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Concerning commuting properties of the derived functors Ext∗ and Tor∗, it was proved by Brown in [10] that Lenzing’s results can be extended to produce more finiteness conditions. In order to state this results, let us recall from [17, p. 103] that a right R module M is F Pn, for a fixed n ≥ 0, if M has a projective which is finitely generated in every dimension ≤ n.

Theorem 1.1. [10, Theorem 1], [22, Theorem A] The following are equivalent for a right R module M and a non-negative integer n:

(1) M is F Pn+1; k (2) ExtR(M, −) commutes with direct limits for all 0 ≤ k ≤ n, R (3) Tork (M, −) commutes with direct products for all 0 ≤ k ≤ n.

The proof of this theorem is based on Lenzing’s characterizations of finitely presented modules via commuting properties of covariant Hom-functors with direct limits, respectively commuting properties of tensor functors with direct products, and these are applied to obtain independently (1)⇔(2) and (1)⇔(3). For the case of Ext-functors, this theorem was refined and completed by Strebel in [22]. We can ask if there is a cyclic proof. A connection is suggested in [22, Corollary 1], where it is proved that a right R-module M of projective dimension at most n has n the property that ExtR(M, −) commutes with direct limits if and only if M has a projective resolution which is finitely generated in dimension n. In the present paper we give such a proof in Theorem 2.4, where modules of projective dimension at most 1 such that the induced covariant Ext1-functor com- mutes with direct limits are characterized via Lenzing’s characterization of finitely presented modules by commuting properties of the tensor product. This theorem is used to obtain some results from Strebel’s paper, [22]. Similar techniques were used by Krause in [18] in order to characterize general coherent functors. In the end of this introduction let us observe that there are commuting properties which are non-trivial for the Hom-functors but they are trivial for Ext-functors. For instance, it is well known that the covariant Hom-functors commute with inverse limits. However, as in [17, Example 3.1.8], we can use [6, Theorem 2] to write every module as an inverse limit of injective modules. Therefore

1 Proposition 1.2. Let M be a module. The functor ExtR(M, −) commutes with inverse limits if and only if M is projective.

Dualizing the definition of small modules we obtain the notion of slender mod- ule. The structure of these modules can be very complicated (see [15] or [17]). Transferring this approach to the contravariant Ext-functor, we say that the co- n travariant functor ExtR(−,M) inverts products if for every countable family F = n (Mi)i<ω of right R-modules the canonical homomorphism i<ω ExtR(Mi,M) → n L ExtR(Qi<ω Mi,M) is an isomorphism. However, in the case when M has the in- jective dimension at most n this property is very restrictive.

Proposition 1.3. Let M be a right R-module of injective dimension at most n ≥ 1. n The functor ExtR(−,M) inverts products with countable many factors if and only if M is of injective dimension at most n − 1.

Proof. By the dimension shifting formula, it is enough to assume n = 1. 1 MODULES M SUCH THAT ExtR(M, −) COMMUTES WITH DIRECT LIMITS 3

In this hypothesis, we start with a module N and with the canonical monomor- phism 0 → N (ω) → N ω. Applying the contravariant Ext-functor and [4, Proposi- tion 2.4] we obtain the natural homomorphism 1 (ω) ∼ 1 ω 1 (ω) ∼ 1 ω ExtR(N,M) = ExtR(N ,M) → ExtR(N ,M) = ExtR(N,M) , which coincides with the canonical homomorphism 1 (ω) 1 ω ExtR(N,M) → ExtR(N,M) , 1  and it is an epimorphism. This is possible only if ExtR(N,M) = 0. We do not know what happens in Proposition 1.3 in case we do not assume any bound on the injective dimension. It is also an open question when the contravariant Ext1-functor preserves products. In [16] the authors provide an answer for abelian groups, and we refer to [8] for the case of contravariant Hom-functors. Other commuting properties of these functors, for the case of abelian groups, are studied in [3] and [21].

2. When Ext commutes with direct limits

Let R be a unital ring and M a right R-module. If F = (Mi,υij )i,j∈I is a direct system of modules and υ : M → limM are the canonical homomorphisms, then i i −→ i for every non-negative integer n there is a canonical homomorphism Φn,M : lim Extn (M,M ) → Extn (M, limM ), F −→ R i R −→ i n the natural homomorphism induced by the family ExtR(M,υij ), i, j ∈ I. For the case of direct sums, if we consider F = (Mi)i∈I a family of modules and we denote by ui : Mi → Li∈I Mi the canonical homomorphisms then we will obtain a natural homomorphism n,M n n ΦF : M ExtR(M,Mi) → ExtR(M, M Mi), i∈I i∈I n induced by the family ExtR(M,ui), i ∈ I. We will use these homomorphisms for the cases n ∈ {0, 1}, and we will denote 0,M M 1,M M 0,M M 1,M M ΦF := ΨF , ΦF =ΦF , respectively ΦF := ΨF , ΦF =ΦF . n We say that ExtR(M, −) commutes with direct limits (direct sums) if the ho- n,M n,M momorphisms ΦF (respectively ΦF ) are for all directed systems F (respectively all families F). M Lenzing’s theorem says us that M is finitely presented if and only if ΨF are isomorphisms for all F. Moreover, using Theorem 1.1 for n = 1, we observe that M M M is an F P2 module if and only if ΨF and ΦF are isomorphisms for all families F. M We will start with a slight improvement of this result, replacing the condition “ΨF is an ” (i.e. M is finitely presented) by the condition “M is finitely generated”. Therefore in the case of finitely generated modules the hypothesis of [17, Lemma 3.1.6] is sharp. Following the terminology used in [7], we will call a module M an fp-Ω1-module (fg-Ω1-module, respectively small-Ω1-module) if there is a projective resolution

α1 (P): ···→ P2 → P1 → P0 → M → 0 1 such that the first syzygy Ω (P) = Im(α1) is finitely presented (finitely generated, respectively small). 4 SIMION BREAZ

Lemma 2.1. If M is fp-Ω1 (fg-Ω1, respectively small-Ω1) right R-modules then there is a L such that M ⊕ L is a direct sum of an F P2-module (finitely presented module, respectively finitely generated module) and a projective module. α β Proof. Let 0 → K → P → M → 0 be an such that P is projective and K is finitely presented (finitely generated, respectively small). If C is a pro- jective module such that P ⊕ C =∼ R(I) is free, then we consider the induced exact α β⊕1C sequence 0 → K → P ⊕ C → M ⊕ C → 0. Since K is small (recall that every finitely generated module is small) as a right R-module there is a finite subset J of I such that Im(α) ⊆ R(J). Then M ⊕ C =∼ R(J)/Im(α) ⊕ R(I\J) = H ⊕ L, where H is an F P2-module (finitely presented, respectively finitely generated) and L is projective.  The closure under direct summands of the class of fg-Ω1-modules can be charac- terized by the commuting property of the covariant Ext1-functor with direct unions (i.e. direct system of monomorphisms). We state this result for sake of complete- ness. Theorem 2.2. [7, Theorem 5] The following are equivalent for a right R-module M: (1) M is a direct summand of an fg-Ω1-module; 1 (2) The functor ExtR(M, −) commutes with direct systems of monomorphisms. For the cases of fp-Ω1-modules and small-Ω1-modules we are able to prove a similar result only in the case of finitely generated modules. In fact, it is easy to observe, using [17, Lemma 3.1.6] that the covariant Ext1-functor induced by an fp- Ω1-modules commutes with direct limits. The converse is true for finitely generated modules, [7, Lemma 1]. A similar result is valid for small-Ω1-modules. Theorem 2.3. Let M be a right R-module M. 1 1 (1) If M is an fp-Ω -module then ExtR(M, −) commutes with direct limits. 1 1 (2) If M is a small-Ω -module then ExtR(M, −) commutes with direct sums. The converses of these statements are true if M is finitely generated. Proof. We will prove (2). The proof for (1) follows the same steps. Using Lemma 2.1, we can suppose that M is finitely generated. If 0 → K → P → M → 0 is an exact sequence with P a projective module then for every family F = (Mi)i∈I of right R-modules, we have the following useful

0 −−−−→ ⊕i∈I HomR(M,Mi) −−−−→ ⊕i∈I HomR(P,Mi) −−−−→

M P ΨF ΨF   y y 0 −−−−→ HomR(M, ⊕i∈I Mi) −−−−→ ⊕i∈I HomR(P,Mi) −−−−→ (♯) 1 −−−−→ ⊕i∈I HomR(K,Mi) −−−−→ ⊕i∈I ExtR(M,Mi) −−−−→ 0

K M ΨF ΦF   y 1 y −−−−→ HomR(K, ⊕i∈I Mi) −−−−→ ExtR(M, ⊕i∈I Mi) −−−−→ 0 whose rows are exact. 1 MODULES M SUCH THAT ExtR(M, −) COMMUTES WITH DIRECT LIMITS 5

M P Since M and P are finitely generated, the arrows ΨF and ΨF are isomorphisms K by [29, 24.10]. Using we observe that ΨF is an isomorphism if and M  only if ΦF is an isomorphism.

We are ready to state the main result of this paper. It states that fp-Ω1-modules of projective dimension at most 1 can be characterized by commuting properties of the induced covariant Ext1-functor. Theorem 2.4. The following are equivalent for a right R-module M of projective dimension 1: (1) M is an fp-Ω1-module. (2) There is a projective module L such that M ⊕ L is a direct sum of an F P2-module (F P1-module) and a projective module. 1 (3) ExtR(M, −) commutes with direct limits. 1 (4) ExtR(M, −) commutes with direct sums of copies of R. Proof. We only need to prove (4) ⇒ (1). 1 Let M be a module of projective dimension at most 1 such that ExtR(M, −) 1 commutes with direct sums of copies of R. Therefore ExtR(M, −) is a right which commutes with direct sums of copies of R. By Watts’s theorem [28, 1 Theorem 1] (and its proof) we conclude that the functor ExtR(M, −) is naturally 1 equivalent to the functor −⊗R ExtR(M, R). It follows that the tensor product func- 1 1 tor −⊗R ExtR(M, R) preserves the products, hence that ExtR(M, R) is a finitely presented left R-module. 1 Let n be a positive integer such that the left R-module ExtR(M, R) is generated by n elements. Using [23, Lemma 6.9] we conclude that there is an exact sequence n 1 of right modules 0 → R → C → M → 0 such that ExtR(C, R) = 0. Starting with this exact sequence we obtain for every cardinal κ a commutative diagram

n (κ) 1 (κ) 1 (κ) Hom(R , R) −−−−→ ExtR(M, R) −−−−→ ExtR(C, R) −−−−→ 0    ,    y y y n (κ) 1 (κ) 1 (κ) Hom(R , R ) −−−−→ ExtR(M, R ) −−−−→ ExtR(C, R ) −−−−→ 0 where the vertical arrows are the canonical homomorphisms induced by the uni- versal property of direct sums. Moreover the first vertical is an isomorphism since every finitely generated module is small, while the second is also an isomorphism by our hypothesis. Therefore the third vertical arrow is also an isomorphism. It fol- 1 (κ) lows that ExtR(C, R ) = 0 for all cardinals κ. Since C is of projective dimension at most 1, it is not hard to see that C is projective. 

Corollary 2.5. Let R be an right hereditary ring and M a right R-module. The following are equivalent: 1 (1) ExtR(M, −) commutes with direct limits; 1 (2) ExtR(M, −) commutes with direct sums; 1 (3) ExtR(M, −) commutes with direct sums of copies of R; (4) M = N ⊕ P , where N is finitely presented and P is projective. Proof. Since every right hereditary ring is right coherent, every finitely presented module is an F P2-module. Therefore only (3) ⇒ (4) requires a proof. 6 SIMION BREAZ

1 Suppose that ExtR(M, −) commutes with direct sums of copies of R. Using Theorem 2.4, we observe that there is a projective module L such that M ⊕ L = F ⊕ U with F finitely presented and U projective. The conclusion follows using the same techniques as in [2]. If πU : F ⊕ U → U is the canonical projection, then πU (M) is projective. Then F ∩ M = Ker(πU|M ) is a direct summand of M. Therefore N = F ∩ M is a direct summand of F ⊕ U = M ⊕ L. Using this, it is not hard to see that N = F ∩ M is a direct summand of F . Hence N is finitely presented and M = N ⊕ P , where P =∼ πU (M) is projective.  Example 2.6. The equivalence (3) ⇔ (4) from Theorem 2.4 is not valid for general finitely generated modules. To see this, it is enough to consider a ring R which has a non-finitely generated small right ideal I. For instance, it is proved in [25] that every of infinitely many rings has such an ideal. Since R/I is a finitely generated right R- 1 module, we can apply Theorem 2.3 to see that ExtR(R/I, −) commutes with direct sums, but it does not commute with direct limits. Remark 2.7. Corollary 2.5 was proved for abelian groups in [9] without the con- 1 ∼ 1 dition that the isomorphisms Li∈I ExtR(M,Mi) = ExtR(M, Li∈I Mi) are the natural ones. Corollary 2.8. [22, Corollary 1] Let M be right R-module of finite projective di- mension ≤ n. Then M admits a projective resolution which is finitely generated n in dimension n if and only if ExtR(M, −) preserves direct sums of copies of R. n If one of the equivalent conditions is fulfilled, then the functors ExtR(M, −) and n −⊗R ExtR(M, R) are naturally isomorphic. Proof. Let (P) be a projective resolution of M. Then the (n−1)-th syzygy Ωn−1(P) is of projective dimension at most 1. Using the dimension shifting formula we n 1 n−1 observe that ExtR(M, −) is naturally isomorphic to ExtR(Ω (P), −), and the conclusion follows from Theorem 2.4 and Watts’s theorem.  Lemma 2.9. Let L be a right R-module of projective dimension at most 1 and K a finitely presented submodule of L. The module M = L/K has the property 1 1 that ExtR(M, −) preserves direct sums of copies of R if and only if M is an fp-Ω - module. Proof. Let I be a set. It induces a commutative diagram (I) 1 (I) 1 (I) HomR(K, R) −−−−→ ExtR(M, R) −−−−→ ExtR(L, R) −−−−→

ΨK ΦM ΦL    y y y (I) 1 (I) 1 (I) HomR(K, R ) −−−−→ ExtR(M, R ) −−−−→ ExtR(L, R ) −−−−→

1 (I) 2 (I) −−−−→ ExtR(K, R) −−−−→ ExtR(M, R) −−−−→ 0

ΦK    y y 1 (I) 2 (I) −−−−→ ExtR(K, R ) −−−−→ ExtR(M, R ) −−−−→ 0 where all arrows represent the canonical homomorphisms. Since K is finitely pre- sented, the arrows ΨK and ΦK are isomorphisms (for ΦK we use Theorem 2.3). Moreover, ΦM is an isomorphism from our hypothesis and it is obvious that the last 1 vertical arrow is also a monomorphism by [22, Lemma 2.2]. Therefore ExtR(L, −) 1 MODULES M SUCH THAT ExtR(M, −) COMMUTES WITH DIRECT LIMITS 7 commutes with direct sums of copies of R. By Theorem 2.4, it follows that there is a projective resolution 0 → Rn → P → L → 0. Using all these data we construct a pullback diagram

0 0

    y y 0 −−−−→ Rn −−−−→ U −−−−→ K −−−−→ 0

    y y 0 −−−−→ Rn −−−−→ P −−−−→ L −−−−→ 0.     y y M M     y y 0 0

Since U is finitely presented, the proof is complete. 

We also obtain the principal ingredient for the proof of [22, Theorem B].

1 Corollary 2.10. [22, Lemma 2.6] Let M be a right R-module such that ExtR(M, −) commutes with direct sums of copies of R. Suppose that there is a projective reso- lution

α1 α0 (P) P2 → P1 → P0 → M → 0 such that Ω2(P) is finitely generated. Then there is a projective resolution

′ ′ ′ (P ) P2 → P1 → P0 → M → 0

′ 2 2 ′ with P1 finitely generated such that Ω (P)=Ω (P ).

(I) Proof. We can suppose that there is a set I such that P1 =∼ R . If J is a finite subset of I such that Ω2(P) ⊆ R(J) and K = R(J)/Ω2(P) then we have an exact sequence

(I\J) α 0 → K ⊕ R → P0 → M → 0,

′ (I\J) where α is induced by α1. If M = P0/α(R ) then we have an exact sequence 0 → K → M ′ → M → 0. Observe that M ′ is of projective dimension at most 1. By Lemma 2.9, the module M is an fp-Ω1-module. Therefore there is an exact sequence 0 → U → Q → M → 0 with Q projective and U finitely presented, and 8 SIMION BREAZ we can construct a commutative diagram with exact rows 0 0 0

      y y y 0 −−−−→ Ω2(P) −−−−→ V −−−−→ U −−−−→ 0

      y y y 0 −−−−→ P1 −−−−→ W −−−−→ Q −−−−→ 0       y y y 1 0 −−−−→ Ω (P) −−−−→ P0 −−−−→ M −−−−→ 0       y y y 0 0 0 Since the middle vertical row is splitting, V is a finitely generated projective module. Moreover we can construct an exact sequence 0 → Ω2(P) → V → Q → M → 0, and the proof is complete.  Acknowldgements: I thank Jan Trlifaj for drawing my attention on Strebel’s work [22]. I also thank to the Referee, whose comments improved this paper.

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Babes¸-Bolyai University, Faculty of Mathematics and Computer Science, Str. Mihail Kogalniceanu˘ 1, 400084 Cluj-Napoca, Romania E-mail address: [email protected]