On the derived category of a graded commutative noetherian ring

ON THE DER IVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOE THERIAN RING IV O DELL’AMBROGIO AND GREG STEVENSON Abstract. F or an y grade d comm uta tiv e noetherian ring, where the grading group is ab elian and where commu tativit y is allow ed to hold in a quite general sense, we est ablish an inclusion-preserving bijection b et w een, on the one hand, the t wist-closed localizing sub categories of the derived cate gory , and, on the other hand, subsets of the homogeneous sp ectrum of prime ideals of the ring. W e provide an application to weigh ted pro jectiv e sch emes. Contents 1. Int ro duction 1 2. Definitions and basic results 3 3. Graded fields 9 4. The small suppor t 10 5. The sp ectrum and loca lizing tensor ideals 13 6. An application to (w eighted) pro jective schemes 15 References 18 1. Introduction In his 1992 pap er [ 22 ], Amnon Neeman ha s sho wn that for a no e ther ian co mm u- tative ring R “one has a co mplete and v ery sa tis fa ctory description of the sp ectral theory of its derived catego ry .” Indeed, after providing a cor rect pro of of Hopkins’ classification of the thick sub categ ories of D b ( R - pro j) by way of s pe c ia lization closed s ubs ets o f Sp ec R , he pro cee ds to sho w that ev en for the un bounded derived category D ( R ) one has an orderly classification, no w in the for m of a bij ection  subsets of Spec R  τ / / o o σ  lo calizing sub categ ories of D ( R )  . This restricts to a bijection b etw een sp ecia lization clo sed subsets on one side and smashing subcateg ories on the other. F rom this one can prov e the telescope c o n- jecture for D ( R ), whic h is the statement that ev ery smas hing subcateg ory of D ( R ) (a lo calizing subcategor y whos e inclusion has a co pro duct preserv ing righ t adjoint) is gener ated by the compact ob jects it con tains. Another proof of the classificatio n for D b ( R - pro j) follows. Neeman’s bijection sends a s ubs e t S ⊆ Sp ec R to the lo calizing subca tegory τ ( S ) = h k ( p ) | p ∈ S i g e ne r ated by the residue fields at the primes in S , and it sends a lo calizing sub ca teg ory L to the set σ ( L ) = { p ∈ Sp ec R | ∃ X ∈ L s.t. k ( p ) ⊗ L X 6 = 2000 Mathematics Subje ct Classific ati on. 13A02, 13D09. Key wor ds and phr ases. Lo calizing subcategories, gr aded ring, weigh ted pro jectiv e scheme. 1 2 IVO DELL’AMBROGIO AND GREG STEVENSON 0 } . In terms of the small supp ort ssupp X = { p ∈ Spec R | k ( p ) ⊗ L X 6 = 0 } of a complex X , it can be refo r mulated as follows: τ ( S ) = { X ∈ D ( R ) | ssupp X ⊆ S } , σ ( L ) = [ X ∈L ssupp X . More recently Dav e Be nson, Srik anth Iyengar and Henning K rause [ 5 , 7 ] hav e int ro duced a no tion of supp or t in the situation whe r e one is given a compactly gen- erated triang ulated categor y T together with a n action by a Z -gra ded commutativ e no etherian ring R . In pr actice it is often the case that T is a rig idly-compactly generated tensor tr ia ngulated categor y , which we now assume (this means: T is also a tensor ca tegory with c opro duct preserving exact tensor pro duct ⊗ and with compact unit ob ject 1 , a nd is suc h that its compact and r ig id ob jects co incide). In this generality the supp ort is given b y supp R X = { p ∈ Sp ec h R | Γ p ( 1 ) ⊗ X 6 = 0 } for each X ∈ T , where Spec h R is the homoge ne o us spectrum of R , and where the Γ p ( 1 ) are suitable tensor idempotent ob jects pr ovided by the theory . A salien t feature of the abstract theory is the iden tification of the follo wing tw o hypotheses whic h together guar antee that the supp ort supp R provides a classi- fication of lo calizing tensor idea ls (here hF i ⊗ denotes the lo calizing tensor ideal generated b y a family of ob jects F ⊆ T ): (1) The (tensor) lo c al-to-glob al principle : h X i ⊗ = h Γ p ( 1 ) ⊗ X | p ∈ Spec h R i ⊗ for each ob ject X ∈ T . (2) Minimality : the loca liz ing t ensor ideal h Γ p ( 1 ) i ⊗ is either zer o or minimal for each p ∈ Sp ec h R . In [ 6 ], this theor y is used to give a clas sification of the lo calizing tensor ideals of the stable mo dule catego r y o f a finite gro up. This sp ectacular success notwithstanding, the methods o f lo c. cit . are for some purposes unsatisfacto r y; for ins ta nce, it is not clear how to f ree o neself from the affine tyrann y of the g raded ring. One would also like to captur e classifications whose parameter space is, say , a no etherian scheme. Some progr ess in this direction has b een made quite recently in t he PhD thes is of the second author (se e [ 26 ]). In this work, the Benso n-Iyengar-Kraus e theory is catego rified as follows: T is a ge neral compactly gener ated category , which is endow ed with a n actio n – in a very natural sense – by a rigidly-co mpactly genera ted category R ; this includes or cour se the case whe r e R = T acts on itself v ia its tensor pro duct. Now the relev ant par ameter space is Spc R c , the sp ectrum (in the sens e of Paul Ba lmer [ 1 ]) o f the rigid-compa ct ob jects of R , and the ob jects Γ x ( 1 ) ( x ∈ Sp c R c ) are provided by the abstract theor y o f genera lized Ric k ard idempo ten ts of P aul Balmer and Giordano F avi [ 4 ]. In this new setting the lo cal-to - global principle alwa ys holds, pr ovided the cate- gory R has a Quillen model a nd the space Sp c R c is no etheria n. Moreover, by its very construction the who le theory is p erfectly compa tible with the exta n t p ow erful metho ds of tens o r triangula r geometry [ 3 ]. Once again, if one a ls o has minimality then one gets a c la ssification fr om the ensuing suppo rt theo r y in terms o f some subsets of Spc R c . Using Neeman’s classification one alre ady understands ho w this theory recov ers the support theory of Benson, Iy engar, and Kraus e when the der ived categ ory of a no etherian ring acts on a c o mpactly g enerated triangulated catego ry . As the theor y of Benson, Iyengar, and Krause w orks in the generality of a Z -gr aded noetherian ring it is thus natural to co nsider what ha ppens when one allows the derived cate- gory o f graded mo dules ov er such a ring to act. This re q uires a c omputation of the sp ectrum o f the c ompact ob jects of such a derived categor y and the main g oal o f the current article is to perform this computation. O f co ur se it is in an y case natural ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 3 to a sk if Neema n’s cla ssification extends, in the obvious wa y , to gr aded mo dules ov er graded rings. The ans wer is yes; this op ens the do or to further exploring such categorie s, an undertaking which seems to ha ve man y p otential applica tions. In Section 6 w e provide one first application to derived categ ories of weight ed pr o jec- tive schemes. ∗ ∗ ∗ W e allo w R to b e graded by any ab elia n group G , po s sibly with torsion, and we allow it to b e comm utative up to any reaso nable sign rule, whic h co vers bo th the strictly c ommut ative case as well as the usual graded co mmu tative one (see Def. 2.4 ). If G is non trivial, D ( R ) is not gene r ated b y the tensor unit and therefore our geometric metho ds require us to restrict atten tion to thos e lo calizing subcategor ies which are tensor idea ls; they are th e same a s those whic h a r e closed under twists by arbitr ary elemen ts g of G . Our classificatio n is a bijection as follows:  subsets of Spec h R  τ / / o o σ  t wist-closed lo calizing sub ca tegories of D ( R )  with cor ollaries similar to Neeman’s (see Theo rem 5.7 ). As a sp ecial case we reprove Neeman’s o riginal clas s ification for ordinary ungr aded rings . That is not to say we giv e what could b e consider ed a new pro of; the idea s inv olved are essentially the sa me. Ho w ever, our approach ma kes it abunda n tly clear whic h parts o f the argument are completely gener al and b elong to the rea lm of tensor triangulated categorie s, and which parts a re instead sp ecific to (gr aded) co mm utative no ether ian rings. Here is a sketc h of the pr o of. First, we notice that D ( R ) has a mo del (Prop. 2.17 ) and acts o n itse lf b y its symmetric tensor product ⊗ L . Then we co nsider the smal l supp ort defined b y the graded r esidue fields k ( p ) ssupp X = { p ∈ Spe c h R | k ( p ) ⊗ L X 6 = 0 } and we establish its pr op erties, us ing only the most elemen tary results o f the the- ory o f injective ob jects in the category of graded R - mo dules. In particular , the small supp or t detects the ob jects o f D ( R ) (Co r. 4.8 ), it is compa tible with the ten- sor pro duct (Lemma 4.6 ) a nd it b ehaves nicely with resp ect to co mpact ob jects (Lemmas 4.11 and 4.12 ). By a quite general cr iterion, this is enough to establish a canonical homeomorphism Spec h R ∼ = Spc D ( R ) c (Theorem 5.1 ). Since R is g raded no etherian, this spa ce is noetherian. Hence the abstract theory can b e applied in its full p ow er, and it remains to verify minimality; this follows easily fro m the identit y h Γ p ( 1 ) i ⊗ = h k ( p ) i ⊗ (Prop. 5.5 ) a nd the “field ob ject” proper ty of the residue field k ( p ) (Lemma 3.5 ). Con v entions. All categor ie s a re Z -categories and all functor s are Z -linear. Ac knowledgemen ts. The a uthors are g rateful to Estanislao Herscovic h for a few helpful comments, as well as a health y session of sign chec king, during which his remark a ble s up er -algebra ic p ow ers pro v ed quite instr uctive. 2. Definitions and basic resul ts Let G denote our gr ading group, which will always be assumed to b e ab elian a nd whose op eratio n will b e written additiv ely . By a gr ade d ring R we alwa ys mean a unital a nd asso ciativ e ring gr aded by G ; in other words, R comes tog ether with a decomp osition R = M g ∈ G R g 4 IVO DELL’AMBROGIO AND GREG STEVENSON such that the mu ltiplication satisfies R g · R h ⊆ R g + h for all g , h ∈ G , and thus also 1 ∈ R 0 . A ( left ) gr ade d mo dule over R is a n R -mo dule M together with a decomp osition M = L g ∈ G M g such th at R g M h ⊆ M g + h . W e denote b y R -GrMo d the catego ry of g r aded R -mo dules and degree-zero homomorphisms , i.e. , those R - linear maps f : M → N such that f ( M g ) ⊆ N g for all g ∈ G . As customary we will write deg m = g to indicate that the degree o f m is g , that is, that m ∈ M g . If M is a n R -mo dule and g and element of G , w e wr ite M ( g ) for M twiste d by g , that is, M e ndow ed with the new G -grading with c omp o nents M ( g ) h := M h + g . W e say that an R -mo dule M is gr ade d fr e e if M is a sum of twists of R . Definition 2.1 . The c omp anion c ate gory o f R , deno ted C R , is the small Z -categ ory whose set of ob jects is o b j( C R ) := { g | g ∈ G } , whose morphism groups are g iven by C R ( g , h ) := R h − g , a nd with comp osition given by restricting the multiplication of R to the appropriate homogeneous components: R ℓ − h × R h − g − → R ℓ − g , ( r , s ) 7→ rs . Lemma 2 .2 ([ 14 , Pro po sition I.1.3]) . Ther e is an e quivalenc e b etwe en R - GrMo d and the add itive functor c ate gory Ab C R , given by the fun ctor R - GrMo d → Ab C R , M 7→ ( g 7→ M g ) . A quasi-inverse is pr ovide d by Ab C R → R - GrMo d , F 7→ M g ∈ G F ( g ) , wher e we endow the ab elian gr oup L g ∈ G F ( g ) with the gr ading on evident display and with the R -action induc e d by functoriality. Under this e quivalenc e, the functor C R ( g , − ) c or epr esente d by g c orr esp onds to the fr e e left R -mo dule R ( − g ) .  In the following, we make the identification R -GrMo d = Ab C R whenever c o nv e- nient . It follo ws from this descr iption that the catego ry R -GrMo d is Grothendiec k ab elian, a nd { R ( g ) | g ∈ G } is a set of pro jective gener a tors. This seco nd statement follows immedia tely from the Y o neda lemma: Lemma 2.3. Ther e is a n atur al iso morphism of ab elian g r oups R - GrMo d( R ( − g ) , M ) ∼ − → M g , f 7→ f (1) for e very M ∈ R - GrMo d and g ∈ G .  Note that, under the iden tification of Lemma 2.2 , the Y o neda embedding be- comes the fully faithful functor R : ( C R ) op − → R -GrMo d with R ( g ) = R ( − g ) on ob jects a nd which sends th e morphism r ∈ C R ( g , h ) to righ t m ultiplication with r , seen a s an R -linear map R ( r ) : R ( − h ) → R ( − g ). Definition 2 .4. Let ǫ : G × G → Z / 2 b e a symmetric Z -bilinear map. W e say that the gra de d ring R is ǫ -c ommutative if r · s = ( − 1) ǫ (deg r, deg s ) s · r holds for all homogeneous elements r and s in R . Examples 2 .5 . (1) If ǫ is identically z e ro, then ǫ -comm utativ e just means comm utative. (2) F or G = Z the in teger s and ǫ : Z × Z · → Z → Z / 2 the multiplication map mo dulo tw o , we recov er the familiar notio n of a ( Z -)g r aded co mmutative ring. F or instance, the graded endomor phism ring of the tensor unit ob ject in a n y reasonable tensor triangulated ca tegory will b e suc h a ring, b y the Eckmann-Hilton ar gument (see [ 27 ]). ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 5 (3) Hov ey , P almieri a nd Stric kland [ 1 6 ] en tertain the notion o f a multigraded unital algebra ic stable homoto p y ca tegory , in which one ha s a finite num b er of genera ting “spheres” S 1 , . . . , S d , th us giving rise to an ǫ -commutativ e Z d -graded endomorphis m ring of the tenso r unit, where the signing form ǫ : Z d × Z d → Z / 2 is given b y ǫ (( n 1 , . . . , n d ) , ( n ′ 1 , . . . , n ′ d )) = n 1 n ′ 1 + . . . + n d n ′ d mo d 2 . (4) A c ommut ative super algebra (o r sup erco mm utative algebra) is an algebra graded o v er Z / 2 whic h is ǫ -co mm utative for the m ultiplication map ǫ : Z / 2 × Z / 2 · → Z / 2. W e will need to use lo ca lization for such ǫ -co mm utative rings and to consider their homogeneo us spec tra; es sentially these are g iven b y the obvious constructions, but let us be (at least a little) explicit ab out what is mean t and let us make a few comments on what happ ens at this level of generality . W e b egin, as in the usual Z -graded case, b y defining the ev en part of such a ring. Definition 2.6. Let R be a n ǫ -commutativ e G -gr a ded ring, as in Definition 2.4 . W e define its even p art , written R ev , to b e the commutativ e G -grade d r ing with comp onents ( R ev ) g :=  R g if ǫ ( g , h ) = 0 for all h ∈ G 0 otherwise and with m ultiplication restricted from R . W e say a homo geneous elemen t is even if it belong s to the even part and w e sa y it is o dd if it is not even. R emark 2.7 . Note that the bilinearity and symmetry of ǫ imply that R ev is indeed a w ell-defined unital s ubring of R , whic h moreo ver is commutativ e. Note also that with this definition o dd elemen ts may still b elong to the center; e.g. , if R 0 is a commutativ e r ing and if we endow R := R 0 [ x ] / ( x 2 ) with the usual Z -g rading where deg x = 1 , then the strictly c ommut ative ring R is a lso ǫ - commutativ e for the pro duct ǫ : Z × Z → Z / 2, for whic h x is odd. Observe that when R is ǫ -c ommut ative all homogeneo us idea ls are auto ma tically t wo-sided. W e denote by Spe c h R the collectio n of all ho mo geneous prime ideals of R , and ca ll it the homo gene ous sp e ct rum of R . W e will consider Spec h R as a top ological spa ce with the Zariski top olo gy . As usual, if we consider instea d the homogeneous pr ime ideals of R ev we would get the same space, since the square of a ny homogeneous element is ev en. W e say that R is ( gr ade d ) no et herian if the ascending chain co nditio n holds f or homogeneous ideals of R . F rom this p oint forward all rings w e consider are assumed to be no ether ian. R emark 2.8 . One may susp ect that the no etherianity of R should requir e the gra d- ing g roup G to b e finitely genera ted. This is no t the case: o ne ca n always artificia lly enlarge the grading group (extending R by zer o). A slightly less trivial ex a mple is if R is a graded field (see the next section) in which case it is no etheria n, indepen- dent ly of G . W e make the following easy o bserv ation abo ut the ho mogeneous sp ectrum of such a ring. Lemma 2.9. If R is a no etherian ǫ -c ommutative G -gr ade d ring, then the sp e ctrum of homo gene ous prime ide als, Sp ec h R , is a no etherian top olo gic al sp ac e.  W e can consider the graded localizatio n of a n ǫ -comm utative ring R at a m ul- tiplicative set S consisting of even (and therefore cen tr a l) ho mogeneous elemen ts. The constructio n of this localiz a tion is the ob vious one a nd it enjoys the usual prop erties; in particular , it is aga in a n ǫ - commutativ e G -gr aded ring. Similar ly , 6 IVO DELL’AMBROGIO AND GREG STEVENSON we can also lo calize any graded R -mo dule a t such a m ultiplicative subset. F or a homogeneous pr ime ideal p ⊆ R a nd a gra ded mo dule M , we denote by M p the lo calization of R at the multiplicativ e set S = R ev ∩ R h ∩ ( R r p ) of the e ven ho- mogeneous elemen ts of R not in p . W e observ e that in this ge nerality it is p ossible for odd ele ments to become inv er tible in suc h a lo calization. Next, w e wan t to define a symmetric monoidal structure for gra ded R -mo dules, where R is a llow ed to b e any ǫ -commutativ e G -graded ring. This can b e done quite explicitly as follo ws. Every left R -mo dule M = R M carries a canonical str uc tur e of right R -mo dule, ma king it in to an R -bimo dule R M R , b y setting (2.10) m • r := ( − 1) ǫ (deg m, de g r ) rm for all homog eneous r ∈ R and m ∈ M . Every morphism of left R -mo dules is a lso a morphism of right mo dules for this actio n. Then the tensor pro duct M ⊗ R N of M with another left mo dule N = R N is given in ea ch comp onent b y t he following quotient of ab elian gro ups: ( M ⊗ R N ) g := L h M h ⊗ Z N g − h h m • r ⊗ n − m ⊗ rn | m ∈ M p , r ∈ R h − p , n ∈ N g − h i ( cf. [ 14 ]). The ring R still acts on R M ⊗ R N on the left. There are ev ident natural asso ciativity a nd righ t and left unit iso morphisms ( L ⊗ R M ) ⊗ R N ∼ = L ⊗ R ( M ⊗ R N ) , M ⊗ R R ∼ = R , R ⊗ R M ∼ = R as w ell as a na tural symmetry isomorphism: τ M ,N : M ⊗ R N ∼ = N ⊗ R M , τ M ,N ( m ⊗ n ) := ( − 1) ǫ (deg m, de g n ) n ⊗ m . Lemma 2.11. The ab ove c onstru ctions ar e wel l- define d and turn the c ate gory of gr ade d left R -mo dules int o a close d s ymm et ric monoidal ab elian c ate gory with tensor unit R = R R . Pr o of. All the verifications are straig htf orward and ar e ther efore omitted. The existence o f the internal Hom follows from the standard fact tha t every c olimit pre- serving functor b etw een Gr othendieck ca tegories, suc h as M ⊗ R ( − ) : R -GrMo d → R -GrMo d, has a right adjoint.  R emark 2.12 . If o ne co nsiders R as a left R -mo dule R R , its canonical right ac- tion ( 2.10 ) used for tensoring is just multip lication in R from the right, by ǫ - commutativit y . F or the twisted left mo dule M = R R ( g ) howev er , b eware that the element m • r in genera l is n ot equal to the pro duct m · r computed in R . Lemma 2.13. Ther e exist two natur al isomorphisms R ( g ) ⊗ R M ∼ − → M ( g ) and M ⊗ R R ( g ) ∼ − → M ( g ) of left R -mo dules f or all g ∈ G and al l M ∈ R - GrMo d . Pr o of. The map R ( g ) ⊗ R M → M ( g ) giv en b y r ⊗ m 7→ ( − 1) ǫ ( g, deg m ) rm is w ell- defined, R -linear , and invertible with inv erse m 7→ ( − 1) ǫ ( g, deg m ) 1 ⊗ m (here m ∈ M deg m = M ( g ) deg m − g ). The second isomorphism is obtain by co mp os ing the fir st one with the switc h is o morphism τ R ( g ) , M .  If M = R ( h ) in Lemma 2.13 , denote b y µ g ,h : R ( − g ) ⊗ R ( − h ) ∼ / / R ( − g − h ) R ( g ) ⊗ R ( h ) ∼ / / R ( g + h ) the first isomorphism of left R -mo dules appe aring in the lemma. ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 7 Prop ositio n 2.14. The c omp anion c ate gory C R of any ǫ -c ommutative ring R c ar- ries a strict symm et ric monoidal structur e ⊗ with unit 0 , given on obje cts by g ⊗ h := g + h and o n mo rphisms r ∈ C R ( g , g ′ ) and s ∈ C R ( h, h ′ ) by the formula r ⊗ s := ( − 1) ǫ ( g, h ′ − h ) rs . With this tensor structur e, the Y one da emb e dding R : ( C R ) op → R - GrMo d t o gether with the identific ations µ g ,h : R ( g ) ⊗ R R ( h ) ∼ − → R ( g ⊗ h ) and µ 0 = id : R (0) = − → R b e c omes a str ong symmetric monoidal functor ( R, µ, µ 0 ) . Mor e over, the t ensor c ate gory C R is rigid. Pr o of. F or the last assertio n, note that the iden tit y C R ( g ⊗ h, ℓ ) = R ℓ − ( g + h ) = R ( − h + ℓ ) − g = C R ( g , − h ⊗ ℓ ) shows that e a ch ob ject h is rigid ( i.e. , strongly dualizable) with dual − h . All other verifications a re straight forward and are therefore omitted.  R emark 2 .1 5 . W e stress that “ s trict” in the last prop o s ition means that the a s so cia- tivit y , left unit, rig ht unit, and symmetry coherence iso morphisms ar e al l identit y maps. R emark 2.16 . It fo llows from the formal theo r y o f Kan extensions – or , in this context, Da y conv olution [ 12 ] – that there exists, up to canonical iso morphism, a unique closed symmetric mo noidal structure o n the functor categor y Ab C R such that the Y o neda e mbedding ( C R ) op → Ab C R is s tr ong symmetr ic monoidal. By uniqueness w e recov er in this wa y the tensor pr o duct of Lemma 2.11 . Indeed, this is how one ca n find the ( rather quain t) form ula for the tenso r pro duct in the companion category: given r ∈ R g ′ − g and s ∈ R h ′ − h , one computes directly t hat the unique dotted map making the follo wing square comm ute R ( − g ′ ) ⊗ R R ( − h ′ ) R ( r ) ⊗ R R ( s )   µ g ′ ,h ′ ∼ / / R ( − g ′ − h ′ )   R ( − g ) ⊗ R R ( − h ) µ g ,h ∼ / / R ( − g − h ) is rig ht multiplication b y ( − 1 ) ǫ ( g, h ′ − h ) rs . Similarly one finds that the symmetr y isomorphism τ R ( g ) , R ( h ) corres p o nds via µ to the identit y map R ( g + h ) → R ( h + g ). Let Ch( R ) := Ch( R -GrMo d) b e the ca tegory of c hain complexes of graded R - mo dules. It has a tensor product in the usual w ay , b y setting ( X ⊗ R Y ) n := M p ∈ Z X p ⊗ R Y n − p ( n ∈ Z ) and by defining the differential with the Leibniz formula, for all complex e s X and Y . The symmetric monoidal catego ry Ch( R ) is again closed, with th e usual Hom com- plexes Hom R ( X, Y ). Prop ositio n 2 . 17. F or every ǫ -c ommutative G -gra de d ring R , the c ate gory Ch( R ) has a (pr op er, c el lular and c ombinatorial) Quil len m o del struct ur e, wher e the we ak e quivalenc es ar e the quasi-iso morphisms and the fibr ations ar e the de gr e ewise sur- je ctions. Mor e over, this mo del is c omp atible with the t ensor pr o duct of c omplexes i n the sense that it turns Ch( R ) into a symmetric monoidal mo del c ate gory (se e [ 15 ] ). 8 IVO DELL’AMBROGIO AND GREG STEVENSON Pr o of. Of t he v arious p ossibilities, we find it most convenien t to cite some results from [ 11 ]. W e r ecall that we have at hand a Grothendieck a belia n c a tegory A := R -GrMo d whic h is equipp ed with a closed symmetric monoidal structure. Moreover, it has a sma ll se t G := { R ( g ) | g ∈ G } of generators which co nt ains the tensor unit R = R (0) and which by Le mma 2.13 is essen tially c losed under the tensor pro duct. It also follows immediately from Lemma 2.13 that each R ( g ) is flat, i.e. , that the functor R ( g ) ⊗ R ( − ) : A → A is exact. With this s et G of generator s, the G -mo del structur e of lo c. cit. ex ists and has the pr op erties listed in the pr op osition. More pr ecisely (a nd adopting the ter mino l- ogy of l o c. cit. ), by [ 11 , Remark 1 .15] it is alw a ys pos sible to c ho os e a family H of complexes such that the pa ir ( G , H ) forms a descent structure, so that by [ 11 , The- orem 1.7] there exists a Quillen mo del str uc tur e on Ch( A ) – which is independent of H other than for the ch oice of gener ating trivial c ofibrations – having quasi- isomorphisms for w eak equiv a lences; the description of fibra tions as the degree wise surjections (which will not be used in this article) follo ws from [ 11 , Corollary 4.9 ] and the fact that ev ery complex X ∈ Ch( A ) is G -loca l, that is, the canonical map K ( A )(Σ n R ( g ) , X ) → D ( A )(Σ n R ( g ) , X ) is bijective for all n ∈ Z , where K ( A ) denotes the homotopy catego ry o f co mplexes and Σ the shift funct or. The facts that t he gener ators G are fla t, include the tensor unit, and are essen- tially closed under tensoring, ensur e that the model is compatible with the g iven symmetric monoidal structure, b y [ 11 , Prop osition 2.8] and [ 11 , Corollary 2.6 ].  R emark 2.18 . Although the ex istenc e o f a mo del for D ( R ) will b e r equired in Section 5 , w e will not hav e to actually w ork with it: the (proba bly) more familiar metho ds of homological algebra will amply suffice, see e. g. [ 17 ]. It follows from Propo sition 2.17 that, b y deriving the tensor pro duct and the in- ternal Hom functors, the derived categ ory of ev er y ǫ - c o mmu tative G -gra ded ring R inherits the structure of a clo sed tensor catego ry ( D ( R ) , ⊗ L R , R, R Hom R ). Mo re- ov er , the tensor structure is compatible with the triangulation in the b est wa y; we refer to [ 16 , Appendix A] fo r precise statemen ts. If the g roup D ( R )(Σ n R ( g ) , X ) v a nis hes for all n ∈ Z and g ∈ G , then X is acyclic. Hence { Σ n R ( g ) | g ∈ G, n ∈ Z } is a set of compact generators for the triang ula ted ca tegory D ( R ). It is not ha rd to see that the ob jects Σ n R ( g ) are also rigid, that is, that the canonical map R Hom R (Σ n R ( g ) , R ) ⊗ L R X → R Hom R (Σ n R ( g ) , X ) obtained by the tensor-Ho m a djunction is a n isomor phism for all X . Hence D ( R ) is a rigid ly-c omp actly gener ate d tensor triangulate d c ate gory , as in [ 16 ] a nd [ 4 ]. In particular, the fu ll subcatego ry D ( R ) c ⊆ D ( R ) of co mpact ob jects coincides with that of rigid ob jects. Notation 2.19 . Since no confusion sho uld ar ise, we will simply write ⊗ for the derived tensor pr o duct ⊗ L R in D ( R ). F or any family of ob jects F ⊆ D ( R ) we w ill use the follo wing notation: hF i := the loca liz ing sub categ ory of D ( R ) generated b y F , hF i ⊗ := the lo ca lizing tensor ideal of D ( R ) generated b y F . The follo wing observ ation will be used rep eatedly . ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 9 Lemma 2.20. L et R b e any ǫ -c ommu tative G -gr ade d ring and F ⊆ D ( R ) any family of obje cts in the derive d c ate gory. Then hF i ⊗ c oincides with the smal lest lo c alizing sub c ate gory of D ( R ) c ontaining F and close d under al l t he twist functors ( − )( g ) , g ∈ G , and also wi th the smal lest lo c alizing sub c ate gory of D ( R ) c ontaining the obje cts { X ( g ) | X ∈ F , g ∈ G } . Pr o of. The equalit y o f the las t t wo sub ca teg ories is o bvious. F or th e first one no te that a loca lizing sub categ ory of D ( R ) is a ⊗ - ideal if and only if it is closed under tensoring with the generators Σ n R ( g ). It suffices there fo re to show that there exist isomorphisms R ( g ) ⊗ X ∼ = X ( g ) fo r all complexes X ∈ D ( R ), but this is an easy consequence of Lemma 2.13 .  3. Graded fields Fix a n a be lia n g roup G together with a Z / 2 Z -v alued s ymmetric bilinear form ǫ . All rings consider e d henceforth ar e ass umed to b e ǫ -commutativ e G -gra ded rings . Let us b egin by recalling that a non-zer o ǫ -commutativ e G -g raded ring K is a gr ade d field if ev ery non- zero ho mogenous element of K is inv ertible. In particular, K 0 is an honest field, and the comp onents M g of every K -mo dule M are K 0 -vector spaces. W e wish to show, in a nalogy with the ungra ded ca se, that ca tegories of mo dules o ver gra de d fields are rather structurally simple; this will pr ovide us with a go o d theory of r esidue ob jects in the der ived categ ory , as in [ 22 ]. W e fix some graded field K throug hout the r est of the section. Definition 3.1. Let M b e a gra de d K -mo dule. W e define the sc affold o f M to b e s ( M ) := { g ∈ G | M g 6 = 0 } . Lemma 3.2. The s c affold, s ( K ) , of K is a sub gr oup of G . Pr o of. As K is unital and 1 6 = 0 w e must ha ve 0 ∈ s ( K ). If g ∈ s ( K ) then there is a non-zero element in K g which, as K is a graded field, must ha ve an in v erse in K − g , s o − g ∈ s ( K ). Finally , supp os e g , g ′ ∈ s ( K ). An y non-zero element of K g gives, via multiplication, an isomorphism K g ′ → K g + g ′ , so g + g ′ ∈ s ( K ).  Lemma 3.3. F or any g ∈ G we have s ( K ( g )) = s ( K ) − g . Pr o of. Just note that s ( K ( g )) = { h ∈ G | K g + h 6 = 0 } = s ( K ) − g .  Lemma 3.4. Every g r ade d K -mo dule M is gr ade d fr e e. Pr o of. If h ∈ s ( M ) and g ∈ s ( K ) then g + h ∈ s ( M ) and M h ∼ = M g + h bec ause K is a graded field. In particular the subgroup s ( K ) of G acts on s ( M ) by translation. Let { h i } i ∈ I be elements of s ( M ) giving a decomp osition o f s ( M ) into disjoint or bits h i + s ( K ). Then ther e is an isomor phism M i ∈ I K ( − h i ) m i − → M , where m i = rank K 0 M h i . Indeed, this is seen easily b y choo sing isomorphisms K ( − h i ) m i h i = K m i 0 − → M h i and extending K -linearly .  This giv es us the next lemma, whic h is the graded ana logue o f [ 8 , Lemma 2 .1 7]. Lemma 3.5 . L et R b e a G -gr ade d ring and R → K a map (of gr ade d rings) into a G -gr ade d field K . Then for al l X ∈ D ( R ) t he obje ct X ⊗ K is a c opr o duct of susp ensions and twists of K . Pr o of. The functor ( − ) ⊗ K : D ( R ) → D ( R ) factors thr o ugh D ( K ) and s o the result is immediate from Lemma 3.4 .  10 IVO DELL’AMBROGIO AND GREG STEVENSON 4. The small suppor t Fix a n a be lia n g roup G and an ǫ -commutativ e noetheria n G -gra ded ring R . W e now define a no tion of s uppo rt in terms of the gr aded residue fields of R . W e pro ve that this supp or t s a tisfies all the desirable proper ties o ne w ould hope for. In this case the v irtues of the s uppo r t ar e not just their own reward: in the next sec tion we see that a complete classifica tion o f the lo calizing ⊗ -ideals of D ( R ) follows in a very straightforward wa y from the r esults of this section and some abstract machinery . Let us begin by defining the ob jects which give rise to the small support. Definition 4.1 . Let p ∈ Sp ec h R b e a homo g eneous prime ideal. W e define the r esidue field at p in the usual w ay: k ( p ) := R p / p R p = ( R/ p ) (0) . Happily it turns out that even in the ǫ -commutativ e case this gives r ise to graded fields. Lemma 4.2. L et p b e a homo gene ous prime id e al of R . Then the r esidue field k ( p ) is a gr ade d field . Pr o of. Let r ∈ R g be a homogeneous elemen t of degr ee g . Then deg r 2 = 2 g and therefore r 2 is e ven. In particular, if r 6∈ p then r 2 ∈ ( R r p ) ∩ R ev bec omes inv erted in k ( p ). But the inv erse r − 2 is also even (of degree − 2 g ). Ther efore in k ( p ) the elemen t r commutes with r − 2 and th us with rr − 2 . Hence r is in vertible with in verse r r − 1 .  Definition 4.3. Let X b e an ob ject of D ( R ). W e define the smal l supp ort of X to be the subset ssupp X := { p ∈ Spe c h R | k ( p ) ⊗ X 6 = 0 in D ( R ) } of the homogeneous spectrum Sp ec h R of R . R emark 4.4 . W e observe that there is no need to twist in this definition since, for every g ∈ G and X ∈ D ( R ), we ha ve k ( p ) ⊗ X 6 = 0 if and only if k ( p )( g ) ⊗ X 6 = 0. Let us b eg in with those prop erties of t he small support whic h a re very obvious from the definition (so ob v ious in fact that w e do not give a pro of ). Lemma 4.5. The s mal l supp ort satisfies the fol lowing p r op erties: (i) F or every X in D ( R ) we have ssupp X = ssupp Σ X . (ii) F or a ny set -indexe d fa mily { X i } i ∈ I of obje cts of D ( R ) we have ssupp  a i ∈ I X i  = [ i ∈ I ssupp X i . (iii) F or a ny t riangle X → Y → Z → Σ X in D ( R ) ther e is a c ontainment ssupp Y ⊆ s supp X ∪ ssupp Z . (iv) ssupp R = Sp ec h R . (v) ssupp 0 = ∅ . Lemma 4.6. The smal l supp ort satisfies the ten sor formula: for any X and Y in D ( R ) we have ssupp( X ⊗ Y ) = ssupp X ∩ ssupp Y . Pr o of. It is clear that ssupp( X ⊗ Y ) is contained in the intersection. T o see the reverse inclus io n just note that if p is in the small suppo rt of both X and Y then k ( p ) ⊗ X ⊗ Y ∼ =  a i Σ m i k ( p )( g i ) α i  ⊗ Y 6 = 0 , ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 11 for some elemen ts g i ∈ G , integers m i , and cardinals α i , where w e use Lemma 4.2 , Lemma 3.5 and the fact that the tensor product commutes with c opro ducts.  W e next wish to chec k that the small support detects the v anishing of ob jects. This is, in so me sense, the most technically unpleasant pro p er t y to verify . How ever, most of the details ar e routine ex tensions of well known facts ab out Z -graded rings to G -graded rings. The catego ry R -GrMo d is a lo cally no etheria n Grothendieck ab elian categ ory . Thu s it has enoug h injectives and every injective is a direct sum of indeco mpo sable injectiv es. T he gene r al form of Matlis’ theory ([ 19 ] and cf. [ 25 , Cha pter V.2]) shows that every indecomp osa ble injective is the injective envelope, E ( R ( g ) /P ), o f R ( g ) / P for some g ∈ G a nd so me irreducible submodule P of R ( g ). As t wisting is an autoequiv ale nc e , it is ea sily seen that it is sufficient to consider only irr educible ideals of R . The arg umen t of [ 19 , Pr o p o sition 3.1 ] then extends in a stra ig htf orward wa y to s how that every indecomposable injective is a twist of the env elop e of R / p where p is a prime idea l, i.e. , is of the form E ( R / p )( g ) ( cf. [ 9 , Theorem 3.6.3]). In particular, they are easily seen to be p -lo cal and p -torsio n in the graded sense. Prop ositio n 4.7. The obje cts k ( p )( g ) , for p ∈ Sp ec h R and g ∈ G , gener ate D ( R ) : D ( R ) = h k ( p )( g ) | p ∈ Sp ec h R , g ∈ G i = h k ( p ) | p ∈ Sp ec h R i ⊗ . Pr o of. Let X b e a no n-zero ob ject of D ( R ) and pic k i ∈ Z such that H i ( X ) 6 = 0. W e may , without loss of genera lity , assume that X is a c omplex of injectives b y taking a K-injective reso lution ([ 24 ]). Pick some non-zer o homo g eneous element of H i ( X ) and obse r ve tha t it is repr e- sented by a mo rphism f : Σ − i R ( g ) → X in D ( R ), which mor eov er may b e assumed to c orresp ond to a morphism of complexes, i.e. , a map R ( g ) → X i . As X i is a direct sum of indecomp osable injectiv es a nd f is determined b y the ima ge of 1, we may assume that the ima ge of f is cont ained in a sing le indecomp osa ble injective E ( R/ p )( g ′ ) ⊆ X i . Indeed R ( g ) → X i factors throug h a finite direct sum o f in- decomp osable injectiv es and if each of the r estrictions of f to these facto r s were nu ll-homotopic, clear ly f would also be nullhomotopic. So we just replace f , if necessary , by the restriction of f to a single indecompo sable summand of X i . Since E ( R/ p )( g ′ ) is p -lo cal, f facto r s via R p ( g ). W e can of course factor R p ( g ) → E ( R/ p )( g ′ ) through its image whic h is finitely generated over R p and p -torsion. Thu s we get a factoriz ation of f via ( R p / p n R p )( g ), for some in teger n . T o summa- rize w e ha v e the follo wing co mm utative diag ram of factorizations of f . R ( g ) / / & & M M M M M M M M M M M   X i R p ( g ) / / & & M M M M M M M M M M E ( R/ p )( g ′ ) O O ( R p / p n R p )( g ) O O T o c o mplete the pr o of, just note that ( R p / p n R p )( g ) is constructed fro m the k ( p )( h ), where h ∈ G , by taking finitely ma n y extensions, so it certa inly lies in the lo ca lizing sub c ategory genera ted by the k ( p )( h ). Hence, in D ( R ), s ome k ( p )( h ) must a lso hav e a non-zero map to X .  That the sma ll supp ort detects ob jects is an easy consequence of the prop o s ition. Corollary 4.8. F or every obje ct X of D ( R ) we have that X ∼ = 0 if and only if ssupp X = ∅ . 12 IVO DELL’AMBROGIO AND GREG STEVENSON Pr o of. One dire c tion is clear . On the other hand, suppos e X ⊗ k ( p ) is zero for all p ∈ Spe c h R . Then the kernel of the functor X ⊗ ( − ) is a lo calizing tensor ideal of D ( R ) cont aining all the residue fields. Hence it m ust be D ( R ) and it is immediate that X ∼ = 0.  Before contin uing, let us note the following important co ns equence of the la st corolla r y . Prop ositio n 4 . 9. F or e ach p ∈ Sp ec h R , the lo c alizing ⊗ -ide al h k ( p ) i ⊗ = h k ( p )( g ) | g ∈ G i is minimal, i.e. , it pr op erly c ontains no non-zer o lo c alizing ⊗ - ide al. Pr o of. Suppo se X ∈ h k ( p ) i ⊗ is a non-zero ob ject. Since k ( p ) ⊗ k ( q ) = 0 whenever q 6 = p , it m ust also hold that X ⊗ k ( q ) is zer o for all q ∈ (Sp ec h R ) r { p } . By the last corolla r y we thus ha ve that X ⊗ k ( p ) is a non-zero ob ject in h X i ⊗ . It follo ws from Lemma 3.5 tha t the ⊗ - ideal g e nerated by X con tains so me twist of the o b ject k ( p ), bec ause lo c a lizing subca tegories are thic k. W e conclude that h X i ⊗ = h k ( p ) i ⊗ .  Next we wish to chec k that the small supp or t of a compac t ob ject of D ( R ) is closed. F or this we need the fo llowing lemma, whose ungr aded ana lo gue is w ell known. Lemma 4.10. L et R b e a ǫ -c ommutative no etherian G -gr ade d ring. (i) An obje ct is c omp act in D ( R ) pr e cisely when it is i somorphic t o a b ounde d c omplex o f finitely gener ate d p r oje ctive gr ade d mo dules. (ii) L et ( R, m , k ) b e gr ade d lo c al (e.g. R p for any homo gene ous prime p ). Then in D ( R ) every right b ounde d c omplex of finitely gener ate d pr oje ctives C has a minimal gr ade d fr e e r esolution f : B → C ; that is, f is a quasi- isomorphi sm, the c omp onents B i ar e finite gr ade d fr e e m o dules, and the differ entials d : B i → B i +1 satisfy d ( B i ) ⊆ m B i +1 . Pr o of. (i) It is easily verified that bounded complexes of finitely generated pro jec- tives a re compact; just use that, for such a co mplex X and any other Y ∈ D ( R ), one computes D ( R )( X , Y ) using homo topy class e s of chain maps. T o show the o ppo site inclusion note that, b y the Thomason-Neeman loca lization theorem [ 20 ], D ( R ) c is the thick s ubca tegory gener ated by the free modules { R ( g ) | g ∈ G } . Hence it suffices to show that mapping cones and direct summands of b ounded co mplexes of finite pro jectives are again of the same form; the first is clea r, and the seco nd follows (for instance) precisely as in [ 10 , Lemma 1.2.1]. (ii) The usual pro of of the ungraded case, by inductio n, still works b eca us e Nak ay ama’s lemma still holds for G -gra ded r ings.  Lemma 4.11. L et C b e a c omp act obje ct of D ( R ) . Then s supp C is a close d subset of Sp ec h R . Pr o of. Assume C p 6 = 0 in D ( R p ). By Lemma 4.10 , the co mplex C p has a minimal graded free resolution ov er R p . T ens o ring with k ( p ) gives a complex whic h is non- zero in at least one degree and has zero different ials and so C ⊗ k ( p ) is certainly non-zero. Conv ersely if C p = 0 then of course C ⊗ k ( p ) = C p ⊗ k ( p ) = 0. Hence ssupp C = V (Ann R H ∗ C ) = S i V (Ann R H i C ), which is clo sed since ther e are o nly finitely many non-v anishing cohomology groups.  Finally , we chec k that there are enough compact ob jects r elative to the small suppo rt. Lemma 4 .12. L et V ⊆ Spec h R b e a close d s u bset. Then t her e exists a c omp act obje ct C of D ( R ) such that ssupp C = V . ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 13 Pr o of. By definition of the Zariski to p o logy V = V ( I ) for some homogeneous ideal I ⊆ R . Since R is no etherian, we may wr ite I = ( f 1 , . . . , f n ) for finitely man y homogeneous elemen ts f i ∈ R g i . Let C i denote the mapping cone o f f i , considered as a mor phism R ( − g i ) → R . Each C i is a c o mpact o b ject, and therefore so is their tensor pr o duct C := C 1 ⊗ · · · ⊗ C n . W e cla im that ssupp C = V . Indeed, we hav e ss upp C = ssupp C 1 ∩ · · · ∩ ssupp C n by the tenso r for m ula (Lemma 4.6 ), so it suffices to show that ssupp C i equals V (( f i )) for each i . By co nsidering the triangle R ( − g i ) → R → C i → Σ R ( − g i ), we see that C i ⊗ k ( p ) 6 = 0 if and only if the morphism f i ⊗ k ( p ) is not inv ertible; th at is, if a nd o nly if the element f i belo ngs to the ideal p . This proves the claim.  5. The spectr um and localizing te nsor ideals Let Spc D ( R ) c denote the sp e ctrum of the tensor triang ulated categor y of com- pact ob jects, in the sens e of Balmer [ 1 ]. W e r ecall tha t this is a sp ectral top olog ical space (defined for every essentially small tensor tria ngulated category ) which comes together with a function X 7→ supp X assigning a closed subset of Sp c D ( R ) c to every ob ject X ∈ D ( R ) c . The supp or t function supp is compatible with the tensor triangular op erations of D ( R ) c , and it is the universal (finest) such . Theorem 5 .1. F or every ǫ -c ommut ative no etherian G -gr ade d r ing R t her e is a unique supp ort pr eserving home omorphism Spec h R → Sp c D ( R ) c . In other wor ds (Sp ec h R, ssupp) is a classifying su pp ort datum ( [ 1 , § 5] ), me aning that t her e a r e inclusio n p r eserving mutual ly i nverse assig nments  sp e cialization close d subsets of Sp ec h R  τ / / o o σ  thick ⊗ -ide als of D ( R ) c  given, for a sp e cializatio n close d su bset V of Spec h R and a thick ⊗ -ide al J , by τ ( V ) = { X ∈ D ( R ) c | ssupp X ⊆ V } and σ ( J ) = { p ∈ Sp e c h R | ∃ X ∈ J s. t. p ∈ ssupp X } . Pr o of. W e wish to apply the rec ognition cr iterion [ 13 , Theorem 3 .1]. The categor y D ( R ) is rigidly - compactly g enerated a nd, by a ser endipitous o ccurrence, w e just happ ened to have prov ed in the last s e ction that (Sp ec h R, ssupp) satisfies all the necessary c onditions to apply this c r iterion (by Coro llary 4.8 and Lemmas 4 .5 , 4.6 , 4.11 and 4.12 ). The second pa rt follo ws fro m the ba sic result of tensor triangular g eometry , [ 1 , Theor em 4.1 0]; in general, on the left hand side o ne would have to consider Thomason subsets, but for the no ether ia n s pace Sp ec h R these coincide with spe- cialization closed subsets , and on the r ight hand side r adic al thic k ⊗ -ideals , but since D ( R ) c is rigid these coincide with ⊗ -tensor idea l, see [ 2 , Pr op osition 2.4].  W e now know that D ( R ) is a rigidly- compactly genera ted tensor tria ngulated category with a mo del and whos e compacts have noetherian sp ectrum. Thus w e can apply all of the machinery o f [ 26 ] to the problem of classifying the lo calizing ⊗ -ideals of D ( R ). W e s ha ll mostly use this machinery , as well as the work of Ba lmer and F avi [ 4 ], as a black b ox; the following pr op osition sp ells out the little we need to kno w. Prop ositio n 5 . 2. F or e ach x ∈ Sp c D ( R ) c ther e exists a ⊗ -idemp otent obje ct Γ x R of D ( R ) such that the assig nment X 7→ supp X := { x ∈ Sp c D ( R ) c | Γ x R ⊗ X 6 = 0 } ( X ∈ D ( R )) 14 IVO DELL’AMBROGIO AND GREG STEVENSON extends the Balmer supp ort of c omp act obje cts and such t hat the fol lowing hold: (i) for x 6 = y we ha ve Γ x R ⊗ Γ y R = 0 ; (ii) for every obje ct X of R ther e is an e quality of ⊗ -ide als h X i ⊗ = h Γ x R ⊗ X | x ∈ Sp c D ( R ) c i ⊗ ; (iii) an obj e ct X of D ( R ) is zer o if and only if supp X = ∅ . Pr o of. The construction and orthogo nality of the idemp otents is due to Balmer and F avi. Given the existence of a mo del and the fact that Sp c D ( R ) c is no etherian the rest is a consequence of [ 26 , Theorem 6.8].  Lemma 5.3. F or e ach p ∈ Spec h R ther e is a u nique x ∈ Spc D ( R ) c such that Γ x R ⊗ k ( p ) is non-zer o. In other wor ds, s upp k ( p ) = { x } . Pr o of. By part (iii) of the proposition w e k now there ex is ts s uch an x . Now sup- po se y is ano ther po int , distinct from x , suc h that Γ y R ⊗ k ( p ) is also no n-zero. Then, using part (i) of the propositio n together with Lemma 3.5 w e g et 0 = Γ y R ⊗ Γ x R ⊗ k ( p ) ∼ = Γ y R ⊗  a i Σ n i k ( p ) m i ( g i )  6 = 0 , which is a contradiction.  Lemma 5.4 . If p 6 = q ar e two homo gene ous prime ide als of R then k ( p ) and k ( q ) have disjoint supp orts. Pr o of. Suppo se k ( p ) and k ( q ) b oth hav e supp ort { x } . Then by the Half ⊗ -Theorem ([ 4 , 7.22]) w e ha v e that, for an y compact ob ject C of D ( R ), supp( k ( p ) ⊗ C ) = s upp k ( p ) ∩ supp C = supp k ( q ) ∩ supp C = supp( k ( q ) ⊗ C ) . Thu s b y Pr op osition 5.2 (iii) w e see that k ( p ) ⊗ C is zero if and only if k ( q ) ⊗ C is zero. But this is clea rly a bsurd as one can see, for example, from L e mma 4.12 .  Prop ositio n 5.5. F or every p ∈ Sp ec h R ther e is an e quality of lo c alizing ⊗ -ide als h Γ x R i ⊗ = h k ( p ) i ⊗ wher e x is t he un ique p oint of Sp ec D ( R ) c such that supp k ( p ) = { x } . Pr o of. The existence and uniqueness of x is Lemma 5.3 . By [ 26 , Prop os ition 5.5 (4)] we deduce from this that Γ x R ⊗ k ( p ) ∼ = k ( p ). O n the other hand, it follows from the la st lemma that Γ x R ⊗ k ( q ) = 0 for every q 6 = p . W e know from Pr op osition 4.7 that the residue fields generate D ( R ) as a tensor ideal. So we ha ve h Γ x R i ⊗ = h Γ x R i ⊗ ⊗ D ( R ) = h Γ x R i ⊗ ⊗ h k ( q ) | q ∈ Sp ec h R i ⊗ = h Γ x R ⊗ k ( q ) | q ∈ Sp ec h R i ⊗ = h k ( p ) i ⊗ where the third equality is an application of [ 26 , Lemma 3.10].  Corollary 5.6. F or al l x ∈ Sp c D ( R ) c the lo c alizing ⊗ -ide al h Γ x R i ⊗ is minimal. F u rthermor e, t he c anonic al home omorphism of Sp c D ( R ) c with Spec h R identifies supp X with ssupp X for al l X in D ( R ) . Pr o of. The first statemen t is immediate fro m the prop osition as the res idue fields generate minimal ⊗ -ideals b y Prop osition 4.9 . The second statement is a trivial consequence of the first.  W e can now ea sily deduce the classification theo rem for lo calizing ⊗ -ideals. ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 15 Theorem 5.7. Ther e ar e inclusion pr eserving mu tual ly inverse bije ctions  subsets of Sp ec h R  τ / / o o σ  lo c alizing ⊗ - ide als o f D ( R )  , and  sp e cialization close d subsets of Sp ec h R  τ / / o o σ  lo c alizing ⊗ -ide als of D ( R ) gener ate d by obje cts of D ( R ) c  wher e for a subset W of Sp ec h R and a lo c alizing ⊗ -ide al L we set τ ( W ) = { X ∈ D ( R ) | ss upp X ⊆ W } and σ ( L ) = { p ∈ Sp ec h R | Γ p R ⊗ L 6 = 0 } . Pr o of. The map τ is a split monomorphis m with left inv er s e σ b y [ 26 , Pr o p o - sition 6.3]. B y the loc a l-to-glo ba l principle (Pro po sition 5.2 (ii)) and its formal consequence [ 26 , Lemma 6.2], for ev ery localizing ⊗ -idea l L we ha ve τ σ ( L ) = τ ( { p ∈ Spe c h R | Γ p R ⊗ L 6 = 0 } ) = h Γ p R | Γ p R ⊗ L 6 = 0 i ⊗ . T o pro v e the first bijection note that, since the Γ p R generate minimal ⊗ -ideals, we m ust ha v e Γ p R ⊗ L = h Γ p R i ⊗ whenever this subca tegory is no n-zero. The result then follows fro m applying the local-to- global principle a gain. The second pa ir of maps are well defined b y Lemma 4.11 , [ 26 , Corolla r y 4.1 2], and the g o o d prop erties of the supp ort. Tha t they give a bijection is immediate from the first bijection.  R emark 5.8 . By Lemma 2.20 , one can reformulate the last theor em in the following wa y: ther e is an inclusion pres erving bijection b etw een subsets of Sp ec h R and lo calizing sub categor ie s of D ( R ) clo sed under all t wists ( − )( g ). Corollary 5.9. The c ate gory D ( R ) satisfies the r elative telesc op e c onje ctur e, i.e. , the se c ond bije ction in The or em 5.7 c ompletely classifies those lo c alizing ⊗ -ide als whose inclusion admits a c opr o duct pr eserving right adj oint. Pr o of. This is an a pplication o f [ 26 , Theo rem 7.14]; one just needs to note that, b y Lemma 4.11 , co mpact ob jects hav e closed supp orts (as we hav e iden tified our t wo notions of supp ort), and th at by Lemma 4.12 an y closed subset of Spec h R can b e realised as the support of a compact ob ject.  6. An applica tion to (w eighted) pr ojective schemes W e now show how one easily o btains from Theorem 5.7 a clas s ification o f the lo- calizing tenso r idea ls of the derived categor y of certa in weight ed pro jective sc hemes. In particula r , if R is a no etherian non-nega tively Z -gr aded ring this giv es a dir ect metho d, from an “affine” p oint of view, of c la ssifying the tenso r ideals in the derived category of Pro j R . Let R b e a commutativ e no etherian G -g raded ring , whe r e G is an ab elian group, and a s previo usly denote by R -GrMo d the categ ory of graded R -mo dules . Let Z be a closed subset o f Sp e c h R and denote b y U its op en complement. W e let R -GrMo d Z denote the Serre subcateg o ry o f R -GrMo d consisting of those o b jects suppo rted on Z in the usual sense: R -GrMo d Z = { M ∈ R -GrMo d | M p = 0 ∀ p ∈ U } . W e write Qcoh X := R -GrMo d /R -GrMo d Z 16 IVO DELL’AMBROGIO AND GREG STEVENSON for the ab elian quotient of R -GrMo d by R -GrMo d Z . W e think o f Qcoh X as the category of qua si-cohere nt sheav es o n a “weigh ted pro jective space X ” (precisely what this mea ns is not impor tant in the sequel, so let us not dwell on it). O bserve that R -GrMo d Z is the smalles t Serre sub categor y of R -GrMo d closed under filter e d colimits and containing all twists of th e residue fields of points in Z . Lemma 6.1. The sub c ate gory R - GrMo d Z is the torsion class, T , of the her e ditary torsion the ory on R - GrMod c o gener ate d by { E ( R/ p )( g ) | p ∈ U, g ∈ G } . Pr o of. Recall that for a ny p ∈ Sp ec h R and g ∈ G the indec o mpo sable injective E ( k ( p ))( g ) = E ( R/ p )( g ) is p -torsion and p -loc a l. Thus, by the universal pro p erty of loca lization, w e m ust have R -GrMo d Z ⊆ T . On the o ther hand no te that a finitely generated mo dule lies in T if and o nly if its injectiv e env elo p e is a (finite) dir ect sum of indecomp osable injectives corres po nding to po int s of Z . As in the ungraded cas e one can easily chec k, using that filtered colimits of injectives in R -GrMo d are injective and lo caliz a tion preserves colimits, this ex tends to all ob jects o f T . Thus every ob ject of T is a subob ject of an o b ject in R -GrMo d Z (namely its injective envelope) and so T ⊆ R -GrMo d Z giving the claimed equality .  It follo ws from th e lemma that we hav e a diagram o f abelian catego r ies (6.2) R -GrMo d Z / / o o R -GrMo d j ∗ / / o o j ∗ Qcoh X where the quotient j ∗ has right adjoint j ∗ . Lemma 6.3 . The sub c ate gory R - GrMo d Z is a tensor ide al, so t hat Qcoh X inherits the t en sor pr o duct of R - GrMo d . Pr o of. This fo llows easily from the definition. Indeed, a mo dule M b elongs to R -GrMo d Z precisely when Supp R M := { p ∈ Spec h R | M p 6 = 0 } is co n tained in Z ; now use that Supp R ( M ⊗ N ) ⊆ Supp R M ∩ Supp R N .  Let us now see what happ ens at the triang ulated level. W e deno te b y D ( R ) c Z the thick sub categor y of compa ct ob jects supp or ted on Z (in the sense of Balmer). W e let Γ Z D ( R ) be the lo calizing subcategor y gener ated b y D ( R ) c Z and note that Γ Z D ( R ) is smas hing as it is generated by compact ob jects of D ( R ) (in fa ct it is precisely the s ubca tegory τ ( Z ) a s in Theor em 5.7 ). Let us b e a little more explicit ab out what all this means. The sub catego ry Γ Z D ( R ) giv es rise to a smas hing lo calization sequence (6.4) Γ Z D ( R ) I ∗ / / o o I ! D ( R ) J ∗ / / o o J ∗ L Z D ( R ) i.e. , all four functors a re exa ct a nd co pro duct preserving, I ∗ and J ∗ are fully faith- ful, I ! is right adjoin t to I ∗ , and J ∗ is right adjoin t to J ∗ . In pa r ticular there are a sso ciated copr o duct preser ving acycliza tion and localizatio n functors giv e n by Γ Z = I ∗ I ! and L Z = J ∗ J ∗ resp ectively . As in [ 16 , Definition 3.3.2] this gives r is e to Ric k ard idempotents Γ Z R and L Z R with the prop erty that I ∗ I ! ∼ = Γ Z R ⊗ ( − ) a nd J ∗ J ∗ ∼ = L Z R ⊗ ( − ) ; it follows that they are ⊗ -orthogo nal to each o ther b y the us ual prop erties of local- ization and a cyclization fu nctors. W e observe that both Γ Z D ( R ) and L Z D ( R ) are tensor triangulated categories with units Γ Z R and L Z R resp ectively . ON THE DERIVED CA TEGOR Y OF A GRADED COMMUT A TIVE NOETHERIAN RING 17 R emark 6.5 . Such lo c alization sequences a re used to construct the idemp otents Γ x R which app eare d in Pro po sition 5.2 . More details can be found in [ 4 ]. Lemma 6.6. F r om the s e quenc e of ab elian c ate gories ( 6.2 ) we obtain a lo c alization se quenc e D R - GrMo d Z ( R ) i ∗ / / o o i ! D ( R ) j ∗ / / o o R j ∗ D (Qcoh X ) , wher e D R - GrMo d Z ( R ) de notes the ful l su b c ate gory of D ( R ) o f c omplexes with c oho- molo gy in R - GrMo d Z . Pr o of. First we sho w that the s equence in the statement is in fac t a lo calization sequence. T o prov e this we need to check that R j ∗ is fully faithful and that the image of i ∗ is the k ernel of j ∗ . W e b egin with the pro of that R j ∗ is fully faithful. If Y is an ob ject of D (Qcoh X ) then j ∗ R j ∗ Y is computed b y ta king a K-injectiv e r esolution ˜ Y of Y a nd applying j ∗ j ∗ . By [ 25 , Chapter X, P rop osition 1.4] an ob ject of Qcoh X is injective if and only if its image under j ∗ is injective in R -GrMo d. Thus j ∗ ˜ Y is just this complex of injectives viewed in D ( R ). In particular, j ∗ R j ∗ Y = j ∗ j ∗ ˜ Y is quasi-iso mo rphic to Y via the natural map. Let us now give the arg ument that D R -GrMo d Z ( R ) is the kernel of j ∗ . The functor j ∗ is exa ct at the level of ab elian c ategories and has kernel equa l to R - GrMod Z ; thus j ∗ commutes with taking co homology and w e see that its kernel consists precisely of those complexes whose cohomology modules lie in R -GrMo d Z .  Lemma 6 .7. The lo c alization se quenc e of the last lemma agr e es, u p to monoidal e quivalenc e, with ( 6.4 ) . Pr o of. First observe that D R -GrMod Z ( R ) is a lo calizing ⊗ -ideal of D ( R ). It is clear that D R -GrMo d Z ( R ) is a lo calizing sub categ ory of D ( R ) which is closed under twist- ing b y all g ∈ G . Thus b y Lemma 2.20 it is a localizing ⊗ -idea l. So by Theorem 5.7 the ⊗ - ideal D R -GrMo d Z ( R ) must corres p o nd to a subset o f Spec h R a nd this subset must contain Z as D R -GrMo d Z ( R ) co n tains the residue field of each p oint in Z . It m ust in fact b e Z as if s ome q / ∈ Z were in σ ( D R -GrMo d Z ( R )) then, again by the classification, w e would have k ( q ) in D R -GrMo d Z ( R ). But this is impo ssible since k ( q ) is not an ob ject of R -GrMo d Z .  Corollary 6.8 . Ther e ar e inclusion p r eserving bije ct ions  subsets of U  τ / / o o σ  lo c alizing ⊗ - ide als o f D (Qco h X )  , and  sp e cialization close d subsets of U  τ / / o o σ  lo c alizing ⊗ - ide als o f D (Qco h X ) gener ate d by obje cts of D ( R ) c  induc e d by the bije ct ions of Th e or em 5.7 . Pr o of. By the last lemma it is s ufficient to pr ov e the result for L Z D ( R ). Note that any ⊗ -ideal of L Z D ( R ) is also a ⊗ -ideal in D ( R ) as a n ob ject X of D ( R ) is in L Z D ( R ) if and only if it is isomor phic to L Z R ⊗ X . Thus the ideals in L Z D ( R ) are precise ly those idea ls of D ( R ) contained in L Z D ( R ). Hence Theo rem 5 .7 tells us t hat they a re in bijection with subsets of U . The restricted bij ection for thos e ideals gener ated by compact o b jects follows dir ectly fro m the first bijection, the fact that the quotien t functor to L Z D ( R ) sends compac ts to compa cts (see f or instance [ 21 , Theorem 5 .1]), and [ 26 , Lemma 7.10] which tells us that co mpacts of L Z D ( R ) hav e closed suppor t in U .  18 IVO DELL’AMBROGIO AND GREG STEVENSON Example 6 .9 (P ro jective sc hemes ) . Supp ose R is a no n-negatively Z -graded no e- therian commutativ e ring which is genera ted b y R 1 ov er R 0 . Then, letting Z be the Za riski closure in Spec h R of the irrelev ant ideal R ≥ 1 , Q coh X is equiv alent to Qcoh(Pro j R ), and by sp ecializing the ab ov e r esult we see that the lo c a lizing tenso r ideals of D (Q coh(Pro j R )) are in bijection with the subsets o f P r o j R . Example 6.10 . Now supp ose R is a no n-negatively Z -gr a ded finitely g enerated com- m utative k - algebra such that R 0 = k , where k is so me field. The gr ading o n R cor- resp onds to an action of k ∗ on Spec R and th e ide a l R ≥ 1 , generated by p ositively graded elements, cor resp onds to a c lo sed fixed p oint 0 of the k ∗ action. Letting Z = { 0 } there is an equiv alence of categories Qcoh X ≃ Q c o h[(Spec R r 0 ) /k ∗ ] , where [(Spec R r 0 ) /k ∗ ] denotes the corresp onding global quotien t stac k, as in [ 23 , Prop o s ition 2 8 ]. Thus we obtain a clas sification of loca lizing ⊗ -ideals of the unbounded derived ca tegory of quasi-co herent sheav es o n the q uotient sta ck [(Spec R r 0 ) / k ∗ ] in terms of subsets of the punctured homo geneous spectr um. In particular, if R is generated by R 1 ov er R 0 the quotient stack is just Pro j R and we are in the situation of the pr evious example. See [ 18 ] for related results. References [1] Paul Balmer, The sp e ctrum of prime i de als in tensor t riangulate d c ate gories , J. Reine Angew. Math. 5 88 (2005), 149–168. 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E-mail addr ess : ambrogio@math.u ni-biele feld.de Universit ¨ at Bielef eld, F aku l t ¨ at f ¨ ur Ma thema tik, BIREP Gruppe, Postf ach 10 01 31, 33501 Bielefeld, Germany. E-mail addr ess : gstevens@math.u ni-biele feld.de