Stratifying triangulated categories

STRA TIFYING TRIANGULA TED CA TEGORIES DA VE BENSON, SRIKANTH B. IYENGAR, A ND HEN N ING K R AUSE Abstract. A notion of stratifica tion is introduced for an y compactly gener- ated triangulated category T endo w ed with an action of a graded commutat ive noetherian ring R . The utili t y of th is notion is demonstrated b y establishing dive rse consequenc es which follo w when T i s stratified by R . Among them are a classificat ion of the l ocali zi ng sub categories of T in terms of subsets of the set of prime ideals i n R ; a classification of the thick subcategories of the subcategory of compac t ob jects in T ; and results concerning the support of the graded R -module of morphisms Hom ∗ T ( C, D ) leading to a nalogues of the t ensor product theorem for supp ort v arieties of mo dular representa tion of groups. Contents 1. Int ro duction 1 2. Lo cal cohomolo gy a nd supp ort 4 3. A lo cal-glo bal principle 9 4. Stratification 13 5. Orthogo nality 14 6. Classifying thick s ubc a tegories 18 7. T ensor tr iangulated categories 19 8. F orma l differen tial graded algebr a s 21 References 23 1. Introduction Over the last few decades, the theory of supp ort v arieties ha s play ed an in- creasingly imp or tant role in v arious asp ects of representation theory . The original context w as Carlson’s supp ort v arieties for mo dular repres e n tations of finite groups [12], but the metho d so on spread to r estricted Lie algebras [14], complete intersec- tions in comm utative algebra [1, 2], Hochsc hild cohomological suppor t for cer tain finite dimensional algebra s [13], and finite group schemes [1 5, 16]. One of the themes in this developmen t has b een the clas sification of thick or lo calizing s ub ca tegories of v arious triang ulated catego ries of r epresentations. This story sta rted with Hopkins’ classification [18] of thick sub categor ies of the p erfect complexes o ver a comm utative No etherian ring R , followed by Neeman’s classifica- tion [25] of loca lizing s ub ca tegories of the full der ived ca tegory o f R ; b oth in volv ed 2010 Mathematics Subje ct Cla ssific ation. 18G99 (prim ary); 13D45, 18E30, 20J06, 55P42. Key wor ds and phr ases. lo calizing sub category , th ick subcategory , triangulat ed category , sup- port, lo cal cohomology . The research of the firs t and second authors wa s undertaken during visits to the Universit y of Pa derborn, eac h supported b y a r esearc h prize fr om the Humboldt F oundation. The research of the second author was partly supp orted by NSF grants, DMS 0602498 and DM S 0903493. 1 2 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE a no tion of supp ort for c o mplexes living in the prime ideal spectrum of R . Some- what la ter ca me the classification by Benson, Carlso n and Rick ard [6] of the thick sub c ategories of the stable mo dule catego ry of finite dimensional representation o f a finite group G in terms of the sp ectrum of its coho mology ring. In [8] we established an analogous cla ssification theorem for the loca lizing sub- categorie s of the stable mo dule category of all representations of G . The strategy of pro of is a series of reductions and in volv es a passag e through v ario us other triangulated categories admitting a tensor structure. T o execute this strategy , it was imp ortant to isolate a pro per ty which w ould p e rmit one to classify lo calizing sub c ategories in tensor triangulated categ ories, and could be track ed easily under changes o f catego ries. This is the no tion of str atific ation in tro duce d in [8] for tens o r triangulated categ ories, in spired by work of Hovey , Palmieri, and Stric kland [19]. F or the stable mo dule category of G , this condition yields a para meterization of lo calizing sub catego r ies reminiscent of, and enhancing , Quillen stratification [30] o f the cohomolog y a lgebra of G , whence the name. In this work we present a no tion of stratifica tio n for any compactly generated triangulated catego ry T , and establish a num b er of conse q uences which follow when this proper t y ho lds for T . The con text is that we ar e given an action of a gr aded commutativ e r ing R on T , namely a map fro m R to the gr aded center of T . W e write Spec R for the set of homogeneo us pr ime idea ls of R . In [7] we developed a theo ry of supp ort for ob jects in T , based on a construction of exact functors Γ p : T → T fo r each p ∈ Sp ec R , which are analo g ous to lo cal cohomolo gy functors from commutativ e algebr a . The support of any ob ject X of T is the s e t supp R X = { p ∈ Sp ec R | Γ p X 6 = 0 } . In this pape r , w e in vestigate in detail what is needed in order to classify loca lizing sub c ategories in this general context, in terms of the set Spec R . W e sepa r ate out t w o essential ingredien ts of the proces s of clas s ifying lo calizing sub c ategories . The first is the lo c al-glob al principle : it states that for each o b ject X of T , the lo caliz ing s ubc a tegory ge nerated by X is the sa me as the lo ca lizing sub c ategory genera ted by the set of ob jects { Γ p X | p ∈ Sp ec R } . W e prove that T has this prop erty when, for example, the dimension of Sp ec R is finite. When the loca l-global principle holds for T the problem of cla ssifying lo calizing sub c ategories of T can be ta ckled o ne prime a t a time. This is the conten t of the following res ult, which is pa rt of Prop o sition 3 .6 . Theorem 1. 1. When t he lo c al-glob al principle holds for T ther e is a one-to-one c orr esp ondenc e b etwe en lo c alizing sub c ate gories of T and functions assigning t o e ach p ∈ Spec R a lo c alizi ng s u b c ate gory of Γ p T . The function c orr esp onding to a lo c al- izing su b c ate gory S sends p to S ∩ Γ p T . The sec o nd ingredient is that in go o d s ituations the sub categ ory Γ p T , which consists of ob jects supp orted at p , is either zero o r co nt ains no prop er lo ca liz ing sub c ategories . If this pr op erty holds for ea ch p and the lo cal-global pr inc iple holds, then we say T is str atifie d b y R . In this case, the map in Theo r em 1 .1 gives a o ne-to- one co rresp o ndence b etw ee n lo calizing subcatego ries o f T and subsets of supp R T , which is the s et of primes p such that Γ p T 6 = 0; see Theorem 4.2. W e draw a num b er of further consequences of stratification. The b est statements are av ailable when T , in addition to b e b eing stratified by R , is no etherian , meaning that the R -mo dule End ∗ T ( C ) is finitely gener ated for each compact o b ject C in T . STRA TIFYING TRIANGULA TED CA TEGORIES 3 Theorem 1 .2. If T is no etherian and st r atifie d by R , then the map describ e d in The or em 1.1 gives a one-to-one c orr esp ondenc e b etwe en the thick sub c ate gories of the c omp act obje cts in T and the sp e cialization close d subsets of supp R T . This result is a rewording of Theo rem 6.1 and can be deduced fr om the classifi- cation o f localizing subca teg ories o f T , using an argument due to Neeman [25]. W e give a differe nt pro o f based on the follo wing result, which is Theorem 5.1. Theorem 1. 3. If T is no etherian and st r atifie d by R , then for e ach p air of c omp act obje cts C, D in T ther e is an e quality supp R Hom ∗ T ( C, D ) = supp R C ∩ supp R D . When in addition R i = 0 holds for i < 0 , one has Hom n T ( C, D ) = 0 for n ≫ 0 if and only if Hom n T ( D , C ) = 0 for n ≫ 0 . The statement of this theorem is inspir ed b y an analogous statemen t for mo d- ules o ver complete in ter section local rings, due to Avramov and Buch weitz [2]. A stratification theorem is not yet a v a ilable in this context; see ho wever [2 1]. The s tratification condition also implies that Rav enel’s ‘telescop e conjecture’ [31], sometimes called the ‘smashing conjecture’, holds for T . Theorem 1 . 4. If T is no etherian and st r atifie d by R and L : T → T is a lo c alization functor that pr eserves arbitr ary c opr o ducts, t hen the lo c alizing sub c ate gory K er L is gener ate d by obje cts that ar e c omp act in T . This result is c ontained in T he o rem 6.3, whic h establishes a lso a cla ssification o f lo calizing subcatego ries of T that are also closed under products. Another applica- tion, Cor ollary 5.7, addr e sses a question o f Rick ard. If S is a localiz ing sub ca tegory of T , write ⊥ S fo r the full subcatego ry of ob jects X s uch that there are no nonzero morphisms from X to any ob ject in S . Theorem 1.5. Supp ose t hat T is no etherian and st r atifie d by R , and that S is a lo c alizing sub c ate gory of T . Then ⊥ S is the lo c alizi ng su b c ate gory c orr esp onding t o the set of primes { p ∈ Sp ec R | V ( p ) ∩ supp R S = ∅ } . In Section 7 we consider the ca se when T has a structure o f a tensor triangulated category co mpatible with the R -action, and discuss a notio n of stratification suit- able for this con text. A noteworthy fea tur e is tha t the analogue of the lo cal-glo bal principle a lwa ys holds, so s tratification concerns o nly whether each Γ p T is minimal as tensor ideal lo calizing sub categor y . When this prop erty holds one has the fol- lowing analog ue of the tensor pro duct theorem of mo dular repres ent ation theory as describ ed in [5, Theorem 10.8 ]; cf. also Theorem 1.3. Theorem 1.6. L et T b e a tensor triangulate d c ate gory with a c anonic al R - action. If R str atifies T , t hen fo r any obje cts X , Y in T ther e is an e quality supp R ( X ⊗ Y ) = supp R X ∩ supp R Y . This result reappe a rs as Theo rem 7.3. One can establish analogues o f other re- sults discussed ab ov e for tensor triangulated categor ies, but w e do not do so; the arguments requir ed are the same, and in any ca se, many of these results app ear already in [8 ], at leas t for tria ngulated categories a s so ciated to mo dular represen- tations of finite gro ups. Most examples of stratified tr ia ngulated categories that appea r in this work are impo rted fro m elsewhere in the litera ture. The one exception is the derived ca tegory of differen tial graded modules over any g raded-co mmutative no etherian ring A . In 4 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE Section 8 we verify that this triangulated c ategory is s tratified by the cano nical A - action, building on arguments from [8, § 5] whic h dealt with the case A is a gr aded po lynomial algebr a ov er a field. There are interesting cla sses of triangulated catego ries whic h cannot b e stratified via a ring action, in the se nse explained ab ove; see Example 4 .6. On the other hand, there a re important contexts where it is reaso nable to exp ect stratification, notably , mo dules ov er coco mmu tative Hopf a lgebras and mo dules over the Steenro d algebra, where analogues of Quillen stratification ha ve b een proved by F riedlander and Pevtsov a [15] and Palmieri [29] r e sp ectively . One goal of [7] and the present work is to pave the wa y to suc h results. Ac knowledgmen ts. It is our pleasure to tha nk Zhi-W ei Li for a critica l reading of an earlier version of this man uscript. 2. Local cohomology and suppor t The foundation for this a rticle is the work in [7] where we co nstructed analo gues of lo cal cohomolo gy functors a nd supp o r t fr o m commutativ e algebr a for triangu- lated catego ries. In this section we further develop these ideas , as r equired, and along the wa y recall basic notions and cons tructions from op. cit. Henc eforth R denotes a gr ade d-c ommutative no et herian ring and T a c omp actly gener ate d R -line ar triangulate d c ate gory with arbitr ary c opr o ducts. W e b egin by explaining what this means. Compact generation. Let T b e a tria ngulated categ ory admitting arbitrary co- pro ducts. A lo c alizing sub c ate gory of T is a full triangulated subcateg ory that is closed under taking copr o ducts. W e write Lo c T ( C ) for the smallest lo c a lizing sub c ategory co nt aining a given c la ss of ob jects C in T , and call it the lo calizing sub c ategory gener ate d by C . An ob ject C in T is c omp act if the functor Hom T ( C, − ) commutes with all copro ducts; we write T c for the full sub catego ry of compact o b jects in T . The category T is c omp actly gener ate d if it is gener ated b y a set of compact ob jects. W e rec a ll some facts concerning lo c a lization functors; see, for example, [7, § 3 ]. Lo calization. A lo c alization functor L : T → T is an exa ct functor that admits a na tural transfor mation η : Id T → L , called adjunction , such that L ( η X ) is an isomorphism and L ( η X ) = η ( LX ) for a ll o b jects X ∈ T . A lo calizatio n functor L : T → T is essentially uniquely determined by the corresp onding full s ubca tegory Ker L = { X ∈ T | L X = 0 } . This means tha t if L ′ is a lo calization functor with Ker L ⊆ K er L ′ and η ′ is its adjunction, then there is a unique morphism ι : L → L ′ such that ιη = η ′ . Given such a loca lization functor L , the natura l trans formation Id T → L induces for eac h ob ject X in T a natural exact lo c alization triangle Γ X − → X − → L X − → This exact triangle gives rise to an exact functor Γ : T → T with Ker L = Im Γ and Ker Γ = Im L . Here Im F , for any functor F : T → T , denotes the essential image : the full sub- category of T formed by ob jects { X ∈ T | X ∼ = F Y for some Y in T } . STRA TIFYING TRIANGULA TED CA TEGORIES 5 The next lemma pro vides the existence of loca lization functors with respect to a fixed lo caliz ing sub categor y; see [26, Theo rem 2.1] for the sp e cial case that the lo calizing sub catego ry is generated b y co mpact ob jects. Lemma 2.1 . L et T b e a c omp actly gener ate d t riangulate d c ate gory. I f a lo c alizing sub c ate gory S of T is gener ate d by a set of obje cts, then ther e exists a lo c alization functor L : T → T with Ker L = S . Pr o of. In [28, Co rollary 4.4.3] it is sho wn that the collection o f morphisms b etw een each pair of ob jects in the V er dier quotien t T / S form a set. The quotient functor Q : T → T / S preserves copro ducts, and a s tandard argument based on Brown’s representabilit y theorem [23, 27] yields an exac t rig h t adjoin t Q ρ . Note that Q ρ is fully faithful; s ee [17, Pr op osition I.1.3]. It follows that the co mpo site L = Q ρ Q is a lo calization functor satisfying Ker L = S ; see [7, Lemma 3.1].  Cen tral ring actions. Let R b e a graded- c ommut ative r ing; th us R is Z -graded and satisfies r · s = ( − 1) | r || s | s · r for each pair o f ho mogeneous elemen ts r , s in R . W e say that the triangulated ca tegory T is R -line ar , o r that R acts on T , if there is a homomor phis m R → Z ∗ ( T ) of gra ded rings, where Z ∗ ( T ) is the graded cen ter of T . In this case, for a ll ob jects X , Y ∈ T the gra ded abelian gro up Hom ∗ T ( X, Y ) = M i ∈ Z Hom T ( X, Σ i Y ) carries the structure of a gr aded R -mo dule. Supp ort. F r om no w on, R deno tes a graded-co mm utative noetheria n ring and T a compactly generated R -linear triangulated category with arbitra r y copro ducts. W e write Spec R for the set o f homogeneous prime ideals of R . Given a homo- geneous ideal a in R , we set V ( a ) = { p ∈ Sp ec R | p ⊇ a } . Let p b e a p oint in Sp ec R and M a graded R -mo dule. W e write M p for the homogeneous lo ca lization of M at p . When the natura l map of R -mo dules M → M p is bijectiv e M is s aid to be p -lo cal. This condition is equiv alent to : supp R M ⊆ { q ∈ Spec R | q ⊆ p } , where supp R M is the supp or t of M . The mo dule M is p - torsion if each e le ment of M is annihilated b y a power of p ; equiv alen tly , if supp R M ⊆ V ( p ); see [7, § 2] for pro ofs of these as sertions. The sp e cializatio n closur e o f a subset U of Sp ec R is the set cl U = { p ∈ Sp ec R | there exists q ∈ U with q ⊆ p } . The subset U is sp e cialization close d if cl U = U ; eq uiv alently , if U is a unio n of Zariski c lo sed subsets o f Sp ec R . F or each sp ecializa tio n closed subset V of Spec R , we define the full s ubca tegory of T of V - torsion ob jects as follows: T V = { X ∈ T | Hom ∗ T ( C, X ) p = 0 for all C ∈ T c , p ∈ Sp ec R \ V } . This is a localizing sub categ ory and there exis ts a localiza tion functor L V : T → T such that K er L V = T V ; see [7, Lemma 4.3, Pr op osition 4.5]. F o r each ob ject X in T the adjunction morphism X → L V X induces the exact lo caliza tion triangle (2.2) Γ V X − → X − → L V X − → . This exa ct triangle giv e s r ise to an exact lo c al c ohomolo gy functor Γ V : T → T . In [7] we established a num b er of pr op erties o f these functors, to facilitate working with them. W e single out one that is used frequen tly in this w o rk: They commute with all copro ducts in T ; see [7, Co rollar y 6.5]. 6 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE F or each p in Spec R set Z ( p ) = { q ∈ Spe c R | q 6⊆ p } , s o V ( p ) \ Z ( p ) = { p } , and X p = L Z ( p ) X for each X ∈ T . The notation is justified by the next result whic h enhances [7, Theorem 4.7]. Prop ositi o n 2.3. L et p b e a p oint in Sp ec R and X , Y obje cts in T . The R -mo du les Hom ∗ T ( X, Y p ) and Hom ∗ T ( X p , Y ) ar e p -lo c al, so the adjunction morphism Y → Y p induc es a unique homomo rphism of R -mo dules Hom ∗ T ( X, Y ) p − → Hom ∗ T ( X, Y p ) . This map is an isomorphism if X is c omp act. Pr o of. The last ass ertion in the sta temen t is [7, Theorem 4 .7]. It implies that the R - mo dule Hom ∗ T ( C, Y p ) is p - lo cal for each compact ob ject C in T . It then follows that Hom ∗ T ( X, Y p ) is p -lo cal for each ob ject X , since X is in the lo calizing sub categ ory generated by the compact ob jects, a nd the sub category of p -lo cal mo dules is clo s ed under taking pro ducts, kernels, cok ernels and extensions; see [7, Lemma 2.5]. A t this p oint w e know that End ∗ T ( X p ) is p -lo cal, and hence so is Hom ∗ T ( X p , Y ), since the R -actio n on it factors through the homomorphism R → E nd ∗ T ( X p ).  Consider the exact functor Γ p : T → T obtained by setting Γ p X = Γ V ( p ) ( X p ) . for e ach ob ject X in T . The essential image of the functor Γ p is denoted by Γ p T , and an o b ject X in T b elong s to Γ p T if and only if Hom ∗ T ( C, X ) is p -lo cal and p -torsio n for every compact ob ject C ; see [7, Corollar y 4.10]. F rom this it follows that Γ p T is a lo calizing sub categor y . The supp ort of an ob ject X in T is a subset of Sp ec R defined as follows: supp R X = { p ∈ Sp ec R | Γ p X 6 = 0 } . In addition to pro p erties of the functor s Γ V and L V , and suppo rt, given in [7], we req uire also the following ones. Lemma 2.4. L et V ⊆ Sp ec R b e a sp e cialization close d subset and p ∈ Sp ec R . Then for e ach obje ct X in T one has Γ p ( Γ V X ) ∼ = ( Γ p X when p ∈ V , 0 otherwise, and Γ p ( L V X ) ∼ = ( Γ p X when p 6∈ V , 0 otherwise. Pr o of. Apply the exact functor Γ p to the exa ct triangle Γ V X → X → L V X → . The assertion then follows from the fa c t that either Γ p ( L V X ) = 0 (and this happ ens precisely when p ∈ V ) or Γ p ( Γ V X ) = 0 ; see [7, Theorem 5.6].  F urther res ults inv olve a useful constr uction from [7, 5.10]. Koszul ob jects. Let r ∈ R b e a homogeneous element of degr e e d and X a n ob ject in T . W e denote X/ /r any ob ject that appear s in an exact triangle (2.5) X r − → Σ d X − → X/ /r − → and ca ll it a Koszul obje ct of r on X ; it is well defined up to (nonunique) iso- morphism. Given a homo geneous ideal a in R we write X/ / a for any Koszul o b ject obtained by iterating the constr uction ab ove with resp ect to so me finite sequence o f generator s for a . This ob ject may de p end on the choice of the genera ting sequence for a , but one has the following uniqueness statement ; see als o Prop osition 2.1 1(2). STRA TIFYING TRIANGULA TED CA TEGORIES 7 Lemma 2.6. L et a b e a homo genous ide al in R . Each obje ct X in T satisfies supp R ( X/ / a ) = V ( a ) ∩ supp R X . Pr o of. W e verify the claim for a = ( r ); an ob vious itera tion gives the gener al result. Fix a p oint p in Sp ec R a nd a compact ob ject C in T . Applying the exact functor Γ p to the exa ct triangle (2.5), and then the functor Hom ∗ T ( C, − ) results in a n exact sequence of R - mo dules Hom ∗ T ( C, Γ p X ) ∓ r − → Hom ∗ T ( C, Γ p X )[ d ] − → − → Hom ∗ T ( C, Γ p ( X/ /r )) − → Ho m ∗ T ( C, Γ p X )[1 ] ± r − → Hom ∗ T ( C, Γ p X )[ d + 1] . Set H = Hom ∗ T ( C, Γ p X ). The R -mo dule H is p -lo cal a nd p - torsion, se e [7, Cor ollary 4.10], and this is us ed as follows. If Hom ∗ T ( C, Γ p ( X/ /r )) 6 = 0 holds, then H 6 = 0 and r ∈ p since H is p -lo cal. On the other ha nd, H 6 = 0 a nd r ∈ p implies tha t Hom ∗ T ( C, Γ p ( X/ /r )) 6 = 0 since H is p - to rsion. This implies the desired equality .  The result b elow is [8, Prop o sition 3 .5], ex cept that ther e G is a ssumed to consist of a single ob ject. The ar gument is how e ver the same, so w e omit the pro of. Prop ositi o n 2.7. L et G b e a set of c omp act obje cts which gener ate T , and let V b e a sp e cializatio n close d s u bset of Spec R . F or any de c omp osition V = S i ∈ I V ( a i ) wher e e ach a i is an ide al in R , ther e ar e e qualities T V = Lo c T ( { C / / a i | C ∈ G , i ∈ I } ) = Lo c T ( { Γ V ( a i ) C | C ∈ G , i ∈ I } ) .  An elemen t r ∈ R d is in vertible on an R -mo dule M if the map M r − → M [ d ] is an isomorphism. In the same v ein, we say r is invertible on an ob ject X in T if the natural morphism X r − → Σ d X is an isomo rphism; equiv alently , if X/ /r is zero. Lemma 2.8. L et X b e an obje ct in T and V ⊆ Sp ec R a sp e cialization close d subset. Each element r ∈ R with V ( r ) ⊆ V is invertible on L V X , and henc e on the R -mo dules Hom ∗ T ( L V X , Y ) and Hom ∗ T ( Y , L V X ) , for any obje ct Y in T . Pr o of. F r om [7, Theore m 5.6] and Lemma 2.6 one gets equa lities supp R L V ( X/ /r ) = V ( r ) ∩ s upp R X ∩ (Sp ec R \ V ( r )) = ∅ . Therefore L V ( X/ /r ) = 0, b y [7, Theo r em 5.2]. Applying L V to the exact trian- gle (2.5) yields an isomo rphism L V X r − → Σ | r | L V X , which is the first part o f the statement. Applying Hom ∗ T ( − , Y ) and Hom ∗ T ( Y , − ) to it giv e s the second pa r t.  Homotopy colimits. Let X 1 f 1 − → X 2 f 2 − → X 3 f 3 − → · · · b e a sequence of morphisms in T . Its homotopy c olimit , denoted ho co lim X n , is defined by an exact triangle M n > 1 X n θ − → M n > 1 X n − → ho colim X n − → where θ is the map (id − f n ); see [11]. Now fix a homogene o us ele ment r ∈ R of deg ree d . F or eac h X in T and each int eger n set X n = Σ nd X and consider the co mmuting diagram X r   X r 2   X r 3   · · · X 1   r / / X 2   r / / X 3   r / / · · · X/ /r / / X/ /r 2 / / X/ /r 3 / / · · · 8 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE where eac h v ertica l sequence is given b y the exact triangle defining X/ /r n , and the morphisms in the last row are the (non-cano nical) o ne s induced by the co mm utativ- it y of the upp er s quares. The g is t of the next re s ult is that the homotopy colimits of the horizontal seq uences in the diagra m compute L V ( r ) X and Γ V ( r ) X . Prop ositi o n 2.9. L et r ∈ R b e a homo gene ous element of de gr e e d . F or e ach X in T t he adjunction morphisms X → L V ( r ) X and Γ V ( r ) X → X induc e isomorphisms ho colim X n ∼ − → L V ( r ) X and ho colim Σ − 1 ( X/ /r n ) ∼ − → Γ V ( r ) X . Pr o of. Applying the functor Γ V ( r ) to the middle ro w of the dia gram ab ove y ields a sequence of morphisms Γ V ( r ) X 1 → Γ V ( r ) X 2 → · · · . F or each compact ob ject C in T , this induces a sequence of mor phisms of R -mo dules Hom ∗ T ( C, Γ V ( r ) X 1 ) g 1 − → Hom ∗ T ( C, Γ V ( r ) X 2 ) g 2 − → · · · Each R -mo dule Hom ∗ T ( C, Γ V ( r ) X n ) is ( r )-tors io n and, identifying this module with Hom ∗ T ( C, Γ V ( r ) X )[ nd ], the map g n is given by multiplication with r . T hus the colimit of the sequence ab ov e, in the category of R -mo dules, satisfies : (2.10) colim Hom ∗ T ( C, Γ V ( r ) X n ) = 0 Applying the functor L V ( r ) to the canonica l mo rphism φ : X → ho co lim X n yields the following co mmutative squa r e. X φ / / ηX   ho colim X n η ho coli m X n   L V ( r ) X L V ( r ) φ / / L V ( r ) ho colim X n The mo r phism η ho colim X n is a n is omorphism since Γ V ( r ) ho colim X n = 0. The equality ho lds b eca use, for ea ch compa ct ob ject C , there is a chain of is omorphisms Hom ∗ T ( C, Γ V ( r ) ho colim X n ) ∼ = Hom ∗ T ( C, ho c olim Γ V ( r ) X n ) ∼ = colim Hom ∗ T ( C, Γ V ( r ) X n ) ∼ = 0 where the second one holds b ecause C is co mpact and the last one is b y (2.1 0). On the other hand, L V ( r ) φ is an isomorphis m, since L V ( r ) ho colim X n ∼ = ho colim L V ( r ) X n and r is in vertible on L V ( r ) X , by Lemma 2.8. Thus hocolim X n ∼ = L V ( r ) X . Now consider the canonical morphism ψ : ho colim Σ − 1 ( X/ /r n ) → X . Applying the functor Γ V ( r ) to it yields a commutativ e square: Γ V ( r ) ho colim Σ − 1 ( X/ /r n ) θ ho colim Σ − 1 ( X/ /r n )   Γ V ( r ) ψ / / Γ V ( r ) X θ X   ho colim Σ − 1 ( X/ /r n ) ψ / / X By [7, L emma 5.11 ], each X/ /r n is in T V ( r ) and hence so is ho colim Σ − 1 ( X/ /r n ). Thu s the morphism θ ho colim Σ − 1 ( X/ /r n ) is an iso morphism. It remains to show that Γ V ( r ) ψ is an iso morphism; equiv alently , that the map Hom ∗ T ( C, Γ V ( r ) ψ ) is an isomorphism for each co mpact ob ject C . STRA TIFYING TRIANGULA TED CA TEGORIES 9 The exact triangle X → X n → X/ /r n → induces an exa ct sequence of R - mo dules: Hom ∗ T ( C, Σ − 1 Γ V ( r ) X n ) − → Hom ∗ T ( C, Σ − 1 ( Γ V ( r ) X/ /r n )) − → − → Hom ∗ T ( C, Γ V ( r ) X ) − → Hom ∗ T ( C, Γ V ( r ) X n ) In view of (2.10), pass ing to their colimits yields that Hom ∗ T ( C, Γ V ( r ) ψ ) is an iso- morphism, as desired.  Prop ositi o n 2 .11. L et a b e an ide al in R . F or e ach obje ct X in T the fol lowing statements hold : (1) X/ / a is in Thic k T ( Γ V ( a ) X ) and Γ V ( a ) X is in Lo c T ( X/ / a ) ; (2) Lo c T ( X/ / a ) = Lo c T ( Γ V ( a ) X ) ; (3) Γ V ( a ) X and L V ( a ) X ar e in Lo c T ( X ) . Pr o of. (1) By construction X/ / a is in Thick T ( X ). As Γ V ( a ) is an e x act functor , one obtains that Γ V ( a ) ( X/ / a ) is in Thic k T ( Γ V ( a ) X ). This justifies the first claim in (1), since X/ / a is in T V ( a ) by [7, Lemma 5 .11]. Now we v er ify that Γ V ( a ) X is in the lo calizing subcatego ry generated by X/ / a . Consider the case where a is gener ated b y a sing le elemen t, say a . Claim : X / /a n is in Thick T ( X/ /a ), for each n ≥ 1. Indeed, this is clear for n = 1. F or any n ≥ 1, the comp osition of maps X a n − → Σ n | a | X a − → Σ ( n +1) | a | X yields, b y the o ctahedr al axiom, an exact triangle X/ /a n − → X/ /a n +1 − → Σ n | a | X/ /a − → . Thu s, when X/ /a n is in Thick T ( X/ /a ), so is X/ /a n +1 . This justifies the claim. It follo ws from Prop os ition 2.9 that Γ V ( a ) X is the homoto py colimit of ob jects Σ − 1 X/ /a n , and hence in Lo c T ( X/ /a ), by the claim above. Now suppose a = ( a 1 , . . . , a n ), and set a ′ = ( a 1 , . . . , a n − 1 ). Then the equality V ( a ) = V ( a 1 ) ∩ V ( a ′ ) yields Γ V ( a ) = Γ V ( a n ) Γ V ( a ′ ) by [7, Prop osition 6.1]. By induction on n one may assume that Γ V ( a ′ ) X is in Lo c T ( X/ / a ′ ). Therefo re Γ V ( a ) X is in Lo c T ( Γ V ( a n ) ( X/ / a ′ )). The basis of the induction implies that Γ V ( a n ) ( X/ / a ′ ) is in the lo calizing sub catego ry genera ted b y ( X/ / a ′ ) / /a n , that is to say , by X/ / a . Therefore, Γ V ( a ) X is in Lo c T ( X/ / a ), as claimed. (2) is an immediate consequence of (1). (3) Since X/ / a is in Thick T ( X ), it follows from (1) that Γ V ( a ) X is in Lo c T ( X ). The lo calizatio n triangle (2.2) then yields that L V ( a ) X is als o in Loc T ( X ).  3. A local-global principle W e in tro duce a lo cal-global princ iple for T and explain ho w, when it holds, the pro blem of classifying the lo calizing subc a tegories can be reduced to one of classifying lo calizing sub catego ries suppor ted at a single po int in Spec R . Recall that T is a compactly gener ated R -linea r triang ulated catego ry . If for each o b ject X in T ther e is an equality Lo c T ( X ) = Lo c T ( { Γ p X | p ∈ Sp ec R } ) we say that the lo c al-glob al princip le holds for T . Theorem 3.1. L et T b e a c omp actly gener ate d R -line ar triangulate d c ate gory. The lo c al-glob al principle is e quivalent to e ach of the following st atemen ts. 10 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE (1) F or any X ∈ T and any lo c alizing s u b c ate gory S of T , one has X ∈ S ⇐ ⇒ Γ p X ∈ S fo r e ach p ∈ Spec R . (2) F or any X ∈ T , one has X ∈ Lo c T ( { Γ p X | p ∈ Sp e c R } ) . (3) F or any X ∈ T and any sp e cializatio n close d subset V of Sp e c R , one has Γ V X ∈ Lo c T ( { Γ p X | p ∈ V } ) . (4) F or any X ∈ T , one has Loc T ( X ) = Lo c T ( { X p | p ∈ Sp ec R } ) . (5) F or any X ∈ T and any lo c alizing s u b c ate gory S of T , one has X ∈ S ⇐ ⇒ X p ∈ S for e ach p ∈ Sp ec R . (6) F or any X ∈ T , one has X ∈ Lo c T ( { X p | p ∈ Sp ec R } ) . The pro of uses some results which ma y also b e useful elsewher e. Lemma 3.2. L et X b e an obje ct in T . Su pp ose that for any sp e cialization close d subset V of Spe c R , one has Γ V X ∈ Lo c T ( { Γ p X | p ∈ V } ) . Then Γ V X and L V X b elong to Lo c T ( X ) for every sp e ciali zation close d V ⊆ Sp ec R . Pr o of. It suffices to prov e that Γ p X is in Lo c T ( X ) for each p , that is to say , that the set U = { p ∈ Sp ec R | Γ p X 6∈ L o c T ( X ) } is empt y . Assume U is not empt y and choose a maximal element, say p , with resp ect to inclusion. This is possible since R is no etheria n. Set W = V ( p ) \ { p } , and consider the lo calization tria ng le Γ W X − → Γ V ( p ) X − → Γ p X − → of Γ V ( p ) X with resp ect to W . The h y po thesis implies the first inclusio n below Γ W X ∈ Lo c T ( { Γ q X | q ∈ W } ) ⊆ L o c T ( X ) , and the second one follows from the choice of p . The ob ject Γ V ( p ) X is in Lo c T ( X ), by Pro p osition 2.1 1, so the exact triang le ab ov e y ields tha t Γ p X is in Lo c T ( X ). This contradicts the c hoice of p , and hence U = ∅ , as desir ed.  Finite dim ension. The di mension o f a subset U of Spec R , denoted dim U , is the supremum of all int egers n such that there exis ts a c hain p 0 ( p 1 ( · · · ( p n in U . The set U is called discr ete if dim U = 0 . Prop ositi o n 3. 3. Le t X b e an obje ct of T and set U = s upp R X . If U is discr ete, then ther e ar e natur al isomo rphisms X ∼ ← − a p ∈U Γ V ( p ) X ∼ − → a p ∈U Γ p X . Pr o of. Arguing as in the pr o of o f [7, Theorem 7 .1] one gets that the morphisms Γ V ( p ) X → X induce the iso morphism on the left, in the s ta tement ab ov e. The isomorphism on the r ight holds since fo r eac h p ∈ U the morphism Γ V ( p ) X → Γ p X is an isomorphis m by Lemma 2.4.  Theorem 3.4. L et T b e a c omp actly gener ate d R -line ar triangulate d c ate gory and X an ob je ct of T . If dim supp R X < ∞ , then X is in Lo c T ( { Γ p X | p ∈ supp R X } ) . Pr o of. Set U = supp R X and S = Lo c T ( { Γ p X | p ∈ U } ). The pro o f is an induction on n = dim U . The cas e n = 0 is co vered by Pro p osition 3.3. F or n > 0 set U ′ = U \ min U , where min U is the set of minimal elements with resp ect to inclusion in U , and set V = cl U ′ . It follows from Lemma 2.4 that s upp R Γ V X = U ′ . Since dim U ′ = dim U − 1, the induction hypothesis y ie lds that Γ V X is in S . O n the other hand, STRA TIFYING TRIANGULA TED CA TEGORIES 11 supp R L V X = min U is discre te and therefore L V X belo ngs to S b y P rop osition 3.3 and Lemma 2.4. Thus X is in S , in view of the lo ca liz ation triangle (2.2).  Pro of of Theorem 3.1. It is eas y to chec k that the lo cal- global principle is equiv- alent to (1). Also, the implications (1) = ⇒ (2) and (4 ) ⇐ ⇒ (5) = ⇒ (6) a re obvious. (2) = ⇒ (3): Fix X ∈ T and a sp e cialization closed subset V o f Sp ec R . Then Γ V X ∈ Lo c T ( { Γ p Γ V X | p ∈ Sp ec R } ) = Lo c T ( { Γ p X | p ∈ V } ) hold, where the last equality follo ws from Lemma 2.4. (3) = ⇒ (1): Since Γ p = Γ V ( p ) L Z ( p ) , it fo llows from condition (3) and Lemma 3 .2 that Γ p X is in L o c T ( X ). This implies Loc T ( X ) ⊇ Lo c T ( { Γ p X | p ∈ Spec R } ) and the reverse inclusion holds by condition (3) for V = Spec R . Th us the loca l-global principle, which is equiv alen t to condition (1), holds. (3) = ⇒ (4): W e have Γ p X = Γ V ( p ) X p ∈ Lo c T ( X p ) for each pr ime idea l p by Prop os itio n 2.1 1 a nd hence the hypothesis implies X ∈ Lo c T ( { X p | p ∈ Sp e c R } ). On the other hand, X p ∈ Lo c T ( X ) for ea ch prime ideal p by Lemma 3.2. (6) = ⇒ (2): Fix X ∈ T . F or every pr ime ideal p , one has, for exa mple from Lemma 2.4, that supp R X p is a subset of { q ∈ Sp ec R | q ⊆ p } . In particula r, it is finite dimensional, s ince R is no etheria n, so X p ∈ Lo c T ( { Γ q X p | q ∈ Spec R } ) holds, b y Theorem 3.4. Thus X ∈ Lo c T ( { X p | p ∈ Sp ec R } ) ⊆ L o c T ( { Γ q X p | p , q ∈ Sp ec R } ) = Lo c T ( { Γ q X | q ∈ Sp ec R } ) , where the last equality follo ws fro m Lemma 2.4.  The result b elow is a n immediate consequence of Theor ems 3.4 and 3.1. Corollary 3. 5. When dim Sp ec R is fin ite the lo c al-glob al principle holds for T .  Classifying lo calizing sub categories. Loca lizing s ubca tegories of T are related to subsets of V = supp R T via the following maps  Lo calizing sub c ategories of T  σ / / τ o o ( F amilies ( S ( p )) p ∈V with S ( p ) a lo calizing subc a tegory of Γ p T ) which ar e defined b y σ ( S ) = ( S ∩ Γ p T ) p ∈V and τ ( S ( p )) p ∈V = Lo c T  S ( p ) | p ∈ V  . The next result expresses the lo cal-g lobal principle in terms of these maps. Prop ositi o n 3.6. The fol lowing c onditions ar e e quivalent. (1) The lo c al-glob al principle ho lds for T . (2) The map σ is bij e ctive, with inverse τ . (3) The map σ is one-to-one. Pr o of. W e rep eatedly use the fact that Γ p is an exact functor pr eserving copro ducts. F or each loca lizing subcateg ory S of T and each p in Spec R there is an inclusion (3.7) S ∩ Γ p T ⊆ Γ p S . W e claim that στ is the iden tity , that is to s ay , tha t for an y family ( S ( p )) p ∈V of lo calizing sub categor ies with S ( p ) ⊆ Γ p T the lo calizing sub c ategory generated by all the S ( p ), call it S , satisfies S ∩ Γ p T = S ( p ) , for ea ch p ∈ V . T o see this, note tha t Γ p S = S ( p ) holds, since Γ p Γ q = 0 when p 6 = q . Hence (3.7) yields an inclusion S ∩ Γ p T ⊆ S ( p ). The reverse inclusion is ob vious. 12 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE (1) = ⇒ (2): It suffices to sho w that τ σ equals the identit y , since σ τ = id holds. Fix a loca liz ing sub ca tegory S of T . It is clear that τ σ ( S ) ⊆ S . As to the reverse inclusion: Fixing X in S , it follows from Theo r em 3.1(1) that Γ p X is in S ∩ Γ p T and hence in τ σ ( S ), for each p ∈ Spe c R . Thus, X is in τ σ ( S ), again by Theorem 3.1(1 ). (2) = ⇒ (3): Clear. (3) = ⇒ (1): Since σ τ = id and σ is one-to- o ne, one g e ts τ σ = id. F or each ob ject X in T there is thus an equality: Lo c T ( X ) = Lo c T ( { Lo c T ( X ) ∩ Γ p T | p ∈ Sp ec R } ) ⊆ Lo c T ( { Γ p X | p ∈ Sp ec R } ) The inclusion follows from (3.7). Now apply Theorem 3.1.  The lo cal-global pr inciple fo c uses attention on the sub categor y Γ p T . Next we describ e some of its pro pe r ties, ev e n though these a re not needed in the sequel. Lo cal structure. Let p b e a po int in Sp ec R . In a nalogy with the cas e of R - mo dules, we sa y that an ob ject X in T is p -lo c al if supp R X ⊆ { q ∈ Sp ec R | q ⊆ p } and that X is p - torsion if supp R X ⊆ { q ∈ Spe c R | q ⊇ p } . The ob jects of Γ p T are pr ecisely those that are both p -lo cal and p -torsio n; see [7, Corollar y 5 .9] for alternative desc riptions. Set X ( p ) = ( X/ / p ) p . The s ub ca tegory Lo c T ( X ( p )) is indep endent of the choice of a genera ting set fo r the ideal p used to construct X/ / p ; this follo ws fro m the result b elow. Lemma 3.8. The fol lowing st atements ho ld for e ach X ∈ T and p ∈ Sp ec R . (1) X ( p ) is p -lo c al and p -torsion. (2) Lo c T ( X ( p )) = Lo c T ( Γ p X ) . (3) Hom T ( W , X ( p )) = 0 for any obje ct W t hat is q -lo c al and q - t orsion with q 6 = p . Pr o of. The argument is based on the fact that the lo calization functor that tak es an ob ject X to X p is exact and pres erves copro ducts . (1) E xactness o f lo ca lization implies ( X/ / p ) p can be re alized as X p / / p . Hence X ( p ) be lo ngs to Thick T ( X p ), so tha t it is p -lo cal; it is p -to r sion by [7, Lemma 5.1 1]. (2) Applying the lo calization functor to the equality Lo c T ( X/ / p ) = Lo c T ( Γ V ( p ) X ) in Prop os ition 2.11 yields (2). (3) If q 6⊆ p holds, then Γ V ( q ) ( X ( p )) = 0 and hence the desired claim follows fro m the adjunction iso morphism Hom T ( W , X ( p )) ∼ = Hom T ( W , Γ V ( q ) X ( p )). If q ⊆ p , then the R -mo dule Hom ∗ T ( W , X ( p )) is q -lo cal, by Pr op osition 2 .3, a nd p -tors ion, by [7, Lemma 5.11], and hence zero since q 6 = p .  Prop ositi o n 3. 9 . F or e ach p in Sp ec R and e ach c omp act obje ct C in T , the obje ct C ( p ) is c omp act in Γ p T , and b oth { C ( p ) | C ∈ T c } and { Γ p C | C ∈ T c } gener ate the triangulate d c ate gory Γ p T . F urthermor e, the R -line ar structur e on T induc es a natur al structur e of an R p -line ar triangulate d c ate gory on Γ p T . Pr o of. Recall tha t Γ p T is a lo calizing sub c a tegory of T , so the co pro duct in it is the same as the one in T . Each o b ject X in Γ p T is p -lo cal, so there is an isomorphism Hom T ( C ( p ) , X ) ∼ = Hom T ( C / / p , X ) . STRA TIFYING TRIANGULA TED CA TEGORIES 13 When C is co mpact in T , so is C / / p . Thus the isomor phism above implies that C ( p ) is compact in Γ p T . F urthermore, the collection of ob jects C / / p with C co mpact in T generates T V ( p ) by Pr op osition 2.7, and hence the C ( p ) gener ate Γ p T . The class of co mpa ct ob jects C generates T hence the ob jects Γ p C generate Γ p T . Prop os itio n 2.3 implies that for eac h pair of ob jects X, Y in Γ p T the R -mo dule Hom ∗ T ( X, Y ) is p -lo cal, so that they admit a natural R p -mo dule structure . This translates to an action of R p on Γ p T .  4. Stra tifica tion In this sectio n w e introduce a notion of stratifica tion for tria ngulated categories with ring actio ns. It is based on the co ncept of a minimal subca tegory introduced by Hov ey , Palmieri, and Stric kla nd [19, § 6]. As befor e T is a c ompactly generated R -linear triangulated catego ry . Minimal sub categories. A loca lizing subcateg ory of T is said to be minimal if it is nonzero and has no prop er nonzer o lo calizing sub ca tegories. Lemma 4.1. A nonzer o lo c alizing sub c ate gory S of T is m inimal if and only if for al l nonzer o obje cts X , Y in S one has Hom ∗ T ( X, Y ) 6 = 0 . Pr o of. When S is minimal and X a no nz e r o o b ject in it Lo c T ( X ) = S , by minimalit y , so if Hom ∗ T ( X, Y ) = 0 for some Y in S , then Hom ∗ T ( Y , Y ) = 0, tha t is to say , Y = 0. Suppo se S c ontains a nonzero prop er lo calizing subca tegory S ′ ; we may a s sume S ′ = Loc T ( X ) for so me nonzero ob ject X . F or each ob ject W in T ther e is then an exact triangle W ′ θ − → W η − → W ′′ → with W ′ ∈ S ′ , Hom ∗ T ( X, W ′′ ) = 0, and θ inv ertible if and only if and W is in S ′ ; see Lemma 2.1. It r emains to pick an o b ject W in S \ S ′ , set Y = W ′′ , and note that Y is in S and nonzer o.  Stratification. W e sa y that T is stra tifie d by R if the following conditions hold: (S1) The lo cal-glo bal principle, discussed in Section 3, holds for T . (S2) F o r ea ch p ∈ Sp ec R the lo calizing s ubca tegory Γ p T is e ither zero o r minimal. The crucial condition here is (S2); for example, (S1) holds when the dimension o f Spec R is finite, b y Cor ollary 3.5. Since the ob jects in Γ p T a re precisely the p - lo cal and p -tor s ion ones in T , co ndition (S2) is that each nonzer o p -lo cal p -torsion ob ject builds every o ther such ob ject. Given a lo ca liz ing subca tegory S of T and a subset U of Sp ec R set supp R S = [ X ∈ S supp R X and supp − 1 R U = { X ∈ T | supp R X ⊆ U } . Observe tha t supp R and supp − 1 R bo th preserve inclusions. Theorem 4.2. L et T b e a c omp actly gener ate d R - line ar triangulate d c ate gory. If T is str atifie d by R , then ther e ar e inclu s ion pr eserving inverse bij e ctions:  L o c alizing sub c ate gories of T  supp R / / supp − 1 R o o n Subsets of supp R T o Conversely, if the map supp R is inje ct ive, then T must b e str atifie d by R . Pr o of. F o r each p ∈ Sp ec R the subcatego ry Ker Γ p is lo c alizing. This implies that for any s ubs et U o f Sp e c R the s ubc a tegory supp − 1 R U is lo ca liz ing, for supp − 1 R U = \ p 6∈U Ker Γ p . 14 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE Moreov er, it is clear that supp R (supp − 1 R U ) = U for each subset U o f supp R T , and that S ⊆ supp − 1 R (supp R S ) holds for any loca lizing subc ategory S . The mo ot p oint is whether S co nt ains supp − 1 R (supp R S ); equiv alen tly , whether supp R is one-to-one. The map supp R factors as σ ′ σ with σ as in Prop osition 3.6 and σ ′ the map ( F amilies ( S ( p )) p ∈ supp R T with S ( p ) a lo calizing sub catego r y of Γ p T ) − → n Subsets of supp R T o where σ ′ ( S ( p )) = { p ∈ Sp ec R | S ( p ) 6 = { 0 }} . E vidently σ ′ is one-to-one if and only if it is bijective, if and only if the minimality condition (S2) holds. The map σ is also one-to-one if and only if it is bijective; moreov er this holds precis ely when the lo cal-glo bal principle holds for T , by Prop ositio n 3.6. The desired result follows.  Corollary 4.3. If R str atifies T and G is a set of gener ators for T , then e ach lo c alizing sub c ate gory S of T is gener ate d by the set S ∩ { Γ p X | X ∈ G , p ∈ Sp ec R } . In p articular, ther e exists a lo c alization functor L : T → T such that S = Ker L . Pr o of. The first asser tion is an immedia te consequence of Theorem 4.2, since S and the lo calizing subca tegory generated by the given set of ob jects ha ve the same suppo rt. Given this, the se c ond one follows fro m Lemma 2.1.  Other c o nsequences of stratificatio n are given in Sections 5 and 6 . Now we provide examples of triangulated catego ries that a re s tratified; see a lso Exa mple 7.4. Example 4.4. Let A be a co mmut ative no etherian ring and D ( A ) the deriv ed category of the catego ry of A -mo dules. The catego r y D ( A ) is co mpactly gener a ted, A -linear, and triang ulated. This exa mple is discussed in [7, § 8], wher e it is proved that the notion of suppo rt intro duce d in [7] coincides with the usual o ne, due to F oxb y and Neeman; see [7, Theorem 9.1]. In view o f Theorem 4 .2, one can refor- m ulate [25, Theorem 2.8] as: The A -linear triang ulated category D ( A ) is stratified by A . This example will b e subsumed in Theorem 8 .1. Example 4. 5. Let k b e a field and Λ an exterior algebra ov er k in finitely many indeterminates o f nega tive o dd degr ee; the grading is upp er . W e view Λ a s a dg algebra, with differential zero . In [8, § 6 ] w e in tro duced the ho motopy categor y o f graded-injective dg Λ-mo dules and prov ed that it is stratified by a natura l action of its cohomolog y algebra, E xt ∗ Λ ( k , k ). The next ex a mple sho ws that there ar e triangulated ca teg ories whic h cannot be stratified by any ring action. Example 4. 6. Let k b e a field a nd Q a quiver of Dynkin t ype ; see, for e xample, [4, Chapter 4]. The pa th alg ebra kQ is a finite dimensional hereditary algebr a o f finite repre sentation type. It is ea s ily chec ked that the g raded center of the deriv ed category D ( k Q ) is isomorphic to k . In fact, ea ch ob ject in D ( kQ ) is a direct sum of indecomposa ble o b jects, and E nd ∗ D ( kQ ) ( X ) ∼ = k for eac h indecomposa ble ob ject X . The lo calizing sub catego ries of D ( k Q ) are parameterize d by the nonc r ossing partitions asso ciated to Q ; this can b e deduced from w o rk of Ingalls and Thomas [20]. Thus the triangulated category D ( k Q ) is str atified by so me ring a cting o n it if and only if the quiver consists of one v ertex and has no arrows. 5. Or thogonality Let X and Y b e ob jects in T . The discuss ion below is motiv ated by the question: when is Ho m ∗ T ( X, Y ) = 0? The orthogona lity pr o p erty [7, Corollar y 5.8] says that if cl(supp R X ) and supp R Y a re disjoint, then one has the v anishing. What we STRA TIFYING TRIANGULA TED CA TEGORIES 15 seek ar e conv e rses to this sta tement, idea lly in terms of the supp orts of X and Y . Lemma 4.1 suggests that one can exp ect satisfactor y answers only when T is stratified. In this section we esta blis h some r esults addr e ssing this question and give e x amples whic h indicate that these ma y be the b es t p o ssible. F or any grade d R -mo dule M set Supp R M = { p ∈ Sp ec R | M p 6 = 0 } . This subset is sometimes referre d to as the ‘big supp o rt’ of M to distinguish it from its ‘homologica l’ support, supp R M . Analogously , for a ny ob ject X in T , we s et Supp R X = [ C ∈ T c Supp R Hom ∗ T ( C, X ) . It follows from [7, Theorem 5.15(1) and Lemma 2.2(1)] that there is an equality: Supp R X = cl(supp R X ) . W e use this equa lit y without further comment. Theorem 5.1. Le t T b e a c omp actly gener ate d R -line ar triangulate d c ate gory. If R str atifies T , then for e ach c omp act obje ct C and e ach obje ct Y , ther e is an e quality Supp R Hom ∗ T ( C, Y ) = Supp R C ∩ Supp R Y . The pro of requires only stratifica tion condition (S2), never (S1). Pr o of. Fix a prime idea l p ∈ Sp ec R . Supp ose Hom ∗ T ( C, Y ) p 6 = 0 ; by definition, one then has p ∈ Supp R Y . Moreover End ∗ T ( C ) p 6 = 0 since the R -a ction on Hom ∗ T ( C, Y ) p factors through it, hence p is als o in Supp R C . Thus there is an inclusion Supp R Hom ∗ T ( C, Y ) ⊆ Supp R C ∩ Supp R Y . Now supp os e Hom ∗ T ( C, Y ) p = 0 . One has to verify that that for any prime ideal q ⊆ p either Γ q C = 0 or Γ q Y = 0. By [7, Theore m 4.7], see also Pro p o sition 2 .3, since C is compact the adjunction mor phism Y → Y q induces an isomor phis m 0 = Hom ∗ T ( C, Y ) q ∼ = Hom ∗ T ( C, Y q ) . As Γ V ( q ) Y is in Lo c T ( Y ), by Prop o s ition 2.11, one obtains that Γ q Y is in Lo c T ( Y q ), hence the calculation ab ov e yields Hom ∗ T ( C, Γ q Y ) = 0. As Γ q Y is q -lo cal the adjunction morphism C → C q induces the isomorphism b elow Hom ∗ T ( C q , Γ q Y ) ∼ = Hom ∗ T ( C, Γ q Y ) = 0 . Using now the fact that Γ q C is in Loc T ( C q ) one gets Hom ∗ T ( Γ q C, Γ q Y ) = 0 . Our hypothesis was that R stratifies T . Thus one of Γ q C or Γ q Y is zero.  The exa mple b elow shows that the conclusio n of the pr eceding theorem need not hold when C is not compact. See also Example 5.9 Example 5. 2. Let A be a commut ative no etheria n ring with Krull dimension at least o ne and m a maximal ideal of A that is not also a minimal prime. F or example, take A = Z and m = ( p ), wher e p is a prime n umber . Let T b e the derived categor y o f A -modules, viewed as a n A -linear category; see Example 4.4. Let E be the injective hull of A/ m . The A -mo dule Hom ∗ T ( E , E ) is then the m -adic completion of A , so it follows that Supp A Hom ∗ T ( E , E ) = { p ⊆ m | p ∈ Sp ec R } ) { m } = Supp A E . Observe tha t supp A Hom ∗ T ( E , E ) = Supp A Hom ∗ T ( E , E ) and supp A E = Supp A E . One drawbac k o f Theorem 5.1 is that it in volves the big suppor t Supp R , while one is mainly interested in supp R . Next w e iden tify a rather natural condition on T under which o ne can obtain results in the desir e d form. 16 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE No etherian categorie s. W e c a ll a compactly genera ted R -linear triangulated cat- egory no etherian if for any compact ob ject C in T the R - mo dule End ∗ T ( C ) is finitely generated. This is equiv alent to the condition that for all co mpa ct ob jects C, D the R -mo dule Hom ∗ T ( C, D ) is finitely g enerated: co nsider End ∗ T ( C ⊕ D ). If C generates T , then T is no etherian if and only if the R -mo dule End ∗ T ( C ) is no etheria n. As a consequence of Theorem 5.1 one gets: Corollary 5. 3. If T is no etherian and str atifie d by R , then for e ach p air of c omp act obje cts C, D in T ther e is an e quality supp R Hom ∗ T ( C, D ) = supp R C ∩ supp R D . When in addition R i = 0 holds for i < 0 , one has Hom n T ( C, D ) = 0 for n ≫ 0 if and only if Hom n T ( D , C ) = 0 for n ≫ 0 . Pr o of. In view of the no ether ian hypo thesis and [7 , Lemma 2.2(1), Theorem 5.5(2)], the desired equality follo ws from Theo r em 5.1. It implies in par ticula r that supp R Hom ∗ T ( C, D ) = supp R Hom ∗ T ( D , C ) . When R i = 0 holds for i < 0 and M is a no e ther ian R -mo dule one has M n = 0 for n ≫ 0 if and only if supp R M ⊆ { p ∈ Spec R | p ⊇ R > 1 } ; see [10, Prop o sition 2.4]. The last part of the cor ollary no w follows from the equality above.  There is a v er sion o f the preceding result where the ob jects C and D need not b e co mpact. This is the topic of the next theorem. As preparatio n for its pro of, and for later applications, we further develop the material in [7, Definition 4.8]. Let C b e a compact ob ject in T . F or each injective R -mo dule I , the Brown representabilit y theorem [23, 27] yie lds an ob ject T C ( I ) in T such that there is a natural isomor phis m: Hom T ( − , T C ( I )) ∼ = Hom R (Hom ∗ T ( C, − ) , I ) . Moreov er, the assig nmen t I 7→ T C ( I ) defines a functor T C : Inj R → T fro m the category of injective R -mo dules to T . Prop ositi o n 5. 4. L et C b e a c omp act obje ct in T . The fun ct or T C : Inj R → T pr eserves pr o ducts. If the R -line ar c ate gory T is no etheria n, e ach I ∈ Inj R satisfies: supp R T C ( I ) = supp R C ∩ s upp R I = supp R End ∗ T ( C ) ∩ supp R I . In p articular, for e ach p ∈ Sp ec R t he obje ct T C ( E ( R/ p )) is in Γ p T . Pr o of. It follows by co nstruction that T C preserves pr o ducts. F or each co mpact ob ject D in T , there is an isomorphism of R -mo dules Hom ∗ T ( D , T C ( I )) ∼ = Hom ∗ R (Hom ∗ T ( C, D ) , I ) . When T is no etheria n, so that the R -mo dule Hom ∗ T ( C, D ) is finitely gener a ted, the isomorphism ab ov e gives the first equalit y b elow: supp R Hom ∗ T ( D , T C ( I )) = supp R Hom ∗ T ( C, D ) ∩ supp R I = s upp R C ∩ supp R D ∩ supp R I . The second equality holds by Co rollar y 5.3. L e mma 5.5 below then yields the fir st of the desired equalities; the sec o nd one holds b y [7, Theorem 5 .5(2)].  The following lemma provides an alterna tive de s cription of the supp or t of a n ob ject in T . Note that T need not be no etheria n. STRA TIFYING TRIANGULA TED CA TEGORIES 17 Lemma 5.5. L et X b e an obje ct in T and U a subset of supp R T . If supp R Hom ∗ T ( C, X ) = U ∩ supp R C holds for e ach c omp act obj e ct C , then supp R X = U . Pr o of. It follows from [7, Theorem 5.2] that supp R X ⊆ U . Fix p in U and choo se a compac t ob ject D with p in supp R D . Then p is in supp R ( D / / p ), s o the h y po thesis yields that p is in supp R Hom ∗ T ( D / / p , X ). Hence p belo ngs to supp R X , by [7, Propo sition 5.12].  F or a co mpact ob ject C , the functor Hom ∗ T ( C, − ) v anishes o n Lo c T ( Y ) if and only if Ho m ∗ ( C, Y ) = 0 . Using this obser v ation, it is ea sy to verify that the theorem below is an extension of Cor ollary 5.3. Compare it also with [7, Cor ollary 5.8]. Theorem 5.6. L et T b e an R - line ar t r iangulate d c ate gory that is no etherian and str atifie d by R . F or any X and Y in T the c onditions b elow ar e e quivalent: (1) Hom ∗ T ( X, Y ′ ) = 0 for any Y ′ in Loc T ( Y ) ; (2) cl(supp R X ) ∩ supp R Y = ∅ . Pr o of. (1) = ⇒ (2): Let p b e a p oint in supp R Y and C a compact ob ject in T . Prop os itio n 5.4 yields that T C ( E ( R/ p )) is in Γ p T , and hence also in Lo c T ( Y ); the last assertion holds by Theorem 4.2. This explains the equality below: Hom ∗ R (Hom ∗ T ( C, X ) , E ( R / p )) ∼ = Hom ∗ T ( X, T C ( E ( R/ p ))) = 0 , while the isomorphism fo llows from the definition of T C . Th us Hom ∗ T ( C, X ) p = 0 . Since C was a rbitrary , this means that p is not in cl(supp R X ). (2) = ⇒ (1): One has supp R Y ′ ⊆ supp R Y for Y ′ in Lo c T ( Y ), s ince the functor Γ p is exa ct and preser ves copro ducts. The orthogo nality pr op erty of s upp o r ts, [7, Corollar y 5 .8] th us implies that if condition (2) holds, then Ho m ∗ T ( X, Y ′ ) = 0.  Recall that the left o rthogona l sub categor y of S , deno ted ⊥ S , is the loca liz- ing sub c a tegory { X ∈ T | Hom ∗ T ( X, Y ) = 0 for all Y ∈ S } . As a straightforward consequence o f Theorem 5.6 one obta ins a desc r iption of the s uppo rt of the left orthogo nal of a localizing categ ory , answering a q uestion rais ed b y Rick ard. 1 Corollary 5.7. F or e ach lo c alizing sub c ate gory S of T the fol lowing e quality holds: supp R ( ⊥ S ) = { p ∈ supp R T | V ( p ) ∩ supp R S = ∅} .  R emark 5.8 . In the con text of Theorem 5.6, for any compact ob ject C one has Hom ∗ T ( C, Y ) = 0 if and only if supp R C ∩ supp R Y = ∅ . The next exa mple shows tha t one cannot do aw ay entirely with the hypothesis that C is co mpact; the p oint b eing that Hom ∗ T ( X, Y ) = 0 do es not imply that Hom ∗ ( X, − ) is zero on Lo c T ( Y ), unless X is compact. Example 5.9. Le t A be a complete lo c a l domain and Q its field o f fractions. F or example, tak e A to be the completion of Z at a prime p . It follows from a r e sult of Jensen [22, Theorem 1] that Ext ∗ A ( Q, A ) = 0 . Thus, with T the de r ived catego ry of A , one gets supp A Hom ∗ T ( Q, A ) = ∅ while supp A Q ∩ supp A A consists of the zero ideal. No te that Q is in Lo c T ( A ), so there is no contradiction with Theorem 5.6. 1 After a talk by Iye ngar at the workshop ‘ H omological metho ds in group theory’, M SRI 2008. 18 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE 6. Classifying thick subca tegories In this section we prov e tha t when T is noether ian and stratified by R its thic k sub c ategories o f compact ob jects ar e parameterized by sp ecializa tion closed subsets of supp R T . As b efore, R is a graded-commutativ e no etherian ring and T is a compactly genera ted R -linea r tr iangulated category . Thic k sub categories. One ca n deduce the next result from the class ification of lo calizing sub catego ries, Theorem 4.2, as in [25, § 3]. W e giv e a differen t proof. Theorem 6.1. L et T b e a c omp actly gener ate d R -line ar t riangulate d c ate gory that is n o etherian and str atifie d by R . The map  Thick sub c ate gories of T c  supp R / /  Sp e cialization close d subsets of supp R T  is bije ctive. T he inverse map sends a sp e cialization close d subset V of Sp ec R to the sub c ate gory { C ∈ T c | supp R C ⊆ V } . Observe that in the pro o f the injectivity of the map supp R requires only that T satisfies the stratification c o ndition (S2), while the surjectivity uses only the hypothesis that T is no etheria n. Pr o of. First we verify that supp R C is sp ecializa tio n c lo sed for any thick sub categ ory C of T c . F or any compact ob ject C the R -mo dule End ∗ T ( C ) is finitely gener ated, and this implies supp R C = supp R End ∗ T ( C ), b y [7, The o rem 5 .5]. Thus supp R C is a closed subset of Sp ec R , and therefore supp R C is sp ecializa tion closed. T o verify tha t the map s upp R is surjective, let V b e a sp ecializatio n closed subset of supp R T and set C = { C / / p | C ∈ T c , p ∈ V } . One then has that Lo c T ( C ) = T V by [7, Theorem 6 .4], and therefore the following equalities hold supp R C = supp R T V = V ∩ s upp R T = V . It remains to prov e that supp R is injectiv e. Let C b e a thick subca tegory o f T and set D = { D ∈ T c | supp R D ⊆ supp R C } . W e need to sho w that C = D . Evidently , an inclusion C ⊆ D holds. T o establish the other inclusio n, let L : T → T b e the lo calization functor with Ker L = L o c T ( C ); see Lemma 2.1 for its existence. Let D be an ob ject in D . Ea ch ob ject C in C satisfies Hom ∗ T ( C, L D ) = 0, so Theorem 5.1 implies Supp R C ∩ Supp R LD = ∅ . Hence LD = 0, that is to say , D b elongs to Lo c T ( C ). It then follows from [25, Lemma 2.2] that D is in C .  Smashing sub categories. Next we pr ove tha t when T is stratified and noether - ian, the telescop e co njecture [31] holds for T . In prepa ration for its pro of, we record an element ary observ a tion. Lemma 6 . 2. L et p ⊆ q b e prime ide als in Sp ec R . The inje ctive hul l E ( R/ p ) of R/ p is a dir e ct summand of a pr o duct of shifte d c opies of E ( R/ q ) . Pr o of. The shifted copies of E ( R/ q ) form a set of injective c o generato rs for the category of q -lo cal mo dules. This implies the desire d res ult.  A subset U of Sp e c R is said to b e close d under gener alization if Spec R \ U is sp ecialization closed. More explicitly: q ∈ U and p ⊆ q imply p ∈ U . Theorem 6.3. L et T b e an R - line ar t r iangulate d c ate gory that is no etherian and str atifie d by R . Ther e is then a bije ction  L o c alizing sub c ate gories of T close d under al l pr o ducts  supp R / /  Subsets of supp R T close d under gener alization  STRA TIFYING TRIANGULA TED CA TEGORIES 19 Mor e over, if L : T → T is a lo c alization functor that pr eserves arbitr ary c opr o ducts, then the lo c alizing su b c ate gory K er L is gener ate d by obje cts that ar e c omp act in T . R emark 6.4 . The inv er se map of supp R takes a generalization closed s ubs e t U of Spec R to the categor y o f ob jects X of T with supp R X ⊆ U ; in other words, the category of L V -lo cal ob jects, where V = Sp ec R \ U . Pr o of. Let S b e a lo ca lizing sub categ o ry of T that is clo sed under arbitrar y pro d- ucts. W e know fr om Theore m 4.2 that S is de ter mined by its s upp o r t supp R S . Thu s w e need to show that it is closed under generalization. Fix prime ideals p ⊆ q in supp R T a nd supp ose that q is in supp R S . I t fol- lows from Theore m 4.2 that Γ q T ⊆ S holds. Pick a compact ob ject C suc h that supp R C contains p ; this is p ossible since supp R T c = s upp R T . Since T is no ether- ian, supp R C is a closed subset of Sp ec R , b y [7, Theorem 5.5], and hence contains also q . Let E ( R/ q ) b e the injective hull of the R -mo dule R / q . Since T is no e- therian, Prop ositio n 5.4 yields that T C ( E ( R/ q )) is in Γ q T and hence in S . The functor T C preserves pro ducts, so Lemma 6.2 implies that T C ( E ( R/ p )) is a direct summand of T C ( E ( R/ q )) and henc e it is also in S , b ecause the la tter is a lo calizing sub c ategory closed under pro ducts. Another application of Propo sition 5.4 shows that supp R T C ( E ( R/ p )) = { p } , so that p ∈ supp R S holds, as desired Next let U be a generalization closed subset of Spe c R and set V = Spec R \ U . Let S b e the categ ory of L V -lo cal ob jects, so that supp R S = U holds, by [7 , Corollar y 5.7]. By construction, the categ ory S is tria ng ulated and close d under arbitrar y pro ducts; it is lo caliz ing b eca use the lo calization functor L V preserves arbitrar y co pr o ducts, by [7, Corollary 6.5]. This completes the pro of that supp R induces the stated bijection. Finally , le t L : T → T be a lo calization functor that pr eserves arbitrary copro d- ucts. The categor y of L -lo cal o b jects, which alw ays is closed under pro ducts, is then also a localizing subcateg ory of T . The first par t of this pro of shows that L ∼ = L V for some specializa tio n closed subs et V of Sp ec R , b eca us e the lo c alization functor L is determined b y the category of L -lo ca l ob jects. It remains to note that Ker L , which is the catego ry T V , is g enerated by compact ob jects, by [7, The o rem 6.4 ].  7. Tensor triangula ted ca tegories In this sec tion we dis c uss spe cial prop er ties of triang ulated categories which hold when they have a tens o r structure. The ma in result her e is Theore m 7.2, which says that the lo cal-global pr inciple holds for such categor ies, when the actio n of the tensor pro duct is also taken in to a ccount. Let T = ( T , ⊗ , 1 ) be a tens o r tria ng ulated catego ry as defined in [7, § 8 ]. In particular, T is a compactly generated triangulated category endo wed with a sym- metric monoidal str uc tur e; ⊗ is its tensor pro duct and 1 the unit of the tensor pro duct. It is assumed that ⊗ is ex act in each v ariable, preser ves copr o ducts, and that 1 is compact. The symmetric monoidal s tructure ensures that the endomorphism ring End ∗ T ( 1 ) is graded commutativ e. This ring acts on T via homomor phisms End ∗ T ( 1 ) X ⊗− − − − − − → End ∗ T ( X ) , In particular , a ny ho momorphism R → End ∗ T ( 1 ) of rings with R graded commu- tative induces an action of R on T . W e say that an R action o n T is c anonic al if it arises from suc h a homomo rphism. In that case there are for each sp ecia lization 20 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE closed subset V a nd point p of Spe c R natural isomor phisms (7.1) Γ V X ∼ = X ⊗ Γ V 1 , L V X ∼ = X ⊗ L V 1 , and Γ p X ∼ = X ⊗ Γ p 1 . These isomorphisms are from [7, Theor em 8.2, Corollary 8.3]. 2 T ensor ideal lo calizing sub categories. A localizing subca teg ory S of T said to be tensor ide al if for each X ∈ T and Y ∈ S , the o b ject X ⊗ Y , hence also Y ⊗ X , is in S . The smallest tensor idea l lo calizing subcateg o ry co ntaining a sub categor y S is denoted Lo c ⊗ T ( S ). Evidently there is alwa ys a n inclusion Lo c T ( S ) ⊆ Loc ⊗ T ( S ); equality ho lds when the unit 1 gener ates T . The following result is prov ed in [8, Theorem 3 .6] under the additional assump- tion that T has a sing le compact generator . The same argument ca r ries over; except that, instead of [8, Pro p o sition 3.5] use Prop osition 2.7 ab ov e. W e omit details. Theorem 7.2. L et T b e a tensor triangulate d c ate gory with a c anonic al R - action. F or e ach obje ct X in T ther e is an e quality Lo c ⊗ T ( X ) = Lo c ⊗ T  Γ p X | p ∈ Sp e c R  . In p articular, when 1 gener ates T , the lo c al glob al princi ple holds for T .  Stratification. F o r each p in Sp ec R , the lo calizing sub catego r y Γ p T , consisting of p -lo cal and p -torsion ob jects, is tensor ideal; this is immediate from (7.1). W e say that T is str atifie d by R when for eac h p , the categ ory Γ p T is either zero or has no prop er tensor ide al lo ca lizing sub c ategories . Note the ana logy with condition (S2) in Section 4; the analo gue of (S1) need not be imp osed thanks to Theorem 7.2. There are a nalogues, for tensor triangulated categories, of results in Sections 5 and 6; the pr o ofs are similar , see also [7, § 11]. One has in addition also the following ‘tensor pro duct theorem’. Theorem 7.3. L et T b e a tensor triangulate d c ate gory with a c anonic al R - action. If R str atifies T , t hen fo r any obje cts X , Y in T t her e is an e quality supp R ( X ⊗ Y ) = supp R X ∩ supp R Y . Pr o of. Fix a p oint p in Spec R . F rom 7.1 it is ea sy to v erify that there are isomor- phisms Γ p ( X ⊗ Y ) ∼ = Γ p X ⊗ Γ p Y ∼ = Γ p X ⊗ Y . These will b e used without further ado. They yield a n inclusion: supp R ( X ⊗ Y ) ⊆ supp R X ∩ supp R Y . When Γ p X 6 = 0 the str atification condition yields Γ p 1 ∈ Lo c ⊗ ( Γ p X ), and henc e also Γ p Y ∈ Lo c ⊗ ( Γ p X ⊗ Y ). Thus when Γ p Y 6 = 0 also holds, Γ p ( X ⊗ Y ) 6 = 0 holds , which justifies the r everse inclusio n.  Example 7.4. Let G b e a finite group, k a field o f characteristic p , where p divides the order of G , and k G the gro up a lg ebra. The homotopy category of complexes of injective k G - mo dules, K ( Inj k G ), is a compa c tly genera ted tensor triangulated category with a ca nonical action of the co ho mology ring H ∗ ( G, k ). One of the main results o f [8], Theorem 9.7, is tha t K ( Inj k G ) is stratified by this action. The same is true also of the stable mo dule categ ory StMo d k G ; see [8, Theor em 10.3]. 2 F or these results to hold, the R action should b e canonical, for the R -linearity of the adjunction isomorphism Hom T ( X ⊗ Y , Z ) ∼ = Hom T ( X, H om ( Y , Z )) is used i n the arguments. STRA TIFYING TRIANGULA TED CA TEGORIES 21 8. Formal differential graded algebras The go al of this sec tion is to prove that the derived categ o ry of differential graded (henceforth abbrevia ted to ‘dg’) mo dules o ver a forma l commutative dg a lgebra is stratified b y its cohomology algebr a, whe n that algebra is no etherian. This result sp ecializes to o ne o f Neeman’s [2 5] conce r ning ring s, which may be viewed as dg algebras concentrated in degree 0. F or basic notio ns conce rning dg alg ebras and dg modules o ver them we refer the reader to Mac Lane [24, § 6.7]. A qu asi-isomorphi sm b etw een dg a lgebras A and B is a morphism ϕ : A → B of dg alg e bras such that H ∗ ( ϕ ) is bijective; A and B are quasi-isomorphic if there is a chain o f quasi-is omorphisms linking them. The m ultiplication on A induces one on its cohomolog y , H ∗ ( A ). W e say that A is formal if it is quasi- is omorphic to H ∗ ( A ), viewed as a dg a lgebra with zero differential. W e write D ( A ) for the derived catego ry of dg modules over a dg alg ebra A ; it is a tria ngulated category , ge ne r ated by the compact ob ject A ; see, for instance, [23]. A dg a lgebra A is said to be c ommut ative if its underlying ring is graded commu- tative. In this case the derived tensor pr o duct of dg modules, denoted ⊗ L , endo ws D ( A ) with a structure of a tensor triang ulated catego ry , with unit A . One is th us in the framework o f Se c tion 7. The next theo r em g eneralizes [8, Theorem 5 .2], whic h deals with the ca se of graded algebr as of the fo rm k [ x 1 , . . . , x n ], where k is a field and x 1 , . . . , x n are indeterminates, of even degree if the c haracteris tic of k is not 2. Theorem 8.1. L et A b e a c ommutative dg algebr a such that the ring H ∗ ( A ) is no etherian. If A is formal, t hen D ( A ) is str atifie d by the c anonic al H ∗ ( A ) -action. In the pro of we use a totaliza tion functor fr om complexes over a gr aded ring to dg mo dules ov er the ring view ed as a dg algebra with differential zero ; see [24, § 10.9], where this functor is called condensa tion, and [23, § 3.3]. T otalization. Let A b e a gra ded a lgebra. F or each graded A -mo dule N and integer d we write N [ d ] for the gr aded A -mo dule with N [ d ] i = N d + i , and multiplication the same as the one on N . Let F b e a complex of grade d A -mo dules with differential δ ; so each F i is a g raded A -mo dule, δ i : F i → F i +1 are morphisms o f graded A -mo dules, and δ i +1 δ i = 0 . W e write F i,j for the co mpo nent of degree j in the g raded mo dule F i . The totalization of F , denoted to t F , is the dg ab elian group with (tot F ) n = M i + j = n F i,j for each n ∈ Z ∂ ( f ) = δ i ( f ) for ea ch f ∈ F i,j W e cons ide r tot F as a gra de d A -mo dule with multiplication defined b y a · f = ( − 1 ) di af for each a ∈ A d and f ∈ F i,j . A r outine c a lculation shows that to t F is then a dg A -mo dule, where A is viewed as dg algebra with zero differential, and that each mor phism α : F → G of co mplexes of gra ded A -mo dules induces a mor phis m tot α : tot F → tot G o f dg A -mo dules . Moreov er, ther e are equalities of dg A -mo dules: • tot A = A ; • tot N [ d ] = Σ d tot N for each g raded A -module N and in teger d ; • tot Σ n F = Σ n tot F . 22 DA VE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE One thus gets an additive functor from the ca tegory of c o mplexes of g raded A - mo dules to the category of dg A -mo dules. It is ea sy to check that if the complex F is acy clic so is tot F . Indeed, fix a cycle z in (tot F ) n , and write z = P i z i where z i ∈ F i,n − i . Since δ ( z ) = P i δ i ( z i ) and δ i ( z i ) ∈ F i +1 ,n − i , each z i is a cycle in F i . Since F is acyclic there exist elements w i ∈ F i − 1 ,j with δ i − 1 ( w i ) = z i ; mor eov er , one may tak e w i = 0 when z i = 0 . Note that the element w = P i w i is in (tot F ) n − 1 and δ ( w ) = z . In conclusion, tot induces an exact functor tot : D ( GrMo d A ) − → D ( A ) . of triang ula ted catego ries; here D ( GrMod A ) is the derived category of graded A - mo dules, while D ( A ) is the derived category of dg A -mo dules. Lemma 8.2. L et E b e the Koszul c omplex on a se quenc e a = a 1 , . . . , a c of homo ge- nous c entr al elements in A . Then tot E ∼ = Σ d A/ / a in D ( A ) , wher e d = P n | a n | . Pr o of. Indeed, since tot preserves ex a ct triangle s , and b oth E and A/ / a c a n b e obtained a s iter ated mapping cones, it suffices to verify the statement for the Ko szul complex on a single element , say a . The desired result is then immediate from the prop erties of tot listed ab ove.  W e requir e a lso some elementary r e sults concerning transfer of str atification along exact functors; a detailed study is taken up in [9, Section 7]. Change of categories. As b efore R is a gr aded commutativ e no etherian ring and T is a compa ctly generated R -linear triang ulated categor y . Le t F : U → T b e an equiv alence of triangulated categories . Obs erve that U is then compactly generated; it is also R -linea r with action given by the isomorphism of graded ab elian groups Hom ∗ U ( X, Y ) ∼ = Hom ∗ T ( F X , F Y ) induced by F , for all X , Y in U . Prop ositi o n 8.3. The ring R str atifies U if and only if it str atifies T . Pr o of. Using [7, Corollar y 5.9], it is eas y to verify that for each p in Spec R and X in U , there is an isomor phism F ( Γ p X ) ∼ = Γ p ( F X ), and that the induced functor Γ p U → Γ p T is an e quiv alence of tria ngulated categ ories. Given this, it is immediate from definitions that R stra tifies U if and only if it stratifies T .  When A → B is a quasi-iso morphism of dg alg ebras, B ⊗ L A − : D ( A ) → D ( B ) is an equiv a lence of categ ories, with q uasi-inv er se the restriction of scalars; s ee, for example, [3, 3.6], or [23, 6.1]. The preceding result thus yields: Corollary 8.4. L et A and B b e quasi-isomorphic dg algebr as. If D ( A ) is str atifie d by an action of R , then D ( B ) is s t r atifie d by the induc e d R -action.  Pro of of Theorem 8.1. Let R = H ∗ ( A ). The catego ry D ( A ) is tensor triangu- lated s o it admits a n R -action induced by the isomorphism R ∼ = Hom ∗ D ( A ) ( A, A ). The dg algebras A and H ∗ ( A ) ar e quasi-is omorphic, as A is formal, so it suffices to prov e that D ( H ∗ ( A )) is stratified b y the induced R -action; see Co rollar y 8.4. It is easy to verify that the ho mo morphism R → Hom ∗ D ( H ∗ ( A )) ( H ∗ ( A ) , H ∗ ( A )) = H ∗ ( A ) induced b y this R -a ction is bijectiv e, and hence that D ( H ∗ ( A )) is stratified by R if and only if it is stratified by the canonical H ∗ ( A )-action. In summary , replacing A by H ∗ ( A ) we ma y thus assume the differential o f A is zero. Set D = D ( A ). Since A is a unit and a generator of this tensor triangulated STRA TIFYING TRIANGULA TED CA TEGORIES 23 category , its lo ca lizing sub categ ories are tensor closed. The lo ca l-global principle then holds for D , b y Theor em 7.2. It remains to verify stratifica tio n conditio n (S2). Fix a p in Spec A . Since A is a compact g enerator for D , a dg A -mo dule M is in Γ p D if and only if the A -mo dule H ∗ ( M ) = Hom ∗ D ( A, M ) is p -lo ca l and p -torsio n. Hence for suc h an M the lo caliza tion map M → M p is an isomorphism; here M p denotes the usual (homogenous) loc a lization of M a t p . Lo caliz ing A at p we may th us assume that it is lo cal with ma ximal idea l p ; s e t k = A/ p , which is a graded field. Setting V = V ( p ), one has an isomorphism of functors Γ p ∼ = Γ V . Evidently , k is in Γ V D , so to verify condition (S2) it suffices to v erify that (8.5) Lo c D ( M ) = Lo c D ( k ) holds for each M in Γ V D with H ∗ ( M ) 6 = 0. It is eno ugh to prov e that (8.5) holds for M = Γ V A . Indeed, applying the functor − ⊗ L A M would then yield the second equality b elow: Lo c D ( M ) = Lo c D ( Γ V A ⊗ L A M ) = Lo c D ( k ⊗ L A M ) , while the first one holds, by (7.1), s ince M ∼ = Γ V M ; in particula r, H ∗ ( k ⊗ L A M ) 6 = 0. Since k is a g raded field and the a c tio n o f A on k ⊗ L A M factor s throug h k , this implies Lo c D ( k ⊗ L A M ) = Lo c D ( k ). Co m bining with the equality a b ov e gives (8.5). Now we verify (8.5) for M = Γ V A . The dg mo dule k is isomorphic to Γ V A ⊗ L A k and he nce in Lo c D ( Γ V A ). It re mains to prove that Γ V A is in Lo c D ( k ). Le t a = a 1 , . . . , a c be a homog eneous set of generators for the idea l p , and let a 2 denote the sequence a 2 1 , . . . , a 2 c . It suffices to prove that (8.6) A/ / a 2 ∈ Thick D ( k ) , for then one has Lo c D ( A/ / p ) = Lo c D ( Γ V A ) = Lo c D ( Γ V ( a 2 ) A ) = Lo c D ( A/ / a 2 ) ⊆ Lo c D ( k ) where the first a nd third equa lities are by Pr op osition 2.11, a nd the s econd holds bec ause the radical of the ideal ( a 2 ) equals p , so that V ( a 2 ) = V . Let tot : D ( GrMo d A ) → D b e the totalizatio n fu nctor descr ib ed above and E in D ( GrMod A ) the Kos zul complex on the sequence a 2 ; note that the elemen ts a i are central in A , since they are of ev en degr ee. The complex E is b ounded, consis ts of finitely genera ted graded A -mo dules, and satisfies ( a 2 ) · H ∗ ( E ) = 0. Since k is a graded field, the sub quo tient s of the filtra tion { 0 } ⊆ ( a ) H ∗ ( E ) ⊆ H ∗ ( E ) ar e thus finite direct sums of shifts of k . Hence there ar e inclusio ns E ∈ Thick( H ∗ ( E )) ⊆ Thick( k ) in D ( GrMod A ); see, for exa mple, [3, Theo rem 6.2(3)]. Since to t is an exac t functor, it follows that tot E is in Thick (tot k ) in D . It remains to note that tot k = k a nd that tot E is isomor phic to a suspensio n of A/ / a 2 , by Lemma 8.2. This justifies (8.6) and hence completes the pro of of the theorem.  References 1. L. L. Avramov, Mo dules of finite virtual pr oje ctive dimension , Inv ent. Math. 9 6 (1989), 71 – 101. 2. L. L. Avramov and R.-O . Buch weitz, Supp ort varieties and c ohomolo gy over c omplete inter- se ctions , Inv ent. Math. 142 (2000), 285–318. 3. L. L. Avramov, R.-O. Buc hw eitz, S. B. Iy engar, and C. Mill er, Homolo gy of p erfe ct c omplexes , Adv. Math. 22 3 (2010), 1731–1781. [C or rigendum: Adv. M ath. 2 25 (2010), 3576–3578.] 4. D. J. 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Palmieri, Quil len str atific ation for t he St e enr o d algebr a , Ann. of Math. (2) 149 (1999), 421–449. 30. D. Quillen, The sp e ctrum of an equivaria nt c ohomolo gy ring. II , Ann. of Math. 94 (1971), 573–602. 31. D. Rav enel, L o ca lization and p e rio dicity in homotopy the ory , Homotop y Theory , Durham 1985, London Math. So c. Lecture Note Ser., vol. 117, Cambridge Univ. Press, 1987, pp. 175– 194. STRA TIFYING TRIANGULA TED CA TEGORIES 25 Da ve Benson, Institute of Mathema tics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K. Srikanth B. Iyenga r, Dep a r tment of Mathema tics, University of Neb raska, Lincoln, NE 6858 8, U.S.A. Henning Krause, Institut f ¨ ur Mathema tik, Universit ¨ at P aderborn, 33095 P aderborn, Germany. Curr ent addr ess : F akult¨ at f ¨ ur Mathematik, U nive rsit¨ at Bi elefeld, 33501 Bi elef eld, Germany