SEQUENTIALLY REFLEXIVE MODULES ALBERTO TONOLO Abstract. We generalize the homological characterization of sequentially Cohen-Macaulay modules over a graded Gorenstein algebra to sequentially reflexive modules over Noetherian, not necessarily commutative rings, with a N -partial cotilting bimodule playing the role of the graded Gorenstein algebra. In such a way we get a complete version of the “Cotilting Theorem”. Finally, conditions are found to insure that the “N -partial cotilting notion” pass through a finite ring extension.
Introduction Let R and S be left and right Noetherian rings, respectively. We say that a faithfully balanced bimodule R VS is N-partial cotilting (N stays for Noetherian) of injective dimension ≤ n if it satisfies the following properties: N1: id R V ≤ n and id VS ≤ n. N2: ExtiR (V, V ) = 0 and ExtiS (V, V ) = 0 for each i ≥ 1. N3: Both R V and VS are finitely generated. We denote by mod both the categories R-mod and mod-S of finitely generated left R- and right S- modules. For each e ≥ 0, let us consider the subclasses Ee (R V ) = {M ∈ R-mod : ExtiR (M, V ) = 0, ∀i 6= e}
and
Ee (VS ) = {N ∈ mod-S : ExtiS (N, V ) = 0, ∀i 6= e}. We will denote briefly by Ee both these classes. In 1986 Miyashita proved (see [15, Theorem 6.1]) that for each module L ∈ Ee • Exte (L, V ) belongs to Ee , and • L∼ = Exte (Exte (L, V ), V ). We call e-reflexive the modules in Ee (R VS ). In 1989 Colby proved (see [5, Proposition 2.2, Theorem 2.4]) that if R VS is N -partial cotilting bimodule of injective dimension ≤ 1, then for all M in R-mod or in mod-S the orthogonality conditions • Hom(Ext1 (M, V ), V ) = 0 and Ext1 (Hom(M, V ), V ) = 0, hold and there are natural transformations γ
Ext1S (Ext1R (−, V ), V ) → 1R-mod
and
γ
Ext1R (Ext1S (−, V ), V ) → 1mod-S
such that • 0
/ Ext1 (Ext1 (M, V ), V )
γM
/M
δM
Hom(Hom(M, U ), U )
/0
is an exact sequence. In particular for each module M in R-mod or in mod-S there exists a filtration 0 = M1 ≤ M0 ≤ M−1 = M i ∼ Ext (Exti (M, V ), V ), i = 0, 1. such that Mi−1 /Mi = Date: January 13, 2004. Key words and phrases. Cotilting bimodule, Duality, Cohen-Macaulay ring, Iwanaga-Gorenstein ring, sequentially Cohen-Macaulay module, Canonical module 2000 Mathematics Subject ClassificationPrimary 16D90; Secondary 16E30. 1
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ALBERTO TONOLO
Let A be a graded Gorenstein commutative algebra of Krull dimension n. The regular bimodule A AA is faithfully balanced and it satisfies properties (N 1 − 3): Therefore it is a N-partial cotilting bimodule of injective dimension n. A finitely generated module M is Cohen-Macaulay of dimension t if dim M = depth M = t, i.e. if ([17, Theorems I.6.3, I.12.3]) ExtiA (M, A) 6= 0 ⇔ i = n − t. Therefore Ee (A AA ) is the class of finitely generated modules which are either CohenMacaulay of dimension n − e or 0. Thus, the Miyashita’s theorem generalizes wellknown results ([4, Theorem 3.3.10]) on the dualities induced by the canonical module. A finitely generated A-module M is sequentially Cohen-Macaulay ([11, Definition 1], [17, Definition III.2.9]) if there exists a finite filtration 0 = Mr < Mr−1 < Mr−2 < . . . < M−1 = M of M by graded submodules of M such that each quotient Mi−1 /Mi is Cohen-Macaulay, and 0 ≤ dim Mr−1 /Mr < dim Mr−2 /Mr−1 < . . . < dim M−1 /M0 . The sequentially Cohen-Macaulay modules can be homologically characterized: These are the finitely generated A-modules M such that for all 0 ≤ i ≤ dim M , Extn−i A (M, A) is either zero or Cohen-Macaulay of dimension i, i.e. such that the orthogonality conditions ExtjA (ExtkA (M, A), A) = 0 hold whenever j 6= k. This theorem, due to Christian Peskine (see [17, Theorem III.2.11] and, for a published proof, [11, Theorem 1.4]), corresponds to the Colby results with the the graded Gorenstein algebra of dimension n playing the role of the N-partial cotilting bimodule of injective dimension ≤ 1. Consider a N-partial cotilting bimodule R VS of injective dimension ≤ n. In the first section we describe as the functors 1mod , Ext1 (Ext1 (−, V ), V ), Ext2 (Ext2 (−, V ), V ), ..., are naturally related in the whole category of finitely generated modules. Hence, generalizing both the Peskine’s and Colby’s theorems, we prove that a module M satisfies the orthogonality conditions Extj (Extk (M, V ), V ) = 0 for each j 6= k if and only if it is sequentially reflexive, i.e. there exists a finite filtration 0 = Mn ≤ Mn−1 ≤ Mn−2 ≤ . . . ≤ M−1 = M of M by submodules of M such that each quotient Mi−1 /Mi is i-reflexive (possibly equal to 0). Any Cohen-Macaulay ring which is a homomorphic image of a Gorenstein ring admits a canonical module; this is a consequence of the following more general result Theorem 0.1 (Theorem 3.3.7, [4]). Let φ : (A, M) → (B, N ) be a local homomorphism of Cohen-Macaulay local rings such that B is a finitely generated A-module. If A admits a canonical module ωA , then there exist a canonical B-module and it is isomorphic to Extm A (B, ωA ), where m = dim A − dim B. In the second section we generalize this important result to our setting. Let R VS be an N -partial cotilting bimodule, and let R → Γ be a ring homomorphism such that R Γ is finitely generated. Substituting the Cohen-Macaulay condition on the new ring with its above quoted homological characterization, we assume that ExtiR (Γ, V ) = 0 for each i 6= m. Then we prove that U = Extm R (Γ, V ) is an N -partial cotilting Γ-EndΓ U -bimodule if and only if ExtiR (U, V ) = 0 for each i 6= m, condition which is automatically satisfied in the commutative setting, because of the duality theory associated to the canonical module. 1. N-partial cotilting bimodules of finite injective dimension Let R and S be left and right Noetherian rings, respectively. Let us fix a natural number n ≥ 1. Throughout the whole paper R VS will denote a N-partial cotilting bimodule of injective dimension ≤ n, and modules will be always
SEQUENTIALLY REFLEXIVE MODULES
3
finitely generated. We denote by E i , i ≥ 0, both the contravariant functors ExtiR (−, V ) and ExtiS (−, V ) associated to R VS : These are functors between the categories R-mod and mod-S of finitely generated left R- and right S- modules. Both the compositions ExtiR (ExtiS (−, V ), V ) and ExtiS (ExtiR (−, V ), V ) will be denoted by (E i )2 . Example 1.1. 1. Let (A, M, k) be a Cohen-Macaulay local ring, i.e. a Noetherian local commutative ring with dim A = depth A, where dim A denotes the Krull dimension of A and depth A is the grade of M in A, i.e. the common lenght of the maximal Asequences in M. A maximal Cohen-Macaulay module ω (i.e. finitely generated with dim ω = depth ω = dim A) is called the canonical module of A [4, Definition 3.3.1] if it has finite injective dimension (necessarily equal to dim A, [4, Theorem 3.1.17]) and dimk ExtiA (k, ω) = 1. The Cohen-Macaulay local ring A admits a canonical module if and only if it is homomorphic image of a Gorenstein ring [4, Theorem 3.3.6]: in such a case the canonical module is unique up to isomorphisms. If A itself is a Gorenstein ring (i.e. a Noetherian local ring with finite injective dimension), then the canonical module ω is isomorphic to A (see [4, § 3.3]). The canonical bimodule is a N-partial cotilting bimodule (see [4, Proposition 3.3.3, Theorem 3.3.4(c)]). 2. Let R be an Iwanaga-Gorenstein ring (see [8, 12, 13]), i.e. R is both left and right Noetherian and it has finite self-injective dimension on both the left and the right. In such a case the injective dimensions of R R and RR coincide [8, Proposition 9.1.8]. Clearly R RR is a N-partial cotilting bimodule. Let R0 be a quasi-Frobenius ring [1, §30] and, for n > 0, Rn be a triangular matrix ring over Rn−1 . Then Rn is an Iwanaga-Gorenstein ring with injective dimension n [20, §5, Lemma B]. Let us fix a module M and a projective resolution P • → M → 0 in the category mod of finitely generated modules. We denote by Bi and Zi , i ≥ 0, the i-boundaries and the i-cycles of the complex E 0 (P • ) := 0 → E 0 P0 → E 0 P1 → E 0 P2 → . . . . Moreover, we denote by cogen V the class of all submodules of finite powers of V . Lemma 1.2. (1) For any projective module P and each j ≥ 1, we have E j E 0 P = 0. (2) For any projective module P , there is a canonical isomorphism δ : P → (E 0 )2 P, p 7→ (φ 7→ φ(p)). (3) For any module L in cogen V , we have Extn (L, V ) = 0. Proof. Suppose P is a finitely generated projective R-module. There exists a splitting exact sequence 0 → P → Rm → P 0 → 0. 1. The module E j E 0 P is a direct summand of E j E 0 Rm = E j V m = 0. 2. Applying twice the functor E 0 we obtain the commutative diagram 0
/P
0
/ Rm δP
/ (E 0 )2 P
δR m
/ (E 0 )2 Rm
/ P0
/0
δP 0
/ (E 0 )2 P 0
/0
Since R VS is faithfully balanced, δRm is an isomorphism. Therefore δP and hence δP 0 are mono and epi. We can argue in the same way if P is a projective right S-module. 3. By definition there exists a monomorphism f : L ,→ V m . Denoted by C the cokernel of f , we have the exact sequence E n V m → E n L → E n+1 C = 0; since n ≥ 1, we have E n V m = 0 and hence E n L = 0 Corollary 1.3. E n (Bi ) = 0 and E n (Zi ) = 0 for each i ≥ 0.
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ALBERTO TONOLO
Proof. If M is in R-mod, we have Bi ≤ Zi ≤ E 0 Pi ≤ E 0 Rni ∼ = V ni . Therefore Bi and Zi belong to cogen V . We conclude by Lemma 1.2. We have B0 = 0, Z0 = E 0 M and the following exact sequences (1) 0 → Zi−1 → E 0 Pi−1 → Bi → 0, i ≥ 1. (2) 0 → Bi → Zi → E i M → 0, i ≥ 1. Applying the functor E 0 we get the two long exact sequences L1. 0 → E 0 Bi → (E 0 )2 Pi−1 → E 0 Zi−1 → E 1 Bi → 0 → . . . . . . → 0 → E j Zi−1 → E j+1 Bi → 0 → . . . L2. 0 → E 0 E i M → E 0 Zi → E 0 Bi → . . . . . . → E j E i M → E j Zi → E j Bi → . . .
i, j ≥ 1. i, j ≥ 1.
From these exact sequences we obtain easily the solid part of the following commutative diagram with exact rows and columns. The dotted arrows are usual factorizations or compositions of morphisms, or easy consequences. (Diagram A) 0O E 1OB1
0
E0Z ∼ = (E 0 )2 M
E 0 B2
0
E1E2M
E i−1 E i M
/ E0E1M
/ E1Z 2
/ E i−1 Z i
0 O l5 ( l l C1 iRi R R / P0 ∼ P1 ∼ = (E 0 )2 P1 = (E 0 )2 P0 _ _ _ _ _ _/ M _ _ _/ 0 6 O l ξ l l l 5C s l l l l5 2 L L% l l l l / E0Z / E0B / (E 1 )2 M / E 1 Z 1 1 1 O
0
/ E1B 2
/ (E 2 )2 M
/ E2Z 2
0 ...
...
...
/ E i−1 B i
/ (E i )2 M
/ EiZ i
Lemma 1.4 (Lemma 1.1, [18]). Given the solid part of the commutative diagram J
J
ψ
φ
/H θ
/K
C
0
/A α /B β C
/0 /0
0
with exact rows and columns, there are unique maps α and β such that the diagram commutes. With these maps the second column is exact; moreover, if θ is monic, then so is α.
SEQUENTIALLY REFLEXIVE MODULES
5
Applying the above Lemma 1.4 to the following commutative triangles from Diagram A φ0
/ (E 0 )2 P0 r9 rrr r r r 0 rrr ξ=θ
(E 0 )2 P1 ψ0
E 0 Z1
(E 0 )2 P0
and
t:
ξ=φ00 ttt
E 0 Z1
t tt tt
ψ 00
O
θ00
/ E0B 1
we get the exact sequences η
E 1 B2 → M → Coker ξ → 0 0 → C2 → Coker ξ → C1 → 0. If = 0, then the map η is injective. Since E j Zi−1 ∼ = E j+1 Bi , i, j ≥ 1 (see L1), composing with morphisms from Diagram A, we obtain the following diagram with exact rows and columns. It describes how the functors 1mod , (E i )2 , i ≥ 0, are canonically related. E0E1M
(Diagram B) E1E2M ...
2 / E1Z ∼ 2 = E B3
/ (E 3 )2 M
/
0 E 1 Z1 ∼ = E 2 B2 o
C2 o
/ M XXX / Coker ξ XXXXX XXXXX XXXXX XX+ δ
η
E 1 B2
(E 1 )2 M o
(E 2 )2 M
0 /0
(E 0 )2 M
E 2 Z2 ∼ = E 3 B3
E 1 B1
0 Using the Schanuel’s Lemma (see [16, Theorem 3.62]) and Lemma 1.2 (1), it’s routine to prove that all modules in Diagram B do not depend on the chosen projective resolution P • of M , and that all maps are natural. In particular we obtain the following diagrams of functors and natural maps (Diagram C)
1mod
/ (E 0 )2
δ
1mod /•o
•
/ (E 1 )2
1mod o
•
/ (E 2 )2
1mod o
•
/ (E 3 )2
...
...
...
The bullets denote suitable functors recognizable in the above Diagram B. We denote by Ee (R V ) the class {L ∈ R-mod : E i L = 0, ∀i 6= e} and by Ee (VS ) the class {N ∈ mod-S : E i N = 0, ∀i 6= e}. We will denote briefly by Ee both the classes Ee (R V ) := {L ∈ R-mod : E i L = 0, ∀i 6= e} and Ee (VS ) = {N ∈ mod-S : E i N = 0, ∀i 6= e}.
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ALBERTO TONOLO
In [15] Miyashita has proved the following Theorem 1.5 (Theorem 6.1, [15]). For each module L ∈ Ee we have • E e L belongs to Ee , and • L∼ = (E e )2 L. By Theorem 1.5, an e-reflexive module M (i.e. a module M belonging to Ee ) satisfies the orthogonal conditions E i E j M = 0 whenever i 6= j. Conversely, if M satisfies the orthogonal conditions E i E j M = 0 ,i 6= j, then E e M is e-reflexive for each 0 ≤ e ≤ n. Let us investigate more closely the interplay between these two notion. Proposition 1.6. Let M be a module satisfying the orthogonality conditions E i E j M = 0 for i 6= j. Then E j Bi = 0 for each j ≥ 1 and i 6= j +1; in particular Diagram B specializes to 0 ...
/ E2B 3
/ (E 3 )2 M
/0
0
(E 1 )2 M (Diagram D) 0
/ E1B 2
(E 2 )2 M
/M R / Cokerξ RRR RRR RRR RRR RRR δM )
/0
(E 0 )2 M
0
0
Proof. We can assume j ≤ n. By Corollary 1.3 we have E n Bi = 0 for each i ≥ 0. Assume E j+1 Bk = 0 for each k 6= j + 2. Let us see that E j Bi = 0, for each i 6= j + 1, j ≥ 1. Consider the exact sequences 0 → Zi → E 0 Pi → Bi+1 → 0
and 0 → Bi → Zi → E i M → 0.
Applying E j to the first exact sequence, we get, by Lemma 1.2, 0 = E j E 0 Pi → E j Zi → E j+1 Bi+1 → E j+1 E 0 Pi = 0; since i + 1 6= j + 2, by hypothesis we have E j Zi ∼ = E j+1 Bi+1 = 0. Applying E j to the second exact sequence, we achieve 0 = E j Zi → E j Bi → E j+1 E i M = 0. Therefore E j Bi = 0 for each i 6= j + 1. We conclude by induction. Finally Diagram D is easily obtained by Diagram B, observed that, since E 0 E 1 M = 0, the map η is a monomorphism. We can prove now our main results. Theorem 1.7. If M belongs to Ei , then it is naturally isomorphic to (E i )2 M . Proof. We have seen in Diagram C that for each i ≥ 0 there exists a “path” of natural maps between the identity functor 1mod and the functor (E i )2 . It follows easily from Diagram D that each of the natural maps of this path becomes an isomorphism in the classes Ei .
SEQUENTIALLY REFLEXIVE MODULES
7
Definition 1.8. We call sequentially reflexive a module M admitting a finite filtration 0 = Mn ≤ Mn−1 ≤ Mn−2 ≤ . . . ≤ M−1 = M with the factors Mi−1 /Mi in Ei . In [5, Proposition 2.2, Theorem 2.4] Colby proved that if R VS is N -partial cotilting bimodule of injective dimension ≤ 1, then any finitely generated module M is sequentially reflexive with filtration 0 ≤ (E 1 )2 M ≤ M and satisfies the orthogonality conditions E i E j M = 0 for i 6= j. We have the following generalization of his result: Theorem 1.9. The module M satisfies the orthogonality conditions E i E j M = 0 for i 6= j if and only if M is sequentially reflexive. In such a case the i-th factor Mi−1 /Mi of the filtration is the i-reflexive module (E i )2 M , for each 0 ≤ i ≤ n Proof. Assume M satisfies the orthogonality conditions E i E j M = 0, i 6= j. Consider Diagram D. Let M0 be the counterimage of (E 1 )2 M through the epimorphism M → Coker ξ. Then we get the filtration M = M−1 ≥ M0 ≥ M1 = E 1 B2 ≥ ... ... ≥ Mn−2 = E n−2 Bn−1 ≥ Mn−1 = (E n )2 M ≥ Mn = 0 with Mi /Mi+1 ∼ = (E i+1 )2 M , for all −1 ≤ i ≤ n − 1. Since E i M belongs to Ei , by Theorem 1.5 also Mi /Mi+1 ∼ = (E i )2 M belongs to Ei . Conversely, assume there exists a filtration 0 = Mn ≤ Mn−1 ≤ Mn−2 ≤ . . . ≤ M−1 = M of M with Mi−1 /Mi in Ei . We need to prove that E i M belongs to Ei for each 0 ≤ i ≤ n. Let us consider the short exact sequences 0 → Mi−1 /Mi → M/Mi → M/Mi−1 → 0,
0 ≤ i ≤ n.
Applying the functor E 0 we get the long exact sequence 0 → E 0 M/Mi−1 → E 0 M/Mi → E 0 Mi−1 /Mi → ... ... → E j−1 Mi−1 /Mi → E j M/Mi−1 → E j M/Mi → E j Mi−1 /Mi → E j+1 M/Mi−1 → ... From this long exact sequence we achieve the exact sequences (1)
E j M/Mj−2 → E j M/Mj−1 → E j Mj−2 /Mj−1 = 0,
(2)
E j M/Mj−1 → E j M/Mj → E j Mj−1 /Mj → E j+1 M/Mj−1 ,
j≥1 j≥0
and the isomorphisms (3)
E j M/Mi−1 ∼ = E j M/Mi ,
i 6= j, j − 1.
From (3) we have that E j M = E j M/Mn ∼ = ... ∼ = E j M/Mj , 0 ≤ j ≤ n. In particular by Theorem 1.5 E 0 M ∼ = E 0 M/M0 = E 0 M−1 /M0 belongs to E0 , since M−1 /M0 belongs to E0 . Assume then j ≥ 1. Again by (3), for each l ≥ 1, we have E l M/Ml−2 ∼ = E l M/Ml−3 ∼ = ... ∼ = E l M/M0 = 0; therefore from (1) we get E j M/Mj−1 = 0, and hence, from (2) we have (E j M ∼ = E j Mj−1 /Mj . =)E j M/Mj ∼ Since Mj−1 /Mj belongs to Ej , by Theorem 1.5 also E j Mj−1 /Mj belongs to Ej . Therefore EiEj M ∼ = E i E j Mj−1 /Mj = 0 whenever i 6= j. Finally, since Mi−1 /Mi is i-reflexive, from E i M ∼ = E i Mi−1 /Mi and Theorem 1.5, since E i M is i-reflexive, it follows that (E i )2 M ∼ = Mj−1 /Mj is i-reflexive.
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ALBERTO TONOLO
2. Dualities and Finite Ring extensions Also in this section we assume that R VS is an N -partial cotilting bimodule of injective dimension ≤ n. Let ξ : R → Γ be a ring homomorphism with R Γ finitely generated. We denote by U the Γ-S-bimodule E m Γ, and by ∆ the endomorphism ring EndΓ (U ). We will prove that Γ U∆ is an N -partial cotilting bimodule, provided R Γ and R U belong to Em (see pag. 5). Many ideas used in this section can be found in [10]. The following proposition describes the image of the functor HomΓ (−, Γ U ) and its derived functors in terms of the functors E i . Proposition 2.1. Assume R Γ belongs to Em . Then for each left Γ module Γ M and each l ≥ 0 we have the following natural isomorphism of S-modules: Extl (M, US ) ∼ = E l+m M. Γ
Proof. Assume l = 0. We want to prove that HomΓ (M, U ) = HomΓ (M, E m Γ) is naturally isomorphic to E m M . If m = 0, it follows by the adjunction between the hom and the tensor functors. Let m ≥ 1. Considered the injective resolution 0 → V → I • of V , we denote by Vi the i-th cosyzygy of V where V0 = V . Applying the functor HomR (Γ, −) to the short exact sequence 0 → Vm−1 → Im−1 → Vm → 0 we get the exact sequence of left Γ-modules / HomR (Γ, Vm−1 )
0
/ HomR (Γ, Im−1 )
/ HomR (Γ, Vm ) 6 mmm m m mmm ) mmmmm m
/ Ext1 (Γ, Vm−1 ) R
/0
L
Applying the functor HomΓ (M, −) we obtain the exact sequences 0 → HomΓ (M, HomR (Γ, Vm−1 )) → HomΓ (M, HomR (Γ, Im−1 )) → HomΓ (M, L) →
(1)
→ Ext1Γ (M, HomR (Γ, Vm−1 )) → Ext1Γ (M, HomR (Γ, Im−1 )) → Ext1Γ (M, L) → → Ext2Γ (M, HomR (Γ, Vm−1 )) (2)
0 → HomΓ (M, L) → HomΓ (M, HomR (Γ, Vm )) → HomΓ (M, Ext1R (Γ, Vm−1 )) → → Ext1Γ (M, L) .
Let us see that (a) Ext1Γ (M, HomR (Γ, Vm−1 )) = 0, and (b) Ext1Γ (M, L) = 0. Observe that by [7, Proposition VI.5.1] we have Exti (M, HomR (Γ, I)) ∼ = HomR (TorΓ (Γ, M ), I) = 0 Γ
i
∀i ≥ 1
for each injective left R-module I. (a) If m = 1, then Vm−1 = V0 = V ; since, by hypothesis, HomR (Γ, V ) = 0 we get the thesis. Let m ≥ 2. Applying HomΓ (M, −) to the short exact sequences 0 → HomR (Γ, Vk−2 ) → HomR (Γ, Ik−2 ) → HomR (Γ, Vk−1 ) → k−1 → Ext1R (Γ, Vk−2 ) ∼ Γ = 0, = Extk−1 R (Γ, V0 ) = E we get for each j ≥ 1
(∗)
k≤m
0 = ExtjΓ (M, HomR (Γ, Ik−2 )) → ExtjΓ (M, HomR (Γ, Vk−1 )) →
j+1 → Extj+1 Γ (M, HomR (Γ, Vk−2 )) → ExtΓ (M, HomR (Γ, Ik−2 )) = 0; hence Ext1Γ (M, HomR (Γ, Vm−1 )) ∼ = Extm Γ (M, HomR (Γ, V )) = 0 since HomR (Γ, V ) = 0.
SEQUENTIALLY REFLEXIVE MODULES
9
(b) Since by hypothesis HomR (Γ, V ) = 0 we have by (∗) Ext2Γ (M, HomR (Γ, Vm−1 )) ∼ (M, HomR (Γ, V )) = 0 = Extm+1 Γ and hence, from the exact sequence (1), we get Ext1Γ (M, L) = 0. Now, composing the exact sequences (1) and (2), and using the adjunction between the hom and the tensor functors, we obtain the following commutative diagram with exact rows HomΓ (M, HomR (Γ, Im−1 ))
/ HomΓ (M, HomR (Γ, Vm ))
∼ =
/ HomΓ (M, Ext1 (Γ, Vm−1 )) R
/0
/ Ext1 (M, Vm−1 ) R
/0
∼ =
/ HomR (M, Vm )
HomR (M, Im−1 )
By Schanuel’s Lemma (see [16, Theorem 3.62]) Ext1R (Γ, Vm−1 ) and Ext1R (M, Vm−1 ) do not depend on the particular injective coresolution. Thus, by the isomorphisms induced by the adjunction, we obtain the natural isomorphism HomΓ (M, E m Γ) ∼ = HomΓ (M, Ext1R (Γ, Vm−1 )) ∼ = Ext1R (M, Vm−1 ) ∼ = E m M. Let l = 1. Consider the short exact sequence of Γ-modules 0 → K → Γn → M → 0. Applying the functors HomΓ (−, U ) and E m we get the following commutative diagram of right S-modules with exact rows HomΓ (Γn , U )
/ HomΓ (K, U )
∼ =
/ Ext1 (M, U ) Γ
/0
/ E m+1 M
/ E m+1 Γn .
∼ =
/ EmK
E m Γn
Since by hypothesis E m+1 Γ = 0, we get the natural isomorphism Ext1Γ (M, U ) ∼ = E m+1 M . Assume our results holds for l = j; then using the diagrams ExtjΓ (K, U )
E m+j K
∼ =
/ Extj+1 (M, U ) Γ
∼ =
/ E m+j+1 M
∼ m+j+1 M . Then we conclude by induction. we get Extj+1 Γ (M, U ) = E Since U is a right S-module, ∆ = EndΓ U is a ∆-S-bimodule. Then we get a ring homomorphism S → ∆. We have the following “∆-version” of Proposition 2.1. Proposition 2.2. Assume R Γ and R U belong to Em . Then for each ∆-module N∆ and each l ≥ 0 we have the following natural isomorphism of R-modules Extl∆ (N, R U ) ∼ = E l+m N. mU Proof. Using Proposition 2.1 and Theorem 1.7, we get that ∆S = HomΓ (U, U ) ∼ = ER belongs to Em . Moreover we have RU
∼ = (E m )2 U ∼ = E m (HomΓ (U, U )) = E m ∆.
Then we can apply Proposition 2.1 with ∆ and R U playing the role of Γ and US . We can now prove our wanted result (compare it with [10, Proposition 4.9] and [4, Theorem 3.3.7] quoted in the introduction).
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ALBERTO TONOLO
Theorem 2.3. Let R VS be an N -partial cotilting bimodule and let R → Γ be a ring homomorphism with R Γ ∈ Em . Denote by U the Γ-module Extm R (Γ, V ). Then Γ UEndΓ U is an N -partial cotilting bimodule if and only if R U belongs to Em . Proof. Set ∆ := EndΓ U and assume R U := E m Γ ∈ Em . We have to prove the following (a) Γ is left Noetherian and ∆ is right Noetherian. (b) Γ U and U∆ are finitely generated. (c) Γ U∆ is faithfully balanced. (d) ExtiΓ (U, U ) = 0 = Exti∆ (U, U ) for each i ≥ 1. (e) id Γ U and id U∆ are finite. (a) The modules in Em are finitely generated, therefore R U and R Γ are finitely generated. Since ∆ = HomΓ (U, US ) = HomR (U, US ) and R is left Noetherian, ∆S is finitely generated. Therefore Γ is left Noetherian and ∆ is right Noetherian, as R and S are. (b) The left R-module structure of U is inherited by the ring homomorphism ξ : R → Γ. Therefore, since R U is finitely generated, also Γ U is. Analogously US = Em Γ is finitely generated and hence U∆ is. (c) It is sufficient to prove that Γ = End∆ (U ). We have by Propositions 2.1, 2.2 the following isomorphisms of left Γ-modules Hom∆ (Γ U, Γ U ) ∼ = Hom∆ (HomΓ (Γ, U ), Γ U ) ∼ = Hom∆ (E m Γ, U ) ∼ = (E m )2 Γ ∼ = ΓΓ Observe that the isomorphism of right S-modules E m Γ ∼ = HomΓ (Γ, U ) defines a structure of right ∆-module on E m Γ. (d) By Propositions 2.1, 2.2 we have ExtiΓ (U, U ) ∼ = E i+m U = 0 and Exti∆ (U, U ) ∼ = i+m E U = 0. (e) By Propositions 2.1, 2.2 we get that max{id Γ U, id U∆ } ≤ n − m. Conversely, assume Γ UEndΓ U is an N -partial cotilting bimodule. By definition Γ U , and hence R U , is finitely generated. Let us see that R U ∈ Em : by Proposition 2.1 E i+m U ∼ = ExtiΓ (U, U ) = 0 for each i > 0. Consider an exact sequence 0 → K → Γn → U → 0; it is also an exact sequence of left R-modules. Applying HomR (−, V ) and remembering that Γ belongs to Em we get E j U = 0 for each 0 ≤ j < m. In the commutative case it is possible further to prove that any Cohen-Macaulay ring admitting canonical module is an homomorphic image of a Gorenstein ring. Something similar can be proved also in our case, but for N -partial cotilting bimodules R VR with the same ring for both its left and right structures. Proposition 2.4. Let R VR be an N -partial cotilting bimodule of injective dimension ≤ n. Then R is an homomorphic image of an Iwanaga-Gorenstein ring of injective dimension ≤ n. Proof. Let us consider the trivial extension R n V . By [9, Theorem 4.32] we have that id RnV R n V = id R V ; analogously id R n VRnV = id VR . Since V is finitely generated and R is Noetherian, also R n V is Noetherian. References [1] F. D. Anderson and K. R. Fuller. Rings and categories of modules, 2nd ed., GTM 13, Springer 1992. [2] I. Assem. Tilting theory – an introduction, Topics in algebra, Part 1. Banach Center Publ. 26, Part 1. PWN, Warsaw 1990. [3] S. Brenner and M. Butler. Generalizations of the Bernstein-Gelfand-Ponomariev reflection functors, in “Proc. ICRA II, Ottawa 1979,” pp.103–169, LNM 832, Springer Verlag, Berlin, 1981. [4] W. Bruns and J. Herzog. Cohen-Macaulay rings, Cambridge studies in advanced mathematics 39, Cambridge University Press, 1993. [5] R. R. Colby. A generalization of Morita duality and the tilting theorem, Comm. Algebra 17(7), (1989), 1709–1722. [6] R. R. Colby and K. R. Fuller. Tilting, cotilting and serially tilted rings, Comm. Algebra 18, (1990), 1585–1615.
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