Dimensions of triangulated categories via Koszul objects

DIMENSIONS OF TRIANGULA TED CA TEGOR IES VIA K OSZUL OBJECTS PETTER ANDREAS BER GH, SRIKANTH B. IYENGAR, HENNING KRAUSE, AND STEFFEN OPPERMANN De dic a te d to Karin Er dm ann on the o c c a sion of her 60th birthday. Abstract. Lo wer bounds for the dimension of a triangulated category are prov ided. T hese b ounds are applied to stable derived ca tegories of Ar tin alge- bras and of commutativ e complete intersection lo cal rings. As a consequence, one obtains b ounds for the representat ion dimensions of certa in Artin alge bras. 1. Introduction A notion of dimension for a triangula ted category was intro duced by Rouquier in [ 26 ]. Roughly sp eaking, it corres po nds to the minimum num ber of steps needed to generate the categ ory from o ne o f its ob jects. Consideratio n of this inv ariant has been cr itical to so me recent developmen ts in algebra and geometry: Using the dimension of the s table category of an exterior algebra o n a d -dimens io nal vector space, Rouquier [ 27 ] pr oved that the r epresentation dimension of the e x terior algebra is d +1 , there by obtaining the first exa mple of an algebr a with repres ent ation dimension more than three. On the other hand, Bondal a nd V an den Bergh [ 11 ] proved that any c ohomologi- cal finite functor o n the b ounded derived category of coherent s heav es on a s mo oth algebraic v a riety ov er a field is r epresentable, b y e s tablishing tha t that triangula ted category ha s finite dimension. In this paper we establish low er bounds for the dimension of a triangulated category and discuss s ome applications. W e make sys tematic use of the g raded- commutativ e structur e o f the tria ngulated categor y – in pa rticular, K oszul ob jects – aris ing fr o m its g raded center; see Sectio n 3 . A triangulated ca teg ory T is by definition an additive Z -categor y equipp ed with a cla ss of exact triangles satisfying v ar io us axioms [ 31 ]. Here, Z -c ate gory simply means that there is a fixed equiv alence Σ : T → T . Given a ny additive Z -categor y T = ( T , Σ), we in tro duce a natural finiteness condition fo r ob jects of T a s follows: Let R = L i > 0 R i be a gra ded-commutativ e ring that acts on T via a homo mo rphism of gra ded rings R → Z ∗ ( T ) to the gra ded center of T . Thus for e ach pair of ob jects X , Y in T , the graded ab elia n group Hom ∗ T ( X, Y ) = M i ∈ Z Hom T ( X, Σ i Y ) is a graded R -mo dule. 2000 Mathematics Subje ct Classific ation. 16E30, 16E40, 18E30. Key wor ds and phr ases. triangulated category , K oszul ob ject, r epresen tation dimension. Bergh and Opp ermann were supp orted by NFR Storforsk grant no. 167130 , and Iyenga r was partly supp orted by NSF grant, D M S 0602498. Iy engar’s work was done at the Unive rsity of Pa derb orn, on a visit supported by a F orsc h ungspreis aw arded by the Humboldt-Stiftung. 1 2 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN Now fix an o b ject X in T and supp ose that fo r e ach Y ∈ T there exists an integer n such that the following prop erties hold: (1) the graded R -mo dule L i > n Hom T ( X, Σ i Y ) is no etherian, a nd (2) the R 0 -mo dule Hom T ( X, Σ i Y ) is of finite length for i ≥ n . In this ca s e, Hom ∗ T ( X, Y ) has finite (Kr ull) dimension over R ev , the subring o f R co nsisting of elements o f even degree, whic h is a co mm utative ring. If X has this finiteness prop er t y also with resp ect to ano ther ring S , then the dimension of Hom ∗ T ( X, Y ) ov er S coincide s with that ov e r R ; see Lemma 4.3 . F or this r e ason, we denote this num b er b y dim Hom ∗ T ( X, Y ). The main result in this w ork is as follows. Theorem 1.1. L et T b e a tr iangulate d c ate gory and X an obje ct with pr op erties as ab ove. One t hen has an ine quality dim T ≥ dim End ∗ T ( X ) − 1 . An in triguing feature of this r esult is that the inv a riant app ear ing on the rig ht hand side of the inequa lit y in volv es only the additive Z -structure of T . Theorem 1.1 is con tained in Theo rem 4.2 . The pr o of is bas ed on a systematic use o f K o szul ob jects and elementary obser v ations concerning ‘eventually no etherian mo dules’; this is inspired by the approach in [ 6 ]. Another imp ortant ingredient is a version of the ‘Gho st Lemma’ from [ 10 ]; see Lemma 3.2 . Our principa l motiv ation for consider ing dimensions of triangulated ca teg ories is that it provides a wa y to o btain low er b ounds on the representation dimension of an Artin algebra . Indeed, for a non- semisimple Artin alg ebra A one ha s an inequality rep . dim A ≥ dim D b st ( A ) + 2 , where D b st ( A ) is the stable derived ca tegory o f A , in the se ns e of B uch weitz [ 13 ]. As one application of the pr eceding result, we b ound the representation dimensio n of A by the Krull dimension of Ho chsc hild cohomolo gy . Corollary 1.2. L et k b e an algeb r aic al ly close d field and A a fi nite dimensional k - algebr a with r adic al r , wher e A is not semi-simple. If Ext ∗ A ( A/ r , A/ r ) is no etherian as a mo dule over the Ho chschild c ohomolo gy algebr a HH ∗ ( A ) of A over k , then rep . dim A ≥ dim HH ∗ ( A ) + 1 . This result is a spe cial ca se of Coro llary 5.12 . In Section 5 we present further applications o f Theorem 4 .2 . 2. E ventuall y noetherian modules Many of the ar guments in this article are based on prop erties of ‘eventually no etherian mo dules’ ov er graded co mm utative rings, intro duced by Avramov and Iyengar [ 6 , § 2]. In this se ction we co llect the requir ed res ults. F or the b enefit of the reader we provide (sketches of ) pr o ofs, although the res ults are well-kno wn, and the arguments bas ed o n standard techniques in commutativ e alg ebra. F or unexpla ined terminology the reader is referred to B runs and Her zog [ 12 ]. Graded-comm utativ e rings. Le t R = L i > 0 R i be a gr ade d-c ommutative ring ; th us R is an N -graded ring with the prop erty that rs = ( − 1) | r || s | sr fo r a ny r, s in R . Elements in a graded ob ject are a ssumed to b e homogeneous. Let M = L i ∈ Z M i be a gr aded R -mo dule. F or any integer n , we set M > n = M i > n M i and R + = R > 1 . DIMENSIONS OF TRIANGULA TED CA TEGORIES 3 Note that M > n is an R -submo dule of M , and that R + is an idea l in R . As in [ 6 , § 2], we say that M is eventu al ly no etherian if the R -mo dule M > n is no etherian for some int eger n ; we wr ite no eth( R ) for the full sub c ategory of the cat- egory of all gra ded R -mo dules, with ob jects the even tually no etheria n mo dules. In this work, the fo cus is on even tua lly no etherian modules M that have the additional prop erty that leng th R 0 ( M n ) is finite for n ≫ 0. The cor resp onding full sub cate- gory of no eth( R ) is denoted no eth fl ( R ). It is ea s y to verify that b oth no eth( R ) and no eth fl ( R ) a re ab elian subcateg ories. Recall that the annihilator of M , which we denote ann R M , is the homogeno us ideal of R c o nsisting of elements r such that r · M = 0. The following rema r k is easily justified. It allows o ne , when consider ing even tually no etherian mo dules, to pass to a situation whe r e the ring itself is noether ian. R emark 2.1 . Suppose that the R -mo dule M > n is no etherian. Set I = ann R ( M > n ). The ring R/I is then no etherian, and M > n is a finitely genera ted and faithful R/I - mo dule. If in addition length R 0 ( M i ) is finite for i ≥ n , then ( R/I ) 0 is a rtinian. One wa y to study mo dules ov er graded-commutativ e ring s is to pass to the subring R ev consisting of elements o f even de g ree, which is then a c ommut ative gr ade d ring: r s = sr for any r, s in R ev . In this work, this passa ge is facilitated by the following o bserv atio n; confer the pro of of [ 12 , Theor em 1.5.5 ]. Lemma 2. 2. L et R b e a gr ade d-c ommu tative ring, and let M b e an R -mo dule. (1) If M is in no e th( R ) , then it is also in no eth( R ev ) . (2) If M is in no e th fl ( R ) , then it is also in no eth fl ( R ev ) . Pr o of. Supp o se M is in no eth( R ). By Remark 2.1 , one ca n a ssume R is itself no etherian a nd M a finitely gener ated R -mo dule. It then suffices to prove that the subring R ev is no etheria n and that R is finitely genera ted as a mo dule ov er it. Observe that there is a decompo sition R = R ev ⊕ R od d as R ev -mo dules. In particular, for a n y ideal I ⊆ R ev one has I R ∩ R ev = I , and hence R no etherian implies R ev no etherian. By the same token, o ne obta ins that R od d , and hence also R , is a no etherian R ev -mo dule.  Dimensio n o ver a commutativ e graded ring. Let R b e a commutativ e graded ring. W e recall so me facts concerning the supp ort of a n R - mo dule M , which we denote Supp R M . It is conv enient to employ also the following notation: Pro j R = { p is a homogeneous prime in R with p 6⊇ R + } Supp + R M = { p ∈ P ro j R | M p 6 = 0 } Ass + R M = { p ∈ P ro j R | Hom R p ( R p / p R p , M p ) 6 = 0 } . Evidently , Supp + R M = Supp R M ∩ Pro j R and Ass + R M = Ass R M ∩ Pro j R , where Ass R M is the set of as so ciated primes of M . The ne x t res ult is readily verified. Lemma 2. 3. L et R b e a c ommutative gr ade d ring and M a gr ade d R -mo dule. (1) F or any inte ger n , one has e qualities Supp + R ( M > n ) = Supp + R M and Ass + R ( M > n ) = Ass + R M . (2) If L ⊆ M is a submo dule, then Supp + R L ⊆ Supp + R M and Ass + R L ⊆ Ass + R M .  A mo dule M is said to b e eventu al ly zer o if M > n = 0 for so me in teger n . The next result is pa r t of [ 6 , § 2.2 ], where it is s tated without pro of. W e g ive details , for the convenience of r eaders. 4 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN Prop ositio n 2.4 . L et R b e a c ommutative gr ade d ring and M an eventual ly no e- therian R -mo dule. The set Ass + R M is finite and the c onditions b elow ar e e quivalent: (i) Ass + R M = ∅ ; (ii) Supp + R M = ∅ ; (iii) M is eventual ly zer o. Pr o of. In view o f Remark 2.1 and Lemma 2 .3 (1), one may as s ume R is no etheria n and that M is a faithful R -mo dule. In this case Ass R M is a finite set and therefor e Ass + R M is finite; see [ 23 , Theor em 6 .5]. An ideal p ∈ Sp ec R b elo ngs to Supp R M if a nd only if there exists q ∈ Ass R M with q ⊆ p ; see [ 23 , Theor em 6.5 ]. F rom this the implications (iii) = ⇒ (ii) and (ii) ⇐ ⇒ (i) a r e o bvious consequences . It rema ins to show (ii) = ⇒ (iii). Since the R -mo dule M is finitely g enerated and faithful, o ne has that Supp R M = Spec R . Thus, Supp + R M = ∅ implies R + ⊆ p for e a ch p ∈ Spe c R , hence the ideal R + is nilp otent. Since R is no etherian, this implies that R is e ven tually zero, and hence also that M is even tually zero.  W e say that an element r ∈ R + is filter-r e gular on M if Ker( M r − → M ) is ev en- tually z ero. This notio n is a minor v a riation o n a well-w orn theme in commutativ e algebra; co nfer , for instanc e , Sc henzel, T rung, and Cuong [ 28 , § 2.1 ]. Lemma 2.5. L et R b e a c ommutative gr ade d ring and M an event ual ly no etherian R -mo dule. Ther e then ex ists an element in R + that is filter-r e gular on M . Pr o of. Prop o s ition 2.4 yields that the set Ass + R M is finite, so by prime avoidance [ 12 , Lemma 1.5.1 0] ther e exists an element r in R + not co ntained in any prime p in Ass + R M . This element is filter-regular on M . Indeed, for K = K er( M r − → M ), one has Ass + R K ⊆ Ass + R M ; se e Lemma 2.3 (2). How ever, for any p in Ass + R M one has K p = 0 , since r 6∈ p , a nd hence Ass + R K = ∅ . Since K is even tually no etherian, being a submo dule of M , P rop osition 2.4 applies and yields that K is even tually zer o .  As usual, the (Krull) dimension of a mo dule M over R is the num b er dim R M = sup  d ∈ N     there e x ists a chain of prime ide a ls p 0 ⊂ p 1 ⊂ · · · ⊂ p d in Supp R M  . When M is in no eth fl ( R ) one c an compute its dimension in terms o f the ra te of growth o f its comp onents. T o make this pr ecise, it is conv enient to introduce the c omplexity of a sequence of no n-negative integers ( a n ) as the num b er cx( a n ) = inf ( d ∈ N      there exists a real num ber c such that a n ≤ cn d − 1 for n ≫ 0 ) . F or basic prop er ties of this notion see, for example, [ 2 , § 2 and Appendix]. As us ual, the set of prime ideals of R containing a g iven ideal I is denoted V ( I ). Prop ositio n 2.6. L et R b e a c ommu tative gr ade d ring and M ∈ no eth fl ( R ) . (1) If r 1 , . . . , r n ar e elements in R + with n < dim R M , then V ( r ) ∩ Supp + R M 6 = ∅ . (2) One has an e quality dim R M = cx(length R 0 ( M n )) . Pr o of. By Remark 2.1 , one ma y assume R is noetheria n, R 0 is artinian, and M is a faithful, finitely genera ted R - mo dule. Part (1) then follows from the Kr ull heig ht theorem; see [ 23 , Theorem 13.5], w hile (2) is cont ained in [ 23 , Theor em 13.2].  DIMENSIONS OF TRIANGULA TED CA TEGORIES 5 Dimensio n ov er a graded-comm utativ e ring. Let R b e a grade d- commutativ e ring. F or each R -mo dule M in no eth fl ( R ), w e introduce its dimension as the num ber dim R M = cx(length R 0 ( M n )) . It follows from Lemma 2.2 a nd Pr op osition 2.6 (2) that this nu mber is finite a nd coincides with the dimension of M as a mo dule over R ev . This remar k will b e used without further commen t. Prop ositio n 2.7. L et R → S b e a homomorphism of gr ade d-c ommutative rings and M an S -mo dule. If M , viewe d as an R -mo dule by r estriction of sc alars, is in no eth fl ( R ) , then dim R M = dim S M . Pr o of. The mo dule M is in no eth fl ( S ) as well and therefor e, by Remark 2.1 , one can pass to a situation where S is no etheria n and M is a faithful S -mo dule that is also no etherian over R . Passing to R/I , wher e I is the kernel of the homomorphism R → S , one may also as sume that the homo morphism is injective. Since one has injectiv e homomorphisms of R - mo dules R ֒ → S ֒ → Hom R ( M , M ) , one th us obtains that the ring R itself is no etherian with R 0 artinian, and that S is a finitely gener ated R -mo dule. This implies that the R 0 -mo dule S 0 is finitely- generated, and hence, for an y S 0 -mo dule N , one has inequalities length S 0 N ≤ length R 0 N ≤ (length R 0 S 0 )(length S 0 N ) . This yields dim R M = dim S M , a s claimed.  3. Koszul objects Let T b e a tria ngulated category . F or any o b jects X and Y in T , we set Hom ∗ T ( X, Y ) = M n ∈ Z Hom T ( X, Σ n Y ) and End ∗ T ( X ) = Hom ∗ T ( X, X ) . The gr ade d c enter o f T , which we denote Z ∗ ( T ), consists in deg ree n o f natural transformatio ns η : id T → Σ n satisfying η Σ = ( − 1 ) n Σ η . Comp osition gives Z ∗ ( T ) a structure of a gra ded-commutativ e ring; see, for instance, [ 14 , § 3], espe cially Lemma 3.2.1 , which explains the signed co mm utation rule, and also [ 22 ]. In what follows, we as s ume that a gra ded-commutativ e ring R acts c entr al ly on T , via a homomorphis m R → Z ∗ ( T ). What this amo unts to is sp ecifying for each X in T a ho momorphism of rings φ X : R → End ∗ T ( X ) such tha t the induced R - mo dule structures o n Hom ∗ T ( X, Y ) c oincide up to the usual sign rule: η ◦ φ X ( r ) = ( − 1) | r || η | φ Y ( r ) ◦ η for any η ∈ Hom ∗ T ( X, Y ) and r ∈ R . W e now recall an elemen tary , a nd extr e mely useful, co nstruction. Koszul ob jects. Let r b e a homog eneous elemen t in R o f degree d = | r | . Given an ob ject X in T , we de no te X/ /r any ob ject that app ears in an exact triang le (3.1) X r − → Σ d X − → X/ /r − → Σ X . It is well-defined up to isomor phism; we call it a Koszul obje ct of r on X . Let Y b e an o b ject in T and set M = Hom ∗ T ( X, Y ). Applying Hom ∗ T ( − , Y ) to the tria ng le ab ov e y ields an exa ct seq uenc e of R -mo dules: M [ d + 1] ∓ r − → M [1] − → Hom ∗ T ( X/ /r, Y ) − → M [ d ] ± r − → M [0] . 6 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN This gives rise to an exac t sequence of gr a ded R -mo dules (3.2) 0 − → ( M /rM )[1] − → Hom ∗ T ( X/ /r, Y ) − → (0 : r ) M [ d ] − → 0 , where (0 : r ) M denotes { m ∈ M | r · m = 0 } . Applying the functor Hom ∗ T ( Y , − ) results in a similar exa ct sequence. Given a s equence of elements r = r 1 , . . . , r n in R , consider ob jects X i defined b y (3.3) X i = ( X for i = 0 , X i − 1 / /r i for i ≥ 1 . Set X/ / r = X n ; this is a Koszu l obje ct of r on X . The result b elow is a straig ht - forward cons equence of ( 3.2 ) and a n induction o n n ; see [ 9 , Le mma 5.11(1 )]. Lemma 3 .1. Le t n ≥ 1 b e an inte ger and set s = 2 n . F or any se quenc e of elements r = r 1 , . . . , r n in R + , and any obje ct X ∈ T one has that r s i · Hom ∗ T ( X/ / r , − ) = 0 = r s i · Hom ∗ T ( − , X/ / r ) for i = 1 , . . . , n.  The next c onstruction qua n tifies the pro cess of ‘building’ ob jects o ut of a given ob ject in the triangulated ca teg ory T . Thic kenings. Given a n ob ject G of T we write thick T ( G ) fo r the thic k sub categ ory of T generated by G . This subca tegory has a filtration { 0 } = thick 0 T ( G ) ⊆ thic k 1 T ( G ) ⊆ · · · ⊆ [ n > 0 thic k n T ( G ) = thic k T ( G ) where thick 1 T ( G ) consists of retracts of finite direct s ums of susp ensions of G , and thic k n T ( G ) co nsists o f retra cts of n - fold extensions of thic k 1 T ( G ). In the literature, the sub categ o ry thick n T ( G ) ha s sometimes b een denoted h G i n . The next result is contained in [ 10 , Lemma 2.1]. Simila r results hav e app ear ed in Kelly [ 20 ], C a rlsson [ 1 5 , Pro o f of Theor em 16 ], Christensen [ 16 , Theor em 3.5], Beligiannis [ 7 , Coro llary 5.5], Rouquier [ 26 , Lemma 4.1 1], and Avramov, Buch weitz, and Iyengar [ 3 , Pro po sition 2.9]. Lemma 3.2 (Ghost Lemma) . L et T b e a triangulate d c ate gory, and let F, G b e obje cts in T . S upp ose ther e ex ist morphisms K c θ c − → K c − 1 θ c − 1 − − − → · · · θ 1 − → K 0 in T such that the fol lowing c onditions hold: (1) Hom n T ( G, θ i ) = 0 for n ≫ 0 and for e ach i = 1 , . . . , c ; (2) Hom n T ( F, θ 1 · · · θ c ) 6 = 0 for infinitely many n ≥ 0 . One then has that F 6∈ thick c T ( G ) .  There is als o a contrav ariant version of the Ghost Lemma, inv olving Hom ∗ T ( − , G ). Theorem 3.3. L et T b e a triangulate d c ate gory and R a gr ade d-c ommutative ring acting c en t r al ly on it. L et X , Y b e obje cts in T with the pr op erty t hat the R -mo dule Hom ∗ T ( X, Y ) is in no eth fl ( R ) . F or any c < dim R Hom ∗ T ( X, Y ) ther e ex ist elements r 1 , . . . , r c in ( R ev ) + with X/ / r 6∈ thick c T ( X ) and Y / / r 6∈ thick c T ( Y ) . R emark 3.4 . In the lang ua ge of levels, introduced in [ 4 , § 2.3], the conclusion of the preceding theor em reads: level X T ( X/ / r ) > c and lev el Y T ( Y / / r ) > c . This formulation is s ometimes more con venien t to use in a rguments. DIMENSIONS OF TRIANGULA TED CA TEGORIES 7 Pr o of. The pla n is to a pply the Gho st Lemma. By Lemma 2.2 one can as s ume that R = R ev , and in par ticular that the gr aded ring R is commutativ e. The R -mo dule Hom ∗ T ( X, Y ) is in no e th fl ( R ) and hence so are Hom ∗ T ( X, Y / / x ) and Hom ∗ T ( X/ / x , Y ), for any finite sequence x of elements in R ; this can b e check ed us ing ( 3.2 ) and a n induction o n the length of x . Set s = 2 c . Using the obs erv ation in the prev ious para g raph and Lemma 2.5 , one can find, by iteration, elements r 1 , . . . , r c in R + such that fo r i = 1 , . . . , c the element r i is filter - regular on the R -mo dule Hom ∗ T ( X, Y / / { r s 1 , . . . , r s i − 1 } ) ⊕ Hom ∗ T ( X/ / { r s 1 , . . . , r s i − 1 } , Y ) Equiv alently , the element r i is filter-r egular on each of the direct summands ab ov e. W e now verify that X/ / r is not in thic k c T ( X ). Set K 0 = Y , se t K i = Σ − i  Y / / { r s 1 , . . . , r s i }  for i = 1 , . . . , c , and let (3.4) K i θ i − → K i − 1 ± r s i − − → Σ s | r i | K i − 1 → Σ K i , be the exa ct triangle obtained (by s uita ble susp ension) from the one in ( 3.1 ). W e claim that fo r each i = 1 , . . . , c the following pr o p e rties hold: (1) Hom n T ( X, θ i ) = 0 for n ≫ 0; (2) Hom ∗ T ( X/ / r , θ i ) is s urjective; (3) Hom n T ( X/ / r , K 0 ) 6 = 0 for infinitely many n ≥ 0. Indeed, for ea ch W ∈ T the triangle ( 3.4 ) induces a n exa ct sequence Hom ∗ T ( W , K i ) Hom ∗ T ( W ,θ i ) − − − − − − − − → Hom ∗ T ( W , K i − 1 ) ± r s i − − → Hom ∗ T ( W , K i − 1 )[ s | r i | ] of gra de d R -mo dules. (1) With W = X in the sequence ab ov e, r i is filter-regular on Hom ∗ T ( X, K i − 1 ), by choice, and hence so is r s i . This proves the claim. (2) Set W = X/ / r in the ex a ct sequence ab ov e, and note that r s i annihilates Hom ∗ T ( X/ / r , K i − 1 ), by L e mma 3.1 . (3) Recall that K 0 = Y . It suffices to prove that one has an equalit y Supp + R Hom ∗ T ( X/ / r , Y ) = V ( r ) ∩ Supp + R Hom ∗ T ( X, Y ) . F or then the choice of c ensures that the set ab ov e is non-empty , by Prop osi- tion 2 .6 (1), a nd hence Hom ∗ T ( X/ / r , Y ) is not ev entually zer o, by Pro po sition 2.4 . The equality ab ove can b e e s tablished as in the pro of of [ 6 , Pr op osition 3.10]: By induction o n the length of the sequence r , it suffices to consider the case wher e r = r . Setting M = Hom ∗ T ( X, Y ), it follows fro m ( 3.2 ) that one has an equality Supp + R Hom ∗ T ( X/ /r, Y ) = Supp + R ( M / rM ) ∪ Supp + R (0 : r ) M . It then remains to note that o ne has Supp + R ( M / rM ) = Supp + R M ∩ V ( r ) and Supp + R (0 : r ) M ⊆ Supp + R M ∩ V ( r ) , where the equality ho lds bec a use one ha s M /r M = M ⊗ R R/R r , while the inclusion holds b ecause (0 : r ) M is a submo dule of M annihilated by r . This justifies claims (1)–(3) above. Observe that (2) a nd (3) imply that Hom ∗ T ( X/ / r , θ 1 · · · θ c ) is not even tually zer o. Therefore, the Ghost Lemma y ie lds X/ / r 6∈ thic k c T ( X ), as desired. A similar ar gument, employing the contrav ariant version of the Ghost L e mma , establishes that Y / / r is no t in thic k c T ( Y ).  8 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN 4. The dimension of a triangula ted ca tegor y The dimension of a triangulated ca tegory T is the num b er dim T = inf { n ∈ N | there exists a G ∈ T with thick n +1 T ( G ) = T } . Evidently , if dim T is finite there exists an ob ject G with thick T ( G ) = T ; we call such a n ob ject G a gener ator for T . The dimension o f T can b e infinite even if it has a g enerator . Lemma 4. 1. L et T b e a triangulate d c ate gory and R a gr ade d-c ommu tative ring acting c ent r al ly on it. If G is a gener ator for T , t hen for e ach obje ct X in T one has e qualities Supp + R ev Hom ∗ T ( X, G ) = Supp + R ev End ∗ T ( X ) = Supp + R ev Hom ∗ T ( G, X ) . Pr o of. W e may assume R = R ev . Using the fact that lo c alization is an ex act functor, it is easy v erify that for any subset U of Spec R the sub catego ry { Y ∈ T | Supp + R Hom ∗ T ( X, Y ) ⊆ U } of T is thick. Since X is in thick T ( G ), o ne thus o btains an inclusio n Supp + R End ∗ T ( X ) ⊆ Supp + R Hom ∗ T ( X, G ) . The reverse inclusion holds bec a use R a cts o n Hom ∗ T ( X, G ) via a homomorphis m of rings R → End ∗ T ( X ). This s ettles the first equality . A similar a rgument gives the second one.  Theorem 4.2. L et T b e a triangulate d c ate gory and R a gr ade d-c ommutative ring acting c ent r al ly on it. If an obje ct X ∈ T is su ch that t he R - m o dule Hom ∗ T ( X, G ) , or Hom ∗ T ( G, X ) , is in no eth fl ( R ) , for some gener ator G , then one has an ine qu ality dim T ≥ dim R End ∗ T ( X ) − 1 . Pr o of. Supp o se that the R -mo dule Hom ∗ T ( X, G ) is in no eth fl ( R ). The full subca t- egory of T with o b jects { Y ∈ T | Hom ∗ T ( X, Y ) ∈ no eth fl ( R ) } is thick. Since it contains G it coincides with T , so one may a ssume that G is an arbitrar y generato r for T . F or c = dim R Hom ∗ T ( X, G ) − 1, Theorem 3.3 yields a Koszul ob ject, G/ / r , not contained in thick c T ( G ). This implies the inequa lit y below: dim T ≥ dim R Hom ∗ T ( X, G ) − 1 = dim R End ∗ T ( X ) − 1 ; the equality is by Lemma 4.1 . The other case is ha ndled in the same wa y .  In the theorem, the num be r dim R End ∗ T ( X ) is indep endent of the ring R , in a sense explained in the following lemma. These re sults tog ether justify Theo rem 1.1 . Lemma 4.3. L et T b e an additive Z -c ate gory and let X , Y b e obje cts in T . Supp ose that ther e ar e gr ade d-c ommu t ative rings R and S acting c entr al ly on T such t hat Hom ∗ T ( X, Y ) is in b oth noe th fl ( R ) and no eth fl ( S ) . One then has an e qu ality dim R Hom ∗ T ( X, Y ) = dim S Hom ∗ T ( X, Y ) . Pr o of. Indeed, the g raded tenso r pro duct R ⊗ Z S is a gra ded-commutativ e r ing, and one has natural homomor phisms of g raded rings R → R ⊗ Z S ← S . The c ent ral actions of R and S o n T extend to one of the r ing R ⊗ Z S , and Hom ∗ T ( X, Y ) is in no eth fl ( R ⊗ Z S ). Propo sition 2.7 , a pplied to the preceding homomor phisms, now yields equalities dim R Hom ∗ T ( X, Y ) = dim R ⊗ Z S Hom ∗ T ( X, Y ) = dim S Hom ∗ T ( X, Y ) .  DIMENSIONS OF TRIANGULA TED CA TEGORIES 9 R emark 4.4 . The preceding re s ult sugg e sts that o ne should consider the full sub- category of ob jects X in T with the prop erty that, for some ring R acting centrally on T and all Y ∈ T , one has Hom ∗ T ( X, Y ) ∈ no eth fl ( R ); let us denote it no eth fl ( T ). Arguing as in the pro o f of L e mma 4.3 , it is not difficult to prov e that no eth fl ( T ) is precisely the subc a tegory noeth fl ( Z > 0 ( T )), where Z > 0 ( T ) is the non-negative par t of the g raded ce nter of T . This implies, for instance, that noeth fl ( T ) is a thick sub c ategory of T , and a lso that one has an ‘intrinsic’ notion of dimensio n for o b- jects in this sub categ ory . Th us, one could sta te the main results of this section without inv olving an ‘exter nal’ ring R . In pra ctice, how ever there ar e usually mor e conv e nient choices than Z > 0 ( T ), for a ring R acting cen trally on T . Cohomolo gical functors. Ther e ar e als o versions of Theorem 4.2 whic h a pply to cohomolog ical functor s. In order to ex plain this, let T b e a tria ng ulated catego ry and H : T → Ab a cohomolo gical functor to the ca teg ory o f ab elian groups. Let R be a gr aded-commutativ e ring that acts centrally on T . The graded a belia n group H ∗ ( Y ) = M n ∈ Z H (Σ n Y ) then has a natural str ucture of a graded R - mo dule. Assume that there exists a genera to r G of T such tha t the R -mo dule H ∗ ( G ) is no etherian and the R 0 -mo dule H i ( G ) has finite length for each i . One can c heck, as in Lemma 4.3 , that in this case, fo r any Y ∈ T , the dimension of the R -mo dule H ∗ ( Y ) is finite and indepe ndent of R ; denote it dim H ∗ ( Y ). Theorem 4. 5. L et T b e a triangulate d c ate gory, and assu me that idemp otents in T split. If H is a c ohomolo gic al functor and G a gener ator of T such that the R - mo dule H ∗ ( G ) is no et herian and the R 0 -mo dule H i ( G ) has finite length for e ach i , then one has an ine quality: dim T ≥ dim H ∗ ( Y ) − 1 for e ach Y ∈ T . Sketch of a pr o of. Under the hypotheses of the theorem, the functor H is repr e - sentable; this can be prov ed by an argument similar to that for [ 11 , Theorem 1.3] due to Bonda l and V an den Berg h. The result is th us contained in Theo rem 4.2 .  The following r e sult is a v ariation on Theorems 4.2 a nd 4.5 which might b e useful in so me con texts. The h yp othesis on T holds, fo r example, when it is algebra ic, in the sense of Keller [ 19 ]. Theorem 4.6 . L et T b e a triangulate d c ate gory with functorial m apping c ones. If H is a c ohomolo gic al functor and G a gener ator for T such that H ∗ ( G ) is in no eth fl ( R ) , for some ring R acting c entr al ly on T , t hen one has an ine quality: dim T ≥ dim H ∗ ( Y ) − 1 for e ach Y ∈ T . Sketch of a pr o of. Since T ha s functorial mapping co nes, for ea ch r ∈ R , the con- struction of the Koszul ob ject Y / /r can b e made functor ia l. Thu s the as sign- men t Y 7→ Y / /r defines an exac t functor on T , a nd therefore the assignment Y 7→ H (Σ − 1 Y / /r ) yields a co ho mological functor; le t us denote it H/ /r , with a cav e at that it is a des uspe ns ion of what is introduced in ( 3.1 ). This functor comes equipp e d with a natural tr a nsformation θ r : H / /r → H . Let G b e a genera tor for T , s e t c = dim H ∗ ( G ) − 1 and s = 2 c . Arguing as in the pro o f of Theo rem 3.3 , one can pick a sequence of elements r 1 , . . . , r c such tha t 10 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN r i +1 is filter-reg ular on H ∗ i ( G ), wher e H 0 = H and H i = H i − 1 / /r s i for i ≥ 1 . One th us has natural transfor mations H c θ r c − − → H c − 1 θ r c − 1 − − − → · · · θ r 1 − − → H 0 satisfying, for each i = 1 , . . . , c , the following conditio ns : (1) θ ∗ r i ( G ) : H ∗ i ( G ) → H ∗ i − 1 ( G ) is ev entually zer o; (2) θ ∗ r i ( G/ / r ) : H ∗ i ( G/ / r ) → H ∗ i − 1 ( G/ / r ) is surjective; (3) H ∗ ( G/ / r ) is no t even tually zer o. It now follows from (an analo gue of ) the Ghost Lemma that G/ / r is no t in thick c T ( G ). This implies the desired r esult.  5. Applica tions Let A b e a no etherian ring. In what follows, D b ( A ) denotes the b ounded derived category of finitely generated A -mo dules , with the usual structure of a triangula ted category . F o llowing Buch weitz [ 13 ], the stable derive d c ate gory of A is the ca tegory D b st ( A ) = D b ( A ) / D p er ( A ) , where D p er ( A ) = thic k( A ) denotes the categor y of pe rfect complexes . Her e the quotient is taken in the sense of V erdier; s ee [ 31 ]. It has a structure of a triangula ted category , for which the canonical functor D b ( A ) → D b st ( A ) is exact. R emark 5.1 . The quotient functor D b ( A ) → D b st ( A ) induces a homo morphism o f graded r ings Z ∗ ( D b ( A )) → Z ∗ ( D b st ( A )). Thus a cen tral a ction o f a g raded commu- tative ring R o n D b ( A ) induces a central action on D b st ( A ). In pa rticular, for any pair of c o mplexes X , Y ∈ D b ( A ) the natural map Ext ∗ A ( X, Y ) = Hom ∗ D b ( A ) ( X, Y ) → Hom ∗ D b st ( X, Y ) is one of R -mo dules. Gorenstein rings. A no etheria n ring A is called Gor enstein if A is of finite in- jective dimens io n both as a left mo dule and a right module ov e r itself. I n the commutativ e ca s e, this is more restrictive than the us ua l definition of a Gor e nstein ring; how ever b oth definitions coincide if A has finite Krull dimension. The following result is [ 13 , Corolla ry 6.3 .4]. Lemma 5.2. L et A b e a n o etherian Gor enstein ring. Then for e ach p air of c om- plexes X , Y ∈ D b ( A ) the natur al map Hom n D b ( A ) ( X, Y ) → Hom n D b st ( A ) ( X, Y ) induc e d by t he quotient functor D b ( A ) → D b st ( A ) is bije ctive for n ≫ 0 .  The notio n of complexity of a sequence of non-negative integers was recalled in the par agraph preceding Prop osition 2.6 . W e define the complexity of a pair X, Y of complexes of A -mo dules to be the num be r cx A ( X, Y ) = cx(length Z ( A ) (Ext n A ( X, Y ))) where Z ( A ) denotes the cen ter of A . Example 5.3. Let A b e a n Artin k -alg ebra, and let r denote its radical. Then every finitely generated A -mo dule M a dmits a minimal pr o jectiv e res olution · · · → P 2 → P 1 → P 0 → M → 0 and one defines the co mplexit y of M as cx A ( M ) = cx(leng th k ( P n )) . DIMENSIONS OF TRIANGULA TED CA TEGORIES 11 It is well-known that cx A ( M ) = cx A ( M , A/ r ); see [ 2 , A.1 3 ] or [ 8 , § 5.3] for details. Recall that a ring A is said to b e a no etherian algebr a if there exists a commu- tative no etheria n ring k such A is a k -alge br a and a finitely generated k -mo dule. Theorem 5 .4. L et A b e a n o etherian algebr a which is Gor enstein. L et X ∈ D b ( A ) b e such that Ex t ∗ A ( X, Y ) is in no eth fl ( R ) , for some gr ade d-c ommutative ring R acting c entr al ly on D b ( A ) , and for al l Y ∈ D b ( A ) . One then has ine qualities dim D b ( A ) ≥ dim D b st ( A ) ≥ cx A ( X, X ) − 1 . Pr o of. The inequality on the left ho lds b ecaus e D b st ( A ) is a quo tien t o f D b ( A ). The R -actio n on D b ( A ) induces an action on D b st ( A ) b y Remark 5.1 , and the finiteness condition on X as an ob ject o f D b ( A ) pa sses to the sta ble ca teg ory D b st ( A ) b eca use of Lemma 5.2 . In par ticular, Lemma 5.2 implies the equalit y dim R End ∗ D b ( A ) ( X ) = dim R End ∗ D b st ( A ) ( X ) . Theorem 1.1 now yields the inequa lit y b elow dim D b st ( A ) ≥ dim R End ∗ D b ( A ) ( X ) − 1 = cx A ( X, X ) − 1 . The eq ua lit y follows from Pr op ositions 2.6 and 2.7 , wher e we use that Ext n A ( X, X ) is finitely generated ov er Z ( A ).  Artin algebras. An Artin algebr a is a no ether ian k -alg ebra where the ring k is artinian; equiv a lent ly the center Z ( A ) of A is an artinian commutativ e r ing and A is finitely genera ted as a mo dule ov e r it. Over such r ings the v ar ious finiteness conditions cons idered in this article coincide. Lemma 5. 5. L et A b e an Artin algebr a and X , Y obje cts in D b ( A ) . If the gr ade d mo dule E xt ∗ A ( X, Y ) is in no eth fl ( R ) for some ring R acting c entr al ly on D b ( A ) , then it is no etherian and de gr e ewise of finite length over the ring R ⊗ Z Z ( A ) . Pr o of. It is e a sy to chec k that the Z ( A )-mo dule Ext i A ( X, Y ) ha s finite length for each i . The desired result is a conse q uence of this observ atio n.  F or Artin algebra s we are able to establish a stro nger version of Theo rem 5.4 , where one do es not hav e to assume b efore hand that the r ing is Gorenstein. This is based on the following o bserv atio n. Prop ositio n 5.6. L et A b e an Artin algebr a with r adic al r . If Ext ∗ A ( A/ r , A/ r ) is no etherian over some gr ade d-c ommutative ring acting c ent r al ly on D b ( A ) , t hen A is Gor enstein. Pr o of. Observe that G = A/ r is a genera tor for D b ( A ). Thus, an A -mo dule X has finite injective dimension if and only if Ex t ∗ A ( G, X ) is eventually zero, and X has finite pro jective dimension if and o nly if E xt ∗ A ( X, G ) is even tually zero . Mo reov er, when Ex t ∗ A ( G, G ) is even tually no etherian over some ring R a cting c ent rally on T , then so are E xt ∗ A ( X, G ) and Ext ∗ A ( G, X ). In view of Lemma 2.2 , one ma y a ssume that R = R ev , so Lemma 4.1 yie lds an equa lit y Supp + R Ext ∗ A ( X, G ) = Supp + R Ext ∗ A ( G, X ) . Applying Pr op osition 2.4 , it follows that X ha s finite pr o jectiv e dimension if and only if it ha s finite injective dimension. The dua lity b etw e e n r ight a nd left mo dules then implies that A is Go renstein.  Recall that the Lo ewy length of an Artin algebra A , with r adical r , is the leas t non-negative integer n such that r n = 0; w e denote it ll( A ). 12 BER GH, IYE NGAR, KRAUSE, AND OP PERMANN Corollary 5. 7. L et A b e an Artin algebr a with r adic al r . If Ext ∗ A ( A/ r , A/ r ) is no etherian as a mo dule over some ring acting c entra l ly on D b ( A ) , then ll( A ) ≥ dim D b ( A ) ≥ dim D b st ( A ) ≥ cx A ( A/ r ) − 1 . Pr o of. The first ineq ua lit y holds b ecause thick ll( A ) ( A/ r ) = D b ( A ); see [ 2 6 , Lemma 7.35]. The rest are obtained by combining Pro p o s ition 5.6 and Theore m 5.4 .  R emark 5.8 . In v iew of results of F riedla nder and Suslin [ 18 ], the preceding result applies, in par ticular, to the cas e when A is a co- c ommut ative Hopf alge bra over a field k . In this case, the k -algebra Ext ∗ A ( k , k ) acts on D b ( A ) via the diagonal action. One may sp ecialize further to the case where k is a field of c harac ter istic p and A = k G is the group algebra o f a finite group G . It follows from a theo r em o f Quillen [ 25 ] tha t cx kG ( k ) equals the p - rank of G . Thus, Cor ollary 5.7 yields the following inequalities ll( k G ) ≥ dim D b st ( k G ) ≥ ra nk p ( G ) − 1 . These estimates were first obtained in [ 24 ] using differen t metho ds. R emark 5.9 . W e should like to note that when A is an Artin k -a lgebra which is also pro jective as a k - mo dule, one has a natural first choice for the ring a cting centrally on D b ( A ), namely , the Hochsc hild cohomo logy HH ∗ ( A ) of A ov er k . Suppo se that A is finite dimensional o ver an a lgebraica lly clo sed field k . In [ 17 , § 2 ], Erdmann et al. in tr o duced the following finiteness condition: There is a no etheria n gr aded subalgebra H of HH ∗ ( A ), with H 0 = HH 0 ( A ), suc h that Ext ∗ A ( A/ r , A/ r ) is finitely generated over H . This condition ha s been inv estigated by v ar ious author s, in particular , in connection with the theory of supp ort v ar ieties. The present work and [ 6 ] suggest that no eth fl ( D b ( A )) = D b ( A ) is the appro- priate finiteness co ndition on A ; s ee Remark 4.4 . In particular, the ring R acting centrally on D b ( A ) is not essential, and the emphas is shifts rather to prop erties of Ext ∗ A ( A/ r , A/ r ) alone. While this p o int of view is more general, is seems also to b e techn ically s impler and more flexible. Complete in tersections. F o r a commutativ e lo cal ring A , with maximal ideal m and r e sidue field k = A/ m , the num b er edim A − dim A is called the c o dimension of A , and denoted co dim A ; here edim A is the emb e dding dimension of A , that is to say , the k -vector spa ce dimension o f m / m 2 . The result b elow holds also without the hypothesis that A is co mplete; see [ 5 ]. F or the definition of a co mplete intersection ring, see [ 12 , § 2.3]. Corollary 5.10. L et A b e a c ommutative lo c al ring, c omplete with r esp e ct to the top olo gy induc e d by its maximal ide al. If A is c omplete interse ction, then dim D b ( A ) ≥ dim D b st ( A ) ≥ co dim A − 1 . Pr o of. Set c = co dim A . The hypo theses on A imply that there is a p olynomial ring A [ χ 1 , . . . , χ c ], where the χ i are indeter minates o f degr ee 2, acting centrally on D b ( A ) with the prop erty tha t, for a ny pair of complexes X , Y in D b ( A ), the graded A -mo dule Ext ∗ A ( X, Y ) is finitely generated ov er A [ χ 1 , . . . , χ c ]; see, for instance, [ 6 , § 7.1]. Since A is complete intersection, it is Gorenstein; see [ 12 , Pro po sition 3.1 .2 0]. Hence Theo r em 5.4 applies and, for the residue field k of A , yields inequalities dim D b ( A ) ≥ dim D b st ( A ) ≥ cx A ( k , k ) − 1 . It r e mains to note that cx A ( k , k ) = co dim A , by a result of T a te [ 30 , Theo rem 6].  W e now apply the preceding results to obtain b ounds on the representation dimension of an Artin a lg ebra. DIMENSIONS OF TRIANGULA TED CA TEGORIES 13 Representa tion di mensio n. Let A b e a n Artin a lgebra. The r epr esentation dimension of A is defined a s rep . dim A = inf  gl . dim End A ( M )     M is a generator and a cogenera tor for mod A  . Auslander has proved that A is semi-simple if and only if rep . dim A = 0, and that rep . dim A ≤ 2 if a nd only if A has finite repr e sentation type; see [ 1 ]. The connection betw een this inv ariant and dimensions for triangulated categories is that, when A is not semi-simple, one has an inequality: rep . dim A ≥ dim D b st ( A ) + 2 . This result is con tained in [ 27 , P rop osition 3.7]. With Theo rem 4.2 , it yields a low er b ound for the representation dimension o f Artin algebras: Theorem 5.11. L et A b e an Artin algebr a that is not semi-simple, and let r b e the ra dic al of A . If Ext ∗ A ( A/ r , A/ r ) is no etherian as a mo dule over some gr ade d- c ommutative ring acting c entr al ly on D b ( A ) , then rep . dim A ≥ cx A ( A/ r ) + 1 .  With a further hypothesis that A is self-injective this r esult was prov ed in [ 10 , Theorem 3.2 ]. Arguing as [ 1 0 , Co rollar y 3.5], which a g ain requir ed that A b e self- injectiv e, one obta ins the following r esult r elating the representation dimension of an algebra to the Kr ull dimension of its Hochsc hild c o homology ring. The h yp othesis on A/ r ⊗ k A/ r holds, for example, if k is alg ebraically closed; s e e [ 21 , XVII, 6.4 ]. Corollary 5. 12. L et k b e a field, and A a fin ite dimensional, non semi-simple, k - algebr a with r adic al r , with A/ r ⊗ k A/ r semi-simple. If Ext ∗ A ( A/ r , A/ r ) is no etherian as a mo dule over the Ho chschild c ohomolo gy algebr a HH ∗ ( A ) of A over k , then rep . dim A ≥ dim HH ∗ ( A ) + 1 . Pr o of. Set R = HH ∗ ( A ). Given Theo rem 5.11 and Prop ositio n 2.6 (2), o ne has only to prov e that Spec R = supp R Ext ∗ A ( A/ r , A/ r ) . This holds b ecause the semi- s implicit y of A/ r ⊗ k A/ r implies that the kernel of the natural map R → Ext ∗ A ( A/ r , A/ r ) is nilp otent; see [ 29 , Pr op osition 4.4].  The inequalit y in the preceding result need not hold if A/ r ⊗ k A/ r is not semi- simple, for then the kernel of the homomorphism fro m Ho chsc hild cohomo logy to the gr aded center of the der ived categ ory need not be nilpotent. This is illustrated by the fo llowing exa mple. Example 5.13. Let k b e a field of characteris tic p > 0. Assume that k is not per fect, so that there is an e le men t a ∈ k that ha s no p th r o ot in k . Let A = k [ a 1 /p ], the extension field o btained b y adjoining the p th ro ot of a . Since A is a field, one has rep . dim A = 0 and dim D b ( A ) = 0. 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Petter Andreas Bergh, Institutt for m a temat iske f ag, NTNU, N-7 4 91 Trondheim, Nor w a y E-mail addr ess : bergh@ma th.ntnu.no Srikanth B. Iyengar, Dep ar tment of Ma thema tics, University of Nebraska, Lincoln NE 68588 , U.S.A. E-mail addr ess : iyengar@ math.unl.ed u Henning Krause, Institut f ¨ ur Ma thema tik, Universit ¨ at P aderborn, 33095 P aderborn, Germany. E-mail addr ess : hkrause@ math.upb.de Steffen Oppermann , Institutt for ma tema tiske f ag, NTNU, N-7491 Trondheim, Nor- w ay E-mail addr ess : opperman @math.ntnu. no