −→ R t − D I ( M, D I ( N )) k −→ R t D I ( M, D I ( N )) · · · . It induces a short exact sequence0 → Im h → R t − D I ( M, D I ( N )) → Im k → . As R i D I ( M, D I ( N )) is finitely generated for all i < t , Im h and TRAN TUAN NAM AND NGUYEN MINH TRI R i D I ( M, D I ( N )) are both finitely generated R -modules for all i < t − R ( R/I, R t − D I ( M, D I ( N ))) is finitelygenerated. Hence Hom R ( R/I, Im k ) is finitely generated.Next, the exact sequence0 → Im k → R t D I ( M, D I ( N )) x −→ R t D I ( M, D I ( N ))deduces a long exact sequence0 → Hom R ( R/I, Im k ) → Hom R ( R/I, R t D I ( M, D I ( N ))) x −→ x −→ Hom R ( R/I, R t D I ( M, D I ( N ))) → · · · As x ∈ I, Hom R ( R/I, R t D I ( M, D I ( N ))) is finitely generated. (cid:3) In [7, 2.5] Tang and Chu proved that if H rI ( M, R/ p ) is Artinian forany p ∈ Supp( N ) and r ≥ pd( M ) , then H iI ( M, N ) is Artinian for all i ≥ r. We show a similar following proposition.
Proposition 2.11.
Let
M, N be two finitely generated R -modules with pd( M ) < ∞ . Assume that t is a positive integer such that t > pd( M ) . (i) If R t D I ( M, R/ p ) is Artinian for all p ∈ Supp( N ) , then R i D I ( M, N ) is also an Artinian R -module for all i ≥ t. (ii) If R t D I ( M, R/ p ) and H tI ( M, R/ p ) are Artinian for all p ∈ Supp( N ) , then Ext iR ( M, N ) is also an Artinian R -module forall i ≥ t. Proof. (i). The proof of (i) is similar to that in the proof of [7, 2.5].(ii). By 2.3 there is an exact sequence0 → H I ( M, N ) → Hom R ( M, N ) → D I ( M, N ) → H I ( M, N ) → · · ·· · · H iI ( M, N ) → Ext iR ( M, N ) → R i D I ( M, N ) → H i +1 I ( M, N ) → · · · . Thus the conclusion follows from (i) and [7, 2.5]. (cid:3)
In the following theorem we study the Artinianness of modules R i D I ( M, N )when N is Artinian or finitely generated. Theorem 2.12.
Let M be a finitely generated R -module. (i) If N is an Artinian R -module, then R i D I ( M, N ) is Artinianfor all i ≥ . (ii) If N is a finitely generated R -module such that p = pd( M ) and d = dim( N ) are finite, then R p + d D I ( M, N ) is an Artinian R -module. ENERALIZED IDEAL TRANSFORMS 9
Proof. (i). It follows from [15, 2.6] that H iI ( M, N ) is Artinian for all i ≥ . On the other hand, Ext iR ( M, N ) is Artinian for all i ≥ . Thusthe claim follows from the exact sequence of 2.3.(ii). When d = dim( N ) = 0. It is clear that N is an Artinian R -module. Hence Ext iR ( M, N ) is Artinian for all i ≥ . By [2, 5.1], H iI ( M, N ) = 0 for all i > pd( M ) and H pI ( M, N ) isArtinian. Now we have the exact sequence by 2.3 · · · R p − D I ( M, N ) → H pI ( M, N ) → Ext pR ( M, N ) → R p D I ( M, N ) → . It follows that R p D I ( M, N ) is an Artinian R -module.Let d = dim( N ) >
0. Since Ext iR ( M, N ) = 0 for all i > pd( M ) ,R p + d D I ( M, N ) ∼ = H p + d +1 I ( M, N ) = 0 . This finishes the proof. (cid:3) Associated primes of the modules R i D I ( M, N )To study some properties of associated primes of R i D I ( M, N ) werecall the concepts of weakly Laskerian modules [9] and
F SF modules[16]. An R -module M is called weakly Laskerian if the set of associatedof prime ideals of any quotient module of M is finite. An R -module M is called a F SF module if there is a finitely generated submodule N of M such that the support of M/N is a finite set. Note that a module M is a weakly Laskerian module if and only if M is a F SF module(see [1, 2.5]).
Lemma 3.1 ([9]) . (i) Let → M ′ → M → M ′′ → be a shortexact sequence. Then M is weakly Laskerian if and only if M ′ and M ′′ are both weakly Laskerian. (ii) If M is a finitely generated R -module and N is a weakly Laske-rian, then Ext iR ( M, N ) and Tor Ri ( M, N ) are weakly Laskerianfor all i ≥ . (iii) Artinian modules and finitely generated modules are weakly Laske-rian modules.
Proposition 3.2.
Let
M, N be finitely generated R -modules. Then Ass( D I ( M, N )) = (Supp( M ) ∩ Ass( N )) \ V ( I ) . Proof.
It follows from [18, 3.1] that Ass( D I ( N )) = Ass( N ) \ V ( I ). Thenwe have by 2.7 Ass( D I ( M, N )) = Ass( D I (Hom R ( M, N )))= Ass(Hom R ( M, N )) \ V ( I )= (Supp( M ) ∩ Ass( N )) \ V ( I )as required. (cid:3) It is well-known that, if
M, N are finitely generated R -modules, then H i m ( M, N ) is Artinian for all i ≥ H i m ( M, N )) is a finite set. When N is a weakly Laskerian R -module we have the following theorem. Theorem 3.3.
Let M be a finitely generated R -module and N a weaklyLaskerian R -module. If m is a maximal ideal of R, then Supp( H i m ( M, N )) and Ass( R i D m ( M, N )) are finite sets for all i ≥ . Proof. As N is weakly Laskerian, there exists a finitely generated sub-module L of N such that Supp( N/L ) is a finite set. Then the shortexact sequence 0 → L → N → N/L → · · · → H i m ( M, L ) f → H i m ( M, N ) g → H i m ( M, N/L ) → · · · Since H i m ( M, L ) is an Artinian R -module, Supp( H i m ( M, L )) is a finiteset and Im f is an Artinian R -module.Note that Supp(Im g ) is a finite set becauseSupp(Im g ) ⊂ Supp( H i m ( M, N/L )) ⊂ Supp(
N/L ) . From the long exact sequence, we obtain a short exact sequence0 → Im f → H i m ( M, N ) → Im g → H i m ( M, N )) = Supp(Im f ) ∪ Supp(Im g ). ThusSupp( H i m ( M, N )) is a finite set and then Ass( H i m ( M, N )) is a finiteset. We now consider the exact sequence0 → Γ m ( M, N ) → Hom R ( M, N ) → D m ( M, N ) → · · ·· · · → Ext iR ( M, N ) → R i D m ( M, N ) → H i +1 m ( M, N ) → · · · Since Ass(Ext iR ( M, N )) is a finite set, we have the conclusion. (cid:3)
Proposition 3.4.
Let M be a finitely generated module and N a weaklyLaskerian module over a local ring ( R, m ) . If dim( N ) ≤ , then R i D I ( M, N ) is weakly Laskerian for all i ≥ . ENERALIZED IDEAL TRANSFORMS 11
Proof.
From 2.3 there is an exact sequence0 → H I ( M, N ) → Hom R ( M, N ) → D I ( M, N ) → H I ( M, N ) → · · ·· · · H iI ( M, N ) → Ext iR ( M, N ) → R i D I ( M, N ) → H i +1 I ( M, N ) → · · · If dim ( N ) ≤
2, then H iI ( M, N ) is weakly Laskerian for all i ≥ iR ( M, N ) is weakly Laskerian for all i ≥ R i D I ( M, N ) is weakly Laskerian for all i ≥ . (cid:3) Theorem 3.5.
Let M be a finitely generated R -module and N an R -module. Then (i) There is a Grothendieck spectral sequence E pq = Ext pR ( M, R q D I ( N )) = ⇒ p R p + q D I ( M, N );(ii) Ass R ( R t D I ( M, N )) ⊆ ( t S i =1 Ass R ( E i,t − it +2 )) S Ass R (Hom( M, R t D I ( N )));(iii) Supp R ( R t D I ( M, N )) ⊆ t S i =0 Supp R (Ext iR ( M, R t − i D I ( N ))) . Proof. (i). Let us consider functors F ( − ) = Hom R ( M, − ) and G ( − ) = D I ( − ) . The functor F ( − ) is left exact. For any injective module E,G ( E ) is also an injective module and then is right F -acyclic. Onthe other hand, there is a natural equivalence by 2.7, D I ( M, − ) ∼ =Hom R ( M, D I ( − )). Thus from [17, 11.38] we have the Grothendieckspectral sequence E pq = Ext pR ( M, R q D I ( N )) = ⇒ p R p + q D I ( M, N ) . (ii). From the spectral of (i) there is a finite filtration Φ of R p + q D I ( M, N )with 0 = Φ t +1 H t ⊂ Φ t H t ⊂ . . . ⊂ Φ H t ⊂ Φ H t = R t D I ( M, N )and E i,t − i ∞ = Φ i H t / Φ i +1 H t , where t = p + q, ≤ i ≤ t. Exact sequences for all 0 ≤ i ≤ t → Φ i +1 H t → Φ i H t → E i,t − i ∞ → i H t ) ⊆ Ass(Φ i +1 H t ) [ Ass( E i,t − i ∞ ) . We may integrate this for i = 0 , , ..., t to conclude thatAss( R t D I ( M, N )) ⊆ t [ i =0 Ass R ( E i,t − i ∞ ) . We now consider homomorphisms of the spectral E i − t − , t − i +1 t +2 −→ E i,t − it +2 −→ E i + t +2 , − i − t +2 . Note that E i − t − , t − i +1 t +2 = E i + t +2 , − i − t +2 = 0 for i = 0 , , ..., t. It follows E i,t − it +2 = E i,t − it +3 = ... = E i,t − i ∞ . In particular, E ,t ∞ = ... = E ,tt +3 = E ,tt +2 ⊆ E ,tt +1 ⊆ ... ⊆ E ,t . ThereforeAss R ( R t D I ( M, N )) ⊆ ( t [ i =1 Ass R ( E i,t − it +2 )) [ Ass R (Hom( M, R t D I ( N ))) . (iii). Analysis similar to that in the proof of (ii) shows thatSupp( R t D I ( M, N )) ⊆ t [ i =0 Supp R ( E i,t − i ∞ )and E i,t − it +2 = E i,t − it +3 = ... = E i,t − i ∞ . Thus E i,t − i ∞ is a subquotient of E i,t − i and thenSupp R ( E i,t − i ∞ ) ⊆ Supp R ( E i,t − i ) = Supp R (Ext iR ( M, R t − i D I ( N )))) . This finishes the proof. (cid:3)
Corollary 3.6.
Let M be a finitely generated R -module, N a weaklyLaskerian R -module and t a non-negative integer. If R i D I ( N ) is weaklyLaskerian for all i < t, then Ass( R t D I ( M, N )) is a finite set.Proof. We have by 3.5Ass R ( R t D I ( M, N )) ⊆ ( t [ i =1 Ass R ( E i,t − it +2 )) [ Ass R (Hom( M, R t D I ( N ))) . As R i D I ( N ) is weakly Laskerian for all i < t, Ext iR ( M, R t − i D I ( N )) isalso weakly Laskerian for all 1 ≤ i < t. Since E i,t − it +2 is a subquotientof E i,t − i = Ext iR ( M, R t − i D I ( N )) , E i,t − it +2 is weakly Laskerian for all 1 ≤ i < t. It follows that t S i =1 Ass R ( E i,t − it +2 ) is finite. Note that R i D I ( N ) ∼ = H i +1 I ( N ) for i > . Thus H iI ( N ) is weakly Laskerian for all i < t + 1 . By[3, 2.2], Ass R ( H t +1 I ( N )) is finite and then Ass R (Hom( M, R t D I ( N ))) isfinite. The proof is complete. (cid:3) ENERALIZED IDEAL TRANSFORMS 13
Corollary 3.7.
Let M be a finitely generated R -module, N an R -module and t a non-negative integer. If Supp R ( R i D I ( N )) is a finiteset for all i ≤ t, then Supp R ( R t D I ( M, N )) is also a finite set.Proof. It follows from 3.5 thatSupp R ( R t D I ( M, N )) ⊆ t [ i =0 Supp R (Ext iR ( M, R t − i D I ( N ))) ⊆ t [ i =0 Supp R ( R t − i D I ( N )) . By the hypothesis we have the conclusion. (cid:3)
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Department of Mathematics-Informatics, Ho Chi Minh Universityof Pedagogy, Ho Chi Minh city, Viet Nam.
E-mail address : [email protected] Department of Natural Science Education, Dong Nai University,Dong Nai, Viet Nam.
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