Hochschild homology and global dimension

HOCHSCH ILD HOMOLOGY AND GLOBAL DIMENSION PETTER ANDREAS BER GH & DA G MADSEN Abstra ct. W e prov e that for certain classes of graded algebras (Koszul, local, cellular), infinite global d imen sion implies that Ho chsc hild homol- ogy does n ot v anish in high d egrees, provided the characteri stic of th e ground field is zero. Ou r pro of uses Igusa’s formula relating the Eu ler chara cteristic of relativ e cyclic homology to the graded Cartan determi- nant. 1. Introduction The homological pr op erties of a finite dimensional algebra are closely re- lated to the b eha vior of the algebra as a bimo du le o v er itself. F or example, if the algebra has finite pro jective d imension as a bimo dule, th en its global dimension is also finite. The con v erse holds if the algebra mo d u lo its Jacob- son radical is separable ov er the ground field, somethin g w hic h automatically happ en s when the field is algebraically closed. In particular, if a finite dimensional algebra o v er an algebraically closed field has finite global dimension, then all its higher Ho chsc hild cohomolgy groups v anish. In [Hap ], follo wing this easy observ ation, Happ el remarked that “the conv erse seems to b e not known”, thus giving birth to wh at subse- quen tly b ecame kn own as “Happ el’s question”: if all the higher Ho c hsc hild cohomolgy group s of a fi nite dimensional alge bra v anish, then is the algebra of fin ite global dimen s ion? As sh o wn in [AvI], the answer is yes when the algebra is comm utativ e. Ho w ev er, it w as sho wn in [BGMS] that the an- sw er in general is no. Namely , giv en a field k and a nonzero element q ∈ k whic h is not a ro ot of un ity , then the total Ho c hsc hild cohomology of the four-dimensional algebra k h X, Y i / ( X 2 , X Y − q Y X , Y 2 ) is fiv e. In p articular, all th e h igher Ho c hsc hild cohomology groups of this algebra v anish, w hereas the algebra, b eing selfinjectiv e, clearly do es n ot h a v e finite global dimension. As sho wn by Han in [Han], the total Ho chsc h ild homology of the ab ov e algebra is infin ite dimens ional. Han then conjectured that the homolog y v ersion of Happ el’s question w ould alwa ys hold, namely that a finite dimen- sional algebra whose higher Ho c hsc hild homology groups v anish must b e of finite global dimension. In the same pap er , h e sho w ed that th e conjecture 2000 Mathematics Subje ct Classific ation. 16E40, 16W50. Key wor ds and phr ases. Hochsc hild homol ogy , cyclic homo logy , graded Cartan determinant. The authors were sup p orted by NFR Storforsk grant no. 167130. 1 2 PETTER AND REAS BERGH & DA G MADSEN holds for m onomial algebras. M oreo v er, as in th e cohomology case, the con- jecture holds if the algebra is commutat iv e, b y [A V-P] (for fi n itely generated but n ot necessarily finite d imensional algebras, see also [V-P]). In this pap er, w e show that Han’s conjecture holds for graded lo cal alge- bras, Koszul algebras and graded cellular algebras, pro vided the c haracter- istic of the ground field is zero. W e do this by exploiting some particular prop erties of the graded Cartan matrix and th e logarithm of its d eterminan t, concepts extensivel y studied in [Igu]. 2. Hochschild homology and cyclic homology Throughout this pap er, w e fix a fi eld k , not necessarily algebraically closed. Let A b e a finite dimensional k -algebra, and den ote b y A e the en v eloping algebra A ⊗ k A op of A . Th e Ho chschild homolo gy of A , denoted HH ∗ ( A ), is defined b y HH ∗ ( A ) = T or A e ∗ ( A, A ). By d efinition, it is obtained b y taking any pro jectiv e bimo du le resolution of A , app lyin g A ⊗ A e − and computing the homology of the resulting complex. Ho w ev er, we shall ex- plore one particular s uc h resolution, w hic h ev en tually leads to th e defin ition of cyclic homology . F or eac h n ≥ 0, denote b y A ⊗ n the n -fold tensor pro duct A ⊗ k · · · ⊗ k A , in wh ic h there are n copies of A . F or n ≥ 2, this is a pr o jectiv e bimo d ule. Define a map A ⊗ ( n +1) b ′ − → A ⊗ n a 0 ⊗ · · · ⊗ a n 7→ n − 1 X i =0 ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n , and consider the complex · · · → A ⊗ 4 b ′ − → A ⊗ 3 b ′ − → A ⊗ 2 in whic h A ⊗ 2 is in degree zero. By [CaE, § IX.6] this complex is exact, and it is therefore a pr o jectiv e b im o dule r esolution of A . This is the standar d r esolution (or Bar r esolution ) of A . App lying A ⊗ A e − to th is resolution, w e obtain the complex · · · → A ⊗ 3 b − → A ⊗ 2 b − → A, in wh ic h the map A ⊗ ( n +1) b − → A ⊗ n is giv en b y a 0 ⊗ · · · ⊗ a n 7→ n − 1 X i =0 ( − 1) i a 0 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n +( − 1) n a n a 0 ⊗ a 1 ⊗ · · · ⊗ a n − 1 . By definition, th e h omology of this complex is the Ho c hsc hild homology of our algebra A . The standard resolution and the Hochsc h ild complex are also the k ey ingredien ts in th e definition of cyclic homology . F or eac h n ≥ 0, define the map A ⊗ ( n +1) t − → A ⊗ ( n +1) a 0 ⊗ · · · ⊗ a n 7→ a n ⊗ a 0 ⊗ · · · ⊗ a n − 1 , HOCHSCHILD HOMOLOGY AND GLOBAL DIMENSION 3 and let N = 1 + t + · · · + t n b e the corresp on d ing norm o p erator. T h en (1 − t ) b ′ = b (1 − t ) and b ′ N = N b (cf. [Lo d, Lemma 2.1.1]), and so . . .   . . .   . . .   A ⊗ 3 b   A ⊗ 3 − b ′   1 − t o o A ⊗ 3 b   N o o · · · 1 − t o o A ⊗ 2 b   A ⊗ 2 − b ′   1 − t o o A ⊗ 2 b   N o o · · · 1 − t o o A A 1 − t o o A N o o · · · 1 − t o o is a first quadran t double complex (in whic h the lo wer left A has degree (0 , 0)). The cyc lic homolo gy of A , denoted HC ∗ ( A ), is the h omology of the resulting total complex. It is closely linked to the Ho c hsc hild cohomology of A via the w ell kno wn long exact sequ en ce · · · → HH n ( A ) I − → HC n ( A ) S − → HC n − 2 ( A ) B − → HH n − 1 ( A ) I − → · · · due to Connes, the SBI se quenc e or Connes’ exact se q u enc e . Both Ho c hsc hild and cyclic homology are fun ctorial, since an algebra homomorphism ind uces a map b et w een the corresp onding Ho chsc hild com- plexes and cyclic double complexes. In particular, if a is a t wosided ideal of A , th en the su rjection A → A/ a ind uces a sur jectiv e map of cyclic double complexes. The kernel of this map is a doub le complex, and the homology of its total complex, d enoted HC ∗ ( A, a ), is the r elative cyclic homolo gy of A with resp ect to a . Supp ose the algebra A is graded, sa y A = A 0 ⊕ · · · ⊕ A s . T h en its grading induces an in ternal grading HH ∗ ( A ) = ⊕ i HH i ∗ ( A ) HC ∗ ( A ) = ⊕ i HC i ∗ ( A ) HC ∗ ( A, a ) = ⊕ i HC i ∗ ( A, a ) on b oth Ho c hsc hild and (relativ e) cycl ic homolog y . Consider th e SBI s e- quence relating the Ho c hsc hild and cyclic homology of A . By a theorem of Go o dwillie (cf. [Go o , Corollary I I.4.6] or [W ei, Theorem 9.9.1]), the image of the map HC i n ( A ) S − → HC i n − 2 ( A ) is ann ihilated b y i (wh er e i is viewe d as an element in k ) for ev ery i > 0. In p articular, if th e c haracteristic of k is zero, then w e obtain a short exact sequence ( † ) 0 → HC i n − 1 ( A ) → HH i n ( A ) → HC i n ( A ) → 0 for eac h i > 0. It follo ws fr om this sequence th at if HC i n ( A ) is nonzero f or some i > 0, then so is HH i n +1 ( A ). Next, supp ose A 0 is a p ro du ct of copies of the groun d fi eld k , s ay A 0 = k × r , and let J d enote the radical A 1 ⊕ · · · ⊕ A s of A . If the charact eristic of k is zero, then it follo es fr om [Igu , Corollary 1.2] that HC m n ( A, J ) = 0 for an y m ≥ 1 and n ≥ m . Therefore, in this case, 4 PETTER AND REAS BERGH & DA G MADSEN the Euler char acteristic of HC m ∗ ( A, J ) is w ell d efined and giv en by χ (HC m ∗ ( A, J )) = m − 1 X n =0 ( − 1) i dim k HC m n ( A, J ) , and we defin e the g r ade d Eu ler char acteristic by χ (HC ∗ ( A, J ))( x ) def = ∞ X m =1 χ (HC m ∗ ( A, J )) x m . The latter is a p o w er series with inte ger co efficien ts. The aim of this p ap er is to establish results concerning the n on-v anishing of Ho c hsc hild homology for certain algebras. T o sim p lify the notation, w e therefore d efine the follo w ing: hhdim A def = sup { n ∈ Z | HH n ( A ) 6 = 0 } c hdim A def = sup { n ∈ Z | HC n ( A ) 6 = 0 } c hdim( A, a ) def = sup { n ∈ Z | HC n ( A, a ) 6 = 0 } . As men tioned in the introd uction, we show in this pap er that if the charac- teristic of k is zero and A h as infin ite global d im en sion, then h hdim A = ∞ when A is graded lo cal, K oszul or grad ed cellular. W e end this section with the f ollo win g result, w hic h shows that we only need to establish the non- v anishing of the relativ e cyclic homology of A with resp ect to the r adical. Lemma 2.1. Supp ose A is gr ade d and that its de gr e e zer o p art is a pr o duct of c op ies of k . F urthermor e, supp ose the char acteristic of k is zer o. Then c hdim( A, J ) = ∞ ⇔ h h dim A = ∞ , wher e J is the r adic al of A . Pr o of. By construction, there is a long exact sequen ce · · · → HC n +1 ( A 0 ) → HC n ( A, J ) → HC n ( A ) → HC n ( A 0 ) → · · · relating relativ e and ordinary cyclic h omology . Since A 0 liv es only in degree zero, its inte rnal p ositiv e degree cyclic h omology v anish es, i.e. HC m ∗ ( A 0 ) = 0 for m > 0. Consequ en tly , there is an isomorp hism HC m n ( A, J ) ≃ HC m n ( A ) for ev ery n and any m > 0. Sup p ose HC n ( A, J ) is nonzero for some n . Sin ce A 0 is a p ro du ct of copies of k , we kno w from ab o v e that HC i n ( A, J ) = 0 for i ≤ n , and therefore th er e must b e an intege r m > n suc h that HC m n ( A, J ) is n onzero. The isomorphism ab ov e then shows th at HC m n ( A ) is nonzero, and from the exact s equence ( † ) we see that the same h olds for HH m n ( A ). This s h o ws the im p lication ⇒ . F or th e rev erse implication, note that fo r n > 0, the group HH 0 n ( A ) v anishes. He nce if HH n ( A ) is nonzero, then from the SBI sequence we s ee that th ere is an m > 0 suc h that either HC m n ( A ) or HC m n − 1 ( A ) is n onzero. T hen either HC m n ( A, J ) or HC m n − 1 ( A, J ) must b e nonzero.  HOCHSCHILD HOMOLOGY AND GLOBAL DIMENSION 5 3. The grade d Car t a n de terminant In this sectio n A denotes a p ositiv ely graded finite dimensional k -algebra A = A 0 ⊕ A 1 ⊕ · · · ⊕ A s . W e assume A 0 ≃ k × · · · × k = k × r as rings. Let 1 A = e 1 + . . . + e r b e the corresp onding decomp osition of the ident it y . In [Igu], the author pr esen ted a form ula relating the graded E u ler c harac- teristic χ (HC ∗ ( A, J ))( x ) of relativ e cyclic homology to the so-called graded Cartan determinant of A . F or 1 ≤ l ≤ s , let C l b e the r × r matrix with en tries C l i,j = dim k e j A l e i . The gr ade d Cartan matrix of A is d efined to b e the r × r matrix C A ( x ) = C 0 + C 1 x + C 2 x 2 + . . . + C s x s with en tries in Z [ x ]. Its determinant d et C A ( x ) is the gr ade d Cartan deter- minant of A . Wit h our assu mptions C 0 is the identit y m atrix, and therefore det C A ( x ) is a p olynomial of degree u ≤ s with in teger co efficien ts and con- stan t term 1. The logarithm of the determinant is then a p o w er series that can b e defined acco rding to the formula log d et C A ( x ) = ∞ X m =1 ( − 1) m +1 (det C A ( x ) − 1) m m . Although this p o w er serie s in general has ratio nal co efficients, its formal deriv ativ e D x (log det C A ( x )) has int eger co efficien ts. W e sa y that a p o w er series is pr op er if it has infinitely many nonzero terms. Lemma 3.1. The p ower series D x (log det C A ( x )) is pr op er and has i nte- ger c o efficie nts { b i } i ≥ 0 . The se q uenc e { b i } i ≥ 0 satisfies a line ar r e curr enc e r elation of or der u with c onstant inte g er c o effici e nts. Pr o of. The c hain rule giv es D x (log det C A ( x )) · d et C A ( x ) = D x (det C A ( x )) , or alternativ ely D x (log det C A ( x )) = (det C A ( x )) − 1 · D x (det C A ( x )) . Since the degree of the p olynomial D x (det C A ( x )) is strictly less than the degree of det C A ( x ), it follo ws from the fir st formula that D x (log det C A ( x )) m ust b e a prop er p ow er series. Since D x (det C A ( x )) has intege r co efficien ts, it follo w s fr om the second formula that D x (log det C A ( x )) has inte ger co ef- ficien ts. If det C A ( x ) = 1 + c 1 x + . . . + c u x u , then for an y m ≥ u , the first form ula giv es b m + c 1 b m − 1 + . . . + c u b m − u = 0. The last statement of the lemma follo w s.  Next we state I gusa’s formula as presente d in [Igu]. Recall that the M¨ obius function µ is the multi plicativ e n um b er th eoretic fu nction defin ed by µ ( n ) =    1 if n = 1, ( − 1) t if n is a pro du ct of t distinct primes, 0 if n has one or more rep eated prime f actors. Theorem 3.2. [Ig u, Theorem 3.5] L et A b e a gr ade d algebr a over a field k of char acteristic zer o, and supp ose A 0 is a pr o duct of c opies of k . Then 6 PETTER AND REAS BERGH & DA G MADSEN (a) χ (HC ∗ ( A, J ))( x ) = ∞ X m =1 log det C A ( x m ) X d | m µ ( d ) d , (b) log d et C A ( x ) = ∞ X m =1 χ (HC ∗ ( A, J ))( x m ) X d | m dµ ( d ) m , wher e µ is the M¨ obius function. An immediate corollary of this theorem is that det C A ( x ) = 1 if and only if χ (HC ∗ ( A, J ))( x ) = 0. Our aim is to show that if d et C A ( x ) 6 = 1, then χ (HC ∗ ( A, J ))( x ) is a pr op er p o w er series. W e would lik e to hav e a f orm ula r elating th e co efficien ts of D x (log det C A ( x )) = ∞ X i =1 b i x i with the co efficien ts of χ (HC ∗ ( A, J ))( x ) = ∞ X i =1 a i x i . F or this purp ose we int ro du ce the num b er theoretic function θ , d efined as follo w s. θ ( m ) = X d | m dµ ( d ) = Y p | m p pr ime (1 − p ) . The fu nction θ is multiplica tiv e a nd has the prop ert y that if n | m , then θ ( n ) | θ ( m ). F rom Theorem 3.2 (b) we get D x (log det C A ( x )) = ∞ X m =1 [ X d | m a d · d · θ ( m d )] x m − 1 . F or con venience let f ( m ) = X d | m a d · d · θ ( m d ) . With th is notation f ( m ) = b m − 1 . If χ (HC ∗ ( A, J ))( x ) has only finitely many non-zero co efficien ts, in other w ords if χ (HC ∗ ( A, J ))( x ) is a p olynomial, th en w e get the follo win g condi- tion on the function f . Prop osition 3.3. If χ (HC ∗ ( A, J ))( x ) is a p olynomial, then for ev ery p air of inte ge rs s , t > 0 ther e ar e s c onse cutive p ositive inte gers N + 1 , . . . , N + s such that 2 t | f ( N + i ) , 1 ≤ i ≤ s . Pr o of. Let v b e the degree of χ (HC ∗ ( A, J ))( x ). Cho ose st different od d prime num b ers p i,j > v , 1 ≤ i ≤ s, 1 ≤ j ≤ t . Let n i = Q t j =1 p i,j for eac h 1 ≤ i ≤ s . The system of congruen ces x ≡ − 1 mo d n 1 , x ≡ − 2 mo d n 2 , . . . HOCHSCHILD HOMOLOGY AND GLOBAL DIMENSION 7 x ≡ − s mod n s has a unique solution mo dulo n 1 n 2 · · · n s . Let N > 0 b e a solution. No w for eac h 1 ≤ i ≤ s , we h a v e n i | N + i . Also n i | N + i d whenev er d ≤ v . Sin ce a d = 0 for d > v , w e h a v e f ( N + i ) = X d | ( N + i ) d ≤ v a d · d · θ ( N + i d ) . Since θ ( n i ) | θ ( N + i d ) w henev er d ≤ v , it follo ws that θ ( n i ) | f ( N + i ). Sin ce θ ( n i ) = Q t j =1 (1 − p i,j ), it follo ws that 2 t | θ ( n i ) | f ( N + i ).  If det C A 6 = 1, then f cannot s atisfy this condition, and as a consequence w e get our theorem. Theorem 3.4. L et A b e a gr ade d finite dimensiona l algebr a over a field k of char acteristic zer o, and supp ose A 0 is a pr o duct of c opies of k . If d et C A 6 = 1 , then χ (HC ∗ ( A, J ))( x ) is a pr op er p ower series. Pr o of. Supp ose for con tradiction that d et C A 6 = 1 and χ (HC ∗ ( A, J ))( x ) is a p olynomial. Since det C A 6 = 1, it follo ws that χ (HC ∗ ( A, J ))( x ) 6 = 0. Let v b e the degree of χ (HC ∗ ( A, J ))( x ), and let l b e the degree of its lo w est degree nonzero term. Let p > v b e a pr ime num b er. T hen f ( l p i ) = a l · l · θ ( p i ) = a l · l (1 − p ) = f ( l p ) 6 = 0 for any i ≥ 1. Let t ≥ 1 b e th e num b er suc h that 2 t − 1 | f ( l p ) but 2 t ∤ f ( l p ). Let u > 0 b e th e degree of d et C A ( x ). By Proposition 3.3 there are u consecutiv e p ositiv e inte gers N + 1 , . . . , N + u suc h that 2 t | f ( N + i ), 1 ≤ i ≤ u . So D x (log det C A ( x )) has u consecutiv e co efficient s b N , . . . , b N + u − 1 whic h are d ivisib le by 2 t . But then b y Lemm a 3.1 we ha v e 2 t | b m for any m ≥ N , so 2 t | f ( m ) for any m ≥ N + 1. This con tradicts the f act that 2 t ∤ f ( l p i ) f or an y i ≥ 0.  F or our in v estigat ions into the v alidit y of Han’s conjecture, the follo wing corollary is imp ortant. Corollary 3.5. L et A b e a gr ade d finite dimensional algebr a over a field k of char acteristic zer o, and su pp ose A 0 is a pr o duct of c opies of k . Supp ose det C A 6 = 1 . Then (a) c hdim( A, J ) = ∞ , (b) hhdim A = ∞ . Pr o of. P arts (a) and (b) are equiv alen t by Lemma 2.1. Since A is finite di- mensional, for eac h n ≥ 0 the group HC n ( A, J ) is finite d imensional and therefore HC m n ( A, J ) 6 = 0 only for finitely man y in ternal degrees m . If χ (HC ∗ ( A, J ))( x ) is a prop er p o w er series, then HC m ∗ ( A, J ) 6 = 0 f or infin itely man y in ternal degrees m , and this is on ly p ossible if chdim( A, J ) = ∞ . So it follo ws from T h eorem 3.4 that if det C A 6 = 1, then chdim( A, J ) = ∞ .  4. Non-v anishing of Hoch schild homology in high degrees In th is section we apply Corollary 3.5 to v arious classes of graded algebras (Koszul, lo cal, cellula r), and p r o v e that Han’s co njecture [Han] hol ds for these classes. Han’s conjecture can b e stated as follo ws. 8 PETTER AND REAS BERGH & DA G MADSEN Conjecture (Han) . Let A b e a finite d imensional algebra o v er a field. I f A has infi nite global dimension, th en hh dim A = ∞ . Let A b e a graded fi nite d imensional k -algebra, and supp ose A 0 ≃ k × r . The matrix C A ( x ) is inv ertible as a matrix ov er Z [ x ] if and only if det C A ( x ) = 1. On the other h and, since det C A ( x ) h as constant term 1, the matrix C A ( x ) is alw a ys in v ertible when consider ed as a matrix o v er Z [[ x ]]. In [Wil] we find the f ollo wing form ula for the en tries in the in verse matrix C − 1 A ( x ). Here Ext gr A denotes graded extensions and X [ u ] denotes th e u th shift of the graded mo dule X . F or eac h 1 ≤ i ≤ r , we let S i denote the degree zero simple mo d ule S i = Ae i /J Ae i . Theorem 4.1. [Wil, Theorem 1.5] The entry c ij in C − 1 A ( x ) is given by c ij = X u ≥ 0 X v ≥ 0 ( − 1) v dim k  Ext v gr A ( S i , S j [ u ])  x u . F or C A ( x ) to be in v ertible as a matrix o v er Z [ x ], all en tries c ij ab o v e ha v e to b e p olynomials (not p rop er p o w er series), and we get the follo wing corollary . Corollary 4.2. det C A ( x ) = 1 i f and only if the e ntries c ij in C − 1 A ( x ) ar e p olynomials for al l 1 ≤ i, j ≤ r . If A has finite global dimension, then d et C A ( x ) = 1. The conv erse is not true, there are algebras A of infin ite global dimension w ith det C A ( x ) = 1. Example 4.3. Let A b e the path algebra A = k Q/I , where Q is the qu iv er 1 α ( ( 2 δ h h β ( ( 3 γ h h and I = h ρ i is the ideal generated b y the set of relations ρ = { β α, γ β , β γ , δ γ , αδ α, δ αδ } . Then d et C A ( x ) = 1, bu t A has infi nite global dimension. S ince A is monomial, Han’s conjecture is kno wn to hold f or this algebra and therefore hhdim A = ∞ . F or some classes of graded algebras, the com b in ation det C A = 1 and infinite global dimension is not p ossible. W e can u s e Corollary 3.5 to pro v e that Han’s conjecture holds for these classes. 4.1. Koszul algebras. Let A b e a graded algebra with A 0 ≃ k × r . S uc h an algebra is called Koszul ([Pri], [BGS]) if Ext v gr A ( S i , S j [ u ]) 6 = 0 imp lies u = v . Theorem 4.4. L et A b e a finite dimensional Koszul algebr a over a field of char acteristic zer o. If A has infinite g lob al dimension, then h hdim A = ∞ . Pr o of. Supp ose det C A ( x ) = 1. T hen C A ( x ) h as an in v erse C − 1 A ( x ) whose en tries are p olynomials and not prop er p o w er series. Since our algebra is Koszul, we kno w that Ext v gr A ( S i , S j [ u ]) is nonzero only when u = v , and so w e may simplify the formula for th e entries in the inv erse matrix and get c ij = X u ≥ 0 ( − 1) u dim k  Ext u gr A ( S i , S j [ u ])  x u . HOCHSCHILD HOMOLOGY AND GLOBAL DIMENSION 9 By assum ption, all the entries in C − 1 A ( x ) are p olynomials, hence all the higher graded extension groups b et w een simple A -mo dules v anish. C onse- quen tly , the global dimension of A is fi nite. This shows th at the d eterminan t of th e graded Cartan matrix of a Koszul algebra of infinite global d imension is not one, and so by Corollary 3.5 we are done.  4.2. Lo cal algebras. If A is graded fi nite dimensional with A 0 = k , then the graded Cartan determinant is b y definition the same as the Hilb ert p olynomial. Theorem 4.5. Supp ose k is of char acteristic zer o, and let A b e a gr ade d finite dimensional k -algebr a with A 0 = k . If A has infinite glob al dimension, then hhd im A = ∞ . Pr o of. If h h dim A < ∞ , then det C A = 1 b y Corollary 3.5. If the Hilb ert p olynomial of A is 1, then A = k and A has finite global dimension.  This th eorem can b e seen as a sp ecial case of the follo wing more general result. Theorem 4.6. Supp ose k is of char acteristic zer o, let Q = ( Q 0 , Q 1 ) b e a finite oriente d quiv e r, and let J b e the ide al in the p ath algebr a k Q gener ate d by the arr ows. F urthermor e, let I b e a homo g e nous ide al in kQ such that J t ⊆ I ⊆ J 2 for some t . If Q c onta ins a lo op, then hhd im k Q/I = ∞ . Pr o of. This follo ws from C orollary 3.5 and the construction of the graded Cartan matrix. T he en tries in C k Q/I ( x ) whic h are n ot on th e diagonal h a v e constan t term zero, so the only contribution to the degree one co efficient in the determinant comes from the pro d uct of the d iagonal en tries. The i th diagonal entry is of the form 1 + n i x + · · · , where n i is the num b er of lo ops at vertex i of Q . Therefore det C k Q/I ( x ) = 1 + | Q 0 | X i =1 n i x + higher terms . Consequent ly , if Q con tains a lo op, then det C k Q/I ( x ) 6 = 1.  4.3. Cellular algebras. C ellular algebras w ere in tro duced in [GL] (see also [KX1]) an d a re finite dimensional algebras whic h admit a sp ecial kind of basis. A k -algebra A is called a c el lular algebr a with cell datum (Λ , M , C, i ) if all of the follo wing th r ee conditions are satisfied. (C1) The set Λ is fi nite and partially ordered. Asso ciated with eac h λ ∈ Λ there is a fin ite set M ( λ ). The algebra A h as a k -basis { C λ S,T } , where ( S, T ) run s through all elemen ts of M ( λ ) × M ( λ ) for all λ ∈ Λ. (C2) The map i is a k -linear an ti-automorphism of A with i 2 = 1 A whic h sends C λ S,T to C λ T ,S . (C3) F or eac h λ ∈ Λ and S, T ∈ M ( λ ) and eac h a ∈ A , th e pro duct a · C λ S,T can b e written as a · C λ S,T = X U ∈ M ( λ ) h a ( U, S ) · C λ U,T + h ′ , 10 PETTER AND REAS BERGH & DA G MADSEN where h ′ is a linear combination of basis elemen ts with upp er index µ strictly smaller than λ , and where the co efficien ts h a ( U, S ) d o n ot dep end on T . Let S 1 , . . . , S r b e a complete set of n on-isomorphic sim p le A -mo du les, and let P 1 , . . . , P r denote th e corresp ondin g in d ecomp osable pr o jectiv e mo dules. The (ung r ade d) Ca rtan matrix of A , d en oted U A , is the r × r matrix o v er Z where for all 1 ≤ i, j ≤ r , the en try ( U A ) ij is equal to the comp osition m ultiplicit y of S j in P i . In [KX2] w e find the fol lo wing c haracterizat ion of cellular algebras of fin ite global dimension. F or the defin ition of quasi- hereditary algebras, see [CPS]. Theorem 4.7. [KX2 , Theorem 1.1] L et A b e a c el lular algebr a over a field. The fol lowing ar e e quivalent. (a) A is quasi-her e ditary, (b) A has finite glob al dimension, (c) det U A = 1 . If A is graded with A 0 ≃ k × r , th en U A can b e obtained from C A ( x ) by ev aluating for x = 1. Therefore det U A = d et C A (1). Theorem 4.8. Supp ose k is of char acteristic zer o, and let A b e a gr ade d c el lular k -algebr a such that A 0 is a pr o duct of c opies of k . If A has infinite glob al dimension, then hhdim A = ∞ . Pr o of. If det U A 6 = 1, then d et C A ( x ) 6 = 1, and so by Corollary 3.5 w e hav e hhdim A = ∞ .  Referen ces [AvI] L. L. Avramov, S. Iyengar, Gaps in Ho chschild c ohomolo gy i m ply smo othn ess for c omm utative algebr as , Math. Res. Lett. 12 (2005), no. 5-6, 789–804. [A V-P] L. L. 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