Characterizing Certain Semidualizing Complexes Via Their Betti and Bass Numbers

arXiv:2103.15531v1 [math.AC] 29 Mar 2021 oe-aalycmlx ulzn ope;smdaiigcmlx ty complex; semidualizing complex; dualizing complex; Cohen-Macaulay e od n phrases. and words Key 2020 lsi ulzn ope o oa ring local a for complex dualizing a is ules nw htahmlgclybuddcomplex d bounded injective homologically finite a with comple that complexes semidualizing known semidualizing precisely and are dimension These shifts projective the complexes. are dualizing complexes and du semidualizing rings of of examples study of Immediate notion the the unify plexes. introduced To [C1] Christensen algebra. modules, commutative semidualizing in and m tool a in venerable modules appeared a Semidualizing were be them algebra. by commutative induced and geometry theory algebraic duality rich the and plexes htthe that r ihrtesit fteudryn igo ulzn,vaterBett their via dualizing, or ring underlying the semidualizin of for shifts criteria the some either establish are We 3.4]. Proposition V, Chapter ic h ieHletpoe i eertdSzg hoe,teimp the theorem, number Syzygy Betti invariants celebrated numerical The his clear. proved becomes resolutions Hilbert time the Since HRCEIIGCRANSMDAIIGCMLXSVIA COMPLEXES SEMIDUALIZING CERTAIN CHARACTERIZING rtedekadHrson nrdcddaiigcmlxs see complexes; dualizing introduced Hartshorne and Grothendieck hogot ( Throughout, ahmtc ujc Classification. Subject Mathematics ye fsmdaiigcmlxsaetesit fteudryn ring underlying the of R shifts Tw the Let are numbers. complexes. s complexes Bass establishing semidualizing and by Betti of their types further, abou via fact, information complexes this of semidualizing highlight amount certain We large a rings. decoding commutative for tools worthwhile are of Abstract. ope for complex OA BLAHBII ARNDVAIAZRADMSODTOU MASSOUD AND DIVAANI-AAZAR KAMRAN BEIGI, ABOLFATH KOSAR d R n set and hBs ubrof number Bass th fadol fthe if only and if n R, R =sup := ti nw httenmrclivrat et ubr n asn Bass and numbers Betti invariants numerical the that known is It m fadol fthe if only and if HI ET N ASNUMBERS BASS AND BETTI THEIR C mltd facmlx asnme facmlx et ubro a of number Betti complex; a of number Bass complex; a of Amplitude k , easmdaiigcmlxfra nltclyirdcbelclring local irreducible analytically an for complex semidualizing a be sacmuaieNehra oa igwt o-eoidentity. non-zero with ring local Noetherian commutative a is ) C n and hBtinme of number Betti th X d =dim := soeadteohrBs ubr of numbers Bass other the and one is d 1. 31;1D2 13D09. 13D02; 13C14; hBs ubrof number Bass th R Introduction C eso that show We . R C 1 soe lo eso that show we Also, one. is fadol fteeeit ninteger an exists there if only and if X ihfiieygnrtdhmlg mod- homology generated finitely with C sone. is C sqaiioopi oashift a to quasi-isomorphic is eo module. a of pe fafiieygenerated finitely a of s m rtrafor criteria ome n dualizing and s C n asnumbers. Bass and i oue over modules t distinguished o H.Daiigcom- Dualizing [H]. eiulzn com- semidualizing X sadualizing a is lzn complexes alizing fteunderlying the of opee that complexes g eas rvnto proven also re r eo e [H, see zero; are n otxsin contexts any mnin tis It imension. rac ffree of ortance e ihfinite with xes umbers SI complex; d such 2 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

R-module M are defined by the minimal free resolution of M. By definition, the ith Betti R number of M, denoted βi (M), is the minimal number of generators of the ith syzygy of M in its minimal free resolution. By knowing the Betti numbers of an R-module, we can decode a lot of valuable information about it. i The ith Bass number of an R-module M, denoted µR(M), is defined by using the minimal injective resolution of M. Similar to the Betti numbers, the Bass numbers also reveal a lot of information about the module. The type of an R-module M is defined dimR M to be µR (M). It is known that local rings with small type have nice properties; see e.g. [B, V, F1, R, CHM, K, Ao, Le1, Le2, CSV]. By the work of Bass [B], every Cohen-Macaulay local ring of type one is Gorenstein. (In [CSV], this is improved to the n statement that if R is a Cohen-Macaulay local ring and µR(R) = 1 for some n ≥ 0, then R is Gorenstein of dimension n.) Vasconcelos [V, 4, p. 53] conjectured that every local ring of type one is Gorenstein. Foxby [F1] solved this conjecture for equicharacteristic local rings. The general case was answered by Roberts [R]. R In this article, we prove that if X is a Cohen-Macaulay complex with βsup X (X) = 1, sup X then X ≃ Σ R/ AnnR X; see Theorem 2.9(a). As an application, we show that if M is a finitely generated R-module of type one, then M is the dualizing module for the ring

R/AnnRM; see Corollary 2.10. This gives an aﬃrmative answer to a conjecture raised by Foxby [F1, Conjecture B], which is already proved; see e.g. [Le1, Corollary 2.7]. We may R improve Theorem 2.9(a), if we consider a semidualizing complex C with βsup C (C) ≤ 2. We do this in Theorems 2.9(b), 2.12 and 2.17. These results assert that:

Theorem 1.1. Let (R, m,k) be a local ring and C a semidualizing complex for R. R sup C (a) If βsup C (C)=1 and Ass R is a singleton, then C ≃ Σ R. R (b) If βsup C (C)=2 and R is a domain, then amp C =0. R (c) If βn (C)=1 for some integer n and either R is Cohen-Macaulay, or C is a module with a rank, then C ≃ ΣnR and n = sup C.

Dually, concerning semidualizing complexes with small type, in Corollaries 2.13 and 2.18, we prove that:

Theorem 1.2. Let (R, m,k) be a local ring and C a semidualizing complex for R. (a) If C is a module of type one, then C is dualizing. (b) If C has type one, then C is dualizing provided either Ass R is a singleton and R admits a dualizing complex, or Ass R is a singleton. (c) If C has type two and R is a domainb admitting a dualizing complex, then C is Cohen-Macaulay. CHARACTERIZING CERTAIN SEMIDUALIZING ... 3

n (d) If µR(C)=1 for some integer n, C is Cohen-Macaulay and R is either Cohen-

Macaulay or analytically irreducible, then C is dualizing and n = dimR C.

Each of parts (a) and (b) extends the above-mentioned result of Roberts. Also, part (c) extends the main result of [CHM]. Finally, part (d) generalizes [CSV, Characterization 1.6]. It is known that every dualizing complex has type one. Also, it is easy to check that if a R complex C is quasi-isomorphic to a shift of R, then βsup C (C) = 1. Thus, we immediately exploit Theorem 1.1(a) and Theorem 1.2(b) to obtain the following corollary:

Corollary 1.3. Let (R, m,k) be an analytically irreducible local ring and C a semidualizing complex for R. Then

R (a) C is quasi-isomorphic to a shift of R if and only if βsup C (C)=1. dimR C (b) C is a dualizing complex for R if and only if µR (C)=1.

2. The Results

In what follows, D(R) denotes the derived category of the category of R-modules. Also, f D✷(R) denotes the full subcategory of homologically bounded complexes with finitely generated homology modules. The isomorphisms in D(R) are marked by the symbol ≃. For a complex X ∈ D(R), its supremum and infimum are defined, respectively, by

sup X := sup{i ∈ Z | Hi(X) =6 0} and inf X := inf{i ∈ Z | Hi(X) =6 0}, with the usual convention that sup ∅ = −∞ and inf ∅ = ∞. Also, amplitude of X is defined by amp X := sup X − inf X. In what follows, we will use, frequently, the endofunctors L n −⊗R −, R HomR(−, −) and Σ (−), n ∈ Z, on the category D(R). For the definitions and basic properties of these functors, we refer the reader to [C2, Appendix]. Theorem 2.9 is the first main result of this paper. To prove it, we need Lemmas 2.5, 2.7 and 2.8. For proving Lemma 2.7, we require to establish Lemmas 2.1, 2.2 and 2.6. Lemma 2.1 improves, slightly, the celebrated New Intersection theorem. Our proofs of Lemmas 2.1, 2.2 are slight modifications of the proofs of [F1, Theorem 1.2] and [F1, Lemma 3.2]. Also, Remark 2.4 serves as a powerful tool throughout the paper.

Lemma 2.1. Let (R, m,k) be a local ring. Let

F =0 −→ Fs −→ Fs−1 −→··· −→ F0 −→ 0

be a non-exact complex of ﬁnitely generated free R-modules and t be a non-negative integer

such that dimR(Hi(F )) ≤ i + t for all 0 ≤ i ≤ s. Then dim R ≤ s + t. 4 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

Proof. Without loss of generality, we may and do assume that R is complete. Let p be a

prime ideal of R with dim R/p = dim R and set F := F ⊗R R/p. Then F is a bounded complex of ﬁnitely generated free R/p-modules. Bye virtue of [C2, Corollarye A.4.16], we have inf F = inf F , and so the complex F is non-exact.

Next, wee show that dimR/p(Hi(F )) ≤ ie+ t for all 0 ≤ i ≤ s. Suppose that the contrary

holds. Then, we may choose ℓ toe be the least integer such that dimR/p(Hℓ(F )) > ℓ + t. e So, there is a prime ideal q/p ∈ SuppR/p(Hℓ(F )) such that dim(R/p)/(q/p) >ℓ + t. From

the choice of ℓ, it turns out that inf((F )q/p)=e ℓ. But, e ∼ (F )q/p = (F ⊗R R/p) ⊗R/p (R/p)q/p ∼ e = F ⊗R Rq/pRq ∼ = (F ⊗R Rq) ⊗Rq Rq/pRq ∼ = Fq ⊗Rq Rq/pRq,

and so applying [C2, Corollary A.4.16] again, yields that inf Fq = ℓ. Hence q ∈

SuppR(Hℓ(F )), and so dimR(Hℓ(F )) >ℓ + t. We arrive at a contradiction. Therefore, in the rest of proof, we assume that R is a catenary local domain. We proceed by induction on d := dim R. Obviously, the claim holds for d = 0. Suppose

that d > 0 and the claim holds for d − 1. If dimR(Hi(F )) ≤ 0 for all 0 ≤ i ≤ s, then by the New Intersection theorem, we deduce that dim R ≤ s. So, we may assume that

dimR(Hj(F )) > 0 for some 0 ≤ j ≤ s. Then, we can take a prime ideal p ∈ SuppR(Hj(F ))

such that dim R/p = 1. Now, Fp is a non-exact complex of ﬁnitely generated free Rp- modules and

dimRp (Hi(Fp)) ≤ dimR(Hi(F )) − 1 ≤ i +(t − 1)

for all 0 ≤ i ≤ s. Since dim Rp = dim R − 1, the induction hypothesis implies that

dim Rp ≤ s +(t − 1). Hence, dim R ≤ s + t.

To present the next result, we have to ﬁx some notation. Let t be a non-negative integer and

F = ··· −→ Fi −→ Fi−1 −→··· −→ Ft+1 −→ Ft −→ 0 be a complex of ﬁnitely generated free R-modules. For each i ≥ t, set

i−t γi(F ) := ri − ri−1 + ··· +(−1) rt,

where rj is the rank of Fj for each j ≥ t.

Lemma 2.2. Let (R, m,k) be a local ring and t ≤ dim R a non-negative integer. Let

F = ··· −→ Fi −→ Fi−1 −→··· −→ Ft −→ 0 CHARACTERIZING CERTAIN SEMIDUALIZING ... 5

be a complex of ﬁnitely generated free R-modules such that dimR(Hi(F )) ≤ i for all t ≤ i ≤ dim R. Then for each integer t ≤ n ≤ dim R, we have:

(a) γn(F ) ≥ 0.

(b) γn(F ) > 0 if Ht(F ) =06 and either dimR(Hn(F )) = n or n< dim R.

Proof. Let the complex

G =0 −→ Fn −→ Fn−1 −→··· −→ Ft −→ 0 be the truncation of F at the spot n and let Zn denote the kernel of the diﬀerential map

Fn −→ Fn−1. Then, Hi(G) = Hi(F ) for all i < n and Hn(G) = Zn. If n = t, then γn(F ) is the rank of the free R-module Ft, and so both assertions are immediate in this case. So, in the rest of the proof, we assume that n > t.

(a) Let p be a prime ideal of R with dim R/p ≥ n. As dimR(Hi(F )) ≤ i for all t ≤ i ≤ n − 1, it follows that the complex

0 −→ (Zn)p −→ (Fn)p −→ (Fn−1)p −→··· −→ (Ft)p −→ 0 is exact. Hence, (Zn)p is a free Rp-module of rank γn(F ). So, γn(F ) ≥ 0.

(b) Assume that Ht(F ) =6 0. Suppose that dimR Zn < n. Then, applying Lemma 2.1 to the complex G, we get dim R ≤ n. Also, we have

dimR(Hn(F )) ≤ dimR Zn < n.

Thus, if either dimR(Hn(F )) = n or n < dim R, then dimR Zn ≥ n. Let p ∈ SuppR Zn be such that dim R/p ≥ n. Then, as we saw in the proof of (a), the non-zero Rp-module

(Zn)p is free of rank γn(F ), and so γn(F ) > 0. Let (R, m,k) be a local ring. For an R-complex X and an integer n, the nth Betti R R L number of X, βn (X), is defined as the rank of the k-vector space Torn (k,X)(:= Hn(k ⊗R n X)). Dually, the nth Bass number of X, µR(X), is defined as the rank of the k-vector n space ExtR(k,X)(:= H−n(R HomR(k,X))). Also, dimension, depth and Cohen-Macaulay defect of an R-complex X are, respectively, defined as follows:

dimR X := sup{dimR(Hi(X)) − i | i ∈ Z},

i depthR X := inf{i ∈ Z | ExtR(k,X) =06 }, and

cmdR X := dimR X − depthR X.

For an R-complex X, it is known that dimR X = sup{dim R/p − inf Xp | p ∈ Spec R}. We set

AsshR X := {p ∈ Spec R | dim R/p − inf Xp = dimR X}. 6 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

dimR X Finally, the type of an R-complex X is deﬁned to be µR (X).

f Deﬁnition 2.3. Let (R, m,k) be a local ring. A non-exact complex X ∈D(R) is called

Cohen-Macaulay if depthR X = dimR X.

f Let (R, m,k) be a local ring. A semidualizing complex for R is a complex C ∈D✷(R)

such that the homothety morphism R −→ R HomR(C,C) is an isomorphism in D(R). If, furthermore, C has ﬁnite injective dimension, then it is called a dualizing complex. It is obvious that any shift of a (semi)dualizing complex is again a (semi)dualizing complex. A dualizing complex D satisfying sup D = dim R, is called a normalized dualizing complex. Given a dualizing complex D, it is immediate that Σdim R−sup DD is a normalized dualizing i complex. For a normalized dualizing complex D, it is known that µR(D) = 0 for all i =06 0 and µR(D) = 1; in particular D has depth zero. An R-module which is a (semi)dualizing complex for R is said to be a (semi)dualizing module. Immediate examples of semidualizing modules are free R-modules of rank one. It is known that the ring R possesses a dualizing complex if and only if it is homomorphic image of a Gorenstein local ring. In particular, in view of the Cohen Structure theorem, every complete local ring admits a (normalized) dualizing complex. Also, by virtue of [BH, Theorem 3.3.6], we can see that the ring R admits a dualizing module if and only if it is Cohen-Macaulay and it is homomorphic image of a Gorenstein local ring.

Remark 2.4. Let (R, m,k) be a local ring and D a normalized dualizing complex for R. There is a dagger duality functor

† f f (−) := R HomR(−,D): D✷(R) −→ D✷(R),

f such that for every X ∈D✷(R), the following assertions hold:

(a) (X†)† ≃ X. † † (b) dimR X = sup X and depthR X = inf X . (c) X is a Cohen-Macaulay complex if and only if amp X† = 0. n R † (d) µR(X)= βn (X ) for all integers n. † (e) idR X = pdR X . (f) X is a semidualizing complex for R if and only if X† is so.

Proof. For parts (a) and (b) see e.g. [C2, Theorem A.8.5]. For part (b), note that

depthR D = 0. (c) is obvious by (b). CHARACTERIZING CERTAIN SEMIDUALIZING ... 7

i (d) As D is a normalized dualizing complex for R, we have µR(D) = 0 for all i =6 0 and 0 µR(D) = 1. Thus, for every integer n, [F2, Theorem 4.1(a)] yields that µn (X)= µn (R Hom (X†,D)) = βR(X†)µn−i(D)= βR(X†). R R R X i R n i∈Z

f (e) For each complex Y ∈ D✷(R), by virtue of [AF, Corollary 2.10.F and Proposition L 5.5], it turns out that pdR Y = sup(k ⊗R Y ) and idR Y = − inf(R HomR(k,Y )), and so

R pdR Y = sup{i ∈ Z | βi (Y ) =06 }, and

i idR Y = sup{i ∈ Z | µR(Y ) =06 }. † Therefore, (d) implies that idR X = pdR X . (f) See [C1, Corollary 2.12].

In the statement and proof of the next result, we use the notions of weak annihilator of a complex and attached prime ideals of an Artinian module. We recall these notions. For an Artinian R-module A, the set of attached prime ideals of A, AttR A, is the set of all prime ideals p of R such that p = AnnR L for some quotient L of A; see e.g. [BS, Chapter 7] for more details. Following [Ap], for a complex X ∈D(R), the weak annihilator of X is deﬁned as AnnR X := AnnR(Hi(X)). iT∈Z

f Lemma 2.5. Let (R, m,k) be a local ring and X ∈ D✷(R) a non-exact complex. Let n := sup X and d := dimR X.

(a) dimR(Hi(X)) ≤ i + d for all i ≤ n.

(b) If X is Cohen-Macaulay, then dimR(Hn(X)) = n + d. (c) If R possesses a normalized dualizing complex D and amp X = 0 = amp X†, then † AnnR X = AnnR X .

Proof. (a) It is obvious, because

d = dimR X = sup{dimR(Hi(X)) − i | i ∈ Z}.

(b) Without loss of generality, we may and do assume that R is complete. Then R possesses a normalized dualizing complex D. By virtue of Remark 2.4, we have amp X† = † † † d † † † 0, sup X = d and dimR X = n. So, X ≃ Σ Hd(X ). Set N := Hd(X ). As dimR X = n, it follows that dimR N = n + d. Thus, in view of Remark 2.4(a) and the Local Duality 8 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI theorem [H, Chapter V, Theorem 6.2], we have the following display of isomorphisms:

dimR N † H (N) ∼= Hom H + N , E (R/m) m R n d R ∼ −d † = HomR Hn Σ N , ER(R/m) † ∼= Hom H ΣdN , E (R/m) R n R † ∼= Hom H X† , E (R/m) R n R ∼ = HomR (Hn(X), ER(R/m)) . Now, [BS, Theorem 7.3.2 and Exercise 7.2.10(iv)] yields that

dimR N AssR(Hn(X)) = AttR(Hm (N)) = {p ∈ AssR N | dim R/p = dimR N}, and so dimR(Hn(X)) = n + d. (c) First of all note that for an R-module L, an R-complex Y and an integer i, we may † i easily verify that AnnR L ⊆ AnnR(Hi(L )) and AnnR Y = AnnR(Σ Y ). Set M := Hn(X) † n † d and N := Hd(X ). Then X ≃ Σ M and X ≃ Σ N. Thus, the claim is equivalent to † n+d † n+d AnnR M = AnnR N. Now, we have M ≃ Σ N and N ≃ Σ M, and so

† AnnR M ⊆ AnnR(M )

= AnnR N † ⊆ AnnR(N )

= AnnR M.

f Lemma 2.6. Let (R, m,k) be a local ring. Let X ∈D✷(R) be a non-exact complex such R sup X that βsup X (X)=1 and amp X =0. Then X ≃ Σ R/ AnnR X. Furthermore:

(a) If X is Cohen-Macaulay, then the local ring R/ AnnR X is also Cohen-Macaulay.

(b) If AnnR X =0, then pdR X < ∞.

n Proof. Set RX := R/ AnnR X, n := sup X and M := Hn(X). Then X ≃ Σ M. Now,

R R −n R β0 (M)= β0 (Σ X)= βn (X)=1.

∼ n Thus M is cyclic, and so M = RX . Therefore, X ≃ Σ RX , as required. Now, the remaining assertions are trivial.

f Lemma 2.7. Let (R, m,k) be a local ring. Let X ∈ D✷(R) be a non-exact complex R with βsup X (X)=1. Suppose that dimR X = 0 and dimR(Hsup X (X)) = sup X. Then sup X X ≃ Σ R/ AnnR X. CHARACTERIZING CERTAIN SEMIDUALIZING ... 9

Proof. In view of Lemma 2.6, the assertion is equivalent to amp X = 0. Let n := sup X and t := inf X. We claim that n = t. On the contrary, assume that n > t. Let

F = ··· −→ Fi −→ Fi−1 −→··· −→ Ft −→ 0

R be the minimal free resolution of X. Then Fi is a free R-module, of ﬁnite rank βi (X), for all integers i. From the deﬁnition of dimension of a complex, we get

0 = dimR X ≥ dimR(Ht(X)) − t,

and so t ≥ 0.

Our assumptions implies that dimR(Hi(X)) ≤ i for all i < n and dimR(Hn(X)) = n. In particular, n ≤ dim R. Thus, by Lemma 2.2, we have γn(F ) > 0 and γn−1(F ) > 0. This yields that R βn (X)= γn(F )+ γn−1(F ) ≥ 2. We arrive at a contradiction.

For every semidualizing complex C for a local ring R, Christensen [C1, Corollary 3.4] has proved the inequality cmd R ≤ cmdR C + amp C. Part (c) of the next result shows that it is an equality, provided Ass R is a singleton.

Lemma 2.8. Let (R, m,k) be a local ring and C a semidualizing complex.

(a) AssR(Hsup C (C)) = {p ∈ Ass R | inf Cp = sup C}.

(b) AsshR C ⊆ Ass R.

Proof. Set n := sup C and d := dimR C.

(a) By [C1, Corollary A.7], a prime ideal p of R belongs to p ∈ AssR(Hn(C)) if and only depth Rp = 0 and inf Cp = n. But, for a prime ideal p of R, it is evident that p ∈ Ass R if and only if depth Rp = 0. Thus,

AssR(Hn(C)) = {p ∈ Ass R | inf Cp = n}.

(b) Let A be an Artinian R-module. Then A can be given a natural structure as an Artinian R-module, and so b AttR A = {p ∩ R | p ∈ AttRb A}. ∼ Also, we know that there is a natural R-isomorphism A ⊗R R = A. By [HD, Proposition d b b 2.1], the R-module Hm(C) is Artinian. This and [Li, Corollary 3.4.4] yield, respectively, 10 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

the following R-isomorphisms: b d ∼ d ∼ d Hm(C) = Hm(C) ⊗R R = Hmb (C ⊗R R). b b By [C1, Lemma 2.6], it follows that C ⊗R R is a semidualizing complex for the local ring R b b and we may check that dimRb(C ⊗R R) = dimR C. Thus, by [HD, Lemma 2.5], we deduce that b d AsshR C = AttR(Hm(C)) d = {p ∩ R | p ∈ AttRb(Hmb (C ⊗R R))} b b = {p ∩ R | p ∈ AsshR(C ⊗R R)}. On the other hand, it is known that b

Ass R = {p ∩ R | p ∈ Ass R}. b Therefore, we may and do assume that R is complete. Then R possesses a normalized dualizing complex D. Now, the Local Duality theorem [H, Chapter V, Theorem 6.2] implies that Hd (C) =∼ Hom H (C†), E (R/m) , m R d R and so d † AsshR C = AttR(Hm(C)) = AssR(Hd(C )). By parts (f) and (b) of Remark 2.4, C† is a semidualizing complex for R with sup C† = d.

Thus, (a) yields that AsshR C ⊆ Ass R. (c) Let p be the unique member of Ass R. Then, by virtue of (b) and (a), we have

AsshR C = {p} = AssR(Hn(C)).

Hence,

dimR C = dim R/p − inf Cp = dim R − n. By [C1, Corollary 3.2], it turns out that

depth R = depthR C + inf C. This implies that

cmdR C + amp C = dim R − n − depthR C + n − inf C = dim R − depth R = cmd R.

Now, we are ready to prove our ﬁrst theorem. CHARACTERIZING CERTAIN SEMIDUALIZING ... 11

f Theorem 2.9. Let (R, m,k) be a local ring. Let X ∈D✷(R) be a non-exact complex with R βsup X (X)=1. sup X (a) Assume that X is Cohen-Macaulay. Then X ≃ Σ R/ AnnR X. (b) Assume that Ass R is a singleton and X is a semidualizing complex for R. Then X ≃ Σsup X R.

dimR X Proof. (a) By replacing X with Σ X, we may further assume that dimR X = 0.

Then, by Lemma 2.5, we have dimR(Hsup X (X)) = sup X. Now, Lemma 2.7 yields the claim. (b) By replacing X with Σdim R−sup X X, we may and do assume that sup X = dim R.

Then, Lemma 2.8(c) implies that dimR X = 0. On the other hand, by Lemma 2.8(a), we deduce that

dimR(Hsup X (X)) = dim R = sup X. sup X Now, Lemma 2.7 yields that X ≃ Σ R/ AnnR X. But, then R/ AnnR X would be a

semidualizing module for R, which in turns implies that AnnR X = 0.

The next result provides an aﬃrmative answer to a conjecture raised by Foxby [F1, Conjecture B].

Corollary 2.10. Let (R, m,k) be a local ring. Let M be a non-zero ﬁnitely generated

R-module of type one. Then M is the dualizing module for the local ring R/AnnRM. In

particular, the local ring R/ AnnR M is Cohen-Macaulay.

Proof. Without loss of generality, we may and do assume that R is complete. Then R † possesses a normalized dualizing complex D. Set X := M and n := dimR M. By using parts (a),(b) and (c) of Remark 2.4, we can see that X is a Cohen-Macaulay complex and R sup X = n. By Remark 2.4(d), we conclude that βn (X) = 1, and so Theorem 2.9(a) yields n that X ≃ Σ R/ AnnR M. Note that Lemma 2.5(c) implies that AnnR X = AnnR M. As † (R/ AnnR M) = R HomR(R/ AnnR M,D)

is a dualizing complex for the local ring R/ AnnR M and

† † † −n † M ≃ (M ) ≃ X ≃ Σ (R/ AnnR M) ,

it follows that M is the dualizing module for the local ring R/AnnRM.

Although the next result is known to the experts, we give a proof for the sake of completeness. 12 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

Lemma 2.11. Let (R, m,k) be a local ring. Let

F F ∂i ∂t+1 F = ··· −→ Fi −→ Fi−1 −→··· −→ Ft −→ 0

F be a complex of ﬁnitely generated free R-modules such that im ∂i ⊆ mFi−1 for all integers i > t.

(a) If F ≃ 0, then Fi =0 for all i ≥ t. F (b) Assume that F is indecomposable in D(R) and ∂j is zero for some j > t. Then

Fi =0 either for all i ≥ j or for all i ≤ j − 1.

Proof. (a) We can split F into the short exact sequences

F im ∂t+2 ֒−→ Ft+1 −→ Ft −→ 0 →− 0 . . F F im ∂i+2 ֒−→ Fi+1 −→ im ∂i+1 −→ 0 →− 0 . .

F Then, it turns out that the R-module im ∂i is projective for all i > t +1 and all of the

above short exact sequences are split. Thus F is split-exact, and so the complex k ⊗R F F is split-exact too. Since, im ∂i ⊆ mFi−1 for all i > t, it follows that all diﬀerential maps

of the complex k ⊗R F are zero. Hence,

0 = Hi(k ⊗R F )= k ⊗R Fi

for all i ≥ t. Now, by Nakayama’s lemma, Fi = 0 for all i ≥ t. (b) Set

Y := ··· −→ Fl −→ Fl−1 −→··· −→ Fj −→ 0 and

Z := 0 −→ Fj−1 −→ Fj−2 −→ ··· −→ Ft −→ 0. Then, clearly we have F ∼= Y ⊕ Z. So, either Y ≃ 0 or Z ≃ 0. Now, the claim follows by (a).

Next, we prove our second theorem.

Theorem 2.12. Let (R, m,k) be a local domain and C a semidualizing complex. If R βsup C (C)=2, then amp C =0.

Proof. Set n := sup C. If pdR C < ∞, then [C1, Proposition 8.3] implies that C ≃ R,

and so amp C = 0. So, in the rest of the proof, we assume that pdR C = ∞. CHARACTERIZING CERTAIN SEMIDUALIZING ... 13

Let

F = ··· −→ Fi −→ Fi−1 −→··· −→ Ft −→ 0 R be the minimal free resolution of C. Then Fi is a free R-module, of ﬁnite rank βi (C),

for all i ≥ t. Since, R is indecomposable and R ≃ R HomR(C,C), it follows that C and,

consequently, F are indecomposable in D(R). Since Fn =6 0, in view of Lemma 2.11(b), F vanishing of the map ∂n+1 implies that Fi = 0 for all i > n. So, aspdR C = ∞, we deduce F that ∂n+1 =6 0. F F Next, we show that ∂n = 0. As im ∂n is torsion-free, we equivalently prove F that rankR(im ∂n ) = 0. Lemma 2.8(a) yields that AssR(Hn(C)) ⊆ Ass R. Hence

AssR(Hn(C)) = {0}, and so Hn(C) is torsion-free. As the R-module Hn(C) is non-zero F and torsion-free, it follows that rankR(Hn(C)) ≥ 1. As, also, rankR(im ∂n+1) is positive, from the short exact sequence

F F 0 −→ im ∂n+1 −→ ker ∂n −→ Hn(F ) −→ 0,

F we get that rankR(ker ∂n ) ≥ 2. Finally, in view of the short exact sequence

F F 0 −→ ker ∂n −→ Fn −→ im ∂n −→ 0,

F we see that rankR(im ∂n ) = 0. F Now as ∂n = 0 and Fn =6 0, Lemma 2.11(b) yields that Fi = 0 for all i ≤ n − 1. In particular, inf C = inf F = n, and so amp C = 0.

Each of parts (a) and (b) of the next result generalizes the main result of [R]. Also, its part (c) extends the main result of [CHM].

Corollary 2.13. Let (R, m,k) be a local ring and C a semidualizing complex for R.

(a) Assume that C is a module of type one. Then C is dualizing. (b) Assume that C has type one. Then C is dualizing; provided either (1) Ass R is a singleton and R admits a dualizing complex; or (2) Ass R is a singleton. (c) Assumeb that R is a domain admitting a dualizing complex and C has type two. Then C is Cohen-Macaulay.

Proof. (a) is immediate by Corollary 2.10. Note that AnnR C = AnnR R = 0.

By [C1, Lemma 2.6], C ⊗R R is a semidualizing complex for R. It is easy to verify that

the R-complex C and the R-complexb C ⊗R R have the same type.b On the other hand, by b b 14 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

[AF, Proposition 5.5], we have

idR C = − inf(R HomR(k,C))

= − inf(R HomR(k,C) ⊗R R) b b = − inf(R HomR(k,C ⊗R R)) b = idRb(C ⊗R R). b So in part (2) of (b), we may also assume that R possesses a dualizing complex. Let D † be a normalized dualizing complex for R. Set n := dimR C. By Remark 2.4(f), L := C is also a semidualizing complex for R. By parts (b) and (d) of Remark 2.4, we have R n sup L = n and βn (L)= µR(C). R (b) We have βn (L) = 1 and Ass R is a singleton. Hence, Theorem 2.9(b) implies that L ≃ ΣnR. So, by Remark 2.4(a), we have C ≃ Σ−nD. R (c) We have βn (L) = 2. Theorem 2.12 yields that amp L = 0, and so, in view of Remark 2.4(c), we conclude that C is Cohen-Macaulay. The next result is a useful tool for reducing the study of Betti numbers of semidualizing modules over Cohen-Macaulay local rings to the same study on Artinian local rings.

Lemma 2.14. Let (R, m,k) be a d-dimensional Cohen-Macaulay local ring and C a semid-

ualizing module for R. Let x = x1, x2,...,xd ∈ m be an R-regular sequence and set R := R/hxi and C := C/hxiC. Then C is a semidualizing module for the Artinian local R R ∼ ring R and βi (C) = βi (C) for all integers i ≥ 0. Furthermore, C = R if and only if C =∼ R.

Proof. By [S, Theorem 2.2.6], C is a semidualizing module for the Artinian local ring R. Let F be a free resolution of the R-module C. By [BH, Proposition 1.1.5], we conclude

that R ⊗R F is a free resolution of the R-module C. Hence, for every non-negative integer i, we have the following natural R-isomorphisms:

R ∼ ∼ ∼ R Tori (k, C) = Hi(k ⊗R (R ⊗R F )) = Hi(k ⊗R F ) = Tori (k,C).

R R Thus βi (C)= βi (C) for all i ≥ 0. The remaining assertion is immediate by [S, Proposi- tion 4.2.18]. We apply the following obvious result in the proof of Theorem 2.17.

Lemma 2.15. Let K be an R-module and 0 −→ L −→ M −→ N −→ 0 an short exact i i sequence of R-homomorphisms such that ExtR(M,K) =0= ExtR(N,K) for all i > 0. Then the sequence

0 −→ HomR(N,K) −→ HomR(M,K) −→ HomR(L, K) −→ 0 CHARACTERIZING CERTAIN SEMIDUALIZING ... 15

i is exact and ExtR(L, K)=0 for all i> 0.

The following result is also needed in the proof of Theorem 2.17. It is interesting in its own right.

Lemma 2.16. Let (R, m,k) be a local ring and M a ﬁnitely generated R-module with a R rank. If the projective dimension of M is inﬁnite, then βn (M) ≥ 2 for all n ∈ N0.

R R Proof. On the contrary, assume that βn (M) < 2 for some n ∈ N0. Then βn (M) = 1, R because pdR M = ∞. For each non-negative integer i, set βi := βi (M). Hence, there is an exact sequence

∂ ∂1 ∂0 F = ··· −→ Rβi −→i Rβi−1 −→··· −→ Rβ0 −→ M −→ 0.

i For each natural integer i, set Ω M := ker ∂i−1. Splitting F into short exact sequences and applying [BH, Proposition 1.4.5], successively, to them, yields that each ΩiM has i i a rank. Since Ω M is torsion-free and non-zero, it follows that rankR(Ω M) > 0 for all i ∈ N. By applying [BH, Proposition 1.4.5] to the short exact sequence

0 −→ Ωn+1M −→ R −→ ΩnM −→ 0,

it turns out that n+1 n rankR(Ω M) + rankR(Ω M)=1. n n+1 Hence, either rankR(Ω M) = 0 or rankR(Ω M) = 0, and we arrive at the desired contradiction.

Finally, we are in a position to prove our third (and last) theorem.

Theorem 2.17. Let (R, m,k) be a local ring and C a semidualizing complex for R such R that βn (C)=1 for some integer n. Assume that either R is Cohen-Macaulay or C is a module with a rank. Then C ≃ ΣnR and n = sup C.

Proof. First, we assume that R is Cohen-Macaulay. By [C1, Corollary 3.4(a)], we have amp C ≤ cmd R, and so we may and do assume that C is a module. Then, it would be enough to show that C ∼= R. On the contrary, suppose that C ≇ R. By Lemma 2.14, we may and do assume that R is Artinian. By [S, Corollaries 2.1.14 and 2.2.8], we ∼ know that if C is cyclic or pdR C < ∞, then C = R. So, we may and do assume that

n ≥ 1 and pdR C = ∞, and look for a contradiction. For each non-negative integer i, set R βi := βi (C). Hence, the minimal free resolution of C has the form

··· −→ Rβn+1 −→ R −→ Rβn−1 −→··· −→ Rβ1 −→ Rβ0 −→ 0. 16 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

Then, we have the following short exact sequences:

(0) 0 −→ Ω1C −→ Rβ0 −→ C −→ 0

(1) 0 −→ Ω2C −→ Rβ1 −→ Ω1C −→ 0 . . (n − 1) 0 −→ ΩnC −→ Rβn−1 −→ Ωn−1C −→ 0 (n) 0 −→ Ωn+1C −→ R −→ ΩnC −→ 0.

∗ We successively apply the contravariant functor (−) := HomR(−,C) to the short exact sequences (0), (1),..., (n) and use Lemma 2.15 to get the following short exact sequences:

(0)∗ 0 −→ R −→ Cβ0 −→ (Ω1C)∗ −→ 0

(1)∗ 0 −→ (Ω1C)∗ −→ Cβ1 −→ (Ω2C)∗ −→ 0 . . (n − 1)∗ 0 −→ (Ωn−1C)∗ −→ Cβn−1 −→ (ΩnC)∗ −→ 0 (n)∗ 0 −→ (ΩnC)∗ −→ C −→ (Ωn+1C)∗ −→ 0.

i As pdR C = ∞, it follows that Ω C =6 0 for all integers i ≥ 0. Suppose that (ΩjC)∗ = 0 for some integer j ≥ 0. Then, by [S, Proposition 3.1.6], (Ωj−1C)∗ ∼= Cβj−1 ∈

BC (R). Applying [S, Proposition 3.1.7(b)], successively, to the short exact sequences ∗ ∗ ∗ (j − 2) ,..., (1) , (0) yields that R ∈ BC (R). Now, [S, Corollary 4.1.7] implies that C =∼ R. Thus (ΩiC)∗ =6 0 for all integers i ≥ 0. n Set h := β0 − β1 + ··· +(−1) βn. Using the short exact sequences (0), (1),..., (n), we deduce that n+1 n+1 (A1): ℓR(C)= hℓR(R)+(−1) ℓR(Ω C) and n+1 (B1): ℓR(Ω C) <ℓR(R). Similarly, by using the short exact sequences (0)∗, (1)∗,..., (n)∗, we may see that

n+1 n+1 ∗ (A2): ℓR(R)= hℓR(C)+(−1) ℓR((Ω C) )

and n+1 ∗ (B2): ℓR((Ω C) ) <ℓR(C).

Assume that n is even. If h ≤ 0, then A1 yields that ℓR(C) < 0. If h = 1, then A1 and

A2, respectively, imply that ℓR(C) < ℓR(R) and ℓR(R) < ℓR(C). If h ≥ 2, then A1 and

B1 imply that ℓR(C) >ℓR(R). On the other hand, A2 and B2 imply that ℓR(R) >ℓR(C). CHARACTERIZING CERTAIN SEMIDUALIZING ... 17

Next, assume that n is odd. If h < 0, then, by A1 and B1, we deduce that ℓR(C) <

0. If h = 0, then A1 and B1 imply that ℓR(C) < ℓR(R) and A2 and B2 imply that

ℓR(R) < ℓR(C). If h ≥ 1, then A1 and A2, respectively, imply that ℓR(C) > ℓR(R) and

ℓR(R) >ℓR(C). Therefore if either n is even or odd, we arrive at a contradiction. Now, assume that C is a module with a rank. Then, Lemma 2.16 implies that C has ﬁnite projective dimension, and so C ∼= R by [S, Corollary 2.2.8]. The next result extends [CSV, Characterization 1.6].

Corollary 2.18. Let (R, m,k) be a local ring and C a semidualizing complex for R such n that µR(C)=1 for some integer n. Assume that either (a) R is Cohen-Macaulay, or (b) R is analytically irreducible and C is Cohen-Macaulay.

Then C is dualizing and n = dimR C.

Proof. We may and do assume that R is complete. Then R possesses a normalized dualizing complex D. By parts (f) and (d) of Remark 2.4, it turns out that L := C† is also a R semidualizing complex for R and βn (L)=1. Set d := dimR C. Then, by Remark 2.4(b), d = sup L. (a) Assume that R is Cohen-Macaulay. Then Theorem 2.17 implies that L ≃ ΣnR and n = d. Thus, by Remark 2.4(a), we deduce that C ≃ Σ−dD. (b) Assume that R is analytically irreducible and C is Cohen-Macaulay. As C is Cohen- Macaulay, Remark 2.4(c) implies that amp L = 0. Since R is a domain, it follows that

Hd(L) is a semidualizing module with a rank, and so Theorem 2.17 implies that

d d L ≃ Σ Hd(L) ≃ Σ R.

Using this, as in the proof (a), we can obtain C ≃ Σ−dD.

References

[Ao] Y. Aoyama, Complete local (Sn−1) rings of type n ≥ 3 are Cohen-Macaulay, Proc. Japan Acad., Ser. A, Math. Sci., 70(3), (1994), 80-83. [Ap] D. Apassov, Annihilating complexes of modules, Math. Scand., 84(1), (1999), 11-22. [AF] L. Avramov and H-B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Al- gebra, 71(2-3), (1991), 129-155. [B] H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82(1), (1963), 8-28. [BS] M. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, (1998). 18 K.ABOLFATHBEIGI,K.DIVAANI-AAZARANDM.TOUSI

[BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, (1998). [C1] L. W. Christensen, Semidualizing complexes and their Auslander categories, Trans. Amer. Math. Soc., 353(5), (2001), 1839-1883. [C2] L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, (2000). [CSV] L. W. Christensen, J. Striuli and O. Veliche, Growth in the minimal injective resolution of a local ring, J. London Math. Soc., 81(2), (2010), 24-44. [CHM] D. Costa, C. Huneke amd M. Miller, Complete local domains of type two are Cohen-Macaulay, Bull. London Math. Soc., 17(1), (1985), 29-31. [F1] H. B. Foxby, On the µi in a minimal injective resolution II, Math. Scand., 41, (1977), 19-44. [F2] H. B. Foxby, Isomorphims between complexes with applications to the homological theory of modules, Math. Scand., 40, (1977), 5-19. [H] R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, 20, Springer-Verlag, Berlin, (1966). [HD] M. Hatamkhani and K. Divaani-Aazar, The derived category analogue of the Hartshorne–Lichtenbaum vanishing theorem, Tokyo J. Math., 36(1), (2013), 195-205. [K] T. Kawasaki, Local rings of relatively small type are Cohen-Macaulay, Proc. Amer. Math. Soc., 122(3), (1994), 703-709. [Le1] K. Lee, Maps in minimal injective resolutions of modules, Bull. Korean Math. Soc., 46(3), (2009), 545-551. [Le2] K. Lee, A note on types of Noetherian local rings, Bull. Korean Math. Soc., 39(4), (2002), 645-652. [Li] J. Lipman, Lectures on local cohomology and duality, Local cohomology and its applications, (Gua- najuato, 1999), Lecture Notes in Pure and Appl. Math., 226, Dekker, New York, (2002), 39-89. [R] P. Roberts, Rings of type 1 are Gorenstein, Bull. London Math. Soc., 15(1), (1983), 48-50. [S] S. Sather-Wagstaﬀ, Semidualizing modules, notes, (2000). [V] W. V. Vasconcelos, Divisor theory in module categories, North-Holland Mathematics Studies, 14, North-Holland Publishing Co., Amsterdam-Oxford, New York, (1974).

K. Abolfath Beigi, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran. Email address: [email protected]

K. Divaani-Aazar, Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran-and-School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran. Email address: [email protected]

M. Tousi, Department of Mathematics, Faculty of Mathematical Sciences, Shahid Be- heshti University, Tehran, Iran. Email address: [email protected]