Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories

GR OTHENDIECK GR OUP AND GENERALIZED MUT A TION R ULE F OR 2-CALABIY A U TRIANGULA TED CA TEGORIES Y ANN P ALU Abstra t. W e ompute the Grothendie k group of ertain 2-CalabiY au tri- angulated ategories app earing naturally in the study of the link b et w een quiv er represen tations and F ominZelevinsky's luster algebras. In this setup, w e also pro v e a generalization of F ominZelevinsky's m utation rule. Intr odution In their study [ 6 ℄ of the onnetions b et w een luster algebras (see [22 ℄) and quiv er represen tations, P . Caldero and B. Keller onjetured that a ertain an tisymmetri bilinear form is w elldened on the Grothendie k group of a lustertilted algebra asso iated with a nitedimensional hereditary algebra. The onjeture w as pro v ed in [19 ℄ in the more general on text of Hom-nite 2-CalabiY au triangulated ate- gories. It w as used in order to study the existene of a luster  harater on su h a ategory C , b y using a form ula prop osed b y CalderoKeller. In the presen t pap er, w e restrit to the ase where C is algebrai (i.e. is the stable ategory of a F rob enius ategory). W e rst use this bilinear form to pro v e a generalized m utation rule for quiv ers of lustertilting sub ategories in C . When the lustertilting sub ategories are related b y a single m utation, this sho ws, via the metho d of [9℄, that their quiv ers are related b y the F ominZelevinsky m utation rule. This sp eial ase w as already pro v ed in [3 ℄, without assuming C to b e algebrai. W e also ompute the Grothendie k group of the triangulated ategory C . In partiular, this allo ws us to impro v e on results b y M. Barot, D. Kussin and H. Lenzing: W e ompare the Grothendie k group of a luster ategory C A with the group K 0 ( C A ) . The latter group w as dened in [1℄ b y only onsidering the triangles in C A whi h are indued b y those of the deriv ed ategory . More preisely , w e pro v e that those t w o groups are isomorphi for an y luster ategory asso iated with a nite dimensional hereditary algebra, with its triangulated struture dened b y B. Keller in [16 ℄. This pap er is organized as follo ws: The rst setion is dediated to notation and neessary ba kground from [8 ℄, [9 ℄, [17℄, [19 ℄. In setion 2, w e ompute the Grothendie k group of the triangulated ategory C . In setion 3, w e pro v e a gener- alized m utation rule for quiv ers of lustertilting sub ategories in C . In partiular, this yields a new pro of of the F ominZelevinsky m utation rule, under the restri- tion that C is algebrai. W e nally sho w that K 0 ( C A ) = K 0 ( C A ) for an y nite dimensional hereditary algebra A . A kno wledgements This artile is part of m y PhD thesis, under the sup ervision of Professor B. Keller. I w ould lik e to thank him deeply for in tro duing me to the sub jet and for his innite patiene. 1 2 Y ANN P ALU Contents In tro dution 1 A  kno wledgemen ts 1 1. Notations and ba kground 2 1.1. F ominZelevinsky m utation for matries 2 1.2. Clustertilting sub ategories 3 1.3. The an tisymmetri bilinear form 3 2. Grothendie k groups of algebrai 2-CY ategories with a lustertilting sub ategory 4 2.1. A short exat sequene of triangulated ategories 4 2.2. In v ariane under m utation 5 2.3. Grothendie k groups 8 3. The generalized m utation rule 9 3.1. The rule 10 3.2. Examples 12 3.3. Ba k to the m utation rule 13 3.4. Cluster ategories 14 Referenes 14 1. Not a tions and ba k gr ound Let E b e a F rob enius ategory whose idemp oten ts split and whi h is linear o v er a giv en algebraially losed eld k . By a result of Happ el [ 10 ℄, its stable ategory C = E is triangulated. W e assume moreo v er, that C is Hom-nite, 2-CalabiY au and has a lustertilting sub ategory (see setion 1.2 ), and w e denote b y Σ its susp ension funtor. Note that w e do not assume that E is Hom-nite. W e write X ( , ) , or Hom X ( , ) , for the morphisms in a ategory X and Hom X ( , ) for the morphisms in the ategory of X -mo dules. W e also denote b y X ˆ the pro je- tiv e X -mo dule represen ted b y X : X ˆ = X (? , X ) . 1.1. F ominZelevinsky m utation for matries. Let B = ( b ij ) i,j ∈ I b e a nite or innite matrix, and let k b e in I . The F omin and Zelevinsky m utation of B (see [8℄) in diretion k is the matrix µ k ( B ) = ( b ′ ij ) dened b y b ′ ij =  − b ij if i = k or j = k , b ij + | b ik | b kj + b ik | b kj | 2 else. Note that µ k  µ k ( B )  = B and that if B is sk ew-symmetri, then so is µ k ( B ) . W e reall t w o lemmas of [9℄, stated for innite matries, whi h will b e useful in setion 3. Note that lemma 7.2 is a restatemen t of [2, (3.2)℄. Let S = ( s ij ) b e the matrix dened b y s ij =  − δ ij + | b ij |− b ij 2 if i = k , δ ij else. Lemma 7.1 ( [9, GeissLelerS hrö er℄ ) : Assume that B is skew-symmetri. Then, S 2 = 1 and the ( i, j ) -entry of the tr ansp ose of the matrix S is given by s t ij =  − δ ij + | b ij | + b ij 2 if j = k , δ ij else. The matrix S yields a on vienen t w a y to desrib e the m utation of B in the diretion k : GR OTHENDIECK GR OUP AND MUT A TION R ULE 3 Lemma 7.2 ( [9, GeissLelerS hrö er℄ , [2 , BerensteinF ominZelevinsky℄ ) : As- sume that B is skew-symmetri. Then we have: µ k ( B ) = S t B S. Note that the pro dut is w ell-dened sine the matrix S has a nite n um b er of non v anishing en tries in ea h olumn. 1.2. Clustertilting sub ategories. A lustertilting sub ategory (see [ 17 ℄) of C is a full sub ategory T su h that a) T is a linear sub ategory; b) for an y ob jet X in C , the on tra v arian t funtor C (? , X ) | T is nitely gen- erated; ) for an y ob jet X in C , w e ha v e C ( X, Σ T ) = 0 for all T in T if and only if X b elongs to T . W e no w reall some results from [17 ℄, whi h w e will use in the sequel. Let T b e a lustertilting sub ategory of C , and denote b y M its preimage in E . In partiular M on tains the full sub ategory P of E formed b y the pro jetiv e-injetiv e ob jets, and w e ha v e M = T . The follo wing prop osition will b e used impliitly , extensiv ely in this pap er. Prop osition [17 , KellerReiten℄ : a) The  ate gory mo d M of nitely pr esente d M -mo dules is ab elian. b) F or e ah obje t X ∈ C , ther e is a triangle Σ − 1 X − → T X 1 − → T X 0 − → X of C , with T X 0 and T X 1 in T . Reall that the p erfet deriv ed ategory per M is the full triangulated sub ate- gory of the deriv ed ategory of D Mo d M generated b y the nitely generated pro- jetiv e M -mo dules. Prop osition [17 , KellerReiten℄ : a) F or e ah X ∈ E , ther e ar e  onations 0 − → M 1 − → M 0 − → X − → 0 and 0 − → X − → M 0 − → M 1 − → 0 in E , with M 0 , M 1 , M 0 and M 1 in M . b) L et Z b e in mo d M . Then Z  onsider e d as an M -mo dule lies in the p erfe t derive d  ate gory per M and we have  anoni al isomorphisms D (per M )( Z , ?) ≃ (p er M )(? , Z [3 ]) . 1.3. The an tisymmetri bilinear form. In setion 3, w e will use the existene of the an tisymmetri bilinear form h , i a on K 0 (mo d M ) . W e th us reall its denition from [6℄. Let h , i b e a trunated Euler form on mo d M dened b y h M , N i = dim Hom M ( M , N ) − dim E xt 1 M ( M , N ) for an y M , N ∈ mod M . Dene h , i a to b e the an tisymmetrization of this form: h M , N i a = h M , N i − h N , M i . This bilinear form desends to the Grothendie k group K 0 (mo d M ) : Lemma [19 , setion 3℄ : The an tisymmetri bilinear form h M , N i a : K 0 (mo d M ) × K 0 (mo d M ) − → Z is w ell-dened. 4 Y ANN P ALU 2. Gr othendiek gr oups of algebrai 2-CY a tegories with a lustertil ting suba tegor y W e x a luster-tilting sub ategory T of C , and w e denote b y M its preimage in E . In partiular M on tains the full sub ategory P of E formed b y the pro jetiv e- injetiv e ob jets, and w e ha v e M = T . W e denote b y H b ( E ) and D b ( E ) resp etiv ely the b ounded homotop y ategory and the b ounded deriv ed ategory of E . W e also denote b y H b E − ac ( E ) , H b ( P ) , H b ( M ) and H b E − ac ( M ) the full sub ategories of H b ( E ) whose ob jets are the E - ayli omplexes, the omplexes of pro jetiv e ob jets in E , the omplexes of ob jets of M and the E -ayli omplexes of ob jets of M , resp etiv ely . 2.1. A short exat sequene of triangulated ategories. Lemma 1. L et A 1 and A 2 b e thik, ful l triangulate d sub  ate gories of a triangulate d  ate gory A and let B b e A 1 ∩ A 2 . Assume that for any obje t X in A ther e is a triangle X 1 − → X − → X 2 − → Σ X 1 in A , with X 1 in A 1 and X 2 in A 2 . Then the indu e d funtor A 1 / B − → A / A 2 is a triangle e quivalen e. Pr o of. Under these assumptions, denote b y F the indued triangle funtor from A 1 / B to A / A 2 . W e are going to sho w that the funtor F is a full, onserv ativ e, dense funtor. Sine an y full onserv ativ e triangle funtor is fully faithful, F will then b e an equiv alene of ategories. W e rst sho w that F is full. Let X 1 and X ′ 1 b e t w o ob jets in A 1 . Let f b e a morphism from X 1 to X ′ 1 in A / A 2 and let Y w A A A A A ~ ~ } } } } } X 1 X ′ 1 b e a left fration whi h represen ts f . The morphism w is in the m ultipliativ e system asso iated with A 2 and th us yields a triangle Σ − 1 A 2 → Y w − → X ′ 1 → A 2 where A 2 lies in the sub ategory A 2 . Moreo v er, b y assumption, there exists a triangle Y 1 → Y → Y 2 → Σ Y 1 with Y i in A i . Applying the o tahedral axiom to the omp osition Y 1 → Y → X ′ 1 yields a omm utativ e diagram whose t w o middle ro ws and olumns are triangles in A Σ − 1 A 2   Σ − 1 A 2   Y 1 / / Y   / / Y 2   / / Σ Y 1 Y 1 / / X ′ 1   / / Z   / / Σ Y 1 A 2 A 2 . Sine Y 2 and A 2 b elong to A 2 , so do es Z . And sine X ′ 1 and Y 1 b elong to A 1 , so do es Z . This implies, that Z b elongs to B . The morphism Y 1 → X ′ 1 is in the m ultipliativ e system of A 1 asso iated with B and the diagram Y 1 A A A A A ~ ~ } } } } } X 1 X ′ 1 GR OTHENDIECK GR OUP AND MUT A TION R ULE 5 is a left fration whi h represen ts f . This implies that f is the image of a morphism in A 1 / B . Therefore the funtor F is full. W e no w sho w that F is onserv ativ e. Let X 1 f − → Y 1 → Z 1 → Σ X 1 b e a triangle in A 1 . Assume that F f is an isomorphism in A / A 2 , whi h implies that Z 1 is an ob jet of A 2 . Therefore, Z 1 is an ob jet of B and f is an isomorphism in A 1 / B . W e nally sho w that F is dense. Let X b e an ob jet of the ategory A / A 2 , and let X 1 → X → X 2 → Σ X 1 b e a triangle in A with X i in A i . Sine X 2 b elongs to A 2 , the image of the morphism X 1 → X in A / A 2 is an isomorphism. Th us X is isomorphi to the image b y F of an ob jet in A 1 / B .  As a orollary , w e ha v e the follo wing: Lemma 2. The fol lowing se quen e of triangulate d  ate gories is short exat: 0 − → H b E − ac ( M ) − → H b ( M ) − → D b ( E ) − → 0 . Remark: This lemma remains true if C is d -CalabiY au and M is ( d − 1) -luster tilting, using setion 5.4 of [ 17 ℄. Pr o of. F or an y ob jet X in H b ( E ) , the existene of an ob jet M in H b ( M ) and of a quasi-isomorphism w from M to X is obtained using the appro ximation onations of KellerReiten (see setion 1.2 ). Sine the one of the morphism w b elongs to H b E − ac ( E ) , lemma 1 applies to the sub ategories H b E − ac ( M ) , H b ( M ) and H b E − ac ( E ) of H b ( E ) .  Prop osition 3. The fol lowing diagr am is  ommutative with exat r ows and  olumns: 0 0 0 / / H b E − ac ( M ) i M / / H b ( M ) / H b ( P ) O O / / E / / O O 0 0 / / H b E − ac ( M ) / / H b ( M ) / / O O D b ( E ) / / O O 0 ( D ) H b ( P ) O O H b ( P ) i P O O / / 0 0 O O 0 O O . Pr o of. The olumn on the righ t side has b een sho wn to b e exat in [18 ℄ and [20 ℄. The seond ro w is exat b y lemma 2. The sub ategories H b E − ac ( M ) and H b ( P ) of H b ( M ) are left and righ t orthogonal to ea h other. This implies that the indued funtors i M and i P are fully faithful and that taking the quotien t of H b ( M ) b y those t w o sub ategories either in one order or in the other giv es the same ategory . Therefore the rst ro w is exat.  2.2. In v ariane under m utation. A natural question is then to whi h exten t the diagram ( D ) dep ends on the  hoie of a partiular lustertilting sub ategory . Let th us T ′ b e another lustertilting sub ategory of C , and let M ′ b e its preimage in E . Let Mo d M (resp. Mo d M ′ ) b e the ategory of M -mo dules (resp. M ′ -mo dules), i.e. of k -linear on tra v arian t funtors from M (resp. M ′ ) to the ategory of k -v etor spaes. Let X b e the M - M ′ -bimo dule whi h sends the pair of ob jets ( M , M ′ ) to the k -v etor spae E ( M , M ′ ) . The bimo dule X indues a funtor F =? ⊗ M ′ X : Mo d M ′ − → Mo d M denoted b y T X in [15 , setion 6.1℄. 6 Y ANN P ALU Reall that the p erfet deriv ed ategory per M is the full triangulated sub ate- gory of the deriv ed ategory D Mo d M generated b y the nitely generated pro jetiv e M -mo dules. Prop osition 4. The left derive d funtor L F : D Mo d M ′ − → D Mo d M is an e quivalen e of  ate gories. Pr o of. Reall that if X is an ob jet in a ategory X , w e denote b y X ˆ the funtor X (? , X ) represen ted b y X . By [15 , 6.1℄, it is enough to  he k the follo wing three prop erties: 1. F or all ob jets M ′ , M ′′ of M , the group Hom D Mo d M ( L F M ′ ˆ , L F M ′′ ˆ [ n ]) v anishes for n 6 = 0 and iden ties with Hom M ′ ( M ′ , M ′′ ) for n = 0 ; 2. for an y ob jet M ′ of M ′ , the omplex L F M ′ ˆ b elongs to per M ; 3. the set { L F M ′ ˆ , M ′ ∈ M ′ } generates D Mod M as a triangulated ategory with innite sums. Let M ′ b e an ob jet of M ′ , and let M 1 / / / / M 0 / / / / M ′ b e a onation in E , with M 0 and M 1 in M , and whose deation is a righ t M -appro ximation (.f. setion 4 of [17 ℄). The surjetivit y of the map M 0 ˆ − → E (? , M ′ ) | M implies that the omplex P = ( · · · → 0 → M 1 ˆ → M 0 ˆ → 0 → · · · ) is quasi-isomorphi to L F M ′ ˆ = E (? , M ′ ) | M . Therefore L F M ′ ˆ b elongs to the sub ategory per M of D Mo d M . Moreo v er, w e ha v e, for an y n ∈ Z and an y M ′′ ∈ M ′ , the equalit y Hom D Mo d M ( L F M ′ ˆ , L F M ′′ ˆ [ n ]) = Hom H b Mo d M ( P , E (? , M ′′ ) | M [ n ]) where the righ thand side v anishes for n 6 = 0 , 1 . In ase n = 1 it also v anishes, sine Ext 1 E ( M ′ , M ′′ ) v anishes. No w, Hom H b Mo d M ( P, E (? , M ′′ ) | M ) ≃ Ker ( E ( M 0 , M ′′ ) → E ( M 1 , M ′′ )) ≃ E ( M ′ , M ′′ ) . It only remains to b e sho wn that the set R = { L F M ′ ˆ , M ′ ∈ M ′ } generates D Mo d M . Denote b y R the full triangulated sub ategory with innite sums of D Mo d M generated b y the set R . The set { M ˆ , M ∈ M} generates D Mo d M as a triangulated ategory with innite sums. Th us it is enough to sho w that, for an y ob jet M of M , the omplex M ˆ onen trated in degree 0 b elongs to the sub ategory R . Let M b e an ob jet of M , and let M / / i / / M ′ 0 p / / / / M ′ 1 b e a onation of E with M ′ 0 and M ′ 1 in M ′ . Sine Ext 1 E (? , M ) | M v anishes, w e ha v e a short exat sequene of M -mo dules 0 − → E (? , M ) | M − → E (? , M ′ 0 ) | M − → E (? , M ′ 1 ) | M − → 0 , whi h yields the triangle M ˆ − → L F M ′ 0 ˆ − → L F M ′ 1 ˆ − → Σ M ˆ .  As a orollary of prop osition 4, up to equiv alene the diagram ( D ) do es not dep end on the  hoie of a lustertilting sub ategory . T o b e more preise: Let G b e the funtor whi h sends an ob jet X in the ategory H b ( M ′ ) to a represen tativ e of ( L F ) X ˆ in H b ( M ) , and a morphism in H b ( M ′ ) to the indued one in H b ( M ) . GR OTHENDIECK GR OUP AND MUT A TION R ULE 7 Corollary 5. The fol lowing diagr am is  ommutative D Mo d M ′ L F / / D Mo d M H b ( M ′ )   *  8 8 p p p p p p p p p p p G / / O O H b ( M )   +  8 8 q q q q q q q q q q H b ( P ) S 3 f f M M M M M M M M M M ?  k K x x q q q q q q q q q q H b ( P ) S 3 f f L L L L L L L L L L ?  O O k K x x r r r r r r r r r r D b ( E ) D b ( E ) and the funtor G is an e quivalen e of  ate gories. W e denote b y per M M the full sub ategory of per M whose ob jets are the om- plexes with homologies in mo d M . The follo wing lemma will allo w us to ompute the Grothendie k group of per M M in setion 2.3 : Lemma 6. The  anoni al t-strutur e on D Mo d M r estrits to a t-strutur e on per M M , whose he art is mo d M . Pr o of. By [13 ℄, it is enough to sho w that for an y ob jet M • of per M M , its truna- tion τ ≤ 0 M • in D Mo d M b elongs to per M M . Sine M • is in per M M , τ ≤ 0 M • is b ounded, and is th us formed from the omplexes H i ( M • ) onen trated in one de- gree b y taking iterated extensions. But, for an y i , the M -mo dule H i ( M • ) atually is an M -mo dule. Therefore, b y [17℄ (see setion 1.2 ), it is p erfet as an M -mo dule and it lies in per M M .  The next lemma already app ears in [21 ℄. F or the on v eniene of the reader, w e inlude a pro of. Lemma 7. The Y one da e quivalen e of triangulate d  ate gories H b ( M ) − → per M indu es a triangle e quivalen e H b E − ac ( M ) − → p er M M . Pr o of. W e rst sho w that the ohomology groups of an E -ayli b ounded omplex M v anish on P . Let P b e a pro jetiv e ob jet in E and let E b e a k ernel in E of the map M n − → M n +1 . Sine M is E -ayli, su h an ob jet exists, and moreo v er, it is an image of the map M n − 1 − → M n . An y map from P to M n whose omp osition with M n → M n +1 v anishes fators through the k ernel E ֌ M n . Sine P is pro jetiv e, this fatorization fators through the deation M n − 1 ։ E . P v v l l l l l l l l   o w     "   0 ( ( P P P P P P P P P P P P P P M n − 1 " " " " F F F F F F F F / / M n / / M n +1 E = = = = | | | | | | | | Therefore, w e ha v e H n ( M ˆ )( P ) = 0 for all pro jetiv e ob jets P , and H n ( M ˆ ) b e- longs to mo d M . Th us the Y oneda funtor indues a fully faithful funtor from H b E − ac ( M ) to per M M . T o pro v e that it is dense, it is enough to pro v e that an y ob jet of the heart mo d M of the t-struture on per M M is in its essen tial image. But this w as pro v ed in [17 , setion 4℄ (see setion 1.2 ).  8 Y ANN P ALU Prop osition 8. Ther e is a triangle e quivalen e of  ate gories per M M ≃ − → p er M ′ M ′ Pr o of. Sine the ategories H b ( P ) and H b E − ac ( M ) are left-righ t orthogonal in H b ( M ) , this is immediate from orollary 5 and lemma 7.  2.3. Grothendie k groups. F or a triangulated (resp. additiv e, resp. ab elian) ategory A , w e denote b y K tri 0 ( A ) or simply K 0 ( A ) (resp. K add 0 ( A ) , resp. K ab 0 ( A ) ) its Grothendie k group (with resp et to the men tioned struture of the ategory). F or an ob jet A in A , w e also denote b y [ A ] its lass in the Grothendie k group of A . The short exat sequene of triangulated ategories 0 − → H b E − ac ( M ) − → H b ( M ) / H b ( P ) − → E − → 0 giv en b y prop osition 3 indues an exat sequene in the Grothendie k groups ( ∗ ) K 0  H b E − ac ( M )  − → K 0  H b ( M ) / H b ( P )  − → K 0  E  − → 0 . Lemma 9. The exat se quen e ( ∗ ) is isomorphi to an exat se quen e ( ∗∗ ) K ab 0  mo d M  ϕ − → K add 0  M  − → K tri 0  E  − → 0 . Pr o of. First, note that, b y [ 21 ℄, see also lemma 7 , w e ha v e an isomorphism b et w een the Grothendie k groups K 0  H b E − ac ( M )  and K 0  per M M  . The t-struture on per M M whose heart is mo d M , see lemma 6 , in turn yields an isomorphism b e- t w een the Grothendie k groups K tri 0  per M M  and K ab 0  mo d M  . Next, w e sho w that the anonial additiv e funtor M α − → H b ( M ) / H b ( P ) indues an isomorphism b et w een the Grothendie k groups K add 0  M  and K tri 0  H b ( M ) / H b ( P )  . F or this, let us onsider the anonial additiv e funtor M β − → H b ( M ) and the triangle funtor H b ( M ) γ − → H b ( M ) . The follo wing diagram desrib es the situation: H b ( M ) H b ( M ) γ o o   M β O O α / / H b ( M ) / H b ( P ) γ g g O O O O O O The funtor γ v anishes on the full sub ategory H b ( P ) , th us induing a triangle funtor, still denoted b y γ , from H b ( M ) / H b ( P ) to H b ( M ) . F urthermore, the funtor β indues an isomorphism at the lev el of Grothendie k groups, whose in v erse K 0 ( β ) − 1 is giv en b y K tri 0  H b ( M )  − → K add 0  M  [ M ] 7− → X i ∈ Z ( − 1) i [ M i ] . As the group K tri 0  H b ( M ) / H b ( P )  is generated b y ob jets onen trated in degree 0 , it is straigh tforw ard to  he k that the morphisms K 0 ( α ) and K 0 ( β ) − 1 K 0 ( γ ) are in v erse to ea h other.  As a onsequene of the exat sequene ( ∗∗ ) , w e ha v e an isomorphism b et w een K tri 0  E  and K add 0  M  / Im ϕ . In order to ompute K tri 0  E  , the map ϕ has to b e made expliit. W e rst reall some results from Iy amaY oshino [12 ℄ whi h generalize results from [4 ℄: F or an y indeomp osable M of M not in P , there exists M ∗ unique up to isomorphism su h that ( M , M ∗ ) is an ex hange pair. This means that M and M ∗ are not isomorphi and that the full additiv e sub ategory of C generated GR OTHENDIECK GR OUP AND MUT A TION R ULE 9 b y all the indeomp osable ob jets of M but those isomorphi to M , and b y the indeomp osable ob jets isomorphi to M ∗ is again a lustertilting sub ategory . Moreo v er, dim E ( M , Σ M ∗ ) = 1 . W e an th us x t w o (non-split) ex hange triangles M ∗ → B M → M → Σ M ∗ and M → B M ∗ → M ∗ → Σ M . W e ma y no w state the follo wing: Theorem 10. The Gr othendie k gr oup of the triangulate d  ate gory E is the quotient of that of the additive sub  ate gory M by al l r elations [ B M ∗ ] − [ B M ] : K tri 0  E  ≃ K add 0  M  / ([ B M ∗ ] − [ B M ]) M . Pr o of. W e denote b y S M the simple M -mo dule asso iated to the indeomp osable ob jet M . This means that S M ( M ′ ) v anishes for all indeomp osable ob jets M ′ in M not isomorphi to M and that S M ( M ) is isomorphi to k . The ab elian group K ab 0  mo d M  is generated b y all lasses [ S M ] . In view of lemma 9 , it is suien t to pro v e that the image of the lass [ S M ] under ϕ is [ B M ∗ ] − [ B M ] . First note that the M -mo dule Ext 1 E (? , M ∗ ) | M v anishes on the pro jetiv es ; it an th us b e view ed as an M -mo dule, and as su h, is isomorphi to S M . After replaing B M and B M ′ b y isomorphi ob jets of E , w e an assume that the ex hange triangles M ∗ → B M → M → Σ M ∗ and M → B M ∗ → M ∗ → Σ M ome from onations M ∗ / / / / B M / / / / M and M / / / / B M ∗ / / / / M ∗ . The splied omplex ( · · · → 0 → M → B M ∗ → B M → M → 0 → · · · ) denoted b y C • , is then an E -ayli omplex, and it is the image of S M under the funtor mo d M ⊂ per M M ≃ H b E − ac ( M ) . Indeed, w e ha v e t w o long exat sequenes indued b y the onations ab o v e: 0 → M (? , M ) → M (? , B M ∗ ) → E (? , M ∗ ) | M → E xt 1 E (? , M ) | M = 0 and 0 → E (? , M ∗ ) | M → M (? , B M ) → M (? , M ) → Ext 1 E (? , M ∗ ) | M → E xt 1 E (? , B M ) | M . Sine B M b elongs to M , the funtor Ext 1 E (? , B M ) v anishes on M , and the omplex: ( C ˆ ) : ( · · · → 0 → M ˆ → ( B M ∗ ) ˆ → ( B M ) ˆ → M ˆ → 0 → · · · ) is quasi-isomorphi to S M . No w, in the notations of the pro of of lemma 9, ϕ [ S M ] is the image of the lass of the E -ayli omplex omplex C • under the morphism K 0 ( β ) − 1 K 0 ( γ ) . This is [ M ] − [ B M ] + [ B M ∗ ] − [ M ] whi h equals [ B M ∗ ] − [ B M ] as laimed.  3. The generalized mut a tion r ule Let T and T ′ b e t w o lustertilting sub ategories of C . Let Q and Q ′ b e the quiv ers obtained from their AuslanderReiten quiv ers b y remo ving all lo ops and orien ted 2-yles. Our aim, in this setion, is to giv e a rule relating Q ′ to Q , and to pro v e that it generalizes the F ominZelevinsky m utation rule. R emark: . Assume that C has lustertilting ob jets. Then it is pro v ed in [3 , Theorem I.1.6℄, without assuming that C is algebrai, that the AuslanderReiten quiv ers of t w o lustertilting ob jets ha ving all but one indeomp osable diret summands in ommon (up to isomorphism) are related b y the F omin Zelevinsky m utation rule. . T o pro v e that the generalized m utation rule atually generalizes the F omin Zelevinsky m utation rule, w e use the ideas of setion 7 of [ 9 ℄. 10 Y ANN P ALU 3.1. The rule. As in setion 2, w e x a lustertilting sub ategory T of C , and write M for its preimage in E , so that T = M . Dene Q to b e the quiv er obtained from the AuslanderReiten quiv er of M b y deleting its lo ops and its orien ted 2- yles. Its v ertex orresp onding to an indeomp osable ob jet L will also b e lab eled b y L . W e denote b y a LN the n um b er of arro ws from v ertex L to v ertex N in the quiv er Q . Let B M b e the matrix whose en tries are giv en b y b LN = a LN − a N L . Let R M b e the matrix of h , i a : K 0 (mo d M ) × K 0 (mo d M ) − → Z in the basis giv en b y the lasses of the simple mo dules. Lemma 11. The matri es R M and B M ar e e qual: R M = B M . Pr o of. Let L and N b e t w o non-pro jetiv e indeomp osable ob jets in M . Then dim Ho m( S L , S N ) − dim Hom( S N , S L ) = 0 and w e ha v e: h [ S L ] , [ S N ] i a = dim Ext 1 ( S N , S L ) − dim Ext 1 ( S L , S N ) = b L,N .  Let T ′ b e another lustertilting sub ategory of C , and let M ′ b e its preimage in the F rob enius ategory E . Let ( M ′ i ) i ∈ I (resp. ( M j ) j ∈ J ) b e represen tativ es for the iso lasses of non-pro jetiv e indeomp osable ob jets in M ′ (resp. M ). The equiv alene of ategories per M M ∼ − → p er M ′ M ′ of prop osition 8 indues an iso- morphism b et w een the Grothendie k groups K 0 (mo d M ) and K 0 (mo d M ′ ) whose matrix, in the bases giv en b y the lasses of the simple mo dules, is denoted b y S . The equiv alene of ategories D Mo d M ∼ − → D Mod M ′ restrits to the iden- tit y on H b ( P ) , so that it indues an equiv alene per M / p er P ∼ − → p er M ′ / p er P . Let T b e the matrix of the indued isomorphism from K 0 (pro j M ) / K 0 (pro j P ) to K 0 (pro j M ′ ) / K 0 (pro j P ) , in the bases giv en b y the lasses [ M (? , M j )] , j ∈ J , and [ M ′ (? , M ′ i )] , i ∈ I . The matrix T is m u h easier to ompute than the matrix S . Its en tries t ij are giv en b y the appro ximation triangles of Keller and Reiten in the follo wing w a y: F or all j , there exists a triangle of the form Σ − 1 M j − → M i β ij M ′ i − → M i α ij M ′ i − → M j . Then, w e ha v e: Theorem 12. a) (Generalized m utation rule) The fol lowing e qualities hold: t ij = α ij − β ij and B M ′ = T B M T t . b) The  ate gory C has a lustertilting obje t if and only if al l its lustertilting sub  ate gories have a nite numb er of p airwise non-isomorphi inde  omp os- able obje ts. ) A l l lustertilting obje ts of C have the same numb er of inde  omp osable dir e t summands (up to isomorphism). Note that p oin t ) w as sho wn in [11 , 5.3.3(1)℄ (see also [3, I.1.8℄) and, in a more general on text, in [7℄. Note also that, for the generalized m utation rule to hold, the lustertilting sub ategories do not need to b e related b y a sequene of m utation. Pr o of. Assertions b) and ) are onsequenes of the existene of an isomorphism b et w een the Grothendie k groups K 0 (mo d M ) and K 0 (mo d M ′ ) . Let us pro v e the equalities a). Reall from [19 , setion 3.3℄, that the an tisymmetri bilinear form GR OTHENDIECK GR OUP AND MUT A TION R ULE 11 h , i a on mo d M is indued b y the usual Euler form h , i E on per M M . The follo wing omm utativ e diagram per M M × p er M M h , i E ' ' O O O O O O O O O O O O O ≃ / / per M ′ M ′ × p er M ′ M ′ h , i E w w n n n n n n n n n n n n n Z , th us indues a omm utativ e diagram K 0 (mo d M ) × K 0 (mo d M ) h , i a ( ( R R R R R R R R R R R R R R R S × S / / K 0 (mo d M ′ ) × K 0 (mo d M ′ ) h , i a v v l l l l l l l l l l l l l l l Z . This pro v es the equalit y R M = S t R M ′ S , or, b y lemma 11 , (1) B M = S t B M ′ S. An y ob jet of per M M b eomes an ob jet of per M / p er P through the omp o- sition per M M ֒ → per M ։ per M / p er P . Let M and N b e t w o non-pro jetiv e indeomp osable ob jets in M . Sine S N v anishes on P , w e ha v e Hom p er M / p er P  M (? , M ) , S N  = Hom p er M  M (? , M ) , S N  = Hom Mo d M  M (? , M ) , S N  = S N ( M ) . Th us dim Ho m p er M / p er P  M (? , M ) , S N  = δ M N , and the omm utativ e diagram per M / p er P × per M / per P R H om ) ) S S S S S S S S S S S S S S S ≃ / / per M ′ / p er P × per M ′ / p er P R H om u u k k k k k k k k k k k k k k k k per k , indues a omm utativ e diagram K 0 (pro j M ) / K 0 (pro j P ) × K 0 (mo d M ) Id ) ) S S S S S S S S S S S S S S S S S T × S / / K 0 (pro j M ′ ) / K 0 (pro j P ) × K 0 (mo d M ′ ) Id u u k k k k k k k k k k k k k k k k k Z . In other w ords, the matrix S is the in v erse of the transp ose of T : (2) S = T -t Equalities (1) and (2) imply what w as laimed, that is B M ′ = T B M T t . Let us ompute the matrix T : Let M b e indeomp osable non-pro jetiv e in M , and let Σ − 1 M − → M ′ 1 − → M ′ 0 − → M b e a KellerReiten appro ximation triangle of M with resp et to M ′ , whi h w e ma y assume to ome from a onation in E . This onation yields a pro jetiv e resolution 0 − → ( M ′ 1 ) ˆ − → ( M ′ 0 ) ˆ − → E (? , M ) | M ′ − → Ext 1 E (? , M ′ 1 ) | M ′ = 0 . so that T sends the lass of M ˆ to [( M ′ 0 ) ˆ ] − [( M ′ 1 ) ˆ ] . Therefore, t ij equals α ij − β ij .  12 Y ANN P ALU 3.2. Examples. 3.2.1. As a rst example, let C b e the luster ategory asso iated with the quiv er of t yp e A 4 : 1 → 2 → 3 → 4 . Its AuslanderReiten quiv er is the Mo ebius strip: 4 ′   ? ? ? ? 4   > > > > >   9 9 9 9 9 9   > > > > > @ @      3 ′   > > > > > @ @       ; ; ; ; ; ; A A         9 9 9 9 9   9 9 9 9 9 B B      2 @ @ @ @ @ > > ~ ~ ~ ~   > > > > > ? ?         9 9 9 9 9 B B        = = = = = C C      1 ? ?     ? ?       2 ′ A A      3 B B      4 ′ . Let M = M 1 ⊕ M 2 ⊕ M 3 ⊕ M 4 , where the indeomp osable M i orresp onds to the v ertex lab elled b y i in the piture. Let also M ′ = M ′ 1 ⊕ M ′ 2 ⊕ M ′ 3 ⊕ M ′ 4 , where M ′ 1 = M 1 , and where the indeomp osable M ′ i orresp onds to the v ertex lab elled b y i ′ if i 6 = 1 . One easily omputes the follo wing KellerReiten appro ximation triangles: Σ − 1 M 1 − → 0 − → M ′ 1 − → M 1 , Σ − 1 M 2 − → M ′ 2 − → M ′ 1 − → M 2 , Σ − 1 M 3 − → M ′ 4 − → 0 − → M 4 and Σ − 1 M 4 − → M ′ 4 − → M ′ 3 − → M 4 ; so that the matrix T is giv en b y: T =     1 1 0 0 0 − 1 0 0 0 0 0 1 0 0 − 1 − 1     . W e also ha v e B M ′ =     0 − 1 1 0 1 0 − 1 0 − 1 1 0 − 1 0 0 1 0     . Let maple ompute T -1 B M ′ T -t =     0 1 0 0 − 1 0 − 1 1 0 1 0 − 1 0 − 1 1 0     , whi h is B M . 3.2.2. Let us lo ok at a more in teresting example, where one annot easily read the quiv er of M ′ from the AuslanderReiten quiv er of C . Let C b e the luster ategory asso iated with the quiv er Q : 1 0 ? ?     ? ?       ? ? ? ?   ? ? ? ? 2 . F or i = 0 , 1 , 2 , let M i b e (the image in C of ) the pro jetiv e indeomp osable (righ t) k Q -mo dule asso iated with v ertex i . Their dimension v etors are resp etiv ely [1 , 0 , 0] , [2 , 1 , 0] and [2 , 0 , 1] . Let M b e the diret sum M 0 ⊕ M 1 ⊕ M 2 . Let M ′ b e the diret sum M ′ 0 ⊕ M ′ 1 ⊕ M ′ 2 , where M ′ 0 , M ′ 1 and M ′ 2 are (the images in C of ) the indeomp osable regular k Q -mo dules with dimension v etors [1 , 2 , 0] , [0 , 1 , 0] GR OTHENDIECK GR OUP AND MUT A TION R ULE 13 and [2 , 4 , 1] resp etiv ely . As one an  he k, using [ 14 ℄, M and M ′ are t w o luster tilting ob jets of C . T o ompute KellerReiten's appro ximation triangles, amoun ts to omputing pro jetiv e resolutions in mo d k Q , view ed as mo d End C ( M ) . One eas- ily omputes these pro jetiv e resolutions, b y onsidering dimension v etors: 0 − → 8 M 0 − → M 2 ⊕ 4 M 1 − → M ′ 2 − → 0 , 0 − → 2 M 0 − → M 1 − → M ′ 1 − → 0 and 0 − → 3 M 0 − → 2 M 1 − → M ′ 0 − → 0 . By applying the generalized m utation rule, one gets the follo wing quiv er 1 (6)           0 (2)   > > > > > > > 2 , (4) O O whi h is therefore the quiv er of End C ( M ′ ) sine b y [5 ℄, there are no lo ops or 2 -yles in the quiv er of the endomorphism algebra of a lustertilting ob jet in a luster ategory . 3.3. Ba k to the m utation rule. W e assume in this setion that the Auslander Reiten quiv er of T has no lo ops nor 2-yles. Under the notations of setion 3.1 , let k b e in I and let ( M k , M ′ k ) b e an ex hange pair (see setion 2.3). W e  ho ose M ′ to b e the luster-tilting sub ategory of C obtained from M b y replaing M k b y M ′ k , so that M ′ i = M i for all i 6 = k . Reall that T is the matrix of the isomorphism K 0 (pro j M ) / K 0 (pro j P ) − → K 0 (pro j M ′ ) / K 0 (pro j P ) . Lemma 13. Then, the ( i, j ) -entry of the matrix T is given by t ij =  − δ ij + | b ij | + b ij 2 if j = k δ ij else. Pr o of. Let us apply theorem 12 to ompute the matrix T . F or all j 6 = k , the triangle Σ − 1 M j → 0 → M ′ j = M j is a KellerReiten appro ximation triangle of M j with resp et to M ′ . W e th us ha v e t ij = δ ij for all j 6 = k . There is a triangle unique up to isomorphism M ′ k − → B M k − → M k − → Σ M ′ k where B M k − → M k is a righ t T ∩ T ′ -appro ximation. Sine the AuslanderReiten quiv er of T has no lo ops and no 2-yles, B M k is isomorphi to the diret sum: L i ∈ I ( M ′ j ) a ik . W e th us ha v e t ik = − δ ik + a ik , whi h equals | b ik | + b ik 2 . Remark that, b y lemma 7.1 of [9 ℄, as stated in setion 1.1 , w e ha v e T 2 = I d , so that S = T t and s ij =  − δ ij + | b ij |− b ij 2 if i = k δ ij else.  Theorem 14. The matrix B M ′ is obtaine d fr om the matrix B M by the F omin Zelevinski mutation rule in the dir e tion M . Pr o of. By [2℄ (see setion 1.1 ), and b y lemma 13 , w e kno w that the m utation of the matrix B M in diretion M is giv en b y T B M ′ T t , whi h is B M , b y the generalized m utation rule (theorem 12 ).  14 Y ANN P ALU 3.4. Cluster ategories. In [1℄, the authors study the Grothendie k group of the luster ategory C A asso iated to an algebra A whi h is either hereditary or anonial, endo w ed with an y admissible triangulated struture. A triangulated struture on the ategory C A is alled admissible in [1℄ if the pro jetion funtor from the b ounded deriv ed ategory D b (mo d A ) to C A is exat (triangulated). They dene a Grothendie k group K 0 ( C A ) with resp et to the triangles indued b y those of D b (mo d A ) , and sho w that it oinides with the usual Grothendie k group of the luster ategory in man y ases: Theorem 15. [BarotKussinLenzing℄ W e have K 0 ( C A ) = K 0 ( C A ) in e ah of the fol lowing thr e e  ases: (i) A is  anoni al with weight se quen e ( p 1 , . . . , p t ) having at le ast one even weight. (ii) A is tubular, (iii) A is her e ditary of nite r epr esentation typ e. Under some restrition on the triangulated struture of C A , w e ha v e the follo wing generalization of ase (iii) of theorem 15 : Theorem 16. L et A b e a nite-dimensional her e ditary algebr a, and let C A b e the asso iate d luster  ate gory with its triangulate d strutur e dene d in [ 16 ℄ . Then we have K 0 ( C A ) = K 0 ( C A ) . Pr o of. By lemma 3.2 in [1 ℄, this theorem is a orollary of the follo wing lemma.  Lemma 17. Under the assumptions of se tion 3.1 , and if mor e over M has a nite numb er n of non-isomorphi inde  omp osable obje ts, then we have an isomorphism K 0 ( C ) ≃ Z n / Im B M . Pr o of. This is a restatemen t of theorem 10 .  Referenes [1℄ M. Barot, D. Kussin, and H. Lenzing. The Grothendie k group of a luster ategory . J. Pur e Appl. A lgebr a , 212(1):3346, 2008. [2℄ Ark ady Berenstein, Sergey F omin, and Andrei Zelevinsky . Cluster algebras. I I I. Upp er b ounds and double Bruhat ells. Duke Math. J. , 126(1):152, 2005. [3℄ Aslak Bakk e Buan, Osam u Iy ama, Idun Reiten, and Jeanne Sott. Cluster strutures for 2-CalabiYau ategories and unip oten t groups. pr eprint arXiv: math/0701557[math.R T℄ . [4℄ Aslak Bakk e Buan, Rob ert Marsh, Markus Reinek e, Idun Reiten, and Gordana T o doro v. Tilting theory and luster om binatoris. A dv. Math. , 204(2):572618, 2006. [5℄ Aslak Bakk e Buan, Rob ert J. Marsh, and Idun Reiten. Cluster m utation via quiv er represen- tations. Comment. Math. Helv. , 83(1):143177, 2008. [6℄ Philipp e Caldero and Bernhard Keller. F rom triangulated ategories to luster algebras. to app e ar in Invent.Math. arXiv: math/0506018[math.R T℄ . [7℄ Raik a Deh y and Bernhard Keller. On the om binatoris of rigid ob jets in 2-CalabiYau ategories. to app e ar in Int. Math. Res. Not. pr eprint arXiv: math/0709.0882[math.R T℄ . [8℄ Sergey F omin and Andrei Zelevinsky . Cluster algebras. I. Foundations. J. A mer. Math. So . , 15(2):497529 (eletroni), 2002. [9℄ Christof Geiÿ, Bernard Leler, and Jan S hröer. Rigid mo dules o v er prepro jetiv e algebras. Invent. Math. , 165(3):589632, 2006. [10℄ Dieter Happ el. T riangulate d  ate gories in the r epr esentation the ory of nite-dimensional alge- br as , v olume 119 of L ondon Mathemati al So iety L e tur e Note Series . Cam bridge Univ ersit y Press, Cam bridge, 1988. [11℄ Osam u Iy ama. Auslander orresp ondene. A dv. Math. , 210(1):5182, 2007. [12℄ Osam u Iy ama and Y uji Y oshino. Mutation in triangulated ategories and rigid Cohen Maaula y mo dules. to app e ar in Invent. Math. pr eprint arXiv: math/0607736[math.R T℄ . [13℄ B. Keller and D. V ossie k. Aisles in deriv ed ategories. Bul l. So . 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(2) , 43(1):3748, 1991. [21℄ Gonçalo T abuada. On the struture of Calabi-Yau ategories with a luster tilting sub ate- gory . Do . Math. , 12:193213 (eletroni), 2007. [22℄ Andrei Zelevinsky . What is . . . a luster algebra? Noti es A mer. Math. So . , 54(11):1494 1495, 2007. Université P aris 7 - Denis Dider ot, UMR 7586 du CNRS, ase 7012, 2 pla e Jussieu, 75251 P aris Cedex 05, Frane. E-mail addr ess : palumath.jussieu.fr