Generalized tilting modules with finite injective dimension

Journal of Algebra 3 (2007) 69 634 www.elsevier.com/locate/jalgebra Generalized tilting modules with finite injective dimension Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 20093, People s epublic of China eceived 25 February 2006 Available online 4 January 2007 Communicated by Luchezar L. Avramov Abstract Let be a left noetherian ring, S a right noetherian ring and U a generalized tilting module with S = End( U). The injective dimensions of U and U S are identical provided both of them are finite. Under the assumption that the injective dimensions of U and U S are finite, we describe when the subcategory {Ext n S (N, U) N is a finitely generated right S-module} is submodule-closed. As a consequence, we obtain a negative answer to a question posed by Auslander in 969. Finally, some partial answers to Wakamatsu Tilting Conjecture are given. 2006 Elsevier Inc. All rights reserved. Keywords: Generalized tilting modules; Injective dimension; U-limit dimension; Submodule-closed; Wakamatsu Tilting Conjecture. Introduction Let be a ring. We use Mod (respectively Mod op ) to denote the category of left (respectively right) -modules, and use mod (respectively mod op ) to denote the category of finitely generated left -modules (respectively right -modules). We define gen ( ) ={X mod there exists an exact sequence P i P P 0 X 0in mod with P i projective for any i 0} (see [W2]). A module U in mod is called selforthogonal if Ext i ( U, U)= 0 for any i. E-mail address: huangzy@nju.edu.cn. 002-8693/$ see front matter 2006 Elsevier Inc. All rights reserved. doi:0.06/j.jalgebra.2006..025

620 Z. Huang / Journal of Algebra 3 (2007) 69 634 Definition.. [W2] A selforthogonal module U in gen ( ) is called a generalized tilting module (sometimes it is also called a Wakamatsu tilting module, see [B]) if there exists an exact sequence: 0 U 0 U U i such that: () U i add U for any i 0, where add U denotes the full subcategory of mod consisting of all modules isomorphic to direct summands of finite sums of copies of U, and (2) after applying the functor Hom (, U) the sequence is still exact. Let and S be any rings. ecall that a bimodule U S is called a faithfully balanced bimodule if the natural maps End(U S ) and S End( U) op are isomorphisms. By [W2, Corollary 3.2], we have that U S is faithfully balanced and selforthogonal with U gen ( ) and U S gen (S S ) if and only if U is generalized tilting with S = End( U) if and only if U S is generalized tilting with = End(U S ). Let and S be Artin algebras and U a generalized tilting module with S = End( U). Wakamatsu proved in [W, Theorem] that the projective (respectively injective) dimensions of U and U S are identical provided both of them are finite. The result on the projective dimensions also holds true when is a left noetherian ring and S is a right noetherian ring, by using an argument similar to that in [W]. In this case, U S is a tilting bimodule of finite projective dimension [M, Proposition.6]. However, because there is no duality available, Wakamatsu s argument in [W] does not work on the injective dimensions over noetherian rings. So, it is natural to ask the following questions: When is a left noetherian ring and S is a right noetherian ring, () Do the injective dimensions of U and U S coincide provided both of them are finite? (2) If one of the injective dimensions of U and U S is finite, is the other also finite? The answer to the first question is positive if one of the following conditions is satisfied: () U S = [Z, Lemma A]; (2) and S are Artin algebras [W, Theorem]; (3) and S are two-sided noetherian rings and U is n-gorenstein for all n [H2, Proposition 7.2.6]. In this paper, we show in Section 2 that the answer to this question is always positive. By the positive answer to the first question, the second question is equivalent to the following question: Are the injective dimensions of U and U S identical? The above result means that the answer to this question is positive provided that both dimensions are finite. On the other hand, for Artin algebras, the positive answer to the second question is equivalent to the validity of Wakamtsu Tilting Conjecture (WTC). This conjecture states that every generalized tilting module with finite projective dimension is tilting, or equivalently, every generalized tilting module with finite injective dimension is cotilting. Moreover, WTC implies the validity of the Gorenstein Symmetry Conjecture (GSC), which states that the left and right self-injective dimensions of are identical (see [B]). In Section 4, we give some partial answers to question (2). Let and S be two-sided artinian rings and U a generalized tilting module with S = End( U).We prove that if the injective dimension of U S is equal to n and the U-limit dimension of each of the first (n )st terms is finite, then the injective dimension of U is also equal to n. Thus it trivial that the injective dimension of U S is at most if and only if that of U is at most. We remark that for an Artin algebra, it is well known that the right self-injective dimension of is at most if and only if the left self-injective dimension of is at most (see [A3, p. 2]). In addition, we prove that the left and right injective dimensions of U and U S are identical if U (or U S ) is quasi-gorenstein, that is, WTC holds for quasi-gorenstein modules.

Z. Huang / Journal of Algebra 3 (2007) 69 634 62 For an ( S)-bimodule U S and a positive integer n, we denote E n (U S ) ={M mod M = Ext n S (N, U) for some N mod Sop }. For a two-sided noetherian ring, Auslander showed in [A, Proposition 3.3] that any direct summand of a module in E ( ) is still in E ( ).He then asked whether any submodule of a module in E ( ) is still in E ( ). ecall that a full subcategory X of mod is said to be submodule-closed if any non-zero submodule of a module in X is also in X. Then the above Auslander s question is equivalent to the following question: Is E ( ) submodule-closed? In Section 3, under the assumption that is a left noetherian ring, S is a right noetherian ring and U is a generalized tilting module with S = End( U) and the injective dimensions of U and U S being finite, we give some necessary and sufficient conditions for E n (U S ) being submodule-closed. As a consequence, we construct some examples to illustrate that neither E ( ) nor E 2 ( ) are submodule-closed in general, by which we answer the above Auslander s question negatively. Throughout this paper, is a left noetherian ring, S is a right noetherian ring (unless stated otherwise) and U is a generalized tilting module with S = End( U). For a module A in Mod (respectively Mod S op ), we use l.id (A), l.fd (A) and l.pd (A) (respectively r.id S (A), r.fd S (A) and r.pd S (A)) to denote the injective dimension, flat dimension and projective dimension of A (respectively A S ), respectively. 2. Some homological dimensions In this section, we study the relations among the U-limit dimension (which was introduced in [H2]) of an injective module E, the flat dimension of Hom(U, E) and the injective dimension of U. Then we show that l.id (U) = r.id S (U) provided both of them are finite. The following result is [W2, Corollary 3.2]. Proposition 2.. The following statements are equivalent. () U is a generalized tilting module with S = End( U). (2) U S is a generalized tilting module with = End(U S ). (3) U S is a faithfully balanced and selforthogonal bimodule. We use add-lim U to denote the subcategory of Mod consisting of all modules isomorphic to direct summands of a direct limit of a family of modules in which each is a finite direct sum of copies of U (see [H2]). Proposition 2.2. () Let V add-lim U. Then Ext i (U (I),V)= 0 for any index set I and i. (2) Ext i (U (I),U (J ) ) = 0 for any index sets I, J and i. Proof. () It is well known that for any i, Ext i (U (I),V) = Ext i (U, V )I. Since U is finitely generated and selforthogonal and V add-lim U, it follows easily from [S, Theorem 3.2] that Ext i (U, V ) = 0 and so Exti (U, V )I = 0. (2) Because a direct sum of a family of modules is a special kind of a direct limit of these modules, (2) follows from () trivially.

622 Z. Huang / Journal of Algebra 3 (2007) 69 634 Definition 2.3. [H2] For a module A in Mod, if there exists an exact sequence U n U U 0 A 0inMod with U i add-lim U for any i 0, then we define the U- limit dimension of A, denoted by U- lim.dim (A), asinf{n there exists an exact sequence 0 U n U U 0 A 0in Mod with U i add-lim U for any 0 i n}. Weset U- lim.dim (A) infinity if no such an integer exists. emark. It is well known that a module over any ring is flat if and only if it is direct limit of a family of finitely generated free modules. So, putting U =, a module in Mod is flat if and only if it is in add-lim ; in this case, the dimension defined as in Definition 2.3 is just the flat dimension of modules. For a module A in Mod (respectively Mod S op ), we denote either of Hom ( U S, A) and Hom S ( U S,A S ) by A. Lemma 2.4. Let E be an injective -module. Then l.fd S ( E) = U- lim.dim (E). Proof. The result was proved in [H2, Lemma 7.3.] when and S are two-sided noetherian rings. The proof in [H2] remains valid in the setting here, we omit it. emark. It is not difficult to see from the proof of [H2, Lemma 7.3.] that for any injective -module E, there exists an exact sequence: U i U U 0 E 0 in Mod with U i in add-lim U for any i 0. So U- lim.dim (E) (finite or infinite) always exists for any injective -module E. Proposition 2.5. Let E be an injective -module. If l.id (U) = n< and l.fd S ( E) <, then l.fd S ( E) n. Proof. Suppose l.fd S ( E) = m<. Then there exists an exact sequence: 0 F m S (I m ) S (I ) S (I 0) E 0 () in Mod S with F m flat and I i an index set for any 0 i m. By [CE, Chapter VI, Proposition 5.3], we have that Tor S ( j U, E ) ( j = Hom Ext S (U, U), E) = 0 for any j. So, by applying the functor U S to the exact sequence (), we get the following exact sequence: 0 U S F m U S S (I m ) U S S (I ) U S S (I 0) U S E 0. By Proposition 2. and [S2, p. 47], we have that U S E = Hom (Hom S (U, U), E) = E. So we get the following exact sequence: d m 0 K m U (I m ) d m d 2 U (I ) d U (I 0) d 0 E 0 (2)

Z. Huang / Journal of Algebra 3 (2007) 69 634 623 where K m = U S F m. Because F m is a flat S-module, it is a direct limit of finitely generated free S-modules. So K m = U S F m add-lim U. Hence, by Proposition 2.2(), Ext j (U (Ii),K m ) = 0 for any j and 0 i m. Since l.id (U) = n, l.id (K m ) n by [S, Theorem 3.2]. If m>n, then Ext m (E, K m) = 0. It follows from the exact sequence (2) that Ext (K m,k m ) = 0, where K m = Coker d m. Thus d m the sequence 0 K m U (Im ) K m 0 splits and U (I m ) = Km K m. In addition, we get an exact sequence: 0 K m U (I m 2) d m 2 U (I ) d U (I 0) d 0 E 0 with K m,u (I m 2),...,U (I 0) add-lim U. Then U- lim.dim (E) m. But l.fd S ( E) = U- lim.dim (E) by Lemma 2.4. Consequently we conclude that l.fd S ( E) l.id (U). We also need the following result, which is [H2, Lemma 7.2.4]. Lemma 2.6. () r.id S (U) = sup{l.fd S ( E) E is injective}. Moreover, r.id S (U) = l.fd S ( Q) for any injective cogenerator Q for Mod. (2) l.id (U) = sup{r.fd ( E ) E S is injective}. Moreover, l.id (U) = r.fd ( Q ) for any injective cogenerator Q S for Mod Sop. We are now in a position to prove one of the main results in this paper. Theorem 2.7. l.id (U) = r.id S (U) provided both of them are finite. Proof. Let Q be an injective cogenerator for Mod. Assume that l.id (U) = n< and r.id S (U) = m<. Then l.fd S ( Q) = m by Lemma 2.6. So m = l.fd S ( Q) l.id (U) = n by Proposition 2.5. Dually, we may prove n m. We are done. Definition 2.8. [AB] Let X be a full subcategory of Mod. For a module A in Mod, if there exists an exact sequence T n T T 0 A 0inMod with T i X for any i 0, then we define X - resol.dim (A) = inf{n there exists an exact sequence 0 T n T T 0 A 0in Mod with T i X for any 0 i n}. WesetX - resol.dim (A) infinity if no such an integer exists. We use Add U to denote the full subcategory of Mod consisting of all modules isomorphic to direct summands of sums of copies of U. Compare the following result with Lemma 2.4. Lemma 2.9. Let E be an injective -module. Then l.pd S ( E) = Add U- resol.dim (E). Proof. We first prove that Add U- resol.dim (E) l.pd S ( E). Without loss of generality, assume that l.pd S ( E) = m<. Then there exists an exact sequence: 0 Q m Q m Q Q 0 E 0 d 2

624 Z. Huang / Journal of Algebra 3 (2007) 69 634 in Mod S with Q i projective for any 0 i m. Then by using an argument similar to that in the proof of Proposition 2.5, we get that Add U- resol.dim (E) m. We next prove that l.pd S ( E) Add U- resol.dim (E). Assume that Add U- resol.dim (E) = m<. Then there exists an exact sequence: 0 U m U U 0 E 0 (3) in Mod with U i Add U for any 0 i m. By Proposition 2.2(2), we have that U i Add S S (that is, U i is a projective left S-module) and Ext j (U, U i) = 0 for any j and 0 i m. So by applying the functor Hom ( U, ) to the exact sequence (3), we get the following exact sequence: 0 U m U U 0 E 0 in Mod S with U i left S-projective for any 0 i m, and hence l.pd S ( E) m. emark. () Put U =. Then Add U- resol.dim (A) = l.pd (A) for any A Mod. (2) Because a direct sum of a family of modules is a special kind of a direct limit of these modules, for any A Mod, wehavethatu- lim.dim (A) Add U- resol.dim (A) if both of them exist. (3) It is not difficult to see from the proof of Lemma 2.9 that for any injective -module E, there exists an exact sequence: U i U U 0 E 0 in Mod with U i in Add U for any i 0. So Add U- resol.dim (E) (finite or infinite) always exists for any injective -module E. The proof of Proposition 2.5 in fact proves the following more general result. Proposition 2.0. Let E be an injective -module. If l.id (U) = n< and l.fd S ( E) <, then Add U- resol.dim (E) n (equivalently, l.pd S ( E) n). Theorem 2.. Let E be an injective -module. If l.id (U) = n<, then the following statements are equivalent. () l.fd S ( E) <. (2) l.pd S ( E) <. (3) U- lim.dim (E) <. (4) Add U- resol.dim (E) <. (5) l.fd S ( E) n. (6) l.pd S ( E) n. (7) U- lim.dim (E) n. (8) Add U- resol.dim (E) n.

Z. Huang / Journal of Algebra 3 (2007) 69 634 625 Proof. The implications that (6) (2) () and (6) (5) () are trivial. The implication of () (6) follows from Proposition 2.0. By Lemma 2.4, we have () (3) and (5) (7). By Lemma 2.9, we have (2) (4) and (6) (8). As an application of the obtained results, we get the following corollary, which gives some equivalent conditions that l.id (U) = n implies r.id S (U) = n. Corollary 2.2. Let Q be an injective cogenerator for Mod.Ifl.id (U) = n(< ), then the following statements are equivalent. () r.id S (U) = n. (2) One of l.fd S ( Q), l.pd S ( Q), U- lim.dim (Q) and Add U- resol.dim (Q) is finite. Proof. Let Q be an injective cogenerator for Mod. Then by Lemmas 2.6(), 2.4 and 2.9, we have that r.id S (U) = l.fd S ( Q) = U- lim.dim (Q) Add U- resol.dim (Q) = l.pd S ( Q). Now the equivalence of () and (2) follows easily from Theorems 2. and 2.7. 3. Submodule-closure of E n (U S ) In this section, we study Auslander s question mentioned in Section in a more general situation. Lemma 3.. For any injective module E and any non-negative integer t, l.fd S ( E) t if and only if Hom (Ext t+ S (N, U), E) = 0 for any module N mod S op. Proof. It is easy by [CE, Chapter VI, Proposition 5.3]. For a module A mod and a non-negative integer n, we say that the grade of A with respect to U, written as grade U A, is at least n if Ext i (A, U) = 0 for any 0 i<n. We say that the strong grade of A with respect to U, written as s.grade U A, is at least n if grade U B n for all submodules B of A (see [H2]). Assume that is a minimal injective resolution of U. α 0 α α i 0 U E 0 E E i Lemma 3.2. Let n be a positive integer and m an integer with m n. Then the following statements are equivalent. () U- lim.dim ( n i=0 E i) n + m. (2) s.grade U ExtS n+m+ (N, U) n for any N mod S op. (3) l.fd S ( E i ) n + m for any 0 i n. Proof. This conclusion has been proved in [H2, Lemma 7.3.2] when and S are two-sided noetherian rings. The argument there remains valid in the setting here, we omit it.

626 Z. Huang / Journal of Algebra 3 (2007) 69 634 For a module A in mod (respectively mod S op ), we call Hom ( A, U S ) (respectively Hom S (A S, U S )) the dual module of A with respect to U S, and denote either of these modules by A. For a homomorphism f between -modules (respectively S op -modules), we put f = Hom(f, U S ).Weuseσ A : A A via σ A (x)(f ) = f(x)for any x A and f A to denote the canonical evaluation homomorphism. A is called U-torsionless (respectively U-reflexive) if σ A is a monomorphism (respectively an isomorphism). Definition 3.3. [H3] Let X be a full subcategory of mod. X is said to have the U-torsionless property (respectively the U-reflexive property) if each module in X is U-torsionless (respectively U-reflexive). We denote U ={M mod Exti (M, U)= 0 for any i } and n U ={M mod Ext i (M, U) = 0 for any i n} (where n is a positive integer). A module M in mod is said to have generalized Gorenstein dimension zero (with respect to U S ), denoted by G-dim U (M) = 0, if the following conditions are satisfied: () M is U-reflexive, and (2) M U and M U S. Symmetrically, we may define the notion of a module in mod S op having generalized Gorenstein dimension zero (with respect to U S ) (see [A2]). We use G U to denote the full subcategory of mod consisting of the modules with generalized Gorenstein dimension zero. It is trivial that U G U. Proposition 3.4. [H3, Proposition 2.3] The following statements are equivalent. () U has the U-torsionless property. (2) U has the U-reflexive property. (3) U = G U. For any n 0, we denote H n ( U) ={M mod Ext i (M, U) = 0 for any i 0 with i n} [W2]. Lemma 3.5. If U has the U-torsionless property, then H n( U) E n (U S ) for any n. Proof. It follows from Proposition 3.4 and [H4, Lemma 3.3]. Lemma 3.6. Assume that n U has the U-torsionless property, where n is a positive integer. If A is a non-zero module in mod with grade U A n, then grade U A = n. Proof. Let 0 A mod with grade U A n. If grade U A>n, then A = 0 and A n U. Since n U has the U-torsionless property, A is U-torsionless and A A = 0, which is a contradiction. Thus grade U A = n. For any n 0, we denote H n ( U) ={M mod any non-zero submodule of M is in H n ( U)}. It is clear that H n ( U) H n ( U). We are now in a position to give the main result in this section. Theorem 3.7. If l.id (U) n and n U has the U-torsionless property, where n is a positive integer, then the following statements are equivalent.

Z. Huang / Journal of Algebra 3 (2007) 69 634 627 () U- lim.dim ( n i=0 E i) n. (2) E n (U S ) is submodule-closed and E n (U S ) = H n ( U). (3) E n (U S ) is submodule-closed and E n (U S ) = H n ( U). Proof. Since n U has the U-torsionless property, H n( U) H n ( U) E n (U S ) by Lemma 3.5. So the implication of (3) (2) is trivial. () (3). Assume that U- lim.dim ( n i=0 E i) n and M is any non-zero module in E n (U S ). Then s.grade U M n by Lemma 3.2. Let A be any non-zero submodule of M in mod. Then grade U A n. By Lemma 3.6, grade U A = n. In addition, l.id (U) n, soa H n ( U) and M H n ( U). It follows that E n (U S ) H n ( U) and E n (U S ) = H n ( U). Notice that H n ( U) is clearly submodule-closed, so E n (U S ) is also submodule-closed. (2) (). We first prove U- lim.dim (E 0 ) n. If U- lim.dim (E 0 )>n, then l.fd S ( E 0 )>n by Lemma 2.4. So by Lemma 3., there exists a module N mod S op such that Hom (Ext n S (N, U), E 0) 0. Hence there exists a non-zero homomorphism f :Ext n S (N, U) E 0. Since U is essential in E 0, f ( U) is a non-zero submodule of Ext n S (N, U). By assumption, H n( U) = E n (U S ) and E n (U S ) is submodule-closed. So f ( U) E n (U S )(= H n ( U)) and hence [f ( U)] = 0, which is a contradiction. Consequently, U- lim.dim (E 0 ) n. We next prove U- lim.dim (E ) n (note: at this moment, n 2). If U- lim.dim (E )> n, then l.fd S ( E )>n by Lemma 2.4. So by Lemma 3., there exists a module N mod S op such that Hom (Ext n S (N,U),E ) 0. Hence there exists a non-zero homomorphism f :Ext n S (N,U) E. Since Ker α is essential in E, f (Ker α ) is a non-zero submodule of Ext n S (N,U). By assumption, f (Ker α ) E n (U S ). Since l.fd S ( E 0 ) = U- lim.dim (E 0 ) n, Hom (f (Ker α ), E 0 ) = 0 by Lemma 3.. From the exact sequence 0 U E 0 Ker α 0 we get the following exact sequence: Hom ( f (Ker α ), E 0 ) Hom ( f (Ker α ), Ker α ) Ext ( f (Ker α ), U ). Since f (Ker α ) E n (U S ) (= H n ( U)), Ext (f (Ker α ), U) = 0. So Hom (f (Ker α ), Ker α ) = 0, which is a contradiction. Hence we conclude that U- lim.dim (E ) n. Continuing this process, we get that U- lim.dim (E i ) n for any 0 i n. If r.id S (U) n, then n U has the U-reflexive property by [HT, Theorem 2.2]. So by Theorem 3.7, we have the following Corollary 3.8. If l.id (U) = r.id S (U) n, then the following statements are equivalent. () U- lim.dim ( n i=0 E i) n. (2) E n (U S ) is submodule-closed and E n (U S ) = H n ( U). (3) E n (U S ) is submodule-closed and E n (U S ) = H n ( U). Let be a two-sided noetherian ring. ecall that is called an Iwanaga Gorenstein ring if the injective dimensions of and are finite. Also recall that is said to satisfy the Auslander condition if the flat dimension of the (i + )st term in a minimal injective resolution of

628 Z. Huang / Journal of Algebra 3 (2007) 69 634 is at most i for any i 0, and is called Auslander Gorenstein if it is Iwanaga Gorenstein and satisfies the Auslander condition (see [Bj]). It is well known that any commutative Iwanaga Gorenstein ring is Auslander Gorenstein. The following corollary gives a positive answer to the Auslander s question for Auslander Gorenstein rings and so in particular for commutative Iwanaga Gorenstein rings. Corollary 3.9. If is an Auslander Gorenstein ring with self-injective dimension n, then E n ( ) is submodule-closed. Proof. Notice that - lim.dim (A) = l.fd (A) for any A Mod, so our assertion follows from Corollary 3.8. Assume that X mod S op g and there exists an exact sequence H H 0 X 0in mod S op. We denote A = Coker g. The following result is a generalization of [HT, Lemma 2.]. The proof here is similar to that in [HT], we omit it. Lemma 3.0. Let X, A, H 0 and H be as above. Assume that H 0 and H are U-reflexive. () If H i i+ U for i = 0,, then we have the following exact sequence: 0 Ext (A, U) X σ X X Ext 2 (A, U) 0. (2) If H i 2 i U S for i = 0,, then we have the following exact sequence: 0 Ext S (X, U) A σ A A Ext 2 S (X, U) 0. Lemma 3.. Let X be a full subcategory of U S which has the U-reflexive property and X a module in mod S op.ifx-resol.dim S (X) = n( ), then grade U Ext n S (X, U) ; if furthermore n 2 and Y 2 U for any Y X, then grade U Extn S (X, U) 2. Proof. Assume that X - resol.dim S (X) = n( ). Then there exists an exact sequence: d n 0 X n X X 0 X 0 in mod S op with X i X ( U S ) for any 0 i n. Set N = Coker d n. Then Ext S (N, U) = Ext n S (X, U). Consider the following commutative diagram with exact rows: 0 X n d n 0 [Ext n S (X, U)] X n σ Xn d n X n X n. σ Xn N 0 Because X has the U-reflexive property, both σ Xn and σ Xn are isomorphisms. So we have that [Ext n S (X, U)] = 0 and grade U Ext n S (X, U).

Z. Huang / Journal of Algebra 3 (2007) 69 634 629 If n 2 and Y 2 U for any Y X, by applying Lemma 3.0() to the exact sequence d n 0 X n X n N 0, we then get the following exact sequence: 0 Ext ( Ext n S (X, U), U ) N σ N N Ext 2 ( Ext n S (X, U), U ) 0. Because X n 2 X and X has the U-reflexive property, X n 2 is U-reflexive. Then N is U-torsionless for it is isomorphic to a submodule of X n 2.Soσ N is monic and Ext (Extn S (X, U), U) = 0. Hence we conclude that grade U Extn S (X, U) 2. For a non-negative integer t, a module N in mod S op is said to have generalized Gorenstein dimension at most t (with respect to U S ), denoted by G-dim U (N) t, if there exists an exact sequence 0 N t N N 0 N 0inmodS op with G-dim U (N i ) = 0 for any 0 i t (see [A2]). Lemma 3.2. Let n 2.Ifl.id (U) = r.id S (U) n, then E n (U S ) = H n ( U). Proof. Because l.id (U) = r.id S (U) n, both n U and nu S have the U-reflexive property by [HT, Theorem 2.2]. It follows from Lemma 3.5 that H n ( U) E n (U S ). Assume that n = and 0 M E (U S ).LetE 0 be the injective envelope of U S. Because r.fd ( E 0 ) l.id (U) by assumption and Lemma 2.6, it follows from the symmetric statements of [H2, Theorem 7.5.5] that grade U M. By Lemma 3.6, grade U M =. Thus M H ( U) and E (U S ) H ( U). The case n = follows. Assume that n = 2 and 0 M E 2 (U S ). Then there exists a module N mod S op such that M = Ext 2 S (N, U) ( 0). Because l.id (U) = r.id S (U) 2, by [HT, Theorem 3.5] we have that G-dim U (N) 2. If G-dim U (N) < 2, then there exists an exact sequence 0 N N 0 N 0inmodS op with G-dim U (N ) = G-dim U (N 0 ) = 0. So Ext 2 S (N, U) = Ext S (N,U) = 0, which is a contradiction. Hence we conclude that G-dim U (N) = 2. Then by Lemma 3., grade U M = grade U Ext 2 S (N, U) 2. By Lemma 3.6, grade U M = 2. Thus M H 2 ( U) and E 2 (U S ) H 2 ( U). The case n = 2 follows. Proposition 3.3. Let n 2. Assume that l.id (U) = r.id S (U) n. () If n =, then E (U S ) is submodule-closed if and only if E 0 add-lim U (that is, U- lim.dim (E 0 ) = 0) if and only if U- lim.dim (E i ) i for i = 0,. (2) If n = 2, then E 2 (U S ) is submodule-closed if and only if U- lim.dim (E 0 E ). Proof. The former equivalence in () and the equivalence in (2) follow from Lemma 3.2 and Theorem 3.7. Notice that U- lim.dim (E ) = l.fd S ( E ) r.id S (U) by Lemmas 2.4 and 2.6, then the latter equivalence in () follows. Let 0 I 0 I I i be a minimal injective resolution of. Putting U S =, by Proposition 3.3 we immediately have the following

630 Z. Huang / Journal of Algebra 3 (2007) 69 634 Corollary 3.4. Let n 2. Assume that l.id () = r.id () n. () If n =, then E ( ) is submodule-closed if and only if I 0 is flat if and only if is Auslander Gorenstein. (2) If n = 2, then E 2 ( ) is submodule-closed if and only if l.fd (I 0 I ). In the following, we give some examples to illustrate that neither E ( ) nor E 2 ( ) are submodule-closed in general. Example 3.5. Let K be a field and a finite dimensional K-algebra which is given by the quiver: 2 3. Then is Iwanaga Gorenstein with l.id () = r.id () = and l.fd (I 0 ) =. By Corollary 3.4, E ( ) is not submodule-closed. Example 3.6. Let K be a field and Δ the quiver: 2 α γ β 3 4. δ If = KΔ/(βα), then is Iwanaga Gorenstein with l.id () = r.id () = 2 and l.fd (E(P 4 )) = 2, where E(P 4 ) is the injective envelope of the indecomposable projective module corresponding to the vertex 4. Since P 4 is a direct summand of, l.fd (I 0 ) = 2. By Corollary 3.4, E 2 ( ) is not submodule-closed. It is clear that mod E ( ) E 2 ( ) E i ( ). From the above argument we know that E n ( ) is submodule-closed for an Auslander Gorenstein ring with self-injective dimension n for any n, and neither E ( ) nor E 2 ( ) are submodule-closed in general. However, we do not know whether E n ( ) (where n 3) is submodule-closed or not in general. 4. Wakamatsu tilting conjecture and (quasi-)gorenstein modules Let be an Artin algebra. ecall that a module T in mod is called a tilting module of finite projective dimension if the following conditions are satisfied: () l.pd (T ) < ; (2) T is selforthogonal; and (3) there exists an exact sequence 0 T 0 T T t 0 in mod with T i add T for any 0 i t. The notion of cotilting modules of finite injective dimension may be defined dually. A generalized tilting module is not necessarily tilting or cotilting. The following conjecture is called Wakamatsu Tilting Conjecture (WTC): Every generalized tilting module with finite projective dimension is tilting, or equivalently, every generalized tilting module with finite injective dimension is cotilting (see [B]). For Artin algebras and S and a generalized tilting module U with S = End( U), by Theorem 2.7 and the dual results of [M, Theorem.5 and Proposition.6], we easily get the following equivalent statements:

Z. Huang / Journal of Algebra 3 (2007) 69 634 63 () WTC holds. (2) If one of l.id (U) and r.id S (U) is finite, then the other is also finite. (3) l.id (U) = r.id S (U). The Gorenstein Symmetry Conjecture (GSC) states that the left and right self-injective dimensions of are identical for an Artin algebra (see [B]). It is trivial from the above equivalent conditions that WTC GSC. As an application of the results obtained in Section 2, we now give some sufficient conditions for the validity of statement (2). In other words, we establish some cases in which WTC holds true. Theorem 4.. [H, Theorem] Let and S be two-sided artinian rings and n and m positive integers. If r.id S (U) n and grade U Ext m (M, U) n for any M mod, then l.id (U) n + m. The following corollary is an immediate consequence of Theorems 4. and 2.7. Corollary 4.2. Let and S be two-sided artinian rings. Then r.id S (U) if and only if l.id (U). and Let 0 U E 0 E E i 0 U S E 0 E E i be minimal injective resolutions of U and U S, respectively. The following Propositions 4.3 and 4.6 generalize some results in [H] and [A4]. Proposition 4.3. Let and S be two-sided artinian rings and n a positive integer. If r.id S (U) = n and U- lim.dim S ( n 2 i=0 E i )<, then l.id (U) = n. Proof. Assume that U- lim.dim S ( n 2 i=0 E i ) = r(< ). It follows from [H2, Lemma 7.3.2] that s.grade U Ext r+ (M, U) n for any M mod. By Theorem 4., l.id (U) r + n(< ). Thus l.id (U) = r.id S (U) = n by Theorem 2.7. The following two results are cited from [H2]. Theorem 4.4. [H2, Theorem 7..] Let and S be two-sided noetherian rings. Then, for a positive integer n, the following statements are equivalent. () s.grade U Ext i (M, U) i for any M mod and i n. () op s.grade U Ext i S (N, U) i for any N mod Sop and i n. U (symmetrically U S ) is called an n-gorenstein module if one of the above equivalent conditions is satisfied, and U (symmetrically U S ) is called a Gorenstein module if it is n-gorenstein for all n.

632 Z. Huang / Journal of Algebra 3 (2007) 69 634 It follows from [H2, Corollary 7..2] that a two-sided noetherian ring satisfies the Auslander condition if and only if is a Gorenstein module. Theorem 4.5. [H2, Theorem 7.5.4] Let and S be two-sided noetherian rings. Then, for a positive integer n, the following statements are equivalent. () s.grade U Ext i+ S (N, U) i for any N mod S op and i n. (2) grade U Ext i (M, U) i for any M mod and i n. U is called a quasi-n-gorenstein module if one of the above equivalent conditions is satisfied, and U is called a quasi-gorenstein module if it is quasi-n-gorenstein for all n. An (n-)gorenstein module is clearly quasi-(n-)gorenstein. But the conserve does not hold in general because the notion of (n)-gorenstein modules is left right symmetric by Theorem 4.4, and that of quasi-(n)-modules is not left right symmetric even in the case U S = (see [H2, Example 7.5.2]). Proposition 4.6. Let and S be two-sided artinian rings. Then l.id (U) = r.id S (U) provided that U (or U S ) is quasi-gorenstein. Proof. Let U be a quasi-gorenstein module. By Theorem 4.5, for any i, we have that grade U Ext i (M, U) i for any M mod and s.grade U Exti+ S (N, U) i for any N mod S op. Then it is easy to see from Theorem 4. that l.id (U) < if and only if r.id S (U) <. Thus l.id (U) = r.id S (U) by Theorem 2.7. Note that Proposition 4.6 generalizes [A4, Corollary 5.5(b)] which asserts that l.id () = r.id () if is an Artin algebra satisfying the Auslander condition. Conjecture 4.7. Let and S be Artin algebras and U a generalized tilting module with S = End( U).If U is (quasi-)gorenstein, then l.id (U) = r.id S (U) < (in fact, under our assumption it has been proved in Proposition 4.6 that l.id (U) = r.id S (U)). Auslander and eiten in [A4] raised the following conjecture, which we call Auslander Gorenstein Conjecture (AGC): An Artin algebra is Iwanaga Gorenstein if it satisfies the Auslander condition (in other words, an Artin algebra satisfies l.id () = r.id () < provided is a Gorenstein module). It is trivial that this conjecture is situated between Conjecture 4.7 and the famous Nakayama Conjecture (NC), which states that an Artin algebra is self-injective if each term in a minimal injective resolution of is projective. That is, we have the following implications: Conjecture 4.7 AGC NC. ecall moreover the Generalized Nakayama Conjecture (GNC): Every indecomposable injective -module occurs as the direct summand of some term in a minimal injective resolution of for an Artin algebra. An equivalent version of GNC is: For an Artin algebra and every simple module T mod, there exists a non-negative integer k such that Ext k (T, ) 0 (see [A]). It is well known that GNC implies AGC. We now show the corresponding result for Conjecture 4.7.

Z. Huang / Journal of Algebra 3 (2007) 69 634 633 Proposition 4.8. Let and S be Artin algebras and U a generalized tilting module with S = End( U). If the following condition is satisfied: for every simple module T mod, there exists a negative-integer k such that Ext k (T, U) 0, then Conjecture 4.7 holds for. Proof. Let {T,...,T t } be the set of all non-isomorphic simple modules in mod. By assumption, for each T i ( i t), there exists a non-negative integer k i such that Ext k i (T i, U) 0. It is easy to verify that Hom (T, E j ) = Ext j (T, U) for any simple -module T and j 0. So Hom (T i,e ki ) 0 for any i t and hence E(T i ) (the injective envelope of T i )isisomorphic to a direct summand of E ki for any i t. Now suppose U is quasi-gorenstein. Then by Theorem 4.5 and Lemma 3.2, we have that l.fd S ( E i ) i + for any i 0. So l.fd S ( [E(T i )]) l.fd S ( E ki ) k i + for any i t. Put E = t i= E(T i ) and k = max{k,...,k t }. Then E is an injective cogenerator for Mod and l.fd S ( E) k +. It follows from Lemma 2.6() that r.id S (U) k +. We are done. Let N be a module in mod S op. ecall that an injective resolution: 0 N δ 0 δ δ 2 V 0 V δ i δ i+ V i is called ultimately closed if there exists a positive integer n such that Im δ n = m j=0 W j, where each W j is a direct summand of Im δ ij with i j <n. By [HT, Theorem 2.4], if U S has a ultimately closed injective resolution (especially, if r.id S (U) < ), then U has the U-reflexive property and the condition in Proposition 4.8 is satisfied. Acknowledgments Part of the paper was written while the author was staying at Universität Bielefeld supported by the SFB 70 Spectral Structures and Topological Methods in Mathematics. The author is grateful to Prof. Claus M. ingel for his warm hospitality and useful comments on this paper. The research of the author was partially supported by Specialized esearch Fund for the Doctoral Program of Higher Education (Grant Nos. 20030284033, 20060284002) and NSF of Jiangsu Province of China (Grant No. BK2005207). The author is grateful to the referee for the careful reading and the valuable and detailed suggestions in shaping this paper into its present version. eferences [A] M. Auslander, Comments on the functor Ext, Topology 8 (969) 5 66. [AB] M. Auslander,.O. Buchweitz, The homological theory of Cohen Macaulay approximations, Mém. Soc. Math. Fr. (2) 38 (989) 5 37. [A] M. Auslander, I. eiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52 (975) 69 74. [A2] M. Auslander, I. eiten, Cohen Macaulay and Gorenstein Artin algebras, in: G.O. Michler, C.M. ingel (Eds.), epresentation Theory of Finite Groups and Finite Dimensional Algebras, Bielefeld, 99, in: Progr. Math., vol. 95, Birkhäuser, Basel, 99, pp. 22 245. [A3] M. Auslander, I. eiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (99) 52. [A4] M. Auslander, I. eiten, k-gorenstein algebras and syzygy modules, J. Pure Appl. Algebra 92 (994) 27. [Bj] J.E. Björk, The Auslander condition on noetherian rings, in: Séminaire d Algèbre Paul Dubreil et Marie-Paul Malliavin, in: Lecture Notes in Math., vol. 404, Springer-Verlag, Berlin, 989, pp. 37 73. [B] A. Beligiannis, I. eiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc., in press.

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