The Integral Cluster Category

THE INTEGRAL CLUSTER CA TEGOR Y BERNHARD KELLER AND SARAH SCHER OTZKE Abstra ct. Integral cluster categories of acyclic quivers h a ve recently b een u sed in the represen tation-theoretic approac h to quantum cluster algebras. W e sho w that ov er a principal ideal domain, such categories b ehav e muc h b etter than one wo uld exp ect: T hey can be d escribed as orbit categories, th eir indecomp osable rigid ob jects do not dep end on the ground ring and the mutatio n op eration is transitive. 1. intr oduct ion Cluster catego ries were in tro duced in [3 ] for acyclic qu iv ers and ind ep en- den tly in [5] for Dynkin quive rs of t yp e A . Th ey hav e pla y ed an imp ortan t rˆ ole in the study of F omin-Zelevinsky’s cluster algebras [9], cf. the sur v eys [2] [13] [15] [20] [21]. In tegral cluster categories app ear naturally in the study of quantum clus- ter algebras as defin ed and s tu died in [4] and [8 ]. Indeed, one would lik e to in terpret the quan tum parameter q as the cardinalit y of a finite field [22] and in order to stu dy the cluster categories ov er all p rime fields simulta ne- ously [19], one considers the clus ter category o v er the ring of intege rs, cf. the app endix to [19 ]. I n this pap er, we con tin ue the study b egun there: F or an acyclic quiver Q and a p rincipal id eal domain R , we constru ct the cluster catego ry C RQ using Amiot’s metho d [1] as a triangle qu otien t of the p erfect deriv ed category of the Ginzbu rg dg algebra [10] asso ciated with the path algebra RQ . On the other han d , we define the category C or b RQ as th e catego ry of orbits o f t he b ounded deriv ed category of RQ un d er the action of th e auto equiv alence Σ − 2 S , where S is the Serre fun ctor and Σ the susp ension functor. In the case where R is a field, cluster ca tegories were originally defined as C or b RQ in [3] and it was shown in [1] that the t wo definitions are equiv alent . Our firs t main r esult is the existence of a natural equiv alence for any prin cipal ideal d omain R C or b RQ ∼ → C RQ . This sh ows in particular t hat the orbit category is triangulated. F or the case where R is a field, this has b een known since [14]; in th e general case, it is quite su r prising since the algebra RQ is of global dimension ≤ 2 and the p ro of in [14] strongly used the fact th at f or a field F , the path algebra F Q is of global d im en sion ≤ 1. Key wor ds and phr ases. Cl uster category , rigid ob jects, q uiver represen tation. 1 2 BERNHARD KELLER AND SARAH SCHER OTZKE Our second main result states that all indecomp osable rigid ob jects of C RQ are either images of rigid indecomp osable R Q -mo dules or direct factors of the image of Σ R Q . If we com bine this with Cra wley-Boevey’s classification [7] of rigid indecomp osable RQ -mo dules, we obtain th at the classification of the rigid indecomp osable ob jects of C RQ is indep enden t of the prin cipal ideal d omain R and that iterated m utation starting from R Q r eac hes all indecomp osable rigid ob jects. Again, this is w ell-kno wn in the fi eld case, cf. [3] [11] [12], but quite sur prising in the general case. The pap er is structured as follo ws: In the second section, w e recall gen- eral adjointness relations b etw een the der ived tensor and the d eriv ed Hom- functor for d g algebras o v er an y comm utativ e ring R . W e d efi ne the (rela- tiv e) Serre fu nctor. In the third se ction, w e consider the d eriv ed ca tegory of an R -algebra A (finitely generated pro jectiv e o ver R ) and an endofun ctor F of the d e- riv ed cat egory of A whic h is give n by the deriv ed tensor pro d u ct with an A -bimo dule complex Θ. Th e tensor d g algebra asso ciated with Θ is a differ- en tial graded algebra wh ic h we denote b y Γ. W e give su fficien t conditions for the orbit category C or b of the p erfect d eriv ed catego ry p er( A ) by F to em- b ed canonically in to the triangulated quotien t category p er( Γ) /D per ( R ) (Γ). Here D per( R ) (Γ) denotes the deriv ed category of the differential graded Γ- mo dules w hose restrictions to R are p erf ect complexes. The metho d s used in this s ection generalize the approac h us ed in [16] to algebras o v er arb itrary comm utativ e rings. In the fourth sectio n, we consider the orbit catego ry of the p erfect de- riv ed category p er( RQ ) under the acti on of th e auto-equiv alence giv en by Σ − 2 S . T h is functor is giv en by the total derived functor asso ciated with the RQ -bimo du le complex Θ = Σ − 2 Hom R ( RQ, R ). By a result of [16], th e differen tial graded tensor algebra of Θ is isomorphic to the Ginzburg algebra Γ asso ciated to the quiver Q with the zero p oten tial. S o using the resu lts of the third section, we can emb ed the orbit catego ry into the int egral cluster catego ry p er(Γ) /D per ( R ) (Γ). F urthermore, w e sho w in t his section that a relativ e 3-Calabi-Y au prop ert y holds f or D (Γ). In the fifth section, under the assumption th at R is a prin cipal ideal domain, we sho w that all rigid ind ecomp osable ob jects in the integ ral cluster catego ry come f rom mo du les and their susp ensions and that th e embedd ing giv en in section 3 is an equiv alence of categories. Hence the orbit category C or b is triangulated and the inte gral cluster category is relativ e 2-Calabi- Y au. Using a result by Crawley-B o ev ey [7] w e establish a bijection b et we en the r igid ind ecomp osable ob jects in the cluster category o v er a field F and those ov er a rin g induced by the triangle functor ? ⊗ L R F . In the last section, we sh o w using the b ijection b etw een rigid ob jects in C RQ and C F Q , th at all cluster-tilting ob jects are related b y m utations. THE INTEGRAL CLUSTER CA TEGOR Y 3 A cknowledgment The second-named author thanks the ‘F ond ation S ciences Math´ ematiques de Pa ris’ for a p ostdo ctoral fello wship durin g wh ic h this p ro ject w as carried out. B oth authors are grateful to the referee for many helpful commen ts. 2. Derived ca t egories over co mmut a tive r ings Let R b e a commutat iv e ring. F or an asso ciativ e differen tial graded R - algebra A whic h is cofibr an t as an R -mo dule (cf. section 2.12 of [18]), w e denote by D ( A ) the derive d c ate gory of dg A -mo dules, by p er ( A ) the p erfe ct derive d c ate gory , i.e. the th ic k sub categ ory of D ( A ) generated by A , and by D per( R ) ( A ) th e full sub cate gory of D ( A ) whose ob jects are the dg A -mo dules whose underlying complex of R -mo dules is p erfect. Thr oughout this article, w e denote by Σ the shift f u nctor in the derived category . If the underlying R -mo du le of A is fi nitely generat ed pro jectiv e o v er R , w e denote b y S R the (relativ e) Serre functor of D ( A ) given b y the total d er ived functor of tensoring w ith the A -bimo d ule Hom . R ( A, R ). Here, for t w o d g A -mo du les L and M , we denote b y Hom . A ( L, M ) the dg R -mo dule whose n th component is the R -mo dule of morp hisms of graded A -mo du les f : L → M homogeneo us of degree n and whose d ifferential sends suc h an f to d M ◦ f − ( − 1) n f ◦ d L . W e denote by RHom the total derived functor of Hom . . W e defin e A e to b e the dg algebra A ⊗ R A op . Th e follo wing well -kno wn isomorphisms will often b e u sed in the r est of the article. Lemma 2.1. L et A and B b e two dg R - algebr as which ar e c ofibr ant over R . (1) L et M ∈ D ( B ⊗ A op ) , L ∈ D ( A ) and N ∈ D ( B ) . Ther e is a bifu nc - torial isomorphism RHom B ( L ⊗ L A M , N ) ∼ → RHom A ( L, RHom B ( M , N )) in D ( R ) . (2) F or P ∈ p er( B ) and M ∈ D ( B ) , ther e is a bifunctorial isomorph ism M ⊗ L B RHom B ( P , B ) ∼ → RHom B ( P , M ) in D ( B op ) . (3) F or al l L and M in D ( A ) , ther e is a bifunctorial isomorphism RHom A e ( A, RHom R ( L, M )) ∼ → RHom A ( L, M ) in D ( A op ) . Pr o of. W e denote b y pM a cofibrant replacement of M and b y iM a fibr an t replacemen t of M . W e hav e L ⊗ L A M ∼ = pL ⊗ A pM . Therefore, we ha v e RHom B ( L ⊗ L A M , N ) = Hom . B ( pL ⊗ A pM , N ) ∼ = Hom . A ( pL, Hom . B ( pM , N )) = RHom A ( L, RHom B ( M , N )) . 4 BERNHARD KELLER AND SAR AH SCHEROTZKE This pr ov es (1). F or p art (2 ), w e ob s erv e that w e ha ve a bifu nctorial morp hism from pM ⊗ B Hom . B ( pP , B ) to Hom . B ( pP , pM ), w hic h is in vertible in D ( R ) for P = A hence for all ob jects in p er( A ). F or part (3), we ha v e RHom A e ( A, RHom R ( L, M )) = Hom . A e ( pA, Hom . R ( pL, iM )) , where pA is a cofib ran t as a d g A e -mo dule. Then pA is also cofibrant as a dg A -mo dule and as a dg A op -mo dule. W e hav e Hom . A e ( pA, Hom . R ( pL, iM )) ⊂ Hom . A op ( pA, Hom . R ( pL, iM )) and by (1), there is a bifu nctorial isomorphism b et w een Hom . A op ( pA, Hom . R ( pL, iM )) a nd Hom . R ( pA ⊗ A op pL, iM ). This isomorphism indu ces a bijection b et we en Hom . A e ( pA, Hom . R ( pL, iM )), whic h consists of all the elements of Hom . A op ( pA, Hom . R ( pL, iM )) that comm ute with the right action of A , and Hom . A ( pA ⊗ A op pL, iM ). Now Hom . A ( pA ⊗ A pL, iM ) is isomorphic to RHom A ( L, M ), whic h finishes the pr o of.  Prop osition 2.2 . Supp ose that the underlying R -mo dule of A is finitely gener ate d pr oje ctive. F or L ∈ D ( A ) and M ∈ p er( A ) , we have the fol lowing c anonic al b ifunctorial isomorphism RHom R (RHom A ( M , S R L ) , R ) ∼ → RHom A ( L, M ) . If Hom R ( A, R ) b elongs to p er( A ) , then S R is an auto-e quivalenc e of p er( A ) with inverse RHom A (RHom(? , R ) , A ) . Pr o of. By applying part (2) of 2.1 twice we obtain RHom A ( M , L ⊗ L A RHom R ( A, R )) ∼ = L ⊗ L A RHom A ( M , RHom R ( A, R )) . By p art (1) of 2.1, we obtain RHom A ( M , RHom R ( A, R )) ∼ = RHom R ( M , R ) . Therefore, we ha v e RHom R (RHom A ( M ,L ⊗ L A RHom R ( A, R ))) , R ) ∼ = RHom R ( L ⊗ L A RHom R ( M , R ) , R ) ∼ = RHom A ( L, RHom R (RHom R ( M , R ) , R )) ∼ = RHom A ( L, M ) b y (1) of 2.1. This prov es the first part. If we c ho ose M = A , we get, by the fi rst statemen t, S R L ∼ = RHom R (RHom A ( L, A ) , R ) . No w we u s e the fact that RHom R (? , R ) and RHom A (? , A ) are dualit y fu nc- tors on p er( A ).  THE INTEGRAL CLUSTER CA TEGOR Y 5 3. Embeddings of orbit c a tegories Let A b e an asso ciativ e R -algebra wh ich is fi nitely generated pr o jectiv e as an R -mo dule and let Θ b e a complex of A - A -bimo dules. W e sup p ose that Θ is cofibr an t as a d g A -bimo dule and that Θ is p erfect as a d g R -mo dule. W e den ote b y F : D ( A ) → D ( A ) the functor ? ⊗ L A Θ. W e define the dg algebra Γ = T A (Θ) to b e the tens or algebra o v er A giv en b y A ⊕ Θ ⊕ (Θ ⊗ A Θ) ⊕ · · · ⊕ (Θ ⊗ A · · · ⊗ A Θ) ⊕ · · · . Then Γ is homological ly smo oth ov er R by [16, 3.7]. F or N ≥ 0, we denote b y Γ >N the ideal L n>N Θ ⊗ A n of Γ and p u t Γ ≤ N = Γ / Γ >N . Th en D per ( R ) (Γ) is con tained in p er(Γ) and Γ ≤ N lies in D per ( R ) (Γ) for all N ∈ N . W e consider the category C (Γ) = p er(Γ) / D per ( R ) (Γ) and compute its morph ism s paces. W e h av e a f unctor ? ⊗ L A Γ : p er ( A ) → C (Γ) and a restriction fun ctor p er(Γ) → D ( A ) induced by the natural em b edd ing of A into Γ. F or any Y ∈ p er(Γ) and an y N ∈ N , let F N ( Y ) = Y ⊗ A Θ ⊗ A N and let m N : F N ( Y ) → Y b e induced by the m ultiplication. W e a ssume that for an y X ∈ D per( R ) ( A ) ther e is an n ∈ N suc h t hat Hom D ( A ) ( F n ( A ) , X ) v anishes. Lemma 3.1. F or Y i n p er(Γ) , we have the fol lowing i somorphisms colim N Hom D (Γ) (Γ >N , Y ) ∼ = Hom C (Γ) (Γ , Y ) ∼ = colim N Hom D ( A ) ( F N +1 ( A ) , Y | A ) . Pr o of. By definition, the sp ace Hom C (Γ) (Γ , Y ) is giv en by colim M Γ Hom D (Γ) (Γ ′ , Y ) , where M Γ denotes the catego ry of all morphisms s : Γ ′ → Γ in D (Γ) such that cone ( s ) lies in D per R (Γ). W e consider the exact sequen ce 0 / / Γ >N e N / / Γ / / Γ ≤ N / / 0 . As Γ ≤ N v anishes in C (Γ), the em b edding e N is an isomorp hism for an y N in N . Th e transition maps in the direct s ystem are in d uced by the em b edding of Γ >N +1 in to Γ >N and the maps from Hom D (Γ) (Γ >N , Y ) to Hom C (Γ) (Γ , Y ) b y comp osing with the inv erse of e N . By a classical result of V erd ier, it is sufficien t to sho w that for ev ery morphism s in M Γ , there is an N ∈ N such that e N factors through s . It is therefore sufficien t to sho w that for ev ery Y ∈ D per( R ) (Γ) there is an N ∈ N suc h that the space Hom D (Γ) (Γ >N , Y ) v anishes. W e h a v e Γ >N = Θ ⊗ A ( N +1) ⊗ A Γ and by adjunction Hom D (Γ) (Θ ⊗ A ( N +1) ⊗ A Γ , Y ) ∼ = Hom D ( A ) (Θ ⊗ A ( N +1) , RHom Γ (Γ , Y )) ∼ = Hom D ( A ) ( F N +1 ( A ) , Y | A ) . 6 BERNHARD KELLER AND SAR AH SCHEROTZKE The transition map s of the direct system colim N Hom D ( A ) ( F N +1 ( A ) , Y | A ) are giv en by applying F and comp osing w ith th e multiplicat ion map m 1 : Y ⊗ L A Θ → Y . If Y ∈ D per( R ) (Γ), then Y | A ∈ D per( R ) ( A ) and by the assumption there is an N ∈ N suc h that Hom D ( A ) ( F N ( A ) , Y | A ) v anishes. By the abov e isomorphism, any m ap fr om Γ >N to Y v anishes. Therefore, the colimit colim N Hom D (Γ) (Γ >N , Y ) ∼ = colim N Hom D ( A ) ( F N +1 ( A ) , Y | A ) computes Hom C (Γ) (Γ , Y ).  Definition 3.2. L et F : C → C b e an endofunctor of an additive c ate gory C . Th e orbit catego ry C / h F i of C by F is the c ate gory with the same obje c ts as C and the sp ac es of morphisms Hom C / h F i ( M , N ) = colim l ∈ N M i ∈ N Hom C ( F i ( M ) , F l ( N )) . Theorem 3.3. L et Y = Y 0 ⊗ L A Γ for some obje ct Y 0 of p er( A ) and su pp ose that Θ b elongs to p er( A ) . Then we have Hom C (Γ) (Γ , Y ) ∼ = colim N M l ∈ N Hom D b ( A ) ( F N ( A ) , F l ( Y 0 )) and ? ⊗ L A Γ induc es a ful ly faithful emb e dding of the orbit c ate gory C or b of p er( A ) by F into C (Γ) . F urthermor e C (Γ) e quals i ts smal lest thick sub c ate- gory c ontaining the orbit c ate gory. Pr o of. By the previous lemma, we ha v e Hom C (Γ) (Γ , Y ) ∼ = colim N Hom D ( A ) ( F N +1 ( A ) , Y | A ) . But Y | A = ( Y 0 ⊗ L A Γ) | A ∼ = M l ∈ N Y 0 ⊗ L A Θ ⊗ A l ∼ = M l ∈ N F l ( Y 0 ) . This pr ov es the fi rst statemen t b ecause F N +1 ( A ) is p erfect in D ( A ). Usin g the fact that Hom C (Γ) (? ⊗ L A Γ , Y ) and colim N M l ∈ N Hom D ( A ) ( F N (?) , F l ( Y 0 )) are h omologic al f unctors on p er( A ), w e ob tain that Hom C (Γ) ( L ⊗ L A Γ , Y ) ∼ = colim N M l ∈ N Hom D ( A ) ( F N ( L ) , F l ( Y 0 )) for all L ∈ p er( A ). Since we ha v e colim N L l ∈ N Hom D ( A ) ( F N ( L ) , F l ( Y 0 )) = Hom C or b ( L, Y 0 ), the functor ? ⊗ L A Γ induces a fully f aithful em b edding of the orbit catego ry in to C (Γ). The functor ? ⊗ L A Γ induces a triangle functor from p er( A ) to p er(Γ) such that A m aps to Γ. The triangle closur e of th e THE INTEGRAL CLUSTER CA TEGOR Y 7 image of p er( A ) is therefore p er(Γ). The last statemen t now follo ws from the comm utativit y of the follo wing d iagram p er( A ) ? ⊗ L A Γ / /   p er(Γ)   C or b ? ⊗ L A Γ / / C (Γ) .  Remark 3.4. N ote that if F is an e qu ivalenc e, t hen the c olimit i n 3.3 is given by M l ∈ Z Hom D ( A ) ( A, F l ( Y 0 )) . Supp ose that F =? ⊗ L A Θ sends p er( A ) to itself. Let F b e a field and π : R → F a rin g homomorph ism. W e denote b y F A the scalar extension A ⊗ R F , b y C ( F Γ) the category p er( F Γ) /D b ( F Γ), by F F the fu nctor ? ⊗ L F A ( F ⊗ L Θ) on p er( F A ) and b y C or b F the orbit category of p er( F A ) b y F F . Corollary 3.5. The fol lowing diagr am c ommutes: p er( A ) / / ? ⊗ L R F   C or b   ? ⊗ L A Γ / / C (Γ)   p er(Γ) o o ? ⊗ L R F   p er( F A ) / / C or b F ? ⊗ L F A F Γ / / C ( F Γ) p er( F Γ) o o Pr o of. W e get the functor from C (Γ) to C ( F Γ) from the fact that ? ⊗ L R F maps D per( R ) (Γ) into D b ( F Γ). Since A is cofibrant o ve r R , every complex that is cofibr an t o v er A or A e is also cofibrant ov er R . Th erefore F ⊗ R Θ is cofibran t as a dg F A e -mo dule and the follo wing diagram comm utes p er( A ) F / / ? ⊗ L R F   p er( A ) ? ⊗ L R F   p er( F A ) F F / / p er( F A ) . This p ro v es the existence of a natural functor f r om C or b to C or b F . The com- m utativit y of the mid d le square follo ws from the diagram in the pro of of 3.3.  4. The integral clu ster ca t egor y Let Q b e a finite qu iv er without oriented cycles. An RQ -lattic e is a finitely generated RQ -mo d ule whic h is free o v er R . W e d en ote b y D ( RQ ) the deriv ed category of R Q -mo dules. W e d enote by Γ the Ginzburg d g algebra 8 BERNHARD KELLER AND SAR AH SCHEROTZKE [10] asso ciated to ( Q, 0) a nd b y D (Γ) th e derived category of differen tial graded Γ -mo dules. W e refer to [18, 2.12] for an introd uction to the Ginzburg algebra and its derived category . The inte g r al cluster c ate gory is d efined as the triangle quotient C RQ = p er(Γ) / D per ( R ) (Γ) . In analogy with [3], we defin e C or b to b e the orbit category of p er( RQ ) b y the auto-equiv alence S R Σ − 2 . Theorem 4.1. The fu nctor ? ⊗ L RQ Γ induc es a ful ly faithful emb e dding of C or b into C RQ . The c ate gory C RQ is the triangulate d hul l of C or b . Pr o of. Let Θ = Σ − 2 RHom RQ e ( RQ, RQ e ) ∼ = Σ − 2 Hom R ( RQ, R ). The tensor functor ? ⊗ L A Θ induces the fu nctor S R Σ − 2 and restricts to an equ iv alence of p er( RQ ) by 2.2 . By [16, 6.3], the tensor algebra T A (Θ) is qu asi-isomorphic to the Ginzbu r g algebra Γ. No w Theorem 3.3 yields the statemen t.  Prop osition 4.2. The c ate gory D (Γ) satisfies the r elative 3-Calabi-Y au pr op erty, i. e. for Y ∈ D (Γ) and X ∈ D per R (Γ) , ther e is a bifunctorial isomorph ism RHom R (RHom D (Γ) ( X, Y ) , R ) ∼ = RHom D (Γ) ( Y , Σ 3 X ) for any Y ∈ D (Γ) and X ∈ D per R (Γ) . Pr o of. By [1 6, 4.8] , the dg mo dule Ω = RHom Γ e (Γ , Γ e ) is isomorph ic to Σ − 3 Γ in D (Γ e ). Now u s ing Lemma 2.1 we obtain the required isomorphism b y the same pr o of as in [16, 4.8].  Corollary 4.3. Supp ose that the ring R is her e ditary. L et L b e an obje ct in D (Γ) and let Z b e an obje ct in the sub c ate gory D per R (Γ) . Then we have Hom R (Hom D (Γ) ( Z, L ) , R ) ⊕ E x t 1 (Hom D (Γ) ( Z, Σ − 1 L ) , R ) ∼ = Hom D (Γ) ( L, Σ 3 Z ) . Pr o of. As R is her ed itary , ev ery ob ject X in D ( R ) is isomorp hic to the sum of its sh ifted homologies L n ∈ Z Σ − n H n ( X ). As H n RHom D (Γ) ( Z, L ) = Hom D (Γ) ( Z, Σ n L ), we hav e RHom D (Γ) ( Z, L ) ∼ = L n ∈ Z Σ − n Hom D (Γ) ( Z, Σ n L ). Therefore RHom R (RHom D (Γ) ( Z, L ) , R ) is isomorph ic to Y n ∈ N Σ n RHom R (Hom D (Γ) ( Z, Σ n L ) , R ) . F urthermore, we ha v e RHom R ( M , R ) ∼ = Hom R ( M , R ) ⊕ Σ − 1 Ext 1 R ( M , R ) for an y R -mo du le M . Therefore, the homology of RHom R (RHom D (Γ) ( Z, L ) , R ) in d egree zero is giv en by Hom R (Hom D (Γ) ( Z, L ) , R ) ⊕ Ext 1 (Hom D (Γ) ( Z, Σ − 1 L ) , R ) . W e obtain the statemen t by comparing the homology in d egree zero in 4.2 and u s ing the fact that the h omology of RHom D (Γ) ( L, Σ 3 Z ) in d egree zero is giv en by Hom D (Γ) ( L, Σ 3 Z ).  THE INTEGRAL CLUSTER CA TEGOR Y 9 5. rigid objec ts W e assume from now on t hat R is a hereditary ring. Our goal in this section is to sho w that, if R is a prin cipal ideal domain, eac h rigid ob ject of the int egral cluster category is either th e image of a rigid ind ecomp osable RQ -mo du le or th e sus p ension of the image of an indecomp osable pro jectiv e RQ -mo du le. W e al so sho w th at the orbit ca tegory C or b and t he in tegral cluster catego ry are equiv alen t, s o that the orbit category is triangulated. By [7, Theorem 1] all rigid ind ecomp osable R Q -mo dules are lattices and there is a bijection b etw een the rigid in decomp osable lattices and the real Sc h ur ro ots of the quiv er Q giv en by the rank v ector. F ollo w ing [1], w e define the fundamental domain to b e the R -linear sub categ ory F = D ≤ 0 ∩ ⊥ D ≤− 2 ∩ p er(Γ) of p er(Γ), w h ere ⊥ D ≤ n denotes the full sub catego ry wh ose ob jects are the X ∈ D (Γ) suc h that Hom D (Γ) ( X, Y ) v anishes for all Y ∈ D ≤ n . Let π : p er(Γ) → C RQ b e th e canonical triangle functor. Theorem 5.1. F or every obje c t Z of C RQ , ther e is an N ∈ Z and an obje c t Y ∈ F [ N ] such that π ( Y ) is isomorphic to Z . Pr o of. F or ev er y ob ject X ∈ p er( Γ), there i s an N ∈ Z and an M ∈ Z suc h that X ∈ D ≤ N and X ∈ ⊥ D ≤ M . This f ollo w s fr om t he facts that Γ ∈ D ≤ 0 (Γ) ∩ ⊥ D ≤− 1 (Γ) and that the p rop ert y is stable u nder taking shifts, extensions and dir ect factors. So let X ∈ ⊥ D ≤ N − 2 for some N ∈ Z . Consider the canonical triangle τ ≤ N ( X ) → X → τ >N ( X ) → Σ τ ≤ N ( X ) . As τ >N ( X ) ∈ D per ( R ) (Γ) and π is a triangle functor, the ob jects π ( τ ≤ N ( X )) and π ( X ) are isomorph ic. It remains to sho w that τ ≤ N ( X ) ∈ ⊥ D ≤ N − 2 , whic h is equiv alen t to the fact that Σ − 1 τ >N ( X ) lies in ⊥ D ≤ N − 2 . By 4.3, for eac h ob ject L of D (Γ), the sum Hom R (Hom D (Γ) (Σ − 1 τ >N ( X ) , L ) , R ) ⊕ Ext 1 R (Hom D (Γ) (Σ − 1 τ >N ( X ) , Σ − 1 L ) , R ) is isomorph ic to Hom D (Γ) ( L, Σ 2 τ >N ( X )). This group v anishes for all L ∈ D ≤ N − 2 . W e fix an ob ject L in D ≤ N − 2 . Th e R -mo d ule H = Hom D (Γ) (Σ − 1 τ >N ( X ) , L ) is left orth ogonal to R an d so has to b e a torsion mo dule. Since Σ L also lies in D ≤ N − 2 , the group Ext 1 R ( H , R ) v anishes and so we ha v e H = 0. Therefore, the ob ject τ ≤ N ( X ) lies in F [ N ] and its image is isomorphic to the image of X in the integral cluster category .  F ollo wing [1], w e define an add(Γ) -r esolution of an ob ject M ∈ p er(Γ) to b e a triangle P 0 → P 1 → M → Σ P 0 10 BERNHARD KELLER AND SAR AH SCHEROTZKE with P 0 , P 1 in add (Γ). Lemma 5.2. An obje ct X ∈ p er(Γ) has an add (Γ) -r esolution if and only if X lies in F . Pr o of. If X lies in F , th en X has an add(Γ)-resolution b y the pr o of of [1, 2.10]. Now let P 1 → P 0 → X → Σ P 0 b e an ad d (Γ)-resolution. App lying the h omology functor to this triangle, w e get a long exact sequence. U sing the fact that Γ has n on-zero homology only in non-p ositiv e degree, we see that X lies in D ≤ 0 . Next we apply the functor Hom D (Γ) (? , Y ) to the add(Γ)-resolution for any ob ject Y ∈ D ≤− 2 . Then we obtain a long exact sequence · · · → Hom(Σ P 0 , Y ) → Hom( X, Y ) → Hom( P 0 , Y ) → Hom( P 1 , Y ) → · · · . All terms in this sequence h a v e to v anish an d therefore X b elongs to ⊥ D ≤− 2 .  W e ha v e Hom C RQ (Γ , Γ) ∼ = Hom C or b ( RQ, RQ ) ∼ = Hom C or b ( RQ, RQ ) ∼ = RQ b y 4.1. Th erefore w e h av e a fun ctor G = Hom C RQ (Γ , ?) fr om C RQ to th e catego ry of RQ -mo dules. Note that G v anish es on Σ Γ, hence G fact ors through the quotien t category C RQ / add(ΣΓ). Example 5.3. We c onsider the qu iver Q : 1 α / / 2 and R = Z . L e t M b e the mo dule given by the quiver r e pr esentation 0 ← Z / 2 Z . Then M has the pr oje ctive r esolution 0 / / P 1 [2 α ] t / / P 1 ⊕ P 2 [ α − 2] / / P 2 / / M / / 0 By applying the functor Hom Z Q (? , M ) to the r esolution we se e that the gr oup of selfextensions of M is isomorphic to Z / 2 Z so that M is not rigid. L et M ′ b e the image of M in p er(Γ) . We have G ( M ′ ) ∼ = M but M ′ do es not lie in the fundamental domain as Hom D ( RQ ) ( M , Σ 2 P 1 ) is isomorphic to Z / 2 Z . But cle arly M ′ b elongs to ⊥ D ≤− 3 (Γ) , henc e by the pr o of of 5.1, we have that τ ≤− 1 ( M ′ ) ∼ = M ′ in C Z Q and Σ − 1 τ ≤− 1 ( M ′ ) ∈ F . Lemma 5.4. L et M b e an obje c t in C RQ . Then G ( GM ⊗ L RQ Γ) and GM ar e isomorp hic in C RQ . If GM is a lattic e, then G ( GM ⊗ L RQ Γ) viewe d as an element of p er(Γ) lies in the fundamental domain. Pr o of. W e hav e G ( GM ⊗ L RQ Γ) = Hom C RQ (Γ , GM ⊗ L RQ Γ) ∼ = Hom C or b ( RQ, GM ) ∼ = Hom RQ ( RQ, GM ) ∼ = GM , THE INTEGRAL CLUSTER CA TEGOR Y 11 as the embed ding of C or b in to C RQ is fully faithful by 4.1. Let n o w GM b e a lattice. Then GM ⊗ L RQ Γ is in D ≤ 0 . W e hav e Hom Γ ( GM ⊗ L RQ Γ , Y ) ∼ = Hom per( RQ ) ( GM , R Hom Γ (Γ , Y )). If Y ∈ D ≤− 2 , then R Hom Γ (Γ , Y ) also lies in D ≤− 2 . As GM is a lattice, w e h a v e that GM lies in ⊥ D ≤− 2 and hen ce Hom per( RQ ) ( GM , R Hom Γ (Γ , Y )) v anish es. Th is fin ish es the pro of.  Definition 5.5. We c al l an inde c omp osable obje ct X in C RQ lattice -lik e , if ther e is a lattic e L such that X is isomorphic to Γ ⊗ L RQ L or X i s isomorph ic to ΣΓ ⊗ L RQ P f or a pr oje ctive inde c omp osable R Q -mo dule P . All lattice-lik e ob jects are images of ob jects in the orbit catego ry C or b . Theorem 5.6. L et M b e an inde c omp osable rigid obje ct of C RQ . Ther e is an N ∈ Z suc h that M is isomorph ic to G (Σ N M ) ⊗ L RQ Σ − N Γ . Pr o of. By 5.1, there is an N ∈ Z and an ob ject M ′ ∈ F [ N ] such that π ( M ′ ) = M . W e assume without loss of generalit y that N = 0. By Lemma 5.2, eac h o b ject of F admits an add(Γ)-resolution. Th erefore, f or N ′ in F , we ha v e, as in Pr op osition 2.1 c) of [17], the isomorphism Hom C RQ / add(ΣΓ) ( π M ′ , π N ′ ) ∼ = Hom RQ ( GM , Gπ N ′ ) . Note also that G ( GM ⊗ L RQ Γ) is isomorphic to GM b y Lemma 5.4. Let Σ − 1 M → P 1 h → P 0 → M b e an add(Γ)-resolution in the inte gral cluster catego ry . Then all m orphisms from P 1 to M factor through h . Applying G to the triangle giv es the start of a pro jectiv e resolution GP 1 → GP 0 → GM → 0 . As P 1 , P 0 and M are all images of ob jects in F , w e ha v e that every morphism from GP 1 to GM factors through Gh . Therefore GM is rigid as an RQ - mo dule and hence is a lattice . If GM v anish es, then M lies in add(ΣΓ). Since we ha v e equiv alences add( RQ ) ∼ → add(Γ) ∼ → add( π (Γ)) ∼ → add( R Q ) , w e obtain M ∼ = G (Σ − 1 M ) ⊗ L RQ ΣΓ. So let u s s upp ose that GM do es not v anish. As GM is a lattice, th e ob ject GM ⊗ L RQ Γ lies in F . It follo ws that there are isomorphisms f ∈ Hom C RQ / add(ΣΓ) ( M , GM ⊗ L RQ Γ) and g ∈ Hom C RQ / add(ΣΓ) ( GM ⊗ L RQ Γ , M ) . W e lift f and g to morp hisms ˜ f and ˜ g in the in tegral cluster categ ory . Then ˜ f ˜ g lies in Hom C RQ ( GM ⊗ L RQ Γ , GM ⊗ L RQ Γ) ∼ = Hom C or b ( GM , GM ) . No w th e fu nctor Hom C or b ( RQ, ?) in duces a sur jectiv e ring homomorphism Hom C or b ( GM , GM ) → Hom RQ ( GM , GM ) 12 BERNHARD KELLER AND SAR AH SCHEROTZKE whose k ernel is a radical id eal. Sin ce f g is an isomorph ism of RQ -mo d ules, ˜ f ˜ g is an isomorp hism in the integral cluster category . But M is ind ecom- p osable and ˜ f ˜ g factors thr ough M , hence the ob jects M and GM ⊗ L RQ Γ are isomorphic.  Note that the pro of of the pr evious theorem also h olds if G (Σ N M ) is a non v anishing lattice. Therefore we ha v e Lemma 5.7. L et M b e an inde c omp osable obje ct in C RQ such that ther e is a Z ∈ F with π ( Z ) is isomorphic to M . If GM is a non vanishing lattic e, then M is isomorphic to GM ⊗ L RQ Γ . The next result is well-kno wn for d eriv ed catego ries of hereditary algebras o v er fields. Lemma 5.8. L et R b e a princip al ide al domain. The Serr e functor S R of p er( RQ ) maps shifts of rigid lattic es to shifts of rigid lattic es. Pr o of. The Serre fu nctor is giv en by the left deriv ed functor of t ensoring with the bimo d ule Θ = Hom R ( RQ, R ). As F Q is hereditary , th e Serre functor S F maps a non-pro jectiv e mo dule L to Σ τ L , where τ d enotes the Auslander-Reiten translation of the c ategory of F Q -mo d ules. Hence the Serre functor applied to non pro jectiv e indecomp osable F Q -mo d ules has non-v anish ing homology only in degree minus one. Moreo ver, the functor S F maps p ro jectiv e mo dules to injectiv e mo du les. The state ment is clear for pro jectiv e latt ices of R Q . Let M b e a non- pro jectiv e ind ecomp osable rigid lattice o v er RQ and 0 → P 1 f → P 0 → M → 0 a p ro jectiv e resolution of M . Then P 0 and P 1 are lattices and f splits as a map of R -mo du les. The ob ject S R ( M ) is isomorphic to the complex · · · → 0 → P 1 ⊗ RQ Θ f ⊗ Θ → P 0 ⊗ RQ Θ → 0 → · · · As Θ is a lattice, so are P 0 ⊗ RQ Θ and P 1 ⊗ RQ Θ. The cok ernel of f ⊗ Θ is giv en b y M ⊗ RQ Θ. W e sho w n ext that f ⊗ Θ is surjectiv e by pr o ving that M ⊗ RQ Θ v anish es. By the pr o of of 3.5, S R comm utes with the functor − ⊗ L R F . Th erefore ( F ⊗ R Θ) ⊗ F Q ( M ⊗ R F ) and (Θ ⊗ RQ M ) ⊗ R F are iso- morphic. By [7, Theorem 2] the mo d ule M ⊗ R F is a rigid indecomp osable non-v anish ing lattice whic h is non-pro jectiv e. Supp ose th at M ⊗ RQ Θ d o es not v anish . Th en there is a field F such that (Θ ⊗ RQ M ) ⊗ R F do es not v anish. Hence S F ( F ⊗ R M ) has n on-v anishin g cohomology in degree zero, which is a con tradiction, as F ⊗ R M is a non-pro jectiv e F Q -mo dule. Therefore f ⊗ Θ is sur j ectiv e. Its kernel h as to b e a lattice as it is a su bmo dule of a lattice. The ob ject S R ( M ) is isomorphic to the one-shift of this lattice.  THE INTEGRAL CLUSTER CA TEGOR Y 13 Next we analyze the relatio nship b etw een the integ ral cluster category and th e cluster category ov er the field F . W e can strengthen [15, A.8]. Theorem 5.9. L et R b e a princip al i de al domain. (1) The rigi d inde c om- p osable obje cts of C RQ ar e lattic e-lik e. (2) The r e duction functor ? ⊗ R F : C RQ → C F Q induc es a bije c tion fr om the set of isomorphism classes of rigi d inde c omp os- able obje cts in C RQ to the set of isomorphism classes of rigi d inde c omp osable obje cts of C F Q . Pr o of. W e denote by F R the functor S R Σ − 2 in p er ( RQ ) and by F the functor S Σ − 2 in p er( F Q ). Let M ∈ C RQ b e a rigid indecomp osable ob ject. By 5.6 and 4.1, we can view M as an ob ject of C or b and M is isomorp hic to the N -shift of the image of a r igid R Q -lattice M ′ . Then M ⊗ L R F is isomorphic to the N -shift of the r igid mod ule M ′ ⊗ R F seen as an ob ject in C F Q . If w e view Σ N M ′ ⊗ L R F as an ob ject in p er( F Q ), we see that there is a rigid indecomp osable mo dule L in mo d F Q or an indecomp osable direct summand P of F Q su c h that Σ N M ′ ⊗ R F and L lie in th e same F -orbit or Σ N M ′ ⊗ L R F and Σ P lie in th e same F -orbit. Let n ∈ Z b e suc h that L ∼ = F n Σ − N M ′ ⊗ R F or Σ P ∼ = F n Σ − N M ′ ⊗ R F . As S R maps the shift of a rigid lattice to the shift of a rigid lattice by 5.8, we hav e that F n R Σ N M ′ is also the k -sh ift of a rigid RQ -lattice, sa y L ′ in p er( RQ ) for some k ∈ Z . By 3.5, w e ha v e that Σ k L ′ ⊗ F and L are isomorphic or Σ k L ′ ⊗ R F ∼ = Σ P in p er( F Q ), h ence k v anish es in the fir st case and k equals one in th e second case. F u rthermore, in the second case L ′ is isomorp hic to a pro jectiv e R Q -mo dule. W e obtain therefore that Σ N M ′ is in the F R -orbit of L ′ in the fi rst case and is in th e F R -orbit of Σ L ′ in the second case. Hence in the orbit category , w e hav e that M is isomorp hic to a lattice or to the one-shift of a pr o jectiv e lattice . This fin ishes the pro of of the fi rst s tatement . Using T h eorem 1 of [7 ] we then immediately obtain the second statemen t.  Note also that all r igid ob jects satisfy the unique decomp ositio n prop ert y , as they are lattice-lik e and the statemen t holds by [7, Th eorem 2] for rigid lattice s in the category of RQ -mo du les. W e can also sho w that the orbit categ ory C or b and the integ ral c luster catego ry coincide and h ence the orbit category is triangulated. Theorem 5.10. The emb e dding of the orbit c ate gory C or b into C RQ is an e qui v alenc e. Ther efor e the orbit c ate g ory C or b is c anonic al ly triangulate d. 14 BERNHARD KELLER AND SAR AH SCHEROTZKE Pr o of. W e consider the comm utativ e diagram of functors p er( RQ ) ? ⊗ L RQ Γ / /   p er(Γ) π   C or b ? ⊗ L RQ Γ / / C RQ . By 3.3, the b ottom fun ctor is fully faithful. Let us sho w that it is essen tially surjectiv e. Let M ∈ C RQ . By 5.1, there is an n ∈ Z and an M ′ ∈ F suc h that Σ n π M ′ ∼ = M . W e assume w ithout loss of generalit y that n = 0 and c hose an add(Γ)-resolution P 1 h → P 0 → M ′ → Σ P 1 . By remark 3.4, the restriction of π to ad d (Γ) is fu lly faithfu l and so is the restriction of p er( RQ ) → p er(Γ) to add( RQ ). Thus, th e morphism h : P 1 → P 0 is the image of a morp hism in p er( RQ ). Since − ⊗ L RQ Γ : p er ( RQ ) → p er(Γ) is a triangle functor, M ′ is also isomorph ic to an image of an ob ject in p er R Q . By the comm utativit y of the diagram, w e d ed uce that M , as an ob ject in C RQ , is isomorphic to the image of an ob ject in C or b . No w the ob jects in C RQ are identica l with the ob jects in p er(Γ), hence ? ⊗ L RQ Γ : C or b → C RQ is essent ially surjectiv e and h ence an equ iv alence.  Corollary 5.11. The inte gr al c luster c ate g ory satisfies the r elative 2-Calabi- Y au pr op erty, i.e. X and Y ∈ C RQ , ther e is a bifunctorial isomorphism RHom R (RHom C RQ ( X, Y ) , R ) ∼ = RHom C RQ ( Y , Σ 2 X ) in D ( R ) . 6. Cluster-til ting m u t a tion Mutations of cluster-tilting ob jects ha v e b een defined f or clus ter cate gories o v er fields in [3], generalizing the mutatio ns of tilting ob jects in hered itary catego ries stud ied in [11]. The m utation of rigid ob jects and cluster-tilting ob jects in the cluster catego ry is used in [6] to giv e an additive categ ori- fication of the cluster algebra asso ciated to the q u iv er Q and its exc hange relations. W e refer to [2] [13] [15] [20] f or o v erviews. Using our classification of rigid ob jects in the in tegral cluster category , w e can generalize the resu lts obtained in [3]. Thr oughout this section, w e assume that R is a principal ideal domain and w e fix a r ing homomorphism from R to a field F . Let Q b e a fin ite quiver without orien ted cycles and let n b e the num b er of its vertice s. Definition 6.1. A cluster-tilting ob ject T is a rigid obje ct in C RQ such that T has n inde c omp osable dir e ct summands which ar e p airwise non-isomorphic. L et T ′ b e another cluster-tilting obje ct. The p air ( T , T ′ ) is c al le d a mutati on pair if T and T ′ have exactly n − 1 isomorphic inde c omp osable summands THE INTEGRAL CLUSTER CA TEGOR Y 15 in c ommon. Then we say that T ′ is c onne cte d to T by a cluster-tilting m utation. By T heorem 5.9 th e r esu lts of [3], every r igid indecomp osable ob ject ap- p ears as a direct summand of a cluster-tilting ob ject. Moreo ver, the functor ? ⊗ R F induces a bijection f r om the set of isomorph ism classes of cluster- tilting ob jects of C RQ on to that of C F Q and this bijection p r eserv es mutation pairs. Lemma 6.2. If X and Y ar e rigid obje cts in C RQ , then Ext 1 C RQ ( X, Y ) is a fr e e R -mo dule and ? ⊗ R F induc es a bije ction b etwe en Ext 1 C RQ ( X, Y ) and Ext 1 C F Q ( F ⊗ R X, F ⊗ R Y ) . Pr o of. Let first X and Y b e tw o rigid RQ lattices. By [7, Th eorem 1], the R -mo d u le Ext 1 RQ ( X, Y ) is free. By applying 5.11 we obtain that th e R -mo du le Ext 1 C RQ ( X, Y ) is isomorph ic to Ext 1 RQ ( X, Y ) ⊕ Hom R (Ext 1 RQ ( Y , X ) , R ) , and hen ce is fr ee. I f we app ly F ⊗ R ?, we obtain, again by [7 , Theorem 1], that it is isomorphic to Ext 1 F Q ( F ⊗ R X, F ⊗ R Y ) ⊕ Hom F Q (Ext 1 F Q ( F ⊗ R Y , F ⊗ R X ) , F ) , whic h is isomorphic to Ext 1 C F Q ( X, Y ). I f Y ∼ = Σ P for some pro jectiv e RQ - mo dule P , then E xt 1 C RQ ( X ⊗ R F , Y ⊗ R F ) is isomorphic to Hom RQ ( X, P ) whic h is also a free R -mo d ule by [7, Theorem 1]. Th e rest of the p ro of is analogous.  Theorem 6.3 (Cluster tilting mutatio n) . L et T b e a cluster tilting obje c t of C RQ and X an inde c omp osable dir e ct summand of T with c omplement X ′ . L et Y b e an inde c omp osable rigid obje c t. Then T ′ := Y ⊕ X ′ is a cluster-tilting obje ct if and only if Ext 1 C RQ ( X, Y ) has r ank one. Pr o of. By 6.2 w e ha v e Ext 1 C RQ ( X, Y ) ⊗ R F ∼ = Ext 1 C F Q ( X ⊗ R F , Y ⊗ R F ). F urthermore b oth ob jects F ⊗ R X and F ⊗ R Y are rigid and indecomp osable. Clearly F ⊗ R T ∼ = F ⊗ R X ⊕ F ⊗ X ′ is a cluster tilting ob ject in C F Q . T h us, b y [3, 7.5] , the ob ject F ⊗ R T ′ is cluster tilting if and only if the extension group Ext 1 C F Q ( X ⊗ R F , Y ⊗ R F ) is one dimensional. As the fu nctor F ⊗ R ? induces a bijection b et ween r igid in d ecomp osable ob jects in C RQ and C F Q , the ob ject T ′ is cluster-tilting if and only if Ext 1 C RQ ( X, Y ) has rank one.  Let X and Y b e rigid indecomp osable ob jects with an extension sp ace of rank one. By the preceding th eorem and the results of [3 ], w e obtain that there is a r igid ob j ect X ′ suc h that Y ⊕ X ′ and X ⊕ X ′ are cluster- tilting ob jects. Let us c ho ose generators ε and ε ′ of the rank one mo dules Ext 1 C RQ ( X, Y ) and E x t 1 C RQ ( Y , X ). W e construct non s plit triangles Y f → E → X ε → Σ Y and X → E ′ g → Y ε ′ → Σ X . 16 BERNHARD KELLER AND SAR AH SCHEROTZKE By Lemma 6.2 these t riangles are mapp ed by the functor F ⊗ R ? to non- split tr iangles in C F Q . By [3, 6.4] the maps F ⊗ R f and F ⊗ R g are m inimal add( F ⊗ R X ′ )-appro ximations. W e call the tr iangles in the inte gral cluster catego ry the exchange triangles of the m utation. By [6 ] they categorify the exc hange relations in the cluster algebra asso ciated to the quiver Q . It w as sho wn in [3] [11], c f. also [12], that all cluster-tilting ob jects of C F Q are related by iterated m utation. Cle arly , as cluster-tilting ob jects and their mutat ions in C RQ are in bijection with cluster-tilting ob jects in C F Q and th eir mutat ions, we obtain the follo wing resu lt. Corollary 6.4. 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Press, Cam bridge, 2007, pp. 413–472. [22] Dylan R up el, On a quantum analo g of the Calder o-Chap oton formula , I nternational Mathematics Researc h Notices (2010), 1–30, 10.1093/imrn/rnq192. B. K. : Uni versit ´ e P aris Dide r ot - P a ris 7, Institut Univ e rsit ai re de France, UFR de Ma th ´ ema tiques, Institut de M a th ´ ema tiques de Jussieu, UM R 7586 du CNRS, C ase 7012, B ˆ atiment C hev aleret, 75205 P aris Cede x 13, France S. S. : Universit ´ e P aris Diderot - P ari s 7, UFR de Ma th ´ ema tiques, Insti- tut de Ma th ´ ema tiques de Jussieu, UMR 7586 d u CNRS, Case 7012, B ˆ atiment Chev aleret, 75205 P aris Ce d ex 13, France E-mail addr ess : keller@math.juss ieu.fr, scherotzke@math.j ussieu.fr