On the vanishing of cohomology in triangulated categories

ON THE V ANISHING OF COHOMO LOGY IN TRIANGULA TED CA TEGORIES PETTER ANDREAS BERGH Abstract. W e study the v anishi ng of cohomology in a triangulated categ ory , in particular v anishing gaps and symmetry . 1. Introduction In this pap er, we study the v anishing of cohomolog y in a triangulated ca teg ory T . Given tw o ob jects X and Y of T , a very natural question arises when lo oking at their coho mo logy: can we detect the v anishing of Hom T ( X, Σ n Y ) for lar ge n by lo oking at fi n ite v anishing g aps? That is , is there a finite set S o f integers such that the implication Hom T ( X, Σ n Y ) = 0 for n ∈ S = ⇒ Hom T ( X, Σ n Y ) = 0 for n ≫ 0 holds? Commutativ e lo cal complete intersection r ing s provide examples wher e this is true. Namely , for such a ring A , the following was shown in [Jor] fo r a mo dule M of complexity d : if N is an A -mo dule, a nd there exists a n in teger n > dim A such that Ext i A ( M , N ) = 0 for n ≤ i ≤ n + d , then Ext i A ( M , N ) v anishes for all i > dim A . Over such a ring A , the complexity of an A -module is at most the co dimension c of A . Therefore, we can always detect v anishing of cohomolo gy over A b y lo o king at g aps of length c + 1 . Another natural question is : do e s symmetry hold in the v anishing of co ho mology in T ? In o ther words, if X and Y are ob jects in T s uch that Ho m T ( X, Σ n Y ) v an- ishes for n ≫ 0 , then do es it neces sarily follow that Hom T ( Y , Σ n X ) also v anishes for n ≫ 0? Again, co mmu tative lo cal complete intersection ring s provide examples where this hold. Namely , it was s hown in [AvB] that if M and N are mo dules over such a ring A , then the implication Ext i A ( M , N ) = 0 for i ≫ 0 = ⇒ Ext i A ( N , M ) = 0 for i ≫ 0 holds. Another class of rings wher e such symmetry holds a re group algebras of finite groups, o r, more gener ally , a s we sha ll se e, sy mmetric a lgebras with “finitely generated” cohomolog y . The tw o questions r aised a re studied in Section 3 and Section 4, resp ectively . W e o bta in some affirmative answers when certain co homology groups a re finitely generated as a mo dule over a ring acting centrally o n our tria ng ulated categ ory , a concept w e define in the following section. In pa r ticular, we show tha t E x t- symmetry holds for s y mmetric p erio dic algebra s. 2. Preliminaries Throughout this pap er, we fix a tria ng ulated ca tegory T with a susp ension func- tor Σ. Th us T is an a dditive Z - c ategory together with a cla s s of dis ting uished triangles satisfying V er dier’s axioms (cf. [V er ]). 2000 Mathematics Subj e ct Classific ation. 16E30, 18E30, 18G15. Key wor ds and phr ases. T riangulate d categories, v anishing of cohomology . The author wa s supp orted b y NFR Storforsk grant no. 167130. 1 2 PETTER ANDREAS BERGH Recall that a thick subcatego ry of T is a full triangulated subca tegory closed under dire ct s ummands. Now let C and D b e sub categor ies of T . W e deno te by thic k 1 T ( C ) the full sub categor y of T consis ting o f all the direct summands of finite direct sums of shifts of ob jects in C . F ur thermore, we denote b y C ∗ D the full sub c ategory of T consisting of ob jects M suc h tha t there exists a distinguished triangle C → M → D → Σ C in T , with C ∈ C and D ∈ D . Now for each n ≥ 2 , define inductively thick n T ( C ) to be thick 1 T  thic k n − 1 T ( C ) ∗ thic k 1 T ( C )  , a nd denote S ∞ n =1 thic k n T ( C ) by thick T ( C ). This is the sma llest thick subc a tegory of T containing C . The aim o f this pa p e r is to study the v anishing of coho mology in tr ia ngulated categorie s satisfying a certain finite genera tion hypothesis. This finite gener ation hypothesis is expressed in terms o f the gr ade d c enter Z ∗ ( T ) of our triangulated category T . Reca ll therefore that for an integer n ∈ Z , the deg ree n comp o nent Z n ( T ) is the s et of natura l tr a nsformations Id f − → Σ n satisfying f Σ X = ( − 1 ) n Σ f X on the level of ob jects. F or such a central element f and o b jects X, Y ∈ T , consider the graded gro up Hom ∗ T ( X, Y ) = ⊕ i ∈ Z Hom T ( X, Σ i Y ). The element f acts from the right on this graded gro up via the mo rphism X f X − − → Σ n X , and fro m the left via the morphism Y f Y − − → Σ n Y . Namely , given a morphism g ∈ Hom T ( X, Σ m Y ), the scalar pr o duct g f is the comp osition X f X − − → Σ n X Σ n g − − − → Σ m + n Y , where a s f g is the comp os ition X g − → Σ m Y Σ m f Y − − − − → Σ m + n Y . How ever, since Id f − → Σ n is a natura l transformatio n, the diagra m X g / / f X   Σ m Y f Σ m Y   Σ n X Σ n g / / Σ m + n Y commutes, and so since f Σ m Y equals ( − 1) mn Σ m f Y we see that g f = ( − 1) mn f g . This shows that Z ∗ ( T ) acts gr aded-commutativ ely o n Hom ∗ T ( X, Y ) for all ob jects X and Y in T . Now let R = ⊕ ∞ i =0 R i be a g raded-co mm utative ring, tha t is , for homog eneous elements r 1 , r 2 ∈ R the eq ua lity r 1 r 2 = ( − 1) | r 1 || r 2 | r 2 r 1 holds. W e say that R acts c entr al ly on T if there exists a graded ring homomorphism R → Z ∗ ( T ). If this is the case, then for every o b ject X ∈ T there is a gra ded r ing homomorphism R ϕ X − − → Hom ∗ T ( X, X ) with the following pr o p erty: for all o b jects Y ∈ T the scalar actions from R on Hom ∗ T ( X, Y ) via ϕ X and ϕ Y are graded equiv alent, i.e. ϕ Y ( r ) f = ( − 1) | r || f | f ϕ X ( r ) for all homo geneous elements r ∈ R and f ∈ Hom ∗ T ( X, Y ). W e say that the R - mo dule Hom ∗ T ( X, Y ) is eventual ly No etherian , a nd write Hom ∗ T ( X, Y ) ∈ No eth R , if there e x ists an integer n 0 ∈ Z such that the R -mo dule Hom ≥ n 0 T ( X, Y ) is No etherian. Moreov er, we say that Hom ∗ T ( X, Y ) is eventu al ly No etherian of fin ite lengt h , a nd write Hom ∗ T ( X, Y ) ∈ No eth fl R , if Hom ∗ T ( X, Y ) ∈ Noe th R , and there exists an int eger n 0 ∈ Z such that ℓ R 0 (Hom T ( X, Σ n Y )) < ∞ for each n ≥ n 0 . Note that if Hom ∗ T ( X, Y ) is even tually No etherian (r esp ectively , even tually Noether ian of finite length), then so is Hom ∗ T ( X ′ , Y ′ ) for a ll ob jects X ′ ∈ thic k T ( X ) a nd Y ′ ∈ thick T ( Y ). In par ticula r, if our categor y T is class ically finitely g enerated, that is, if there exists a n ob ject G such that T = thick T ( G ), then Hom ∗ T ( X, Y ) ∈ No eth R (respe ctively , Hom ∗ T ( X, Y ) ∈ No eth fl R ) for all X , Y ∈ T if a nd only if Hom ∗ T ( G, G ) ∈ No e th R (resp ectively , Hom ∗ T ( G, G ) ∈ No eth fl R ). ON THE V ANISHING OF COHOMOLOGY IN TRIANGULA TED CA TEGORIES 3 Definition. Given ob jects X and Y of T , we define the c omplexity of the ordered pair ( X, Y ) a s cx T ( X, Y ) def = dim R ev Hom ∗ T ( X, Y ) , where R ev denotes the commutativ e graded s ubalgebra ⊕ ∞ i =0 R 2 i of R . W e define the complexity cx T X of the single ob ject X a s cx T X def = cx T ( X, X ). When studying v anishing of coho mology in T , we will o nly b e dealing with ob jects X , Y ∈ T with the prop erty that the R -mo dule Hom ∗ T ( X, Y ) is even tually No etherian of finite length. Th is motiv ates the choice of terminology . Namely , it follows from [BIKO, Prop ositio n 2.6] that if Hom ∗ T ( X, Y ) ∈ No eth fl R , then the Krull dimension o f the R ev -mo dule Hom ∗ T ( X, Y ) equals the infimum of all non- negative integers t with the following pr op erty: there exists a real n um ber a such that ℓ R 0 (Hom T ( X, Σ n Y )) ≤ an t − 1 for n ≫ 0. A prior i, the co mplexity of a pair is not finite. How ever, whe n Hom ∗ T ( X, Y ) is even tually No e therian of finite length, then the finiteness of cx T ( X, Y ) follows from the ab ove together with [BIKO , Remark 2.1] a nd [AtM, Theorem 11.1]. It follows from the a bove alternative descr iption o f complexity that if X and Y are ob jects of T with Hom ∗ T ( X, Y ) ∈ No eth fl R , then cx T ( X, Y ) = 0 if and only if Hom ∗ T ( X, Y ) is even tually zero, that is, if Ho m T ( X, Σ n Y ) = 0 for n ≫ 0. Now digress for a momen t, and let Λ b e a ring. Then Λ s atisfies Auslander’s c ondition if for every finitely genera ted mo dule M , there exists a n in teger d M , dep ending only on M , satisfying the following: if N is a finitely generated Λ-mo dule a nd Ext n Λ ( M , N ) = 0 for n ≫ 0 , then Ext n Λ ( M , N ) = 0 for n ≥ d M . Motiv ated b y this, w e define a full sub categor y C of T to b e a left Auslander sub c ate gory if for every ob ject X ∈ C , ther e exists a n in teger d X , dep ending o nly o n X , such that the following holds: if Hom ∗ T ( X, Y ) is even tually zero for some ob ject Y ∈ T , then Hom T ( X, Σ n Y ) = 0 for n ≥ d X . It is e a sy to see that this holds if and only if for all o b jects X ∈ C and Y ∈ T , the implication Hom T ( X, Σ n Y ) = 0 for n ≫ 0 = ⇒ Hom T ( X, Σ n Y ) = 0 for all n ∈ Z holds. Dually , we can define right Ausla nder sub categor ies. Note that if an ob- ject X ∈ T b elongs to a left or rig ht Auslander subca tegory and Hom ∗ T ( X, X ) is even tually zero , then X = 0 . 3. V an ishing of cohomol ogy W e start with the following result, the key ingredient in the main theorem. It shows that we can a lwa y s reduce the complexity of a n ob ject whos e endomor phism ring is even tually No etherian of finite length. How ever, recall first the following notion. Let R b e a g raded-commutative ring acting centrally on T , and let X ∈ T be an ob ject. Then, given a homogene o us ele ment r ∈ R , we can c o mplete the map X ϕ X ( r ) − − − − → Σ | r | X into a triangle X ϕ X ( r ) − − − − → Σ | r | X → X/ /r → Σ X . The ob ject X/ /r is well defined up to iso mo rphism, and is called a Koszu l obje ct o f r o n X . Prop ositi o n 3.1. L et R b e a gr ade d-c ommutative ring acting c entr al ly on T , and let X ∈ T b e an obje ct such that Hom ∗ T ( X, X ) ∈ No eth fl R . Th en if cx T X is 4 PETTER ANDREAS BERGH nonzer o, ther e exists a homo gene ous element r ∈ R , of p ositive de gr e e, whose Koszul obje ct X/ /r in t he triangle X ϕ X ( r ) − − − − → Σ | r | X → X/ / r → Σ X satisfies cx T X/ /r = c x T X − 1 . Pr o of. Supp os e cx T X > 0. By [BIKO , Lemma 2.5 ], there exists an integer i 0 and a ho mogeneous element r ∈ R , of po sitive degree, such that scalar multip lication Hom T ( X, Σ i X ) · r − → Hom T ( X, Σ i + | r | X ) is injective for i ≥ i 0 . Applying Hom T ( X, − ) to the tr iangle X ϕ X ( r ) − − − − → Σ | r | X → X/ /r → Σ X , we o btain a long ex a ct s e q uence · · · → Hom T ( X, Σ i X ) · ( − 1) i r − − − − − → Hom T ( X, Σ i + | r | X ) → Hom T ( X, Σ i X/ /r ) → · · · in cohomolog y . This lo ng exact sequence induce s a sho r t ex a ct se quence 0 → Hom ≥ i 0 T ( X, X ) · r − → Hom ≥ i 0 + | r | T ( X, X ) → Ho m ≥ i 0 T ( X, X/ /r ) → 0 of even tually No etherian R -mo dules o f finite length, a sequence fro m which we de- duce that cx T ( X, X/ /r ) = cx T X − 1. F rom this exact sequence we also se e that the element r a nnihilates Hom ≥ i 0 T ( X, X/ /r ). Therefore, when a pplying Hom T ( − , X/ /r ) to o ur tr iangle, we o btain a short exac t sequence 0 → Hom ≥ i 0 + | r | T ( X, X/ /r ) → Hom ≥ i 0 + | r | +1 T ( X/ /r, X/ /r ) → Ho m ≥ i 0 +1 T ( X, X/ /r ) → 0 of eventually No etherian R - mo dules of finite leng th. This g ives the inequality cx T X/ /r ≤ cx T ( X, X/ /r ). Since Hom ∗ T ( X, X/ /r ) is even tually No etherian, there exists a n integer n 0 such that the R -mo dule Hom ≥ n 0 T ( X, X/ /r ) is finitely generated. The R -sca lar action factors through the r ing homomorphism ϕ X/ /r , and there fo re Hom ≥ n 0 T ( X, X/ /r ) is also finitely genera ted a s a mo dule over Hom ∗ T ( X/ /r, X/ /r ). Consequently , the rate of growth of the sequence { ℓ R 0 (Hom T ( X, Σ n X/ /r )) } ∞ n =1 is at most the rate o f growth o f { ℓ R 0 (Hom T ( X/ /r, Σ n X/ /r )) } ∞ n =1 , i.e. cx T ( X, X/ /r ) ≤ cx T X/ /r .  Now let X b e an ob ject of T , a nd let r 1 , . . . , r c be a sequence of homogeneous elements belo nging to a gra ded-commutativ e r ing R a cting cen trally on T , with c ≥ 2. Then we define the K oszul element X / / ( r 1 , . . . , r c ) inductiv ely a s X/ / ( r 1 , . . . , r c ) def = ( X / / ( r 1 , . . . , r c − 1 )) / /r c . Suppo se the R -mo dule Hom ∗ T ( X, X ) is even tually No etherian of finite leng th, and denote the complex ity of X b y c . If c > 0, then the previous r esult guarantees the existence of a homogeneo us element r 1 ∈ R , of p ositive degree, such tha t cx T X/ /r 1 = c − 1. Since X/ /r 1 belo ngs to thic k T ( X ), the R -mo dule Hom ∗ T ( X/ /r 1 , X/ /r 1 ) is also even tually Noether ia n o f finite length. Therefore, if c − 1 > 0, then we may use the ab ove r esult aga in; there exists a homogeneous ele- men t r 2 ∈ R , of p os itive deg ree, such that cx T X/ / ( r 1 , r 2 ) = c − 2. Contin uing like this, we o btain a sequence r 1 , . . . , r c of homog eneous elements of R , all of p os itive degree, together with tr iangles X → Σ | r 1 | X → X 1 → Σ X X 1 → Σ | r 2 | X 1 → X 2 → Σ X 1 . . . X c − 1 → Σ | r c | X c − 1 → X c → Σ X c − 1 ON THE V ANISHING OF COHOMOLOGY IN TRIANGULA TED CA TEGORIES 5 in which X i = X/ / ( r 1 , . . . , r i ) and cx T X i = c − i for 1 ≤ i ≤ c . W e say that the s equence r 1 , . . . , r c r e duc es the c omplexity of the ob ject X . Note that such a sequence is not unique in g eneral, and tha t the order matters. Note also that the sequence reduces the complexity of Σ n X for a ny n ∈ Z , since cx T Y = cx T Σ n Y for all o b jects Y in T . W e now pr ov e our main result. It shows that, for tw o o b jects X and Y , if Hom ∗ T ( X, Y ) con tains a large enough “gap” , then Hom ∗ T ( X, Y ) is actually zero. The length of the ga p dep ends o n the sum o f the degrees of a sequence r e ducing the complexity of X . Theorem 3.2. L et X and Y b e obje cts of T with Hom ∗ T ( X, X ) ∈ No eth fl R for some gr ade d-c ommu tative ring R acting c entr al ly on T , and supp ose thick T ( X ) is either a left or right Auslander sub c ate gory of T . L et r 1 , . . . , r c b e a se quenc e of p ositive de gr e e homo gene ous elements of R r e ducing the c omplexity of X , wher e c = cx T X . Then the fol lowing ar e e quivalent: (i) Ther e exists an int e ger n ∈ Z s u ch that Hom T ( X, Σ i Y ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . (ii) Hom T ( X, Σ i Y ) = 0 for al l i ∈ Z . Pr o of. W e a rgue by induction on c that (i) implies (ii). If c is zero, then Hom ∗ T ( X, X ) is even tually zero, and so X = 0 since thic k T ( X ) is either a left or right Auslander subca tegory of T . If c > 0, consider the tria ngle X → Σ | r 1 | X → X / / r 1 → Σ X . Applying Hom T ( − , Y ) to this triangle gives the lo ng exact sequence · · · → Hom T ( X, Σ i − 1 Y ) → Hom T ( X/ /r 1 , Σ i Y ) → Hom T ( X, Σ i −| r 1 | Y ) → · · · in cohomolog y , from whic h we s ee that Hom T ( X/ /r 1 , Σ i Y ) = 0 for ( n + | r 1 | ) ≤ i ≤ ( n + | r 1 | ) + | r 2 | + · · · + | r c | − ( c − 1) . The co mplexity of X/ /r 1 is c − 1, and the sequence r 2 , . . . , r c is a complexity reducing sequence for this ob ject. Therefo r e, by the induction hypothesis, w e co nclude that Hom T ( X/ /r 1 , Σ i Y ) = 0 for all i ∈ Z . The long exact seq ue nce then shows that Hom T ( X, Σ i Y ) is iso morphic to Hom T ( X, Σ i + | r 1 | Y ) for all int egers i , and from (i) we then see that Hom T ( X, Σ i Y ) must v anish for all i ∈ Z .  If we interc hange the ob jects X and Y in the theorem, then the corresp onding result of course ho lds. W e sta te this without pro of. Theorem 3.3. L et X and Y b e obje cts of T with Hom ∗ T ( X, X ) ∈ No eth fl R for some gr ade d-c ommu tative ring R acting c entr al ly on T , and supp ose thick T ( X ) is either a left or right Auslander sub c ate gory of T . L et r 1 , . . . , r c b e a se quenc e of p ositive de gr e e homo gene ous elements of R r e ducing the c omplexity of X , wher e c = cx T X . Then the fol lowing ar e e quivalent: (i) Ther e ex ist s an inte ger n ∈ Z such that Hom T ( Y , Σ i X ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . (ii) Hom T ( Y , Σ i X ) = 0 for al l i ∈ Z . W e now use these res ults to study the v anishing of coho mology over Artin al- gebras. L et Λ b e a Noe ther ian ring, a nd denote the b ounded der ived ca tegory of finitely g enerated Λ-mo dules by D b (Λ). F ur thermore, let D p erf (Λ) b e the thick sub c ategory of D b (Λ) generated by Λ; it consists of the p erfect complex es, that is, ob jects isomorphic to b ounded complexe s of finitely g enerated pro jective Λ- mo dules. The stable derive d c ate gory o f Λ, denoted D b st (Λ), is the V erdier quotient D b st (Λ) def = D b (Λ) /D p erf (Λ) . 6 PETTER ANDREAS BERGH This is a triangula ted categor y whose suspension functor co rresp onds to that in D b (Λ). Moreov er, b y [BIK O, Remark 5 .1], the central a ction of a graded- commutativ e ring R on D b (Λ) carries ov er to D b st (Λ) v ia the ring homomorphism Z ∗ ( D b (Λ)) → Z ∗ ( D b st (Λ)) induced b y the natur al quotient functor . Th us if X and Y are complexes in D b (Λ), then the natural map Hom ∗ D b (Λ) ( X, Y ) → Ho m ∗ D b st (Λ) ( X, Y ) is a n R -mo dule homomorphism. If Λ is also Gorenstein, that is, if the injectiv e dimension of Λ b oth a s a left a nd a s a right mo dule o ver itself is finite, then by [Buc, Co rollar y 6 .3.4] this homomor phism is even tually bijective. Tha t is, if Λ is a No etherian Gorenstein ring, then the natural map Hom D b (Λ) ( X, Σ n Y ) → Ho m D b st (Λ) ( X, Σ n Y ) is bijective for n ≫ 0. Consequently , for any complexe s X and Y in D b (Λ), if Hom ∗ D b (Λ) ( X, Y ) is an event ually No ether ian R -mo dule o f finite length, then s o is Hom ∗ D b st (Λ) ( X, Y ). Suppo se Λ is an Artin alg e bra, that is, the center Z (Λ) of Λ is a co mmu tative Artin ring over whic h Λ is finitely genera ted as a mo dule. Denote by mo d Λ the category of finitely generated left Λ-mo dules . If Λ is Gorenstein, then denote by MCM(Λ) the catego ry of finitely generated maximal Cohen- Ma caulay Λ-mo dules , i.e. MCM(Λ) = { M ∈ mo d Λ | Ext i Λ ( M , Λ) = 0 for a ll i > 0 } . It follows from genera l cotilting theory that this is a F rob enius exact category , in which the pro jectiv e injectiv e ob jects are the pro jective Λ-mo dules, and the in- jective envelopes are the le ft add Λ-approximations. Ther efore the s table ca teg ory MCM (Λ), which is obtained by facto ring o ut all morphisms which factor through pro jective Λ-mo dules, is a triangulated categ ory . Its shift functor is given by cok- ernels o f left add Λ- approximations, the inv erse shift is the usua l syzyg y functor. It follows fro m work by Buch w eitz, Happel and Rick ard (cf. [Buc], [Ha p], [Ric]) that MCM (Λ) and the quotient catego ry D b (Λ) /D p erf (Λ) ar e equiv a lent as triang ulated categorie s. If M and N are maximal Cohen-Macaulay mo dules in mod Λ, then there is an iso mo rphism Ext n Λ ( M , N ) ≃ Hom MCM (Λ) (Ω n Λ ( M ) , N ) for every n > 0 . W e use this isomorphism to prov e the following result. It shows that when a cer tain finiteness condition holds, then the thick sub catego ry in MCM (Λ) generated by a mo dule is a left and right Auslander sub catego ry . Prop ositi o n 3.4. L et Λ b e an A rt in Gor en stein algebr a with Jac obson r adic al r , and let M b e a maximal Cohen-Mac aulay mo dule. If either Ext ∗ Λ ( M , Λ / r ) or Ext ∗ Λ (Λ / r , M ) b elongs to No eth fl R for some gr ade d-c ommutative ring R acting c en- tr al ly on D b (Λ) , then thic k MCM (Λ) ( M ) is a left and right Auslander sub c ate gory of MCM(Λ) . Pr o of. Supp os e that Ext ∗ Λ ( M , Λ / r ) ∈ No eth fl R . Let X and Y b e maximal Cohen-Macaulay mo dules in mod Λ with X ∈ thick MCM(Λ) ( M ), and suppos e that Hom ∗ MCM (Λ) ( X, Y ) is even tually z ero. W e prov e by induction o n cx MCM (Λ) X tha t Hom MCM (Λ) ( X, Σ n Y ) = 0 for all n ∈ Z . Suppo se cx MCM(Λ) X = 0. The finiteness conditio n implies tha t the R -mo dule Ext ∗ Λ ( M , N ) is even tually No etherian of finite length for all N ∈ mo d Λ, and so the same holds for Hom ∗ MCM (Λ) ( X, X ). Therefor e Hom ∗ MCM(Λ) ( X, X ) is even tually zero, in particular Ext n Λ ( X, X ) = 0 for n ≫ 0. No w consider the R -mo dule E xt ∗ Λ ( X, Λ / r ). Since it b elo ngs to No eth fl R , and the R -mo dule structure facto r s throug h the r ing ON THE V ANISHING OF COHOMOLOGY IN TRIANGULA TED CA TEGORIES 7 homomorphism R ϕ X − − → Ext ∗ Λ ( X, X ), we see tha t it must b e even tually zero. The Λ-mo dule X therefore has finite pro jectiv e dimension, and is is omorphic to the zero ob ject in MCM (Λ). Co ns equently Hom MCM(Λ) ( X, Σ n Y ) = 0 for all n ∈ Z . If cx MCM(Λ) X > 0, then let r ∈ R b e a homogeneous element of po sitive degree such that cx MCM (Λ) X/ /r = cx MCM(Λ) X − 1. Since Hom ∗ MCM (Λ) ( X, Y ) is e ventually zero, we see from the tria ngle X → Σ | r | X → X/ / r → Σ X that the same holds for Hom ∗ MCM (Λ) ( X/ /r, Y ). The Ko s zul ob ject X/ /r be lo ngs to thic k MCM (Λ) ( M ), hence b y induction Hom MCM(Λ) ( X/ /r, Σ n Y ) = 0 for a ll n ∈ Z . F r om the tr iangle we o btain the is o morphism Hom MCM (Λ) ( X, Σ n Y ) ≃ Ho m MCM(Λ) ( X, Σ n + | r | Y ) for all integers n , a nd this implies that Hom MCM (Λ) ( X, Σ n Y ) = 0 for all n ∈ Z . W e hav e now proved that if Ex t ∗ Λ ( M , Λ / r ) ∈ No eth fl R , then thick MCM (Λ) ( M ) is a left Auslander subca tegory of MCM(Λ). Virtually the same pro of sho ws that thic k MCM (Λ) ( M ) is a lso a right Auslander sub categor y of MCM(Λ). Mor eov er, an analogo us pr o of shows that the same holds if E xt ∗ Λ (Λ / r , M ) ∈ No eth fl R .  Before proving the next re s ult, we recall the following. Let Λ b e an Artin alge br a with Jaco bson ra dical r , and let M ∈ mo d Λ be a mo dule with minimal pro jectiv e and injective resolutions · · · → P 2 → P 1 → P 0 → M → 0 and 0 → M → I 0 → I 1 → I 2 → · · · resp ectively . Then the c omplexity and plexity of M , denoted cx Λ M and px Λ M , resp ectively , are defined as cx Λ M def = inf { t ∈ N ∪ { 0 } | ∃ a ∈ R s uch that ℓ Z (Λ) ( P n ) ≤ an t − 1 for n ≫ 0 } , px Λ M def = inf { t ∈ N ∪ { 0 } | ∃ a ∈ R s uch that ℓ Z (Λ) ( I n ) ≤ an t − 1 for n ≫ 0 } , where Z (Λ ) is the cen ter of Λ. Now let R b e a gra ded-commutativ e ring acting centrally on D b (Λ). If E xt ∗ Λ ( M , Λ / r ) be longs to No eth fl R , then cx Λ M coincides with cx D b (Λ) M , and Ext ∗ Λ ( M , M ) a lso belo ngs to No e th fl R . Ther efore there ex- ists a sequence r 1 , . . . , r cx Λ M of homo geneous elements o f R , all of p o sitive degr ee, reducing the co mplexity of M a s an ob ject in D b (Λ). Similar ly , if Ex t ∗ Λ (Λ / r , M ) belo ngs to Noeth fl R , then px Λ M = cx D b (Λ) M . In this case, there exists a homo- geneous sequence in R of length p x Λ M reducing the complexity of M as an ob ject in D b (Λ). Using this and Prop os itio n 3.4, we o bta in the following v anishing r esults on cohomology over Gorenstein a lgebras. W e prov e only the firs t o f these res ults; the pro of of the other res ult is simila r. Theorem 3.5. Le t Λ b e an Artin Gor enst ein algebr a with Jac obson r adic al r , and let M ∈ mo d Λ b e a maximal Cohen-Mac aulay mo dule. S upp ose Ext ∗ Λ ( M , Λ / r ) ∈ No eth fl R for some gr ade d-c ommutative ring R acting c ent r al ly on D b (Λ) , and let r 1 , . . . , r c b e a se quenc e of p ositive de gr e e homo gene ous elements of R r e ducing the c omplexity of M , wher e c = cx Λ M . Then for any N ∈ mo d Λ , the implic ations (i) ⇔ (ii) and (iii) ⇔ (iv) hold for the fol lowing statements: (i) Ther e exists a numb er n > id Λ such that Ext i Λ ( M , N ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . (ii) Ext i Λ ( M , N ) = 0 for al l i > id Λ . (iii) Ther e exist s a numb er n > id Λ su ch that Ext i Λ ( N , M ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . 8 PETTER ANDREAS BERGH (iv) Ext i Λ ( N , M ) = 0 for al l i > id Λ . Pr o of. B y [AuB , Theo r em 1 .8], there ex is ts an exact s equence 0 → Q → C → N → 0 in mod Λ, in which Q has finite pro jective dimension and C is maximal Cohen- Macaulay . Since Q a lso has finite injective dimension and id Q is at most id Λ, there are isomorphisms Ext i Λ ( M , N ) ≃ Ex t i Λ ( M , C ) for i > id Λ. More ov e r , the mo dule Ω id Λ Λ ( N ) is ma ximal Cohen-Macaulay , and E xt i Λ ( N , M ) ≃ E xt i − id Λ Λ (Ω id Λ Λ ( N ) , M ) for i > id Λ. W e may therefore without loss o f genera lity assume that N itself is maximal Cohen-Ma caulay , and repla c e id Λ by 0 in the statements. The implica- tions now follow from Theorem 3.2, Theo rem 3.3, P rop ositio n 3.4 and the fact that Ext i Λ ( X, Y ) ≃ Hom MCM (Λ) ( X, Σ i Y ) when X and Y are maximal Cohen-Ma caulay and i > 0 .  Theorem 3.6. Le t Λ b e an Artin Gor enst ein algebr a with Jac obson r adic al r , and let M ∈ mo d Λ b e a maximal Cohen-Mac aulay mo dule. Su pp ose E xt ∗ Λ (Λ / r , M ) ∈ No eth fl R for some gr ade d-c ommutative ring R acting c ent r al ly on D b (Λ) , and let r 1 , . . . , r c b e a se quenc e of p ositive de gr e e homo gene ous elements of R r e ducing the c omplexity of M , wher e c = p x Λ M . Then for any N ∈ mo d Λ , the implic ations (i) ⇔ (ii) and (iii) ⇔ (iv) hold for the fol lowing statements: (i) Ther e exists a numb er n > id Λ such that Ext i Λ ( M , N ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . (ii) Ext i Λ ( M , N ) = 0 for al l i > id Λ . (iii) Ther e exist s a numb er n > id Λ su ch that Ext i Λ ( N , M ) = 0 for n ≤ i ≤ n + | r 1 | + · · · + | r c | − c . (iv) Ext i Λ ( N , M ) = 0 for al l i > id Λ . 4. Symmetr y F o r a group alge bra k G of a finite group G over a field k , the group cohomo logy ring H ∗ ( G, k ) is gr aded-commutativ e and acts centrally on D b ( k G ). Mor eov er, by a classical re s ult of Evens a nd V enk ov (cf. [E ve ], [V e1], [V e 2]), the coho mology ring is No ether ian, and Ext ∗ kG ( M , N ) is a finitely generated H ∗ ( G, k )-mo dule for a ll M and N in mo d k G . There fore the v anishing results from the previo us section apply to g roup algebras . Commutativ e lo cal complete in tersection r ings also hav e finitely gene r ated co- homology . F or such a ring A , it was shown in [Avr] that there exists a certa in po lynomial ring b A [ χ 1 , . . . , χ c ] acting centrally o n D b ( b A ), where b A denotes the com- pletion of A with resp ect to its maximal ideal. Again, for a ll finitely generated A -mo dules M a nd N , the b A [ χ 1 , . . . , χ c ]-mo dule Ext ∗ b A ( c M , b N ) is finitely generated. Consequently , v anishing results similar to thos e in the previous section a lso ho ld in this case. A fas cinating asp ect o f the v anishing o f co homology ov er b oth g roup algebr as and co mmu tative lo cal complete intersections is symmetry . In [AvB ] it was shown that for finitely generated mo dules M a nd N ov er a commutative lo cal complete int ersection A , the v anishing of Ext i A ( M , N ) for i ≫ 0 implies the v anishing of Ext i A ( N , M ) for i ≫ 0. The pro of inv olves the theo r y o f ce r tain supp or t v arieties attached to each pair o f A -mo dules . Denote b y c the co dimensio n of A and b y K the a lgebraic closure o f its residue field. A cone V ∗ A ( M , N ) in K c is asso ciated to the ordered pair ( M , N ), with the following prop erties: V ∗ A ( M , N ) = { 0 } ⇔ Ext i A ( M , N ) = 0 for i ≫ 0 , V ∗ A ( M , N ) = V ∗ A ( M , M ) ∩ V ∗ A ( N , N ) . ON THE V ANISHING OF COHOMOLOGY IN TRIANGULA TED CA TEGORIES 9 The symmetry in the v anishing of cohomo logy follows immediately fr o m these pro p- erties. Similarly , the theo r y of s uppo rt v ar ieties for mo dules ov er group algebra s of finite groups can be used to show that symmetry holds also for such algebr as (cf. [Ben]). W e shall see in this s ection that in gener al there is no symmetry in the v anishing of cohomolog y ov er an Artin a lgebra, even when the alge br a is selfinjectiv e and has finitely g enerated coho mo logy in the sense of gr oup algebra s. But firs t, we study situations where symmetry holds. Let k b e a commutativ e Artin ring, and suppo se T is a Hom-finite tria ngulated k -category . In other w ords, for a ll ob jects X , Y , Z ∈ T the gro up Hom T ( X, Y ) is a k -module of finite length, a nd compo sition Hom T ( Y , Z ) × Hom T ( X, Y ) → Ho m T ( X, Z ) is k -bilinear, where D = Hom k ( − , k ). A Serr e fu n ctor on T is a triangle equiv alence T S − → T , together with functorial isomorphisms Hom T ( X, Y ) ≃ D Hom T ( Y , S X ) of k -modules for a ll o b jects X , Y ∈ T . B y [B oK], such a functor is unique if it exists. F ollo wing [Kel], for an integer d ∈ Z , the categ ory T is said to b e we akly d - Calabi-Y au if it a dmits a Serre functor which is is o morphic as a k -linea r functor to Σ d . If, in a ddition, this isomorphism is an isomorphism o f tr iangle functors, then T is d -Calabi-Y au . How ever, w e will only b e dealing with weakly d -Cala bi-Y au categorie s. When T is such a category , then for all ob jects X , Y ∈ T there is an isomorphism Hom T ( X, Y ) ≃ D Hom T ( Y , Σ d Y ) of k - mo dules. It follows immediately that if this holds, then Hom T ( X, Σ n Y ) = 0 for n ≫ 0 if and only if Hom T ( Y , Σ n X ) = 0 for n ≪ 0 . Now let Λ b e an Artin Go renstein algebr a. F ollo wing [Mor], w e say that Λ is stably symmetric if MCM(Λ) is w eakly d -Cala bi- Y a u for some integer d ∈ Z . It was shown in that pap er that if Λ in addition satisfies Auslander’s condition, then symmetry holds in the v anishing o f cohomolo g y o f Λ- mo dules. The following res ult shows that sy mmetry ho lds for mo dules with finitely g enerated cohomolo gy . Theorem 4.1. L et Λ b e a st ably symmetric Artin Gor enst ein algebr a with J a- c obson r adic al r . L et M ∈ mo d Λ b e a mo dule such that either Ext ∗ Λ ( M , Λ / r ) or Ext ∗ Λ (Λ / r , M ) b elongs to No eth fl R for some gr ade d-c ommutative ring R acting c en- tr al ly on D b (Λ) . Then for every N ∈ mod Λ , t he fol lowi ng ar e e qu ivalent: (i) Ext i Λ ( M , N ) = 0 for i ≫ 0 . (ii) Ext i Λ ( M , N ) = 0 for i > id Λ . (iii) Ext i Λ ( N , M ) = 0 for i ≫ 0 . (iv) Ext i Λ ( N , M ) = 0 for i > id Λ . Pr o of. As in the pr o of o f Theo r em 3 .5, there ex is ts a n exact sequence 0 → Q N → C N → N → 0 in mo d Λ, in which Q N has finite pr o jectiv e (and injective) dimension, and C N is maximal Cohen-Mac aulay . Th us there is an iso morphism E xt i Λ ( M , N ) ≃ Ext i − id Λ Λ (Ω id Λ Λ ( M ) , C N ) for every i > id Λ . Moreover, since either Ext ∗ Λ ( M , Λ / r ) or Ext ∗ Λ (Λ / r , M ) b elongs to No eth fl R , so do either Ext ∗ Λ (Ω id Λ Λ ( M ) , Λ / r ) or Ext ∗ Λ (Λ / r , Ω id Λ Λ ( M )). Therefor e, as shown in the pro of of Theorem 3.5, and b y Theorem 3.6, the implicatio n Ext i Λ (Ω id Λ Λ ( M ) , C N ) = 0 for i ≫ 0 ⇒ Ext i Λ (Ω id Λ Λ ( M ) , C N ) = 0 for i > 0 10 PETTER ANDREAS BERGH holds, showing that (i) implies (ii). T o show that (iii) implies (iv), fix an exa ct sequence 0 → Q M → C M → M → 0 in mo d Λ , in which Q M has finite pro jective (and injective) dimension, and C M is maximal Co hen-Macaulay . There is an isomorphism E xt i Λ ( N , M ) ≃ Ext i Λ ( N , C M ) for every i > id Λ. Also, as ab ove, since either Ext ∗ Λ ( M , Λ / r ) or Ext ∗ Λ (Λ / r , M ) belo ngs to No eth fl R , so does one of E xt ∗ Λ ( C M , Λ / r ) and Ext ∗ Λ (Λ / r , C M ). Hence (iii) implies (iv) by Theorem 3.5 a nd Theo rem 3.6. By Theor em 3 .4, the s ub ca tegory thic k MCM (Λ) ( C M ) of MCM(Λ) is a left and right Auslander subcatego ry . Moreover, by a ssumption M CM(Λ) is w eakly d - Calabi-Y au for some integer d ∈ Z . Therefore the implications Ext i Λ ( M , N ) = 0 for i ≫ 0 ⇔ E xt i Λ ( C M , C N ) = 0 for i ≫ 0 ⇔ Ho m MCM(Λ) ( C M , Σ i C N ) = 0 for i ≫ 0 ⇔ Ho m MCM (Λ) ( C M , Σ i C N ) = 0 for i ∈ Z ⇔ Ho m MCM (Λ) ( C N , Σ i C M ) = 0 for i ∈ Z ⇔ Ho m MCM (Λ) ( C N , Σ i C M ) = 0 for i ≫ 0 ⇔ E xt i Λ ( C N , C M ) = 0 for i ≫ 0 ⇔ E xt i Λ ( N , M ) = 0 for i ≫ 0 hold, a nd the pro of is co mplete.  F o r an Artin algebra Λ with radica l r , if E xt ∗ Λ (Λ / r , Λ / r ) ∈ No eth fl R for some g raded-co mm utative ring R acting centrally on D b (Λ), then Ext ∗ Λ ( M , N ) ∈ No eth fl R for all mo dules M , N ∈ mo d Λ. Moreov er, if this holds, then Λ is a u- tomatically Gorenstein by [BIKO, Prop os ition 5.6 ]. Consequently , w e obtain the following “ global version” of Theorem 4 .1. Theorem 4.2. L et Λ b e a st ably symmetric A rtin algebr a with Jac obson r adic al r , and supp ose that Ex t ∗ Λ (Λ / r , Λ / r ) b elongs t o No eth fl R for some gr ade d-c ommutative ring R acting c ent ra l ly on D b (Λ) . Then for al l mo dules M , N ∈ mo d Λ , t he fol low - ing ar e e quivalent: (i) Ext i Λ ( M , N ) = 0 for i ≫ 0 . (ii) Ext i Λ ( M , N ) = 0 for i > id Λ . (iii) Ext i Λ ( N , M ) = 0 for i ≫ 0 . (iv) Ext i Λ ( N , M ) = 0 for i > id Λ . Next, we include a sp ecial case o f this theorem. Recall that for a co mmutative Artin ring k , a n Ar tin k -algebra Λ is symmetric if there is an is omorphism Λ ≃ Hom k (Λ , k ) of Λ-Λ- bimo dules. Such an a lgebra is neces sarily selfinjective. Corollary 4.3 . L et Λ b e a symmetric Artin algebr a with Jac obson r adic al r , and supp ose that Ext ∗ Λ (Λ / r , Λ / r ) b elongs to No eth fl R for some gr ade d-c ommutative ring R acting c entr al ly on D b (Λ) . Then for al l mo dules M , N ∈ mo d Λ , the fol lowing ar e e quivalent: (i) Ext i Λ ( M , N ) = 0 for i ≫ 0 . (ii) Ext i Λ ( M , N ) = 0 for i > 0 . (iii) Ext i Λ ( N , M ) = 0 for i ≫ 0 . (iv) Ext i Λ ( N , M ) = 0 for i > 0 . Pr o of. B y [Mo r, Cor ollary 4.4], a symmetric Artin algebra is stably symmetric.  W e turn now to a particular class of algebras having finitely gener ated coho- mology in the sense of Theor em 4 .2 and Corolla ry 4.3. Details concerning the ON THE V ANISHING OF COHOMOLOGY IN TRIANGULA TED CA TEGORIES 11 following can b e found in [SnS] and [Sol]. L e t k be a field and Λ a finite dimen- sional k -algebra, and denote the enveloping algebra Λ ⊗ k Λ op of Λ by Λ e . F or n ≥ 0, the n th Ho chschild c ohomolo gy gro up of Λ, denoted HH n (Λ), is the vector space Ext n Λ e (Λ , Λ). The graded vector s pa ce HH ∗ (Λ) = Ext ∗ Λ e (Λ , Λ) is a graded- commutativ e ring with Y oneda pro duct, and for every M ∈ mo d Λ the tensor pro duct − ⊗ Λ M induces a homomorphism HH ∗ (Λ) ϕ M − − → E x t ∗ Λ ( M , M ) of gr aded k -algebr as. If N ∈ mo d Λ is a no ther mo dule a nd η ∈ HH ∗ (Λ) and θ ∈ Ext ∗ Λ ( M , N ) ar e homoge ne o us elements, then the relation ϕ N ( η ) ◦ θ = ( − 1) | η | | θ | θ ◦ ϕ M ( η ) holds, where “ ◦ ” deno tes the Y oneda product. Therefore the Ho chsc hild cohomolog y r ing HH ∗ (Λ) acts cent rally o n D b (Λ). Suppo se now in addition t hat Λ is indecomp osa ble as an algebr a, and that Λ / r ⊗ k Λ / r is semisimple (as happ ens for exa mple when k is algebraica lly closed). F ur thermore, suppo se that Λ is a p erio dic algebr a . That is, there exists a num b er p > 0 suc h that Λ is isomor phic to Ω p Λ e (Λ) as a left Λ e -mo dule (i.e. as a bimo d- ule). By [ErH], [E HS] and [E Sn], this ha pp e ns for example when Λ is a s elfinjective Nak ay ama alg ebra, a M¨ obius algebr a o r a prepr o jectiv e alg ebra (see a lso [ESk]), and by [GSS] the condition implies that Λ is selfinjective. Letting Q n denote the n th mo dule in the minimal pro jective Λ e -resolution of Λ , w e hav e an exa ct sequence 0 → Λ → Q p − 1 → · · · → Q 0 → Λ → 0 of bimo dules, and w e denote this b y µ . T his extension is an element of HH p (Λ). If θ is a n element of HH n (Λ) for some n > p , then θ = ¯ θµ i for some i a nd a homo geneous element ¯ θ of degree not more than p . Hence the Ho chschild coho mo logy ring HH ∗ (Λ) is g enerated over HH 0 (Λ) by the finite set of k -g enerator s in HH 1 (Λ) , . . . , HH p (Λ), and ther efore is No ether ia n. If S is a simple no n-pro jective Λ-mo dule, then µ ⊗ Λ S is the b eginning of the minimal pro jective resolution of S , since Λ / r ⊗ k Λ / r is semisimple. Ther e fore S must b e p er io dic with p erio d dividing p . If N is an y finitely g enerated Λ -mo dule and ω is an elemen t of E xt n Λ ( S, N ) for some n > p , then, as ab ov e , ω = ¯ ω ( µ ⊗ Λ S ) for some element ¯ ω ∈ Ext m Λ ( S, N ) with m ≤ p . Therefore Ext ∗ Λ ( S, N ) is finitely g enerated as a mo dule over HH ∗ (Λ), and this shows that E x t ∗ Λ (Λ / r , Λ / r ) is a finitely generated HH ∗ (Λ)-mo dule. The following result is therefore an application of Co rollar y 4.3. Theorem 4.4. L et k b e a field, let Λ b e a symm et ric p erio dic k -algebr a with J a- c obson r adic al r , and supp ose that Λ / r ⊗ k Λ / r is semisimple. Then for al l mo dules M , N ∈ mo d Λ , the fol lowing ar e e quivalent: (i) Ext i Λ ( M , N ) = 0 for i ≫ 0 . (ii) Ext i Λ ( M , N ) = 0 for i > 0 . (iii) Ext i Λ ( N , M ) = 0 for i ≫ 0 . (iv) Ext i Λ ( N , M ) = 0 for i > 0 . W e finish this pap er with a n example in which we lo o k a t selfinjective Nak ay ama algebras . As w e have seen, these a lgebras are p erio dic and ther efore hav e finitely generated coho mology . How ever, the exa mple shows that unless the algebra is symmetric, symmetry do es not necessarily hold in the v a nishing of cohomolog y . 12 PETTER ANDREAS BERGH Example. Let Γ be the cir cular quiver 1 α 1 / / 2 α 2   > > > > > > > t α t = = z z z z z z z z z 3 α 3   t − 1 α t − 1 O O 4 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ b b F F F F F F F F where t ≥ 2 is an integer. Let k be a field, denote by k Γ the path algebra of Γ ov e r k , and let J ⊂ k Γ b e the ideal gener ated by the a rrows. Fix an integer n ≥ 1, let Λ b e the quotient algebra k Γ /J n +1 , and denote by r the Jacobson radica l o f Λ. Then Λ is a finite dimensional indeco mpo sable selfinjective Nak ay ama algebra, and Ext ∗ Λ (Λ / r , Λ / r ) is a finitely gener ated HH ∗ (Λ)-mo dule (the ring structure of HH ∗ (Λ) was studied and deter mined in [BLM] a nd [Er H]). W r ite n = q t + r , where 0 ≤ r < t . Let S i be the simple mo dule corresp onding to the vertex i , a nd P i its pro jective cov er. There is a n exa ct s e q uence 0 → Ω 2 Λ ( S i ) → P i +1(mod t ) · α i − − → P i → S i → 0 , and it is ea sy to see that Ω 2 Λ ( S i ) is isomorphic to S i +1+ r (m od t ) . Ther efore the minimal pro jective reso lution o f S i is · · · → P i +3+2 r → P i +2+2 r → P i +2+ r → P i +1+ r → P i +1 → P i → S i → 0 , with Ω 2 j Λ ( S i ) = S i + j + j r (all the indices a re taken mo dulo t ). A n um ber of co m- pletely different situations may o ccur, dep ending on the v alues o f the parameter s t and r . F or example, if r = 0, then w e s e e that all the simple mo dules app e ar infin- itely many times as ev en syz y gies in the minimal pro jective re s olution of any s imple mo dule. Ther e fore, in this c a se, if S and S ′ are simple mo dules, then E xt n Λ ( S, S ′ ) is no nzero for infinitely many n . Note that when r = 0, then Λ is sy mmetr ic , a nd so by Theor em 4.4 symmetry holds in the v anis hing of Ext. How ev er, symmetry do es not hold for all Nak a yama algebras . F or example, suppo se t ≥ 3 and r = t − 1. Then the exa ct se q uences 0 → S 1 → P 2 → P 1 → S 1 → 0 0 → S 2 → P 3 → P 2 → S 2 → 0 are the first pa rts of the minimal pr o jectiv e reso lutio ns of S 1 and S 2 , and therefore Ext n Λ ( S 1 , S 2 ) 6 = 0 whenever n is odd, wher eas Ext n Λ ( S 2 , S 1 ) = 0 for all n . Thus in this situation there is no sy mmetr y in the v anishing of E xt over Λ . Ackno wledgements I would like to thank Steffen Opp er mann and Idun Reiten fo r v aluable c o mment s on this pa p e r . References [AtM] M. F. Atiy ah, I. G. Macdonald, Intr o duction to c ommutative algebr a , A ddison-W esley , 1969. [AuB] M. Auslander, R.-O. Buc h w eitz, The homolo gic al the ory of maximal Cohen-Mac aulay appr oximations , M´ em. So c. Math. F rance 38 (1989), 5-37. [Avr] L. 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