Dimension Estimates for Representable Equivalences of Module Categories

193, 660]676 Ž1997. JA967005

JOURNAL OF ALGEBRA ARTICLE NO.

Dimension Estimates for Representable Equivalences of Module Categories Jan Trlifaj* Katedra algebry MFF UK, Sokolo¨ ska ´ 83, 186 00 Praha 8, Czech Republic Communicated by Kent R. Fuller Received July 22, 1996

Let R be a ring, P be a )-module, and S s End P. Inspired by the tilting theory, we investigate relations between the global dimensions gl dim R and gl dim S, and between the structure of the Grothendieck groups K 0 Žmod-R. and K 0 Žmod-S .. We prove that gl dim S F gl dim R q D, where D s 1, e.g., for P almost tilting, but the symmetry fails: for each n - v , there are almost tilting modules with gl dim R y gl dim S ) n. If R is right artinian and S right noetherian, then there is an isomorphism K 0 Žmod-S . ( K 0 Žmod-R., where R s RrAnn R Ž P .. We also prove that if R and R9 are right artinian rings such that there is a torsion theory counter equivalence between Mod-R and Mod-R9, then K 0 Žmod-R. ( K 0 Žmod-R9.. Q 1997 Academic Press

INTRODUCTION Tilting theory may be viewed as a far-reaching generalization of the Morita theory of equivalences between module categories. For example, by the Brenner]Butler Theorem, large subcategories of module categories over a finite dimensional algebra, and over its tilted algebra, are equivalent wHRx. A different way of generalization of the Morita theory was presented by Fuller by introducing the notion of a quasi-progenerator wFx. Later, Menini and Orsatti found a common point by discovering the general notion of a )-module wMOx. Colpi then proved that tilting modules coincide with the )-modules that generate all injectives, while quasi-progenerators are just the )-modules which generate all of their submodules wC2x. *Research supported by Grant GAUK-44. E-mail address: [email protected]. 660 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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Other aspects of the tilting theory of finite dimensional algebras have recently been considered in the general setting of Žinfinitely generated. modules over arbitrary associative rings with unit. For example, the Bongartz approach using partial tilting modules applies also to the general case of Žinfinitely generated. tilting modules wCT, Sect. 1x, and so do parts of the Assem]Smalø characterization of tilting torsion theories wCT, Sect. 2x. For finite dimensional algebras, the linear space duality enables us to introduce Žpartial. cotilting modules and obtain their properties as duals of those of the Žpartial. tilting ones. Surprisingly, some of the results persist in the general setting. The point is that one can define the basic notions using associated categories of modules rather than the individual modules themselves. The dualization is made possible by category-theoretic and homological methods despite the fact that there is no module-theoretic duality available in general wCTTx. One of the remarkable properties of )-modules is the following: if R is an arbitrary ring, P is a )-module, and Q is an injective cogenerator for Mod-R, then the ‘‘dual’’ module Hom R Ž P, Q . is a cotilting right End Pmodule Žthis result is implicit already in wCM, Proposition 1.2x.. So )-modules are, in a sense, counterparts to cotilting modules. This suggests that further results of the tilting theory might be extended to )-modules. From this point of view, we look at the following two parts of the tilting theory: the global dimension estimates, and the structure of Grothendieck groups. By a result of Miyashita wMx, if T is a ‘‘tilting module of finite projective dimension F r ’’ then the Grothendieck groups of R and End T are isomorphic, and
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In Section 2, we compare the Grothendieck groups of R and S s End P for a )-module P. Since w dim S P F 1 ŽLemma 2.1., there are an infinite cardinal, l, and an abelian group homomorphism, w , such that w maps the Grothendieck group of the class of all - l-generated right S-modules, K Ž l, S ., into K Ž l, R . ŽLemma 2.4.. If R is right artinian, S right noetherian, and R s RrAnn R Ž P ., then w induces an isomorphism of K 0 Žmod-S . onto K 0 Žmod-R. ŽCorollary 2.7.. Nevertheless, w need not be injective if S is not right noetherian ŽExample 2.8., and w may even be zero if R is not right artinian ŽExample 2.11Žii... We also obtain isomorphisms of Grothendieck groups in the case when R and R9 are right artinian rings such that there is a torsion theory counter equivalence between Mod-R and Mod-R9 in the sense of wCF2x Žsee Theorem 2.9.. Let R be an associative ring with unit. Denote by Mod-R, and by mod-R, the category of all, and all finitely generated, respectively Žunitary right R-. modules. Further, simp-R denotes a representative set of all simple modules. Let M g Mod-R. Then genŽ M . denotes the minimal cardinality of an R-generating subset of M. Further, End M denotes the endomorphism ring of M Žwe often view M also as a left End M-module.. Denote by Gen M ŽCogen M . the class of all modules generated Ž cogenerated . by M, that is, of all M9 g Mod-R such that there exist a cardinal l f

f

and an epimorphism M Ž l. ª M9 ª 0 Ža monomorphism 0 ª M9 ª M l .. Similarly, Pres M denotes the class of all modules presented by M, that is, of all M9 g Mod-R such that there exist cardinals l, l9 and homomorf9

f

phisms f , f 9 such that the sequence M Ž l9. ª M Ž l . ª M9 ª 0 is exact. Further, Add M Žadd M . denotes the class of all direct summands of a direct sum of arbitrarily many Žfinitely many. copies of M. For a module X, Tr X Ž M . s Ý f g Hom R Ž X , M . ImŽ f . is the trace of X in M. We denote by proj dimŽ M . and w dimŽ M . the projective, and the weak, dimension of M, respectively. The right global dimension of the ring R is denoted by gl dim R. Further, R is said to be finitely cogenerated if each set, A, of right ideals satisfying F A s 0 contains a finite subset, B : A, such that F B s 0. Let P g Mod-R and S s End P. Let Q be an injective cogenerator for Mod-R and P* s Hom R Ž P, Q . g Mod-S. Then Gen P and Cogen P* are the largest subcategories of Mod-R and Mod-S, respectively, between which Hom R Ž P, y. and ymS P may induce a category equivalence wCMx. If this is the case then P is said to be a )-module. Let P be a )-module. Then P is finitely generated wTrx, and Gen P s Pres P wCMx. Moreover, the functor Hom R Ž P, y. preserves short exact sequences consisting of elements of Gen P, and Cogen P* s Ker Tor1S Žy, P . is a torsion free class in Mod-S wCMx Žin fact, Cogen P* is a cotilting torsion free class in the sense of wCTTx..

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A module P is a quasi-progenerator if P is a )-module and Cogen P* s Mod-S Žor, equivalently, P is a quasi-projective finitely generated self-generator wFx.. Further, P is a tilting module if P is a )-module and Gen P contains all injective modules. Tilting modules can equivalently be defined as the finitely generated modules satisfying Gen P s P H 1 Žwhere P H i s Ker Ext iR Ž P, y., i - v .. They also coincide with the finitely generated modules P satisfying proj dim P F 1, Ext 1R Ž P, P . s 0, and ‘‘Hom R Ž P, M . s 0 s Ext 1R Ž P, M . implies M s 0 for all M g Mod-R’’ wCTx. The group of all integers is denoted by Z. For further notation, we refer to wAF, ARS, Bux.

1. UPPER BOUNDS FOR THE GLOBAL DIMENSION Let R be a ring and P g Mod-R be a )-module. Put S s End P. Let M g Gen P Žs Pres P .. For M s 0 put d M s y1. Assume M / 0. Put d M s m provided that m is the smallest integer G 0 such that there is an exact sequence in Mod-R 0 ª Mm ª ??? ª M0 ª M ª 0,

Ž 1.

where Mi g Add P for all i s 0, . . . , m. If there is no such integer m, put d M s `. LEMMA 1.1.

d M s proj dim Hom R Ž P, M ..

Proof. Since P is a )-module, the functors F s Hom R Ž P, y. and G s ymS P are inverse category equivalences between Gen P and Cogen P*, where P* s Hom R Ž P, Q . and Q g Mod-R is an injective cogenerator wCMx. By wTr, Theorem 1x, P is finitely generated. By wAF, Lemma 29.4x, F and G restrict to an equivalence between the category of all projective right S-modules and the category Add P. In particular, if N g Gen P, then Hom R Ž P, N . is a projective right S-module iff N g Add P. Consider the exact sequence Ž1. with Mi g Add P for each i s 0, . . . , m. Then the images Žand kernels. of all mappings in Ž1. belong to Gen P. By wCM, Theoremx, an application of Hom R Ž P, y. to Ž1. gives an exact sequence in Mod-S. 0 ª Nm ª Nmy1 ª ??? ª N0 ª Hom R Ž P , M . ª 0,

Ž 2.

where Ni is a projective right S-module for each i s 0, . . . , m. Conversely, consider an exact sequence of the form Ž2. with Ni a projective right S-module for each i s 0, . . . , m. Then the kernels Žand images. of all mappings in Ž2. belong to Cogen P*. By wCM, Proposition

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1.2x, an application of ymS P to Ž2. gives an exact sequence in Mod-R of the form Ž1., with Mi g Add P for each i s 0, . . . , m. This shows that d M s proj dim Hom R Ž P, M .. EXAMPLE 1.2. Let R s Z and P g Mod-Z be a )-module. Then P is a quasi-progenerator wCM, Corollary 2.5x. So either P is a progenerator or P s [p g F ŽZŽ p n p ..Ž k p . for a finite set of primes, F, and for 0 - n p - v , 0 - k p - v Ž p g F .. In the former case, d M s proj dim M F 1 for each M g Gen P s Mod-Z. In the latter, Cogen P* s Mod-S, where S s End P ( Ł p g F Mk pŽZŽ p n p ... Since S is a QF-ring, for each N g Mod-S either proj dim N F 0 or proj dim N s `. It follows that d M s ` iff M g Gen P _ Add P Žthis happens, e.g., when 0 / M ; ZŽ p n p ... In particular, gl dim S s ` provided that 1 - n p for some p g F, while gl dim R s 1. In Example 1.2, the modules that make gl dim S much bigger than gl dim R are those lying in  M g Gen P < M f Add P and proj dim M F proj dim P 4 . This can be generalized to arbitrary )-modules: THEOREM 1.3. Let R be a ring and P be a )-module. Let S s End P, d s proj dim P, A P s  M g Gen P < M f Add P and proj dim M F d4 , and D s maxŽsup M g A P d M y d ., 14 . Then gl dim S F gl dim R q D. Proof. W.l.o.g., we assume that d - ` and D - `. Let 0 / N g Mod-S. Take an exact sequence in Mod-S, 0 ª X ª F ª N ª 0, such that 0 / F is projective, and X s 0 provided that N is projective. By wBu, Sect. 8.1, Corollary 2x, proj dim X s proj dim N y 1. Let Q be an injective cogenerator for Mod-R and put P* s Hom R Ž P, Q .. By wCM, Proposition 1.2x, Cogen P* is a torsion free class containing S, whence X g Cogen P*. Since P is a )-module, we have X ( Hom R Ž P, M ., where M s X mS P g Gen P. It remains to show that proj dim X F proj dim M q D y 1. We prove by induction on e s proj dim M. Since D G 1, the assertion holds if M g Add P Žand hence if e F 0.. By the choice of A P and by Lemma 1.1, it also holds when e F d. Assume that M f Add P and e ) d. By wCM, Theoremx, there is a non-splitting short exact sequence in Mod-R 0 ª M1 ª P Ž k . ª M ª 0, where k is a cardinal and M1 g Gen P s Pres P. By wBu, Sect. 8.1, Corollary 2x, proj dim M1 s e y 1. By wCM, Theoremx, the induced sequence in Mod-S is exact, but non-splitting, and has the form 0 ª Hom R Ž P , M1 . ª S Ž k . ª X ª 0.

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By wBu, Sect. 8.1, Corollary 2x, proj dim X s proj dim Hom R Ž P, M1 . q 1. By the induction premise, proj dim Hom R Ž P, M1 . F e y 1 q D y 1, whence proj dim X F e q D y 1. Theorem 1.3 can be strengthened in two ways: by restricting the class of )-modules considered, or by restricting the class of rings considered. For the first way, the classes of quasi-tilting, and almost tilting, modules play a crucial role: DEFINITION 1.4. Let R be a ring and P g Mod-R. Then P is quasi-tilting if P is a )-module such that Gen P : P H 1 wCDT, Definition 2.2x; P is almost tilting if P is a )-module such that Gen P : F1F i- v P H i . v

v

Clearly, any tilting module is almost tilting Žof proj dim F 1., and any almost tilting module is quasi-tilting. The converse is not true in general Žsee Example 1.5Žii. below.. Nevertheless, if proj dim P F 1 then P is almost tilting iff P is quasi-tilting. In particular, the two notions coincide if R is right hereditary. Also, if proj dim P s n for some n - v then P is almost tilting iff P is quasi-tilting and Gen P : P H i for all 2 F i F n. EXAMPLE 1.5. Ži. Assume R is an artin algebra and P g Mod-R. Then P is quasi-tilting iff P is a tilting right RrAnn R P-module wCM, Proposition 1.10; C2, Corollary 6x. Similarly, P is almost tilting iff P is a tilting right RrAnn R P-module, and Gen P : F2 F i- v P H i . The next part, Žii., shows that even in this particular case, the two notions do not coincide. Žii. For each 1 - d - v , there exists a finite dimensional algebra of finite representation type, R, and P, P9 g Mod-R such that d s proj dim P s proj dim P9, and P is almost tilting while P9 is quasi-tilting, but not almost tilting. Note that it suffices to find a P such that P is a simple module, proj dim P s d and Ext Rk Ž P, P Ž k . . s 0 whenever k is a cardinal and 1 F k - v . ŽThen P is a quasi-progenerator, so P provides the required example.. In order to construct R, P, and P9, we modify wCDT, Example 5.4x: Let 2 - n - v and K be a field. Let R s UTnŽ K .rI, where UTnŽ K . denotes the upper triangular matrix ring of degree n over K and I is the square of the Jacobson radical of UTnŽ K .. In other words, R is the path algebra of the quiver A n with oriented arrows a i , i s 1, . . . , n y 1, and relations a i a iq1 s 0 for all i s 1, . . . , n y 2, ???

a ny2

ny1

a ny1

n.

6

a2

6

2

6

a1

6

1

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Denote by P Ž i . and SŽ i . the indecomposable projective and the simple, respectively, module corresponding to the vertex i, 1 F i F n. Take 1 F m - n y 1 and put P s SŽ m. and P9 s Ž[i/ mq1 P Ž i .. [ P. By wCDT, Example 5.4x, the Auslander]Reiten quiver of R is P Ž n y 1.

P Ž 1. 6

6

6

S Ž n y 1.

6

6

6

P Ž n. s S Ž n.

S Ž 2.

S Ž 1.

It follows that proj dim P9 s proj dim P s n y m and gl dim R s n y 1 Žcf. wCDT, Example 5.4Ža.x.. Moreover, the modules SŽ1. and P Ž i ., i s 1, . . . , n y 1, are injective. Let k be a cardinal and 1 F k F n. Then Ext Rk Ž P, P Ž k . . ( Ext Rky 1 Ž P, S Ž m y 1.Ž k . . ( ??? ( Ext 1R Ž P, SŽ m y k q 1.Ž k . . ( Hom R Ž S Ž m q 1., S Ž m y k q 1 . Ž k . . s 0 for 1 F k F m, and Ext Rk Ž P , P Ž k . . ( Ext Rky mq1 Ž P, SŽ1.Ž k . . s 0 for m - k F n. Since gl dim R s n y 1, we see that Ext Rk Ž P, P Ž k . . s 0 for all 1 F k - v and all cardinals k . It follows that P is an almost tilting module. Let R s RrAnn R Ž P9.. As in wCDT, Example 5.4x we get that P9 is a tilting right R-module. By part Ži., P9 is quasi-tilting. We also have the isomorphisms Ext Rny m Ž P9, P9. ( Ext Rny m Ž P, P Ž n.. [ Ext Rny m Ž P, P . ( Ext Rny my1 Ž SŽ mq1., P Ž n.. (???( Ext 1R Ž SŽ n y 1., P Ž n.. / 0. In particular, P9 is not almost tilting. In Example 1.5Žii., End P ( K, whence gl dim End P s 0. So gl dim R s gl dim End P q proj dim P q m y 1. In particular, the difference gl dim R y gl dim End P can be arbitrarily big even for almost tilting modules. On the other hand, we have THEOREM 1.6. Let R be a ring and P be an almost tilting module. Put S s End P. Then gl dim S F gl dim R q 1. Proof. W.l.o.g., we assume that d s proj dim P - `. By Theorem 1.3, it suffices to show that D s 1. Take M g Gen P such that M f Add P and proj dim M F d. Since Gen P s Pres P, we have the exact sequences in Mod-R 0 ª M1 ª P Ž k 1 . ª M 0 ª 0 ??? 0 ª Mdq 1 ª P Ž k dq 1 . ª Md ª 0,

Ž 3.

where k i are cardinals, Mi g Gen P for all i s 1, . . . , d q 1, and M0 s M.

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By induction on 0 F i F d, we prove that Ext Rdq1yi Ž Mi , Mdq1 . s 0. This is clear for i s 0. Since Gen P : P H dq 1y i , we have 0 s Ext Rdq 1yi Ž P Ž k i . , Mdq1 . ª Ext Rdq 1yi Ž Mi , Mdq1 . ª Ext Rdq 1yŽ iy1. Ž Miy1 , Mdq1 . , and the induction works. For i s d, we get that the sequence Ž3. splits. So Md g Add P, whence d M F d by Lemma 1.1. It follows that D s 1. COROLLARY 1.7. Let R be a ring and P be a quasi-tilting module such that proj dim P F 1. Put S s End P. Then gl dim S F gl dim R q 1. EXAMPLE 1.8. Ži. If T g Mod-R is tilting and S s End T, then T is also a tilting left S-module Žsee wM, Theorem 1.5x or wCF1, Proposition 1.1x., and Corollary 1.7 yields
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A module P is a )-module iff PR is a right R-)-module wCM, Proposition 1.10x, where R s RrAnn R Ž P .. Since S s End P s End PR , Theorem 1.3 can also be employed to estimate gl dim S by means of gl dim R q D, the number D being computed in Mod-R. Restricting to right artinian rings, we get thus a much more precise estimate for gl dim S. This is achieved by means of the following LEMMA 1.9. Let R be a ring, P g Mod-R, and R s RrAnn R Ž P .. Assume that R is finitely cogenerated. Then P is a )-module iff P is a tilting right R-module. Proof Ž Colpi .. By wCDT, Corollary 2.4x, Mod-R coincides with the class of all submodules of elements of Gen P. By wCM, Proposition 1.10x, P is a )-module iff PR is a right R-)-module. If this is the case, then Gen PR s PRH 1 by wC1, Proposition 4.5x. So P is a tilting right R-module by wCT, Proposition 1.3x. COROLLARY 1.10. Let R be a ring, P be a )-module, R s RrAnn R Ž P ., and S s End P. Assume that R is finitely cogenerated Ž e. g., assume that R is right artinian.. Then gl dim R y 1 F gl dim S F gl dim R q 1. Proof. This is by Example 1.8Ži. and Lemma 1.9.

2. RELATIONS BETWEEN THE GROTHENDIECK GROUPS LEMMA 2.1. Let R be a ring, P be a )-module, and S s End P. Then w dim S P F 1. Moreo¨ er, w dim S P s 0 iff PR is a quasi-progenerator. Proof. Let Q g Mod-R be an injective cogenerator and P* s Hom R Ž P, Q .. Let N g Mod-S. There exists a cardinal k and an exact sequence 0 ª N9 ª S Ž k . ª N ª 0 in Mod-S. The induced long exact sequence in Mod-R gives 0 ª Tor2S Ž N, P . ª Tor1S Ž N9, P . ª 0 ª Tor1S Ž N, P . ª N9 mS P ª P Ž k . ª N mS P ª 0. Since S g Cogen P*, we have N9 g Cogen P* s Ker Tor1S Žy, P ., and Tor2S Ž N, P . ( Tor1S Ž N9, P . s 0. Moreover, P is a quasi-progenerator iff Cogen P* s Mod-S iff P is a flat left S-module. DEFINITION 2.2. Let R be a ring and C : Mod-R be a class of modules closed under isomorphic images. For each C g C denote by w C x the isomorphism class of C. Assume that w C x< C g C 4 is a set.

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Define the Grothendieck group K Ž C . by K Ž C . s FrG, where F is the free abelian group generated by the set w C x< C g C 4 , and G is the subgroup of F generated by all elements of the form w C2 x y w C1 x y w C3 x, where C1 , C2 , C3 g C and there is an exact sequence 0 ª C1 ª C2 ª C3 ª 0 in Mod-R. For C g C denote by w C x the coset of w C x in K Ž C .. In particular, if l is an infinite cardinal and Cl s  M g Mod-R N genŽ M . - l4 , we define K Ž l, R . s K Ž Cl ., so that K Ž v , R . s K 0 Žmod-R.. For example, if R is a right artinian ring, then K 0 Žmod-R. coincides with the Grothendieck group of the class of all finite length modules over R in the sense of wARS, p. 5x. In this particular case, K 0 Žmod-R. can be identified with the free group generated by simp-R. Moreover, for a finitely generated module M, the coset w M x corresponds to the sequence formed by multiplicities of all elements of simp-R in a composition series of M wARS, I, Theorem 1.7x. DEFINITION 2.3. Let R be a ring, P g Mod-R, and S s End P. Denote by rnŽ P . the smallest infinite cardinal l such that Ži. genŽ I . - l for each right ideal I of S, Žii. genŽ P . - l, Žiii. if M g Gen P and genŽ M . - l then genŽ M9. - l for each submodule M9 of M. For example, rnŽ P . s v provided that P is finitely generated, and R and S are right noetherian rings. LEMMA 2.4. Let R be a ring, P be a )-module, S s End P, and l s rnŽ P .. Then the mapping w : K Ž l, S . ª K Ž l, R . defined by w w N x s w N mS P x y Tor1S Ž N, P . is an abelian group homomorphism. Proof. Take N g Mod-S such that genŽ N . - l. There is a cardinal k - l and an exact sequence 0 ª N9 ª S Ž k . ª N ª 0 in Mod-S. The induced long exact sequence in Mod-R gives 0 ª Tor1S Ž N, P . ª N9 mS P ª P Ž k . ª N mS P ª 0. Clearly, genŽ P Ž k . . - l, so genŽ N mS P . - l. By condition Ži. of Definition 2.3, genŽ N9. - l, so a similar argument gives genŽ N9 mS P . - l. Since N9 mS P g GenŽ P ., condition Žiii. gives genŽTor1S Ž N, P .. - l. Assume that N1 , N2 , N3 are - l-generated right S-modules such that there is an exact sequence 0 ª N1 ª N2 ª N3 ª 0 in Mod-S. By Lemma 2.1, the induced long exact sequence in Mod-R gives 0 ª Tor1S Ž N1 , P . ª Tor1S Ž N2 , P . ª Tor1S Ž N3 , P . g

ª N1 mS P ª N2 mS P ª N3 mS P ª 0.

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Clearly, the cosets of the elements wTor1S Ž N2 , P .x y wTor1S Ž N1 , P .x y wTor1S Ž N3 , P .x q wKer g x and w N2 mS P x y w N3 mS P x y w N1 mS P x q wKer g x are zero in K Ž l, R .. This proves that w is an abelian group homomorphism. The canonical isomorphism of K 0 Žmod-R. and K 0 Žmod-S . for tilting modules over finite dimensional algebras was proved in wHR, Proposition 3.2x Žcf. wBo, Theorem 1.6x.. The result was extended to ‘‘tilting modules of finite projective dimension’’ over artin algebras in wM, Theorem 1.19x. Considering different Grothendieck groups, Valenta obtained a similar result for tilting modules over semiperfect rings wV, Theorem 1.5x. The following is a generalization to )-modules over right noetherian rings: THEOREM 2.5. S s End P.

Let R be a right noetherian ring, P be a )-module, and

Ži. Assume that S is right noetherian. Then the mapping w : K 0 ŽmodS . ª K 0 Žmod-R. defined by w w N x s w N mS P x y Tor1S Ž N, P . is an abelian group homomorphism. Žii. Assume that either R s RrAnn R Ž P . is finitely cogenerated or P is a tilting module. Then the mapping c : K 0 Žmod-R. ª K 0 Žmod-S . defined by c w M x s Hom R Ž P , M . y Ext 1R Ž P , M . is an abelian group homomorphism. Žiii. Assume that S is right noetherian, and either R s RrAnn R Ž P . is finitely cogenerated or P is tilting. Then c and the mapping w : K 0 Žmod-S . ª K 0 Žmod-R., defined by w w N x s w N mS P x y Tor1S Ž N, P . , are in¨ erse abelian group isomorphisms. Proof. Ži. This is by Lemma 2.4 Žcf. wBa, Remark to IX.2.2x.. Žii. By Lemma 1.9, P is a tilting right R-module. In particular, there is an exact sequence in mod-R, 0 ª R ª P9 ª P Ž n. ª 0, with P9 g add P and n - v . Applying Hom R Ž P, y., we get S Ž n. ª Ext 1R Ž P, R . ª 0. Let M g mod-R, so R Ž m. ª M ª 0 for some m - v . Then Ext 1R Ž P, R Ž m. . ª Ext 1R Ž P, M . ª 0. It follows that Ext 1R Ž P, M . is a finitely generated right S-module. Further, Hom R Ž P, M . ( Hom R Ž P, TrP Ž M ... Since R is right noetherian, TrP Ž M . g mod-R. So there are p - v and an exact sequence in mod-R, 0 ª K ª P Ž p. ª TrP Ž M . ª 0. Applying Hom R Ž P, y., we get S Ž p. ª Hom R Ž P, Tr P Ž M .. ª Ext 1R Ž P, K . ª 0. It follows that Hom R Ž P, M . is a finitely generated right S-module. Since proj dim PR F 1, we infer that c is a well-defined group homomorphism.

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Žiii. By part Ži. Žwith R replaced by R ., w is a group homomorphism. By Lemma 1.9, P is a tilting right R-module. By wCF1, The Tilting Theoremx we get cw s id K 0 Žmod- S . and wc s id K 0 Žmod- R. Žcf. the proof of wCF3, Proposition 3.4x., so w and c are inverse group isomorphisms. Remark 2.6. The main application of Theorem 2.5 is to the right artinian case Žsee Corollary 2.7 below., but there are applications to other settings. For example, take an infinite ordinal s and consider the opposite ring, R9, of the skew-polynomial ring K w xb , rb , 0 - b - s x from wJ, Sect. 4x. Denote by S9 the set of all q g R9 such that the Žright. standard form of q contains a non-zero element of K Žsee wJ, Theorem 2.6x.. Let R s RXS9 be the right localization of R9 at S9. By wJ, Theorem 4.6x, each right ideal of R is two-sided, and of the form qR for a monic monomial q in the standard form. Moreover, all Žright. ideals are well-ordered by reverse inclusion. Take a monic monomial in the standard form, q, such that q s q1 x 1 for a monomial q1 and put R s RrqR. By wJ, p. 436x, R is finitely cogenerated. Note that R is right artinian iff all variables occurring in q1 belong to  x n < n - v 4. Let P be a free right R-module of rank m. Since Ann R Ž P . s qR, P is a )-module by wCM, Proposition 1.10x. Since R is right noetherian, the ring S s End P ( MmŽ R . is right noetherian as well Žbut not right artinian if some variables occurring in q1 do not belong to  x n < n - v 4.. So Theorem 2.5 applies, giving just an isomorphism of Grothendieck groups of the Morita equivalent rings R and S. COROLLARY 2.7. Let R be a right artinian ring, P be a )-module, R s RrAnn R Ž P ., and S s End P. Assume that S is right noetherian. Then the mapping w : K 0 Žmod-S . ª K 0 Žmod-R. defined by w w N x s w N mS P x y Tor1S Ž N, P . is an abelian group isomorphism. Note that the module P in Corollary 2.7 is of finite length, so the condition ‘‘S is right noetherian’’ is equivalent to ‘‘S is right artinian’’ wAF, Corollary 29.3x. Also, K 0 Žmod-R. ( K 0 Žmod-S . and K 0 Žmod-R. are free groups of rank equal to cardŽsimp-R. and cardŽsimp-R., respectively. So rankŽ K 0 Žmod-S .. F rankŽ K 0 Žmod-R... This may fail if S is not right noetherian: EXAMPLE 2.8. Let F, G be skew-fields such that F is a subring of G, the left dimension of G over F is 2, but the right dimension is l G v . Let R s Ž UF G0 . where G UF s Hom F ŽF G, F F .. Let P s Ž GF GG .rŽ G0 00 . and S s Ž GF G0 .. By wTa, Example to 3.7; V, pp. 6052]6053x, P is a tilting module, R is right artinian, but S s End P is not right noetherian.

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Clearly, K 0 Žmod-R. ( Z Ž2. is of rank 2. We prove that K 0 Žmod-S . is of infinite rank. First, S is semiprimary, right hereditary, and of right Loewy length 2. Also, simp-S s  X, Y 4 , where X s Ž F0 00 . is a projective right S-module and Y s Ž G0 G0 .rŽ G0 00 . is a Ý-injective right S-module. The set  X, Z 4 is an irredundant set of representatives of indecomposable projective right S-modules, where Z s Ž G0 G0 .. Let N g mod-S. Since N has Loewy length 2, we infer that N ( X Ž m. [ Y Ž n. [ M where m, n - v , M s Ý i- k Mi for some k - v , Tr Y Ž M . s 0, MrTrX Ž M . ( [i- k Ž Mi q Tr X Ž M ..rTr X Ž M ., and Mi s ZrIi for some right ideals Ii n Rad S, for all i - k. Since Tr Y Ž M . s 0, the Mi ’s can actually be taken to form a direct sum: M s [i - k Mi . A similar argument shows that if N9 is an S-submodule of N such that N9 is isomorphic to ZrI for some I : Rad S, then N9 is a direct summand of N. Moreover, the endomorphism ring of each of the direct summands in the decomposition N ( X Ž m. [ Y Ž n. [ Ž [i- k Mi .

Ž 4.

is local. Note that N is of finite length iff gen S ŽŽRad S .rIi . - v for all i - k. By wAF, Corollary 12.7x, the decomposition Ž4. complements direct summands. It follows that each short exact sequence in mod-S is of the form F [ G [ Ž [i- k Mi .

G [ Ž [i- k MiX . ª 0,

6

0 F[id G[ p

id F[ n

6

0 ª F [ X Ž p.

Ž 5.

where F and G are finitely generated, k, p - v , Mi and MiX are elements n of the set  ZrI < I : Rad S4 for all i - k, and the sequence 0 ª X Ž p. ª p X [i- k Mi ª [i- k Mi ª 0 is exact. Define an equivalence relation on the set of all right S-submodules of Rad S by I ; I9 iff there exist finitely generated right S-submodules C, D of Rad S such that I q C s I9 q D. Denote by A a set of representatives for ; . Since gen S ŽRad S . s l G v , A is infinite. Define a mapping t : K 0 Žmod-S . ª Z Ž A. as follows. For each I g A, let p I be the I-th projection of Z Ž A. onto Z. Define p I Žt w N x . as the number of the direct summands in the decomposition Ž4. that are isomorphic to ZrI9 for some I9 ; I. By Azumaya’s Theorem wAF, Theorem 12.6x and by Ž5., t is a well-defined surjective abelian group homomorphism. In particular, K 0 Žmod-S . possesses a direct summand isomorphic to Z Ž A., and K 0 Žmod-S . is of infinite rank.

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The module P in Example 2.8 is a noetherian module such that S s Hom R Ž P, P . is not a noetherian right S-module. Though P is a tilting module, the mapping c from Theorem 2.5Žii. does not induce a welldefined mapping between the Grothendieck groups of the classes of all noetherian modules. A remark before wCF3, Proposition 3.4x claiming the opposite is mistaken. In the right artinian case, we also obtain isomorphisms of Grothendieck groups of rings bound by a torsion theory counter equivalence in the sense of wCF2, Sect. 2x Žcf. wCF3, Proposition 3.4x.: THEOREM 2.9. Let R and S be right artinian rings. Assume that there is a torsion theory counter equi¨ alence Ž T , F . l l Ž X , Y . between Mod-R and Mod-S. Let I s Ann R Ž T ., J s Ann S Ž Y ., K s Ann R Ž F ., L s Ann S Ž X ., I9 s I q K, J9 s J q L, R s RrI, S s SrJ, R˜ s RrK, S˜ s SrL, R9 s RrI9, and S9 s SrJ9. Then there is a torsion theory counter equi¨ alence Ž T 9, F 9. l l Ž X 9, Y 9. between Mod-R9 and Mod-S9. Moreo¨ er, there are abelian group isomor˜. ( K 0 Žmod-S˜., K 0 Žmod-R9. ( phisms K 0 Žmod-R. ( K 0 Žmod-S ., K 0 Žmod-R K 0 Žmod-S9., and K 0 Žmod-R. ( K 0 Žmod-S .. Proof. Let S PR and R PSX be bimodules inducing the torsion theory counter equivalence Žsee wCF2, Theorem 2.5x.. By wCDT, Corollary 4.3x, Ž S, P, R . and Ž R, P9, S . are quasi-tilting triples in the sense of wCDT, Sect. 2x. By wCDT, Lemma 3.1x, Ann R Ž P . s I, Ann S Ž P . s J, Ann R Ž P9. s K, ˜ P9, S˜. are and Ann S Ž P9. s L. By wCDT, Corollary 2.4x, Ž S, P, R . and Ž R, w x tilting triples in the sense of CF1, p. 1588 . So Corollary 2.7 gives the ˜. ( K 0 Žmod-S˜.. isomorphisms K 0 Žmod-R. ( K 0 Žmod-S . and K 0 Žmod-R ˜ ˜ and X 9 s Let T 9 s T l Mod-R, F 9 s F l Mod-R, Y 9 s Y l Mod-S, Ž . X l Mod-S. Then T 9, F 9 is a torsion theory in Mod-R9, and Ž X 9, Y 9. is a torsion theory in Mod-S9. Since Ž R, P9, S . is a quasi-tilting triple, wCDT, Theorem 4.1x implies that the categories T 9 and Y 9 are equivalent Žvia Tor1R Žy, P9. and Ext 1S Ž P9, y... Similarly, X 9 and F 9 are equivalent Žvia Tor1S Žy, P . and Ext 1R Ž P, y... In other words, there is a torsion theory l Ž X 9, Y 9. between Mod-R9 and Mod-S9. counter equivalence Ž T 9, F 9. l Since R and S are right artinian, there are 0 - m, n - v such that K 0 Žmod-R. ( Z Ž m. and K 0 Žmod-S . ( Z Ž n.. Since any simple module is either torsion or torsion-free, we infer that m s cardŽsimp-R. q ˜. y cardŽsimp-R9. and n s cardŽsimp-S . q cardŽsimp-S˜. y cardŽsimp-R cardŽsimp-S9.. We have already proved that K 0 Žmod-R. ( K 0 Žmod-S . and ˜. ( K 0 Žmod-S˜., so cardŽsimp-R. s cardŽsimp-S . and cardŽsimpK 0 Žmod-R R˜. s cardŽsimp-S˜.. It follows that m s n iff K 0 Žmod-R9. ( K 0 Žmod-S9..

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By induction on the sum of the composition lengths of R and S, we prove that m s n. First, if R is a skew-field, then P and P9 are free modules of finite rank. Moreover, either P9 s 0 and P is a progenerator, or P s 0 and Gen PSX s Mod-S. In both cases, R and S are Morita equivalent, so K 0 Žmod-R. ( K 0 Žmod-S .. Similarly, K 0 Žmod-R. ( K 0 ŽmodS . provided that S is a skew-field. Further, if Ann R Ž T . s 0 s Ann S Ž Y . or Ann R Ž F . s Ann S Ž X . s 0, then clearly m s n. Otherwise, R9 has composition length strictly less than R, or S9 has composition length strictly less than S, and the induction works. PROPOSITION 2.10. Let R be a ring, P be a quasi-progenerator, S s End P, and l G v . Then the mapping f : K Ž l, S . ª K Ž l, R . defined by f w N x s w N mS P x is a group homomorphism. Moreo¨ er, if R is right artinian and l s v , then f is a group embedding. Proof. Take N g Mod-S such that genŽ N . - l. There is k - l and an epimorphism S Ž k . ª N ª 0 which induces an epimorphism P Ž k . ª N mS P ª 0. Clearly, genŽ P Ž k . . - l, so genŽ N mS P . - l. Let Pl s  M g Gen P N genŽ M . - l4 . By Lemma 2.1, the mapping c : K Ž l, S . ª K Ž Pl . defined by c w N x s w N mS P x is a group homomorphism. Let n : K Ž Pl . ª K Ž l, R . be the canonical group homomorphism induced by identity. Then f s nc is a group homomorphism. Moreover, define a mapping p : K Ž Pl . ª K Ž l, S . by p w M x s Hom R Ž P , M . . By wCM, Theoremx, p is a group homomorphism. Since P is a quasi-progenerator, N ( Hom R Ž P, N mS P . for all N g Mod-S, and M ( Hom R Ž P, M . mS P for all M g Pl . So c and p are inverse abelian group isomorphisms. Assume that R is right artinian and l s v . As above, f s nc . Since P is a quasi-progenerator, the coset w M x g K 0 Žmod-R. for a finitely generated module M g Gen P corresponds to the sequence of multiplicities of all simple modules from Gen P in a composition series of M. This shows that n , and hence f , is injective. EXAMPLE 2.11. Ži. If R is right artinian, then K 0 Žmod-R. is isomorphic to the free group generated by isomorphism classes of simple modules, so rankŽ K 0 Žmod-R.. s card simp-R. Take P g simp-R. Then P is a quasi-progenerator and S s End P is a skew-field, so K 0 Žmod-S . ( Z. It follows that even if the homomorphism f in Proposition 2.10 is an embedding, it is far from being an isomorphism in general. Moreover, Example 1.5Žii. provides an example of this phenomenon where P is an almost tilting module.

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Žii. If R is right noetherian, but not right artinian, then the group homomorphisms w and f from Theorem 2.5Ži. and Proposition 2.10 may even be zero. To see this let l s v and R be a principal ideal domain which is not a field. Let P g simp-R and S s End P. Clearly, P is a quasi-progenerator, S is a field, and Mod-S s Gen P : Mod-R. Let Pv s  M g Gen P N genŽ M . - v 4 and let n : K Ž Pv . ª K 0 Žmod-R. be the canonical group homomorphism induced by identity. Then n s 0. Indeed, if M g Pv then M ( P Ž m. for some m - v . From the exact sequence 0 ª R ª R ª P ª 0, we get w P x s w R x y w R x s 0 in K 0 Žmod-R.. Then w M x s 0, and n s 0. So f s nc s 0. Since P is a quasi-progenerator, we conclude that w s f s 0.

ACKNOWLEDGMENTS The author thanks Riccardo Colpi and Gabriella D’Este for many stimulating discussions concerning the topic of this paper.

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