On cotilting cotorsion and torsion pairs

COTORSION P AIRS, TORSION P AIRS, AND Σ -PURE INJECTIVE COTIL TING MODULES RICCARDO COLPI, FRANCESCA MANTESE, ALBER TO TONOLO to the memory of our frie nd and c ol le ague Silvia L ucido Abstract. In this paper we stu dy cotorsion and torsion pair s induced by cotilting modules. W e prov e the existence of a strong relationship bet we en the Σ-pure i njectivit y of the cotilting module and the property of the in- duced cotorsion pair to b e of finite type. In particular for cotilting mo dules of injective dimension at most 1, or for noetherian r ings, the tw o notions are equiv alent . On the o ther hand w e prov e that a torsion pair is cogenerate d by a Σ-pure injective cotilting mo dule if and only if its heart is a lo cally noetherian Grothendiec k category . Moreov er we prov e that any ring admitting a Σ-pure injectiv e cotilting mo dule of injective dimension at most 1 is necessarily co- heren t. Finally , f or no etherian rings, we c haracterize cotilting torsi on pairs induced by Σ-pure injective cotilting m odules. Introduction The class of mo dules over an arbitrary asso ciative ring R is too complex to admit any sa tisfactory classification. F or this rea son, usually one restr icts to study particular, p os sibly la r ge and representative, classes of mo dules. In the r ecent literature, the theory of modules widely uses the notions of tor - sion a nd cotorsion pair s. T orsion a nd cotorsion pair s are couples ( L , M ) of clas ses o f mo dules which ar e maximal with r esp ect to the o rthogona lity conditions Hom( L , M ) = 0 and E xt( L , M ) = 0, res pec tively . These pairs are par tially order ed b y inclusion of their first components, forming complete lattices. The s tudy of their prop erties allows a n approximation of the whole catego ry of modules. In this pap er we concentrate on torsion and cotorsio n pa irs induc e d by a cotilting R -mo dule, in the sequel briefly called cotilting torsion and cotors ion pairs. In particular we study finiteness proper ties of cotilting to rsion and co torsion pa irs. An y cotilting mo dule is pure injective; fo llowing a suggestion o f E nrico Grego rio, we will fo cus o n torsio n and cotor sion pa irs induced by Σ-pure injective cotilting mo dules. In the first section we compare an arbitrary cotorsion pair ( L , M ) with the cotorsio n pair s ge ne r ated by the ℓ -presented mo dules in L . In the second s e c tion we a nalyze co tilting cotors ion pairs, o btaining new charac- terizations of those of fi n ite typ e (see Theorem 2.2). In par ticular, in the no etheria n case, these are exactly those induced by a Σ-pure injective cotilting mo dule (see Corollar y 2.4). Given a torsion pair ( X , Y ) in the catego ry of right R -mo dules, the he art of the torsion pair ( X , Y ) is an ab elian sub ca teg ory H ( X , Y ) o f the derived categor y of Researc h supp orted by grant CPDA07 1244/07 of Pado v a Universit y . 1 2 RICCARDO COLPI, FRANCESCA M ANTESE, ALBER TO TONOLO right R -mo dules (see se c tion 3 for mo r e details). Recently , in [8] it has b een prov ed that H ( X , Y ) is a Grothendieck ca teg ory if and o nly if ( X , Y ) is a co tilting tor s ion pair. In the third section we prove (see Theorem 3.3) that H ( X , Y ) is a lo c al ly no etherian Grothendieck category if and only if ( X , Y ) is co generated by a Σ- pur e injectiv e cotilting mo dule. Moreov er, we get that a ny ring R admitting a Σ-pure injectiv e cotilting mo dule of injective dimension at most 1 is neces sarily co herent (see Cor ollary 3.4). Finally , in the fourth section, we study cotilting tors io n pair s over a no etherian ring, giving a complete characterization of those induced by a Σ- pure injective cotilting mo dule. In particular we prove (see Theor em 4.3) that these are exactly the cotilting torsio n pairs which satisfy the R eiten- Ringel c ondition (see Condition 3.1). This w as originally in tro duced for finite dimensional k -a lg ebras in [1 3] as a sufficient condition to guarantee that the closure under direct limits of a splitting tors ion pair in the ca tegory of finitely gener ated modules is a splitting torsion pair in the category of all mo dules. Not a tion and terminology Let R be a ring. W e denote b y Mo d- R the category of right R -mo dules and by FP ℓ the sub categ ory of the rig ht ℓ -presented R - mo dules, i.e. the mo dules M in Mo d- R whic h admit a r esolution P ℓ → ... → P 1 → P 0 → M → 0 where P i is a finitely generated pro j ective mo dule for 0 ≤ i ≤ ℓ . Denote b y mod- R the intersection ∩ ℓ ∈ N FP ℓ . In particular FP 0 and FP 1 are the c ategories of finitely generated a nd of finitely prese nted right R -mo dules, res pe c tively . F or any mo dule C ∈ Mo d- R , P ro d C denotes the cla ss of a ll dir ect summands of direct pro ducts of copies of C . Given a cla ss C ⊆ Mo d- R , w e define the following: C ⊥ = { M ∈ Mo d- R | Ext 1 R ( C, M ) = 0 for any C ∈ C } C ⊥ ∞ = { M ∈ Mo d- R | Ext n R ( C, M ) = 0 for any C ∈ C and for an y n > 0 } . Similarly w e define ⊥ C and ⊥ ∞ C . A mo dule U ∈ Mo d- R is an n -c otilting mo dule if Ext i R ( U α , U ) = 0 for any i > 0 and all cardinals α , U has injective dimension a t most n , and there exis ts a long exact sequenc e 0 → U n → · · · → U 0 → W → 0 where W is an injective cog enerator of Mo d- R and U i ∈ P r o d U for i = 0 , . . . , n . An y cotilting mo dule U is pure-injective [3, 15], so the class Y = ⊥ ∞ U is closed under direct limits. Mo reov er, if U ha s injectiv e dimension at mos t one, then Y coincides w ith the class o f mo dules cogener a ted by U ; it is a torsion-free cla ss a nd the co rresp onding to rsion pair ( X , Y ) is called the c otilting torsion p air c o gener ate d by U . Notice that the latter is a faithful torsion pa ir, i.e. R R belo ngs to Y . Let A , B ⊆ Mo d- R . The pa ir ( A , B ) is ca lled a c otorsion p air if A = ⊥ B and B = A ⊥ . If B = C ⊥ for a class of mo dules C , we say that the cotor s ion pair is gener ate d by C . Similarly , if A = ⊥ C we sa y that the cotorsion pair is c o gener ate d by C . Moreov er ( A , B ) is said of finite typ e if it is gener ated by a set of mo dules in mo d- R . A cotorsio n pa ir is called her e ditary if A = ⊥ ∞ B and B = A ⊥ ∞ . A cotorsio n pair ( A , B ) is hereditar y if and only if A is a re solving class , i.e. it is closed under extensions, kernels of epimorphisms and it contains the pro jectives, or equiv alently , if B is a c or esolving class , i.e., it is closed under extensions, c okernels of COTORSION P AIRS, TORSION P AIRS, AND Σ-PURE INJECTIVE COTIL TING MODULES 3 monomorphisms and it contains the injectiv es [9]. A cotorsion pair is ca lled c omplete if A provides sp ecia l precov ers or, equiv alently , if B provides sp e cial preenv elop es (see [10, Ch. 2]). If U is a cotilting mo dule a nd Y = ⊥ ∞ U , then the co torsion pair ( Y , Y ⊥ ) gener- ated by Y is he r editary and complete [10, Ch. 8]; its kernel Y ∩ Y ⊥ coincides w ith Pro d U [10, Lemma 8.1.4]. In the s equel we will refer to ( Y , Y ⊥ ) as the c otilting c otorsion p air induc e d by U . Finally , tw o cotilting mo dules are ca lled e quivalent if they induce the same cotor sion pair. Let σ b e a n or dina l. An increasing chain of submo dules ( M α | α ≤ σ ) of a mo dule M is ca lled a filtr ation of M provided that M 0 = 0, M α = ∪ β <α M β for any limit o r dinal α ≤ σ , and M σ = M . Given a c lass of modules C , a C -filtr ation of M is a filtra tion such that, for a ny α < σ , M α +1 / M α is isomorphic to so me element of C . Filtrations play an impo rtant role in the study of cotorsion pairs, a s widely de- scrib ed in [10]. In pa rticular, in the sequel we will often refer to the following v ersion of Hill Lemma, stated and prov ed in a more general form in [1 0, Theo r em 4.2.6]. Theorem 0.1 (Hill Lemma) . L et R b e a ring and C a set of finitely pr esente d mo dules. L et M b e a mo dule with a C - filtr ation M = ( M α | α ≤ σ ) . Then ther e exists a family F of submo dules of M such that: (1) M ⊆ F ; (2) F is close d under arbitr ary sums and interse ctions; (3) F or any F 1 and F 2 in F su ch that F 1 ≤ F 2 , the m o dule F 2 /F 1 admits a C -filtra tion. (4) F or any finitely gener ate d submo dule L of M , ther e exists F ∈ F such that L ≤ F and F admits a finite C -fi lt r ation. In p articular, F is finitely pr esente d. 1. Com p aring cotorsion p airs Given a coto rsion pair A := ( A , A ⊥ ), we denote by A i the class of mo dules in A which b elo ng to FP i , i ≥ 0 , and by A ∞ the cla ss o f mo dules in A w hich b elo ng to mo d- R . Bes ide A = ( A , A ⊥ ), we co nsider the co torsion pairs A i = ( ⊥ ( A ⊥ i ) , A ⊥ i ) generated by the se t A i , i ≥ 0, and A ∞ = ( ⊥ ( A ⊥ ∞ ) , A ⊥ ∞ ) g enerated by the set A ∞ . Considering the partial order induced by the inclusio n o f their first comp onents, we hav e the following chain of co to rsion pair s: A ∞ ≤ ... ≤ A i +1 ≤ A i ≤ ... A 1 ≤ A 0 ≤ A . The c o torsion pairs A i , 0 ≤ i ≤ ∞ , ar e co mplete, and their cotorsion classes ⊥ ( A ⊥ i ) consist of all direct summands of A i -filtered mo dules (see [10, The o rem 3.2.1 and Corollar y 3.2.4]). Moreover, by definition, the cotor sion pair A ∞ is of finite t y pe. Prop ositi on 1.1. L et A = ( A , A ⊥ ) b e a c otorsion p air, and S b e any set of mo dules in A . Assume A is a r esolving class. Then the fol lowing ar e e quivalent: (1) S ⊥ = A ⊥ ; (2) S ⊥ ∩ A = A ⊥ ∩ A . Pr o of. Clearly 1 implies 2. Assume 2 holds; S ⊥ ⊇ A ⊥ is a lwa ys true. Therefore S = ( ⊥ ( S ⊥ ) , S ⊥ ) ≤ ( A , A ⊥ ). Let M b e in A . By [10, Theor e m 3.2.1] S is a complete co to rsion pair , therefor e the class ⊥ ( S ⊥ ) g ives sp ecial precovers. Let 0 → S 1 → S 2 → M → 0 4 RICCARDO COLPI, FRANCESCA M ANTESE, ALBER TO TONOLO be a ⊥ ( S ⊥ )-sp ecial precover o f M . Then S 2 belo ngs to ⊥ ( S ⊥ ) ⊆ A and S 1 belo ngs to S ⊥ . Since A is resolving , the mo dule S 1 belo ngs to S ⊥ ∩ A = A ⊥ ∩ A . Then the a bove exa ct sequence splits and M is a direct summand o f S 2 , so M belo ngs to ⊥ ( S ⊥ ). Therefor e S = ( A , A ⊥ ) a nd we conclude S ⊥ = A ⊥ .  Corollary 1.2. Le t A b e a r esolving class and 0 ≤ i ≤ ∞ . Then A i = A if and only if A ⊥ i ∩ A = A ⊥ ∩ A . Prop ositi on 1. 3. L et i ≥ 0 . Then A i = A i +1 if and only if A i = A i +1 . If mor e over A is a r esolving class, t hen A i = A ∞ . Pr o of. By [10, Corolla ry 3.2.4 ], ⊥ ( A ⊥ i +1 ) = ⊥ ( A ⊥ i ) consists of all direct summands of A i +1 -filtered mo dules. In pa rticular any mo dule A in A i is a direct summand of a A i +1 -filtered mo dule. Since A is finitely generated, b y Theorem 0.1 it is a direct summand of a finitely presented module A ′ admitting a finite A i +1 -filtration. Therefore, since A ′ is a finite extension of mo dules in A i +1 and A is a direc t summand of A ′ , bo th A a nd A ′ belo ng to A i +1 . Then A i is contained in A i +1 and hence A i +1 = A i . Finally , let us as sume A res o lving and let M b elo ng to A i +1 ; then there exists an exa ct sequence P i +1 → P i → ... → P 0 → M → 0 where P ℓ is a finitely g enerated pro jectiv e mo dule for 0 ≤ ℓ ≤ i + 1. L e t us denote by I ℓ the image o f P ℓ → P ℓ − 1 ; since A is clo sed under kernels of epimorphisms, it is easy to pr ov e r ecursively that I ℓ belo ngs to A i +1 − ℓ . In particular I 1 belo ngs to A i = A i +1 , a nd hence the finitely g enerated pro jectiv e r esolution o f M can b e contin ued with one more step o n the left. Rep eating this ar gument, we conclude that M be longs to A ∞ .  Prop ositi on 1.4 . L et i ≥ 1 ; if A = A i , then A ⊆ { dir e ct summ ands of lim − → A i } . Pr o of. If A = A i , then A = ⊥ ( A ⊥ i ). Therefore a ny ob ject A in A is a direct summand of a A i -filtered mo dule A . The mo dule A is the dir ect limit o f its finitely generated submo dules: A = lim − → ℓ ∈ Λ F ℓ . By Theor e m 0.1, for ea ch ℓ ∈ Λ there exists a submodule G ℓ of A in F P i , and containing F ℓ , such that G ℓ is A i -filtered. This gives A = lim − → ℓ ∈ Λ F ℓ = lim − → ℓ ∈ Λ G ℓ , and hence the thesis.  2. Cotil ting cotorsion p airs In all this section we ass ume that Y = ( Y , Y ⊥ ) is a co torsion pair induced by an n -cotilting module U . Lemma 2.1. L et 0 ≤ i ≤ ∞ . If U R is Σ -pur e-inje ctive and Y = lim − → Y i , then Y i = Y . In p articular if U R is a Σ -pur e-inje ctive 1-c otilting mo dule, then Y 0 = Y . Pr o of. Let M be in Y ⊥ i ∩ Y ; there exists a shor t exact sequence 0 → M → U β → U ′ → 0 COTORSION P AIRS, TORSION P AIRS, AND Σ-PURE INJECTIVE COTIL TING MODULES 5 with U ′ in Y (see [1 0, Prop o sition 8.1 .5]). By assumption, U ′ = lim − → λ ∈ Λ U ′ λ with the U ′ λ in Y i . Consider the pullbac k diag ram 0 / / M / / U β / / U ′ / / 0 0 / / M / / P λ / / O O U ′ λ / / O O 0 Since M ∈ Y ⊥ i the lower exact s e quence splits. Therefo re the upp er exact sequence is a direct limit of splitting exact seq uences, and hence it is pure. Since U R is Σ-pure-injective, by [1 2, Corollar y 8 .2], a lso the upp er short exact s equence splits; then M b elongs to Pro d U = Y ⊥ ∩ Y . B y Prop osition 1 .1 , w e c o nclude Y ⊥ i = Y ⊥ , and hence the thesis.  Theorem 2.2 . The fol lowing ar e e quivalent for a c otorsion p air Y induc e d by an n -c otilting mo dule U : (1) Y 1 = Y ; (2) Y ∞ = Y , i.e. Y is of finite typ e; (3) Y i = Y for some i ≥ 1 ; (4) U is Σ -pur e-inje ct ive and Y = lim − → Y 1 ; In p articular in such a c ase F P n = F P n +1 = mo d- R . Pr o of. 1 ⇒ 2: since Y is a reso lving class, by P rop osition 1 .3 we hav e Y 1 = Y ∞ . 2 ⇒ 3 : is clear. 3 ⇒ 4: Since U is pur e injective, every mo dule in Pro d U is pure injective. Therefore, Y ⊥ 1 ∩ Y = Pro d U b eing closed under arbitra ry direct sums, U ( α ) is pure injective for ea ch cardina l α . Th us U is Σ-pure-injective. More over, s ince Y is closed under dir ect limits a nd direct summands, by Pr o p osition 1 .4 we have Y = lim − → Y 1 . 4 ⇒ 1 : it fo llows immedia tely b y Lemma 2 .1. Finally , let M b e a mo dule in F P n . Consider the finitely gener ated pr o jectiv e resolution P n f n → P n − 1 → · · · → P 0 f 0 → M → 0 Since the injective dimension of U is ≤ n , b y dimension shifting for any i ≥ 1 Ext i R (Ker f n − 1 , U ) ∼ = Ext i + n − 1 R (Ker f 0 , U ) ∼ = Ext i + n R ( M , U ) = 0 . Therefore the finitely gener ated mo dule K er f n − 1 belo ngs to ⊥ ∞ U = Y . Since Y 0 = Y 1 , the mo dule Ker f n − 1 is finitely presented; thus Ker f n is finitely genera ted and M b elo ng s to F P n +1 .  Problem 2.3. Ar e ther e Σ -pur e-inje ctive n -c otilting mo dules inducing a c otorsion p air Y which is not of finite typ e, i.e. Y 1 6 = Y ? In [1, Cor ollary 4 .11] there is a negative answer in the case R is right no ether ian. The pro ble m has a nega tive answer also in the general ca se if n = 1: Corollary 2. 4. If n = 1 , then Y is of finite typ e if and only if the 1 - c otilting mo dule U is Σ -pur e inje ctive. Pr o of. The proo f is an immedia te conseque nc e of the forthco ming Cor ollary 3.4.  6 RICCARDO COLPI, FRANCESCA M ANTESE, ALBER TO TONOLO 3. Cotil ting torsion p airs Beyond inducing the co torsion pair ( Y , Y ⊥ ), a 1-cotilting r ight R -mo dule cog e n- erates a torsio n pair ( X , Y ). Among tors io n pairs, the cotilting ones are precis ely those for which the torsio n-free class gives sp ecial precov e r s [2, Theor em 2.5]. Note that tw o equiv alent 1-c o tilting mo dules cogenerate the same torsion pair . Studying splitting torsion pa irs ov er finite dimensional k -algebr as, Reiten and Ringel in [13] in tro duced the following finiteness condition on a torsio n pair ( X , Y ): Reiten-Ri ngel Condi tion 3.1. If Y ∈ Y has a fin itely gener ate d submo dule 0 6 = Y 0 ≤ Y such that Y / Y 0 ∈ X , then Y is fin itely gener ate d. This condition tur ns out to have in teresting applications also in a more genera l setting, as we will show in the next results. Prop ositi on 3.2. If a c otilting torsion p air ( X , Y ) satisfies t he R eiten- Ringel c on- dition, then the c otorsion p airs Y and Y 0 c oincide. Pr o of. Let us prove that Y ⊥ = Y ⊥ 0 . Let M b elongs to Y ⊥ 0 ; we hav e to show that any shor t exact sequence 0 → M → E → Y → 0 with Y ∈ Y , s plits. Set D = { D ≤ E | M ∩ D = 0 , E / ( M ⊕ D ) ∈ Y } . Since Y is clos ed under direct limits, any ascending ch ain in D has union in D , so that D contains a maximal element. Let’s ca ll it D max : the goal consists in proving that M ⊕ D max = E . Supp ose that this is not the ca s e, and let M ⊕ D max < E ′ ≤ E , with E ′ / ( M ⊕ D max ) finitely gener ated. Let E ′′ /E ′ = t X ( E /E ′ ) ∈ X , where t X denotes the torsio n radical asso ciated to X . F rom the exa ct sequenc e 0 → E ′ / ( M ⊕ D max ) → E ′′ / ( M ⊕ D max ) → E ′′ /E ′ → 0 since E ′′ / ( M ⊕ D max ) ≤ E / ( M ⊕ D max ) ∈ Y , the mo dule E ′′ / ( M ⊕ D max ) is finitely g enerated by the Reiten-Ringel condition. By assumption, the exact sequence 0 → M ∼ = ( M ⊕ D max ) /D max → E ′′ /D max → E ′′ / ( M ⊕ D max ) → 0 splits, with E ′′ / ( M ⊕ D max ) 6 = 0. Thus there exists a mo dule D ′ , with D max < D ′ ≤ E ′′ , s uch that E ′′ /D max = (( M ⊕ D max ) /D max ) ⊕ ( D ′ /D max ). In particula r M ∩ D ′ ≤ M ∩ D max = 0 and M ⊕ D ′ = E ′′ . Finally , E / ( M ⊕ D ′ ) = E /E ′′ ∼ = ( E /E ′ ) / t X ( E /E ′ ) ∈ Y , con trary to the ma ximality of D max .  The he art H ( X , Y ) o f a tor sion pair ( X , Y ) is the ab elian sub catego ry of the derived categor y of Mo d- R who se o b jects are the complexes which have zero coho- mologies everywhere, except for degrees 0 and − 1 where they hav e co homologies in X and in Y , r esp ectively . In [6] it is pr ov ed that, if ( X , Y ) is faithful, the stalk com- plex V := R [1 ] is a tilting ob ject in H ( X , Y ) with endomorphisms r ing R ; mor eov er it determines a torsio n pair ( T , F ) and a pa ir of eq uiv alences H V : T − − → ← − − Y : T V and H ′ V : F − − → ← − − X : T ′ V where H V = Hom( V , − ), H ′ V = Ext( V , − ) and T V , T ′ V are their adjoint functor s. In [8 ] a nd [7] it has b een prov ed that a faithful torsio n pair is cotilting if and only if the asso cia ted heart is a Gro thendieck category . In the nex t theorem we show that a 1-cotilting mo dule is Σ-pure injective if and only if the co rresp onding faithful tor sion pa ir has a lo cally no etherian he a rt (see [14, Se c tio n V.4]). COTORSION P AIRS, TORSION P AIRS, AND Σ-PURE INJECTIVE COTIL TING MODULES 7 Theorem 3 .3. A faithful torsion p air ( X , Y ) in Mo d- R is c o gener ate d by a 1- c otilting Σ -pur e inje ctive right R -mo dule if and only if the he art H ( X , Y ) is a lo c al ly no etherian Gr othendie ck c ate gory. Pr o of. By [8] and [7], H := H ( X , Y ) is a Grothendieck ca tegory if and only if ( X , Y ) is cog enerated by a 1-cotilting mo dule C R . Let us assume that H is locally no etheria n. First let us sho w that the to rsion pair ( X , Y ) satisfies the Reiten-Ringel condition. Indeed, let 0 → Y 0 → Y → X → 0 b e an exact sequence with Y ∈ Y , X ∈ X and Y 0 finitely generated. W e g et the exac t sequence 0 → T ′ V X → T V Y 0 → T V Y → 0. Since Y 0 is finitely g enerated over R , w e see that T V Y 0 is a factor of V n , for some n ∈ N . F ollowing [5, Coro lla ry 4.3 ], we hav e that V is finitely presented, so that T V Y 0 is finitely gener ated. Thus T V Y is finitely generated. Since the functor H V carries finitely gener ated ob jects of H to finitely g enerated R -mo dules [5, Lemma 6 .1], the mo dule Y ∼ = H V T V Y is finitely generated. So the Reiten-Ringel conditio n is satisfied and, by Pr op osition 3.2 we get that Y = Y 0 . Finally , let us pr ov e that Y 0 = Y 1 ; then Y = Y 1 and we conclude by Theore m 2.2 . Indee d, let F ∈ Y 0 and 0 → K → R n → F → 0 b e an exa ct sequence in Y . Then we obtain the exa ct sequence 0 → T V K → V n → T V F → 0 and, since H is lo cally no etherian, T V K is finitely gener ated. Thus K ∼ = H V T V K is finitely generated and so F is finitely pr esented. The pro of of the conv ers e implication follows the same arg ument s used in [8]. First note that if C is Σ-pur e injectiv e, then P r o d C is close d under direct s ums. Moreov er, in the Gro thendieck ca tegory H , an ob ject I is injective if and only if I = T V ( C ′ ), for s ome C ′ ∈ Pro d C [7, Pr op osition 3.8]. F rom these easy obser v ations, it follows that in H , the class of injective ob jects is clo sed under co pro ducts. Indeed let I λ , λ ∈ Λ, a set of injective ob jects: then ⊕ I λ = ⊕ T V ( C ′ λ ) = T V ( ⊕ C ′ λ ), since T V commutes with copr o ducts. Hence, following the pro of of [14, P rop ositio n V.4 .3], one gets that any small o b ject in H is no e ther ian. Finally , in [7, Lemma 3.4] it is is shown that the set { Z ≤ V n , n ∈ N } gener ates H : sinc e V is small and so no etherian, H admits a s et o f no etheria n gener ators.  The following corolla ry s hows that the assumption that a ring admits a Σ-pure injectiv e 1- cotilting mo dule is v ery s trong, since it implies the ring is coher ent . Corollary 3. 4. L et C b e a 1-c otilting right R -m o dule and Y t he c orr esp onding c otorsion p air. If C is Σ -pur e-inje ctive, then: (1) Y = Y 1 ; (2) the ring R is right c oher ent. Pr o of. F ollowing the pro of of Theor em 3.3, we get that Y = Y 0 = Y 1 and so , by Theorem 2.2 , the r ing R is right coherent.  Problem 3.5. Is a ring admitting a Σ -pur e-inje ctive n -c otilting mo dule ne c essarily c oher ent? It follows also a nice applicatio n to the tilting setting: any class ical 1-tilting mo dule (for the definition s ee [1 0, Ch. 5]) ov er a right no etheria n ring has right coherent endomo rphism r ing. Corollary 3 .6. L et T S b e a classic al 1 -tilting mo dule. The fol lowing ar e e quivalent: (1) S is no etherian; 8 RICCARDO COLPI, FRANCESCA M ANTESE, ALBER TO TONOLO (2) Hom S ( T , W ) is a Σ -pur e inje ctive 1-c otilting End S ( T ) -mo dule for any in- je ctive c o gener ator W of Mo d- S . In such a c ase, End S ( T ) is a right c oher ent ring. Pr o of. It is known [7, Theorem 2.3] that the ca tegory Mo d- S is equiv alent to the heart of the co tilting tors io n pair ( X , Y ) cogenera ted b y Hom S ( T , W ). Then the result follows fro m Theor em 3.3.  4. Cotil ting torsion p airs for noetherian rings As we hav e s een in the pr evious sections , the notio n of Σ-pure injectivity for a cotilting mo dule is closely r elated to finiteness conditions on the rings and on the classes inv o lved. The a im of this sec tio n is to characterize the cotilting torsion pair s cogenera ted by a Σ-pur e injective 1-cotilting mo dule in the no etherian setting, a nd inv estiga te their finiteness prop er ties. F or the r est of this sec tion, R deno tes a right no etherian ring . Buan and Kra us e in [4] prov ed that cotilting torsio n pairs pla y a relev a nt role pass ing fro m the cat- egory o f finitely gener ated R -mo dules to the whole catego ry of R -mo dules: indeed there is a bijective cor r esp ondence b etw een c otilting to rsion pairs ( X , Y ) in Mo d- R and faithful torsion pa irs ( X 0 , Y 0 ) in mo d- R . The corr e sp ondence is given by the m utually inv erse as signments X 7→ X 0 = X ∩ mo d- R , X 0 7→ X = lim − → X 0 and Y 7→ Y 0 = Y ∩ mo d- R, Y 0 7→ Y = lim − → Y 0 . In ca se the torsion pair ( X , Y ) is co generated by a Σ-pure injective 1-cotilting mo dule, this corr e s po ndence preserves the following relev ant (see the notio ns of quasitilted a r tin alg e br as [11] and of quasitiled rings [6]) prop erties: Prop ositi on 4.1. L et C R b e a Σ -pur e inje ctive 1-c otilting mo dule and let Y = ⊥ C . Then: (1) ( X 0 , Y 0 ) splits if and only if ( X , Y ) splits; (2) pro j dim Y 0 ≤ 1 if and only if pr o j dim Y ≤ 1 . Pr o of. 1. Le t X ∈ X and Y 0 ∈ Y 0 . Since Y 0 is finitely pres ent ed and X = lim − → X α for a directed family of submo dules X α ∈ X 0 of X , we have Ext 1 R ( Y 0 , X ) = Ext 1 R ( Y 0 , lim − → X α ) ∼ = lim − → Ext 1 R ( Y 0 , X α ) . Now, if Ext 1 R ( Y 0 , X 0 ) = 0, we der ive that Ext 1 R ( Y 0 , X ) = 0. Since C is Σ-pure injectiv e, by Lemma 2.1 w e hav e Y ⊥ = Y ⊥ 0 . Therefor e Ext 1 R ( Y , X ) = 0. 2. Similar ly , if Ext 2 R ( Y 0 , Mo d- R ) = 0, by dimension shifting and using the fact that Y ⊥ 0 = Y ⊥ we se e that Ext 2 R ( Y , Mo d- R ) = 0 .  As we men tioned in the previous section, in or der to gua rantee that a split- ting torsion pair ( X 0 , Y 0 ) in mo d- R gives rise to a splitting tors ion pair ( X , Y ) = (lim − → X 0 , lim − → Y 0 ), Reiten and Ringel in [13] introduced the Condition 3.1. Here we prov e that, for a no e ther ian ring , the Reiten-Ringel co ndition completely charac- terizes to rsion pair s co g enerated by a Σ-pure injective 1-c otilting mo dule. Lemma 4.2. The class of al l Y 0 -filter e d mo dules is close d under submo dules. COTORSION P AIRS, TORSION P AIRS, AND Σ-PURE INJECTIVE COTIL TING MODULES 9 Pr o of. Suppo s e that Y is Y 0 -filtered by ( Y λ | λ ≤ µ ). Then any Y ′ ≤ Y is Y 0 - filtered b y ( Y ′ λ = Y λ ∩ Y ′ | λ ≤ µ ). Indeed this is a contin uous c hain star ting from 0 and ending to Y ′ such that for ev ery λ < µ Y ′ λ +1 Y ′ λ = Y λ +1 ∩ Y ′ Y λ ∩ Y λ +1 ∩ Y ′ ∼ = Y λ + ( Y λ +1 ∩ Y ′ ) Y λ ≤ Y λ +1 Y λ ∈ Y 0 , so that Y ′ λ +1 / Y ′ λ ∈ Y 0 .  Theorem 4.3 . Le t R b e a right n o etherian ring. F or a c otilting torsion p air ( X , Y ) in Mo d- R the fol lowing c onditions ar e e qu ivalent: (1) Y satisfies the R eiten-Ringel c ondition; (2) any 1-c otilting mo dule c o gener ating ( X , Y ) is Σ -pur e inje ctive; (3) ( X , Y ) is c o gener ate d by a Σ - pur e inje ctive 1-c otilting mo dule. Pr o of. 1 ⇒ 2 F ollows from P rop osition 3 .2 and Corollar y 2.4. 2 ⇒ 3 is obvious. 3 ⇒ 1 Let us ass ume tha t C is Σ-pur e injective. By Co rollar y 2.4 , we hav e Y = ⊥ ( Y ⊥ 0 ). By [10, Coro llary 3.2.4] a nd Lemma 4 .2, any mo dule in Y is Y 0 - filtered. Let Y be a mo dule in Y a nd F a finitely g enerated s ubmo dule of Y such that Y / F is tors io n. By Theorem 0.1, F is co nt ained in a finitely gener ated submo dule F of Y suc h that Y / F b elongs to Y . Since Y /F is a quotient of Y /F , it is also torsion and hence Y = F is finitely generated.  The Σ-pure injectivity of C is eq uiv alent to some o ther interesting finiteness condition o n Y . Prop ositi on 4.4. L et C b e a 1-c otilting m o dule and ( X , Y ) the torsion p air c o gen- er ate d by C . Then the fol lowing c onditions ar e e quivalent: (1) C is Σ -pur e inje ctive; (2) if Y ∈ Y has a finitely gener ate d submo dule 0 6 = Y 0 ≤ Y such that Y / Y 0 ∈ X , then Y is Y 0 -filter e d; (3) ther e ar e no infinite st rictly asc ending chains Y 0 < Y 1 < · · · < Y i < Y i +1 < . . . in Y 0 such that Y i +1 / Y i ∈ X 0 (e quiv. Y i +1 / Y 0 ∈ X 0 ) for al l i ∈ N ; (4) if Y 0 is a fin itely gener ate d submo dule of Y ∈ Y , then t he torsion p art of Y / Y 0 is finitely gener ate d; (5) any non-zer o mo dule Y ∈ Y has a fi nitely gener ate d n on-zer o submo dule Y + ≤ Y such that Y / Y + ∈ Y . Pr o of. 1 ⇒ 2: it is clear, since any mo dule in Y is Y 0 -filtered. 2 ⇒ 3: assume that ( Y i ) i ∈ N is a infinite strictly ascending chains Y 0 < Y 1 < · · · < Y i < Y i +1 < . . . in Y 0 such that Y i +1 / Y i ∈ X 0 for all i ∈ N . Then Y = ∪ Y i = lim − → Y i ∈ Y is not finitely generated and Y / Y 0 = (lim − → Y i ) / Y 0 ∼ = lim − → ( Y i / Y 0 ) ∈ X . Therefore Y is Y 0 -filtered a nd by Theorem 0.1, Y 0 is contained in a finitely gener ated submo dule Y 0 of Y such that Y /Y 0 is in Y . Since Y /Y 0 is a quo tient of Y / Y 0 , and hence it is also tors ion, we hav e Y = Y 0 , co ntradicting the fa c t that I is not finitely generated. 3 ⇒ 4: supp ose that for a given Y in Y , and a finitely generated submodule F of Y , the quotient Y /F has torsio n par t Y /F which is not finitely gener a ted. Then Y /F is a direct limit of mo dules X i , i ∈ I , in X 0 . Denoting by Y i /F the homomorphic imag es in Y /F of the mo dules X i , by means of the Y i ’s one can construct a n infinite strictly ascending chain in Y 0 , c ontradicting 3 . 10 RICCARDO COLPI, FRANCESCA M ANTESE, ALBER TO TONOLO 4 ⇒ 5 : g iven any finitely generated submo dule Y 0 of Y , take as Y + the submo d- ule of Y containing Y 0 such that Y + / Y 0 is the tors ion part of Y / Y 0 . 5 ⇒ 1 : we will prove that any mo dule in Y is Y 0 -filtered. By [10, Coro llary 3.2.4] and Lemma 4.2, we will get Y = ⊥ ( Y ⊥ 0 ), so that C is Σ-pure injective b y Cor ol- lary 2.4 . Let Y in Y a nd µ = 2 | Y | . W e construct, by transfinite induction, a contin uo us chain ( Y λ | λ ≤ µ ) of submodules of Y such that, for ev ery λ < µ , i) Y / Y λ ∈ Y , ii) Y λ +1 / Y λ ∈ Y 0 , iii) if Y λ  Y , then Y λ  Y λ +1 . Set Y 0 = 0, and set Y λ +1 = Y λ in case Y λ = Y , o r Y λ +1 / Y λ = ( Y / Y λ ) + if Y λ  Y (if this is the case, then Y / Y λ +1 ∼ = ( Y / Y λ ) / ( Y / Y λ ) + ∈ Y ). Then co nditions i), ii) and iii) are clearly satisfied. Finally , if λ is a limit ordina l, w e set Y λ = S k<λ Y k , and condition iii) holds since Y / Y λ = lim − → k<λ Y / Y k ∈ Y . This defines a Y 0 -filtration ( Y λ | λ ≤ µ ). Finally , the ch oice µ = 2 | Y | and the prop erty iii) gua rantee that Y µ = Y .  R emark 4.5 . 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[13] Reiten, I. and Ringel, C. M. Infinite dimensional r epr esent ations of c anonic al algebr as , Canad. J. Math. 58 (2006), 180–224. [14] Stenstr¨ om, B. Rings of Quotients , Grund. m ath. Wiss. 217 , Springer, New Y ork (1975). [15] ˇ St’o v ´ ıˇ cek, J. All n - c otilting mo dules ar e pur e injective , Pr o c. Amer. Math. So c. 134 (2006), 1891 –1897. COTORSION P AIRS, TORSION P AIRS, AND Σ-PURE INJECTIVE COTIL TING MODULES 11 (R. Colpi) Dip. Ma tem atic a Pura ed Applicat a, Universit ` a degli studi di P adov a, via Trieste 63 , I-35121 P adov a It a l y E-mail addr ess : colpi@math .unipd.it (F. Mantese) Dip ar timento di Informa tica, Un iversit ` a degli Studi di Verona, strada Le Grazie 15, I-37134 Verona - It al y E-mail addr ess : francesca. mantese@un ivr.it (A. T onolo) Dip. M a tema tica Pura ed Applica t a, Universit ` a degli studi di P a dov a, via Trieste 63 , I-35121 P adov a It a l y E-mail addr ess : tonolo@mat h.unipd.it