Natural Dualities

NATURAL DUALITIES FRANCESCA MANTESE, ALBERTO TONOLO Abstract. Let S be an arbitrary associative ring and S W be a left S-module. Denote by R the ring End S W and by ∆ both the contravariant functors HomS (−, W ) and HomR (−, W ). A module M is reflexive if the evaluation map δM : M → ∆2 M is an isomorphism. Any direct summand of finite direct sums of copies of S W and of RR is reflexive. Increasing in a minimal way the classes of reflexive modules, a “cotilting condition” on finitely generated R-modules naturally arises.

Introduction Tilting theory was introduced in the context of finitely generated modules over artin algebras by Brenner and Butler [5] and Happel and Ringel [15]. The dual of a tilting module with respect to the usual artin algebra duality is called a cotilting module. This notion, step by step, has been extended to infinitely generated modules over arbitrary rings, gaining a proper independent role. Many papers are involved in this generalization process, and different cotilting-type notions are considered (see [2, 6, 7, 10, 11, 12, 13, 17, 19, 20]). Notation. Let WR be a right R-module. We introduce the following classes Cogen WR = {M ∈ Mod-R | M ,→ WRX , for some set X}; ⊥ W = {N ∈ Mod-R | Ext1 (N, W ) = 0}; R R add WR = {M ∈ Mod-R | M is a summand of a finite direct sum of copies of WR }; The injective cogenerators have a key role in the theory of Morita dualities. A right R-module C is an injective cogenerator if Mod-R = Cogen C ⊆ ⊥ C. All cotilting-type notions, considered in the above quoted papers, rotate around natural generalizations of this concept. For all W among them, it is (\)

mod-R ∩ Cogen W ⊆ ⊥ W ;

for many W among them (\\)

mod-R ∩ Cogen W = mod-R ∩ ⊥ W.

In the first section we show how much naturally the natural conditions (\) and (\\) come out. They are characterized (see Theorems 1.4, 1.7) through extremely weak closure properties of reflexive modules. In the second section, we begin studying the relationship between the (\\) condition and the costar modules, introduced by Colby and Fuller (see [8]). The two notions result to be deeply connected: in particular, they coincide for modules which are faithful, finitely generated and finendo. Observe that Colpi in [9, Theorem 3] proved dually that tilting modules and ∗-modules coincide for modules which are faithful, finitely generated and finendo. Next we finish proving as, in the original setting of finitely generated modules over artin algebras, cotilting modules are characterized by the (\\) condition. 0

Research partially supported by GNSAGA - Istituto Nazionale di Alta Matematica , Italy. 1

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FRANCESCA MANTESE, ALBERTO TONOLO

1. Characterizing the (\) and the (\\) conditions Let S W be a left S-module. We denote by R the ring (End S W )op and by ∆ both the contravariant functors HomS (−, W ) and HomR (−, W ). By ∆2 we mean both the compositions of the two functors. Denoted by δ : 1 → ∆2 the evaluation map, we say that a module M is torsionless (resp. reflexive) if δM is a monomorphism (resp. an isomorphism). Clearly a module is torsionless if and only if it is cogenerated by W . We denote by ReflR and ReflS the classes of reflexive R- and S- modules. All left S-modules in add S W and all right R-modules in add RR are reflexive. We are interested in understanding what happens increasing in a “minimal way” the class of reflexive left S-modules. The following two lemmas proved by Colby and Fuller will be often useful in the sequel. f

Lemma 1.1 (Lemma 2.2, [8]). Suppose 0 → M → X → L → 0 is exact where X is reflexive and ∆(f ) is an epimorphism. Then M is reflexive if and only if L is torsionless. f

Lemma 1.2 (Lemma 2.5, [8]). Suppose 0 → M → X → L → 0 is exact. Denote the image of ∆(f ) by I. If X is reflexive and L is torsionless, then ∆(f ) is epic if and only if I is reflexive. tl

We denote by add S W the closure of add S W under torsionless homomorphic images and by tl add RR the closure of add RR under submodules with torsionless quotients. In tl general the class add S W is not contained in the class of reflexive left S-modules. Example 1.3. Let S denote the K-algebra given by the quiver 1 → 2 → 3, S W the 2 1 tl module ⊕ and R = End S W . The simple left S-module 2 belongs to add S W , but 3 2 1 it is not reflexive, since ∆R ∆S (2) = . 2 tl

Theorem 1.4. The class add S W is contained in ReflS if and only if WR satisfies (\). tl In this case there is a duality between the classes add S W and tl add RR . Proof. (⇒) Let MR be a module in Cogen WR ∩ mod-R. Consider the exact sequence f

0 → L → Rn → M → 0. Applying ∆ we obtain the exact sequence 0 → ∆M → S W n → I → 0 where I is the homomorphic image of ∆(f ). Since I is contained in ∆L, it belongs to tl add S W and hence it is reflexive. Then, by Lemma 1.2, ∆(f ) is an epimorphism and hence Ext1R (M, W ) = 0. tl (⇐) Let S M be in add S W . Consider the exact sequence f

0 → L → S W n → M → 0. Applying ∆ we obtain the exact sequence 0 → ∆M → ∆W n ∼ = Rn → I → 0 where I embeds in ∆L. Since by hypothesis Ext1R (I, W ) = 0, applying again ∆ we have the following commutative diagram with exact rows: 0

/L

/ Wn ∼ = δW n

0

/ ∆I



/ ∆2 W n

/ M _ 

/0

δM

/ ∆2 M

/0

NATURAL DUALITIES

3

It follows easily that δM is an epimorphism and so M is reflexive. tl Clearly for each module M in add S W , ∆M belongs to tl add RR . Let N in tl add RR ; then there exists an exact sequence f

0 → N → Rn → C → 0 with C cogenerated by WR . Since WR satisfies condition (\), Ext1R (C, W ) = 0 and hence tl ∆N belongs to add S W . Moreover by Lemma 1.1, since ∆(f ) is an epimorphism, the module N is reflexive. It is easy to verify that in Example 1.3, the right R-module WR is finitely generated but it does not belong to ⊥ WR . A module UR is said to be quasi cotilting (see [17]) if Cogen U ⊆ ⊥ U . In the above theorem we have characterized the left S-modules which satisfy the quasi cotilting condition on finitely generated right End S W -modules. tl We denote by ∆← (add S W ) the class of all torsionless right R-modules M such that tl ∆M belongs to add S W . Assume WR satisfies condition (\). From Theorem 1.4 we have tl add R ⊆ ∆← (add W tl ). We are now interested in understanding when the duality R S tl −−−→tl ∆ : add S W − ←−−−− add RR : ∆ tl

involves the as large as possible class of right R-modules, i.e. when the classes ∆← (add S W ) and tl add RR coincide. In fact, generally, the class tl add RR is properly contained in tl ∆← (add S W ). a

b

Example 1.5. Let S denote the K-algebra given by the quiver 1 → 2 → 3, with relation 2 1 ⊕2⊕ and R = End S W . The ring R is isomorphic ba = 0, S W the module 3 ⊕ 3 2 c d e to the K-algebra given by the quiver 4 → 5 → 6 → 7, with relation ed = 0, and WR ∼ = 6 7 ⊕ 5 ⊕ 4. It is easy to verify that Cogen WR ∩ mod-R ( ⊥ WR ∩ mod-R and hence 6 4 condition (\) is satisfied. The simple right R-module 6 is, of course, finitely generated tl and cogenerated by WR . The left S-module 3 = ∆(6) belongs to add S W , but 6 is not 7 tl reflexive, since ∆2 (6) = . Therefore 6 belongs to ∆← (add S W ) but not to tl add RR . 6 tl

Lemma 1.6. A right R-module M belongs to ∆← (add S W ) if and only if there is a monomorphism f : M → Rn such that ∆(f ) is an epimorphism. Proof. (⇒) Let M be a torsionless right R-module such that there exists the following exact sequence of left S-modules π

0 → L → S W n → ∆M → 0. Applying ∆ we get the injective map ∆(π) : ∆2 M → Rn . Let f : M → Rn be the monomorphism ∆(π) ◦ δM ; since ∆(f ) ◦ δW n = ∆(δM ) ◦ ∆2 (π) ◦ δW n = ∆(δM ) ◦ δ∆M ◦ π = π, ∆(f ) is an epimorphism. f

(⇐) Applying ∆ to the exact sequence 0 → M → Rn → C → 0 we get 0 → ∆C → SW

) n ∆(f →

tl

∆M → 0. Since M is torsionless, M belongs to ∆← (add S W ).

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FRANCESCA MANTESE, ALBERTO TONOLO

Theorem 1.7. The following statements are equivalent: tl

∆

(1) ∆← (add S W ) − ←− −− −− −→ − add S W

tl

is a duality;

∆

f

(2) for any exact sequence 0 → M → Rn → L → 0 in Mod-R, the module L is torsionless if and only if ∆(f ) is an epimorphism. (3) WR satisfies (\\). tl (4) WR satisfies (\) and tl add RR = ∆← (add S W ). tl (5) WR satisfies (\) and ∆← (add S W ) ⊆ ReflR . Proof. (1) ⇒ (3) By Theorem 1.4 it is enough to prove that if the sequence f

0 → L → Rn → M → 0 is exact and Ext1R (M, W ) = 0, then M is cogenerated by WR . Applying the functor ∆, we obtain the exact sequence ∆(f )

0 → ∆M →S W n → ∆L → 0. By (1) L is reflexive, so by Lemma 1.1 M is torsionless. (2) ⇔ (3) It is easy. tl (2) ⇒ (1) By Lemma 1.6 a right R-module M belongs to ∆← (add S W ) if and only if there is an exact sequence 0 → M → Rn → L → 0 tl

with L in Cogen WR . Thus, given M in ∆← (add S W ), applying Lemma 1.1 to the exact sequence 0 → M → Rn → L → 0 with L in Cogen WR , we get that M is reflexive. Now, by Theorem 1.4, each module tl tl N in add S W is reflexive. Clearly, if M belongs to ∆← (add S W ), then ∆M belongs tl tl to add S W . Conversely, let N ∈ add S W . Since N ∼ = ∆2 N = ∆∆N , ∆N belongs to tl ∆← (add S W ). (1) ⇔ (4) and (4) ⇒ (5) They follow from Theorem 1.4. (5) ⇒ (4) Since WR satisfies the (\) condition, by Theorem 1.4 we have tl add RR ⊆ tl tl ∆← (add S W ). Conversely, let M belong to ∆← (add S W ). Again by Theorem 1.4, M∼ = ∆2 M belongs to tl add RR . A module UR is said to be cotilting (see [11]) if Cogen U = ⊥ U . In the above theorem we have characterized the left S-modules which satisfy the cotilting condition on finitely generated right End S W -modules. Let us see that in condition (2) of Theorem 1.7 the module Rn can be replaced by any tl module in ∆← (add S W ). Proposition 1.8. The following statements are equivalent (1) WR satisfies (\\); f

tl

(2) For any exact sequence 0 → M → X → L → 0 in Mod-R with X ∈ ∆← (add S W ), the module L is torsionless if and only if ∆(f ) is an epimorphism. tl

If these conditions are satisfied, the class ∆← (add S W ) is closed with respect to submodules with torsionless cokernel. f

Proof. (1) ⇒ (2) Assume 0 → M → X → L → 0 is exact in Mod-R with X ∈ tl ← ∆ (add S W ). Let ∆(f ) be an epimorphism. Clearly ∆X and hence ∆M belong to

NATURAL DUALITIES

5

tl

add S W . Then M is reflexive and, by Lemma 1.1, L is cogenerated by WR . Coversely, let L be cogenerated by WR . Since condition (\\) holds, by Lemma 1.6 we can construct the following commutative diagram with exact rows and columns

0

/M

f

0

0

 /X

 /L

i



/0



Rn _ _ _ / W B 

C 

0 where C is cogenerated by WR . The dashed arrow is obtained from Hom(i, W B ) : HomR (Rn , W B ) → HomR (X, W B ) which is an epimorphism by Theorem 1.7 (2). Then we have an exact sequence i◦f

0 → M → Rn → C × W B . tl

Since condition (\\) holds, by Lemma 1.6 we get that M belongs to ∆← (add S W ). By Proposition 1.8 (2), ∆(i ◦ f ) = ∆(f ) ◦ ∆(i) is epic and hence ∆(f ) is an epimorphism too. (1) ⇐ (2) It follows by Theorem 1.7 (2) ⇔ (3). f

tl

Consider an exact sequence 0 → L → M → N → 0 with M ∈ ∆← (add S W ) and N torsionless. By condition (2) the map ∆(f ) is an epimorphism. Therefore L belongs to tl ∆← (add S W ). Corollary 1.9. If WR satisfies the (\\) condition, then the functor ∆ preserves the extl tl actness of the short exact sequences in add S W and ∆← (add S W ). g

tl

Proof. Let 0 → L → M → N → 0 be an exact sequence in add S W . Then we obtain tl the exact sequence 0 → ∆N → ∆M → I → 0, where ∆M ∈ ∆← (add S W ) and I is torsionless since I ,→ ∆(L). Thus by Proposition 1.8 we can construct the commutative diagram with exact rows 0

0

/L

/M





/ ∆I

δM

/ ∆2 M

/N 

/0

δN

/ ∆2 N

/0

tl Since ∆I ∼ = L, then I ∈ ∆← (add S W ) is reflexive. From Lemma 1.2 it follows that ∆(g) is tl epic. If the short exact sequence belongs to ∆← (add S W ), the proof follows immediately from Proposition 1.8.

Denote by B the ring BiEnd S W of biendomorphisms of S W . While the classes add S W tl tl and add B W coincide, in general we have only the inclusion add B W ⊆ add S W . Proposition 1.10. Let S W be a left S-module, R = End S W , B = BiEnd S W . If WR tl tl satisfies the (\) condition, then add B W ∼ = add S W .

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FRANCESCA MANTESE, ALBERTO TONOLO

Proof. Since R = End B W , by Theorem 1.4 the classes add B W both dual to tl add RR .

tl

and add S W

tl

are

The above proposition says that, under the (\) condition, the classes involved in the duality are the same if we change to the faithfully balanced case. 2. The (\\) condition versus costar and cotilting modules Let WR be a right module over an arbitrary ring R. The module WR is costar ([8, f

Theorem 2.7] if for any exact sequence 0 → L → W n → M → 0 in Mod-R, ∆(f ) is epic if and only if M is cogenerated by WR . In this section, we compare on WR the (\\) condition and the costar and cotilting properties. Proposition 2.1. Let WR ∈ Mod-R be a finitely generated module satisfying (\\) condition. Then WR is costar. f

Proof. Let 0 → L → WRn → M → 0 be an exact sequence. Since WR is finitely generated and it satisfies (\\), WRn belongs to Cogen WR ∩ mod-R = ⊥ WR ∩ mod-R. Therefore ∆(f ) is epi if and only if Ext1R (M, W ) = 0; since M is finitely generated this is equivalent to say that M belongs to Cogen WR . Proposition 2.2. Let WR ∈ Mod-R be a finendo and faithful module. If WR is costar, then WR satisfies (\\). f

Proof. Let 0 → L → Rn → M → 0 be an exact sequence in Mod-R. We have to prove that M is cogenerated by WR if and only if M belongs to ⊥ WR . The latter is equivalent to say that ∆(f ) is epi. Since WR is faithful, R is cogenerated by WR . Since WR is finendo, then HomR (Rn , End(WR ) WR ) = End(WR ) W n is finitely generated. Now [8, Proposition 2.9] applies, proving the (\\) condition. Corollary 2.3. Let WR be a faithful, finitely generated and finendo module. Then WR satisfies the (\\) condition if and only if WR is costar. In [9, Theorem 3, (a) ⇔ (d)] Colpi proved that a finitely generated, faithful and finendo module is tilting if and only if it is a ∗-module. Then Corollary 2.3 can be regarded as the dual version of this result. The (\\) condition arose in the previous section as consequence of weak closure properties of the classes of reflexive modules. Nevertheless, in the original setting of finitely generated modules over artin algebras, it is sufficient to completely characterize the cotilting modules. This follows by the main result of a paper of Smalø (see [18, Theorem, (ii ⇔ iii)]), since if the (\\) condition holds, the pair (Ker ∆ ∩ mod-R, ⊥ W ∩ mod-R) is a torsion theory on finitely generated R-modules. Anyway, we prefer to give a direct proof of it. Theorem 2.4. Let R be an artin algebra over k and WR a finitely generated R-module. Then WR is cotilting if and only if it satisfies the (\\) condition. Proof. The necessity of the (\\) condition is clear. Let us prove its sufficiency. By [13] and [3], denoted by Q an injective cogenerator of Mod-R, we have to prove that (i) id WR ≤ 1. (ii) Ext1R (W α , W ) = 0 for each cardinal α. (iii) There exists an exact sequence 0 → W1 → W2 → Q → 0 with W1 , W2 direct summands of arbitrary direct products of copies of W .

NATURAL DUALITIES

7

(i) By the Baer’s criterion it is sufficient to prove that Ext2R (M, W ) = 0 for any finitely generated module M . Let 0 → K → Rn → M → 0 be an exact sequence. By the (\\) condition, Rn is cogenerated by WR . Since M is finitely presented, K belongs to Cogen WR ∩ mod-R and hence Ext1R (K, W ) = 0. Therefore Ext2R (M, W ) = 0. (ii) Since Ext1R (W, W ) = 0, we can conclude proving that WR is product complete, i.e. every product of copies of W is a direct summand of a direct sum of copies of W . As it is wellknown (see [4, Proposition II.1.1]) End(WR ) is an artin algebra over k and W is a finitely generated left End(WR )-module. Therefore we conclude by [16, Proposition 3.9]. (iii) Since R is artinian, every finitely generated module M in Cogen WR is finitely cogenerated by WR (see [1, Corollary 10.3]). Moreover there exists an exact sequence 0 → M → Wn → C → 0 with C ∈ Cogen WR ∩ mod-R. In fact, since by (ii) the ring End WR is noetherian and End WR W is finitely generated, HomR (M, W ) is a finitely generated End WR -module. Let f1 , . . . , ft generate HomR (M, W ). Consider the monomorphism η : M → W t , η(m) = (f1 (m), . . . , ft (m)); its cokernel C is clearly finitely generated. By construction ∆(η) is epic and then Ext1R (C, W ) = 0. Therefore C is cogenerated by WR . Let Q be a finitely generated injective cogenerator of Mod-R. Let ϕ : Rn → Q be an epimorphism. Since Rn is clearly projective and finitely generated, then it is finitely cogenerated by WR . Therefore ϕ extends to an epimorphism f : W m → Q with kernel K. For what we have observed above, there exists an exact sequence 0 → K → Wr → C → 0 with C in Cogen WR ∩ mod-R. Consider the following pushout diagram with exact rows and columns 0 0

0

 /K

 / Wm

0

 / Wr

 /E

0

 /C

C



0

f

/Q

/0

/Q

/0

 

0

The module E is finitely generated and Ext1R (E, W ) = 0. Therefore E is finitely cogenerated by W . Then there exists an exact sequence (∗)

0 → E → Ws → Y → 0

with Y in Cogen WR ∩ mod-R. Since Ext1R (Y, W r ) = 0 = Ext1R (Y, Q), it follows that Ext1R (Y, E) = 0. Thus the exact sequence (∗) splits and the second row of the above diagram satisfies condition (iii). Acknowledgement We wish to thank our colleagues and friends Robert Colby, Gabriella D’Este and Kent Fuller, for for the useful discussions and suggestions. We thank also the Department of Mathematics of the University of Iowa for its hospitality during the preparation of this note.

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