Rouquiers cocovering theorem and well-generated triangulated categories

Rouquier’s Co co v ering Theorem and W ell-gen erated T riangulated Cate gories Daniel Murfet Abstract W e study coco ve rings of triangulated cat egories, in the se nse of Rouquier, and pro v e that for an y regular cardinal α the condition of α -compactness, in the sense of Neeman, is lo cal with resp ect to suc h coco v erings. This was established for ordinary compact ness by Rouquier. Our result yields a new tec hniqu e for pro ving that a giv en triangulated category is w ell-generate d. As an application we describ e th e α -compact ob jects in the u n b ounded deriv ed cate gory of a qu asi-compact s emi-separated sc heme. 1. Introduction Let T b e a triangulated category with copro ducts, and recall th at an ob ject Y of T is c omp act if the functor T ( Y , − ) comm utes with copro du cts. When T = D ( X ) is the u n b ounded deriv ed category of quasi-coheren t shea v es on a reasonable sc heme X , the condition of compactness in T is lo cal: giv en an op en co v er { U 1 , . . . , U n } of X , an ob ject F is compact in D ( X ) if and only if F | U i is compact in D ( U i ) for 1 6 i 6 n . F or arb itrary T , Rouquier introd uces in [Rou08, § 5] a suitable generalisation: he defines a c o c overing of T to b e a sp ecial family of Bousfield sub categories F = {I 1 , . . . , I n } (the precise defi n ition is r ecalled b elo w). The analogue of restriction to U i is then passage to the quotien t T − → T / I i , an d un der some natur al hypotheses on F , compactness in T is lo cal: an ob ject Y is compact in T if and only if the image of Y is compact in T / I i for 1 6 i 6 n . This article concerns the large cardinal generalisa tion. Let α b e a regular card inal, th at is, α is an infi nite cardinal whic h is not the sum of fewer than α cardinals, all smaller than α . In his b o ok [Nee01] Neeman asso ciates to α a class T α ⊆ T of α - c omp act ob jects. Th e defin ition is not so easily stated, b ut in t ypical examples, say the homotop y categ ory of sp ectra or the derive d category of an associativ e r ing, the condition of α -compactness is very natural; see Section 4. In particular, the ℵ 0 -compact ob jects are precisely the compact ob j ects. Ou r m ain theorem sa ys, among other things, that the condition of α -compactness is lo cal: giv en a co co v ering F o f T as ab o v e, satisfying some natural hyp otheses, an ob j ect Y is α -compact in T if and only if the image of Y is α -compact in T / I i for 1 6 i 6 n . In order to giv e the pr ecise statemen ts, we need some n otation: recall that a lo c alising sub c ate gory S of T is a tria ngulated su b category closed und er small coprod ucts, and S is Bousfield if the inclusion S − → T has a righ t adjoint. Giv en a class S of ob jects in T , we write h S i for the smallest lo calising sub category of T conta ining S . Let α b e a regular cardinal. If T α is essen tially small and hT α i = T , then T is said to b e α -c omp actly gener ate d , and T is called wel l-gener ate d if it is α -compactly generated for some regular cardin al α . If T is α -compactly generated, then a lo calising sub catego ry S ⊆ T is α -c omp actly gener ate d in T if there is a set S ⊆ T α suc h that S = h S i . In this case S is α -compactly generated, and S α = S ∩ T α (see Theorem 5). Tw o Bousfield su b categories I 1 , I 2 of T are said to interse ct pr op erly if, for every p air I ∈ I 1 and J ∈ I 2 , an y morphism I − → J or J − → I factors through an ob ject of I 1 ∩ I 2 [Rou08, (5.2 .3)]. D aniel Murfet Finally , a c o c overing of T is a finite family of Bousfield sub categ ories F = {I 1 , . . . , I n } of T which are pairwise pr op erly intersec ting, su c h that T n i =1 I i = 0; see [Rou08, (5.3.3)]. The α = ℵ 0 case of the follo wing theorem is the aforemen tioned result of Rouquier, namely [Rou08, Theorem 5.15 ]. Theorem 1 . Let T b e a triangulated category with copro ducts and α a r egular cardinal. Sup p ose that F = {I 1 , . . . , I n } is a co cov ering of T with the follo wing prop erties: (1) T / I is α -compactly generated for ev ery I ∈ F . (2) F or ev ery I ∈ F and nonempty subset F ′ ⊆ F \ {I } the essentia l image of the comp osite \ I ′ ∈F ′ I ′ inc − → T can − → T / I is α -compactly generated in T / I . Then T is α -compactly generated, and an ob ject X ∈ T is α -compact if an d only if the im age of X is α -compact in T / I for ev ery I ∈ F . Let S b e a Bousfield sub category of T in tersecting prop erly with eac h I ∈ F , suc h that: (3) S / ( S ∩ I ) is α -compactly generated in T / I for every I ∈ F . (4) F or ev ery I ∈ F and nonempty subset F ′ ⊆ F \ {I } the essentia l image of the comp osite S ∩ \ I ′ ∈F ′ I ′ inc − → T can − → T / I is α -compactly generated in T / I . Then S is α -compactly generated in T . T o return to the geometric examp le: if T = D ( X ) and we are giv en an op en co v er as ab o v e, then for eac h 1 6 i 6 n denote by I i = D X \ U i ( X ) the fu ll su b category of D ( X ) consisting of complexes with cohomology sup p orted on X \ U i . Th er e is a canonical equiv alence T / I i ∼ = D ( U i ), the quotien t functor T − → T / I i corresp onds to r estriction, and th e family F = {I 1 , . . . , I n } is a co co vering of D ( X ) satisfying the hypotheses (1) , (2) of the theorem for α = ℵ 0 [Rou08, § 6.2 ]. F or this c h oice of T an d F the h yp otheses are v ery natural, and easily v erified; for the f u ll elaboration, see Section 5. Applying the theorem (recall that, since α = ℵ 0 , this is jus t Rouquier’s [Rou08, Th eorem 5.15]) one obtains a pr o of of the fact, due originally to Neeman [Nee96], th at the compact ob jects in D ( X ) are pr ecisely the p erf ect complexes; see [Rou08 , T heorem 6.8]. Using the α > ℵ 0 case of the theorem w e obtain in Section 5 a d escription of the α -compact ob jects in D ( X ). W e hav e another application in mind , whic h will app ear in the forthcoming [Mur08]. Let A b e an asso ciativ e ring with ident it y , K (Pro j A ) and K (Flat A ) the homotop y catego ries of pro jectiv e and flat left A -mo d ules, resp ectiv ely . A complex of left A -mo du les F is pur e acyclic if it is acyclic, and N ⊗ A F is acyclic for ev ery right A -mo du le N . Let K pac (Flat A ) denote the fu ll su b category of pure acyclic complexes in K (Flat A ). Th is is a triangulated sub categ ory , and Neeman pro v es in [Nee08] that the composite K (Pro j A ) inc − → K (Flat A ) can − → K (Flat A ) / K pac (Flat A ) (1) is an equiv alence. No w let X b e a qu asi-compact semi-separated scheme. Unless X is affine, pro jec- tiv e qu asi-coheren t shea ves on X are rare, and the homotop y category of pro jectiv e quasi-coheren t shea v es on X is often the zero catego ry . In this case, the equiv alence (1) suggests a suitable replace- men t. Let K (Flat X ) b e the homotop y catego ry of flat qu asi-coherent sh ea v es on X , and denote b y K pac (Flat X ) the full sub catego ry of acyclic complexes F with the prop erty that F ⊗ O X A is acyclic for ev ery quasi-coheren t sheaf A . Define N (Flat X ) := K (Flat X ) / K pac (Flat X ) , 2 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories and let { U 1 , . . . , U n } b e an affine op en co v er of X , with sa y U i ∼ = Sp ec( A i ) for 1 6 i 6 n . W e sho w in [Mur08] that there is a co co ve ring of N (Flat X ) by Bousfield sub categories { N X \ U i (Flat X ) } 1 6 i 6 n , where N X \ U i (Flat X ) is the kernel of a n atural restriction functor N (Flat X ) − → N (Flat U i ). More- o v er, there are canonical equiv alences N (Flat X ) / N X \ U i (Flat X ) ∼ = N (Flat U i ) ∼ = K (Pro j A i ) . Neeman pro v es in lo c.cit. that K (Pro j A i ) is ℵ 1 -compactly generated, and even compactly generated when A i is coherent. In [Mur08] w e com bine Neeman’s results with Th eorem 1 to see that the global catego ry N (Flat X ) is ℵ 1 -compactly generated, and compactly generate d when X is noetherian. The p ro of of Th eorem 1 is by induction on the size n = |F | of the co co v ering. The real cont ent is in th e in itial step of th e induction, which we separate into its own section. The p ro of of the theorem is completed in S ection 3. Our basic reference for tr iangulated categories is [Nee0 1 ], whose notation w e follo w with one exceptio n: give n a class C of ob j ects in T , w e write C ⊥ = { Y ∈ T | Hom T (Σ n X, Y ) = 0 f or all X ∈ C and n ∈ Z } , ⊥ C = { X ∈ T | Hom T ( X, Σ n Y ) = 0 for all Y ∈ C and n ∈ Z } for the orthogonals, whic h are triangulated sub categories of T . F or furth er information on the theory of w ell-generated triangulated categories, the reader is referred to [Nee05, Kra07]. In this article, all triangulated categories ha ve “small Homs”. A cknow le dgements. I would like to thank Amnon Neeman for suggesting improv ements to an ear- lier v ersion and comm un icating the pr o of of T heorem 19, an d Henn ing Kr ause for helpf u l d iscussion on the sub ject of this pap er. 2. Initia l step of the induction Throughout, T is a triangulated categ ory with copro ducts. Tw o Bousfield sub categories I 1 , I 2 of T are ortho gonal if I 1 ⊆ I ⊥ 2 and I 2 ⊆ I ⊥ 1 . In this situation the comp osite I a − → T − → T / I b is fully faithful for { a, b } = { 1 , 2 } and I a ma y b e ident ified with a Bousfield sub categ ory of T / I b . L et u s state the n = 2 case of the Theorem 1 as a prop osition: Pr opo sition 2 . Let I 1 , I 2 b e orthogonal Bousfield sub categories of T , and sup p ose that for s ome regular cardinal α , w e ha ve: (1) T / I a is α -compact ly generated for a ∈ { 1 , 2 } , (2) I a is α -compact ly generated in T / I b for { a, b } = { 1 , 2 } . Then T is α -compactly generated, and an ob ject X ∈ T is α -compact if an d only if the im age of X is α -compact in b oth T / I 1 and T / I 2 . Let S b e a Bousfield sub catego ry of T int ersecting prop erly with I 1 and I 2 , and supp ose that: (3) S / ( S ∩ I a ) is α -compactly generated in T / I a for a ∈ { 1 , 2 } , (4) S ∩ I a is α -compact ly generated in T / I b for { a, b } = { 1 , 2 } . Then S is α -compactly generated in T . W e dev elop the pro of as a series of lemmas. Since the α = ℵ 0 is handled by [Rou08, Prop osition 5.14], w e assume that α > ℵ 0 . Throughout this section the notation of the prop osition is in force. F or a ∈ { 1 , 2 } w e write i a ∗ : I a − → T for the inclusion, i ! a for the r igh t adjoin t of i a ∗ , j ∗ a : T − → T / I a for the quotien t functor an d j a ∗ for the right adjoin t of j ∗ a . 3 D aniel Murfet T o p ro v e that T is α -compactly generated w e need to pro d uce, in the language of [Nee01, Ch.8], an α -p erfect set of α -small ob jects, wh ich generates. Th e condition of α -smallness is very simple: an ob ject X ∈ T is α -smal l if for eve ry family { Y i } i ∈ I of ob j ects of T , any morphism X − → M i ∈ I Y i factors through a s ub copr o duct L i ∈ J Y i for some su b set J ⊆ I of cardinalit y | J | < α . An ob ject X is ℵ 0 -small if and only if T ( X, − ) comm utes with copro du cts, and in this case one says that X is c omp act . W e refer the r eader to [Nee01, Ch.3] for the defin ition of α -p erfect classes, and restrict ourselv es here to one trivial fact: an y triangulated sub category of T is an ℵ 0 -p erfect class. By hyp othesis the quotients T / I a are α -compactly generated, and I b is α -compactly generate d in T / I a for { a, b } = { 1 , 2 } . Hence these catego ries all p ossess α -p erfect classes of α -small ob jects whic h generate. T h e strategy emplo y ed by Rouquier [Rou08] in the α = ℵ 0 case is to take generating sets E and E ′ for I 2 , T / I 2 resp ectiv ely , use a gluing argument to lift E ′ to class E ′′ of compact ob jects in T , and take the union E ∪ E ′′ . This is a generating set of compact ob j ects for T . In the α > ℵ 0 case w e tak e a different approac h, in whic h it s eems easier to manage the p erfection condition (whic h is trivial for α = ℵ 0 ). T o p r o ceed, we fi r st recall how to rephrase the cond ition on our generating set in terms of a prop erty of a certain exact functor b et we en ab elian categories. A triangulated sub category S ⊆ T is said to b e α -lo c alising if the copro du ct of few er than α ob jects of S lies in S . F or example, the class T α of α -compact ob jects is an α -localising triangulated sub category of T . Giv en an α -lo calising su b category S of T we denote by Add α ( S op , A b ) th e ab elian catego ry of all functors S op − → A b which preserv e pro d ucts of fewer than α ob jects, wh ere A b is the category of ab elian group s. Th ere is a canonical homological functor T − → Add α ( S op , A b ) , X 7→ T ( − , X ) | S . Let T − → A ( T ) b e F reyd’s un iv ersal homologica l f unctor, where A ( T ) is the ab elianisation of T [Nee01, Ch . 5]. F r om the unive rsal prop erty of this construction, w e d educe an exact fu nctor π : A ( T ) − → Add α ( S op , A b ) , and Neeman prov es in [Nee01, Theorem 1.8] that S is an α -p erfect class of α -small ob jects precisely when π preserves copro d u cts. Here, then, is th e strategy of our pr o of: in Definition 3 b elo w w e tak e the ob vious candidate for a generating set S = T | α | of T . W e ha v e to pro v e tw o things: firs tly , that this is an α -p erfect class of α -small ob j ects, and secondly , that it generates. The second condition is easily v erified, and for the first w e ju st need to pr o v e that π p reserv es copro d ucts. The co cov ering {I 1 , I 2 } of T leads to a pair of lo calisations of Add α ( S op , A b ), whic h w e may think of as a “co v er” of this ab elian category . Chec king that π p reserv es copro ducts then b ecomes a “lo cal” prob lem with resp ect to th is cov er. The lo cal pieces in the co v er corresp ond to the quotient s T / I a , an d w e can use the fact that these cate gories are α -compactly generated to complete th e pro of. Definition 3 . W e d efine a full sub category of T by T | α | = { X ∈ T | j ∗ a ( X ) ∈ ( T / I a ) α for a ∈ { 1 , 2 }} . Lemma 4 . T | α | is an α -lo calising sub category of T . Pr o of. F ollo ws from the fact that j ∗ a preserve copro ducts, and ( T / I a ) α is α -localising. Let u s recall the state ment of the Neeman-Ra ve nel-Thomason lo calisation theorem. Theorem 5 . Let R b e a trangulated category with copro du cts whic h is α -compactl y generated, and let S ⊆ R b e a lo calising sub categ ory α -compactly generated in R . Then S is α -compactly 4 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories generated, and S α = R α ∩ S . The canonical functor R − → R / S preserves α -compactness and the induced fu nctor R α / S α − → ( R / S ) α is an equiv alence ( recall that α > ℵ 0 ) . Pr o of. See [Nee01 , T heorem 4.4.9]. The fu ll sub category of α -compact ob jects in I a is denoted b y I α a . O ne n eeds to b e careful to distinguish b et wee n ob jects X ∈ I a whic h are α -compact in I a , and those that are α -compact in the larger catego ry T . At this p oin t, we do n ot kn o w that these classes are the same. It follo ws from h yp otheses (1) and (2) of Prop osition 2, and Theorem 5, that I α a ⊆ I a is precisely the class of ob j ects X ∈ I a with th e prop erty that j ∗ b ( X ) ∈ ( T / I b ) α , where { a, b } = { 1 , 2 } . Moreov er, I a = hI α a i . Lemma 6 . Th ere is an inclusion I α 1 ∪ I α 2 ⊆ T | α | . Pr o of. If X ∈ I α a then by (2), j ∗ b ( X ) ∈ ( T / I b ) α . Sin ce j ∗ a ( X ) = 0, it f ollo ws that X ∈ T | α | . Lemma 7 . Given X ∈ T | α | and Y ∈ I a for a ∈ { 1 , 2 } , any morph ism f : X − → Y in T factors as X − → I − → Y for some I ∈ I α a . Pr o of. Let b ∈ { 1 , 2 } b e su c h th at b 6 = a . By hyp othesis (2) there is a set Q ⊆ ( T / I b ) α ∩ I a suc h that h Q i = I a . By [Nee01, Theorem 4.3.3] the morp hism j ∗ b f : j ∗ b X − → j ∗ b Y factors in T / I b as j ∗ b X − → N − → j ∗ b Y for some N ∈ h Q i α = I α a . Since I := j b ∗ N b elongs to I α a , the comp osite X can − → j b ∗ j ∗ b X − → j b ∗ N − → j b ∗ j ∗ b Y ∼ = Y pro vides the desired factorisatio n of f . W e use sev eral facts ab out prop er intersecti on of sub categories dev elop ed b y Rouquier [Rou08, § 5]. F or the reader’s con v enience, the n ecessary f acts are recalled here in App endix B. F or example, since I 1 , I 2 are prop erly in tersecting the V erdier su m op eration is comm utativ e: I 1 ⋆ I 2 = I 2 ⋆ I 1 . It follo ws that I 1 ⋆ I 2 is a Bousfi eld sub categ ory of T and, follo wing the notation of [Rou08, § 5], w e write i ∪ ∗ : I 1 ⋆ I 2 − → T for the in clusion, i ! ∪ for its r igh t adjoint, j ∗ ∪ : T − → T / ( I 1 ⋆ I 2 ) for the quotien t, and j ∪ ∗ for its righ t adjoint. Note that in lo c.cit. Rouqu ier writes hI 1 ∪ I 2 i ∞ for I 1 ⋆ I 2 , to reflect the fact that this is the smallest triangulated sub catego ry of T conta ining I 1 ∪ I 2 . F or { a, b } = { 1 , 2 } the quotien t j ∗ ∪ induces a functor j ∗ a ∪ : T / I a − → T / ( I 1 ⋆ I 2 ) fitting into a sequence 0 − → I b − → T / I a − → T / ( I 1 ⋆ I 2 ) − → 0 whic h is exact , in the sense that I b − → T / I a is fu lly faithful and j ∗ a ∪ is, up to natural equiv alence, the V erd ier quotient of T / I a b y I b . W e write j a ∪ ∗ for the right adjoin t of j ∗ a ∪ . Lemma 8 . Given a ∈ { 1 , 2 } and Y ∈ ( T / I a ) α , there is X ∈ T | α | suc h that j ∗ a ( X ) ∼ = Y . Pr o of. W e u se the argument giv en in the pro of of [Rou08, Prop osition 5.14]. Let b ∈ { 1 , 2 } b e such that { a, b } = { 1 , 2 } . F rom hyp otheses (1) , (2) and Theorem 5 we deduce that the quotien t fu nctor j ∗ a ∪ preserve s α -compactness. Hence, if we set D a = Y , then the ob ject D ∪ := j ∗ a ∪ D a is α -compact in T / ( I 1 ⋆ I 2 ). Also by Theorem 5, the canonica l fun ctor j ∗ b ∪ : ( T / I b ) α − →  T / ( I 1 ⋆ I 2 )  α 5 D aniel Murfet is a V erd ier quotien t, so w e can fin d D b ∈ ( T / I b ) α and an isomorph ism j ∗ b ∪ D b ∼ = D ∪ . There are unit morphisms η 1 : D 1 − → j 1 ∪ ∗ D ∪ , η 2 : D 2 − → j 2 ∪ ∗ D ∪ and w e defi ne δ to b e the morphism induced out of the copro duct j 1 ∗ D 1 ⊕ j 2 ∗ D 2 b y j 1 ∗ ( η 1 ) − j 2 ∗ ( η 2 ). I f we defin e X by extendin g δ to a triangle X − → j 1 ∗ D 1 ⊕ j 2 ∗ D 2 δ − → j ∪ ∗ D ∪ + − → , then one c hec ks that j ∗ a X ∼ = D a = Y , and that j ∗ b X ∼ = D b , so X ∈ T | α | as requir ed . Lemma 9 . T | α | is essentia lly s m all. Pr o of. F or X ∈ T | α | there is a canonical triangle [Rou08, Pr op osition 5.10] X − → j 1 ∗ j ∗ 1 X ⊕ j 2 ∗ j ∗ 2 X − → j ∪ ∗ j ∗ ∪ X + − → . (2) By h yp othesis j ∗ a X ∈ ( T / I a ) α for a ∈ { 1 , 2 } . No w, since T / I a is α -compactly generated, ( T / I a ) α is essen tially small. It f ollo w s that there is, up to isomorphism, only a “set” of p ossible ob jects X in a triangle of the form (2), whence T | α | is essen tially s m all. Lemma 10 . F or a ∈ { 1 , 2 } the canonical fu n ctor j ∗ a : T | α | / ( T | α | ∩ I a ) − → ( T / I a ) α (3) is an equiv alence. Pr o of. The comp osite T | α | inc − → T j ∗ a − → T / I a factors, b y definition, through the inclusion ( T / I a ) α − → T / I a . Th e factorisation T | α | − → ( T / I a ) α v anishes on T | α | ∩ I a , and induces a fu nctor (3). T o verify that (3) is fully faithful, we use a stand ard argum ent. Let s : X − → Y b e a morphism in T with cone in I a and Y ∈ T | α | . Extend to a triangle X s − → Y f − → I + − → . The map f factors, by Lemm a 7, as Y − → I ′ − → I with I ′ ∈ I α a . F rom the o ctahedral axiom, applied to the p air of morphisms in this factorisation, we obtain ob j ects C , D and triangles Y − → I ′ − → C + − → , (4) I ′ − → I − → D + − → , (5) C − → Σ X − → D + − → . (6) Since I ′ ∈ I α a ⊆ T | α | w e find that C b elongs to T | α | and D to I a . Th us Σ − 1 C − → X is a morphism with domain in T | α | , the comp osite of which with s : X − → Y has cone in I a . It now follo ws easily that (3) is fu lly faithful. T o see th at it is su rjectiv e on ob j ects, w e use Lemma 8. Fix an index a ∈ { 1 , 2 } . By Lemma 10 the canonical functor j ∗ a : T | α | − → ( T / I a ) α is a V erdier quotien t whic h p reserv es α -copro ducts. By [Kra07, Lemma B.8] the (exact) restriction functor q ∗ a : Add α ( { ( T / I a ) α } op , A b ) − → Add α ( {T | α | } op , A b ) , q ∗ a ( F ) = F ◦ j ∗ a has an exac t left adjoint q a ∗ . The righ t adjoin t q ∗ a is fully faithful, s o q a ∗ is a Gabriel localisation of 6 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories Add α ( {T | α | } op , A b ). As w e will see, it is reasonable to think of the pair of lo calisations Add α ( { ( T / I 1 ) α } op , A b ) Add α ( {T | α | } op , A b ) q 1 ∗ 1 1 q 2 ∗ - - Add α ( { ( T / I 2 ) α } op , A b ) as a co v ering of Add α ( {T | α | } op , A b ). T o make this pr ecise, we sho w that a functor F which is sen t to zero b y b oth lo calisations, must already b e zero. Lemma 11 . Ker( q 1 ∗ ) ∩ Ker( q 2 ∗ ) = 0 . Pr o of. Fix a ∈ { 1 , 2 } . By [Kra07, L emma B.8] a functor F ∈ Add α ( {T | α | } op , A b ) b elongs to Ker( q a ∗ ) if and only if for an y C ∈ T | α | , ev ery morphism T | α | ( − , C ) − → F factors via T | α | ( − , γ ) : T | α | ( − , C ) − → T | α | ( − , C ′ ) for some morph ism γ : C − → C ′ in T | α | with j ∗ a ( γ ) = 0 in T / I a . F rom Lemma 10 we deduce that j ∗ a ( γ ) = 0 if an d only if γ factors, in T | α | , via an ob ject of T | α | ∩ I a . W e conclude that F b elongs to Ker( q a ∗ ) if and only if every morphism T | α | ( − , C ) − → F factors via T | α | ( − , I ) for some I ∈ T | α | ∩ I a . Assume no w that F b elongs to K er( q 1 ∗ ) ∩ K er( q 2 ∗ ) and let x : T | α | ( − , C ) − → F b e any morphism . By the ab ov e, this must f actor as T | α | ( − , C ) − → T | α | ( − , I ) − → F for some I ∈ T | α | ∩ I 1 . Since F also b elongs to Ker( q 2 ∗ ), the morph ism T | α | ( − , I ) − → F factors as T | α | ( − , I ) − → T | α | ( − , I ′ ) − → F for some I ′ ∈ T | α | ∩ I 2 . But since I 1 and I 2 are orthogonal, the morp h ism T | α | ( − , I ) − → T | α | ( − , I ′ ) v anishes, and we conclude that x = 0. It follo ws that F = 0, as claimed. Pr opo sition 12 . T | α | is an α -p erfect class of α -small ob jects in T . Pr o of. W e h a v e α -localising sub categories T | α | ⊆ T , ( T / I 1 ) α ⊆ T / I 1 and ( T / I 2 ) α ⊆ T / I 2 and, as discussed at the b eginning of this section, there are canonical exact fu nctors π : A ( T ) − → Add α ( {T | α | } op , A b ) , π 1 : A ( T / I 1 ) − → Add α ( { ( T / I 1 ) α } op , A b ) , π 2 : A ( T / I 2 ) − → Add α ( { ( T / I 2 ) α } op , A b ) . W e claim that for a ∈ { 1 , 2 } the diagram A ( T ) π   A ( j ∗ a ) / / A ( T / I a ) π a   Add α ( {T | α | } op , A b ) q a ∗ / / Add α ( { ( T / I a ) α } op , A b ) (7) comm utes up to n atural equiv alence, w here A ( j ∗ a ) is the in duced fu nctor b et w een the ab elianisati ons. By the unive rsal prop erty of the ab elianisations, it su ffi ces to pro v e that the related diagram T ρ   j ∗ a / / T / I a ρ a   Add α ( {T | α | } op , A b ) q a ∗ / / Add α ( { ( T / I a ) α } op , A b ) (8) comm utes, where ρ and ρ a are the restricted Y oneda functors. T o do this, w e recycle an argumen t of Krau s e from the pro of of [Kra07, Th eorem 6.3]. The fi r st thin g to observ e is that the comp osite 7 D aniel Murfet q a ∗ ◦ ρ v anishes on I a : one uses the descrip tion in [Kra07, Lemma B.8] of the k ernel of q a ∗ , toge ther with Lemm a 7. F or C ∈ T | α | and X ∈ T / I a there is an adjunction isomorp hism T / I a ( j ∗ a C, X ) ∼ = T ( C, j a ∗ X ) , and it follo ws that there is a natural equiv alence q ∗ a ◦ ρ a ∼ = ρ ◦ j a ∗ . Comp osing with q a ∗ w e obtain a natural equiv alence ρ a ∼ = q a ∗ ◦ q ∗ a ◦ ρ a ∼ = q a ∗ ◦ ρ ◦ j a ∗ and consequently ρ a ◦ j ∗ a ∼ = q a ∗ ◦ ρ ◦ j a ∗ ◦ j ∗ a . F rom the unit η : 1 − → j a ∗ ◦ j ∗ a w e obtain a natural tr an s formation q a ∗ ◦ ρ ( q a ∗ ◦ ρ ) η / / q a ∗ ◦ ρ ◦ j a ∗ ◦ j ∗ a ∼ = ρ a ◦ j ∗ a . This is the desired natural equiv alence, b ecause f or ev er y X ∈ T the cone of η X : X − → j a ∗ j ∗ a ( X ) is an ob ject of I a , on whic h q a ∗ ◦ ρ v anishes. Since ( T / I a ) α is an α -p erfect class of α -small ob jects, w e infer from [Nee01, T heorem 1.8] that π a preserve s copro d ucts. Let { x λ } λ b e a family of ob jects in A ( T ), and let ξ : L λ π ( x λ ) − → π ( L λ x λ ) b e the canonical morphism in Add α ( {T | α | } op , A b ). Extend on b oth sides to an exact sequence 0 − → Ker( ξ ) − → M λ π ( x λ ) ξ − → π  M λ x λ  − → Cok er( ξ ) − → 0 , whic h maps un der q a ∗ to an exact sequence 0 − → q a ∗ Ker( ξ ) − → q a ∗ M λ π ( x λ ) q a ∗ ( ξ ) − → q a ∗ π  M λ x λ  − → q a ∗ Cok er( ξ ) − → 0 . Here is where w e us e comm utativit y of (7). Both π a and A ( j ∗ a ) preserve copro d ucts, whence q a ∗ ◦ π ∼ = π a ◦ A ( j ∗ a ) pr eserves copro d ucts. Sin ce q a ∗ preserve s copro d ucts (it has a r igh t adjoint), we conclude that q a ∗ ( ξ ) is an isomorphism , and th us q a ∗ Ker( ξ ) and q a ∗ Cok er( ξ ) b oth v anish. Since a ∈ { 1 , 2 } w as arbitrary , it follo ws f r om Lemma 11 that Ker( ξ ) = Cok er( ξ ) = 0, whence ξ is an isomorphism and π pr eserv es copro ducts. By [Nee01, Th eorem 1.8], T | α | is an α -p erfect class of α -small ob jects. Pr o of of Pr op osition 2. First w e pro ve that T | α | is an α -compact generating set 1 for T , in the sen se of [Nee01, Definition 8.1.6]. In ligh t of Pr op osition 12, it su ffices to prov e th at if an ob ject x ∈ T satisfies T ( y , x ) = 0 for all y ∈ T | α | then x = 0. Note that I α 1 ⊆ T | α | , so x ∈ ( T | α | ) ⊥ ⊆ ( I α 1 ) ⊥ = I ⊥ 1 , since hI α 1 i = I 1 . Let t ∈ ( T / I 1 ) α b e giv en, and choose b y Lemma 8 a t ′ ∈ T | α | with j ∗ 1 ( t ′ ) ∼ = t . Then 0 = T ( t ′ , x ) ∼ = T / I 1 ( j ∗ 1 t ′ , j ∗ 1 x ) ∼ = T / I 1 ( t, j ∗ 1 x ) . But T / I 1 is α -compactly generated and t wa s arbitrary , so j ∗ 1 x = 0. Hence x b elongs to b oth I 1 and I ⊥ 1 , wh ich is only p ossible if x = 0. It n o w follo ws from [Nee01, Prop osition 8.4.2] that T = hT | α | i , and from [Nee01, Theorem 4.4.9] that T α is the smallest α -lo calising sub categ ory of T conta ining T | α | . Hence T α = T | α | , whic h settles the fi rst state ment of the th eorem. The second statemen t of the theorem deals with a Bousfield sub category S . The intersectio ns S ∩ I 1 , S ∩ I 2 are orth ogonal Bousfield sub catego ries of S . W e wan t to apply the fir st part of the theorem to S an d this pair of sub categories. Condition (1) is certainly satisfied, s ince by hyp othesis (3) the qu otien ts S / ( S ∩ I a ) are α -compactly generated. F or condition (2) w e m ust sh o w that S ∩ I a is α -compactly generated in S / ( S ∩ I b ) for { a, b } = { 1 , 2 } . It follo w s from hyp othesis (4) that ( S ∩ I a ) α = ( S ∩ I a ) ∩ ( T / I b ) α (9) and from h yp othesis (3) that  S / ( S ∩ I b )  α =  S / ( S ∩ I b )  ∩ ( T / I b ) α . (10) 1 Strictly sp eaking T | α | is an essentially small class, not a set, b ut let us replace T | α | by a represen tative set of ob jects and ignore th e distinction. 8 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories This implies that th e inclusion S ∩ I a − → S / ( S ∩ I b ) pr eserv es α -compactness, f rom whic h we deduce th at the former category is α -compactly generated in the latte r. Now, u sing the first p art of the theorem, we conclude that S is α -compactly generated and that an ob ject X ∈ S is α -compact in S if and only if the image u nder S − → S / ( S ∩ I a ) is α -compact for eac h a ∈ { 1 , 2 } . By (10) these are precisely the X ∈ S th at are α -compact in T , so S is α -compactly generated in T . 3. Pro of of the Theorem Let us briefly recall the setup of Theorem 1. W e are give n a triangulated category T with copro ducts, a regular cardinal α , and a co co v ering F = {I 1 , . . . , I n } satisfying some conditions (1) , (2), and we wish to p ro v e that T is α -compactly generated. O nce again, since the α = ℵ 0 case is han d led b y [Rou08, Theorem 5.1 5], we r estrict to the case α > ℵ 0 . In what follo ws we make implicit use of the prop erties of prop er in tersection describ ed in App endix B, particularly L emm a 38. Pr o of of The or e m 1. Th e pro of is by induction on the n um b er n > 2 of elemen ts in the co co v er F (to b e clear, the induction includes the second statemen t of th e theorem, ab out S ). The n = 2 case is giv en b y Prop osition 2, and for th e in ductiv e step the argument is iden tical to the inductive step in the pro of of [Rou08, Theorem 5.15]. F or th e reader’s con v enience, let u s rep eat the argumen t here. Assu me that n > 2 and set I ∩ = I 2 ∩ · · · ∩ I n . Then {I 1 , I ∩ } is an orthogonal pair of Bousfi eld sub categ ories of T . By hyp othesis T / I 1 is α - compactly generated, and I ∩ is α -compactly generated in T / I 1 , so in ord er to apply the n = 2 case of the Theorem to the p air I 1 , I ∩ it r emains to c hec k that (i) T / I ∩ is α -compactly generated, and (ii) I 1 is α -compactly generated in T / I ∩ . Set T = T / I ∩ and for I ∈ F define I = I / ( I ∩ I ∩ ). This is a Bousfield sub category of T , and { I 2 , . . . , I n } is a co cov ering of T (Lemma 38). Moreo v er: – F or I ∈ F \ {I 1 } th e category T / I ∼ = T / I is α -compactly generated. – F or I ∈ F \ {I 1 } and a n onempt y s ubset F ′ ⊆ F \ {I , I 1 } th e image of the canonical functor \ I ′ ∈F ′ I ′ inc − → T can − → T / I ∼ = T / I is ju st the essen tial image of the comp osite \ I ′ ∈F ′ I ′ inc − → T can − → T / I , whic h is, by hypothesis, α -compactl y generated in T / I . F rom the in ductiv e h yp othesis, w e dedu ce that T is α -compactly generated, and that X ∈ T is α -compact if and only if the images of X in T / I ∼ = T / I are α -compact for eac h I ∈ F \ {I 1 } . This v erifies condition ( i ) ab o ve , and it remains to chec k ( ii ). Iden tify I 1 as a s ub category of T via th e em b edd ing I 1 − → T − → T / I ∩ . Th en I 1 is a Bousfield sub category , prop erly intersecting I for I ∈ F \ {I 1 } . Moreo ve r: – F or I ∈ F \ {I 1 } th e sub category I 1 / ( I 1 ∩ I ) of T / I is id entified, u nder the equ iv alence T / I ∼ = T / I , with I 1 / ( I 1 ∩ I ), whic h is α -compactly generated in T / I b y hypothesis. 9 D aniel Murfet – F or ev ery I ∈ F \ {I 1 } and nonempty subset F ′ ⊆ F \ {I , I 1 } the image of I 1 ∩ \ I ′ ∈F ′ I ′ inc − → T can − → T / I ∼ = T / I is ju st the essen tial image of the comp osite I 1 ∩ \ I ′ ∈F ′ I ′ inc − → T can − → T / I whic h is, by hypothesis, α -compactl y generated in T / I . F rom the inductive hyp othesis (with S = I 1 ) we conclude that I 1 is α -compactly generated in T . Ha ving no w established b oth ( i ) and ( ii ) ab o v e, we deduce from the n = 2 case of the T heorem that T is α -compactly generated, and that X ∈ T is α -compact if and only if X is α -compact in b oth T / I 1 and T . But the image of X in T is α -compact if and only if th e images of X in T / I ∼ = T / I are α -compact for I ∈ F \ {I 1 } , w h ic h gives the desired criterion for α -compact ness in T . T o complete th e inductiv e step, it remains to treat the second statement : we are giv en a Bousfield sub category S prop erly in tersecting ev ery I ∈ F , satisfying conditions (3) , (4). By hyp othesis, then, S / ( S ∩ I 1 ) and I ∩ ∩ S are α -compactly generated in T / I 1 , and to apply the n = 2 case of the Theorem to S and the co co v er {I 1 , I ∩ } it remains to c hec k that (i)’ S / ( S ∩ I ∩ ) is α -compactly generated in T , and (ii)’ I 1 ∩ S is α -compactly generated in T . Set S = S / ( S ∩ I ∩ ). T his is a Bousfield sub categ ory of T prop erly intersecting every elemen t of the co co v ering { I 2 , . . . , I n } of T . Moreo ve r: – F or I ∈ F \ {I 1 } the sub categ ory S / ( S ∩ I ) of T / I is id en tified un der th e equiv alence T / I ∼ = T / I w ith the sub catego ry S / ( S ∩ I ), whic h is α -compactly generated in T / I b y h yp othesis. – F or ev ery I ∈ F \ {I 1 } and nonempty subset F ′ ⊆ F \ {I , I 1 } the image of S ∩ \ I ′ ∈F ′ I ′ inc − → T can − → T / I ∼ = T / I is ju st the essen tial image of the comp osite S ∩ \ I ′ ∈F ′ I ′ inc − → T can − → T / I , whic h is, by hypothesis, α -compactl y generated in T / I . F rom the indu ctive hyp othesis, we conclude that S is α -compactly generated in T , whic h is ( i ) ′ ab o v e. A s imilar argumen t v erifies ( ii ) ′ , and from the n = 2 case of th e Theorem w e conclude that S is α -compactly generated in T . This completes the in ductiv e step, and thus the pr o of. Corollar y 13 . In the situation of Th eorem 1, for any regular card inal β > α an ob ject X ∈ T is β -compact if and on ly if the image of X is β -compact in T / I for every I ∈ F . Pr o of. If a triangulated category Q is α -compactly generated, or a sub catego ry S is α -compactly generated in some larger triangulated category , then the same is true for an y regular cardin al β > α . Hence, if the co co v er F satisfies the h yp otheses of T heorem 1 for α , it satisfies the same conditions for β > α , whence th e claim. 4. Deriv ed Categories of Rings In the next section w e obtain a c haracterisation of the α -compact ob jects in the deriv ed category of a sc heme. W e will u s e a r eduction to the affine case, s o in this section we pr ep are the ground with a 10 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories review of some facts ab out the deriv ed cate gory of a r ing. Throughout, a ring is a (not necessarily comm utativ e) asso ciativ e ring with iden tit y , and all mo d ules are left m o dules. Giv en a ring R we denote b y D ( R ) the u n b ounded d eriv ed category of R -mo du les. If α is a regular cardinal, then D ( R ) α denotes the full sub category of α -compact ob jects in D ( R ). A complex of R -mo dules P is called K -pr oje ctive if, for ev ery acyclic complex X of R -mo du les, the complex of ab elian grou p s Hom R ( P , X ) is acyclic [S pa88]. F or example, an y b ounded ab ov e complex of pro jectiv e R -mo d ules is K -pro jectiv e. The K -pr oje ctive r esolution of a complex of R - mo dules M is a quasi-isomorphism P − → M , wh er e P is K -pro jectiv e. In this case, P is the unique (up to homotop y equiv alence) K -pro jectiv e complex isomorph ic to M in D ( R ). Theorem 14 (Neeman) . Let R b e a ring. The deriv ed category D ( R ) is compactly generated, and giv en a regular cardin al α > ℵ 0 a complex of R -mo dules is α -compact in D ( R ) if and only if it is quasi-isomorphic to a K -pro jectiv e complex of free R -mo du les of rank < α . Remark 15 . Since we are allo win g n oncomm utativ e rings R , one has to b e a bit careful ab out the meaning of “rank”; w e d irect the reader to [Nee08, (5.2 )]. Pr o of. Pa rt of this criterion is stated w ithout pr o of in [Nee01], and th e full statemen t can b e d educed from th e more general r esults of [Nee08, § 7]. T o b e p recise: taking K -pro jectiv e resolutions d efines a fully f aithful functor D ( R ) − → K (Pro j R ), an d we identify D ( R ) as a sub category of K (Pro j R ) via this emb ed ding. Neeman pr o v es in [Nee 08 ] that K (Pro j R ) is ℵ 1 -compactly generated. Since D ( R ) is a lo calising sub category of K (Pro j R ) generate d by R , w h ic h is compact in K (Pro j R ), it follo ws from Theorem 5 that for any regular cardinal α > ℵ 0 w e ha v e D ( R ) α = K (Pr o j R ) α ∩ D ( R ) , that is, a complex M of R -mo d ules is α -compact in D ( R ) if and only if the K -pro jectiv e resolution P of M is α -compact in K (Pro j R ). But by [Nee08, Prop osition 7.4] and [Nee08, Pr op osition 7.5], P is α -compact in K (Pr o j R ) if and only if it is homotopy-e quiv alen t to a complex of free R -mo dules of rank < α . Remark 16 . In the con text of the th eorem, it is natural to ask if the condition of K -pro jectivit y can b e dropp ed, that is: are the α -compact ob jects in D ( R ) precisely the complexes quasi-isomorphic to a complex of free mo d ules of rank < α ? W e will see in Th eorem 19 that this is tru e pro vided that R is either left no etherian or has cardinality < α . In general, h o w ev er, the answer is negativ e: w e construct a counterexa mple in App endix A, consisting of a rin g B and a complex of free B -mo dules of rank < α that is not α -compact in D ( B ). Definition 17 . Let R b e a ring and α an infinite cardinal. An R -mo dule M is said to b e α - gener ate d if it can b e generated as an R -mo d ule b y a sub set of cardinalit y α . W e say th at M is sub- α -g ener ate d if it can b e generated by a sub set of cardinalit y < α . W e in clude a pro of of th e follo wing standard fact: Lemma 18 . Let R b e a rin g, α an infinite cardinal, and supp ose that R is either left no etherian, or has cardin alit y 6 α . Then if an R -mo dule M is α -generated, every sub mo dule of M is α -generated. Pr o of. Let κ b e a fix ed regular card inal with κ > α . W e p ro v e the follo wing statemen t by transfinite induction: if x is an ordinal < κ , and M is an y R -mo du le generated by a set of cardin ality | x | , then ev ery submo d ule of M is generated by a set of cardinalit y < κ . Call th is statemen t B ( x ). T aking x = α and κ to b e the successor card in al of α (whic h is regular) giv es th e result. Successor ordinals: if x is an ordinal < κ then either | x | = | x + | , in whic h case the statemen t B ( x + ) is just B ( x ) and the inductiv e step is trivial, or these t w o cardinals are distinct, in which 11 D aniel Murfet case x is a finite cardin al. If x is finite, then since R is either left no etherian or h as cardinalit y 6 α , it is straigh tforw ard to v erify that B ( x ) holds. Limit ordinals: assu me that x is a limit ordinal < κ , and that B ( β ) holds for all β < x . W e m a y assume that x is a cardinal, since otherwise the inductiv e s tep is trivial. Let M b e an R -mo dule generated by a set of cardinality x , sa y by { m t | t < x } , and for t < x let M ℵ 0 a r egular cardinal. Supp ose that R is either left no etherian, or has cardinalit y < α . Then for a complex F of R -mo du les the follo wing are equiv alent: (i) F is α -compact in D ( R ) . (ii) F is isomorphic, in D ( R ) , to a complex of free R -mo dules of rank < α . (iii) H i ( F ) is a sub - α -generated R -mo du le for all i ∈ Z . Pr o of. The implication ( i ) ⇒ ( ii ) is a consequence of Theorem 14, while ( ii ) ⇒ ( iii ) follo w s from Lemma 18, so it remains to pro v e th at ( iii ) ⇒ ( i ). Let D α ( R ) d enote the f u ll sub categ ory of D ( R ) consisting of complexes w ith su b- α -generated cohomology . Using Lemm a 18 this is easily c hec ke d to b e an α -lo calising triangulated sub category . W e already know that D ( R ) α ⊆ D α ( R ), and w e wa nt to p ro v e the reve rse inclusion. Let u s b egin with a tec hnical observ ation. Giv en a complex F in D α ( R ), w e claim that it is p ossible to construct a morphism of complexes φ : F − → F ′ with F ′ ∈ D α ( R ) suc h that the mapp ing cone of φ is α -compact and φ is a ghost , that is, the induced m aps H i ( φ ) : H i ( F ) − → H i ( F ′ ) are zero for all i ∈ Z . F or eac h i ∈ Z there exists a surjectiv e map P i − → H i ( F ) from some free R -mo dule P i of rank < α , and this lifts to a morphism of complexes g i : Σ − i P i − → F . The coprod uct P = L i ∈ Z Σ − i P i is α -compact in D ( R ), and the sum g : P − → F of the g i ’s is a morphism of complexes surjectiv e on cohomology . Extending to a triangle P g − → F φ − → F ′ − → Σ P w e obs er ve that φ is a ghost with α -compact mapp ing one, and F ′ has su b- α -generated cohomology . If we b egin with a fixed F = F 0 in D α ( R ) and iterate this p ro cess to construct a sequence F 0 φ 0 − → F 1 φ 1 − → F 2 φ 2 − → · · · of gh ost maps φ i in D α ( R ) with α -compact cones, then th e homotop y colimit of this s equ ence in D ( R ) is acyclic (that is, zero). Since D α ( R ) is closed un der counta ble copro du cts, this also calculates the homotop y colimit in D α ( R ), and th us in the quotien t D α ( R ) / D ( R ) α . But in this qu otien t eac h φ i is an isomorphism (as it has zero cone), s o in this case the homotopy colimit is equal to F . T ogether, th ese obs er v ations imply that F ∈ D ( R ) α , as claimed. 12 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories Remark 20 . W e learn fr om the theorem that when α > | R | , the α -compacts in D ( R ) can b e c har- acterised by the “size” of their cohomolo gy . In fact, this is a general p henomenon in w ell-generated triangulated categories, as explained by Kraus e in [Kr a01, Theorem B] and [Kra02, Th eorem C]. 5. Derived Ca tegories of Sc hemes In this section we apply Th eorem 1 to the d er ived category of quasi-coheren t shea v es on a scheme. W e refer the reader to [Lip09, Nee96, AJL97, AJPV08] for bac kground on unb ou n ded deriv ed catego ries of s c hemes. A sc heme is semi-sep ar ate d if it admits a co vering by affine op en sub sets { V i } i ∈ I with all pairwise in tersections V i ∩ V j affine; see [AJPV08 , TT90]. Separated sc hemes are semi-separated, and semi-separated sc hemes are qu asi-separated. Giv en a sc heme X we denote by D ( X ) the unb ound ed d er ived category of quasi-coheren t s hea v es on X . If X is quasi-compact and semi-separated, then D ( X ) is equiv alen t to the full su b category of complexes with quasi-coheren t cohomology in the deriv ed category of shea ves of O X -mo dules [BN93]. Let X b e a quasi-compact semi-separated sc heme, and { U 1 , . . . , U n } a co v er of X by affin e op en subsets. F or 1 6 i 6 n set Z i = X \ U i and denote by D Z i ( X ) the full sub category of D ( X ) consisting of complexes with cohomolog y su pp orted on Z i . Th e inclusion D Z i ( X ) − → D ( X ) and restriction ( − ) | U i fit into a sequence of functors 0 / / D Z i ( X ) inc / / D ( X ) ( − ) | U i / / D ( U i ) / / 0 whic h is exact, in the sense that ( − ) | U i induces an equiv alence D ( X ) / D Z i ( X ) ∼ − → D ( U i ). I n fact this is a localisation sequen ce: the r igh t adjoin t of D Z i ( X ) − → D ( X ) is Grothendiec k’s lo cal cohomology functor R Γ Z i ( − ), and the right adjoin t of ( − ) | U i is the d eriv ed direct image R f ∗ , wh ere f : U i − → X is the inclusion. In particular, eac h D Z i ( X ) is a Bousfield sub categ ory of D ( X ). The family F = { D Z 1 ( X ) , . . . , D Z n ( X ) } is a cocov ering of D ( X ), and Rouq u ier p ro v es in [Rou08, § 6.2] that this co co v ering satisfies the h yp otheses (1) , (2) of Theorem 1 f or the cardinal α = ℵ 0 . Let us examine the con tent of these hyp otheses, and sket c h Rouquier’s argumen t in eac h case: (1) requ ires that D ( X ) / D Z i ( X ) ∼ = D ( U i ) b e compactly generated for 1 6 i 6 n . But U i ∼ = Sp ec( A i ) is affine, and th us D ( U i ) ∼ = D ( A i ) is kno wn to b e compactly generated. (2) requ ires, giv en an index 1 6 j 6 n and n onempt y su b set I ⊆ { 1 , . . . , n } not con taining j , that the essentia l image of the comp osite T i ∈ I D Z i ( X ) inc / / D ( X ) ( − ) | U j / / D ( U j ) b e compactly generated in D ( U j ). Bu t this image is just D Z ( U j ), where Z is the complemen t of U j ∩ S i ∈ I U i in U j . Sin ce U j is affine one can generate this su b category b y a Koszul complex, whic h is compact in D ( U j ); s ee [Rou08, Prop osition 6.6] or [BN93 , Prop osition 6.1]. In particular, Rouquier pro v es that if Z is a closed subset of X with quasi-compact complemen t U , then D Z ( X ) is compactly generated in D ( X ). S ince the r estriction functor ( − ) | U : D ( X ) − → D ( U ) factors as D ( X ) − → D ( X ) / D Z ( X ) ∼ = D ( U ) we infer from [Nee01 , Theorem 4.4 .9] that the functor ( − ) | U preserve s α -compactness for any regular cardinal α . Let u s record the follo wing sp ecial case of Corollary 13, with T = D ( X ) and F as ab o v e. Pr opo sition 21 . Let X b e a q u asi-compact semi-separated s cheme and α a r egular cardinal. Giv en a co v er { U 1 , . . . , U n } of X by affine op en subsets, a complex F of quasi-coheren t shea v es on X is α -compact in D ( X ) if and only if F | U i is α -compact in D ( U i ) for 1 6 i 6 n . 13 D aniel Murfet The prop osition reduces the problem of und erstanding the α -compacts in D ( X ) to the problem of un derstanding the α -compacts in the deriv ed categ ory of a rin g, which w as settled in Section 4. Pr opo sition 22 . Let X b e a qu asi-compact semi-separated scheme, and α > ℵ 0 a r egular cardinal. A complex F of quasi-coherent sh eav es on X is α -compact in D ( X ) if and only if, for ev ery x ∈ X , there is an affine op en n eigh b orho o d U of x suc h that Γ( U, F ) is quasi-isomorphic as a complex of A = Γ( U, O X ) -mo dules to a K -pro jectiv e complex of free A -mo dules of rank < α . Pr o of. Using P rop osition 21 we reduce to the case of affine X , whic h is Theorem 14. Let D ( X ) α denote the fu ll sub category of α -compact ob jects in D ( X ). S ub categ ories of D ( X ) are t ypically d efined by imp osing conditions on h omology , so it is comforting to hav e suc h a description of D ( X ) α . W e b egin with some defin itions. Definition 23 . Let α b e an infin ite cardin al. A quasi-coheren t sheaf F on a sc heme X is said to b e lo c al ly α -gener ate d if f or ev ery x ∈ X , th ere exists an op en neighborh o o d U of x together with an epimorphism L j ∈ J O X | U − → F | U , for some index set J of cardinalit y α . If for eac h x ∈ X we can arrange for the set J to b e of cardinalit y < α , then F is lo c al ly sub- α -gener ate d . Lemma 24 . Let R b e a commutat ive ring, α an infi nite cardinal and F an R -mo dule. Th en the follo w ing are equiv alen t: (i) F is an α -generated R -mo du le. (ii) Th e complex F of q u asi-coheren t shea v es on Sp ec( R ) asso ciate d to F is lo cally α -generated. Pr o of. ( i ) ⇒ ( ii ) is clear. F or ( ii ) ⇒ ( i ), set X = Sp ec( R ) and supp ose that F is lo cally α - generated. W e may fi nd generators f 1 , . . . , f r of the un it ideal of R , with the pr op erty that on eac h U i = D ( f i ) there is an epimorphism ⊕ j ∈ J i O X | U i − → F | U i for some set J i of cardinalit y α . That is, F [ f − 1 i ] can b e generated as an R [ f − 1 i ]-mo dule by a subset { a j /f n j i } j ∈ J i of cardinalit y | J i | = α . F orm a set J consisting of the u nion, o v er eac h 1 6 i 6 r , of the set of numerato rs { a j } j ∈ J i . This set has cardinalit y α and generates F as an R -mo dule. Finally , w e arriv e at a c h aracterisatio n of α -compactness in terms of cohomology shea v es. Corollar y 25 . Let X b e a quasi-compact semi-separated sc heme and α > ℵ 0 a regular cardinal. Supp ose th at X is either (a) no etherian, or (b) admits a co ve r by op en affines { U i } 1 6 i 6 n with Γ( U i , O X ) of cardinalit y < α for 1 6 i 6 n . Then a complex F of quasi-coheren t shea v es on X is α -compact in D ( X ) if and only if H i ( F ) is lo cally su b - α -generated for ev ery i ∈ Z . Pr o of. Under either h yp othesis on X there is an affine op en co ver { U 1 , . . . , U n } of X such th at the conclusion of Th eorem 19 app lies to eac h of th e r ings Γ( U i , O X ). By Lemma 24, a quasi-coheren t sheaf G on X is lo cally sub- α -generated if and on ly if Γ( U i , G ) is a su b- α -generated Γ( U i , O X )- mo dule for 1 6 i 6 n , so the result f ollo w s from Pr op osition 21. App endix A. Coun terexample Let R b e a rin g and α > ℵ 0 a regular cardinal. W e pro v ed in Theorem 19 that when R is either left no etherian or has cardin alit y < α , the α -compact ob jects in D ( R ) are the ob jects isomorphic to a complex of free R -mo dules of rank < α . In this app endix w e sho w that this charact erisation of the α -compact ob jects cannot hold for arbitrary r ings, by constru cting f or any giv en regular cardinal 14 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories α > ℵ 0 a commutat ive lo cal ring B and a complex of fr ee B -mo du les of rank one whic h is not α -compact. Needless to sa y , B is n on -n o etherian and has cardinalit y > α . Let k b e a field and β an infi nite cardinal, and I = { x i } i ∈ β a s et of v ariables indexed by β . W e denote by k [[ I ]] the r ing of formal p o we r series in the set of v ariables I . More precisely , let N ( I ) b e the set of all functions γ : I − → N with finite supp ort, and define k [[ I ]] := { fun ctions f : N ( I ) − → k } , with the usual addition ( f + g )( γ ) = f ( γ ) + g ( γ ) and pro du ct ( f · g )( γ ) = P α + β = γ f ( α ) g ( β ). T hen k [[ I ]] is a comm utativ e lo cal rin g, with maximal ideal giv en by th e ideal of p ow er s eries with zero constan t term . W e sa y that a m onomial in the set of v ariables I is pur e if it is of the form x n i for some i ∈ β and n > 0, with x 0 i understo o d to b e the iden tit y in k [[ I ]]. Let a denote th e ideal of p o wer series in which the co efficient of ev ery pure monomial is zero (e.g. x i x j for i 6 = j ) and defin e B ′ := k [[ I ]] / a . Eac h residue class of B ′ con tains a unique p o we r series f in which only pur e monomials ha v e nonzero co efficien ts, and su c h f can b e written as a f ormal sum f = λ · 1 + X i ∈ β X n > 1 f i,n · x n i , λ, f i,n ∈ k . (11) W e sa y that a p o wer series f involves a v ariable x i if the co efficient of x n i in f is n onzero f or some n > 1. Finally , w e may defin e the desired rin g B as a su b ring of B ′ . Definition 26 . Giv en a field k and an infin ite cardin al β , w e d efi ne a commutat ive k -algebra B b y B := { f ∈ B ′ | f in v olv es on ly finitely man y v ariables in I } . Concretely , this is the r in g of formal p o wer s er ies of the form (11) in the set of v ariables { x i } i ∈ β , with eac h p o w er series in v olving only a fin ite num b er of v ariables. P o w er series are m ultiplied according to the relations x n i x m i = x n + m i and x i x j = 0 for i 6 = j . T his is a commutat ive lo cal ring, with maximal ideal m B giv en b y the set of p o w er series with zero constant term, and resid ue fi eld k = B / m B . Remark 27 . The follo wing prop erties of B are immediate: – Giv en a nonempt y subset J ⊆ β , the id eal ( x j ) j ∈ J in B consists pr ecisely of those p o we r series f with zero constant term, whic h d o not in vo lv e any v ariable x i with i ∈ β \ J . – Giv en i ∈ β , the k ernel of B x i − → B is the ideal ( x j ) j ∈ β \{ i } . – F or an y nonempty subset J ⊆ β there is an internal direct sum ( x j ) j ∈ J ∼ = L j ∈ J ( x j ). W e w ill n eed the follo wing consequence of Theorem 14. Lemma 28 . Let A b e a comm utativ e lo cal r ing with residu e field k , and α > ℵ 0 a regular cardinal. If M is an A -mo dule b elonging to D ( A ) α then r ank k T or n ( M , k ) < α for all n > 0 . Pr o of. If M b elongs to D ( A ) α then by Theorem 14 it admits a K -pro jectiv e resolution b y a complex P of free A -mo d ules of rank < α . Hence P ⊗ A k = M ⊗ L A k is a complex of k -v ector sp aces of d imension < α , and the claim follo ws. The key pathology of the ring B b ecomes app arent in the next lemma. Lemma 29 . Giv en i ∈ β and the corresp onding ideal ( x i ) in B , we hav e rank k T or n (( x i ) , k ) = ( 1 n = 0 , β n > 0 . Consequent ly , if α > ℵ 0 is a regular cardinal such that α 6 β , then ( x i ) do es not b elong to D ( B ) α . 15 D aniel Murfet Pr o of. T o construct a fr ee resolution of ( x i ) we b egin with the epimorphism B x i − → ( x i ), which has k ernel ⊕ i 1 ∈ β \{ i } ( x i 1 ). O ne p r o ceeds to construct the follo wing resolution of ( x i ), call it F : · · · − → M i 1 ∈ β \{ i } i 2 ∈ β \{ i 1 } i 3 ∈ β \{ i 2 } B − → M i 1 ∈ β \{ i } i 2 ∈ β \{ i 1 } B − → M i 1 ∈ β \{ i } B − → B with m o dules F − n = L i 1 ∈ β \{ i } ,...,i n ∈ β \{ i n − 1 } B for n > 1 (w e set i 0 = i ) and differen tials ∂ − n F = M i 1 ∈ β \{ i } ,...,i n − 1 ∈ β \{ i n − 2 } ( x i n ) i n ∈ β \{ i n − 1 } , where ( x i n ) i n ∈ β \{ i n − 1 } denotes an in finite ro w matrix ⊕ i n ∈ β \{ i n − 1 } B − → B . Notice that for n > 1 the mo dule F − n is free of r ank β , and the d ifferen tials in F are annihilated by − ⊗ A k , so th e v ector space T or n (( x i ) , k ) = H − n ( F ⊗ A k ) has the desired rank for n > 0. I t follo ws from Lemma 28 that ( x i ) is not α -compact in D ( B ). Pr opo sition 30 . Let α > ℵ 0 b e a regular cardinal and supp ose that α 6 β . T hen the complex T : · · · − → 0 − → 0 − → B x 0 − → B x 1 − → B x 0 − → B x 1 − → · · · do es n ot b elong to D ( B ) α . Pr o of. T o b e clear, the complex T is zero in degrees < 0 and B in degrees > 0, with different ials given b y alternating pr o ducts with x 0 and x 1 . It is n ot d ifficult to see that there is a quasi-isomorphism ( x i ) i> 0 ⊕ M k > 1 Σ − k ( x i ) i> 1 ∼ = T , and an isomorphism of B -mo dules ( x i ) i> 0 ∼ = L i> 0 ( x i ), so that for an y i > 0 in β the ideal ( x i ) is a d irect summand of T in D ( B ). If T b elonged to D ( B ) α then, since this sub category is thic k , we w ould ha v e ( x i ) ∈ D ( B ) α . But this p ossibilit y is excluded b y Lemma 29, whence T / ∈ D ( B ) α . T o conclude, if α > ℵ 0 is a regular cardinal, and w e tak e any fi eld k and β = α in the ab o v e, we obtain a comm utativ e lo cal rin g B and complex T of free B -mo d ules of rank one, with T / ∈ D ( B ) α . App endix B. Prop er Intersection Throughout this app endix, T is a triangulated category . W e say that a su b category C of T is strictly ful l if it is full, and whenev er X ∈ T is isomorph ic to an ob j ect of C , then X ∈ C . Definition 31 . Let A , B b e strictly full su b categories of T closed u nder Σ , Σ − 1 . Th en A ⋆ B denotes the fu ll sub category of T giv en by the ob jects X ∈ T for wh ic h there exists a tr iangle A − → X − → B − → Σ A with A ∈ A and B ∈ B . Th is is a strictly full s ub category closed un d er Σ , Σ − 1 . Remark 32 . Let A , B , C b e strictly full sub catego ries closed un der Σ , Σ − 1 . It is a sim p le exercise in the o ctahedral axiom to see that A ⋆ ( B ⋆ C ) = ( A ⋆ B ) ⋆ C . Clearly if A is a tr iangulated sub category , then A ⋆ A = A . Definition 33 . Tw o triangulate d s u b categories A , B of T are said to interse ct pr op erly if there is an equalit y of sub categories A ⋆ B = B ⋆ A . In the next lemma w e v erify that this defin ition of prop er in tersection agrees w ith the one given b y Rouquier [Rou08, (5.2. 3)] for Bousfield sub categories. 16 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories Lemma 34 . Let A , B b e triangulated sub categ ories of T . The f ollo win g are equ iv alen t: (i) A and B in tersect prop erly , that is, A ⋆ B = B ⋆ A . (ii) Give n A ∈ A and B ∈ B , every morph ism A − → B an d ev ery morphism B − → A factors through an ob ject of A ∩ B . Pr o of. ( i ) ⇒ ( ii ) Let f : A − → B b e giv en, with A ∈ A and B ∈ B , and extend to a triangle A f − → B − → X − → Σ A. Since X b elongs to B ⋆ A = A ⋆ B , th er e is a triangle A ′ − → X − → B ′ − → Σ A ′ with A ′ ∈ A and B ′ ∈ B . App lying the o ctahedral axiom to the p air B − → X, X − → B ′ w e obtain an ob ject D and triangles B − → B ′ − → D γ − → Σ B , A δ − → Σ − 1 D − → A ′ − → Σ A, suc h that γ ◦ Σ δ = Σ f . F rom the fi rst triangle w e d educe that D ∈ B , and from the second triangle w e conclude that D ∈ A , whence f factors via A ∩ B . The factorisation argument for a morphism B − → A is du al. ( ii ) ⇒ ( i ) Let X ∈ A ⋆ B b e giv en, so that there is a triangle A − → X − → B s − → Σ A. By hyp othesis s factors as B − → D − → Σ A for some D ∈ A ∩ B , and fr om the o ctahedral axiom applied to the p air of morphisms in this f actorisatio n of s , w e conclude that X ∈ B ⋆ A . This sh ows that A ⋆ B ⊆ B ⋆ A , and the rev erse inclusion follo ws similarly . Lemma 35 . Let A , B b e pr op erly in tersecting triangulated sub categ ories of T . Then A ⋆ B is a triangulated su b category of T . Pr o of. It suffi ces to p r o v e that A ⋆ B is closed un der mapping cones. Let f : X − → Y b e a morp hism in T with X , Y ∈ A ⋆ B , and fix triangles X f − → Y − → C − → Σ X , A X − → X − → B X − → Σ A X , A Y − → Y − → B Y − → Σ A Y , where A X , A Y ∈ A and B X , B Y ∈ B . App lying the o ctahedral axiom to the pair A Y − → Y , Y − → C yields an ob ject D and triangles A Y − → C − → D − → Σ A Y , (12) B Y − → D − → Σ X − → Σ B Y . (13) Using the prop er in tersection prop erty , w e infer f r om (13) that D ∈ B ⋆ ( A ⋆ B ) = B ⋆ ( B ⋆ A ) = ( B ⋆ B ) ⋆ A = B ⋆ A = A ⋆ B , whence by (12) w e ha v e C ∈ A ⋆ ( A ⋆ B ) = ( A ⋆ A ) ⋆ B = A ⋆ B . Hence A ⋆ B is closed und er mapping cones, and therefore triangulated. Remark 36 . Let A , B b e p rop erly in tersecting triangulated su b categories of T . Th en A ⋆ B is clearly the smallest triangulated sub catego ry of T con taining A ∪ B . Notice th at if A and B are lo calising, then so is A ⋆ B . W e w ill n eed the follo wing results from [Rou08]. 17 R ouquier ’s Cocovering Theorem and Well-gene ra ted Triangula ted Ca tegories Lemma 37 . Let A , B b e prop erly in tersecting Bousfield sub categ ories of T . Then A ∩ B an d A ⋆ B are Bousfield sub catego ries of T . Pr o of. See [Rou08, Lemma 5.8] . Lemma 38 . Let F b e a fin ite f amily of Bousfield sub categories of T , an y tw o of wh ic h in tersect prop erly . Giv en a sub s et F ′ ⊆ F , th e in tersection ∩ I ∈F ′ I is a Bousfield sub category of T intersecting prop erly with any sub category in F . Giv en I 1 , I 2 , I ∈ F , the quotien ts I 1 / ( I 1 ∩ I ) and I 2 / ( I 2 ∩ I ) are prop erly int ersecting Bousfield sub categ ories of T / I . Pr o of. See [Rou08, Lemma 5.9] . Referen ces AJL97 Leovigildo Alonso T arr ´ ıo, Ana Je r em ´ ıas L´ op ez, a nd Jo seph L ipma n, L o c al homolo gy and c ohomol o gy on schemes , Ann. Sci. ´ Ecole Nor m. Sup. (4) 30 (1 997), no. 1, 1–39 . AJS00 Leovigildo Alonso T arr ´ ıo, Ana Jerem ´ ıas L´ op ez, and Mar ´ ıa Jos´ e Souto Salorio , L o c alization in c ate- gories of c omplexes and unb ounde d r esolutions , Ca nad. J . Math. 52 (2000), no . 2, 22 5–247 . AJPV08 Leovigildo Alonso T ar r ´ ıo, Ana Jerem ´ ıas L´ opez, Mar ta P´ erez Rodr ´ ıguez, and Ma r ´ ıa J. V ale Go n- salves, The derive d c ate gory of quasi-c oher ent she aves and axiomatic stable homotopy , Adv. Ma th. 218 (2008), no . 4 , 1224 –125 2 . BN93 Marcel B¨ okstedt and Amnon Neeman, Homotopy limits in t riangulate d c ate gories , Comp ositio Math. 86 (1 993), no. 2, 209– 234. Kra01 Henning Kr ause, On Ne eman ’s wel l gener ate d triangulate d c ate gories , Do cument a Math. 6 (2001 ), 121–1 26. Kra02 , A Br own r epr esentability the or em via c oher en t functors , T o p o logy 41 (2 0 02), 853–86 1. Kra07 , L o c alization the ory for triangulate d c ate gories , a rXiv:0806 .1324 . Lip09 Joseph Lipman, Notes on derive d c ate gories and derive d fun ctors , Lecture notes in Ma th., no. 1960, Spr ing er (2009 ), 1– 2 59. Av ailable online at: http:// www.m ath.purdue.edu/$\sim$lipman/ Dualit y.pdf . Mur07 Daniel Murfet, The mo ck homotopy c ate gory of pr oje ctives and Gr othendie ck duality , Ph.D. thesis, 2007. Av ailable from: http:/ /www.t herisingsea.org/thesis.pdf . Mur08 , The pur e derive d c ate gory of flat she aves and Gr othendie ck duality , preprint. Nee96 Amnon Neeman, The Gr othendie ck duality t he or em via Bousfield’s te chniques and Br own r epr e- sentability , J. Amer. Math. So c. 9 (1 996), no. 1, 205– 236. Nee01 , T riangulate d c ate gories , Annals of Mathematics Studies, vol. 148, Pr inceton Universit y Pre s s, Princeton, NJ, 2001. Nee05 , A su rvey of wel l gener ate d triangulate d c ate gories , Represe nt ations of algebr as and rela ted topics, Fields Inst. Commun., vol. 45 , Amer. Math. So c., Pr ovidence, RI, 20 05, pp. 307– 329. Nee08 , The homotopy c ate gory of flat mo dules, and Gr othendie ck duality , Inv ent. Math. 174 (2008), 255–3 08. Rou08 Rapha¨ el Rouquier, Dimensions of triangulate d c ate gories , J. K-Theory 1 (2008), no. 2, 193– 256. Spa88 N. Spaltenstein, Reso lutions of unb ounde d c omplexes , Comp os itio Ma th. 65 (1988), no . 2, 12 1–15 4 . TT90 R. W. Tho mason and Thomas T r obaugh, Higher algebr aic K -the ory of schemes and of derive d c at- e gories , The Gr othendieck F estschrift, V ol. I I I, Pr ogr. Math., vol. 8 8, B ir kh¨ auser Bosto n, Boston, MA, 1 990, pp. 24 7–43 5 . Daniel Mur f et murfet@math.uni-b onn.de Hausdorff Cente r for Mathematic s, Univ ersit y of Bonn 18