On U-dominant dimension

Transcription

1 Journal of Algebra 285 (2005) On U-dominant dimension Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 20093, PR China Received 20 August 2003 Available online February 2005 Communicated by Kent R. Fuller Abstract Let Λ and Γ be artin algebras and Λ U Γ a faithfully balanced selforthogonal bimodule. We show that the U-dominant dimensions of Λ U and U Γ are identical. As applications of the results obtained, we give some characterizations of the double U-dual functors preserving monomorphisms and being left exact respectively Elsevier Inc. All rights reserved. Keywords: U-dominant dimension; Flat dimension; Faithfully balanced selforthogonal bimodules; Double U-dual functors. Introduction For a ring Λ, weusemodλ (respectively mod Λ op ) to denote the category of finitely generated left Λ-modules (respectively right Λ-modules). Definition.. Let Λ and Γ be rings. A bimodule Λ T Γ is called a faithfully balanced selforthogonal bimodule if it satisfies the following conditions: () Λ T mod Λ and T Γ mod Γ op. (2) The natural maps Λ End(T Γ ) and Γ End( Λ T) op are isomorphisms. (3) Ext i Λ ( ΛT, Λ T)= 0 and Ext i Γ (T Γ,T Γ ) = 0 for any i. address: huangzy@nju.edu.cn /$ see front matter 2004 Elsevier Inc. All rights reserved. doi:0.06/j.jalgebra

2 670 Z. Huang / Journal of Algebra 285 (2005) Definition.2. Let U be in mod Λ (respectively mod Γ op ) and n a non-negative integer. For a module M in mod Λ (respectively mod Γ op ), () M is said to have U-dominant dimension greater than or equal to n, written U- dom.dim( Λ M) (respectively U-dom.dim(M Γ )) n, if each of the first n terms in a minimal injective resolution of M is cogenerated by Λ U (respectively U Γ ), that is, each of these terms can be embedded into a direct product of copies of Λ U (respectively U Γ ) [0]. (2) M is said to have dominant dimension greater than or equal to n, written dom.dim( Λ M) (respectively dom.dim(m Γ )) n, if each of the first n terms in a minimal injective resolution of M is Λ-projective (respectively Γ op -projective) [2]. Assume that Λ is an artin algebra. By [4, Theorem 3.3], Λ I and each of its direct summands are projective for any index set I. So, when Λ U = Λ Λ (respectively U Γ = Λ Λ ), the notion of U-dominant dimension coincides with that of (ordinary) dominant dimension. Tachikawa in [2] showed that if Λ is a left and right artinian ring then the dominant dimensions of Λ Λ and Λ Λ are identical. Hoshino then in [6] generalized this result to left and right noetherian rings. Kato in [0] characterized the modules with U-dominant dimension greater than or equal to one. Colby and Fuller in [5] gave some equivalent conditions of dom.dim( Λ Λ) (or 2) in terms of the properties of the double dual functors (with respect to Λ Λ Λ ). The results mentioned above motivate our interests in establishing the identity of U- dominant dimensions of Λ U and U Γ and characterizing the properties of modules with agivenu-dominant dimension. Our characterizations will lead a better comprehension about U-dominant dimension and the theory of selforthogonal bimodules. Throughout this paper, Λ and Γ are artin algebras and Λ U Γ is a faithfully balanced selforthogonal bimodule. The main result in this paper is the following Theorem.3. U-dom.dim( Λ U)= U-dom.dim(U Γ ). Put Λ U Γ = Λ Λ Λ, we immediately get the following result, which is due to Tachikawa (see [2]). Corollary.4. dom.dim( Λ Λ) = dom.dim(λ Λ ). Let M be in mod Λ (respectively mod Γ op ) and G(M) the subcategory of mod Λ (respectively mod Γ op ) consisting of all submodules of the modules generated by M. M is called a QF-3 module if G(M) has a cogenerator which is a direct summand of every other cogenerator [3]. By [3] Proposition 2.2 we have that a finitely cogenerated Λ-module (respectively Γ op -module) M is a QF-3 module if and only if M cogenerates its injective envelope. So by Theorem.3 we have Corollary.5. Λ U is QF-3 if and only if U Γ is QF-3. We shall prove our main result in Section 2. We study the case that the double U-dual functors ( ) preserves monomorphisms by the language of Lambek torsion theory, show the left right symmetry of the fact that ( ) preserves monomorphisms, and then prove

3 Z. Huang / Journal of Algebra 285 (2005) the main result. It should be pointed out that this strategy is similar to that of Hoshino [7]. As applications of the results obtained in Section 2, we give in Section 3 some characterizations of the double U-dual functors ( ) preserving monomorphisms and being left exact respectively. The results of this paper are natural generalizations of (ordinary) dominant dimension and of several author s approach to dominant dimension (see Tachikawa [2], Colby Fuller [5] and Hoshino [6,7]). In fact, most of the results here are the U-dual versions of the results in [6,7]. 2. The proof of main result Let E 0 be the injective envelope of Λ U. Then E 0 defines a torsion theory in mod Λ. The torsion class T is the subcategory of mod Λ consisting of the modules X satisfying Hom Λ (X, E 0 ) = 0, and the torsionfree class F is the subcategory of mod Λ consisting of the modules Y cogenerated by E 0 (equivalently, Y can be embedded in E0 I for some index set I ). A module in mod Λ is called torsion (respectively torsionfree) if it is in T (respectively F). The injective envelope E 0 of U Γ also defines a torsion theory in mod Γ op and we may give in mod Γ op the corresponding notions as above. Let X be in mod Λ (respectively mod Γ op ) and t(x) the torsion submodule, that is, t(x) is the submodule X such that Hom Λ (t(x), E 0 ) = 0 (respectively Hom Γ (t(x), E 0 ) = 0) and E 0 (respectively E 0 ) cogenerates X/t(X) (cf. [9]). Let A be in mod Λ (respectively mod Γ op ). We call Hom Λ ( Λ A, Λ U Γ ) (respectively Hom Γ (A Γ, Λ U Γ )) the dual module of A with respect to Λ U Γ, and denote either of these modules by A. For a homomorphism f between Λ-modules (respectively Γ op -modules), we put f = Hom(f, Λ U Γ ).Letσ A : A A via σ A (x)(f ) = f(x) for any x A and f A be the canonical evaluation homomorphism. A is called U-torsionless (respectively U-reflexive) if σ A is a monomorphism (respectively an isomorphism). The following result is analogous to [7, Lemma 4]. Lemma 2.. For a module X in mod Λ (respectively mod Γ op ), t(x)= Ker σ X if and only if Hom Λ (Ker σ X,E 0 ) = 0(respectively Hom Γ (Ker σ X,E 0 ) = 0). Proof. The necessity is trivial. Now we prove the sufficiency. We have the following commutative diagram with the upper row exact: 0 t(x) X σ X π X/t(X) σ X/t(X) 0 X π [X/t(X)] Since Hom Λ (t(x), E 0 ) = 0, [t(x)] = 0 and π is an isomorphism. So π is also an isomorphism and hence t(x) Ker σ X. On the other hand, Hom Λ (Ker σ X,E 0 ) = 0by assumption, which implies that Ker σ X is a torsion module and contained in X. Sowe conclude that Ker σ X t(x) and Ker σ X = t(x).

4 672 Z. Huang / Journal of Algebra 285 (2005) Remark. From the above proof we always have t(x) Ker σ X. Suppose that A mod Λ (respectively mod Γ op f ) and P P 0 A 0 is a (minimal) projective resolution of A. Then we have an exact sequence 0 A P 0 f P Coker f 0. We call Coker f the transpose (with respect to Λ U Γ )ofa, and denote it by Tr U A. The following result is the U-dual version of [7, Theorem A]. Proposition 2.2. The following statements are equivalent. () t(x)= Ker σ X for every X mod Λ. (2) is monic for every monomorphism f : A B in mod Λ. () op t(y)= Ker σ Y for every Y mod Γ op. (2) op g is monic for every monomorphism g : C D in mod Γ op. Proof. By symmetry, it suffices to prove the implications of () (2) op () op. () (2) op.letg: C D be monic in mod Γ op. Set X = Coker g. We have that Ker σ TrU X = Ext Γ (X, U) and Tr U X mod Λ by [8, Lemma 2.]. By () and Lemma 2., Hom Λ (Ext Γ (X, U), E 0) = 0. Since Coker g can be imbedded in Ext Γ (X, U), Hom Λ (Coker g,e 0 ) = 0. But (Coker g ) Hom Λ (Coker g,e 0 ),so(cokerg ) = 0 and hence Ker g = (Coker g ) = 0, which implies that g is monic. (2) op () op.lety be in mod Γ op and X any submodule of Ker σ Y and f : X Ker σ Y the inclusion. Assume that f is the composition: X f Ker σ Y Y. Then σ Y f = 0 and f σy = (σ Y f) = 0. But σy is epic by [, Proposition 20.4], so f = 0 and = 0. By (2) op, is monic, so X = 0 and X = 0. Since X is isomorphic to a submodule of X by [, Proposition 20.4], X = 0. We claim: Hom Γ (Ker σ Y,E 0 ) = 0. Otherwise, there exists 0 α Hom Γ (Ker σ Y,E 0 ). Then Im α U Γ 0 since U Γ is an essential submodule of E 0.Soα (Im α U Γ ) is a non-zero submodule of Ker σ Y and there exists a non-zero map α (Im α U Γ ) U Γ, which implies that (α (Im α U Γ )) 0, a contradiction with the former argument. Hence we conclude that t(y)= Ker σ Y by Lemma 2.. Let A be a Λ-module (respectively a Γ op -module). Denote either of Hom Λ ( Λ U Γ, Λ A) and Hom Γ ( Λ U Γ,A Γ ) by A, and the left (respectively right) flat dimension of A by l.fd Λ (A) (respectively r.fd Γ (A)). We give a remark as follows. For an artin algebra R and a left (respectively right) R-module A, we have that the left (respectively right) flat dimension of A and its left (respectively right) projective dimension are identical; especially, A is left (respectively right) flat if and only if it is left (respectively right) projective.

5 Z. Huang / Journal of Algebra 285 (2005) Lemma 2.3. Let Λ E (respectively E Γ ) be injective and n a non-negative integer. Then l.fd Γ ( E) (respectively r.fd Λ ( E) n) if and only if Hom Λ (Ext n+ Γ (A, U), E) (respectively Hom Γ (Ext n+ Λ (A, U), E) = 0) for any A mod Γ op (respectively mod Λ). Proof. It is trivial by [3, Chapter VI, Proposition 5.3]. The following result is similar to [7, Proposition B]. In fact, we obtain the first two statements of this result by replacing E( R R) is flat and E is flat of [7, Proposition B] by E 0 is flat and E is flat respectively. The third statement is analogous to the corresponding one of [7, Proposition B]. Proposition 2.4. The following statements are equivalent. () E 0 is flat. (2) There is an injective Λ-module E such that E is flat and E cogenerates E 0. (3) t(x)= Ker σ X for any X mod Λ. Proof. () (2). It is trivial. (2) (3).LetX mod Λ. Since Ker σ X = Ext Γ (Tr U X, U) with Tr U X mod Γ op by [8, Lemma 2.]. By (2) and Lemma 2.3, Hom Λ (Ext Γ (Tr U X, U), E) = 0. Since E cogenerates E 0, there is an exact sequence 0 E 0 E I for some index set I. So ( ) ( Hom Λ Ext Γ (Tr U X, U), E 0 HomΛ Ext Γ (Tr U X, U), E I ) = [ Hom Λ ( Ext Γ (Tr U X, U), E )] I = 0 and Hom Λ ( Ext Γ (Tr U X, U), E 0 ) = 0. By Lemma 2., t(x)= Ker σ X. (3) (). LetN mod Γ op. Since Ker σ TrU N = Ext Γ (N, U) with Tr U N mod Λ by [8, Lemma 2.], By (3) and Lemma 2. we have Hom Λ (Ext Γ (N, U), E 0) = Hom Λ (Ker σ TrU N,E 0 ) = 0, and so E 0 is flat by Lemma 2.3. Dually, we have the following Proposition 2.4. The following statements are equivalent. () E 0 is flat. (2) There is an injective Γ op -module E such that E is flat and E cogenerates E 0. (3) t(y)= Ker σ Y for any Y mod Γ op. Corollary 2.5. E 0 is flat if and only if E 0 is flat. Proof. By Propositions 2.2, 2.4 and 2.4.

6 674 Z. Huang / Journal of Algebra 285 (2005) Let A mod Λ (respectively mod Γ op ) and i a non-negative integer. We say that the grade of A with respect to Λ U Γ, written grade U A, is greater than or equal to i if Ext j Λ (A, U) = 0 (respectively Extj Γ (A, U) = 0) for any 0 j<i. Lemma 2.6. Let X be in mod Γ op and n a non-negative integer. If grade U X n and grade U Ext n Γ (X, U) n +, then Extn Γ (X, U) = 0. Proof. Since X is U-torsionless, X = 0 if and only if X = 0. Then the case n = 0 follows. Now let n and P n P P 0 X 0 be a projective resolution of X in mod Γ op. Put X n = Coker(P n+ P n ). Then we have an exact sequence 0 P 0 P n f Xn Extn Γ (X, U) 0 in mod Λ with each Pi add Λ U. Since grade U Ext n Γ (X, U) n +, Ext i Λ( Ext n Γ (X, U), U ) = 0 for any 0 i n. So Ext i Λ (Extn Γ (X, U), P j ) = 0 for any 0 i n and 0 j n, and hence Ext Λ (Extn Γ (X, U), Im f) = Ext n Λ (Extn Γ (X, U), P 0 ) = 0, which implies that we have an exact sequence Hom Λ (Ext n Γ (X, U), X n ) Hom Λ(Ext n Γ (X, U), Extn Γ (X, U)) 0. Notice that Xn is U-torsionless and Hom Λ(Ext n Γ (X, U), U) = 0. So Hom Λ(Ext n Γ (X, U), Xn ) = 0 and Hom Λ(Ext n Γ (X, U), Extn Γ (X, U)) = 0, which implies that Extn Γ (X, U) = 0. Remark. We point out that all of the above results (from 2. to 2.6) in this section also hold in the case Λ and Γ are left and right noetherian rings. For a module T in mod Λ (respectively mod Γ op ), we use add Λ T (respectively add T Γ ) to denote the subcategory of mod Λ (respectively mod Γ op ) consisting of all modules isomorphic to direct summands of finite direct sums of copies of Λ T (respectively T Γ ). Let A be in mod Λ. If there is an exact sequence U n U U 0 A 0in mod Λ with each U i add Λ U for any i 0, then we define U-resol.dim Λ (A) = inf{n there is an exact sequence 0 U n U U 0 A 0inmodΛ with each U i add Λ U for any 0 i n}. WesetU-resol.dim Λ (A) infinity if no such an integer exists. Dually, for a module B in mod Γ op, we may define U-resol.dim Γ (B) (see [2]). Lemma 2.7. Let E be injective in mod Λ (respectively mod Γ op ). Then l.fd Γ ( E) (respectively r.fd Λ ( E) n) if and only if U-resol.dim Λ (E) (respectively U-resol.dim Γ (E) n).

7 Z. Huang / Journal of Algebra 285 (2005) Proof. Assume that E is injective in mod Λ and l.fd Γ ( E) n. Then there is an exact sequence 0 Q n Q Q 0 E 0 with each Q i flat (and hence projective) in mod Γ for any 0 i n. By [3, Chapter VI, Proposition 5.3] Tor Γ j (U, E) = Hom Λ (Ext j Γ (U, U), E) = 0 for any j. Then we easily have an exact sequence: 0 U Γ Q n U Γ Q U Γ Q 0 U Γ E 0. It is clear that U Γ Q i add Λ U for any 0 i n. By [, p. 47], U Γ E = Hom Λ (Hom Γ (U, U), E) = E. Hence we conclude that U-resol.dim Λ (E) n. Conversely, if U-resol.dim Λ (E) n then there is an exact sequence 0 X n X X 0 E 0 with each X i in add Λ U for any 0 i n. Since Ext j Λ (U, X i) = 0 for any j and 0 i n, 0 X n X X 0 E 0 is exact with each X i (0 i n) Γ -projective. Hence we are done. Corollary 2.8. Let E be injective in mod Λ (respectively mod Γ op ). Then E is flat in mod Γ (respectively mod Λ op ) if and only if Λ E add Λ U (respectively E Γ add U Γ ). From now on, assume that 0 Λ U f 0 f f 2 E 0 E f i f i+ E i is a minimal injective resolution of Λ U. The following result is the U-dual version of [6, Lemma 2.2]. Lemma 2.9. Suppose U-dom.dim( Λ U). Then, for any n 2, U-dom.dim( Λ U) n if and only if grade U M n for any M mod Λ with M = 0. Proof. For any M mod Λ and i, we have an exact sequence Hom Λ (M, E i ) Hom Λ (M, Im f i ) Ext i Λ (M, U) 0. ( ) Suppose U-dom.dim( Λ U) n. Then E i is cogenerated by Λ U for any 0 i n. So, for a given M mod Λ with M = 0 we have that Hom Λ (M, E i ) = 0 and Hom Λ (M, Im f i ) = 0 for any 0 i n. Then by the exactness of ( ),Ext i Λ (M, U) = 0 for any i n, and so grade U M n. Now we prove the converse, that is, we will prove that E i add Λ U for any 0 i n. First, E 0 add Λ U by assumption. We next prove E add Λ U. For any 0 x Im f, we claim that M = Hom Λ (M, U) 0, where M = Λx. Otherwise,we have Ext i Λ (M, U) = 0 for any 0 i n by assumption. Since E 0 add Λ U, Hom Λ (M, E 0 ) = 0. So from the exactness of ( ) we know that Hom Λ (M, Im f ) = 0, which is a contradiction. Then we conclude that Im f, and hence E, is cogenerated by ΛU. Notice that E is finitely cogenerated, so E add Λ U. Finally, suppose that n 3 and E i add Λ U for any 0 i n 2. Then by using a similar argument to that above we have E n add Λ U. The proof is finished.

8 676 Z. Huang / Journal of Algebra 285 (2005) Dually, we have the following Lemma 2.9. Suppose U-dom.dim(U Γ ). Then, for any n 2, U-dom.dim(U Γ ) n if and only if grade U N n for any N mod Γ op with N = 0. We now are in a position to prove the main result in this paper. Proof of Theorem.3. We only need to prove U-dom.dim( Λ U) U-dom.dim(U Γ ). Without loss of generality, suppose U-dom.dim( Λ U)= n. The case n = follows from Corollaries 2.5 and 2.8. Let n 2. Notice that U-dom.dim( Λ U) and U-dom.dim(U Γ ). By Lemma 2.9 it suffices to show that grade U N n for any N mod Γ op with N = 0. By Lemmas 2.3 and 2.7, for any i, Hom Λ (Ext i Γ (N, U), E 0) = Tor Γ i (N, E 0 ) = 0, so [Ext i Γ (N, U)] = 0. Then by assumption and Lemma 2.9, grade U Ext i Γ (N, U) n for any i. It follows from Lemma 2.6 that grade U N n. 3. Some applications As applications of the results in above section, we give in this section some characterizations of ( ) preserving monomorphisms and being left exact respectively. Assume that f 0 0 U Γ E 0 f E f 2 E i f i f i+ is a minimal injective resolution of U Γ. We first have the following Proposition 3.. The following statements are equivalent for any positive integer k. () U-dom.dim( Λ U) k. (2) 0 ( Λ U) 0 E0 f E f2 f k Ek is exact. () op U-dom.dim(U Γ ) k. (2) op 0 (U Γ ) (f 0 ) (E 0 (f ) ) (E (f ) 2 ) (f k ) (E k ) is exact. Proof. By Theorem.3 we have () () op. By symmetry, we only need to prove () (2). If U-dom.dim( Λ U) k, then E i is in add Λ U for any i k. Notice that Λ U and each E i (0 i k ) are U-reflexive and hence we have that 0 ( Λ U) 0 E 0 E f2 f k E k

9 Z. Huang / Journal of Algebra 285 (2005) is exact. Assume that (2) holds. We proceed by induction on k.byassumptionwehavethe following commutative diagram with exact rows: 0 ΛU f 0 σ U E 0 0 ( Λ U) f0 E 0 f σ E0 E E f 2 σ E 2 f k k E k E k σ Ek Since σ U is an isomorphism, σ E0 f 0 = f0 σ U is a monomorphism. But f 0 is essential, so σ E0 is monic, that is, E 0 is U-torsionless and E 0 is cogenerated by Λ U. Moreover, E 0 is finitely cogenerated, so we have that E 0 add Λ U (and hence σ E0 is an isomorphism). The case k = is proved. Now suppose that k 2 and E i add Λ U (and then σ Ei is an isomorphism) for any 0 i k 2. Put A 0 = Λ U, B 0 = ( Λ U), g 0 = f 0, g 0 = f 0 and h 0 = σ U. Then, for any 0 i k 2, we get the following commutative diagrams with exact rows: 0 A i g i h i 0 B i g i E iσei A i+ h i+ 0 E i B i+ 0 and 0 A i+ g i+ h i+ 0 B i+ g i+ E i+ E i+ σ Ei+ where A i = Im f i and A i+ = Im f i+, B i = Im fi and B i+ = Im fi+, g i and g i+ are essential monomorphisms, h i and h i+ are induced homomorphisms. We may get inductively that each h j is an isomorphism for any 0 j k. Because σ Ek g k = g k h k is a monomorphism, by using a similar argument to that above we have E k add Λ U. Hence we conclude that U-dom.dim( Λ U) k. The following result develops [5, Theorem ] and [6, Proposition 3.]. Proposition 3.2. The following statements are equivalent. () U-dom.dim( Λ U). (2) ( ) :modλ mod Λ preserves monomorphisms.

10 678 Z. Huang / Journal of Algebra 285 (2005) (3) 0 ( Λ U) E0 is exact. () op U-dom.dim(U Γ ). (2) op ( ) :modγ op mod Γ op preserves monomorphisms. (3) op 0 (U Γ ) (f 0 ) (E 0 ) is exact. Proof. By Theorem.3 we have () () op. By symmetry, we only need to prove that the conditions of (), (2) and (3) are equivalent. () (2). IfU-dom.dim( Λ U) then t(x)= Ker σ X for any X mod Λ by Corollary 2.8 and Proposition 2.4. So ( ) preserves monomorphisms by Proposition 2.2. (2) (3) is trivial and (3) () follows from Proposition 3.. The following result except (3) and (3) op is the U-dual version of [7, Proposition E], which develops [5, Theorem 2]. Proposition 3.3. The following statements are equivalent. () U-dom.dim( Λ U) 2. (2) ( ) :modλ mod Λ is left exact. (3) 0 ( Λ U) 0 E0 f E is exact. (4) ( ) :modλ mod Λ preserves monomorphisms and Ext Γ (Ext Λ (X, U), U) = 0 for any X mod Λ. () op U-dom.dim(U Γ ) 2. (2) op ( ) :modγ op mod Γ op is left exact. (3) op 0 (U Γ ) (f 0 ) (E 0 (f ) ) (E ) is exact. (4) op ( ) :modγ op mod Γ op preserves monomorphisms and Ext Λ (Ext Γ (Y, U), U)= 0 for any Y mod Γ op. Proof. By Theorem.3 we have () () op and by Proposition 3. we have () (3). So, by symmetry we only need to prove that () (2) and () (4) () op. () (2). Assume that ( ) :modλ mod Λ is left exact. Then, by Proposition 3.2, we have that U-dom.dim( Λ U) and E 0 add Λ U. Let K = Im(E 0 E ) and v : K E be the essential monomorphism. By assumption and the exactness of the sequences 0 U E 0 K 0 and 0 K v E,wehave the following exact commutative diagrams: 0 U σu E 0σE0 KσK 0 0 U E 0 K

11 Z. Huang / Journal of Algebra 285 (2005) and 0 K v E σ E σ K 0 K v E where σ U and σ E0 are isomorphisms. By applying the snake lemma to the first diagram we have that σ K is monic. Then we know from the second diagram that σ E v = v σ K is a monomorphism. However, v is essential, so σ E is monic, that is, E is U-torsionless and E is cogenerated by Λ U. Moreover, E is finitely cogenerated, so we conclude that E add Λ U. α β C 0isanex- Conversely, assume that U-dom.dim( Λ U) 2 and 0 A B act sequence in mod Λ. By Proposition 3.2, α is monic. By assumption, Corollary 2.8 and Lemma 2.3 we have Hom Γ (Ext Λ (C, U), E 0) = 0. Since Coker α is isomorphic to a submodule of Ext Λ (C, U), Hom Γ (Coker α,e 0 ) = 0 and Hom Γ (Coker α,u) = 0. Then, by Theorem.3 and Lemma 2.9, grade U Coker α 2. It follows easily that 0 A α β B C is exact. () (4). Suppose U-dom.dim( Λ U) 2. By Proposition 3.2, ( ) :modλ mod Λ preserves monomorphisms. On the other hand, we have that U-dom.dim(U Γ ) 2 by Theorem.3. It follows from Corollary 2.8 and Lemma 2.3 that Hom Γ (Ext Λ (X, U), E 0 ) = 0 for any X mod Λ. So[Ext Λ (X, U)] = 0 and hence Ext Γ (Ext Λ (X, U), U) = 0 by Lemma 2.9. (4) () op. Suppose that (4) holds. Then U-dom.dim(U Γ ) by Proposition 3.2. Let A be in mod Λ and B any submodule of Ext Λ (A, U) in mod Γ op. Since U-dom.dim(U Γ ), Hom Γ (Ext Λ (A, U), E 0 ) = 0 by Corollary 2.8 and Lemma 2.3. So Hom Γ (B, E 0 ) = 0 and hence Hom Γ (B, E 0 /U) = Ext Γ (B, U). On the other hand, Hom Γ (B, E 0 ) = 0 implies B = 0. Then by [8, Lemma 2.] we have that B = Ext Λ (Tr U B,U) with Tr U B in mod Λ. By (4), Hom Γ (B, E 0 /U) = Ext Γ (B, U) = Ext Γ (Ext Λ (Tr U B,U),U) = 0. Then by using a similar argument to that in the proof (2) op () op in Proposition 2.2, we have that Hom Γ (Ext Λ (A, U), E ) = 0 (note: E is the injective envelope of E 0 /U). Thus E add U Γ by Lemma 2.3 and Corollary 2.8, and therefore U-dom.dim(U Γ ) 2. Finally we give some equivalent characterizations of U-resol.dim Λ (E 0 ) as follows, which is the U-dual version of [7, Proposition D]. Proposition 3.4. The following statements are equivalent. () U-resol.dim Λ (E 0 ). (2) σ X is an essential monomorphism for any U-torsionless module X in mod Λ. (3) is a monomorphism for any monomorphism f : X Y in mod Λ with YU-torsionless.

12 680 Z. Huang / Journal of Algebra 285 (2005) (4) grade U Ext Λ (X, U) (that is, [Ext Λ (X, U)] = 0) for any X in mod Λ. Proof. () (2). Assume that X is U-torsionless in mod Λ. Then Coker σ X = Ext 2 Γ (Tr U X, U) by [8, Lemma 2.]. By Lemmas 2.7 and 2.3 we have Hom Λ (Coker σ X,E 0 ) = Hom Λ ( Ext 2 Γ (Tr U X, U), E 0 ) = 0. Then Hom Λ (A, Λ U)= 0 for any submodule A of Coker σ X, which implies that any nonzero submodule of Coker σ X is not U-torsionless. Let B be a submodule of X with X B = 0. Then B = B/(X B) = (X + B)/X is isomorphic to a submodule of Coker σ X. On the other hand, B is clearly U-torsionless. So B = 0 and hence σ X is essential. (2) (3). Letf : X Y be monic in mod Λ with Y U-torsionless. Then σ X = σ Y f is monic. By (2), σ X is an essential monomorphism, so is monic. (3) (4).LetX be in mod Λ and 0 Y g P X 0 an exact sequence in mod Λ with P projective. It is easy to see that [Ext Λ (X, U)] = Ker g. On the other hand, g is monic by (3). So Ker g = 0 and [Ext Λ (X, U)] = 0. (4) (). LetM be in mod Γ op and P P 0 M 0 a projective resolution of M in mod Γ op. Put N = Coker(P 2 P ). By [8, Lemma 2.], Ext 2 Γ (M, U) = Ext Γ (N, U) = Ker σ TrU N. On the other hand, since N is U-torsionless, Ext Λ (Tr U N,U) = Ker σ N = 0. Let X be any finitely generated submodule of Ext 2 Γ (M, U) and f : X Ext 2 Γ (M, U) ( = Ker σ TrU N ) the inclusion, and let f be the composition: X f Ext 2 Γ g (M, U) Tr U N, where g is a monomorphism. By using the same argument as that in the proof of (2) op () op in Proposition 2.2, we get that f = 0. Hence, by applying Hom Λ (,U)to the exact sequence 0 X f Tr U N Coker f 0, we have X = Ext Λ (Coker f,u). Then X = [Ext Λ (Coker f,u)] = 0 by (4), which implies that X = 0 since X is a direct summand of X (= 0) by [, Proposition 20.24]. Also by using the same argument as that in the proof of (2) op () op in Proposition 2.2, we get that Hom Λ (Ext 2 Γ (M, U), E 0) = 0. It follows from Lemma 2.3 that l.fd Γ ( E 0 ). Therefore U-resol.dim Λ (E 0 ) by Lemma 2.7. Remark. By Theorem.3, we have that E 0 add Λ U if and only if E 0 add U Γ, that is, U-resol.dim Λ (E 0 ) = 0 if and only if U-resol.dim Γ (E 0 ) = 0. However, in general, we don t have the fact that U-resol.dim Λ (E 0 ) if and only if U-resol.dim Γ (E 0 ) even

13 Z. Huang / Journal of Algebra 285 (2005) when Λ U Γ = Λ Λ Λ.WeuseI 0 and I 0 to denote the injective envelope of ΛΛ and Λ Λ, respectively. Consider the following example. Let K be a field and the quiver: α β 2 γ 3. () If Λ = K /(αβα). Then l.fd Λ (I 0 ) = and r.fd Λ (I 0 ) 2. (2) If Λ = K /(γα,βα). Then l.fd Λ (I 0 ) = 2 and r.fd Λ (I 0 ) =. Acknowledgments The author thanks Professor Kent R. Fuller and the referee for their helpful comments. The research of the author was partially supported by National Natural Science Foundation of China (Grant No ) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant No ). References [] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, second ed., Grad. Texts in Math., vol. 3, Springer-Verlag, Berlin Heidelberg New York, 992. [2] M. Auslander, R.O. Buchweitz, The homological theory of maximal Cohen Macaulay approximations, Soc. Math. France 38 (989) [3] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 956. [4] S.U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (960) [5] R.R. Colby, K.R. Fuller, Exactness of the double dual, Proc. Amer. Math. Soc. 82 (98) [6] M. Hoshino, On dominant dimension of Noetherian rings, Osaka J. Math. 26 (989) [7] M. Hoshino, On Lambek torsion theories, Osaka J. Math. 29 (992) [8] Z.Y. Huang, G.H. Tang, Self-orthogonal modules over coherent rings, J. Pure Appl. Algebra 6 (200) [9] J. Lambek, Torsion Theories, Additive Semantics, and Rings of Quotients (with an Appendix by H.H. Storrer on Torsion Theories and Dominant Dimension), Lecture Notes in Math., vol. 77, Springer-Verlag, Berlin Heidelberg New York, 97. [0] T. Kato, Rings of U-dominant dimension, Tôhoku Math. J. 2 (969) [] B. Stenström, Rings of Quotients, Grundlehren Math. Wiss. Einz., vol. 27, Springer-Verlag, Berlin Heidelberg New York, 975. [2] H. Tachikawa, Quasi-Frobenius Rings and Generalizations (QF-3 and QF- Rings), Lecture Notes in Math., vol. 35, Springer-Verlag, Berlin Heidelberg New York, 973. [3] R. Wisbauer, Decomposition properties in module categories, Acta Univ. Carolin. Math. Phys. 26 (985)