Reflexivity in Derived Categories

REFLEXIVITY IN DERIVED CA TEGORIES FRANCESCA MANTESE AND ALBER TO TONOLO to the memory of our friend and co l le ague Silvia Lucido Abstract. An adjoint pair of con tra v ariant functors bet we en abelian cate- gories can b e extended to the adjoint pair of their derived functors in the associated derived categories. W e describ e the reflexive complexes and i n ter- pret the achiev ed results in terms of ob jects of the ini tial abelian categories. In particular w e prov e that, for functors of an y finite cohomological dimension, the ob jects of the ini tial ab elian categories whic h are reflexive as stalk com- plexes form t he largest cl ass where a Cotilting Theorem in the sense of Colby and F ull er [CbF1, Ch. 5] w orks . Introduction Adjoin t pa ir s of functor s in derived categories are deeply s tudied by several authors (see for instance [CPS, Hap, Har, R]). A larg e class of thes e pairs is obtained extending ([K, L e mma 13.6 ]) adjoint functors betw een ab elian categor ies to their deriv ed functors. In this paper w e focus on the a djunctions and on the corres p onding dualities obtained by extending contrav a rian t adjoin t functors. Wide classes of examples arise in mo dule and sheaf catego r ies. Both the adjunctions, the one in the ab elian categorie s a nd the one in the a sso- ciated derived categor ies, determine a differ en t notion of reflexivity: to distinguish them, we will call D -r eflexive the complexes which a re re flexiv e with resp ect to the adjunction in the derived categories. An ob ject of the starting ab elian categorie s is ca lled D -r eflexive if it is D -reflexive a s stalk complex, simply r eflexive if it is reflexive with resp ect to the adjunction in the abelian catego ries. Our main aim is to describ e the D -reflexive complexes and to study the D - reflexive ob jects o f the initial ab elian categ o ries. Reading on the under ly ing ab elian categorie s all the effects of the dua lit y in the corr esponding derived categories , we prov e that for an adjoin t pair of functors o f co homological dimension 1, the D - reflexive ob jects are e x actly thos e for whic h a Cotilting The or em in the sense of Colby and F uller ([CbF1, Ch. 5 ]) holds (Theorem 4.3). Our approa c h allows o n one side to read this celebrated r e sult in its traditional framework as a natural consequence of the duality betw een derived categor ies induced by contra v ariant Hom-functors asso ciated to a cotilting bimodule. On the other s ide it p e rmits to generalize the Cotilting The o rem to ar bitrary ab e lian categor ies and adjoin t pair s of co n trav ariant functors of an y finite cohomo logical dimension. Thus, on the o ne hand we get a general a nd unitary version of all the several c a ses considered in the literature of dualities induced b y cotilting bimodules of injectiv e dimension 1 (see [Cb, Cb1, CbF, CbF1, Cp, CpF, Ma, T]). On the other hand, under coho mological conditions automatically sa tisfied in the tr a ditional settings , we succeed in finding Researc h supported by gran t CPDA071 244/07 of Pado v a Uni v ersity . 1 2 F. MANTESE AND A. TONOLO po sitiv e results which generalize to ar bitrary abe lia n categories and adjoint functors of cohomologic a l dimension n the res ults obtained by Miyashita [M] for a cotilting bimo dule o f injective dimensio n n in the noetherian cas e. In the first section w e recall some preliminaries on deriv e d functors and their prop erties; par ticular a ttention is dedicated to the notion of way-out functor. In the s e c ond s ection we describ e the adjoint pa irs w e are interested in, and we compare the related notions of reflexivity in the a belian and in the asso c ia ted de- rived categorie s. In particular we give exa mples of D -reflexive ob jects in the starting ab elian categor ies which are not reflex iv e and, conv erse ly , of r eflexiv e ob jects which are not D -reflexive. In the third section we inv estig a te the relation b et ween the D -reflexivit y o f a complex and of its terms or its cohomo logies. W e show that if the cohomolog ie s or the terms are D -reflexive, then so is the complex itself. The c o n verse in genera l is not true (see Examples 3 .2, 3.3). Assuming that the functors ha ve cohomological dimension at most o ne, we prov e that a complex is D -reflexive if and only if its cohomolog ies are D -reflexive (s e e Coro llary 3.6). In the fourth a nd fifth sections we study in details the D -reflexive o b jects in the ab e lian ca teg ories w e start from. The fourth is dedicated to the fav o rable case of functors of c ohomological dimensio n ≤ 1. W e prove that a Cotilting Theo - rem [CbF1] for the classes of D -reflexive ob jects (Theo rem 4.3) ho lds. Finally the fifth section is devoted to the ca se of functors with arbitrary finite cohomolo g ical dimension, ass uming the ab elian categories ha ve eno ug h pro jectives. The la tter hypothesis permits us to use the standard to ol of sp ectral seq uences. F or a spectr al sequence ana lysis in the co v ariant case see [BB1]. This appr oac h allows us to reveal the coho mological conditions (Condition I, II page 22) necessary to generalize the results obtained in cohomolo gical dimension ≤ 1 (Theor ems 5.2, 5 .4). In particular we completely r e c o ver the results obtained for mo dule ca tegories by Miyashita [M] in the no etherian ca se and in [A T] for arbitrar y asso ciativ e ring s. Several examples o ccur along the whole pa p er describing patholo gies and p ositive results. F or the unexpla ined notations in module theory we refer to [AF], for thos e in sheaf theo ry to [Har2]. W e follow [Ha r , W] for definitio ns and results reg arding derived categor ies, derived functors a nd sp ectral sequences. 1. Prelimin a ries Given an ab elian catego ry A , we denote b y K ( A ) (r esp. K + ( A ), K − ( A ), K b ( A )) the homotopy ca tegory of unbounded (resp. bo unded b elo w, b ounded above, bo unded) co mplexes o f ob jects of A and b y D ( A ) (resp. D + ( A ), D − ( A ), D b ( A )) the asso ciated derived c a tegory . In the sequel with D ∗ ( A ) or D † ( A ) we will denote any of these der iv ed categor ies. Moreov er, with D ∗ ≤ n ( A ), n ∈ Z , we mean all the complexes in D ∗ ( A ) whose cohomologies are zero in any degree greater than n . Similarly , we define D ∗ ≥ n ( A ). All the considered functors betw een der ived c a tegories ar e assumed to be δ - functors , i.e. they commut e with the shift functor and send triangles to triangles . Given an o b ject M ∈ A , we contin ue to denote by M also the stalk c omplex in D ( A ) a ssocia ted to M , i.e. the complex with M concentrated in deg r ee zer o. REFLEXIVITY IN DERIVED CA TEGORIES 3 Let X : · · · → X − 1 g − 1 → X 0 g 0 → X 1 → . . . b e a complex in D ( A ). F o r any integer n ∈ Z w e define the following tr uncations: τ >n X : · · · → 0 → X n +1 → X n +2 → . . . τ ≤ n X : · · · → X n − 1 → X n → 0 → . . . σ >n X : · · · → 0 → X n / ker g n → X n +1 → . . . σ ≤ n X : · · · → X n − 1 → ker g n → 0 → . . . In particular, for any n ∈ Z ther e a re the following triangles: τ >n X → X → τ ≤ n X → τ >n X [1] σ ≤ n X → X π X → σ >n X → σ ≤ n X [1] . In this section we study the b eha vior o f the comp osition of contrav ar ian t way-out functors a nd the relations among the w ay-out conditio ns, the finite cohomolo gical dimension and the closure prop erties of the acyclic ob jects asso ciated to a c o n- trav ariant functor. Let us first recall the definition of way-out functors, as in [Har, Chp. I § 7] and [L, Chp. I § 11]. Definition 1.1. Let A and B be a belian categor ies and let F : D ∗ ( A ) → D ( B ) b e a cov a rian t (resp. contra v ariant) functor. (1) The functor F is way-out left if there ex is ts n ∈ Z such that F ( D ∗ ≤ 0 ( A )) ⊆ D ≤ n ( B ) (resp. F ( D ∗ ≥ 0 ( A )) ⊆ D ≤ n ( B )); in such a c a se we de fine the u pp er dimension o f F setting dim + F = inf { n : F ( D ∗ ≤ 0 ( A )) ⊆ D ≤ n ( B ) } (resp. = inf { n : F ( D ∗ ≥ 0 ( A )) ⊆ D ≤ n ( B ) } ) . (2) The functor F is way-out right if there ex is ts n ∈ Z such that F ( D ∗ ≥ 0 ( A )) ⊆ D ≥ n ( B ) (resp. F ( D ∗ ≤ 0 ( A )) ⊆ D ≥ n ( B )); in such a c a se we de fine the lower dimension of F setting dim − F = s up { n : F ( D ∗ ≥ 0 ( A )) ⊆ D ≥ n ( B ) } (resp. = sup { n : F ( D ∗ ≤ 0 ( A )) ⊆ D ≥ n ( B ) } ) . Remark 1.2. Let F : D ∗ ( A ) → D ( B ) b e a cov ariant (res p. co n trav ariant) functor . If F is w ay-out left a nd dim + F = m , then for a n y k ∈ Z F ( D ∗ ≤ k ( A )) ⊆ D ≤ k + m ( B ) (resp. F ( D ∗ ≥ k ( A )) ⊆ D ≤ m − k ( B )) Analogously , if F is wa y - out right and dim − F = m , then for any k ∈ Z F ( D ∗ ≥ k ( A )) ⊆ D ≥ k + m ( B ) (resp. F ( D ∗ ≤ k ( A )) ⊆ D ≥ m − k ( B )) . Clearly , if F is both wa y -out left and right , then it is b oun de d , i.e. it se nds bo unded complexes in D ∗ ( A ) to b ounded complexes in D ( B ). The following easy pr oposition will b e useful in the sequel. Prop osition 1. 3. L et G 1 : D ∗ ( A ) → D † ( B ) and G 2 : D † ( B ) → D ( C ) b e two c ontr avariant functors and F = G 2 G 1 their c omp osition. (1) If G 1 is way-out left with dim + G 1 = m 1 and G 2 is way-out right with dim − G 2 = m 2 , then F is way-out rig ht with dim − F = m 2 − m 1 ; (2) if G 1 is way-out right with dim − G 1 = m 1 and G 2 is way-out left with dim + G 2 = m 2 , then F is way-out lef t with dim + F = m 2 − m 1 . F rom no w on w e denote by Φ : A → B and Ψ : B → A tw o a dditive non zero co n- trav ariant functors betw een the ab elian categories A and B . F ollo wing [Har, The o - rem 5.1], to guarantee the existence of the deriv ed functors R ∗ Φ : D ∗ ( A ) → D ( B ) and R † Ψ : D † ( B ) → D ( A ), we assume the existence of triangulated sub categories P of K ∗ ( A ) and Q of K † ( B ) such that: 4 F. MANTESE AND A. TONOLO • every ob ject o f K ∗ ( A ) and every ob ject of K † ( B ) admits a quasi- isomorphism from ob jects of P and Q , resp ectively; • if P a nd Q are exact complexes in P and Q , then also Φ( P ) and Ψ( Q ) are exact. Given co mplexes X ∈ D ∗ ( A ) and Y ∈ D † ( B ), we hav e R ∗ Φ X = Φ P and R † Ψ Y = Ψ Q , where P is a co mplex in P quasi-isomor phic to X , and Q is a complex in Q quasi-iso morphic to Y . If Φ( K ∗ ( A )) ⊆ K † ( B ) and Φ( P ) ⊆ Q , then there exists a lso R ∗ (ΨΦ) and it is isomorphic to R † Ψ R ∗ Φ [Har, Prop osition 5 .4]. Definition 1.4. (1) An ob ject A in A is called Φ - acy clic if H i ( R ∗ Φ A ) = 0 for any i 6 = 0 . (2) The category A has enough Φ -acyclic obj e cts if any ob ject in A is image of a Φ-acy clic ob ject. (3) The functor Φ has c ohomolo gic al dimension ≤ n if, for each A in A , w e hav e H i ( R ∗ Φ A ) = 0 for | i | > n Remark 1 . 5. If A has enoug h Φ-ac yclic ob jects, then the rig h t derived functor R − Φ : D − ( A ) → D ( B ) is defined and it may b e computed using Φ-acyclic re s o- lutions: given a complex X ∈ D ≤ n ( A ), we hav e that R − Φ X = Φ L , where L is a complex in D ≤ n ( A ) with Φ-acyclic terms qua si-isomorphic to X . In particular, if the categ ory A has eno ugh pro jectives, R − Φ : D − ( A ) → D ( B ) is defined, and for each ob ject A in A , H n ( R − Φ A ) coincides with the usual right n th -derived functor of Φ ev aluated in A . If Φ has finite cohomolo gical dimension n and A has enough Φ-acyclics, then any complex X ∈ D ( A ) is quasi-is o morphic to a complex L with Φ-acyclic terms; th us the tota l der iv ed functor R Φ ex is ts a nd R Φ X = Φ L (se e [Ha r, Corolla r y 5.3, γ .]). Notice that if R ∗ Φ : D ∗ ( A ) → D ( B ) is wa y-out in b oth directions, then it has finite cohomolog ical dimension. Under the hypothesis that A has enough Φ-acyc lic ob jects, also the converse holds: Prop osition 1.6. L et A and B b e ab elian c ate gories, and Φ : A → B a c ontr avari- ant fun ctor. Assume A has enough Φ -acyclic obje cts; t hen (1) R − Φ is way-out righ t of lower dimension 0 ; (2) if Φ has finite c ohomolo gic al dimension n , then R Φ is way-out left of upp er dimension n . Pr o of. 1. It is clear, since R − Φ may b e computed on Φ-acyc lic resolutions. 2. Let X := 0 → X 0 → X 1 → ... b e an ob ject in D ≥ 0 ( A ). Since Φ has finite cohomolog ical dimension, there exists a complex of Φ- acyclic ob jects L := ... → L − 1 → L 0 → L 1 → ... qua s i isomorphic to X . Denote by C the cokernel of L − 1 → L 0 ; then C is quasi isomor phic to τ ≤ 0 L . Since H i ( R Φ C ) = 0 fo r i > n , then for each i > n we ha ve 0 = H i ( R Φ( τ ≤ 0 L )) = H i (Φ( τ ≤ 0 L )) = H i (Φ L ) = H i ( R Φ X ) .  Prop osition 1.7 . Assu me R ∗ Φ is way-out right of lower dimension ≥ 0 and way- out left of upp er dimension ≤ n . If 0 → X 0 → X 1 → · · · → X n is an exact c omplex REFLEXIVITY IN DERIVED CA TEGORIES 5 wher e the X i , i > 0 , ar e Φ -acyclic obje cts of A , then also X 0 is Φ -acyclic. In p articular, if n = 1 the class of Φ -acy clic obje cts is close d under submo dules. Pr o of. Let X := 0 → X 0 → X 1 → · · · → X n → 0 ; we hav e to prov e tha t H i ( R ∗ Φ( X 0 )) = 0 for ea c h i 6 = 0. Since the stalk complex X 0 belo ngs to D ∗ ≤ 0 ( A ), and R ∗ Φ has lower dimension ≥ 0, R ∗ Φ( X 0 ) belong s to D ≥ 0 ( B ). Ther efore it is sufficient to prove that H i ( R ∗ Φ( X 0 )) = 0 for each i > 0. Consider the tr iangle τ > 0 X → X → τ ≤ 0 X → τ > 0 X [1] and obser v e that τ ≤ 0 X is the stalk complex X 0 ; then for each i > 0 w e have the exact sequence H i − 1 ( R ∗ Φ( τ > 0 X )) → H i ( R ∗ Φ( X 0 )) → H i ( R ∗ Φ( X )) . Now, since the terms in τ > 0 X are Φ-acyclics , R ∗ Φ( τ > 0 X ) has non-zero terms only in negative degr ees, a nd therefore it has the ( i − 1) th cohomolog y equal to zer o . Since X belo ngs to D ∗ ≥ n ( A ), and dim + ( R ∗ Φ) ≤ n , w e get that R ∗ Φ( X ) be longs to D ∗ ≤ 0 ( B ). So also the i th cohomolog y of R ∗ Φ( X ) v anishes, and we conclude.  Definition 1.8. Assume that Φ( K ∗ ( A )) ⊆ K † ( B ). W e say that an o b ject L ∈ A is Ψ - Φ -acycli c if L is Φ-acyclic a nd Φ( L ) is Ψ-acyclic. W e say that the abelian category A has enough Ψ - Φ -acyclic obje cts if a n y A ∈ A is image of a Ψ-Φ-a c y clic ob ject. Prop osition 1.9. Assum e that Φ( K ∗ ( A )) ⊆ K † ( B ) . (1) If A has en ou gh Ψ - Φ -acyclic obje cts, then R † Ψ R ∗ Φ is way-out left of upp er dimensio n ≤ 0 . (2) If Φ has c ohomolo gic al dimension n , and A and B have enough Φ -acyclics and Ψ -acyclics, r esp e ctively, then R † Ψ R ∗ Φ is way-out righ t of lower di- mension ≥ − n . Pr o of. 1. It follows since a ny co mplex in D ∗ ≤ 0 ( A ) is quasi-iso morphic to a complex in D ∗ ≤ 0 ( A ) with Ψ-Φ-acy clic terms. 2. It follows by Prop ositions 1.3, 1.6.  2. Ad junctio n and Reflexive objects F rom now on, we are interested in the situation when (Φ , Ψ) is a rig h t adjoint pair; in pa rticular Φ and Ψ are left exact. The following result has a key role in our analysis. Lemma 2. 1 ([K, Lemma 13 .6]) . L et (Φ , Ψ) b e a right adjoint p air. Assume that R ∗ Φ( X ) and R † Ψ( Y ) b elong to D † ( B ) and D ∗ ( A ) , for a ny X in D ∗ ( A ) and Y in D † ( B ) , r esp e ct iv ely. Then ( R ∗ Φ , R † Ψ) is a right ad joint p air. Thu s, under s uitable ass umptions on th e existence of the derived functors, an y adjunction in abelia n c ategories can be extended to the asso ciated der iv ed ca te- gories. In this section we c ompare these t w o adjunctions. In particular we describ e the relationship be t ween the units of the tw o adjunction, and we s how the inde- pendenc e of the r e lated notions of reflexivit y . Example 2. 2. 1. Let ( X, O X ) be a lo cally noetherian scheme suc h that every coherent sheaf on X is a quotient of a lo cally free sheaf. Consider the ab elian category Mod X of sheav es of O X -mo dules, a nd the thick s ubcategor y Coh X of 6 F. MANTESE AND A. TONOLO coherent sheav es. Let G b e a coherent sheaf o f finite injective dimensio n; consider the functor H om ( − , G ) : Mod X → Mod X . The pair ( H om ( − , G ) , H om ( − , G )) is a right adjunction. By [Har2, Chp. I II] there exists the der iv ed functor R b H om ( − , G ) : D b ( Coh X ) → D b ( Coh X ) . Therefore, by Lemma 2.1, ( R b H om ( − , G ) , R b H om ( − , G )) is a rig h t adjoint pair. Moreov er, by [Har , Co r. I.5.3], there exists als o the total derived functor R H om ( − , G ) : D ( Coh X ) → D ( Coh X ) . and so ( R H om ( − , G ) , R H om ( − , G )) is a right adjo int pair. 2. Let R b e a ring, R U a left R - module a nd S the endomo rphism ring o f R U . The pair (Hom R ( − , U ) , Hom S ( − , U )) is a r igh t adjunction. By [S, Theo- rem C], the derived functors R Hom R ( − , U ) and R Hom S ( − , U ) alw ays exist and so ( R Hom R ( − , U ) , R Hom S ( − , U )) is a right adjoint pair. If b oth R U and U S hav e finite injectiv e dimension, then R Ho m R ( − , U ) and R Ho m S ( − , U ) ar e b ounded, since they are wa y-o ut in b oth directions. It follows that also ( R b Hom R ( − , U ) , R b Hom S ( − , U )) is a right adjoint pair . In the sequel, we assume that (Φ , Ψ ) is an a djoin t pair inducing the adjoint pair ( R ∗ Φ , R † Ψ). Deno ted by η and ξ the units o f the rig ht adjoint pair (Φ , Ψ), we indicate with ˆ η and ˆ ξ the units of the rig h t adjoint pair ( R ∗ Φ , R † Ψ), i.e. the natural maps ˆ η : id D ∗ ( A ) → R † Ψ R ∗ Φ , ˆ ξ : id D † ( B ) → R ∗ Φ R † Ψ such that R ∗ Φ( ˆ η X ) ◦ ˆ ξ R ∗ Φ X = 1 R ∗ Φ X and R † Ψ( ˆ ξ Y ) ◦ ˆ η R † Ψ Y = 1 R † Ψ Y for each X in D ∗ ( A ) and each Y in D † ( B ). Suppo se that A has enoug h Ψ-Φ-acyclic ob jects. Let X ∈ D − ( A ) and L b e a complex in K − ( A ) of Ψ- Φ - acyclics quasi-isomor phic to X . Then R † Ψ R ∗ Φ X = ΨΦ L , ˆ η X is iso morphic to ˆ η L in D − ( A ), a nd the latter coincides with the term to term extension of the unity η to K − ( A ). Prop osition 2. 3. Assum e that A has enough Ψ - Φ -acyclic obje cts. L et A b e an obje ct of A and ι b e the c anonic al map of c omplexes σ ≤ 0 R ∗ Φ A → R ∗ Φ A . If Φ ( A ) admits a Ψ -acyclic r esolution, we have η A = H 0 ( R † Ψ( ι )) ◦ H 0 ( ˆ η A ) = H 0 ( R † Ψ( ι ) ◦ ˆ η A ) . Pr o of. Consider a Ψ-Φ-acyclic reso lutio n L : ... → L − 1 d − 1 → L 0 → 0 of A with augmentation f : L 0 → A ; we have the commutativ e diag ram: L 0 f / / ˆ η L 0 = η L 0   A ˆ η A   R † Ψ R ∗ Φ L 0 = Ψ Φ( L 0 ) R † Ψ R ∗ Φ( f ) / / R † Ψ R ∗ Φ A R † Ψ( ι ) / / R † Ψ( σ ≤ 0 R ∗ Φ A ) = R † Ψ(Φ A ) REFLEXIVITY IN DERIVED CA TEGORIES 7 Applying the co homology functor H 0 , the solid part o f the following diagram com- m utes: L 0 f / / ˆ η L 0 = η L 0   A η A " " ^ [ W T P K F H 0 ( ˆ η A )   ΨΦ( L 0 ) ΨΦ( f ) 4 4 U W Y Z \ ] _ a b d e g i H 0 ( R † Ψ R ∗ Φ( f )) / / H 0 ( R † Ψ R ∗ Φ A ) H 0 ( R † Ψ( ι )) / / Ψ(Φ A ) Let us see that H 0 ( R † Ψ( ι )) ◦ H 0 ( R † Ψ R ∗ Φ( f )) = ΨΦ( f ); then, for the natur alit y of η w e will have η A ◦ f = ΨΦ( f ) ◦ η L 0 = ( H 0 ( R † Ψ( ι )) ◦ H 0 ( R † Ψ R ∗ Φ( f )) ◦ ˆ η L 0 = ( H 0 ( R † Ψ( ι )) ◦ H 0 ( ˆ η A )) ◦ f ; since f is an epimorphism, we will conclude. Let Q b e a Ψ -acyclic resolution of Φ( A ). Consider the diagr a m 0 / / Φ( L 0 ) / / 0 R ∗ Φ L 0 0 / / O O Φ( L 0 ) / / Φ( L − 1 ) O O / / ... R ∗ Φ A R ∗ Φ( f ) O O 0 / / O O Φ( A ) Φ( f ) O O / / 0 O O / / ... Φ A O O Q − 1 / / O O Q 0 O O / / 0 O O Q qiso O O Applying Ψ we get the co mmutative diagra m 0   ΨΦ( L 0 ) o o 0   o o R † Ψ R ∗ Φ( L 0 ) R † Ψ R ∗ Φ( f )   0   ΨΦ( L 0 )   ΨΦ( f ) y y s s s s s s s s s s o o ΨΦ( L − 1 )   ΨΦ( d − 1 ) o o ... o o R † Ψ R ∗ Φ( A ) R † Ψ( ι )   ΨΦ( A )  s % % K K K K K K K K K K Ψ( Q − 1 ) Ψ( Q 0 ) o o 0 o o R † Ψ(Φ( A )) 8 F. MANTESE AND A. TONOLO Therefore, having observed that ΨΦ( f ) ◦ ΨΦ( d − 1 ) = 0, the induced maps on the 0 th -cohomolo gies are obtained as follows: H 0 ( R † Ψ R ∗ Φ L 0 ) H 0 ( R † Ψ R ∗ Φ( f ))      ΨΦ( L 0 ) Coker(ΨΦ( d − 1 ) = H 0 ( R † Ψ R ∗ Φ A ) H 0 ( R † Ψ( ι ))      ΨΦ( L 0 ) o o o o ΨΦ( f )   H 0 ( R † Ψ(Φ A )) ΨΦ( A ) i.e. ΨΦ( f ) = H 0 ( R † Ψ( ι )) ◦ H 0 ( R † Ψ R ∗ Φ( f )).  Both the adjoint pairs (Φ , Ψ) and ( R ∗ Φ , R † Ψ) define on the co r respo nding categorie s the cla sses of r eflexive obje cts , i.e. the classes where the unit y maps induce iso morphisms. T o distinguish, we call simply r eflexive the ob jects A in A or B in B such that the natural maps η A or ξ B are iso morphisms; instead we say D -r eflexive the complexes whic h are r eflexiv e with r espect to the adjoin t pair ( R ∗ Φ , R † Ψ). Observe that any ob ject A in A is also, in a natural way , a n ob ject in D ∗ ( A ). Bo th the maps η A and ˆ η A can b e considered; therefore A can be reflexive or D -r eflexiv e . The tw o notions are indep enden t: Example 2. 4. In this a nd all future exa mples k denotes an algebra ic ally closed field. F o r any finite-dimensional k -alg ebra given by a quiv er with relations, if i is a vertex, we denote by P ( i ) the indeco mp osable pr o jective asso ciated to i , b y E ( i ) the indecomp osable injectiv e asso ciated to i , a nd by S ( i ) the s imple to p of P ( i ) o r, equiv alently , the simple so cle o f E ( i ). Let Λ denote the k -algebra given by the quiver · 1 a → · 2 b → · 3 c → · 4 with re la tions ba = 0 = cb . (1) Let Λ W = S (1) ⊕ S (3); then S = End Λ W is k ⊕ k . Since Λ W and W S hav e finite injective dimensio n, we hav e the tw o right adjoint pairs (Hom Λ ( − , W ) , Hom S ( − , W )) a nd ( R b Hom Λ ( − , W ) , R b Hom S ( − , W )) . An easy co mputation p ermits to v erify that S (1) is r e fle x iv e. Regarding S (1) a s a stalk complex, it is quasi is omorphic to its pro jective res olution P := 0 → P (3) → P (2 ) → P (1 ) → 0. Since P has Hom S ( − , W )-Hom Λ ( − , W )-a cyclic terms, R b Hom S ( R b Hom Λ ( S (1) , W ) , W ) = Hom S (Hom Λ ( P, W ) , W ) is the complex 0 → S (3) → 0 → S (1) → 0, whic h is not quasi- is omorphic to P . Then S (1) is not D -reflexive. (2) Let Λ Λ Λ be the r egular bimo dule. Since the left and the rig h t regular mo dules hav e finite injective dimensio n, we ha ve the tw o right a djoin t pairs (Hom Λ ( − , Λ) , Ho m Λ ( − , Λ)) and ( R b Hom Λ ( − , Λ) , R b Hom Λ ( − , Λ)) . It is straightforward to verify that the simple mo dule S (2) ∈ Λ -mod is not reflex- ive. Since a ll indecomp osable pro jectiv e mo dules a re reflexive and Ho m Λ ( − , Λ)- Hom Λ ( − , Λ)-acyclic, the simple mo dule S (2) is D -reflexive. Prop osition 2.5. If A ∈ A is r eflexive and Ψ - Φ - acyclic, then A is D -r eflex iv e. REFLEXIVITY IN DERIVED CA TEGORIES 9 Pr o of. W e hav e to prov e that ˆ η A is a quasi-isomo rphism, i.e. H i ( ˆ η A ) are isomor- phisms for each i . Since A is Ψ-Φ-a c yclic, then R † Ψ R ∗ Φ A is the stalk complex ΨΦ A . Clearly H i ( ˆ η A ) = 0 for each i 6 = 0 a re isomorphisms; since A is reflexive, also H 0 ( ˆ η A ) = η A is an isomorphism.  The catego ry of D -reflexive complexes is a tria ngulated s ubcategor y of D ∗ ( A ). In particular the sub category of stalk D - r eflexiv e complexes is thic k, i.e, if tw o terms of a short exact sequence in A are D -r eflexiv e, then a lso the third is. This follows eas ily s ince any s hort exact se quence in A gives rise to a tr iangle in D ∗ ( A ). Note that, fr o m the adjunction formulas, it follows that if a co mplex X is D - reflexive, then also R ∗ Φ X is D -reflexive. Definition 2.6. [A C, Sect. 2] L et R be a r ing. A left mo dule R U is p artial c otilting if it satisfies the following c o nditions: (1) injdim R U < ∞ ; (2) Ext i R ( U α , U ) = 0, for ea c h i > 0 and any car dinal α . The module R U is c otilting if mor eo ver the following condition is satisfied (3) there exists n ∈ N a nd an exact sequence 0 → U n → · · · → U 1 → U 0 → Q → 0 wher e Q is a n injective cogenerator of R -Mo d and U i are direct summands of pro ducts of copies of U . A bimo dule R U S is (p artial) c otilting if b oth R U and U S are (partial) cotilting. Partial co tilting mo dules g ive rise to an interesting class of examples o f ad- joint pairs o f contrav ar ian t functors. If R U S is a partia l cotilting bimodule, the functors in the adjoint pair (Hom R ( − , U ) , Hom S ( − , U )) ha ve finite cohomo logical dimension; thus the der iv ed functors R b Hom R ( − , U ) and R b Hom S ( − , U ) for m a right adjoin t pair in D b ( R -Mo d) and D b (Mo d- S ). If P is a pro jectiv e mo dule in R -Mo d, Hom R ( P, U ) is a direct summand of U α S for a suitable cardinal α , a nd so, b y co ndition (2) in Definition 2.6, Hom R ( P, U ) is Hom S ( − , U )-acyclic. Thu s R -Mo d, and s imilarly Mo d- S , hav e enough Hom S ( − , U )-Hom R ( − , U )-acyclic ob- jects. Conversely , it is interesting to o bserv e that, given a bimo dule R U S , to a ssume bo th the finite cohomologica l dimension of Hom R ( − , U ) a nd Hom S ( − , U )), and the Hom S ( − , U )-Hom R ( − , U )-acyclicity of the pr o jectives, implies that R U S is a partia l cotilting bimo dule. 3. Reflexive complexes Let us now inv estigate the relation betw een the D -reflexivity of a co mplex in D ∗ ( A ) and the D -reflexivit y of its terms or its co homologies. This analysis will hav e an essential ro le in order to obta in our main results in the fourth a nd fifth sections. W e alwa ys assume that (Φ , Ψ) is an adjoint pair inducing the adjoint pa ir ( R ∗ Φ , R † Ψ). Theorem 3.1. L et X b e an obje ct in D ∗ ( A ) . (1) If X ∈ D b ( A ) and any term of X is D -r eflexive, then X is D -r eflexive; (2) if X ∈ D b ( A ) and H i ( X ) is D -r eflexive for e ach i ∈ Z , then X is D - r eflexive. Assume R † Ψ R ∗ Φ is way-out lef t (r esp. right). (3) If X ∈ D − ( A ) (r esp. D + ( A ) ) and any term of X is D -r eflexive, t he n X is D - r eflexive; 10 F. MANTESE AND A. TONOLO (4) if X ∈ D − ( A ) (r esp. D + ( A ) ) and H i ( X ) is D -r eflexive for any i , t hen X is D -r eflexive. Assume R † Ψ R ∗ Φ is way-out on b oth dir e ctions. (5) If any term of X is D -r eflexive, then X is D -r eflexive; (6) if H i ( X ) is D -r eflexive for any i , then X is D -r eflexive. Pr o of. 1. W e can assume X := 0 → X − n → X − n +1 → ... → X 0 → 0. The thesis follows easily , by inductio n o n the le ng th n of X , cons idering the triangles τ > − 1 ( X ) → X → τ ≤− 1 ( X ) → τ > − 1 ( X )[1] . 2, 4, 6. The r esults follows applying [Har, Chp. I, Prop. 7.1] to the mor phism ˆ η : 1 D ∗ ( A ) → R † Ψ R ∗ Φ and consider ing the thick sub category o f D -reflexive ob jects of A . 3. F or shor t we denote by Γ the comp osition R † Ψ R ∗ Φ. W e prov e the result for Γ wa y- out left; the righ t cas e is ana logous. F ollowing the pro of of [Har, I.7.1 ], for each j ∈ Z , it is po ssible to find a suitable n ∈ Z such that H j ( τ >n X ) ∼ = H j ( X ) and H j ( Γ τ >n X ) ∼ = H j ( Γ X ) . Then the conclusion follows s ince H j ( Γ τ >n X ) ∼ = H j ( τ >n X ) by par t 1. 5. Let X ∈ D ( A ); consider the triangle τ > 0 X → X → τ ≤ 0 X → τ > 0 X [1]; F rom 3 we know that τ ≤ 0 X is D -reflexive since R † Ψ R ∗ Φ is w ay-out left and tha t τ > 0 X is D -reflexive since R † Ψ R ∗ Φ is wa y-out right. Th us we conclude that X is D - reflexiv e.  The con v erse of the pr evious theorem is not in general true: in the following examples we show that there exist D -reflexive complexes with not D -reflexive terms or not D -reflexive cohomologies . Example 3.2. Le t Λ denote the k -alge br a given by the quiver · 3   > > > > > > > · 1 ? ?          > > > > > > > · 4 / / · 5 o o · 2 ? ?        with relations s uc h that the left pr o jective mo dules are 1 2 3 4 , 2 4 , 3 4 5 , 4 5 and 5 3 . Consider the mo dule Λ U = 5 ⊕ 3 4 5 ⊕ 1 2 3 4 and let S = E nd Λ ( U ). The alge- bra S is given b y the quiver · 6 → · 7 → · 8 with right pro jectives 8 7 , 7 6 and 6 , and U S = 8 7 ⊕ 7 6 ⊕ 7 6 ⊕ 6 ⊕ 6 . Since Λ U S is a partial co tilting bimo d- ule, ( R b Hom Λ ( − , U ) , R b Hom S ( − , U )) is a rig h t adjunction and the pro jective Λ- mo dules ar e Hom S ( − , U )-Hom Λ ( − , U )-acyclic o b jects. Consider the complex with pro jectiv e terms P : 0 / / 4 5 / / 3 4 5 / / 5 3 / / 3 4 5 / / 1 2 3 4 / / 0 REFLEXIVITY IN DERIVED CA TEGORIES 11 and the ob vious non-zero differentials. It is easy to chec k that the mor phism ˆ η P , given by the dia gram 0 / / 4 5 / / η P (4)   3 4 5 / / η P (3)   5 3 / / η P (5)   3 4 5 / / η P (3)   1 2 3 4 / / η P (1)   0 0 / / 3 4 5 ∼ = / / 3 4 5 / / 5 / / 3 4 5 / / 1 2 3 4 / / 0 is a qua si-isomorphism. Nevertheless the terms P (5) and P (4) are not D - reflexiv e. Example 3.3. Le t Λ b e the k -algebra given by the quiver · 1 / / · 2 / / z z · 3 / / · 4 / / · 5 x x with rela tions such that the left pro jective mo dules ar e 1 2 1 , 2 1 3 4 , 3 4 5 , 4 5 and 5 3 . Let us consider the module Λ U = 2 1 ⊕ 1 2 1 ⊕ 5 3 and let S = End Λ ( U ). Then S is given by the quiver · 6 · 7 / / · 8 z z with relations suc h tha t the right pro jectives ar e 8 7 , 7 8 7 , 6 , and U S = 8 7 ⊕ 7 8 7 ⊕ 6 ⊕ 6 . Since Λ U S is a pa rtial co tilting bimo dule, ( R b Hom Λ ( − , U ) , R b Hom S ( − , U )) is a right adjunction and the pro jective Λ-mo dules ar e Hom S ( − , U )-Hom Λ ( − , U )- acyclic ob jects. Let us conside r the complex X ∈ D b ( A ) with pro jective terms 0 → 1 2 1 f → 1 2 1 f → 1 2 1 → 0 where Im f = so c P (1) a nd Ker f = rad P (1). This complex is D -reflex ive: indeed R b Hom S ( R b Hom Λ ( X, U ) , U ) = Hom S (Hom Λ ( X, U ) , U ) a nd 1 2 1 = Hom S (Hom Λ ( 1 2 1 , U ) , U ), so the ev aluation map ˆ η X is tr iv ially a quasi-is omorphism. Nevertheless the co ho- mology mo dule K er f / Im f = S (2) is not D -reflexive. In fact, let us consider a pro jectiv e res olution of S (2) P : 0 → 4 5 → 3 4 5 → 5 3 ⊕ 4 5 → 3 4 5 ⊕ 3 4 5 → 2 1 3 4 ⊕ 5 3 → 1 2 1 ⊕ 3 4 5 → 2 1 3 4 → 0 . An easy computation shows that R b Hom S ( R b Hom Λ ( S (2) , U ) , U ) = Hom S (Hom Λ ( P, U ) , U ) is the complex 0 → 0 → 5 3 ∼ = → 5 3 → 5 3 ⊕ 5 3 → 2 1 ⊕ 5 3 → 1 2 1 ⊕ 5 3 → 2 1 → 0 which has non zer o cohomolo gies in deg rees 0 and − 3. So ˆ η S (2) is not a quasi- isomorphism and the mo dule S (2) is not D -reflexive. Given a finitely generated co tilting mo dule of injective dimension ≤ 1 ov er a n Artin algebr a, in [CbCpF] is proved, using our ter minology (see the forthco m- ing Theo rem 4 .3 ) that the class of D -reflexive mo dules coincides with the class of finitely genera ted o nes. This can b e generalized to cotilting mo dules of arbitra ry finite injective dimensio n; in par ticula r we obtain, in this setting, a conv ers e of Theorem 3.1. Theorem 3 .4. L et Λ b e an A rt in algebr a, Λ U a finitely gener ate d c otilting mo dule and S = E nd Λ U . Consider the adjoint p air ( R b Hom Λ ( − , U ) , R b Hom S ( − , U )) . 12 F. MANTESE AND A. TONOLO (1) A c omplex X ∈ D b (Λ-Mo d) is D -r eflexive if and only if the c ohomolo gies H i ( X ) , i ∈ Z , ar e finitely gener ate d. (2) The sub c ate gory of b oun de d D -r eflexive c omplexes is e quivalent to D b (Λ-mo d) . In p articular a c omplex X ∈ D b (Λ-Mo d) is D -re flexive if and only if t he c ohomolo- gies H i ( X ) , i ∈ Z , ar e D - r eflexive. Pr o of. W e r ecall that the assumptions imply that Λ U S is a faithfully balanced cotilting bimo dule ([M , Theor em 1.5]). 1. Let ∗ = Hom( − , W ) be the usual duality betw een mo d-Λ and Λ-mo d , where W is the minimal injectiv e cogenera tor. Then Λ U ∗ = V Λ is a finitely generated tilting mo dule (see [M]). Recall that a Λ -module is reflex iv e with resp ect to the adjoint pair ( ∗ , ∗ ) if and only if it is finitely g enerated, and that the a djoin t pa ir ( −⊗ L S V , R Hom Λ ( V , − )) defines an equiv alenc e b et ween D b (Mo d-Λ) and D b (Mo d- S ) (see [C P S, Hap]). Let now X ∈ D b (Λ-Mo d) b e a D -reflexiv e complex a nd let P ∈ K − (Λ-Mo d) b e a complex o f pro jective modules quasi-iso morphic to X . Then P is quasi-isomorphic to Hom S (Hom Λ ( P, U ) , U ). Using the standard adjunction formulas, since U = V ∗ , we get that Hom S (Hom Λ ( P, U ) , U ) = Hom S (Hom Λ ( P, V ∗ ) , U ) ∼ = Hom S (Hom Λ ( V , P ∗ ) , U ) = = Hom S (Hom Λ ( V , P ∗ ) , V ∗ ) ∼ = Hom S (Hom Λ ( V , P ∗ ) ⊗ S V , W ) = = Hom Λ (Hom Λ ( V , P ∗ ) ⊗ S V , W ) . Moreov er Hom Λ ( V , P ∗ ) = R Hom Λ ( V , P ∗ ) and, since Hom Λ ( V , I ) is ( − ⊗ S V )- acyclic for any injective Λ-mo dule I [M, Lemma 1.7 ], we obtain that Hom Λ ( V , P ∗ ) ⊗ S V = R Hom Λ ( V , P ∗ ) ⊗ L S V ∼ = P ∗ . Hence P is quasi-is omorphic to P ∗∗ . F or ∗ is an ex act functor, we conclude that H i ( P ) is isomorphic to H i ( P ) ∗∗ for an y i . Thus all the cohomologies of X ar e finitely generated. Co n versely , if all the coho mologies of X are finitely g e nerated, they a r e D -reflexive: indeed all finitely genera ted pro jective Λ-mo dules a re reflexive with resp ect to the adjoint pair (Hom R ( − , U ) , Hom S ( − , U )). Then we conclude by Prop osition 2 .5 and Theor em 3.1. 2. It is w ell known tha t the sub category of complexes in D b (Λ-Mo d) with finitely generated cohomologies is equiv alent to D b (Λ-mo d) (see [Har, Prop osition I.4.8]).  Limiting str o ngly the way-out dimensio ns, it is p ossible to prove that a co mplex is D - reflexiv e if and only if its cohomolog ies a r e D - reflexiv e in a mo re general setting. Prop osition 3.5. Le t X b e an obje ct of D ∗ ( A ) . Supp ose the functor R † Ψ R ∗ Φ to b e way-out left of upp er dimensio n ≤ 0 and way-out right of lower dimension ≥ − 1 . Then X is D -r eflexive if and only if its c ohomolo gies ar e D -r eflexive. Pr o of. F or short, let us denote by Γ the compo sition R † Ψ R ∗ Φ. First let us sup- po se X ∈ D − ( A ) to be a D -r eflexiv e complex. W e can assume X ∈ D ≤ 0 ( A ) is of the form X : . . . → X − 1 → X 0 → 0 . Let us first prov e that H 0 ( X ) is a D -reflexive ob ject. Consider the triangle ( ∗ ) σ ≤− 1 X → X → σ > − 1 X → σ ≤− 1 X [1] . The co mplex σ > − 1 X is quasi isomorphic to the stalk complex H 0 ( X ), and the com- plex σ ≤− 1 X has zero cohomolo gies in degrees grea ter than − 1 . Since dim + Γ ≤ 0, we REFLEXIVITY IN DERIVED CA TEGORIES 13 hav e H i ( Γ X ) = 0, H i ( Γ σ > − 1 X ) = 0 and H i ( Γ σ ≤− 1 X [ − 1]) = H i − 1 ( Γ σ ≤− 1 X ) = 0 for i > 0. Applying to the triang le ( ∗ ) first Γ and then the cohomology functor, w e get the commutativ e diagr am with exact rows H − 1 ( σ ≤− 1 X ) ∼ = / /   H − 1 ( X ) / / ∼ =   0 / /   0 / /   H 0 ( X ) / / ∼ =   H 0 ( σ > − 1 X ) / /   0 H − 1 ( Γ σ ≤− 1 X ) / / H − 1 ( Γ X ) / / H − 1 ( Γ σ > − 1 X ) / / 0 / / H 0 ( Γ X ) / / H 0 ( Γ σ > − 1 X ) / / 0 Thu s we deduce that H − 1 ( Γ σ > − 1 X ) = 0. Since dim − Γ ≥ − 1, we have H i ( Γ σ > − 1 X ) = 0 for i ≤ − 2. Hence σ > − 1 X ∼ = H 0 ( X ) is D -reflexive. Then, from the triangle ( ∗ ) we deduce that the complex σ ≤− 1 X is D - reflexiv e. Rep eating the same argument for σ ≤− 1 X [ − 1], w e get that H − 1 ( σ ≤− 1 X ) ∼ = H − 1 ( X ) is D - reflexiv e. Contin uing in such a wa y , w e conclude that H i ( X ) is a D -reflexive ob ject fo r any i ≤ 0. Suppo se no w X to be a D -reflexive co mplex in D ( A ). Consider the triangle σ ≤ 0 X → X → σ > 0 X → σ ≤ 0 X [1] F or the way-out dimensions of Γ , we have H i ( Γ σ > 0 X ) = 0 for i < 0 a nd H i ( Γ σ ≤ 0 X ) = 0 for i > 0. So we get the co mm utative exact diagr am H − 1 ( X ) / / ∼ =   0 / /   H 0 ( σ ≤ 0 X ) / /   H 0 ( X ) / / ∼ =   0 / /   0 / /   H 1 ( X ) / / ∼ =   H 1 ( σ > 0 X ) / /   0 H − 1 ( Γ X ) / / 0 / / H 0 ( Γ σ ≤ 0 X ) / / H 0 ( Γ X ) / / H 0 ( Γ σ > 0 X ) / / 0 / / H 1 ( Γ X ) / / H 1 ( Γ σ > 0 X ) / / 0 from which we conclude that σ ≤ 0 X and σ > 0 X ar e D -reflexive complexes. Since the complex σ ≤ 0 X belo ngs to D − ( A ), for wha t w e hav e alre ady prov ed w e ge t that H i ( X ) is D -reflexive fo r an y i ≤ 0. Similarly , considering the trunca tio n in degree i > 0 and the triangle σ ≤ i X → X → σ >i X → σ ≤ i X [1] , we conclude that H i ( X ) is D -reflexive for any index i .  Corollary 3.6. Assume Φ has c ohomo lo gic al dimension at most one and A has enough Ψ - Φ -acycli c obje cts. If B has enough Ψ -acyclics, then a c omplex X ∈ D ( A ) is D -r eflexive if and only if its c ohomol o gies H i ( X ) ar e D -r eflexive. Pr o of. By Propo sition 1.9, R † Ψ R ∗ Φ is wa y- out le ft of upper dimension ≤ 0 and wa y-o ut right of low er dimension ≥ − 1. So we can apply Pro position 3.5.  Example 3.7. As in Ex ample 2.2, let ( X , O X ) b e a lo c a lly noetheria n scheme such that every coherent sheaf on X is a quotient of a lo cally free shea f of finite rank. Assume the structur e sheaf O X has injective dimension one. Cons ide r the ab elian ca tegory Mod X of sheav es of O X -mo dules and the thic k sub category Co h X of coherent s hea ves. Then ( R H om ( − , O X ) , R H om ( − , O X )) is a right adjoint pair in D ( Coh X ) which satisfies the assumptions o f the previous corollar y . Indeed, let L be the cla ss of lo cally free sheav es of finite rank. An y ob ject in L is H om ( − , O X )- acyclic and any F ∈ Coh X is image of a lo cally free shea f of finite ra nk. Moreover, for any G lo cally free of finite rank , H om ( G , O X ) is lo cally a finite dire c t sum of copies of O X and so it is H om ( − , O X )-acyclic. Thus L satisfies the ass umption of Definition 1.8. Finally R H om ( − , O X ) has cohomolo gical dimension one. Applying Prop osition 2 .5 we get that a n y loca lly free shea f of finite rank is D - reflexive. Thus, considering lo cally free resolutions, by Theor em 3.1 we o bta in that any coherent sheaf is D -reflexive and so any complex in D ( Coh X ) is D -reflexive. Note that Cor ollary 3.6 is trivially verified: indeed, if Y is a D -reflexive complex 14 F. MANTESE AND A. TONOLO in D ( Coh X ), then its cohomologies ar e D -reflexive ob jects, being coherent sheav es (cfr. [Har, Prop. V. 2.1]). The following tec hnica l result will b e us eful in the fifth section; its pro of follows the same arg uments used proving P ropo s ition 3.5. Lemma 3. 8. L et X b e a D -r eflexive obje ct in D ∗ ≤ n ( A ) , n ∈ Z . Supp ose the functor R † Ψ R ∗ Φ to b e way-out left of upp er dimension ≤ 0 and that R ∗ Φ H j ( X ) is a stalk c omplex fo r e ach j ∈ Z . F or e ach j ∈ Z , let ρ ( j ) b e an inte ger such t hat H i ( R ∗ Φ H j ( X )) = 0 fo r e ach i 6 = ρ ( j ) . Then t he c ohomolo gies of X ar e D -re flexive if and only if H i ( R † Ψ H ρ ( j ) ( R ∗ Φ H j ( X ))) = 0 f or e ach i 6 = ρ ( j ) , ρ ( j ) − 1 . Pr o of. Consider the triangle ( ∗ ) σ ≤ n − 1 X → X → σ >n − 1 X → σ ≤ n − 1 X [1] . The co mplex σ >n − 1 X is quasi isomorphic to the stalk co mplex H n ( X )[ − n ], and the complex σ ≤ n − 1 X has zero co homologies in deg rees greater than n − 1 . By hy- po thesis, H i ( R ∗ Φ H n ( X )) = 0 for each i 6 = ρ ( n ); therefor e R ∗ Φ( H n ( X )[ − ρ ( n )]) = R ∗ Φ H n ( X )[ ρ ( n )] is quasi isomor phic to the sta lk complex H ρ ( n ) ( R ∗ Φ H n ( X )). Let us denote by Γ the composition R † Ψ R ∗ Φ; then we have Γ ( σ >n − 1 X ) = Γ ( H n ( X )[ − n ]) = R † Ψ( R ∗ Φ H n ( X )[ n ]) = = R † Ψ( H ρ ( n ) ( R ∗ Φ H n ( X ))[ n − ρ ( n )]) = R † Ψ H ρ ( n ) ( R ∗ Φ H n ( X ))[ ρ ( n ) − n ] . Since dim + Γ ≤ 0, we hav e H i ( Γ X ) = 0, H i ( Γ σ >n − 1 X ) = 0 and H i ( Γ σ ≤ n − 1 X [ − 1]) = H i − 1 ( Γ σ ≤ n − 1 X ) = 0 for i > n . Applying to the triangle ( ∗ ) first Γ and then the cohomolog y functor, we get the commutativ e diagram with exact rows H n − 1 ( σ ≤ n − 1 X ) ∼ = / /   H n − 1 ( X ) / / ∼ =   0 / /   0 / /   H n ( X ) / / ∼ =   H n ( σ >n − 1 X ) / /   0 H n − 1 ( Γ σ ≤ n − 1 X ) / / H n − 1 ( Γ X ) / / H n − 1 ( Γ σ >n − 1 X ) / / 0 / / H n ( Γ X ) / / H n ( Γ σ >n − 1 X ) / / 0 Thu s we deduce that H n − 1 ( Γ σ >n − 1 X ) = 0 and that H n ( Γ σ >n − 1 X ) ∼ = H n ( σ >n − 1 X ). Since H n − i ( σ >n − 1 X ) = 0 for ea c h i > 0, the complex σ >n − 1 X is D -reflexive, and hence the n -th cohomology of X is D -r e flexiv e, if and only if for each i > 1 we have 0 = H n − i ( Γ σ >n − 1 X ) = H n − i ( Γ ( H n ( X )[ − n ])) = = H n − i ( R † Ψ H ρ ( n ) ( R ∗ Φ H n ( X ))[ ρ ( n ) − n ]) = H ρ ( n ) − i R † Ψ H ρ ( n ) ( R ∗ Φ H n ( X )) . Next, fro m the tr ia ngle ( ∗ ), σ >n − 1 X is D -reflexive if and only if σ ≤ n − 1 X is D - reflexive. Applying the same argument to σ ≤ n − 1 X , we prov e that H n − 1 ( X ) is D - reflexiv e if a nd o nly if the c ohomologies H i ( R † Ψ H ρ ( n − 1) ( R ∗ Φ H n − 1 ( X ))) = 0 for each i 6 = ρ ( n − 1) , ρ ( n − 1) − 1. Iterating this pro cedure, w e conclude.  Remark 3 .9. Observe that if the functor Φ has cohomologica l dimension ≤ 1, and there are enough Ψ-a cyclic ob jects, under the hypotheses of Lemma 3.8 the condition H i ( R † Ψ H ρ ( j ) ( R ∗ Φ H j ( X ))) = 0 for eac h i 6 = ρ ( j ) , ρ ( j ) − 1 is alw ays satisfied (compare with Corolla ry 3.6). The key p oint is that R Ψ H ρ ( j ) R Φ H j X ∼ = ( R Ψ R Φ H j X )[ − ρ ( j )] . W e hav e H i R Ψ H ρ ( j ) R Φ H j X = 0 for i < 0 (b ecause there ar e enough Ψ -acyclic ob jects) and H i ( R Ψ R Φ H j X )[ − ρ ( j )] = H i − ρ ( j ) ( R Ψ R Φ H j X ) = 0 for i > ρ ( j ) REFLEXIVITY IN DERIVED CA TEGORIES 15 (beca use R Ψ R Φ is wa y out left of upp er dimens io n ≤ 0 ). Since Φ has cohomolog ical dimension ≤ 1 , it is | ρ ( j ) | ≤ 1 and so we co nclude. Let us see now the connection b et ween the cohomolog ical dimension of Φ and the closure of the clas s of D -reflexive ob jects under k ernels and co kernels. Theorem 3.10. Ass u me that A has enough Ψ - Φ -acyclic obje cts and B has enough Ψ -acyclic obje cts. If Φ has c ohomolo gic al dimension ≤ n , then, in any exact se- quenc e M 1 f 1 → M 2 f 2 → ... f n → M n +1 of D -r eflexive obje cts of A , the kernels and the c okernels of the morphisms f i , i = 1 , ..., n , ar e D -r eflexive. In p articular, if Φ has c ohomolo gic al dimension at most one, the class of D - r eflexive obje ct s in A is an exact ab elian sub c ate gory of A . Pr o of. First observe that by Prop osition 1.9, for any ob ject A in A , the ob ject R † Ψ R ∗ Φ( A ) b elongs to D ≥− n ( A ) ∩ D ≤ 0 ( A ). Denoted b y K i the kernel o f the morphism f i , i = 1 , ..., n , by K n +1 the image of f n , and by K n +2 the cokernel of f n , let us co nsider the following triangles in D b ( A ): K i → M i → K i +1 → K i [1] , i = 1 , ..., n + 1 . Consider the maps H 0 ( ˆ η M i ) : M i → H 0 ( R † Ψ R ∗ Φ M i ), a nd H 0 ( ˆ η K j ) : K j → H 0 ( R † Ψ R ∗ Φ K j ), 1 ≤ i ≤ n , 1 ≤ j ≤ n + 2. Since M i are D -reflexive ob jects, clearly H 0 ( ˆ η M i ) are isomo rphisms. W e will prov e that H j ( R † Ψ R ∗ Φ K i ) = 0 for each j 6 = 0 a nd ea c h 1 ≤ i ≤ n + 2 and that a ll H 0 ( ˆ η K j ) are isomorphisms. Because of the wa y-out dimension of R † Ψ R ∗ Φ, H j ( R † Ψ R ∗ Φ K i ) = 0 for each j > 0. Applying the co ho mology functor we get the long exact sequence s 0 → H − n ( R † Ψ R ∗ Φ K i ) → 0 → H − n ( R † Ψ R ∗ Φ K i +1 ) → H − n +1 ( R † Ψ R ∗ Φ K i ) → 0 → ... ... → 0 → H − 2 ( R † Ψ R ∗ Φ K i +1 ) → H − 1 ( R † Ψ R ∗ Φ K i ) → 0 → → H − 1 ( R † Ψ R ∗ Φ K i +1 ) → H 0 ( R † Ψ R ∗ Φ K i ) → H 0 ( R † Ψ R ∗ Φ M i ) → H 0 ( R † Ψ R ∗ Φ K i +1 ) → 0 , for i = 1 , ..., n + 1. In particula r H − n ( R † Ψ R ∗ Φ K i ) = 0 for i = 1 , ..., n + 1; therefore for j = 1 , ..., n , since n − j + 1 < n − j + 2 ≤ n + 1 , we have H − j ( R † Ψ R ∗ Φ K 1 ) ∼ = H − n ( R † Ψ R ∗ Φ K n − j +1 ) = 0 , and H − j ( R † Ψ R ∗ Φ K 2 ) ∼ = H − n ( R † Ψ R ∗ Φ K n − j +2 ) = 0 . W orking a little on diagr ams 0 / / K i / / H 0 ( ˆ η K i )   M i / / ∼ =   K i +1 / / H 0 ( ˆ η K i +1 )   0 ... / / H 0 ( R † Ψ R ∗ Φ K i ) / / H 0 ( R † Ψ R ∗ Φ M i ) / / H 0 ( R † Ψ R ∗ Φ K i +1 ) / / 0 with i = 1 , 2, we get that H 0 ( ˆ η K 1 ) and H 0 ( ˆ η K 2 ) are isomo r phisms. Therefor e K 1 and K 2 are D -reflexive. W o rking with the triangles K i → M i → K i +1 → K i [1] , i = 2 , ..., n + 1 we get that also K 3 , ..., K n +2 are D -reflexive.  16 F. MANTESE AND A. TONOLO 4. The 1-dimensional cas e In the prev ious se ction we hav e seen that more precise results a re av ailable when the inv olved fu nctors hav e cohomologica l dimens ion at most one. This section is dedicated to study in detail this fa vorable case. O ur aim is to characterize the D - reflexiv e ob jects in the ab elian categories A and B , pro ducing a general for m of the Cotilting Theor em in the s ense of Colby and F uller (see [CbF1, Ch. 5]), a contra v ariant version of the celebrated Brenner and Butler Theo rem [BB] . W e assume A has eno ugh Ψ-Φ-ac yclic ob jects and B has enough Φ-Ψ-acyclic ob jects, resp ectiv ely , and (Φ , Ψ) is an adjoint pa ir o f contra v ariant functors of cohomolog ical dimension at most one. In particular, under these a ssumptions, • there exist the total der iv ed functors R Φ and R Ψ, and they hav e both low er dimension ≥ 0 and upper dimension ≤ 1, • the co mposition R Ψ R Φ results to be way-out left of upper dimension ≤ 0 and wa y-out r igh t of low er dimension ≥ − 1 (Prop osition 1.9), a nd it is isomorphic to R (ΨΦ) ([Har, P ropo sition 5.4]), • the families of Φ-acyclic a nd Ψ -acyclic o b jects a re closed under submo dules (Prop osition 1.7), • a complex is D -reflexive if and only if its cohomolog ies a re D - reflexiv e (Corollar y 3.6), • the cla sses of D -reflexive ob jects in A and B are exa ct ab elian sub categories of A and B (Theorem 3.10). In this setting w e deal with the unbounded derived categories D ( A ) and D ( B ) and the total de r iv ed functors R Φ and R Ψ: for any complex X in D ( A ) (resp. D ( B )), we denote by R i Φ X (resp. R i Ψ X ), i ∈ Z , the i th -cohomolo gy H i ( R Φ X ) (resp. H i ( R Ψ X )). Obs erv e that R 0 Φ A = Φ A and R 0 Ψ B = Ψ B for eac h A in A and B in B . Lemma 4.1. Any obje ct in Im Φ is Ψ -acycl ic. Pr o of. Let A b e an ob ject in A . Consider an epimorphism L → A → 0 where L is a Ψ-Φ- a cyclic o b ject. Applying Φ we get the monomorphism 0 → Φ A → Φ L . Since Φ L is Ψ-a cyclic, and the family of Ψ-acyclic ob jects is clo sed under submo dules, we conclude that Φ A is Ψ- a cyclic.  Observe that by the previous lemma an y Φ-acyclic ob ject is a lso Ψ- Φ-acyclic. Prop osition 4. 2. An ob je ct A ∈ A is D -r eflexive if and only if Ψ R 1 Φ A = 0 and the map H 0 ( ˆ η A ) : A → R 0 (ΨΦ) A is an isomorp hism. Pr o of. Since R Ψ R Φ is w ay-out of upper dimension ≤ 0 and low e r dimension ≥ − 1, the ob ject A is D -reflexive if and only if H 0 ( ˆ η A ) and H − 1 ( ˆ η A ) are isomorphisms, the latter b eing equiv alent to H − 1 ( R (ΨΦ) A ) = 0. Let us consider the tria ngle σ ≤ 0 R Φ A → R Φ A → σ > 0 R Φ A → σ ≤ 0 R Φ A [1]; taking in account the way-out dimensions of R Φ, this triang le is isomor phic to Φ A → R Φ A → R 1 Φ A [ − 1] → Φ A [1] Applying R Ψ w e get, using Lemma 4.1, the tria ngle R Ψ( R 1 Φ A [ − 1]) → R Ψ R Φ A = R (ΨΦ) A → ΨΦ A → R Ψ R 1 Φ A [2] . REFLEXIVITY IN DERIVED CA TEGORIES 17 Considering the asso ciated cohomology sequence, we get the exa ct sequence 0 → H − 1 ( R Ψ( R 1 Φ A [ − 1])) → H − 1 ( R (ΨΦ) A ) → 0; Then w e conclude since H − 1 ( R Ψ( R 1 Φ A [ − 1])) = H 0 ( R Ψ R 1 Φ A ) = Ψ R 1 Φ A.  Theorem 4.3. An obje ct A in A is D -r eflexive if and only if (1) Φ( A ) and R 1 Φ( A ) ar e D -r eflexive; (2) R i Ψ R j Φ( A ) = 0 if i 6 = j ; (3) ther e exists a natu r al map γ A and an exact se quenc e 0 → R 1 Ψ R 1 Φ( A ) γ A → A η A → ΨΦ( A ) → 0 In such a c ase, when denoting by π R Φ A the natura l map R Φ A → σ > 0 R Φ A , we have γ A = H 0 ( ˆ η A ) − 1 ◦ R 0 Ψ( π R Φ A ) . Pr o of. Assume A is D -reflexive. Since R Φ A is D -reflexive, from Cor ollary 3.6 it follows that its cohomologies Φ( A ) and R 1 Φ A are D -reflexive. By Lemma 4.1 Φ( A ) is Ψ-acyclic; therefore R 1 Ψ( R 0 Φ( A )) = R 1 Ψ(Φ( A )) = 0 . By P ropos ition 4 .2, we know also that Ψ R 1 Φ( A ) = 0, and so R 0 Ψ R 1 Φ( A ) = 0. T o prov e (3), let us cons ider the tria ngle σ ≤ 0 R Φ A ι → R Φ A π R Φ A → σ > 0 R Φ A → σ ≤ 0 R Φ A [1]; taking in account the way-out dimensions of R Φ, this triang le is isomor phic to Φ( A ) ι → R Φ A π R Φ A → R 1 Φ( A )[ − 1] → Φ( A )[1] . Applying R Ψ, b y Lemma 4.1 we ge t the tria ngle R Ψ( R 1 Φ( A ))[1] R Ψ( π R Φ A ) → R Ψ R Φ A R Ψ( ι ) → ΨΦ( A ) → R Ψ( R 1 Φ( A ))[2] . Considering the asso ciated co homology sequence, we get the natural short exact sequence 0 → R 1 Ψ( R 1 Φ( A )) R 0 Ψ( π R Φ A ) → H 0 ( R Ψ R Φ A ) = H 0 ( R (ΨΦ)( A )) R 0 Ψ( ι ) → Ψ(Φ( A )) → 0 . Since A is D -reflex iv e, H 0 ( ˆ η A ) : A → H 0 ( R (ΨΦ)( A )) is an isomorphism. Denote by γ A the comp osition H 0 ( ˆ η A ) − 1 ◦ R 0 Ψ( π R Φ A ); we ca n apply Prop osition 2.3 to get R 0 Ψ( ι ) ◦ H 0 ( ˆ η A ) = η A and hence the natura l exa ct sequence 0 → R 1 Ψ R 1 Φ( A ) γ A → A η A → ΨΦ( A ) → 0 . Conv er s ely , assume conditions (1), (2) and (3) hold. Applying (1) to Φ( A ) and R 1 Φ( A ), we get that R 1 Ψ R 1 Φ( A ) and ΨΦ( A ) a re D -reflexive. Therefor e, by (3) also A is D -reflexive.  The same result holds for an y D - reflexiv e ob ject B in B , with the ma p θ B : R 1 Φ R 1 Ψ( B ) → B , θ B = H 0 ( ˆ ξ B ) − 1 ◦ R 0 Φ( π R Ψ B ), which plays the role o f the natural map γ . W e are now re a dy to give a Co tilting Theor em in the s ense of [CbF1, Ch. 5], betw een the clas ses of D -r eflexiv e ob jects induced by the pair o f adjoin t functors (Φ , Ψ ). 18 F. MANTESE AND A. TONOLO Corollary 4. 4. Consider the fol lowi ng su b classes of the ab elian sub c ate gories D A and D B of D -r eflex ive obje cts in A and B : T A = K er Φ ∩ D A , F A = K er R 1 Φ ∩ D A T B = Ker Ψ ∩ D B , F B = K er R 1 Ψ ∩ D A Then t he fol lowing c onditions ar e satisfie d: (1) Φ : D A → F B , R 1 Φ : D A → T B , Ψ : D B → F A , R 1 Ψ : D B → T A . (2) for e ach obje ct A in D A and B in D B we have the fol low ing exact se quenc es of natur al maps 0 → R 1 Ψ R 1 Φ( A ) γ A → A η A → ΨΦ( A ) → 0 0 → R 1 Φ R 1 Ψ( B ) θ B → B ξ B → ΦΨ( B ) → 0 (3) the r estrictions Φ : F A − − → ← − − F B : Ψ and R 1 Φ : T A − − → ← − − T B : R 1 Ψ define c ate gory e quivale nc es. Mor e over these ar e the lar gest p ossible classes wher e su ch a duali ty arises Our star ting p oin t w a s that (Φ , Ψ) is an adjoin t pair of functors b etw een the ab elian categories A and B . Now we show that ( R 1 Φ , R 1 Ψ) is a n adjoin t pair o f functors b et ween the a belian ca tegories of D -reflexive ob jects o f A a nd B . Theorem 4.5. In the classes D A and D B of D -r eflexive obje cts of A and B , the p air ( R 1 Φ , R 1 Ψ) is left adjo int with the natu r al maps γ and θ as units. Pr o of. In o rder to prov e that ( R 1 Φ , R 1 Ψ) is a left a djo in t pair in the classes D A and D B , it is enough to show tha t θ R 1 Φ A ◦ R 1 Φ( γ A ) = id R 1 Φ A for any A ∈ D A and, analogo usly , γ R 1 Ψ B ◦ R 1 Ψ( θ B ) = id R 1 Ψ B for any B ∈ D B . Note that, from the adjunction formula R Φ( ˆ η A ) ◦ ˆ ξ R Φ A = id R Φ A , w e ge t that R 1 Φ( ˆ η A ) ◦ H 1 ( ˆ ξ R Φ A ) = id R 1 Φ A . W e will pr o ve that θ R 1 Φ A = H 1 ( ˆ ξ R Φ A ) − 1 and R 1 Φ( γ A ) = R 1 Φ( ˆ η A ) − 1 . First, let us consider the diagram Φ A / / ˆ ξ Φ A   R Φ A π R Φ A / / ˆ ξ R Φ A   R 1 Φ A [ − 1] / / ˆ ξ R 1 Φ A [ − 1]   Φ A [1] ˆ ξ Φ A [1]   R Φ R Ψ(Φ A ) / / R Φ R Ψ( R Φ A ) / / R Φ R Ψ( R 1 Φ A [ − 1]) / / R Φ R Ψ(Φ A [1]) Applying the cohomolog y functor H 1 we get 0 / / R 1 Φ A H 1 ( ˆ ξ R Φ A )   R 1 Φ A H 1 ( ˆ ξ R 1 Φ A [ − 1] )   / / 0 ( ∗ ) 0 / / H 1 ( R Φ R Ψ( R Φ A )) H 1 ( R Φ R Ψ( π R Φ A )) / / H 1 ( R Φ R Ψ( R 1 Φ A [ − 1])) / / 0 REFLEXIVITY IN DERIVED CA TEGORIES 19 Let us pro ve that H 1 ( R Φ R Ψ( π R Φ A )) is the identit y map. Consider a Φ-acyclic resolution P of A ; then w e hav e R Φ A := π R Φ A   0 / / Φ( P 0 ) p   α / / Φ( P 1 ) / / ... σ > 0 R Φ A := 0 / / Φ( P 0 ) / Ker α / / Φ( P 1 ) / / ... Since the terms in b oth the complexes are Φ-Ψ- acyclic, we ge t R Φ R Ψ R Φ A := R Φ R Ψ( π R Φ A )   0 / / ΦΨΦ( P 0 ) ΦΨ( p )   ΦΨ( α ) / / ΦΨΦ( P 1 ) / / ... R Φ R Ψ( σ > 0 R Φ A ) = 0 / / ΦΨ(Φ( P 0 ) / Ker α ) / / ΦΨΦ( P 1 ) / / ... Since the functor ΦΨ is ex a ct on the shor t exact sequence of Φ-Ψ-acyclic ob jects 0 → Ker α → Φ( P 0 ) p → Φ( P 0 ) / Ker α → 0 , the map ΨΦ( p ) is surjective; it is now clear that H 1 ( R Φ R Ψ( π R Φ A )) = 1 H 1 ( R Φ R Ψ R Φ A ) . Since π R Ψ( R 1 Φ A ) : R Ψ( R 1 Φ A ) → σ > 0 R Ψ( R 1 Φ A ) is the identit y map, by Theo- rem 4.3 and diagra m ( ∗ ) we hav e θ R 1 Φ A = H 0 ( ˆ ξ R 1 Φ A ) − 1 = H 1 ( ˆ ξ R 1 Φ A [ − 1] ) − 1 = H 1 ( ˆ ξ R Φ A ) − 1 . Second, thinking at γ A : R 1 Ψ R 1 Φ A → A as a map b et ween s talk complexes, let us consider the following commutative diagra m (see Theorem 4.3) R Ψ( σ > 0 R Φ A ) R Ψ( π R Φ A ) / / ∼ = qiso   R Ψ R Φ A ˆ η − 1 A / / A R Ψ( R 1 Φ A [ − 1]) R Ψ( R 1 Φ A )[1] ∼ = qiso   R 1 Ψ R 1 Φ A γ A : : u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u Applying R Φ we get the commutativ e diagram R Φ R Ψ R Φ A R Φ R Ψ( π R Φ A ) / / R Φ( ˆ η A )   R Φ R Ψ( σ > 0 R Φ A ) R Φ A R Φ( γ A ) 4 4 h h h h h h h h h h h h h h h h h h h h Applying the cohomolog y functor H 1 we get R 1 Φ( γ A ) ◦ H 1 ( R Φ( ˆ η A )) = H 1 ( R Φ R Ψ( π R Φ A )) = 1 H 1 ( R Φ R Ψ R Φ A ) .  20 F. MANTESE AND A. TONOLO Example 4.6. Let R a nd S be a rbitrary ass o ciative rings . Consider a par- tial cotilting bimo dule R U S of injective dimension ≤ 1. Then the adjunction (Hom R ( − , U ) , Hom S ( − , U )) satisfies the a ssumptions of Theor em 4.3 a nd Theo- rem 4 .5 . In par ticula r the cla sses o f D -reflexive mo dules a re abe lian sub categories of R -Mo d a nd Mo d- S and the re s triction of the functors Ext 1 ( − , U ) to these class e s forms a left adjoint pair . Mor eo ver, following Corolla r y 4.4, the functors Ext 1 ( − , U ) induce a duality b et ween the sub c ategories of D -r eflexiv e mo dules in Ker Hom( − , U ) and the functors Hom( − , U ) b etw een the sub categories of D - reflexiv e mo dules in Ker Ext 1 ( − , U ) (compare with [Cb, Cb1, CbF, Cp, CpF, Ma , T]). Example 4.7. As in Ex ample 2.2, let ( X , O X ) b e a lo c a lly noetheria n scheme such that every coher en t sheaf on X is a quotient of a lo cally free shea f. Assume the structur e sheaf O X has injectiv e dimension o ne and consider the adjunction ( R H om ( − , O X ) , R H om ( − , O X )) in D ( Coh X ). As we have already seen, any co - herent s heaf is D -r eflexiv e. Since any lo cally free shea f of finite rank is H om ( − , O X )- H om ( − , O X )-acyclic, the ass umption of Theor e m 4.3 a nd Theorem 4.5 are satisfied. Denoted as E xt ( − , O X ) the first derived functor of H om ( − , O X ) (see [Har2, Chp. II I]), we get that ( E xt ( − , O X ) , E xt ( − , O X )) is a left adjoin t pair in Coh X , the func- tors E xt ( − , O X ) induce a duality b et ween the coher e nt sheav es in Ker H om ( − , O X ) and the functors H om ( − , O X ) b et ween the coherent sheav es in Ker E xt ( − , O X ). 5. The n -dimensional cas e In this section w e re co ver a Co tilting Theorem in the ca se of functor s of co homo- logical dimension greater than one. W e assume A and B have eno ugh pro jectives, (Φ , Ψ ) is an adjoint pair of contra v ariant functors of cohomolo gical dimension at most n , Φ( P ) is Ψ -acyclic for each pro jective P in A , and Ψ( Q ) is Φ-acyclic for ea c h pro jectiv e Q in B . F or ins tance this is the cas e when Φ a nd Ψ are the cont rav ariant Hom-functors asso ciated to a pa rtial co tilting bimo dule. Let P be a pro jective resolution of an ob ject A in A . Denote by Q ∗∗ a Carta n- Eilenberg r esolution of Φ( P ); applying Ψ to the bicomplex Q ∗∗ , w e get the bico m- plex ... ... ... ... ... / / Ψ Q 2 , − 1 O O / / Ψ Q 1 , − 1 O O / / Ψ Q 0 , − 1 O O / / 0 ... / / Ψ Q 2 , 0 O O / / Ψ Q 1 , 0 O O / / Ψ Q 0 , 0 O O / / 0 0 O O 0 O O 0 O O T o this bico mplex we asso ciate tw o sp ectral sequences I E pq 2 and I I E pq 2 : I E pq 2 = H p h ( H q v (Ψ Q ∗∗ )) =  0 if q 6 = 0 H p h (ΨΦ P ) = R p (ΨΦ)( A ) if q = 0 I I E pq 2 = H p v ( H q h (Ψ Q ∗∗ )) = H p v (Ψ( H − q h ( Q ∗∗ ))) = H p v ( R Ψ( R − q Φ A )) = R p Ψ R − q Φ( A ) . Observe that I E pq 2 = 0 for e ither p > 0 or q 6 = 0 and I I E pq 2 = 0 for e ither p < 0 or q > 0. REFLEXIVITY IN DERIVED CA TEGORIES 21 Both these spe c tral seque nc e s conv erge to the hypercoho mology R q + p Ψ( R Φ( A )). The first sp ectral sequence I E pq 2 collapses to yield R n Ψ( R Φ( A )) = R n (ΨΦ)( A ) , which is zero for n > 0. The second sp ectral sequence I I E pq 2 lies o n the fourth quadrant: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ R 0 Ψ R 0 Φ( A ) R 1 Ψ R 0 Φ( A ) R 2 Ψ R 0 Φ( A ) ... R 0 Ψ R 1 Φ( A ) R 1 Ψ R 1 Φ( A ) R 2 Ψ R 1 Φ( A ) ... R 0 Ψ R 2 Φ( A ) R 1 Ψ R 2 Φ( A ) R 2 Ψ R 2 Φ( A ) ...                ... ... ... together with maps d pq 2 : I I E pq 2 = R p Ψ R − q Φ( A ) → I I E p +2 ,q − 1 2 = R p +2 Ψ R 1 − q Φ( A ) . Since the cohomological dimension of Φ and Ψ is at most n , w e ha ve I I E pq n +1 = I I E pq n + i = I I E pq ∞ for each p , q . F or each s ≤ 0, R s (ΨΦ)( A ) has a finite filtra tio n 0 = F n +1+ s R s (ΨΦ)( A ) ⊆ F n + s R s (ΨΦ)( A ) ⊆ ... ... ⊆ F 1 R s (ΨΦ)( A ) ⊆ F 0 R s (ΨΦ)( A ) = R s (ΨΦ)( A ) with [ F i R s (ΨΦ)( A )] / [ F i +1 R s (ΨΦ)( A )] ∼ = I I E i,s − i ∞ . If n = 1 we have I I E pq 2 = I I E pq ∞ for each p a nd q ; therefore 0 = R 1 (ΨΦ)( A ) = I I E 10 2 = R 1 Ψ R 0 Φ( A ) . If A is D -reflexive, R − 1 (ΨΦ)( A ) = 0 and R 0 (ΨΦ)( A ) ∼ = A ; hence using the edge homomorphisms, it is easy to get (1) I I E 0 − 1 2 = R 0 Ψ R 1 Φ( A ) = 0; (2) there e x ists the following s hort exa ct sequence with natur al maps 0 → I I E 1 − 1 2 = R 1 Ψ R 1 Φ( A ) → A → I I E 00 2 = R 0 Ψ R 0 Φ( A ) → 0 . It is not hard now to r eco ver Pr oposition 4.2 and partially Theo rem 4.3, (3). If n = 2 , we hav e I I E pq 2 = I I E pq ∞ for ( p, q ) = (1 , 0), ( p, q ) = (2 , 0), ( p, q ) = (0 , − 2) and ( p, q ) = (1 , − 2 ). Since R 1 (ΨΦ)( A ) = R 2 (ΨΦ)( A ) = 0, we g et R 1 Ψ R 0 Φ( A ) = R 2 Ψ R 0 Φ( A ) = 0. If A is D -reflexive, R − 2 (ΨΦ)( A ) = R − 1 (ΨΦ)( A ) = 0 a nd R 0 (ΨΦ)( A ) ∼ = A ; hence using the edge homo morphisms, one gets (1) R 0 Ψ R 2 Φ( A ) = R 1 Ψ R 2 Φ( A ) = 0; (2) there e x ist the following exact sequences with natura l maps 0 → R 0 Ψ R 1 Φ( A ) → R 2 Ψ R 2 Φ( A ) → A → A/ I I E 2 , − 2 ∞ → 0 0 → R 1 Ψ R 1 Φ( A ) → A/ I I E 2 , − 2 ∞ → R 0 Ψ R 0 Φ( A ) → R 2 Ψ R 1 Φ( A ) → 0 22 F. MANTESE AND A. TONOLO Example 5.1. Le t Λ denote the k -alge br a given by the quiver · 1   > > > > > > >          · 2   > > > > > > > · 3          · 4 with relations such that the left pro jective mo dules are 1 2 3 , 2 4 , 3 4 , and 4 . Let us consider the regular bimo dule Λ Λ Λ ; it is eas y to verify that it is a cotilting bimo dule of pr o jective dimension 2 . Consider the D -reflexive left Λ-module A := 2 3 4 ⊕ 1 . The seco nd sp e ctral sequence at the se c ond stage I I E pq 2 and at the third and stable stage I I E pq 3 = I I E pq ∞ assumes the following as pects: 1 2 3 ⊕ 1 2 3 ) ) S S S S S S S S S S S S S S S 0 0 1 2 3 ) ) S S S S S S S S S S S S S S S S S 4 1 2 ⊕ 1 3 0 0 1 2 ⊕ 1 3 2 ⊕ 3 0 0 0 4 0 0 0 1 Therefore we get the following exact sequences (see the previous conditio n (2) in the case n = 2): 0 → 1 2 3 → 1 2 ⊕ 1 3 → A = 1 ⊕ 2 3 4 → 2 3 4 → 0 0 → 4 → 2 3 4 → 1 2 3 ⊕ 1 2 3 → 1 2 ⊕ 1 3 → 0 Note that, passing from n = 1 to n > 1, the sp ectral sequence I I E pq 2 stabilizes at the n + 1 th stage; therefore we lo ose in general the po ssibilit y to describ e the D - reflexivit y of a n ob ject A in terms of prop erties of the ob jects R i Ψ R j Φ( A ). Resuming, the key pro perties which conse nt us to giv e a “nice” Co tilting Theo- rem in the 1-dimensio nal case are: Condition I: the s p ectral se q uence I I E pq 2 stabilizes at the se c o nd stage , Condition I I: the cohomo logies of a D -reflexive complex ar e D - reflexiv e. Both thes e pro perties are in g eneral false (see Examples 5.1, 3.3 ). The technical conditions assumed in the fo llo wing theorems gua ran tee both Condition I a nd I I. First, generalizing Theorem 4.3 we hav e Theorem 5.2. Assume Φ and Ψ have c ohomolo gic al dimension n and 1, r esp e c- tively. An obje ct A in A is D -r eflexive if and only if (1) Φ( A ) and R 1 Φ( A ) ar e D -r eflexive; (2) R i Ψ R j Φ( A ) = 0 fo r e ach i 6 = j ; (3) ther e exists a short exact se qu enc e 0 → R 1 Ψ R 1 Φ( A ) → A → ΨΦ( A ) → 0 . In su ch a c ase R i Φ( A ) = 0 fo r e ach i > 1 . REFLEXIVITY IN DERIVED CA TEGORIES 23 Pr o of. The sp e ctral se quence I I E pq 2 stabilizes at the second stage: for only t wo columns surv ive. Therefore if A is D -reflexive, we get immediately the or tho gonal relations R i Ψ R j Φ( A ) = 0 for each i 6 = j . The filtration of A ∼ = R 0 (ΨΦ( A )) pro duces the short exact sequence 0 → R 1 Ψ R 1 Φ( A ) → A → R 0 Ψ R 0 Φ( A ) = ΨΦ( A ) → 0 . By the a djunction, also R Φ( A ) is D -reflexive; then, by P r opositio ns 3 .5 and 1.9, its cohomolog ies R i Φ( A ) are also D -reflexive. Then R Ψ( R i Φ( A )) are D -reflexive: r e- mem be r ing the orthogo na l relations, b oth the co mplexes R Ψ(Φ( A )) = Ψ Φ( A ) and R Ψ( R 1 Φ( A )) ∼ = R 1 Ψ R 1 Φ( A )[ − 1] are D -reflexive; mor eo ver, since 0 ∼ = R Ψ( R i Φ( A )), for i ≥ 2 the ob jects R i Φ( A ) are eq ua l to z ero. Conv ersely , co nsider the triangle asso ciated to the short exa ct sequence (3). By (1), (2) and the adjunction, b oth the complexe s R Ψ(Φ( A )) = ΨΦ( A ) and R Ψ( R 1 Φ( A )) ∼ = R 1 Ψ R 1 Φ( A )[ − 1] a re D - reflexive; thus one gets the D -reflexivity of A from the D -reflexivity of the other t wo terms in the se quence of 3).  Let us give an exa mple wher e the prev ious theorem applies. Example 5.3. Le t Λ denote the k -alge br a given by the quiver · 0 / / · 1 / / · 2 / / · 3 / / · 4 with rela tions such that the left pro jective mo dules are 0 1 2 , 1 2 3 , 2 3 4 , 3 4 and 4 . Cons ide r the left Λ-mo dule Λ U = 2 3 4 ⊕ 3 ⊕ 1 2 3 ⊕ 1 ; it is easy to verify that it ha s injectiv e dimension 2. The endomorphism r ing S := E nd( Λ U ) is the k -alg ebra given b y the quiver · 7 · 5 / / · 6 / / ? ?        · 8 with rela tions such that the right pr o jective mo dules a re 7 6 , 8 6 , 6 5 , and 5 . The right S -mo dule U S = 7 ⊕ 7 8 6 ⊕ 7 6 ⊕ 6 5 has injective dimension 1. It is easy to verify tha t Λ U S is a partial cotilting bimo dule. The pro jective module 0 1 2 and its pro jections 0 1 , 0 are the o nly not D -reflexive indecompo sable Λ-mo dules, while all the inde- comp osable S -mo dules are D -reflexive. In particula r, consider the indecompo sable left Λ-mo dule 1 2 ; it is a D -reflexive module of pro jective dimension 2. It s a tisfies the three conditions of Theo r em 5.2: (1) Φ( 1 2 ) = 5 and R 1 Φ( 1 2 ) = 8 are D -reflexive; (2) R i Ψ R j Φ( 1 2 ) = 0 for each i 6 = j ; (3) there e x ists a shor t exact sequence 0 → R 1 Ψ R 1 Φ( 1 2 ) = 2 → 1 2 → 1 = ΨΦ( 1 2 ) → 0 . Moreov er R 2 Φ( 1 2 ) = 0. A second p ossibilit y to obtain partia l re s ults is to characterize the D -reflexive ob jects inside a suita ble sub class of A . Both Theo r ems 4.3 and 5.2 suggest to consider the sub class of ob jects A in A such that R i Ψ R j Φ( A ) = 0 for each i 6 = j ; in such a wa y , the spectra l s equence I I E pq 2 stabilizes at the second stage. T o ha ve also that the cohomolog ies of a D - reflexiv e co mplex are still D -reflexive, Lemma 3 .8 suggests to restrict further o ur clas s (compare with [A T, Theorem 2.7]). 24 F. MANTESE AND A. TONOLO Theorem 5. 4. L et (Φ , Ψ) b e an adjoint p air of c ontr avariant funct ors of c oho- molo gic al dimensions ≤ n . A n obje ct A in A , such that R i Ψ R j Φ( A ) = 0 and R i Φ R j Ψ R j Φ( A ) = 0 if i 6 = j , is D -r eflex ive if and only if for i = 0 , 1 , ..., n (1) R i Φ( A ) ar e D -r eflexive; (2) ther e exists a filtra tion 0 = A n +1 ≤ A n ≤ ... ≤ A 0 = A such that A i / A i +1 ∼ = R i Ψ R i Φ( A ) . In su ch a c ase the obje cts R i Ψ R i Φ( A ) r esult to b e D - r eflexive. Pr o of. If A is D -reflexive, also R Φ( A ) is D -reflexive; by Lemma 3.8, choosing as ρ the iden tity function , the co homologies R i Φ( A ), i = 1 , ..., n , are D -reflexive to o. Since R i Ψ R j Φ( A ) = 0 for each i 6 = j , the spec tral seq uence I I E pq 2 stabilizes at the second stage. Therefore R s (ΨΦ) A = 0 for each s 6 = 0 a nd R 0 (ΨΦ) A ∼ = A has a finite filtration 0 = A n +1 ≤ A n ≤ ... ≤ A 0 = A with factors A i / A i +1 ∼ = I I E i, − i ∞ = I I E i, − i 2 = R i Ψ R i Φ( A ). Conv er s ely , let us assume condition (1) and (2) are satisfied. Since R i Φ( A ) is D -reflexive, the complex R Ψ R i Φ( A ) is D -reflexive; we want to prov e that its cohomolog y R i Ψ R i Φ( A ) is D -reflexive to o. By hypo theses R j Φ R i Ψ R i Φ( A ) = 0 for any j 6 = i ; mo reo ver, since R i Φ A ∼ = R Φ R Ψ( R i Φ A ) = R Φ( R i Ψ R i Φ A [ − i ]) = R Φ( R i Ψ R i Φ A )[ i ] , we hav e R i Φ R i Ψ R i Φ A = H 0 ( R Φ( R i Ψ R i Φ A )[ i ]) ∼ = R i Φ A. Therefore R j Ψ R i Φ R i Ψ( R i Φ A ) = 0 for eac h j 6 = i and by Lemma 3.8 the coho- mologies of R Ψ R i Φ( A ) are D -reflexive. C o nsider now the triangles R n Ψ R n Φ( A ) = A n → A n − 1 → A n − 1 / A n = R n − 1 Ψ R n − 1 Φ( A ) → A n [1] ....... A 1 → A 0 = A → A 0 / A 1 = R 0 Ψ R 0 Φ( A ) → A 1 [1] Since R n Ψ R n Φ( A ) and R n − 1 Ψ R n − 1 Φ( A ) a re D -reflexive, also A n − 1 is D -reflexive. Iterating this pro cedure on the other triangles, using the D -reflexivity o f A i − 1 / A i , i = n, n − 1 , ..., 1 , we pr o ve the D -reflexivity of A .  W e are now re a dy to give a Co tilting Theor em in the s ense of [CbF1, Ch. 5], betw een the clas ses of D -r eflexiv e ob jects induced by the pair o f adjoin t functors (Φ , Ψ ) in the n -dimens io nal cas e. Corollary 5.5. Consider the fol lowing sub classes of the classes D A and D B of D - r eflexive obje cts in A and B : D A = ∩ i 6 = j  Ker R i Ψ R j Φ ∩ Ke r R i Φ R j Ψ R j Φ  and D B = ∩ i 6 = j  Ker R i Φ R j Ψ ∩ Ker R i Ψ R j Φ R j Ψ  . Then set ting E i Φ = ( ∩ j 6 = i Ker R j Φ) ∩ D A and E i Ψ = ( ∩ j 6 = i Ker R j Ψ) ∩ D B the fo l lowing c onditions ar e satisfie d: (1) R i Φ : D A → E i Ψ and R i Ψ : D B → E i Φ . REFLEXIVITY IN DERIVED CA TEGORIES 25 (2) for e ach obj e ct A in D A and B in D B ther e exists filtr ations 0 = A n +1 ≤ A n ≤ ... ≤ A 0 = A and 0 = B n +1 ≤ B n ≤ ... ≤ B 0 = B such that A i / A i +1 ∼ = R i Ψ R i Φ( A ) and B i /B i +1 ∼ = R i Φ R i Ψ( B ) . (3) the r estrictions R i Φ : E i Φ − − → ← − − E i Ψ : R i Ψ define c ate gory e quivale nc es. Example 5.6. L e t Λ deno te the k -alg ebra given b y the quiver A 8 with relations such that the left pro jective mo dules are 1 2 3 4 , 2 3 4 , 3 4 , 4 5 6 7 8 , 5 6 7 8 , 6 7 8 , 7 8 , 8 . Co nsider the cotilting mo dule Λ U = 1 2 3 4 ⊕ 1 ⊕ 3 4 ⊕ 4 5 6 7 8 ⊕ 5 6 7 8 ⊕ 6 7 8 ⊕ 7 8 ⊕ 7 of injective dimension 2 and let S = End Λ ( U ). Applying Theorem 3.4 we get that any co mplex in D b (Λ-mo d) and in D b (mo d- S ) is D -reflexive w.r.t the adjunction ( R Hom Λ ( − , U ) , R Hom S ( − , U )). The Λ-mo dule X = 1 2 3 satisfies the ass umptions o f Theorem 5.4, indeed: • Ex t i S (Ext j Λ ( X, U ) , U ) = 0 for i 6 = j • Hom S (Hom Λ ( X, U ) , U ) = 1, Ext 1 S (Ext 1 Λ ( X, U ) , U ) = 2 , E xt 2 S (Ext 2 Λ ( X, U ) , U ) = 3 • Ex t 1 Λ (1 , U ) = Ext 2 Λ (1 , U ) = 0, Ho m Λ (2 , U ) = Ex t 2 Λ (2 , U ) = 0, Hom Λ (3 , U ) = Ext 1 Λ (3 , U ) = 0 W e co nc lude that X a dmits a filtration 0 ≤ X 1 ≤ X 0 ≤ X , where X 1 = 3 , X 0 /X 1 = 2, X/X 0 = 1. Consider no w the simple mo dule 4; since Ext 2 S (Ext 1 Λ (4 , U ) , U ) = 3, the assumptions o f Theorem 5.4 fail. Moreov e r Hom S (Hom Λ (4 , U ) , U ) = 3 4 , Ext 1 S (Ext 1 Λ (4 , U ) , U ) = 0 , Ext 2 S (Ext 2 Λ (4 , U ) , U ) = 0 a nd so 4 do es no t admit any filtration with D -reflexive factors. Remark 5.7. If R is no etherian and R U S is a finitely genera ted co tilting bimo dule, then any finitely generated pro jective mo dule is r eflexiv e and Hom( − , U )-acyclic, so D -reflexive. It follows that any finitely generated mo dule is D -reflexive. Let now M ∈ R -mo d such that Ext j R ( M , U ) = 0 for i 6 = j . Then, for M a nd E xt i R ( M , U ) are D -r eflexiv e, it follows that Ext j S (Ext i R ( M , U )) = 0 and Ext j R (Ext i S (Ext i R ( M , U ) , U ) , U ) = 0 for i 6 = j , so we are in the ass umption of Theor em 5.4. Thus we g et from Corolla r y 5.5 that M ∼ = Ext i (Ext i R ( M , U ) , U ) , recov ering the Miyashita result (cfr. [M, Theorem 1.16]). Ackno wledgements W e wish to thank the referee fo r her/his useful sug gestions which improv ed the pap er. 26 F. MANTESE AND A. TONOLO References [AC] L. A nge leri-H ¨ ugel and F. U. Coel ho, Infinitely gener ate d tilting mo dules of finite pr oje ctive dimension , F orum Math 13 (2001), 239-250. [AF] F. W. Anderson and K. R. F uller, Rings and c ate gories of mo dules , Graduate texts in Mathematics, Springer, New Y ork, Heidelb erg, Berli n, 1973. [A T] Maria Archet ti and Alberto T onolo, Partial tilting and c otilting bimo dules J. A lgebra 305 (2006), 321–359. [BB] S. Brenne r e M. Butler, Gener alizations of the Bernstein- Gelfand -Ponomar ev r efle ction functors , Lecture Notes in M ath . 832 , Spri ng er-V erlag, Berl i n, 1980, 103–169. [BB1] S. Brenner e M. Butler, A sp e c tr al se quenc e analysis of classic al tilt ing functors , Handb ook of Tilting Theory , London Math. Soc. LNS 332 , 2007. [Cb] R. R. Colby , A gener alization of Morita duality and the t ilting the or em , Comm . Algebra 17(7) , 1989, pp. 1709–1722. [Cb1] R. R. Colby , A c otilting the or em for rings , Metho ds in Mo dule Theory , M. Dekker, New Y ork, 1993, pp. 33–37. [CbCpF] R. R. Colby , R. Colpi and K. R. F uller, A note on c otilting mo dules and gener alize d Morita dualities , Lecture N ot es in Pure and Appl. M at h., 236 , Dekker, New Y or k, 2004, 85–88. [CbF] R. R. Colby and K. R. F ul l er, Tilting, c otilt i ng and serial ly tilted rings , Comm . Algebra. 18(5) (1990 ), 1585–1615 . [CbF1] R. R. Col b y and K. R. F uller, Equivalenc e and Duality for Mo dule Cate gories , Cambridge T racts in Mathematics 16 1 , Cambridge Unive rsity Press, 2004. [Cp] R. C ol pi, Cotilting bimo dules and their dualities , Pr oc. Euro conf. Murcia ‘ 98, LNP AM 210 , Dekk er, New Y ork 2000, 81–93. [CpF] R. Colpi and K. R. F uller, Cotilting mo dules and bimo dules , Pa cific J. Math. 192 (2000), 275–291. [CPS] E. Cl ine, B. Parshall and L. Scott, De ri ve d c ate gories and Morita the ory , J. A lgebra 10 4 (1986), 397–409. [L] J. Lipman, Notes on Derive d F unctors and Gr othendie ck Duality , h ttp://www.math.purdue.edu/ ∼ lipman/ [Har] R. Hartshorne, R esidues and duality , Lecture Notes in Mathematics 20 , Springer-V erlag, Berlin-New Y ork, 1966. [Har2] R. Hartshorne, Algebr aic ge ometry , Gr adu ate T exts in Mathematics 52 , Springer-V erlag, New Y ork-H eidelberg, 1977. [Hap] D. Happel, T riangula te d ca te gories in the R epr ese ntation The ory of Finite Dimensional Algebr as , Cambridge U niv. Press, Cam bridge, 1988. [K] B. Keller, Derive d Catego ries and t heir use , Hand bo ok of Algebra. V ol. 1. Edited by M. Hazewinkel, North-Holland Publishi ng Co., Amsterdam, 1996. h ttp://www.math.jussieu.fr/ ∼ keller/ [Ma] F. Mant ese, Gener alizing c otilting dualities , J. Algebra 2 3 6 (2001), 630–644. [M] Y. Miyashita, Tilting mo dules of finite pr oje ctive dimension , Math. Z. 19 3 (1986), 113–146. [R] J. Rick ard, M orita the ory for Derived Cate gories , J. London M at h. Soc. 39 (1989), 436–456. [S] N. Spaltenstein, R esolutions of unb ounde d c omplexes , Compositio Math. 65 (1988), no. 2, 121–154. [T] A. T onolo, Ge ner alizing Morita Dualities: a homolo gic al appr o ach , J. Algebra 232 (2000), 282–298. [W] C. W eibel, An intr o duction to homolo g ic al algebr a , Cambridge Studies in Adv anced M ath - ematics 3 8 , 1994. (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 : francesca.mant ese@univr.it (A. T onolo) Dip. Ma temat ica Pura ed Applica t a, Universit ` a degli studi di P a do v a, via Trieste 63, I-35121 P adov a It al y E-mail addr ess : tonolo@math.un ipd.it