On the Matlis duals of local cohomology modules and modules of generalized fractions

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1 Proc. Indan Acad. Sc. (ath. Sc.) Vol. 120, No. 1, February 2010, pp Indan Acadey of Scences On the atls duals of local cohoology odules and odules of generalzed fractons KAZE KHASHYARANESH Departent of Pure atheatcs, Center of Excellence n Analyss on Algebrac Structures, Ferdows Unversty of ashhad, P.O. Box , ashhad, Iran E-al: Khashyar@p.r S receved 7 Septeber 2009; revsed 3 Noveber 2009 Abstract. Let (R, ) be a coutatve Noetheran local rng wth non-zero dentty, a a proper deal of R and a fntely generated R-odule wth a. Let D( ) := Ho R (,E) be the atls dual functor, where E := E(R/) s the njectve hull of the resdue feld R/. In ths paper, by usng a coplex whch nvolves odules of generalzed fractons, we show that, f x 1,...,x n s a regular sequence on contaned n a, then H(x n 1,...,xn)R D(H a n())) s a hooorphc age of D(), where H b ( ) s the -th local cohoology functor wth respect to an deal b of R. By applyng ths result, we study soe condtons on a certan odule of generalzed fractons under whch D(H(x n 1,...,xn)R (D(H a n()))) = D(D()). Keywords. Local cohoology odule; atls dual functor; odule of generalzed fractons; flter regular sequence. 1. Introducton For an deal a of a coutatve Noetheran local rng (R, ), we denote the n-th local cohoology functor wth respect to a by Ha n( ) and the atls dual functor Ho R(,E) by D( ), where E s the njectve hull of the feld R/. Also, for a sequence := x 1,...,x n of eleents of R, weuser or (x 1,...,x n )R to denote the deal n =1 x R of R. Recently, there has been soe work on odules D(Ha n(r)) and H R n (D(H a n(r))), where := x 1,...,x n s a sequence of eleents of a, and on soe probles related to these odules (see for exaple, conjecture ( ) n [3] and [4] and Queston 3.8 n [5]). Also, Hellus and Stückrad, n [5], showed that studyng the atls dual of local cohoology odules s a useful tool for the descrpton of the endoorphs rng of local cohoology odules. oreover, they rased the followng queston: f R s a coutatve Noetheran coplete local rng and := x 1,...,x n s a regular sequence on R contaned n a, when exactly s J,a,R := D(HR n (D(H a n (R)))) zero? On the other hand, Sharp and Zaker, n [12], over an arbtrary coutatve rng, ntroduced the concept of odules of generalzed fractons. It was shown that ths concept has any nteractons wth topcs of recent and current nterest n coutatve algebra. In partcular, there are strong lnks between odules of generalzed fractons and local cohoology odules (cf. [13] and [7]). 35

2 36 Kaze Khashyaranesh In ths paper, for a fntely generated R-odule, we show that whenever := x 1,...,x n s a regular sequence on n a, there exsts an exact sequence HR n 1 (D(K)) D() H R n (D(H a n ())) 0, where K s the kernel of a dfferental ap n a certan coplex whch nvolves odules of generalzed fractons. oreover, we show that J,a, := D(H n R (D(H n a ()))) = D(D()) f H n R (D(U n 1 )) = 0, where U n 1 s a odule of generalzed fractons of wth respect to a certan trangular subset U of R. In ths paper, we study the vanshng of J,a, (wthout any restrcton on R). Our orgnal goal of ths paper s to fnd soe relatons between the theory of the generalzed fractons and local cohoology theory. Although we can gan soe techncal results such as Theore 3.4, we hope that, these can be useful n future. Throughout ths paper, we wll generally assue that R s a coutatve rng wth nonzero dentty and a s an deal of R. We shall use N 0 (respectvely N) to denote the set of non-negatve (respectvely postve) ntegers. Our ternology follows the textbook [1] on local cohoology. 2. atls dual of local cohoology odules Let be an R-odule. The constructon of a odule of generalzed fractons of requres a (postve nteger n and a) trangular subset (see Defnton 2.1 of [12]) U R n ; the constructon produces a odule U n, called the odule of generalzed fractons of (u 1,...,u n ), wth respect to U, whose eleents, called generalzed fractons, have the for where and (u 1,...,u n ) U. The concept of a chan of trangular subsets on R s explaned n p. 420 of [8]. Such a chan U = (U ) N deternes a coplex of odules of generalzed fractons 0 d 1 d0 U 1 1 U d U 1 +1, n whch d 0 () = /(1) for all and d (/(u 1,...,u )) = /(u 1,...,u, 1) for all N, and (u 1,...,u ) U. We shall denote ths coplex by C(U,). The reader s referred to 2 of [12] and [8] for ore detals of the above constructons. In the rest of the paper, we assue that s a fntely generated R-odule. Lea 2.1 (See Proposton 3.3 of [2]) () Let U be a trangular subset of R such that (1) U. Then U {1} s a trangular subset of R 2, and there s an exact sequence d0 U 1 ω (U {1}) 2 0, n whch d 0 s the natural hooorphs and ω ( ) (u) = (u) U. (u,1) for each and

3 atls duals of local cohoology odules 37 () Let U = (U n ) n N be a chan of trangular sets on R. Choose n N. Then U {1} s a trangular subset of R n+2, and there s an exact sequence Un n dn U n 1 ω (U {1}) n 2 0, n whch d n ( ) s defned as above and ω (u 1,...,u ) = (u 1,...,u,1) for each and (u 1,...,u ) U. Notaton 2.2. Let := x 1,...,x n be a sequence of eleents of R. For each N, set U() :={(x α 1 1,...,xα ) : thereexsts j wth 0 j such that α 1,...,α j N and α j+1 = =α = 0}, where x r s nterpreted as 1 whenever r>n. It s easy to see that, for each N, U() s a trangular subset of R.WeuseR() to denote the faly (U() ) N. Hence R() s a chan of trangular subsets on R. Wrte the assocated coplex C(R(), ) as 0 d 1 d,, 0 U() 1 1 U() It wll be convenent to allow U() 0 0 d, U() and U() ( 1) 1 to denote and 0 respectvely. Now, suppose that (R, ) s a local rng and E := E R (R/) denotes an njectve hull of the feld R/.ByD( ) we denote the atls dual functor Ho R (,E)(cf. [10]). Lea 2.3. () Let a be an deal of R and = x 1,...,x n be a sequence of eleents of R such that x n a. Then Ha ) n n (D(U( )) = 0 for all N 0. () Let := x 1,...,x n be a sequence of eleents of R. Then, for every x R, H n R (D(Ker dn x 1,...,x,)) = 0. Proof. () Set V := {(x α 1 1,...,xα n n ) : α 1,...,α n N}. Then V s a trangular subset of R n. Consder the R-onoorphs ϕ: V n U() n n such that ϕ( (x α ) = 1 1,...,xαn n ) (x α for all and α 1 1,...,α n N. 1,...,xαn n ) We show that ϕ s surjectve. Let ξ = U() n for soe and β 1,...,β n (x β 1 1,...,xβn n ) n N 0. In vew of Reark 3.3() of [12], we ay assue that

4 38 Kaze Khashyaranesh β 1,...,β n 1 N and β n N 0. Hence ξ = x n 1,...,xβ n 1 n 1 ( x β 1 V n = U() n. Therefore t s enough to show that n H a (D(V n )) = 0 ) I ϕ. Thus,xβ n for all N 0. In vew of Lea 2.1 of [13], the ultplcaton by x n provdes an autoorphs on V n. Hence, by applyng the functor D( ), t s easy to see that the ultplcaton by x n provdes agan an autoorphs on D(V n ). Snce x n a, t follows that H a (D(V n )) = 0 for all N 0. () Let x R and use the exact sequence 0 Ker d n ȳ, to deduce the exact sequence U(ȳ) n n I dn ȳ, 0 0 D(I dȳ, n ) D(U(ȳ) n n ) D(Ker dn ȳ, ) 0, ( ) where ȳ := x 1,...,x. By usng the constructon of odules of generalzed fractons, U(ȳ) n n = U() n n. Hence t follows fro () that H (D(U(ȳ) n n )) = 0 for all N 0. Now, applyng the functor ƔR( ) on the exact sequence ( ) and Corollary of [1] copletes the proof. Note that the frst part of the above lea and ts proof are rather closed to Proposton 2.4 of [6] and ts proof. In the next theore, we use a natural generalzaton of regular sequences whch s called flter regular sequences (cf. [11], [15] and [7]). We say that a sequence x 1,...,x n of eleents of a s an a-flter regular sequence on f Supp R ( (x1,...,x 1 ) : x (x 1,...,x 1 ) ) V(a) for all = 1,...,n, where V(a) denotes the set of pre deals of R contanng a. Also, we say that an eleent x a s an a-flter regular eleent on f Supp R (0: x) V(a). The concept of an a-flter regular sequence on s a generalzaton of the one of a flter regular sequence whch has been studed n [11], [15] and has led to soe nterestng results. Both concepts concde f a s an -prary deal of a local rng wth axal deal. Note that x 1,...,x n s a weak -sequence f and only f t s an R-flter regular sequence on. It s easy to see that the analogue of Appendx Lea 2() of [15] holds true whenever R s Noetheran, s fntely generated and s replaced by a; so that, f x 1,...,x n s an a-flter regular sequence on, then there s an eleent y a such that x 1,...,x n,y s an a-flter regular sequence on. Thus, for a postve nteger n, there exsts an a-flter regular sequence on of length n. Theore 2.4. Let (R, ) be a Noetheran local rng and a be a proper deal of R. Let x := x 1,...,x n (n > 0) be a regular sequence on contaned n a. Then there s an exact sequence Hx n 1 R (D(Ker dn ȳ, )) D() H n R (D(H a n ())) 0, for every x a such that ȳ := x 1,...,x n,x s an a-flter regular sequence on.

5 atls duals of local cohoology odules 39 Proof. Let x be an eleent of a such that ȳ := x 1,...,x n,x s an a-flter regular sequence on. Note that the exstence of such eleent s explaned n the above paragraph. Snce x 1,...,x n s a regular sequence on, by usng the exactness theore for generalzed fractons (see Theore 3.1 of [14] or Theore 3.3 of [8]), we have the exact sequence Note that U(ȳ) d 1 d ȳ, ȳ, 0 d1 d n 2 0 U(ȳ) 1 1 ȳ, ȳ, U(ȳ) n 1 I dn 1 ȳ, 0. = U() and the followng dagra coutes: d U(ȳ) +1 1 ȳ, U(ȳ) = = U() +1 d 1, U() for all N 0 wth n. Now, by breakng the above exact sequence nto short exact sequences and applyng the exact functor D( ) on the, n conjuncton wth Lea 2.3(), we have the followng soorphss. H n 1 R (D(I dn 1 ȳ, )) = HR n 2 (D(I dn 2 =... ȳ, )) = H 1 R (D(I d1 ȳ, )) = H 0 R (D()) Snce D() s Artnan, H 0 R (D()) = D() and so H n 1 R (D(I dn 1 ȳ, )) = D(). ( ) Next, t follows fro Consequences 1.3() of [7] that Ha n() = Ker dȳ, n /I dn 1 ȳ,. Hence, we can obtan the exact sequence 0 I dȳ, n 1 Ker dn ȳ, H a n () 0, whch nduces the exact sequence 0 D(H n a ()) D(Ker dn ȳ, ) D(I dn 1 ȳ, ) 0. Therefore, by applyng the functor ƔR( ) to t, we have the exact sequence HR n 1 (D(Ker dn n 1 ȳ, )) HR (D(I dn 1 ȳ, )) H R n (D(H a n ())) H n R (D(Ker dn ȳ, )). The result now edately follows fro Lea 2.3() and ( ).

6 40 Kaze Khashyaranesh The followng corollary s an edate consequence of Theore 2.4. COROLLARY 2.5 (Cop. Theore 2.5 of [6]) Let (R, ) be a Noetheran local rng and a be a proper deal of R. Let := x 1,...,x n be a regular sequence on R contaned n a. Then H n R (D(H a n (R))) s a hooorphc age of E. 3. On the queston of Hellus and Stückrad In ths secton, we study a slght generalzaton of the deal J,a,R of R. In ths regard we need the followng defnton. DEFINITIONS 3.1 (Cop. Defnton 3.3 of [5]) Let a be a proper deal of a Noetheran local rng R such that a. Let := x 1,...,x n be a regular sequence on contaned n a. Set J,a, := D(H n R (D(H n a ()))). In [5], Hellus and Stückrad studed the deal J,a,R n the case that (R, ) s a Noetheran coplete local rng wth respect to -adc topology and = x 1,...,x n a s a regular sequence on R and they asked when J,a,R := D(HR n (D(H a n (R)))) s exactly zero. In ths secton, by usng the theory of odules of generalzed fractons, we study the R-odule J,a, (wthout any restrcton on R). In fact we show that f ȳ := x 1,...,x n,x (n > 0) s an a-flter regular sequence on such that := x 1,...,x n s a regular sequence on and H R n (D(U(ȳ) n 1 )) = 0, then J,a, = D(D()). The followng proposton s a consequence of Theore 2.4. PROPOSITION 3.2 Let a be a proper deal of a Noetheran local rng R. Let := x 1,...,x n (n > 0) be a regular sequence on contaned n a. Then there exsts an exact sequence 0 J,a, D(D()) D(H n 1 x R (D(Ker dn ȳ, ))) for every x a such that ȳ := x 1,...,x n,x s an a-flter regular sequence on. Lea 3.3. Let n N 0 and z := z 1,...,z be a sequence of eleents of R. Then there exsts the exact sequence H n (z 1,...,z n )R (D(Hz ())) H (z n 1,...,z n )R ) n 1 (D(U(z )) Proof. In vew of Lea 2.1, there s an exact sequence 0 I d n z, R U(z ) n 1 H n 1 (z 1,...,z n )R (D(Ker dn,)) 0. z U(z ) n 2 0. ( ) n+2

7 atls duals of local cohoology odules 41 On the other hand, by Theore of [1], the (n + 1)-th local cohoology odule z R () can be nterpreted as a drect lt of Koszul hoology odules, and so, () = l/ wth the ap H H z R t N / j=1 j=1 zt j zj t / j=1 zj t+1 beng nduced by ultplcaton by z 1...z. Thus, n vew of Theore 2.4 of [9], Hz R () = U(z ) n 2 n+2. So ( ) ples the exact sequence 0 I d n z, whch nduces the exact sequence U(z ) n 1 Hz R () 0, 0 D(Hz R ()) D(U(z ) n 1 ) D(I dn z, ) 0. Therefore, by applyng the functor Ɣ (z1,...,z n )R( ) on the above exact sequence together wth Corollary of [1], one can obtan the exact sequence H n (z 1,...,z n )R Now, use the exact sequence 0 Ker d n z, to deduce the exact sequence Set z (D(Hz R ())) H (z n 1,...,z n )R (D(U(z ) n 1 )) H(z n 1,...,z n )R (D(I dn z, )) 0. ( ) U(z ) n n I dn z, 0 0 D(I dz, n ) D(U(z ) n n ) D(Ker dn z, ) 0. := z 1,...,z n. Note that U(z ) = U(z U(z ) +1 1 d z, U(z ) = = ) +1, ) ) d 1 z U(z U(z coutes for all N 0 wth n. Thus, by Lea 2.3, and the dagra H n (z 1,...,z n )R (D(I dn z, )) = H n 1 (z 1,...,z n )R (D(Ker dn z, )). The result now follows fro ( ).

8 42 Kaze Khashyaranesh Theore 3.4. Let (R, ) be a Noetheran local rng and a be a proper deal of R. Let x := x 1,...,x n (n > 0) be a regular sequence on n a. Suppose that there exsts x a such that ȳ := x 1,...,x n,x s an a-flter regular sequence on and H n R (D(U(ȳ) n 1 )) = 0. Then J,a, = D(D()). Proof. It follows fro Lea 3.3 and Proposton 3.2. The followng corollary s a natural generalzaton of the plcaton () (v) of Theore 3.7 n [5] n a ore general case. COROLLARY 3.5 (Cop. Theore 3.7 of [5]) Let (R, ) be a Noetheran local rng and a be a proper deal of R. Let := x 1,...,x n be a regular sequence on contaned n a such that a = R. Then,a, J = D(D()). Proof. Let x be an eleent n a such that ȳ := x 1,...,x n,x s an a-flter regular sequence on. We show that U(ȳ) n 1 = 0. To acheve ths, let (x α 1 1,...,xα be an ) arbtrary eleent of U(ȳ) n 1 for soe and α 1,...,α N 0. Snce a = R = (x α 1 1,...,xα n n )R, there exsts r 1,...,r n R such that x β = r 1x α r n x α n n for soe β N. So, n vew of Reark 3.3() of [12], (x α 1 1,...,xα n n,x α The result now follows fro Theore 3.4. β ) = x (x α 1 1,...,xα n n,x α +β ) = 0. In the followng proposton, we study the R-odule H n R (D(U(ȳ) n 1 )). PROPOSITION 3.6 Let (R, ) be a Noetheran local rng and n be a postve nteger. Suppose that x 1,...,x s a sequence of eleents of R. Set := x 1,...,x n and ȳ := x 1,...,x. Then we have the followng soorphss: () n R (D(U(ȳ) n 1 )) = H n R (R) R Ho R (U(ȳ) n 1 R,D()). () D(H n R (D(U(ȳ) n 1 ))) = Ho R (H n R (R), D(D(U(ȳ) n 1 ))). Proof. In vew of Lea 3.16 of [14], U(ȳ) n 1 = U(ȳ) n 1 R R. Hence H n R (D(U(ȳ) n 1 )) = H n R (R) R D(U(ȳ) n 1 ) Also, ths s a fact that = HR n (R) R Ho R (U(ȳ) n 1 R, D()). D(HR n (R) R D(U(ȳ) n 1 )) = Ho R (HR n (R), D(D(U(ȳ) n 1 ))), whch provdes the soorphs ().

9 atls duals of local cohoology odules 43 Reark 3.7. In vew of Proposton 3.6 and Reark () n [1], one can replace the condton HR n (D(U(ȳ) n 1 )) = 0 by each of the followng condtons: () HR n (D(U(ȳ) n 1 )) = 0. () HR n (R) R Ho R (U(ȳ) n 1 R,D()) = 0. () Ho R (HR n (R), D(D(U(ȳ) n 1 ))) = 0. Acknowledgents The author s deeply grateful to the referee for helpful suggestons. Ths research was supported by a grant fro Ferdows Unversty of ashhad (No. P87101KHA). References [1] Brodann and Sharp R Y, Local cohoology an algebrac ntroducton wth geoetrc applcatons, Cabrdge Studes n Advanced atheatcs 60 (Cabrdge Unversty Press) (1998) [2] Gbson G J and O Carroll L, Drect lt systes, generalzed fractons and coplexes of Cousn type, J. Pure Appl. Algebra 54(2 3) (1988) [3] Hellus, On the assocated pres of atls duals of top local cohoology odules, Co. Algebra 33(11) (2005) [4] Hellus, Local Cohoology and atls dualty (Lepzg: Habltatonsschrft) (2006) [5] Hellus and Stückrad J, On endoorphs rngs of local cohoology, Proc. A. ath. Soc. 136 (2008) [6] Khashyaranesh K, On the atls duals of local cohoology odules, Arch. ath. (Basel) 88(5) (2007) [7] Khashyaranesh K, Salaran Sh and Zaker H, Characterzatons of flter regular sequences and uncondtoned strong d-sequences, Nagoya ath. J. 151 (1998) [8] O Carroll L, On the generalzed fractons of Sharp and Zaker, J. London ath. Soc. (2) 28(3) (1983) [9] O Carroll L, Generalzed fractons, deternantal aps, and top cohoology odules, J. Pure Appl. Algebra 32(1) (1984) [10] atls E, Injectve odules over Noetheran rngs, Pacfc J. ath. 8 (1958) [11] Schenzel P, Trung N V and Cuong N T, Verallgeenerte Cohen-acaulay-oduln, ath. Nachr. 85 (1978) [12] Sharp R Y and Zaker H, odules of generalzed fractons, atheatka 29(1) (1982) [13] Sharp R Y and Zaker H, Local cohoology and odules of generalzed fractons, atheatka 29(2) (1982) [14] Sharp R Y and Zaker H, odules of generalzed fractons and balanced bg Cohen- acaulay odules, Coutatve algebra: Durha 1981 (Durha, 1981) pp , London ath. Soc. Lecture Note Ser. 72 (New York: Cabrdge Unv. Press, Cabrdge) (1982) [15] Stückrad J and Vogel W, Buchsbau rngs and applcatons (Berln: VEB Deutscher Verlag der Wssenschaften) (1986)