On classes defining a homological dimension

ON CLASSES DEFINING A HOMOLOGICAL DIMENSION FRANCESCA MANTESE AND ALBER TO TONOLO Abstract. A class F of ob jects of an ab elian category A is said to define a homolo gica l dimension if for an y ob ject in A the length of any F -resolution is uniquely determined. In the presen t pap er w e in v estigate classes satisfying this property . Introduction In gene r al the class o f the ob jects of a given ab elian category A is to o complex to admit any satisfacto r y classification. Starting from a known sub class F of A , one may try to a ppro ximate arbitrar y ob jects by the ob jects in F . This appro a c h has succes s fully be en follow ed ov er the past few decades for ca teg ories of mo dules through the theor y of precov ers and preenv elopes, or left and rig ht approximations (see [6] or [8] for a detailed list o f r eferences). Another p oint o f v ie w c o uld be to measur e the “distance” o f a n y ob ject in A from the class F , int ro ducing a notion o f dimension with r espect to the clas s F , computed by means of F -resolutions. In this framework, the notio ns of pro jective dimension, weak dimension, Gor enstein dimension of modules hav e bee n deeply studied. Our aim is to define a g o o d concept of dimension with resp ect to a wide family of classes o f ob jects. W e say that a cla ss F of ob jects of a n ab elian categor y A defines a homolo gic al dimension if for any ob ject in A , the length of a n y F -resolution is uniquely determined (see Definition 1 .6). In such a wa y to each ob ject in A one can asso ciate a n F -inv a rian t num b er which r epresen ts lo cally the relev a nce of F . In the fir st section we s tudy several pro perties o f c la sses defining a homo lo gical dimension; in pa rticular we discuss their closure prop erties and the co nnec tio n with precov er cla sses and cotorsion pairs. In the second sectio n, using to ols from derived ca tegories, we g eneralize the Auslander notion of Gorenstein dimension to ar bitrary ab elian categ ories. W e consider a homological dimension asso ciated to an adjoint pair ( Φ, Ψ ) of contra v ariant functors, obta ining again the classical Gorenstein dimension on R -mo dules in case Φ = Ψ = Hom( − , R ) for a commutativ e no etherian ring R . 1. Homological dimension Definition 1. 1 (conf. [2]) . Let F b e a class of ob jects in a n ab elian catego ry A . W e say that an ob ject M in A has left F -dimension ≤ α , α ∈ N ∪ {∞} , if ther e Key wor ds and phr ases. homological dimensi on, abeli an categories, cotorsion pairs AMS classification 18G20, 16E10. Researc h of the second author supp orted by grant CDP A 0483 43 of Pado v a U ni v ersi t y . 1 2 F. MANTESE AND A. TONOLO exists a long exact sequence ... → F i → F i − 1 → ... → F 1 → F 0 → M → 0 with F i ∈ F ∪ { 0 } , and F i = 0 for i > α . W e denote b y F α , the cla ss of ob jects M of left F -dimens io n ≤ α (sho r tly F dim M ≤ α ), and by F < ∞ the class of ob jects of finite left F -dimension. In general there exist o b jects whic h hav e not a left F -dimension: in particular all ob jects whic h are not quotien ts of ob jects in F . W e denote by F the class of all ob jects in A w hich are homomorphic image of ob jects in F . Remark 1.2. If A ha s enough pro jectives and F is clo sed under direct summands, then F = A if and only if F contains all pr o jective ob jects. In particular, if A = R -Mo d, denoted by P and F l the classes of pro jectiv e and flat mo dules res pectively , then P = R -Mo d and F l = R -Mo d, and le ft P - a nd left F l - dimensio ns a re the usual pro jective and fla t (or weak) dimensions of a mo dule. Definition 1.3. W e say that A ha s glob al left F -dimension ≤ α (r esp. < ∞ ), α ∈ N ∪ {∞} , if for ea c h ob ject M in A we hav e F dim M ≤ α (resp. < ∞ ). Clearly A has glo bal left F -dimension ≤ ∞ if and only if A = F . In a n y ab elian ca tegory A it is p ossible (see [13, Ch. VII]) to define, for any pair of ob ject A , B ∈ A , the family Ex t i A ( A, B ) of equiv a lence c lasses o f exa ct seq ue nc e s of length i with left end B and r igh t end A , with resp ect to the Y oneda equiv alence relation. The family Ext i A ( A, B ) in general is not a set (see [7, Ch. VI]); nev ertheless it can be equipp ed with an additiv e structure and b ecome a big ab elian gr oup . The big ab elian gro ups are defined in the same way as ordinar y ab elian groups , except than the under lying class need not be a set. Quoting [1 3], “[...] w e are preven ted from talking ab out the category of big abelia n g roups beca use the class of morphisms b e tw een a given pair of big groups nee d not b e a set. Nevertheless this will no t keep us from talking abo ut kernels, cokernels, imag es, ex act sequences, etc., for big ab elian gr oups.” If A has enoug h injectives or pro jectives, then Ext i A ( A, B ) is an ab e lia n group for each A , B ∈ A . Given a cla s s of ob jects G , we denote by G ⊥ m = { M ∈ A : E x t i A ( G, M ) = 0 , ∀ 1 ≤ i ≤ m, G ∈ G } ; the intersection T m ≥ 1 G ⊥ m will be denoted b y G ⊥ ∞ . Dually , we denote by ⊥ m G = { M ∈ A : Ext i A ( M , G ) = 0 , ∀ 1 ≤ i ≤ m, G ∈ G } ; the intersection T m ≥ 1 ⊥ m G will b e denoted by ⊥ ∞ G . Definition 1.4 ([13, Ch. VI.6]) . Let A be an ob ject o f an abelian category A . The c ohomolo gic al dimension ch . dim A of A is the least integer n such that the one v ariable functor Ext n A ( − , A ) is not zero . If A has enough injective o b jects (e.g., if A is a Grothendiec k category) the cohomolog ical dimension of an ob ject co incides with its injectiv e dimension. Prop osition 1.5. Assume that A has enough pr oje ctives. (1) If gl F dim A ≤ n , n ∈ N , t hen ch . dim Y ≤ n for e ach Y ∈ F ⊥ n +1 . (2) If F = ⊥ m G for a class G of mo dules of c ohomolo gic al dimension less or e qual than n ∈ N , then gl F dim A ≤ n . ON CLASSE S DEFINING A HOMOLOGICAL DIMENS ION 3 Pr o of. Let M be a n ar bitrary ob ject in A . 1) Since gl F dim A ≤ n , there exists an exact sequence 0 → F n → F n − 1 → · · · → F 1 → F 0 → M → 0 . Applying the contrav aria n t functor Hom( − , Y ), since Y ∈ F ⊥ n +1 , b y dimension shift w e get Ext n +1 A ( M , Y ) ∼ = Ext 1 A ( F n , Y ) = 0. Since Ext n +1 A ( M , Y ) = 0 for e a c h ob ject M in A , and the latter has enough pro jectives, then Ex t n + i A ( M , Y ) = 0 for each i ≥ 1, i.e. c h . dim Y ≤ n . 2) Consider an exa ct sequence 0 → K n → P n − 1 → · · · → P 1 → P 0 → M → 0 with P i pro jective for i = 0 , . . . n − 1. Since P i ∈ F it is eno ug h to prove that K n belo ngs to F . So let G ∈ G ; then Ext i A ( K n , G ) ∼ = Ext n + i A ( M , G ) = 0 for 1 ≤ i . Therefore K n ∈ ⊥ ∞ G ⊆ ⊥ m G = F .  In order to introduce a go o d measure o f the distance b et ween an ob ject of A and a given class F , the length o f a F -r esolution has to b e uniquely determined. Definition 1. 6. W e say that the left F -dimension asso ciated to a class F is ho- molo gic al (or that the class F defines a homolo gic al dimension ) if (1) for any short exa ct sequence 0 → K → F → M → 0 with F ∈ F and M ∈ F ∞ , the ob ject K b elongs to F ∞ ; (2) for any exact sequence 0 → K n → F n − 1 → · · · → F 1 → F 0 → X → 0 with F i ∈ F , i = 0 , 1 , ..., n − 1, and X ∈ F n , the ob ject K n belo ngs to F . Clearly if A = F we have A = F ∞ , and the first condition is empty . Example 1.7. If A = R - Mo d, the classes P a nd F l define a homological dimension. The class o f free mo dules defines a homolog ical dimension if and only if it coincides with the class of pro jective mo dules (see Prop osition 1 .9), e.g . if R is lo cal. If A is the catego r y of co heren t sheav es on a no etherian scheme X , the classes of the lo cally free sheaves LF and o f the in vertible sheav es I b oth define a ho mo logical dimension (see [9, Chp. 2 § 5, Chp.3 § 6]). If X is quasi-pr o jective ov er Sp ec R , wher e R is a no etherian commutativ e ring, then LF = A . Note that the notio n of homological dimension can b e easily dualized obtaining a notion of homolo gic al c o dimension ; for insta nc e , if A = R -Mo d, the class I of injectiv e modules defines a homolo gical co dimension. Most of the res ults we obtain in this pap er could be reformulated for this dual co nc e pt. In the sequel we study clo sure pr operties of classes defining a homologica l di- mension. Let F be a c lass of mo dules and 0 → A → F → C → 0 b e an exact sequence with F ∈ F . Thus, for any i ≥ 1 in N , if A ∈ F i − 1 then C ∈ F i . Lemma 1.8 . L et F b e a class of obje cts in A and 0 → A → F → C → 0 b e an exact se quenc e with F ∈ F . If F defines a homolo gic al dimension and C ∈ F i , t hen A ∈ F i − 1 . In p articular F is close d under kernels of epimorph isms. Pr o of. By the definition of homolo g ical dimension, A belongs to F ∞ . Therefore consider an exact sequence 0 → K i − 1 → F i − 2 → · · · → F 1 → F 0 → A → 0 4 F. MANTESE AND A. TONOLO with F j ∈ F . Since 0 → K i − 1 → F i − 2 → · · · → F 1 → F 0 → F → C → 0 is an F - resolution for C and F dim C ≤ i , we get that K i − 1 ∈ F .  Prop osition 1. 9. L et F b e a class of obje cts defining a homolo gic al dimension. If F is close d under c ount able dir e ct sums, then F is close d under dir e ct summands. Pr o of. Let L ⊕ M = F ∈ F ; consider the short exact sequence 0 → L → L ⊕ ( M ⊕ L ) ( ω ) → ( M ⊕ L ) ( ω ) → 0; since b oth ( M ⊕ L ) ( ω ) and L ⊕ ( M ⊕ L ) ( ω ) ∼ = ( L ⊕ M ) ( ω ) belo ng to F , also L b elongs to F .  In the next theorem we compar e the F -dimension of ob jects in a sho rt exact sequence. Theorem 1.10. Assume F defin es a homolo gic al dimension and it is close d under finite dir e ct sums. Le t 0 → A → B → C → 0 b e a short exact se quenc e. Then for e ach i ∈ N we have t ha t (1 i ) if B and C b elong t o F i then A b elongs to F i ; (2 i ) if A and B b elong to F i then C b elongs to F i +1 . If F is close d un der extensions, then (3 i ) if A and C b elong to F i +1 , then B b elongs t o F i +1 ; (4 i ) if B ∈ F i and C ∈ F i +1 , then A b elongs to F i . Pr o of. (1) - (2): I f i = 0, 2 0 is clearly true by definition and 1 0 follows by F dim C = 0 ≤ 1 a nd the fact that F defines a homo logical dimension. Ass ume 1 i − 1 and 2 i − 1 true for i − 1 ≥ 0. Let us consider the pullback dia gram ( ∗ ) 0 0 0 / / A / / O O B / / O O C / / 0 0 / / P B / / O O F B / / O O C / / 0 K B O O K B O O 0 O O 0 O O with F B in F . 1 i : by Lemma 1.8 b oth K B and P B in diag r am ( ∗ ) b elong to F i − 1 , and so by induction A ∈ F i . ON CLASSE S DEFINING A HOMOLOGICAL DIMENS ION 5 2 i : Let now A and B b e in F i ; there ex ist F B ∈ F and a n epimorphism π : F B → B . Consider the following pullbac k diagram 0 0 0 / / A / / B g / / O O C / / O O 0 0 / / A / / P C p / / O O F B / / g ◦ π O O 0 K C O O K C O O 0 O O 0 O O Since P C is a pullback, there ex is ts j : F B → P C such that p ◦ j = 1 F B . Then the middle exa ct sequence splits, and ther efore P C = A ⊕ F B ; since F is closed under finite direct sums, P C belo ngs to F i . Therefore by 1 i we have K C ∈ F i and hence C b elongs to F i +1 . (3) - (4): If i = 0, 4 0 follows by the definition of homologica l dimension. Since F is closed under extensions, if A a nd C are in F , a lso B belo ngs to F . Then, if A a nd C b elong to F 1 , we can cons ider the pullback diagra m ( ∗ ) with F B in F . Since C b elongs to F 1 , then P B belo ngs to F ; since A b elongs to F 1 , then also K B belo ngs to F , and therefor e B b elongs to F 1 . Assume 3 i − 1 and 4 i − 1 true for i − 1 ≥ 0. 4 i : Let us consider the pullback diagr am 0 0 0 / / A / / B / / O O C / / O O 0 0 / / A / / P C / / O O F C / / O O 0 K C O O K C O O 0 O O 0 O O with F C ∈ F ; then K C belo ngs to F i . Since B b elongs to F i , by 3 i − 1 we hav e that P C ∈ F i , and hence, by 1 i , A b elongs to F i . 3 i : Since F is clos e d under extensions, we can consider the pullback dia g ram ( ∗ ) with F B in F . By L emma 1.8, P B belo ngs to F i ; then K B ∈ F i by 4 i , and hence B b elongs to F i +1 .  6 F. MANTESE AND A. TONOLO Remark 1.11. It follows that if F is closed under finite direct s ums and F is closed under extensions, then • the cla ss F < ∞ is clos ed under extensio ns, kernels of epimorphisms and cokernels of mo nomorphisms; • the class e s F i , i ≥ 0, are closed under kernels of epimo rphisms; if i ≥ 1, they are clos ed also under ex tens io ns. Prop osition 1.12. As s u me F defines a homol o gic al dimension, it is clo se d un- der fin ite dir e ct sums, and F = A . Then also F i and F < ∞ define a homolo gic al dimension for any i ≥ 1 . Pr o of. Since F = A , a lso F i = A = F < ∞ . Therefore co nditio n 1 in Definition 1.6 is empt y in b oth the cases. Let M be a n ob ject admitting an F i -resolution 0 → F i,n → F i,n − 1 → · · · → F i, 0 → M → 0 . Consider an exact s equence 0 → K → F ′ i,n − 1 → · · · → F ′ i, 0 → M → 0 with F ′ i,j ∈ F i . F rom the firs t sequence, applying recur siv ely Theo rem 1.10, 2), we get that M ∈ F n + i . Applying recurs iv ely Theorem 1.1 0, 4) to the sec o nd exact sequence we obtain that K ∈ F i . Since each finite F < ∞ resolution is actually a n F m resolution fo r a suita ble m ∈ N , we conclude that als o F < ∞ defines a homological dimension.  In ca se the ab elian categ ory A has e no ugh pr o jectives, a relev ant family of cla sses defining a homologica l dimension is given b y the left orthogo na l of any class . Prop osition 1 .13. Assume A has enough pr oje ctives, and let G b e a class of obje cts in A . Then F = ⊥ m G , 1 ≤ m ∈ N , defines a homolo gic al dimension if and only if F = ⊥ ∞ G . In su ch a c ase A = F . Pr o of. Assume F = ⊥ m G defines a homo logical dimension. Let us prov e that F = ⊥ m +1 G ; then we conclude inductiv ely . Co nsider an arbitra ry ob ject F ∈ F . Consider a short exact sequence 0 → K → P → F → 0 with P pro jective; since P b elongs to F , by Lemma 1.8 we hav e that also K ∈ F . Therefore for each G ∈ G we have Ext m +1 A ( F, G ) ∼ = Ext m A ( K, G ) = 0 , bec ause K ∈ F . Conv er sely , let us prove that F = ⊥ ∞ G defines a ho mological dimensio n. Clea rly , containing F the pro jectives, each ob ject has left F - dimens io n ≤ ∞ . Let M be an ob ject with F dim M ≤ n , n ∈ N . Then there exists an exact sequence 0 → F ′ n → F ′ n − 1 → · · · → F ′ 1 → F ′ 0 → M → 0 with F ′ i ∈ F for i = 0 , . . . , n . Le t us co ns ider a n exa ct sequence 0 → K n → F n − 1 → · · · → F 1 → F 0 → M → 0 with F i ∈ F for i = 0 , . . . n − 1 . Let us show that K n ∈ F . In fact, let X ∈ G . Then Ext i A ( K n , X ) ∼ = Ext n + i A ( M , X ) ∼ = Ext i A ( F ′ n , X ) = 0 for each i ≥ 1 .  ON CLASSE S DEFINING A HOMOLOGICAL DIMENS ION 7 Example 1.1 4. (1) Since Z has glo bal dimension 1, the class W = ⊥ 1 Z = ⊥ ∞ Z of Whitehead ab elian groups defines a homologica l dimension. By Prop osition 1.5, (ii) we hav e gl W dim Z ≤ 1. (2) An y tors io n free cla ss in a category of mo dules defines a homologica l di- mension, s ince it is closed under submo dules. In ge neral it is not the left orthogo nal of any class. Consider for exa mple the c la ss R o f reduced ab elian groups; since R ⊥ ∞ is the cla ss of divisible groups, ⊥ ∞ ( R ⊥ ∞ ) is the whole class o f a belian g roups. Ther efore R canno t be the left or tho gonal of a class, otherwise ⊥ ∞ ( R ⊥ ∞ ) would b e equal to R . In the following results we ar e interested in giving necessar y or sufficient condi- tions for a class defining a ho mo logical dimension to b e a left o r thogonal. Lemma 1.15. Assume A has en ough pr oje ctives. If F defines a homolo gic al di- mension and it c ontains the pr oje ctives, then F ⊥ 1 = F ⊥ ∞ . Pr o of. Let M b e an o b ject in F ⊥ 1 and F ∈ F . Consider a sho rt exact seq uence 0 → F ′ → P → F → 0 with P pro jective; since F defines a homo logical dimension also F ′ belo ngs to F . Applying Hom A ( − , M ) we get Ext i +1 A ( F, M ) ∼ = Ext i A ( F ′ , M ); then Ext 2 A ( F, M ) = 0 and we conclude by induction.  Theorem 1.1 6 . A ssume A has enough pr oje ctives, and let F b e a sp e cial pr e c over class. Then F defines a homolo gic al dimension if and only if F = ⊥ ∞ ( F ⊥ ∞ ) . Pr o of. If F = ⊥ ∞ ( F ⊥ ∞ ), by P ropo sition 1.13 we get that F defines a homologica l dimension. Conv er sely , suppo s e that F defines a homolog ical dimensio n. Let us prov e that F = ⊥ 1 ( F ⊥ ∞ ). Of cour se F ⊆ ⊥ 1 ( F ⊥ ∞ ). Let now M ∈ ⊥ 1 ( F ⊥ ∞ ); consider a sp ecial F -precover 0 → K → F → M → 0. Since by the pr evious lemma K ∈ F ⊥ 1 = F ⊥ ∞ , we get Ext 1 R ( M , K ) = 0. Since the specia l precov er clas s es are closed under direct summands [8, Section 2.1], then M ≤ ⊕ F belo ngs to F . Again by Prop osition 1.13 we conclude that F = ⊥ ∞ ( F ⊥ ∞ ).  Most of the examples o f classe s defining a homolog ical dimensio n give sp ecial precov ers. Nevertheless obser v e that this is not alw ays the case : E klof and Shela in [5] prov ed that, consistently with ZFC, the class of Whitehead abelia n gr oups, which defines a homolog ical dimension (see E xample 1 .1 4), do es no t pr o vide pr eco vers. In particular they prov ed that Q , which has W - dimension 1, do e s not a dmit W - precov er. Remark 1. 17. If F is a sp ecial precov er clas s and it defines a homological dimen- sion, then for each mo dule M it is p ossible to get an F -r esolution · · · → F i → · · · → F 1 → F 0 → M → 0 such that, denoted by Ω i F ( M ) the i -th F sy zygy of M , the induced map F j → Ω j − 1 F ( M ) is a sp ecial F -prec over of Ω j − 1 F ( M ). Therefore, in such a case o ur defini- tion o f F -dimension coincides with the definition g iv en by Eno chs a nd Jenda (see [6, Definition 8.4.1]). Other sig nificativ e classes defining a homo logical dimension ar e those studied by Auslander-B uc hw eitz in [2]. In that pap er they introduced the notion of E xt - inje ctive c o gener ator for an additively closed exact sub category F of A : an addi- tively closed subcateg ory ω ⊆ F is a n Ext-injective cogener ator for F if ω ⊆ F ⊥ ∞ 8 F. MANTESE AND A. TONOLO and for any F ∈ F there exists an exact sequence 0 → F → X → F ′ → 0 where F ′ ∈ F a nd X ∈ ω . Prop osition 1.18. [2, Prop ositions 2.1, 3.3 ] L et F b e an additivel y close d ex- act sub c ate gory of A close d under kernels of epimorphisms. If F admits an E xt - inje ctive c o gener ator ω , t hen F defines a homolo gic al dimension. Mor e over, if any obje ct has finite F -dimension, then F = ⊥ ∞ G , wher e G is the class of obje cts in A of fin ite ω -dimension. W e conclude this section remar king the connec tion b et w een class es defining a homologica l dimension and cotorsion pairs in categor ies of mo dules. So we assume A = R -Mo d, the category of left R -mo dules ov er a ring R . Definition 1.19. Let A and B b e tw o classes of mo dules. The pair ( A , B ) is called a c otorsion p air if A = ⊥ 1 B and A ⊥ 1 = B . The pa ir ( A , B ) is ca lle d an her e ditary c otorsion p air if A = ⊥ ∞ B or equiv alently A ⊥ ∞ = B . W e stress that, b y P r opos ition 1.13, the her editary cotors io n pairs a re exactly the cotorsio n pa irs ( A , B ) such tha t A defines a homologica l dimension. Example 1.20. Let R b e a commut ative doma in. A module M is Matlis c otorsion provided that Ext 1 R ( Q, M ) = 0, where Q is the quotient fie ld of R . Since Q is flat, the clas s MC of Matlis cotorsion mo dules cont ains the class E C := F l ⊥ 1 of Eno chs c otorsion mo dules . D enoted b y T F the cla ss of tors ion-free mo dules, the la tter class E C contains the class W C := T F ⊥ of W a rfield cotors ion mo dules. Th us we hav e the fo llowing chain of c otorsion pairs, or dered with resp ect to the inclusion on the first class: ( ⊥ 1 MC , MC ) ≤ ( F l = ⊥ 1 E C , E C ) ≤ ( T F = ⊥ 1 W C , W C ) . The mo dules in ⊥ 1 MC are called st r ongly flat . The Eno c hs and W arfield cotorsio n pairs ( F l , E C ) a nd ( T F , W C ) are hereditar y and the classes of flat and torsion free mo dules , as well known, define a homo logical dimensio n. In genera l the Matlis cotorsio n pair ( ⊥ 1 MC , MC ) is not here dita ry and therefo r e strongly flat mo dules do not define a homolo gical dimension; precis ely , the Matlis cotorsion pair is heredita ry , and s o s trongly fla t mo dules define a ho mological dimensio n, if and only if the quotient field Q of R has pro jective dimension ≤ 1, i.e. R is a Ma tlis domain [12, Section 10]. 2. Generalizing the G o renstein dimension Auslander in [1] introduced the notion of Gorenstein dimension for finite mo d- ules over a comm utative no etherian ring . More precisely , let R be a commutativ e no etherian r ing; following [4, Definition 1 .1.2] w e say tha t a finite R -mo dule M belo ngs to the G-class G ( R ) if : (1) Ext m R ( M , R ) = 0 for m > 0 (2) Ext m R (Hom R ( M , R ) , R ) = 0 for m > 0 (3) the ca no nical mor phism δ M : M → Hom R (Hom R ( M , R ) , R ), δ M ( x )( ψ ) = ψ ( x ), is an isomor phism. An y finite mo dule a dmitting a G ( R )-resolutio n o f length n is said to have Gor en - stein dimension at mos t n . In [4, Theo rem 1.2.7 ] it is shown that G ( R ) defines a homologica l dimensio n on the category of finite R -mo dules. ON CLASSE S DEFINING A HOMOLOGICAL DIMENS ION 9 Given an ab elian categor y A , w e deno te by K ( A ) (resp. K + ( A ), K − ( A ), K b ( A )) the homotopy catego ry of unbo unded (r esp. bo unded b elow, b ounded ab ov e, bo unded) c o mplexes o f ob jects of A and by D ( A ) (resp. D + ( A ), D − ( A ), D b ( A )) the asso ciated derived ca tegory . In the sequel with D ∗ ( A ) or D † ( A ) we will denote any of these derived ca tegories. Consider a r igh t adjoint pair o f contra v ariant functor s ( Φ, Ψ ) b e tween the ab elian categorie s A and B , with the natural morphisms η a nd ξ as unities. F ollowing [9, Theorem 5.1], to guara ntee the existence of the derived functors R ∗ Φ : D ∗ ( A ) → D ( B ) and R † Ψ : D † ( B ) → D ( A ), w e assume the existence of triangulated subc ate- gories P o f K ∗ ( A ) and Q of K † ( B ) s uc h tha t: • every ob ject of K ∗ ( A ) a nd every ob ject of K † ( B ) admits a quasi-isomo rphism int o ob jects of P a nd Q , r espectively; • if P and Q are exact complexes in P and Q , then also Φ ( P ) and Ψ ( Q ) are exact. Given complex e s X ∈ D ∗ ( A ) and Y ∈ D † ( B ), we have R ∗ ΦX = ΦP and R † Ψ Y = Ψ Q , where P is a complex in P qua si-isomorphic to X , and Q is a complex in Q quasi-iso morphic to Y . The functor Φ has c ohomolo gic al dimension ≤ n if, for each A in A , w e hav e H i ( R ∗ ΦA ) = 0 for | i | > n . An ob ject A in A is called Φ -acycl ic if H i ( R ∗ ΦA ) = 0 for an y i 6 = 0. Similarly , Ψ -acyclic ob jects in B are defined. Definition 2.1. W e say that an ob ject A ∈ A b elongs to the class G ΦΨ if (1) A is Φ -acyclic; (2) Φ ( A ) is Ψ -a c yclic (3) the morphism η A : A → Ψ Φ ( A ) is an iso mo rphism. Note that, since the catego ry of mo dules ov er a r ing R has enough pro jectives, the total de r iv ed functor R Hom( − , R ) alw ays exists (see [14]). Thus the class G ΦΨ for the adjoint pair ( Φ, Ψ ) = (Hom( − , R ) , Hom( − , R )) in the categor y of finite R - mo dules, coincides with the G ( R )-class in tro duced a bov e if R is a commutativ e no etherian ring. W e want to prov e that the class G ΦΨ asso ciated to the right a djoin t pair ( Φ, Ψ ) alwa y s defines a homological dimension. First we pr ove that the G ΦΨ -dimension can b e computed using the cohomolog y groups H i ( R ∗ Φ ). As a c onsequence it follows that, when the c ategory A has enough pro jectives, the G ΦΨ -dimension can b e compa red with the pro jective dimension (conf. [4, P ropo sition 1.2.10 ]). Prop osition 2.2. L et A b e an obje ct in A of fin ite G ΦΨ -dimension. Then (a) G ΦΨ -dim A = sup { i : H i ( R ∗ ΦA ) } 6 = 0 (b) If A has enough pr oje ctives, then G ΦΨ -dim A ≤ p d A Pr o of. (a) Let G ΦΨ -dim A = n . Ther e fore there exis ts an exact sequence 0 → G n → G n − 1 ... → G 0 → A → 0 with G i ∈ G ΦΨ , i = 0 , 1 , ..., n . By shift dimension we get H i ( R ∗ ΦA ) = 0 for each i > n . If sup { i : H i ( R ∗ ΦA ) 6 = 0 } < n , let K b e the cokernel of G n → G n − 1 . W e will prov e that K b elongs to G ΦΨ contradicting the assumption G ΦΨ -dim A = n . Indeed, K is Φ -acyclic sinc e H i ( R ∗ ΦK ) ∼ = H ( i + n − 1) ( R ∗ ΦA ) = 0 for each i > 0 ; applying Ψ to the short exact sequence 0 → ΦK → ΦG n − 1 → ΦG n → 10 F. MANTESE AND A. TONOLO 0 and comparing it with the short ex act sequence 0 → G n → G n − 1 → K → 0, we get that ΦK is Ψ -acyclic and the unity η K is an isomor phism. (b) If A ha s enough pro jectives, then any ob ject A in A admits a pr o jective resolution P . Since the pro jectives a re Φ -acyclic, we hav e R ∗ ΦA = ΦP and then sup { i : H i ( R ∗ ΦA ) 6 = 0 } = sup { i : H i ( ΦP ) 6 = 0 } ≤ p d A.  Observe that, differently from the G ( R )-dimension, the inequa lity betw een the G ΦΨ -dimension and the pro jective dimension c a n b e strict also for ob jects of finite pro jective dimension (cf. [4 , Pr o position 1.2.10 ]). Example 2.3 . Let Λ b e the path algebr a of the quiver 1 / / 2 / / 3 Let us consider the mo dule Λ U = 1 2 3 ⊕ 2 3 ⊕ 2 and let S = End Λ ( U ). Co nsider the adjoint pair (Hom Λ ( − , U ) , Hom S ( − , U )): since Ext 1 Λ ( U, U ) = 0, Ext 1 S ( S, U ) = 0 and U ∼ = Hom S (Hom Λ ( U, U ) , U ), the Λ-mo dule U belo ngs to G ΦΨ , where ( Φ, Ψ ) = (Hom Λ ( − , U ) , Hom S ( − , U )). Thus U has pro jective dimension one, but ob viously G ΦΨ -dimension 0. In or der to prove that the class G ΦΨ defines a homologica l dimension, we also need to reca ll some no tions and results on derived catego r ies. By [10, Lemma 13.6 ] we know that, in our a ssumptions, ( R ∗ Φ, R † Ψ ) is a r ig h t a djo int pair in the der iv ed categorie s D ∗ ( A ) and D † ( B ), with unities ˆ η a nd ˆ ξ natura lly inherited from the unities η a nd ξ . In [1 1] a complex X ∈ D ∗ ( A ) is called D -r eflexive if the morphism ˆ η X is an isomorphism in D ∗ ( A ). An ob ject A ∈ A is called D -r eflexive if it is D - reflexive as a stalk complex. Lemma 2.4 . L et X ∈ A such that X is Φ -acyclic and Φ ( X ) is Ψ - acy clic. Then ˆ η X is a quasi-isomo rphism if and only if η X is an isomorphism. In p articular any obje ct in G ΦΨ is D -r eflexive. Pr o of. In general, if C ∈ D ∗ ( A ) and L is a complex q uasi-isomorphic to C such that a n y term L i of L is Φ -acyclic and Φ ( L i ) is Ψ -acyclic, then ˆ η C coincides with η L , where η L is the ter m-to-term extension of the unit y η to the triangula ted ca tegory K ∗ ( A ) (cf. [11]). Then we easily g et the statemen t.  Corollary 2.5. Any obje ct A in A of finite G ΦΨ -dimension is D -r eflexive. Pr o of. Let G ΦΨ -dim A = n . Therefore there exists an exact sequence 0 → G n → G n − 1 ... → G 0 → A → 0 with G i ∈ G ΦΨ , i = 0 , 1 , ..., n . Therefore in the b ounded derived categ ory D b ( A ), A is q uasi-isomorphic to the co mplex G := 0 → G n → G n − 1 ... → G 0 → 0. Since G is a complex with D -reflexive ter ms by Lemma 2.4, we conclude by [11, Theor em 3.1,(1 )] that A is D -reflexive.  Prop osition 2.6. If X ∈ A is Φ -acyclic and D -re flexive, t hen X b elongs t o G ΦΨ . Pr o of. Since X is Φ -a cyclic, R ∗ ΦX is q ua si isomorphic to the stalk complex Φ ( X ). Moreov er, for X is D -r eflexiv e, w e get that R † Ψ ( ΦX ) ∼ = R † Ψ ( R ∗ ΦX ) is quasi- isomorphic to X . Thus H i ( R † Ψ ( ΦX )) = 0 for an y i 6 = 0 and so ΦX is Ψ -acyclic. Finally we conclude since , by the pr evious lemma, η X is an isomor phism.  Theorem 2.7. The class G ΦΨ defines a homolo gic al dimension. ON CLASSE S DEFINING A HOMOLOGICAL DIMENS ION 11 Pr o of. Let us consider a long exact seq ue nc e 0 → G n → G n − 1 → · · · → G 0 → X → 0 with G i ∈ G ΦΨ . B y Corolla r y 2.5, X is D -reflexive. Consider now a long exa ct sequence 0 → X n → F n − 1 → · · · → F 0 → X → 0 with F i ∈ G ΦΨ . The D -reflexive o b jects ar e a thick sub category of A (see [11]), i.e, if tw o terms of a short exact s equence in A are D - reflexive, then also the third is D -reflexive. Therefore, by induction, it follows that X n is D -reflexive. Since by Pr opos ition 2.2 H i ( R ∗ ΦX ) = 0 fo r ea c h i > n , b y shift dimensio n X n is Φ -acyclic, and so w e conclude that X n belo ngs to G ΦΨ .  In [11], the authors were interested in characterizing the D -r eflexiv e ob jects asso ciated to a given a djoin t pair ( Φ, Ψ ). Assume A is a module categor y and denote b y F P n the class of mo dules A whic h hav e an exa ct reso lution P n − 1 → ... → P 1 → P 0 → A → 0 , where the P i ’s are finitely generated pro jectives. In particular F P 1 is the cla ss o f finitely gener ated mo dules. Then the D -reflexive mo dules in F P n can b e charac- terized through their G ΦΨ -dimension. Theorem 2. 8. Le t A = R -Mod for an arbitr ary ring R . Assume R R to b e D - r eflexive and Φ of c ohomolo gic al dimension ≤ n . Then a mo du le M ∈ F P n is D - r eflexive if and only if it has G ΦΨ -dimension ≤ n . Pr o of. The sufficiency of the finiteness of the G ΦΨ -dimension is pro v ed in Corol- lary 2.5. Conversely , supp ose M to be a D -r eflexiv e mo dule in F P n . Let 0 → K → P n − 1 → · · · → P 0 → M → 0 be an exa ct s equence with the P i ’s finitely generated pro jectives. Since R R is a ssumed to b e D -reflexive, any P i is D -reflexive, a nd so we g et that K is D -reflexive. Since Φ has cohomolog ical dimension ≤ n , by shift dimension we ge t that K is Φ -a c y clic. Then, by P r opos ition 2.6, we conclude that K b elongs to G ΦΨ .  References [1] M. Auslander, A nne aux de Gor enst e in, et torsion en alg` ebr e c ommutative. S´ eminaire d’Alg` ebre Comm utativ e dirig´ e par Pi erre Samuel, 1966/67. T exte r´ edig´ e, d’apr` es des expos´ es de Mauri ce Ausl ande r, Mar querite Mangeney , Christian Peskine et Lucien Szpiro. ´ Ecole Nor- male Sup´ erieure de Jeunes Fill es Secr ´ etariat math´ ematique, Paris 1967. [2] M. Auslander, R.O. Buch weitz, The homolo gic al the ory of maximal Cohen-Mac aulay appr ox- imations , M m. Soc. Math. F rance (N.S.) N o. 38 (1989), 5–37. [3] S. Bazzoni, L. Salce, On str ongly flat mo dules over integr al domains , Ro c kyMountain J. Math. 34 (2004) , no. 2, 417–43 9. [4] L. W. Chri stense n, Gor enstein Dimension . Lecture Notes in Mathematics, 1747. Springer- V erl ag, Berlin. [5] P . C. Eklof , S. Shelah, O n the existenc e of pr e co vers , Illinois J. Math. 47 (2003), no. 1- 2, 173–188. [6] E. E. Eno c hs, O. M. G. Jenda. R elative homolo g ic al algebr a. Exp osition in M ath. 30, de Gruyter 2000. [7] P . F reyd, A b elian c ate gories. Harp er’s Series in Mo dern M ath ematics, Harp er and Row, New Y ork- Ev anston-London , 1964. [8] R. G¨ obel, J. T rlif a j, Appr oximations and e ndo morphism algebr as of mo dules. Exposition in Math. 41, de Gruyter 2006. [9] R. Hartshorne, A lgebr aic ge ometry . Graduate T exts in M athematics 52, Springer-V erlag, New Y ork- Heidelberg, 1977. [10] B. Keller, Derive d Cate gories and their use , Handbo ok of Algebra. V ol. 1. Edited by M. Hazewink el, N orth-Holland Publishing Co., Amsterdam, 1996. [11] F.Mante se, A . T onolo, R eflexiv ity in derive d c ate g ories , arXiv: 0705.2537 12 F. MANTESE AND A. TONOLO [12] E. Matlis, T orsion-fr e e mo dules. Chicago Lectures in Mathematics. The Unive rsity of Chicago Press, Chicago-London, 1972. [13] B. M itc hell, The ory of cate gories. P ur e and Applied Mathematics, Academic Press, New Y ork- Londo n, 1965. [14] N. Spalte nstein, R esolutions of unb ounde d c omplexes , Comp ositio Math. 65 (1988), no. 2, 121–154. (F. Man tese) Dip ar timento di Informa tica, Universit ` a degli Studi di Verona, strada Le Grazie 15, I-37 134 Verona - It al y E-mail addr ess : mantese@s ci.univr .it (A. T onolo) Dip. Ma tema tica Pura ed Applica t 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 : tonolo@ma th.unipd .it