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LOCAL HOMOLOGY, FINITENESS OF TOR MODULES AND COFINITENESS KAMRANDIVAANI-AAZAR,HOSSEINFARIDIANANDMASSOUDTOUSI Abstract. LetabeanidealofacommutativenoetherianringRwithunityand 7 M anR-modulesupportedatV(a). Letnbethesupermumoftheintegersifor 1 0 which Hia(M) 6= 0. We show that M is a-cofinite if and only if the R-module 2 TorRi (R/a,M) is finitely generated for every 0≤ i≤n. This provides a hands- onandcomputablefinitely-many-stepscriteriontoexaminea-confiniteness. Our n a approachreliesheavilyonthe theoryoflocalhomologywhichdemonstrates the J effectiveness andindispensabilityofthistool. 6 2 ] C 1. Introduction A Throughout this note, R denotes a commutative noetherian ring with unity. . h t The theory of local cohomology has been developed tremendously during the six a m decades of investigation after its introduction by Grothendieck. However, its dual [ theory, i.e. the theory of local homology has not been developed much. The theory of local homology was initiated by Matlis [Ma] in 1974, and its study was continued 1 v by Simon in [Si1] and [Si2]. After the groundbreaking works of Greenlees and May 1 [GM], and Alonso Tarr´ıo,Jerem´ıas Lo´pez and Lipman [AJL], a new era in the study 2 7 of this theory has begun; see e.g. [Sc], [CN1], [CN2], [Fr] and [Ri]. 7 To prove Theorem 1.2 below, we deploy the theory of local homology as the main 0 1. tool, which in effect illustrates the conspicuous role of this theory in the study of 0 cofiniteness, and provides along the way results of independent interest for a deeper 7 1 insight into the theory of local homology. v: It is proved in [BA, Lemma 2.3] that if a is an ideal of R that can be generated i by n elements and M is an R-module, then the R-module TorR(R/a,M) is finitely X i r generated for every i ≥ 0 if and only if it is finitely generated for every 0 ≤ i ≤ n. a Recall that the cohomologicaldimension of M with respect to a is cd(a,M):=sup{i≥0|Hi(M)6=0}, a andnotethatcd(a,M)≤ara(a). Fromthisangle,thefollowinggeneralizationofthis result is posed as a question in [PAB, Question 2]. 2010 Mathematics Subject Classification. 13D45;13E05. Keywords and phrases. Cofinitemodules;localcohomology;localhomology. The research of the first and third authors are supported by grants from IPM (No. 94130212 and No. 92130211; respectively). 1 2 K.DIVAANI-AAZAR,H.FARIDIANANDM.TOUSI Question 1.1. Let a be an ideal of R. If the R-module TorR(R/a,M) is finitely i generated for every 0 ≤ i ≤ cd(a,R), then is the R-module TorR(R/a,M) finitely i generated for every i≥0? Weanswerthisquestionintheaffirmativeinanevenstrongerform. Givenanideal a of R and an R-module M, we define the homological dimension of M with respect to a by hd(a,M):=sup{i≥0|Ha(M)6=0}. i It follows from [GM, Theorem 2.5 and Corollary 3.2], that hd(a,M) ≤ cd(a,R). As a matter of fact, we derive the following result. Theorem 1.2. (SeeTheorem 2.9.) Letabean ideal of R andM an R-module. Then the following assertions are equivalent: (i) Ha(M) is a finitely generated Ra-module for every i≥0. i (ii) TorR(R/a,M) is a finitely generated R-module for every i≥0. i b (iii) TorR(R/a,M) is a finitely generated R-module for every 0≤i≤hd(a,M). i In his algebraic geometry seminar of 1962, Grothendieck asked whether the R- modulesHom R/a,Hi(M) werefinitelygeneratedforeveryidealaofRandevery R a (cid:16) (cid:17) finitely generated R-module M; see [Gr, Conjectures 1.1 and 1.2]. Two years later, Hartshorne provided a counterexample in [Ha, Section 3], to show that this question does not have an affirmative answer in general. He then proposed the following definition. Definition 1.3. Let a be an ideal of R. An R-module M is said to be a-cofinite if Supp (M)⊆V(a), and Exti (R/a,M) is finitely generated for every i≥0. R R As an immediate application of Theorem 1.2 , we deduce the following result. It is worthnoting that Tormodules are easierto compute andmorehandy to dealwith than local homology modules. Moreover, the next result sets forth a finitely-many- steps criterion for cofiniteness. Corollary 1.4. (See Corollary 2.11.) Let a be an ideal of R and M an R-module with Supp (M)⊆V(a). Then the following assertions are equivalent: R (i) M is a-cofinite. (ii) TorR(R/a,M) is a finitely generated R-module for every 0≤i≤hd(a,M). i 2. Main Results We need to equip ourselveswith a handful of lemmas andpropositions in advance. We beginwith recallingthe definitionoflocalhomologymodules. Givenanideal aof R, we let Λa(M):=Ma =lim(M/anM) for an R-module M, and Λa(f):=f for an ←− R-homomorphismf :M →N. This provides us with the so-calleda-adic completion c b functor Λa(−) on the category of R-modules. Moreover, we define the ith local LOCAL HOMOLOGY, FINITENESS OF TOR ... 3 homology module of an R-module M with respect to a to be Ha(M):=H Λa(P) i i for every i≥0, where P is any projective resolution of M. (cid:0) (cid:1) Lemma 2.1. Let a be an ideal of R and M an R-module. If M/aM is a finitely generated R-module, then Ma is a finitely generated Ra-module. Proof. As M/aM is finitecly generated, there is a finibtely generated submodule N of M such that M = N +aM. Let ι : N → M be the inclusion map. Then by [Si1, Lemma 1.2], the Ra-homomorphism ι : Na → Ma is surjective. On the other hand, since Na ∼= Ra⊗RbN, it is clear thatbNabis a ficnitely generated Ra-module. It then follows that Ma is a finitely generated Ra-module. (cid:3) b b b b We recallcthat an R-module M is saibd to be a-adically quasi-complete if the completion map θ :M →Ma is an epimorphism. M Lemma 2.2. Let a be acn ideal of R. Then the following assertions hold: (i) Any submodule of an a-adically separated R-module is a-adically separated. (ii) Anyhomomorphicimageofana-adicallyquasi-completeR-moduleisa-adically quasi-complete. (iii) If f : M → N is a homomorphism of a-adically complete R-modules, then both kerf and imf are a-adically complete. Proof. (i): As kerθ = ∞ aiM for any R-module M, the assertion is clear. M i=1 (ii): If M is a-adicallyTquasi-complete and f : M → N is an epimorphism, then it follows from [Si1, Lemma 1.2] that f : Ma → Na is surjective. Therefore, the commutative diagram b c b f M N θ θ M N f Ma Na b c b shows that θ is surjective, i.e. N is a-adically quasi-complete. N (iii): As N is a-adically complete, it can be seen by inspection that ∞ kerf = kerf +aiM , i\=1(cid:16) (cid:17) i.e. kerf is a closed submodule of M in the a-adic topology. It now follows from [Si1, Proposition 1.3 (ii)] that kerf is a-adically complete. On the other hand, imf is both a submodule of N and a homomorphic image of M, so using (i) and (ii), we infer that imf is a-adically complete. (cid:3) Lemma2.3. LetabeanidealofRandM anR-module. SupposethatRisa-adically complete,M isa-adicallyquasi-complete, andM/aM isafinitelygeneratedR-module. Let θ :M →Ma be the completion map with K =kerθ . Then Ka =0=Ha(K). M M 0 c b 4 K.DIVAANI-AAZAR,H.FARIDIANANDM.TOUSI Proof. Since M/aM is a finitely generated R-module, Lemma 2.1 implies that Ma is a finitely generated R-module. From the short exact sequence c 0→K →M −θ−M→Ma →0, c we get the exact sequence TorR R/a,Ma →K/aK →M/aM, 1 (cid:16) (cid:17) which implies that K/aK is a finitecly generated R-module. Subsequently, from the short exact sequence 0→K/aK →M/aK →Ma →0, we deduce that M/aK is a finitely generatedR-mocdule. It then follows that the zero submodule of M/aK has a minimal primary decomposition n aK/aK = Q /aK, i i\=1 which in turn gives a minimal primary decomposition n aK = Q i i\=1 ofaK. WeprovethatK =aK,byshowingthatK ⊆Q forevery1≤i≤n. Assume i to the contrary that there is an integer 1 ≤ j ≤ n and an element x ∈ K\Q . Then j we have ax ⊆ aK ⊆ Q . This means that the homomorphism M/Q −→a M/Q is j j j not injective for every a∈a. As Q is a primary submodule of M, we conclude that j a⊆ (Q : M). Therefore, there is an integer t≥1 such that atM ⊆Q . But j R j p ∞ K = aiM ⊆atM ⊆Q , j i\=1 which is a contradiction. Therefore, K =aK, and so by [Si1, Lemma 5.1 (ii)], Ka = 0=Ha(K). (cid:3) 0 b The next result may be of independent interest. Lemma 2.4. Let a be an ideal of R and M an R-module. Then for any j ≥ 0 we have: a H (M) if i=0 Ha Ha(M) ∼= j i j (cid:16) (cid:17) 0 if i≥1.  Proof. Let ∂F ∂F F =···→F −−2→F −−1→F →0 2 1 0 be a free resolution of M. Then by definition Ha(M)=ker∂F/im∂F , j j j+1 c d LOCAL HOMOLOGY, FINITENESS OF TOR ... 5 for any j ≥ 0. By Lemma 2.2 (iii), both im∂F and ker∂F are a-adically complete. j+1 j Therefore,invoking[Si1,Lemmas 5.1(i), andd5.2(i)], therecarenaturalisomorphisms α:Ha im∂F →im∂F , 0 (cid:18) j+1(cid:19) j+1 d d and β :Ha ker∂F →ker∂F. 0 (cid:18) j (cid:19) j c c Now the short exact sequence (2.4.1) 0→im∂F −→ι ker∂F −→π Ha(M)→0, j+1 j j d c yields the following commutative diagram with exact rows Ha(ι) Ha(π) Ha im∂F 0 Ha ker∂F 0 Ha Ha(M) 0 0 (cid:18) j+1(cid:19) 0 (cid:18) j (cid:19) 0 (cid:16) j (cid:17) d c α ∼= β ∼= ∃ γ 0 im∂jF+1 ι ker∂jF π Hja(M) 0 d c from which we deduce that Ha(ι) is injective and γ is an isomorphism. On the other 0 hand, the short exact sequence (2.4.1) yields an exact sequence 0=Ha ker∂F →Ha Ha(M) →Ha im∂F =0, i+1(cid:18) j (cid:19) i+1(cid:16) j (cid:17) i (cid:18) j+1(cid:19) c d for every i≥1, where the vanishing follows from[Si1, Lemma 5.2 (i)]. It follows that Ha Ha(M) =0 for every i≥1. It further yields the exact sequence i+1 j (cid:16) (cid:17) 0=Ha ker∂F →Ha Ha(M) →Ha im∂F −H−−0a−(→ι) Ha ker∂F . 1 (cid:18) j (cid:19) 1 (cid:16) j (cid:17) 0 (cid:18) j+1(cid:19) 0 (cid:18) j (cid:19) c d c As Ha(ι) is injective, we conclude that Ha Ha(M) =0. (cid:3) 0 1 j (cid:16) (cid:17) Lemma 2.5. Let a be an ideal of R, M an R-module, and s≥0 an integer. Suppose that R is a-adically complete, and (R/a)⊗ Ha(M) is a finitely generated R-module. R s Then Ha(M) is a finitely generated R-module. s Proof. LetL=Ha(M). AsL/aLisfinitelygenerated,Lemma2.1impliesthatLa is s a finitely generated R-module. Lemma 2.2 yields that L is a-adically quasi-complete, b soLemma2.3impliesthatHa(kerθ )=0,whereθ :L→La isthecompletionmap. 0 L L Hence from the short exact sequence b 0→kerθ →L−θ→L La →0, L b we get the exact sequence a a a a 0=H (kerθ )→H (L)→H L →0, 0 L 0 0 (cid:16) (cid:17) b 6 K.DIVAANI-AAZAR,H.FARIDIANANDM.TOUSI implying that Ha(L) ∼= Ha La . By [Si1, Lemma 5.2 (i)], we have Ha La ∼= La. 0 0 0 (cid:16) (cid:17) (cid:16) (cid:17) On the other hand, Lemma 2b.4 implies that Ha(L) ∼= L. Hence L ∼= La isba finitebly 0 generated R-module. (cid:3) b Lemma 2.6. Let a be an ideal of R and M an R-module. Then there is a convergent spectral sequence E2 =TorR R/a,Ha(M) ⇒TorR (R/a,M). p,q p q p+q (cid:16) (cid:17) p Proof. Let F = (R/a)⊗ −, and G = Λa(−). Then F is right exact, and G(P) R is left F-acyclic for every projective R-module P, since the a-adic completion of a flat R-module is flat by [B, 1.4.7]. Therefore, by [Ro, Theorem 10.48], there is a Grothendieck spectral sequence E2 =L F L G(M) ⇒L (FG)(M). p,q p q p+q p (cid:0) (cid:1) LetF be afreeresolutionofM. By[B,Theorem1.3.1]or[Ma,Theorem15],wehave (R/a)⊗RΛa(F)∼=(R/a)⊗RF, whence Lp+q(FG)(M)=Hp+q (FG)(F) ∼=TorRp+q(R/a,M). (cid:0) (cid:1) (cid:3) Lemma 2.7. Let a be an ideal of R, M an R-module, and s ≥ 0 an integer. If TorR R/a,Ha(M) is finitely generated for every i ≥ 0 and 0 ≤ j ≤ s−1, and i j (cid:16) (cid:17) TorR(R/a,M) is finitely generated, then (R/a)⊗ Ha(M) is finitely generated. s R s Proof. By Lemma 2.6, there is a convergent spectral sequence E2 =TorR R/a,Ha(M) ⇒TorR (R/a,M). p,q p q p+q (cid:16) (cid:17) p Therefore, there is a finite filtration 0=U−1 ⊆U0 ⊆···⊆Us =TorR(R/a,M), s such that Up/Up−1 ∼=Ep∞,s−p for every p∈Z. As TorRs(R/a,M) is finitely generated, we conclude that E∞ ∼=U0/U−1 =U0 0,s is finitely generated. Let r ≥2, and consider the differentials Er −d−rr,−s−−r−+→1 Er −d−r0→,s Er =0, r,s−r+1 0,s −r,s+r−1 where the vanishing comes from the facts that E2 = 0 and Er is a −r,s+r−1 −r,s+r−1 subquotientofE−2r,s+r−1 foreveryr ≥2. Ontheotherhand,ass−r+1≤s−1,the hypothesis implies that Er and consequently imdr are finitely generated r,s−r+1 r,s−r+1 for every r ≥2. We thus obtain E0r,+s1 ∼=kerdr0,s/imdrr,s−r+1 =E0r,s/imdrr,s−r+1, LOCAL HOMOLOGY, FINITENESS OF TOR ... 7 and consequently a short exact sequence (2.7.1) 0→imdrr,s−r+1 →E0r,s →E0r,+s1 →0. There is an integer r ≥ 2, such that E∞ = Er+1 for every r ≥ r . It follows that 0 0,s 0,s 0 Er0+1 is finitely generated. Now the short exact sequence (2.7.1) implies that Er0 0,s 0,s is finitely generated. Using the short exact sequence (2.7.1) inductively, we conclude that E2 =(R/a)⊗ Ha(M) is finitely generated as desired. (cid:3) 0,s R s The next result is of grave importance in the theory of cofinite modules, which is included here for the convenience of the reader. Lemma 2.8. Let a be an ideal of R and M an R-module. Then the following condi- tions are equivalent: (i) Exti (R/a,M) is a finitely generated R-module for every i≥0. R (ii) Exti (N,M) is a finitely generated R-module for every finitely generated R- R module N with Supp (N)⊆V(a) and every i≥0. R (iii) TorR(R/a,M) is a finitely generated R-module for every i≥0. i (iv) TorR(N,M) is a finitely generated R-module for every finitely generated R- i module N with Supp (N)⊆V(a) and every i≥0. R (v) Ha(M) is a finitely generated Ra-module for every i≥0. i b Proof. Apply [WW, Propositions 7.1, 7.2, and Theorem 7.4] to the module case, bearing in mind that Ha(M)=0 for every i>cd(a,R). (cid:3) i Now, we are ready to prove our main result. Theorem 2.9. Let a be an ideal of R and M an R-module. Then the following assertions are equivalent: (i) Ha(M) is a finitely generated Ra-module for every i≥0. i (ii) TorR(R/a,M) is a finitely generated R-module for every i≥0. i b (iii) TorR(R/a,M) is a finitely generated R-module for every 0≤i≤hd(a,M). i Proof. (i) ⇒ (ii): Holds by Lemma 2.8. (ii) ⇒ (iii): Obvious. (iii) ⇒ (i): We claim that we may assume that R is a-adically complete. Indeed, if TorR(R/a,M) is a finitely generated R-module, then by [Si2, Lemma 2.3], i TorRiba Ra/aRa,Ra⊗RM ∼=TorRi (R/a,M) (cid:16) (cid:17) is a finitely generated R-mobduleb, anbd thus a finitely generated Ra-module. If the result is proved in the complete case, then by [Si2, Lemma 2.3], b Hia(M)∼=HiaRba Ra⊗RM (cid:16) (cid:17) is a finitely generated Ra-module. In addition,bthe latter isomorphism indicates that b hd(a,M)=hd aRa,Ra⊗ M . R (cid:16) (cid:17) b b 8 K.DIVAANI-AAZAR,H.FARIDIANANDM.TOUSI Having assumed that R is a-adically complete, we proceed to do induction on s to show that if TorR(R/a,M)is a finitely generatedR-module for every 0≤i≤s, then i Ha(M) is a finitely generated R-module. Clearly, this establishes the implication. If s s=0, then M/aM is finitely generated. Now [Si1, Theorem1.1and Lemma 5.1 (iv)] implies that Ha(M)/aHa(M) ∼=H\a(M)a/aH\a(M)a 0 0 0 0 ∼=Ma/aMa ∼=Mc/aMc. Therefore, the R-module Ha(M)/aHa(M) is finitely generated. Hence Lemma 2.5 0 0 implies that Ha(M) is finitely generated. Now assume that s > 0 and the claim is 0 true for s−1. Hence Ha(M) is finitely generated for every 0≤i ≤s−1. It follows i that TorR R/a,Ha(M) is finitely generated for every i ≥ 0 and 0 ≤ j ≤ s−1, i j (cid:16) (cid:17) and TorR(R/a,M)is finitely generated. Lemma 2.7 implies that (R/a)⊗ Ha(M) is s R s finitely generated, and subsequently Lemma 2.5 yields that Ha(M) is finitely gener- s ated as desired. (cid:3) Corollary 2.10. Let a be an ideal of R and M an R-module. If the R-module hd(a,R) TorR(R/a,M) i Mi=0 is finitely generated, then TorR(N,M) is a finitely generated R-module for every i finitely generated R-module N with Supp (N)⊆V(a) and every i≥0. R Proof. The proof is clear in light of Theorem 2.9 and Lemma 2.8. (cid:3) The next result provides an applicable criterion for cofiniteness. Corollary 2.11. Let a be an ideal of R and M an R-modulewith Supp (M)⊆V(a). R Then the following assertions are equivalent: (i) M is a-cofinite. (ii) TorR(R/a,M) is a finitely generated R-module for every 0≤i≤hd(a,M). i Proof. The proof is clear in view of Theorem 2.9 and Lemma 2.8. (cid:3) Corollary 2.12. Let N be a finitely generated R-module and M an R-module. If TorR R/Ann N,M isafinitelygeneratedR-moduleforevery0≤i≤hd(Ann N,M), i R R then T(cid:0)orR(N,M) is a(cid:1)finitely generated R-module for every i≥0. i Proof. The proof is clear in light of Theorem 2.9 and Lemma 2.8. (cid:3) References [AJL] L. Alonso Tarr´ıo, A. Jerem´ıas López and J. Lipman, Local homology and cohomology on schemes,Ann.Sci.EćoleNorm.Sup.,(4)30(1), (1997), 1-39. [B] J.Bartijn,Flatness, completion, regular sequences, un ménage à trois,Thesis.Utrecht(1985). LOCAL HOMOLOGY, FINITENESS OF TOR ... 9 [BA] K. Bahmanpour and M. Aghapournahr, A note on cofinite modules, Comm. Algebra, 44(9), (2016), 3683-3691. 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Math., 36(5),(2006), 1679-1703. [Ro] J.J.Rotman,Anintroductiontohomological algebra,Universitext.Springer,NewYork,second edition,2009. [Si1] A.M.Simon,Some homological properties ofcomplete modules,Math.Proc.Camb.Phil.Soc., 108(2),(1990), 231-246. [Si2] A.M. Simon, Adic-completion and some dual homological results, Publ. Mat. Camb., 36(2B), (1992), 965-979. [WW] S. Sather-Wagstaff and R. Wicklein, Support and adic finiteness for complexes, Arxive: 1401.6925v3, [math.AC]5Jun.2015. [Sc] P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand., 92(2), (2003), 161-180. K. Divaani-Aazar, Department of Mathematics, Alzahra University, Vanak, Tehran, Iran,ZipCode19834;andSchoolofMathematics,InstituteforResearchinFundamental Sciences(IPM),Tehran,Iran,P.O. Box 19395-5746. E-mail address: [email protected] H.Faridian,DepartmentofMathematics,ShahidBeheshtiUniversity,G.C.,Evin,Tehran, Iran,Zip Code 1983963113. E-mail address: [email protected] M.Tousi,DepartmentofMathematics,ShahidBeheshtiUniversity,G.C.,Evin,Tehran, Iran,Zip Code 1983963113. E-mail address: [email protected]