Perverse coherent t-structures through torsion theories

PER VERSE COHERENT T-STR UCTURES THROUGH TORSION THEORIES JORG E VITÓRIA Abstract. Bezru k a vnik o v, later togethe r with Arinkin, reco v ered Deli gne’s wo rk defining perv erse t-structur es in the der ive d category of c oheren t shea v es on a pro jectiv e scheme. W e pro v e that these t-structure s can b e obtained through tilting with respect to torsion theories, as in the work of Happ el, Reiten and Smalø. This approa ch allows us to define, in the quasi- coheren t setting, simil ar perver se t-structur es for certain noncommutat ive pro jectiv e planes. 1. Introduction A t-structure in a triangulated category D ([9]) is a pair of strict full s ub ca te- gories, ( D ≤ 0 , D ≥ 0 ) , such that, for D ≤ n := D ≤ 0 [ − n ] and D ≥ n := D ≥ 0 [ − n ] ( n ∈ Z ), (1) Hom( X , Y ) = 0 , ∀ X ∈ D ≤ 0 , ∀ Y ∈ D ≥ 1 ; (2) D ≤ 0 ⊆ D ≤ 1 ; (3) F or all X ∈ D , there are A ∈ D ≤ 0 , B ∈ D ≥ 1 and a triangle A − → X − → B − → A [1] . The in tersection D ≤ 0 ∩ D ≥ 0 is an a be lia n category ([9]), c alled the heart. Also, it is well known ([21]) that D ≤ 0 , called the aisle, determines the t-structure by setting D ≥ 0 = ( D ≤ 0 ) ⊥ [1] . A t-structure ( D ≤ 0 , D ≥ 0 ) has as so ciated truncation functors τ ≤ i : D → D ≤ i , τ ≥ i : D → D ≤ i and cohomolo gical functors H i : D → D ≤ 0 ∩ D ≥ 0 , for a ll i ∈ Z (see [9] for details). If A is a n a b e lia n categ ory , its derived catego ry D ( A ) has a standa r d t-s tructure, denoted throughout by ( D ≤ 0 0 , D ≥ 0 0 ) , defined by D ≤ 0 0 := { X ∈ D : H i 0 ( X ) = 0 , ∀ i > 0 } , D ≥ 0 0 := { X ∈ D : H i 0 ( X ) = 0 , ∀ i < 0 } , where H i 0 is the usual complex cohomo logy functor. W e denote the a sso ciated truncation functor s by τ ≤ i 0 and τ ≥ i 0 and the asso ciated coho mological functor is precisely the complex cohomology functor H i 0 , for all i ∈ Z . The standard t- structure r estricts to the b ounded derived categor y D b ( A ) a nd we sha ll use the same notations for the r estriction, when appropria te. Let K b e an algebra ically closed field. F or a scheme X ov er K , Arinkin and Bezruk a vnik ov ([4],[1 0]) constr ucted p erverse coherent t-s tr uctures in D b ( coh ( X )) Most of this work wa s deve loped at the Unive rsit y of W arwick and supp orted by F CT (Por tu- gal), researc h grant SFRH/BD/28268/2006. Later , this pro ject was also supported by DFG (SPP 1388) in Stuttgar t and by SFB 701 in Bielefel d. The author would like to thank Steffen Koenig, Qunh ua Liu, Dmitriy Rumynin, Jan Šťovíček and the anon ymous referee for v aluable comments on the previous ve rsions of this pap er. 1 2 JORG E VITÓRIA as follows. Let X top denote the set of generic points o f all closed ir r educible sub- schemes of X . A p erversit y is a map p : X top − → Z satisfying (1.1) y ∈ ¯ x ⇒ p ( y ) ≥ p ( x ) ≥ p ( y ) − ( dim ( x ) − dim ( y )) . Note that the imag e of p has a t mos t dim ( X ) + 1 elements. The p erverse co herent t-structure ass o ciated with p ([4], [10]) is defined b y the aisle D p, ≤ 0 = n F • ∈ D b ( coh ( X )) : ∀ x ∈ X top , Li ∗ x ( F • ) ∈ D ≤ p ( x ) 0 ( O { x } - mod ) o , where Li ∗ x is the left der ived functor of the pullback by the inclusion of schemes i x : { x } − → X . In our notation, we iden tify mo dules ov er the residue field k ( x ) with qua si-coherent s heav es over { x } . Still, we cho ose to use the notation O { x } - mod for coherent s heaves ov er { x } to b e co nsistent with the notation in [10]. O ur main theorem gives a n alternative descr iption of this aisle . Theorem (Theorem 4.6) L et X b e a smo o th pr oje ctive scheme over K , R = Γ ∗ ( X ) its homo gene ous c o or dinate ring and p a p erve rsity on X . Supp ose that R is a c o mmutative c onne ct e d, no etherian, p ositive ly gr ade d K - algebr a gener ate d in de gr e e 1. L et T i denote t he torsion class c o gener ate d in T ail s ( R ) by πE i , wher e E i = Y { x ∈ X top : p ( x ) ≤ i } E g ( R/I x ) , with I x standing for the de fining ide al of x ∈ X top in R . The n we have: D p, ≤ 0 =  F • ∈ D b ( T ai l s ( R )) : H i 0 ( F • ) ∈ T j , ∀ i > j  ∩ D b ( tail s ( R )) . W e clarify some notation. Denoting b y O X the structure sheaf of X and b y Γ the functor o f glo bal sections, the homogeneous co or dina te ring R is defined by R = Γ ∗ ( X ) := M n ∈ Z Γ( X , O X ( n )) . Throughout, R will b e assumed to b e no e ther ian. W e denote the injectiv e enve- lop e of a graded module M in the categor y of gr aded (righ t) R -mo dules, Gr ( R ) , b y E g ( M ) . The categ ory T ail s ( R ) is the quotient Gr ( R ) /T ors ( R ) (we denote the pro jection functor to this quotient by π ) where T or s ( R ) is the full subca tegory of mo dules M in Gr ( R ) such that for a ll x in M there is N ≥ 0 with xR j = 0 , for all j > N . This categor y is equiv a lent to Qcoh ( X ) , the category of quasi- coherent sheav es over X , as shown by Ser r e in [2 9]. When wr itten in the lower case, tail s ( R ) = gr ( R ) / tors ( R ) denotes the sub categor y of finitely ge ner ated ob- jects in T ail s ( R ) , thus b eing equiv alent to coh ( X ) . Throughout we will use these equiv a lences without mention. W e will show in section 3 that, for s ome rings R ,  X • ∈ D b ( T ai l s ( R )) : H i 0 ( X • ) ∈ T j , ∀ i > j  is an aisle of D b ( T ai l s ( R )) , obtained throug h a suitable iteration o f tilting with re- sp e ct to tor s ion theories (as in the work of Happel, Reiten and Smalø, [18]), for s ome torsion classes { T a , ..., T a + s } (see a lso [3], [22] and [3 0] for similar co ns tr uctions). The ca tegories T ail s ( R ) ar e also defined as ab ove for noncommutativ e r ings R , providing a framework to noncommutativ e pr o jective g eometry ([8]). Our theo rem motiv a tes similar constructions of t-structures in this new setting (see section 5). Given a (nonco mm utative) graded K - algebra R , the nonco mm utative pro jective scheme a sso ciated with R ca n b e though t of as a n abstra ct space P r oj ( R ) whose PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 3 category of quasi-c oher ent she aves (resp ectively , c oh er ent she aves ) is the category T ai l s ( R ) (resp ectively , tail s ( R ) ) with structure sheaf π R . Analo g ously to the com- m utative case, π admits a right adjoint Γ ∗ . A rtin and Sc helter defined in [5] the class of alg e br as whose catego r ies o f tails play the ro le of coherent sheaves ov er non- commut ative pro jective planes. These , ca lled Artin-Schelter (AS for shor t) regular algebras of dimension 3 , a re alge br as of global dimension 3, finite Gelfand-Kirillov dimension (in fac t equal to 3 ) and Gor e ns tein. These a lgebras have be e n classified ([5],[6]) by a sso ciating to ea ch o ne a triple ( E , σ, L ) , where E is a scheme, σ is an automor phism of E a nd L is an inv ertible s heaf on E . In section 5 we fo cus on AS-regular a lgebras of dimension 3 with 3 genera tors such that E is a divisor of degr ee 3 in P 2 , σ is an automorphism o f finite or der and L is the restriction o f O P 2 (1) . These a re no etherian a lgebras a nd finite ov er their cent res ([7]) - hence fully bo unded no etheria n. In this setting, we provide an example o f a new c onstruction of p erverse q ua si-coherent t-structures . The pap er is o utlined as follows. Section 2 presen ts some bas ics on torsion theories for ca tegories of gra ded modules . In sectio n 3 we show ho w to obtain a t-structure by adequately iterating a result of [1 8], using certain torsion theories of an AB4 ab elian catego ry . Section 4 shows how to rsion theories come into play when describing perverse coher ent t-structur e s and section 5 ap plies section 3 to define per verse quasi- coherent t-structures on some noncommutative pro jective pla nes. 2. Torsion theories for graded modules Let R b e a no e ther ian gr aded ring (not necessa rily commutativ e). The category Gr ( R ) is a Grothendieck catego ry , admitting injective env elopes which, for a gra ded mo dule M , w e denote by E g ( M ) . The homomorphisms b etw een mo dules M and N in Gr ( R ) ( R -linear, gr ading preserving) are denoted by Hom Gr ( R ) ( M , N ) . The subset of homogeneous element s of M is denoted by h ( M ) . It is clear that M is generated by h ( M ) . Also, for a pr ime ideal P , define C g ( P ) = C ( P ) ∩ h ( R ) , where C ( P ) is the set of regular elemen ts mo dulo P , i.e., the set o f elements x of R such that x + P is neither left nor rig ht zero divisor in R/ P . If R is commutativ e, then C ( P ) = R \ P . The following remar k pr ov es to b e useful. R emark 2 .1 . Giv en a connected p ositively graded ring R generated in degree o ne and a homo geneous pr ime ideal P 6 = R + := L i ≥ 1 R i , we have P n 6 = R n for all n > 1 . Recall that an ideal P of a ring R is pr ime if and only if for all x, y ∈ R , whenever xR y ⊂ P , either x or y m ust b elo ng to P . Supp ose that there is n 0 > 1 such that P n 0 = R n 0 . Note that, since the ring is genera ted in deg ree o ne, we have P n = R n for all n > n 0 . Let x 1 be an element in R 1 \ P 1 . Since P is prime, there is r 1 ∈ R such that x 2 = x 1 r 1 x 1 / ∈ P . Now, deg ( x 2 ) ≥ 2 since R is p ositively g raded. Thu s we can inductively construct a sequence o f element s ( x n ) n ∈ N none of them lying in P and such that deg ( x n ) > deg ( x n − 1 ) , yielding a contradiction with the assumption that P n = R n for all n > n 0 . W e recall the definition of torsio n theor y . Definition 2.2. Let A be an ab elian ca tegory . A pair of full sub ca teg ories ( T , F ) is said to b e a tors ion theory if: (1) Hom ( T , F ) = 0 , for a ll T ∈ T and F ∈ F ; (2) F or all M ∈ A there is a n exa ct sequence 0 − → τ ( M ) − → M − → M / τ M − → 0 , 4 JORG E VITÓRIA where τ ( M ) ∈ T and M /τ ( M ) ∈ F . W e say that ( T , F ) is a hereditary tors io n theo r y if T is closed under sub ob jects. W e are particular ly int erested in (hereditar y) torsion theories defined as follows. Definition 2.3 . A torsion theory (or its tor sion clas s ) in Gr ( R ) is said to be cogenera ted by a n injectiv e ob ject E if the torsion ob jects are precisely those M satisfying Hom Gr ( R ) ( M , E ) = 0 . Since R is no ether ia n, g r ( R ) is closed under taking sub ob jects. Th us, torsion theories in Gr ( R ) res trict to torsion theo ries in g r ( R ) . F or this it is enoug h to observe that, g iven a mo dule M in g r ( R ) and τ the torsio n radical functor induced b y a torsion theory in Gr ( R ) , b oth τ ( M ) and M / τ ( M ) a ls o lie in g r ( R ) . The following lemma pro ves a useful criterion for g raded mo dules to b e tor - sion with resp ect to the torsion theory cogenera ted by an injective o b ject. The arguments of the pro of mimic the ung raded cas e (see [24], lemma 2.5 ). Lemma 2.4. Given gr ade d mo dules T and F ove r a gr ade d ring R , the fo l lowing c o nditions ar e e quiva lent: (1) Hom Gr ( R ) ( T , E g ( F )) = 0 ; (2) ∀ t ∈ h ( T ) , ∀ f ∈ h ( F ) \ 0 , deg ( f ) = deg( t ) , ∃ r ∈ h ( R ) : tr = 0 ∧ f r 6 = 0 . Pr o of. Supp ose Ho m Gr ( R ) ( T , E g ( F )) 6 = 0 . Let α b e one of its nonzer o elements. Cho ose u ∈ h ( T ) suc h that α ( u ) 6 = 0 . Now, F is a g raded essential submo dule of E g ( F ) , i.e., g iven any nontrivial graded submo dule of E g ( F ) , its intersection with F is non trivial. H ence there is s ∈ h ( R ) such that 0 6 = α ( u ) s = α ( us ) ∈ F . If w e choose t = u s and f = α ( us ) , they a re homogeneo us o f the same degree and clearly , given r ∈ R , if tr = 0 then f r = 0 . Suppos e now that (2 ) is false, i.e., there a r e t ∈ T and f ∈ F \ { 0 } homogeneo us of the same degr ee such that for a ll r ∈ h ( R ) , if tr = 0 then f r = 0 . Then, ther e is a well defined nonzero g raded homomor phism tR − → F, tr 7→ f r since h h ( R ) i = R . Since E g ( F ) is an injective ob ject in the categ ory o f graded mo dules, we ca n find a nonzer o graded homomor phism from T to E g ( F ) .  The following co rollar y shows how to r eformulate a statement a b o ut graded lo calisation in terms of tor sion. Corollary 2.5. L et R b e a c ommutative gr ade d ring, P a homo ge ne ous prime ide al in R and S = h ( R \ P ) . Given M a gr ade d R -mo dule then ( S − 1 M ) 0 = 0 if and only if Hom Gr ( R ) ( M , E g ( R/P )) = 0 . Pr o of. This follows fro m the fact ( S − 1 M ) 0 = 0 is equiv alent, by definition of graded lo calisation, to co ndition (2) o f the ab ov e lemma with T = M a nd F = R/ P .  W e will now lo ok at rigid tor sion theories ([26 ]). W e s hall consider the fo llowing subset of Hom R ( M , N ) : Hom( M , N ) := L i ∈ Z Hom Gr ( R ) ( M , N ( i )) . Definition 2. 6. W e s ay that a torsio n theor y in Gr ( R ) is rigid if the clas s of tors io n mo dules (equiv alen tly , the class of torsio n-free mo dules) is closed under shifts of the gr ading. The rigid torsion the ory c o gener ate d by an inje ctive obje ct E in Gr ( R ) is defined such that a mo dule M is tors ion if Hom( M , E ) = 0 . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 5 R emark 2 .7 . Observ e that, given g raded modules T and F ov er a graded r ing R , Hom( T , E g ( F )) = 0 if a nd only if Hom Gr ( R ) ( T ( j ) , E g ( F )) = 0 for all j ∈ Z . This allows us to get a nalogues of le mma 2.4 a nd cor ollary 2.5 as follows: (1) The following conditions ar e equiv alen t: • Hom( T , E g ( F )) = 0 • ∀ t ∈ h ( T ) , ∀ f ∈ h ( F ) \ 0 , ∃ r ∈ h ( R ) such that tr = 0 and f r 6 = 0 . (2) If R is commutativ e, P a homog eneous prime ideal in R , S = h ( R \ P ) and M a graded R -mo dule, then we hav e that S − 1 M = 0 if and only if Hom( M , E g ( R/P )) = 0 . It is, in fac t, po s sible to get a more g eneral s tatemen t, including some non- commut ative r ing s, by co mpa ring this rigid torsion theory with the torsio n theory asso ciated to a mu ltiplicative set. F or a homogeneo us right ideal J of a gra ded ring R we use the notation J ⊳ r g R and, given r ∈ R , we define a right idea l r − 1 J := { a ∈ R : r a ∈ J } . Recall the following result ([26 ], Prop o s ition A.I I.9.1 1). Prop ositio n 2.8. L et R b e a gr ade d ring, S a mult iplic ative subset c ontaine d in h ( R ) . Then the class of mo dules M such that ther e is J ∈ L S with M J = 0 , wher e L S =  J ⊳ r g R : r − 1 J ∩ S 6 = ∅ , ∀ r ∈ h ( R )  , is a torsion cla ss for a rigid torsion the ory in Gr ( R ) . L S as ab ove is said to be a graded Gabriel filter for that to rsion theory . If S = C g ( P ) for some homogeneous prime ideal P , then we denote the filter by L P . The r ig id torsion theory a sso ciated to an injective gr aded mo dule E a lso has an asso ciated gra ded Ga briel filter given by: L r E =  J ⊳ r g R : Hom( R/J, E ) = 0  . In fact, hereditary rigid tor sion theories ar e in bijection with graded Gabriel filters ([26], Lemma A.I I.9.4). Thus the gr aded Gabriel filter determines the torsio n theory and vice- versa. The following t w o suppo rting lemmas will be useful in proving the main theorem o f this s e c tio n. Lemma 2 . 9. L et R = ⊕ i ≥ 0 R i b e a no etheri an gr a de d ring, P a homo gene ous prime ide al and J a homo gene ous right ide al of R . If J ∩ C ( P ) 6 = ∅ then J ∩ C g ( P ) 6 = ∅ . Pr o of. Recall that, in a prime no etherian ring, an element is left regular if and only if it is right reg ular (see [19] for details). Therefore w e can reg ard C ( P ) as the set of elements x ∈ R such that x + P i s r ight regula r in R/P . Let c ∈ J ∩ C ( P ) and consider its homogeneo us deco mp os ition in J : c = c i 1 + c i 2 + ... + c i n where c i j ∈ J \ { 0 } ∩ R i j . If c i 1 ∈ C ( P ) , we are do ne. If not, by definition, there is r 1 ∈ R \ P such that c i 1 r 1 ∈ P . Moreover, the choice of r 1 can b e made in h ( R \ P ) , since P is a homogeneous ideal. Clearly cr 1 + P is right regular in R/ P and, thus, c (1) := cr 1 − c i 1 r 1 / ∈ P . W e iterate this arg umen t by lo oking at the first homogeneo us comp onent o f c (1) (whic h is c i 2 r 1 ). Assume now that this n -step iteration do es not yield a homogeneo us element in J ∩ C ( P ) . Then, this ar gument gives a sequence r 1 , ..., r n of elements in R such that cr 1 ...r n ∈ P , which is a contradiction to c being regular mo dulo P .  6 JORG E VITÓRIA Lemma 2. 10. L et R b e a p osi tively gr ade d no etherian ring, J a right ide al of R and M a gr ade d right R -mo dule. Then J lies in L r E g ( M ) if and only if m ( x − 1 J ) 6 = 0 for al l m in h ( M ) \ { 0 } and x in h ( R ) . Pr o of. Note that J ∈ L r E g ( M ) if and only if, for every cyclic graded submo dule C of R/J , Hom( C, E g ( M )) = 0 . Now, it is e a sy to see that the graded cy clic submo dule generated by x + J , for some x ∈ h ( R ) , is isomor phic to R/x − 1 J . No w, of co urse, Hom( R/x − 1 J, M ) = 0 if and only if, for all m ∈ h ( M ) \ { 0 } , m ( x − 1 J ) 6 = 0 .  The following theorem is a g raded version of the main result in [24]. Theorem 2.11. L et P b e a homo gene ous prime ide a l of a gr ade d ring R and R /P no etheri an. L et M b e a gr ad e d right R -mo dule. Then M is t orsion with r esp e ct to the rigid torsion the ory asso ciate d with C g ( P ) if and only if M is torsion with r esp e ct to the rigid torsion the ory asso ciate d to E g ( R/P ) . Pr o of. W e will pr ov e that the g raded Gabr iel filters of b oth tor s ion theories coincide. Let E = E g ( R/P ) and J ∈ L r E . B y lemma 2.10 this is equiv a lent to say that for all 0 6 = a + P ∈ h ( R /P ) and for all x ∈ h ( R ) , ( a + P )( x − 1 J ) 6 = 0 . This mea ns that, for any choice of a + P ∈ h ( R/P ) , the tw o-sided ideal K := ( R /P )( a + P )(( x − 1 J + P ) /P ) ⊳ R /P is nonzer o. Since R/P is a t wo-sided no etheria n ring, it is w ell-known that every t wo-sided ideal is ess ent ial as a rig h t ideal and, therefor e , by Goldie’s theor em for graded rings (see [16], theorem 4), we hav e that K contains a ho mo genous regula r element c + P . Now c + P = ( b a + P )( t + P ) where t ∈ x − 1 J . Clear ly , t + P is right regular and, since R/P is a prime no etherian ring, it is regular. Therefor e, we conclude that t ∈ x − 1 J ∩ C ( P ) . By lemma 2.9, we also hav e that x − 1 J ∩ C g ( P ) 6 = ∅ and thus J ∈ L P . Conv ersely , supp o se J ∈ L P and let a, b ∈ h ( R ) , b / ∈ P . By hypothesis, a − 1 J ∩ C g ( P ) 6 = ∅ . Let z b e one of its element s. Then, clea rly , az ∈ J and bz / ∈ P . Again, b y remar k 2.7, the result follows.  In the comm utativ e p ositively g raded case, ho wev er, the torsion theory a sso ci- ated with the injective mo dule E g ( R/P ) coincides with the o ne asso ciated to the m ultiplicative set h ( R \ P ) b y the following well-kno wn result, the proof of which we, thus, omit. Prop ositio n 2.12. L et R b e a c ommutative no etherian p ositively gr ade d c onne cte d K -algebr a gener ate d in de gr e e 1, P a homo gene ous prime ide al in R not e qual to the irr elevant ide al and M a gr ade d R -mo dule. Then, for S = h ( R \ P ) , S − 1 M = 0 if and only if ( S − 1 M ) 0 = 0 . This shows tha t under the conditions of the pro po sition a b ove, the torsion theor y cogenera ted b y E g ( R/P ) is automatically rigid. This statement, how ev er, c a n b e prov ed without assuming commutativit y . Lemma 2. 13. L et R b e a no etheria n p ositively gr ad e d c onne cte d K -algebr a gener- ate d in de gr e e 1, P a homo gene ous prime ide a l in R not e qual to the irr elevant ide al and M a right gr ade d R -mo dule. Then, Hom Gr ( R ) ( M , E g ( R/P )) = 0 if and only if Hom( M , E g ( R/P )) = 0 . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 7 Pr o of. One direction is clear. Supp o se Hom( M , E g ( R/P )) 6 = 0 . Then by remark 2.7 there is m ∈ h ( M ) such that Ann ( m ) ∩ C g l ( P ) = ∅ , where Ann ( m ) stands for right annihilator of m a nd C g l ( P ) stands for homogeneo us left regula r elements mo d P . W e wan t to pro ve Hom Gr ( R ) ( M , E g ( R/P )) 6 = 0 . By remark 2.1, there is a homo geneous element in R \ P i n eac h p ositive degr ee and th us, by lemma 2.4, Hom Gr ( R ) ( M , E g ( R/P )) 6 = 0 is equiv ale nt to the ex istence of ˜ m ∈ h ( M ≥ 0 ) such that Ann ( ˜ m ) ∩ C g l ( P ) = ∅ . Note that the irrelev an t ideal, R + , is a homogeneous maximal idea l con taining P . Clear ly , R + /P is an essential idea l in the graded prime Goldie ring R/P a nd so , b y gra ded Goldie’s theorem (see [16], theorem 4), it co nt aining a r egular elemen t. This means that there is a homo geneous reg ula r element o f p ositive deg ree in R/P and thus C g ( P ) ≥ k 6 = ∅ for a ll k ∈ N . Cho ose s ∈ C g ( P ) suc h that deg ( ms ) ≥ 0 . Note that if there is a ∈ Ann ( ms ) ∩ C g l ( P ) , then sa ∈ Ann ( m ) ∩ C g l ( P ) yielding a contradiction. Ther efore, take ˜ m = ms and we a re done.  R emark 2.14 . W e summarise the results of this section. If R is a no etherian p osi- tiv ely graded connected K -algebr a genera ted in degr ee 1, P a homogeneous prime ideal no t equa l to the ir relev an t ideal R + and M a gra ded R -mo dule, then the following are equiv a lent : (1) Hom Gr ( R ) ( M , E g ( R/P )) = 0 ; (2) Hom( M , E g ( R/P )) = 0 ; (3) M is tors io n with resp ect to C g ( P ) . If, furthermore, R is co mm utative and S = h ( R \ P ) , then (1), (2) and (3) a re equiv a lent to S − 1 M = 0 and to ( S − 1 M ) 0 = 0 . 3. t-structures via torsion theories Recall that an ab elian ca teg ory A is said to b e AB4 if it admits arbitra ry co- pro ducts a nd they are exact. It is well known that, under this assumption, D ( A ) admits ar bitrary copr o ducts as w ell. In this section w e will sho w that, for a ∈ Z , n ∈ N a nd ce r tain ordered sets (indexed by a string of integers of length n s tarting at a ) of hereditary torsion classes in an AB4 ab elian category A S = {T a , T a +1 , ..., T a + n − 1 } with T a ⊇ T a +1 ⊇ T a +2 ⊇ ... ⊇ T a + n − 1 = 0 , the following s ubca tegory is the aisle of a t-structure in D b ( A ) , D S, ≤ 0 :=  X • ∈ D b ( A ) : H i 0 ( X • ) ∈ T j , ∀ i > j  . R emark 3.1 . Clearly such a ca tegory is a sub categ ory of D ≤ a + n − 1 0 , a shift of the standard aisle. This follows from the ass umption that T a + n − 1 = 0 . Our pro o f relies o n a suitable iteratio n o f a well-kno wn theor em, originally due to Happ el, Reiten and Smalø ([18], Prop osition 2.1). W e present here a slightly mo dified version of that result, as stated by Bridgela nd ([12]). Recall that a t- structure ( D ≤ 0 , D ≥ 0 ) in a triang ulated c a tegory D is s a id to be b ounded if [ n ∈ Z D ≤ n = D = [ n ∈ Z D ≥ n . Theorem 3.2 (Happ el, Reiten, Smalø, [18], Br idgeland, [12]) . L et A b e the he art of a b ounde d t-structure in a triangulate d c ate gory D . Supp ose that ( T , F ) is a 8 JORG E VITÓRIA torsion the ory in A and t hat H i denotes the i-th c ohomolo gy functor with r esp e ct to A . Then ( D ≤ 0 , D ≥ 0 ) is a t-structur e in D , wher e D ≤ 0 =  E ∈ D : H i ( E ) = 0 , ∀ i > 0 , H 0 ( E ) ∈ T  D ≥ 0 =  E ∈ D : H i ( E ) = 0 , ∀ i < − 1 , H − 1 ( E ) ∈ F  . Mor e over, ( F [1 ] , T ) is a torsion the ory in D ≤ 0 ∩ D ≥ 0 . The new t-structure (or its heart) o btained in the theorem will b e called the HRS-tilt of A with r esp e ct to ( T , F ) . W e will need a few tec hnical lemmas ab o ut this new hea rt. W e start with an useful obser v a tio n a b o ut some of its morphisms. Lemma 3.3. L et A b e t he he art of a b ounde d t-st ructur e in D , a t riangulate d c a te gory. Supp ose that ( T , F ) is a her e ditary torsion the ory in A and that B is the c o rr esp onding he art of t he HRS-t ilt. F or an obje ct T in T we have: (1) a morphism f : T − → N is an epimorp hism in B if and only if N lies in T ⊂ A and f is an epimo rphism in A ; (2) a morphism f : M − → T is a monomorphism in B if and only if M lies in T ⊂ A and f is a monomorp hism in A . Pr o of. (1) Let T be an ob ject in T , N in B and f an epimor phism in Hom B ( T , N ) . Let C ∈ T a nd F [1] ∈ F [1] b e such that we hav e a shor t exact sequence in B 0 − → F [1] − → N − → C − → 0 since ( F [1] , T ) is a tor sion theory in B . Consider the following commutativ e diagra m T f / /   N / /   K [1] / / T [1 ]   T / / f   C / /   L [1] / / T [1 ]   N / / C / / F [2] / / N [1] where the r ows are triangles in D and where K stands for the kernel of f in B and L for the kernel in B of the co mpo s ition o f f with the epimorphism N → C in B . Note, how ev er, that s ince T ∈ T and T is a tor sion-free class in B (thus c lo sed under sub ob jects in B ), b oth K and L lie in T . The o c tahedral axiom applied to the ab ove diagram yields (after an adequate ro tation) the triangle F − → K − → L − → F [1] which induces a shor t exact sequence in A , wher e F is, therefore, a subo b ject of K in A . Since T is a hereditary torsio n class in A and K lies in T , we co nclude that F lies in T and hence it is zero, proving that N is isomorphic to C , a n ob ject of T . Conv ersely , if N lies in T and f in Hom A ( T , N ) is an e pimo r phism, then its kernel in A also lies in T ( T is here dita r y). Thus, the shor t exact sequence defined b y f is a lso a sho rt exact s equence in B a nd f is an epimorphism in B . (2) Giv en a monomorphism g ∈ Hom B ( M , T ) for so me M ∈ B and T ∈ T , we easily see that M ∈ T (since T is a torsion-free cla ss in B ) a nd, by (1), that the cokernel of g in B lies in T . Thus, we hav e a short exact sequence in A : 0 − → M − → T − → cok er ( g ) − → 0 . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 9 Conv ersely , if M lies in T and f in Hom A ( T , M ) is a mono mo rphism, then its cokernel in A a lso lies in T ( T is a torsion cla ss). Thus, the short exa ct sequence defined by f is also a short exact sequence in B and f is a monomorphism in B .  Definition 3.4. Let A b e an abelian category . W e say that the hear t B of a bo unded t-structure in D b ( A ) (or the t-structure itself ) is uniformly b ounde d if there are m, n ∈ Z such that B ⊆ D ≤ m 0 ∩ D ≥ n 0 . A family o f ob jects ( Z k ) k ∈ K in D b ( A ) is uniformly b ounde d with r esp e ct to a t-stru ctur e ( D ≤ 0 , D ≥ 0 ) if there are m, n ∈ Z such that Z k ∈ D ≤ m ∩ D ≥ n for all k ∈ K . Note that, in a n AB4 ab elian category A , a family ( Z k ) k ∈ K is uniformly b ounded with resp ect to the standar d t-structure if and only if its copro duct lies in D b ( A ) . This follows fr o m the fact that the standard co ho mology commutes with co pro ducts, since copro ducts in A a re ex a ct. W e now show similar statements for certain hearts. Lemma 3.5. L et A b e an AB4 ab elia n c ate gory, ( D ≤ 0 , D ≥ 0 ) a u n iformly b ounde d t-structu r e in D b ( A ) and B its he art. Then, B is c o c ompl ete if and only if existing c o pr o ducts ar e t-exact in D b ( A ) . Pr o of. Since B is uniformly b o unded a nd co pr o ducts commute with standard co ho- mologies, small copro ducts of elements in B exist in D b ( A ) . If existing copro ducts are t-exact in D b ( A ) then, c lea rly , B is co complete. Conv ersely , we first o bs erve that right t-exactness is automatic. This follows from the fact that D ≤ n is left Hom-or tho g onal to D ≥ n +1 (see, for example, lemma 1.3 in [2]). T o prov e left t-exactness, let ( Y k ) k ∈ K a family of ob jects in D ≥ 0 such that its copro duct, call it Y , lies in D b ( A ) . W e shall prov e that Y ∈ D ≥ 0 . As usua l, τ ≤ n , τ ≥ n denote the tr unca tion functors a nd H n the coho mological functors, fo r all n ∈ Z , with resp ect to the fixed t-str uctur e ( D ≤ 0 , D ≥ 0 ) . Denote by Y i k := τ ≥ i ( Y k ) and by B i k := H i ( Y k ) , for any k ∈ K and i ∈ Z , the resp ective truncation a nd cohomolog y functors. W e hav e the following sequence of triangles (often c a lled a P ostniko v tow er o r, in certain co nt exts, a Harder-Nar a simham filtra tion) for each Y k , where m k ≥ 0 is the maximal degr ee for whic h cohomo lo gy do es not v anish. Y k = Y 0 k / / Y 1 k [1] ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ / / . . . / / Y m k k / / [1] } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ 0 [1] { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ B 0 k c c ● ● ● ● ● ● ● ● ● . . . _ _ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ B m k k [ − m k ] e e ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ . Observe that, s ince Y ∈ D b ( A ) and B is uniformly bo unded, the s e t { m k : k ∈ K } has a maxim um - call it m . By extending trivially eac h o f these s e q uences of triangles to sequences with m tr iangles a nd since the c o pro duct o f triangles is a triangle, we may consider the copro duct o f these sequences comp onent wise and this yields a (finite) Postnik ov to wer for Y , since B is co complete. Th us, the Postnik ov tow er lo oks a s follows: Y / / ` k ∈ K Y 1 k [1] | | ① ① ① ① ① ① ① ① ① / / . . . / / ` k ∈ K Y m k / / [1] ~ ~ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ 0 [1] | | ② ② ② ② ② ② ② ② ② ② ② ② ` k ∈ K B 0 k _ _ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ . . . ` ` ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ` k ∈ K B m k [ − m ] e e ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ . 10 JORG E VITÓRIA Now, it is clear that a k ∈ K Y m k ∼ = a k ∈ K B m k [ − m ] ∈ B [ − m ] ⊂ D ≥ m , since B is co complete. The triangle a k ∈ K B m − 1 k [ − m + 1 ] − → a k ∈ K Y m − 1 k − → a k ∈ K Y m k − → ( a k ∈ K B m − 1 k [ − m + 1 ])[1] shows that a k ∈ K Y m − 1 k ∈ D b ( A ) and a k ∈ K Y m − 1 k ∈ D ≥ m − 1 . Iterating this a rgument we get that Y lies in D ≥ 0 .  R emark 3 .6 . Note that a uniformly bounded family of ob jects ( Z k ) k ∈ K in D b ( A ) with resp ect to a uniformly bounded t-structure ( D ≤ 0 , D ≥ 0 ) is also uniformly bo unded with resp ect to the standard t-structure in D b ( A ) . This can be chec k ed using Postnik ov tow ers, in a similar arg ument to the one used in the pr o of ab ov e. Corollary 3. 7 . L et A b e an AB4 ab elian c ate gory and ( D ≤ 0 , D ≥ 0 ) a uniformly b o unde d t-structur e in D b ( A ) with c o c omplete he art B and c ohomolo gic al functors H i , for al l i ∈ Z . If ( Z k ) k ∈ K is a uniformly b ounde d fami ly of obje cts with re sp e ct to ( D ≤ 0 , D ≥ 0 ) , then H i ( a k ∈ K Z k ) = a k ∈ K H i ( Z k ) . Pr o of. Let Z denote the co pro duct of the family ( Z k ) k ∈ K . By re ma rk 3.6, Z lies in D b ( A ) . The ar gument in the pro of of lemma 3.5 shows that a Postnik o v tower of Z can b e obtained as the copro duct ov er K of P ostniko v tow ers of each Z k . Since the low er vertices of a Postniko v tower ar e unique up to isomorphism (they are shifts of the co homologies with r esp ect to the fixed t-s tructure), the result follows.  W e will consider torsion classes with a sp ecial prop erty . Recall that an ob ject X in D ( A ) is compact if the functor Hom D ( A ) ( X, − ) commutes with copr o ducts. Definition 3.8 . A sub categ ory T o f a hear t B of D b ( A ) is c omp actly gener ate d in D b ( A ) if e very ob ject in T is the colimit in B of a family of sub ob jects in B which are compact when rega rded as ob jects in D b ( A ) . Similar results to the follo wing lemma ha ve been obtained b y Colpi and F uller in [13] for HRS-tilts o f the standard heart. Lemma 3.9. L et A b e an AB4 ab elia n c ate go ry. S upp o se that ( T , F ) is a her e d itary torsion t he ory in the he art B of a uniformly b ounde d t-structu r e in D b ( A ) , with B c o c ompl ete and T a su b c ate gory of B c omp actly gener ate d in D b ( A ) . Then the HRS-tilt of B with r esp e ct t o ( T , F ) is uniformly b ounde d and c o c omplete. Pr o of. First we show that F is closed under copro ducts. Let ( Y i ) i ∈ I be a family of ob jects in F a nd let X ∈ T . Let ( X j ) j ∈ J be a family of compact sub ob jects o f X in B (and, hence, in T , by lemma 3.3) such that X = lim − → j ∈ J X j . Now, Hom B (lim − → j ∈ J X j , a i ∈ I Y i ) = lim ← − j ∈ J Hom B ( X j , a i ∈ I Y i ) = lim ← − j ∈ J a i ∈ I Hom B ( X j , Y i ) = 0 since the X i are co mpact in D b ( A ) a nd B is a full subca tegory of D b ( A ) . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 11 Let C b e the HRS-tilt o f B with resp ect to ( T , F ) , ( Z k ) k ∈ K a family of ob jects in C a nd Z its co pro duct. W e denote H i ( i ∈ Z ) the co homologica l functors defined b y the t-structure of which B is the hea rt. F or any C ∈ C , there is a triangle H − 1 ( C )[1] − → C − → H 0 ( C ) − → H − 1 ( C )[2] which shows tha t, since B is uniformly b ounded, then s o is C . This s hows that Z lies in D b ( A ) . Since B is co complete, Coro lla ry 3.7 shows that copr o ducts in D b ( A ) co mm ute with H i , for all i ∈ Z . It is then clea r that H i ( Z ) = 0 fo r all i 6 = 0 , − 1 and, s ince b oth T a nd F are closed under co pro ducts, H − 1 ( Z ) ∈ F and H 0 ( Z ) ∈ T , showing that Z ∈ C and, thus, completing the pro of.  The next result ca n b e found in Dic kson’s work [14] o r in Stens tr öm’s b o ok [3 1], usually also assuming that the underlying ab elia n category is complete and well- powered (i.e., the class of sub ob jects o f a given ob ject form a set). It is c lea r from the pro o f, howev er, tha t the completeness assumption is not necessary and that the well-pow ered condition o n the ab elian ca tegory can b e made weak er. Given a full subc a tegory X of an ab elia n categor y A , we will say that A is X - wel l-p ower e d if for any g iven ob ject o f A , the clas s of its sub ob jects lying in X form a se t. Lemma 3.10. L et A b e a c o c omplete ab eli an c ate gory and T a ful l sub c a te gory of A . Assume that every obje ct in T is the c olimit of a dir e cte d set of sub obj e cts lying in a sub c ate gory X ⊆ T such that A is X -wel l-p owe r e d. T hen T is a torsion class if and only if it is close d under extensions, images and c opr o ducts. The a s sumptions in the lemma ensure that a n y ob ject Y in A has a maximal subo b ject lying in T : it is the co limit of all sub ob jects o f Y that lie in X ( which form a s et by hypothesis). The following lemma is cr ucial in the pro of of theorem 3.13. Althoug h we need the technical assumptions in the lemma for this abstra ct setting, they are har mless for the purp ose of o ur applicatio ns (see rema rk 3.15 at the end of this section). Lemma 3.1 1. L et A b e an AB4 ab eli an c ate gory and c onsider a uniformly b ounde d t-structu r e in D b ( A ) with a c o c omple te he art B . Supp ose that ( T , F ) is a her e ditary torsion the ory in B su ch that T is a sub c ate gory of B c omp actly gener ate d in D b ( A ) and such that the c omp act obje cts of D b ( A ) lying in T form a set. L et C denote the HRS-tilt of B with r esp e ct to ( T , F ) . Then, a sub c ate gory T 1 of T is a her e ditary torsion class in C if and only if it is a her e ditary t orsion class in B , in which c ase the sub c ate gory T 1 of C is, mor e over, c omp actly gener ate d in D b ( A ) . Pr o of. Supp ose first that T 1 is a hereditar y tor sion class in B . W e first show that T 1 is closed under extensio ns, co pro ducts and epimorphic ima ges in C . The fir s t t wo hold trivially (exact sequences in C are pr e c isely the distinguished tria ng les of D ( A ) that lie in C and if the tw o outer terms lie in B then so do es the middle one) since T 1 is a torsion class in B . T o see it is closed under epimor phisms , we use lemma 3.3. Indeed, if f : T − → C is an epimorphism in C with T ∈ T 1 ⊆ T a nd C ∈ C , we ha ve that, by lemma 3 .3, C ∈ T a nd f is an epimor phism in B . Since T 1 is a to rsion class in B , C must lie in T 1 . Finally , o bserve that if g : C ′ − → T is a monomorphism in C with C ′ ∈ C and T ∈ T 1 , then, by lemma 3.3, C ′ lies in T and g is a mono mo rphism in B . Ther efore, since T 1 is hereditary in B , C ′ lies in T 1 . W e furthermore observe that T 1 is compa ctly generated in D b ( A ) as a s ub ca teg ory of C . Indeed, for X in T 1 , co nsider a family ( X j ) j ∈ J of sub ob jects of X in B , compac t 12 JORG E VITÓRIA in D b ( A ) , such that X = lim − → j ∈ J X j . Since T 1 is a hereditary torsion class in B , each X j lies in T 1 and it is a sub ob ject o f X in C . Now, X lies in C and s o do es the copro duct of the fa mily ( X j ) j ∈ J (more precisely it lies in T 1 ). The colimit of this family in C is the cokernel o f an endomor phism of the copro duct (see, fo r example, [31], IV.8.4) and, thus, it is s till X , as wan ted. Note that, by lemma 3.9, C is co complete. T o complete the pro of tha t T 1 is a hereditar y torsion class in C using lemma 3.10, we just need to show that C is X -well-powered, where X is the sub categ o ry of T 1 formed by thos e ob jects which are co mpact in D b ( A ) . This follows from the fact that T 1 is a sub c a tegory of C compactly gener ated in D b ( A ) and from our assumption that the co mpact ob jects of D b ( A ) lying in T form a set (and, th us, so do the ones lying in T 1 ). It is then clear that, for an y ob ject Y in C , the family of sub ob jects of Y lying in X lies in the pro duct o f the sets H om C ( X, Y ) , where X runs over the set X . Conv ersely , supp ose T 1 is a tor s ion class in C . As a sub categor y of B it is ag ain obviously closed under extensions and copr o ducts. Suppo se that f : T − → B is a n epimorphism in B with T ∈ T 1 ⊆ T a nd B ∈ B . Then, clear ly B ∈ T a nd, by le mma 3.3, f is a n epimorphism in C . Hence, since T 1 is a torsion class in C , B m ust lie in T 1 . Mo reov er, observe that if g : B ′ − → T is a monomorphism in B with B ′ ∈ B and T ∈ T 1 , then, clear ly B ′ ∈ T and, by lemma 3.3, g is a monomorphism in C . Therefore, since T 1 is her editary in C , B ′ lies in T 1 . Finally , since B is co co mplete and T 1 is X - well-pow ered, where X is the set o f co mpact o b jects of D b ( A ) lying in T 1 , lemma 3.10 concludes the pro o f.  W e need o ne more s imple but useful lemma (in ligh t of r emark 3.1) a bo ut the relation b etw een truncations of t-structures whos e a isles a r e related b y inclusion. Lemma 3 .12. Su pp ose ( D ≤ 0 A , D ≥ 0 A ) and ( D ≤ 0 B , D ≥ 0 B ) ar e t wo t- structur es in a tri- angulate d c ate go ry D with trun c ation functors τ ≤ 0 A and τ ≤ 0 B , r esp e ctive ly. If D ≤ 0 A ⊂ D ≤ 0 B , then for all X ∈ D ther e is a t riangle: (3.1) τ ≤ 0 A ( X ) − → τ ≤ 0 B ( X ) − → Y − → τ ≤ 0 A ( X )[1] such that Y ∈ D ≥ 1 A ∩ D ≤ 0 B . Pr o of. First no te that, since D ≤ 0 A ⊂ D ≤ 0 B , we hav e D ≥ 1 B ⊂ D ≥ 1 A . The triang le τ ≤ 0 B ( X ) − → X − → τ ≥ 1 B ( X ) − → τ ≤ 0 B ( X )[1] then shows that the na tur a l map τ ≤ 0 A ( X ) − → X must factor through τ ≤ 0 B ( X ) (since Hom ( τ ≤ 0 A ( X ) , τ ≥ 1 B ( X )) = 0 ). Let Y b e defined by the following triang le τ ≤ 0 A ( X ) − → τ ≤ 0 B ( X ) − → Y − → τ ≤ 0 A ( X )[1] . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 13 Since aisles a re closed under taking cones and τ ≤ 0 A ( X ) ∈ D ≤ 0 B , we hav e that Y ∈ D ≤ 0 B . W e wan t to prov e Y ∈ D ≥ 1 A . Consider the diagra m τ ≤ 0 A ( X ) / /   τ ≤ 0 B ( X ) / /   Y / / τ ≤ 0 A ( X )[1]   τ ≤ 0 A ( X ) / /   X / /   τ ≥ 1 A ( X ) / / τ ≤ 0 A ( X )[1]   τ ≤ 0 B ( X ) / / X / / τ ≥ 1 B ( X ) / / τ ≤ 0 B ( X )[1] where rows are tria ng les and the squares commut e by the obs e r v ation ab ov e. Then, the o ctahedr al axiom gives us a new triangle Y − → τ ≥ 1 A ( X ) − → τ ≥ 1 B ( X ) − → Y [1 ] . Since τ ≥ 1 B ( X ) ∈ D ≥ 1 A , so is τ ≥ 1 B ( X )[ − 1] . By the long exact sequence of coho mology induced from this triang le, it is easy to see that this shows that Y ∈ D ≥ 1 A .  W e say that a hear t B is obtained b y iter ate d H RS-tilts in D b ( A ) if there is a finite sequence o f hearts A = A 0 , A 1 , ..., A n = B and a sequence of s ubca tegories T 0 , ..., T n − 1 such that T i is a torsion clas s in A i and A i +1 is the HRS-tilts of A i with res pec t to the tors ion theory given by T i , for all 0 ≤ i ≤ n − 1 . W e now prove the main r esult of this section. Theorem 3.1 3 . L et A b e an AB4 ab elian c ate gory and S = {T a , T a +1 ..., T a + n − 1 } a set of her e ditary torsion classes of A , c o mp ac tly gener ate d in D b ( A ) , such that T a ⊇ T a +1 ⊇ T a +2 ⊇ ... ⊇ T a + n − 1 = 0 and such that the c omp act obje cts of D b ( A ) lying in T a form a set. Then, the ful l sub c ate gory given by D S, ≤ 0 =  X • ∈ D b ( A ) : H i 0 ( X • ) ∈ T j , ∀ i > j  is the aisle of a uniformly b ounde d t-structu r e in D b ( A ) with a c o c omplete he art B and it is obtai ne d by iter ate d HRS-tilts with r esp e ct t o the se quenc e S . Pr o of. Without loss of generality , w e assume that a = − n + 1 . W e use induction on the num ber n of elements of S to show that D S, ≤ 0 can b e obtained by itera ted HRS-tilts with res pec t to a sequence o f tor s ion class es given by the following ch ain T − n +1 ⊇ T − n +2 ⊇ ... ⊇ T − 1 ⊇ T 0 = 0 Suppos e n = 1 , i.e., S = {T 0 = 0 } . Then, w e hav e D S, ≤ 0 =  X ∈ D b ( A ) : H i 0 ( X ) = 0 , ∀ i > 0  = D ≤ 0 0 , the standar d a isle in D b ( A ) , which clearly satisfies all the desired pro p er ties. Suppos e the r esult is v alid for sequences S of n torsion classes s atisfying the assumptions of the theorem. Let S be a s e q uence with n + 1 hereditary torsio n classes of A which are compactly generated in D b ( A ) , S = {T − n , T − n +1 , ..., T 0 } such that T − n ⊇ T − n +1 ⊇ ... ⊇ T − 1 ⊇ T 0 = 0 . 14 JORG E VITÓRIA Let us consider the s e q uence S =  T − n +1 , T − n +2 , ..., T 0  where T i = T i − 1 , ∀ i < 0 , T 0 = 0 . Clearly , S is a lso a decreasing chain of hereditary torsion classes of A , compactly generated in D b ( A ) . W e fall into the case of n torsion class e s and by the induction h yp othesis w e have a n asso ciated uniformly b ounded t-struc ture with a co complete heart given by the aisle obtained by iterated HRS-tilts with r esp ect to S , D S , ≤ 0 = { X ∈ D b ( A ) : H i 0 ( X ) ∈ T j , ∀ i > j } = = { X ∈ D b ( A ) : X ∈ D ≤ 0 0 , H i 0 ( X ) ∈ T j − 1 , ∀ i > j } . W e deno te the corr esp onding hear t by B and asso ciated cohomolog ical functor by H 0 S := τ ≥ 0 S τ ≤ 0 S , where the τ S ’s are the asso ciated trunca tio n functors. Observe now that T − 1 is a s ubca tegory o f B . This follows fro m the fa c t that it is contained o n every torsio n cla ss in S and that B is o btained by itera ted HRS-tilts. By applying iteratively lemma 3.11, we can also co nclude that T − 1 is a hereditar y torsion cla ss in B , c o mpactly g enerated in D b ( A ) . By lemma 3.9 we get that the HRS-tilt o f B with r esp ect to T − 1 yields a unifor mly b ounded t-s tructure with a co complete heart. It rema ins to show that the aisle of the HRS-tilt can b e expresse d in terms of the to rsion classes in S as wan ted. That follows from the following lemma . Lemma 3.14. A n obje ct X lies in D S, ≤ 0 if and only if H 0 S ( X ) lies in T − 1 and H i S ( X ) = 0 , for al l i > 0 . Pr o of. Supp ose that X lies in D S, ≤ 0 . It is clear from the definition of S that D S, ≤ 0 ⊆ D S , ≤ 0 , thus proving that X lies in D S , ≤ 0 , whi ch is equiv alent to the condition that H i S ( X ) = 0 , for all i > 0 . Note that H 0 S ( X ) fits in the tr iangle τ ≤− 1 S ( X ) − → τ ≤ 0 S ( X ) − → H 0 S ( X ) − → τ ≤− 1 S ( X )[1] , which , aga in due to the fact tha t D S, ≤ 0 ⊆ D S , ≤ 0 , amounts to the tria ngle (3.2) τ ≤− 1 S ( X ) − → X − → H 0 S ( X ) − → τ ≤− 1 S ( X )[1] . Now, lemma 3.12 applied to D S , ≤− 1 ⊂ D ≤− 1 0 (see remark 3.1) shows that there is Y ∈ D S , ≥ 0 ∩ D ≤− 1 0 and a triangle τ ≤− 1 S ( X ) − → τ ≤− 1 0 ( X ) − → Y − → τ ≤ 0 S ( X )[1] . Since X lies in D S, ≤ 0 , we hav e that τ ≤− 1 0 ( X ) lies in D S , ≤− 1 , by c o nstruction of S and, thus, we get Y = 0 a nd τ ≤− 1 S ( X ) ∼ = τ ≤− 1 0 ( X ) . Since tw o of the vertices of a triangle determine the thir d o ne up to iso morphism, the triangle 3.2 shows that H 0 S ( X ) = H 0 0 ( X ) , which, by definition of D S, ≤ 0 , tells us that H 0 S ( X ) lies in T − 1 . Conv ersely , s upp os e X ∈ D S , ≤ 0 and H 0 S ( X ) ∈ T − 1 . As befor e, we hav e a triangle τ ≤− 1 S ( X ) − → X − → H 0 S ( X ) − → τ ≤− 1 S ( X )[1] . W e now apply to it the standard co homology functor, getting a long e x act sequence of o b jects in A . On one hand, since D ≤− 1 S is con tained in D ≤− 1 0 , we see that H i 0 ( τ ≤− 1 S ( X )) = 0 for all i > − 1 . This shows, in particular, that (3.3) H 0 0 ( X ) ∼ = H 0 0 ( H 0 S ( X )) PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 15 and, therefo r e, H 0 0 ( X ) lies in T − 1 (hence, it lies in T j for all j < 0 ). Let us now lo ok at the negative standard cohomo logies of X . Since, by h ypo thesis, H i 0 ( H 0 S ( X )) = 0 for all i 6 = 0 , the s ame long exact sequence also yields the following isomorphisms (3.4) H i 0 ( τ ≤− 1 S ( X )) ∼ = H i 0 ( X ) , ∀ i < 0 . Since D S , ≤− 1 = D S , ≤ 0 [1] , we hav e D S , ≤− 1 = { Y ∈ D b ( A ) : H i 0 ( Y ) ∈ T j , ∀ i > j − 1 } and, thus, the isomo rphisms (3.4) show that, in particular, H i 0 ( X ) ∼ = H i 0 ( τ ≤− 1 S ( X )) ∈ T i = T i − 1 , ∀ i < 0 . This proves that, for all 0 > i > j , H i 0 ( X ) lies in T j , since the torsion classes for m a chain. This fact, tog ether with the isomor phism (3 .3) prov es that X lies in D S, ≤ 0 .  D S, ≤ 0 can thus b e obtained as the aisle of the HRS-tilt of the hear t B , defined earlier in this pro of, with respect to the tor sion theory whose torsio n cla ss is T − 1 , hence finishing the pro o f.  R emark 3.15 . In the following sections of the pap er we will use this construction in categor ies of the for m D b ( T ai l s ( R )) , where R is a p ositively gra ded, connected, no etherian K - algebra generated in degree 1 such that one o f the following holds: • R is co mm utative (and, th us, D b ( T ai l s ( R )) ∼ = D b ( Qcoh ( P r oj ( R ))) ); • R is a 3-dimensional Artin-Schelter regular K -a lg ebra, which is finitely gen- erated as a mo dule ov er its centre. The technical assumptions o n the torsio n classes will b e s atisfied in these contexts. (1) The catego ry T ail s ( R ) is AB4 (it is, in fa ct, a Grothendieck categ ory), where every ob ject is a colimit of its sub ob jects lying in ta i l s ( R ) . The sub- category o f compact ob jects of D b ( T ai l s ( R )) is equiv alen t to D b ( tail s ( R )) ([11], lemmas 4.3.2 and 4.3.3) and, thus, ev ery hereditary tor sion class in T ai l s ( R ) is compactly generated in D b ( T ai l s ( R )) . (2) The functor s π and Γ ∗ induce equiv alences b etw een the a dditiv e ca tegories of tor sion-free injectiv e o b jects in Gr ( R ) a nd injective o b jects in T ail s ( R ) (see [15], coro llaries 2 and 3, pp.375, and also lemma 4.1 below). In b oth cases cons idered, injectiv e ob jects in Gr ( R ) (or in T ail s ( R ) ) are direct sums of indecomp osable injective ob jects ([28], s ee also prop os ition 5.4 below), which form a se t I nj ( Gr ( R )) (or I nj ( T ails ( R )) ) para metrised (up to iso- morphism a nd shifts) b y ho mo geneous prime ideals ([25],[26],[32]). Any tor- sion theo ry ( T , F ) co nsidered is cog enerated by a subset X o f I nj ( T ail s ( R )) and, thu s, injective env elopes o f ob jects in T are direct sums of o b jects in I nj ( T ai l s ( R )) \ X . In our cases, the adjoint pair ( π , Γ ∗ ) restr icts to an adjoint pair b etw een g r ( R ) a nd tail s ( R ) (see [8]) and, th us, the ob jects of T which are compa ct in D b ( T ai l s ( R )) hav e injective env elop es given by finite direct sums of ob jects in I nj ( T ail s ( R )) \ X , thus forming a set. 16 JORG E VITÓRIA 4. Per verse coherent t-structures through torsio n theories In this section, we will prove o ur main theorem. W e s tart by fixing some nota tion. Let X be a smo oth pr o jective scheme ov er a n algebr aically clo s ed field K such that its homogeneous co ordinate ring R = Γ ∗ ( X ) is a commut ative no etherian p ositively graded K -algebra g enerated in deg ree 1. W e deno te by π the pr o jection functor from Gr ( R ) to its q uotient T ail s ( R ) (and the co rresp onding restr iction to g r ( R ) ). It has a rig ht adjoint given by Γ ∗ ( π M ) = M i ∈ Z Hom T ails ( R ) ( π R, π M ( i )) . F or more details on the formalism of these q uotient categor ie s chec k [8], for instance. Let p : X top − → Z b e a p er versit y as defined in the int ro duction. Suppose that the p e rversit y has n v a lues and that, witho ut loss of g enerality , the maximal v alue of the p erversity is zero . Let I x be the homogeneous ideal of functions v a nishing at an element x of X top and, for i in I m ( p ) , define E i = Y { x ∈ X top : p ( x ) ≤ i } E g ( R/I x ) . Lemma 4.1. L et R b e a ring and A and B gr ad e d R -mo dules. If B is t orsion-fr e e and inje ctive, t hen we have an isomorphism Hom Gr ( R ) ( A, B ) ∼ = Hom T ails ( R ) ( π A, π B ) . Pr o of. This follows fro m [1 5] (lemme 1, prop osition 3, pp. 370– 371). Since Γ ∗ is right a djoint to π , we hav e Hom T ails ( R ) ( π A, π B ) ∼ = Hom Gr ( R ) ( A, Γ ∗ π B ) . The unit of the adjunction φ : B → Γ ∗ π B has a tor sion kernel and a torsion cokernel and, since B is torsion- fr ee, φ must b e and injective map. Since B is an injective ob ject, the shor t exa ct sequence induced by φ splits and, since Γ ∗ π B is torsion- fr e e , we conclude that φ is a n iso morphism.  Recall that a n injectiv e ob ject I of an abelia n category A cogenerates A if, for any X in A , Hom A ( X, I ) = 0 implies X = 0 , i.e., the a sso ciated torsion class, T I , is zero (since T I is the kernel of the functor Hom A ( − , I ) ). R emark 4.2 . Note that given R p ositively g raded no ether ia n connected K -a lgebra, R/P is tors ion-free for a n y homogeneous prime ideal P not equa l to the irr elev a n t ideal. Indeed, for x / ∈ P , if xR ≥ n = 0 , then ( RxR )( R ≥ n ) ⊆ P a nd hence , by 2.1, RxR ⊆ P , which yields a co nt radiction. Note that a ll mo dules E i are torsion- fr e e b y the r emark a bove. Indeed, since the torsion-fre e class of a hereditary torsion theory is clo sed under taking injective env elopes, E g ( R/P ) is torsion-fre e. Also, π E i is injective in T ails ( R ) , for all i , since π is essentially sur jectiv e and Γ ∗ is left exact. Corollary 4.3. The obje ct π E 0 c o gener ates T ail s ( R ) , wher e R = Γ ∗ ( X ) is as ab ove. Pr o of. Supp ose that M is not torsion, i.e., that there is an element m ∈ h ( M ) such that Ann ( m ) 6 = R ≥ n for a ny n > 1 . W e prov e that Ann ( m ) is contained in a homogeneous prime ideal. It is clear that, since m is not torsio n, the radical of Ann ( m ) , which w e denote by p Ann ( m ) , is not the augmen tation ideal R + . Thu s we ca n choose f ∈ R 1 such that f / ∈ p Ann ( m ) . Applying Zorn’s lemma to the set S = n J ⊃ Ann ( m ) ho mogeneous : f / ∈ √ J o (whic h is no nempt y since PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 17 Ann ( m ) ∈ S ) we get a ma ximal element - call it P . W e prov e that P is prime. In fa ct, for a, b ∈ h ( R ) , if ab ∈ P and a / ∈ P , then there is an in teger l such that f l ∈ aR + P (since P is maximal in S ). If there is an in teger s such that f s ∈ bR + P , then f l + s ∈ ( aR + P )( bR + P ) ⊂ P , a contradiction. Hence b ∈ P . This prov es that P is a homogeneous gr-prime ideal. T o complete the pro of we need the lemma b elow. Recall that in n oncommutativ e ring theory , primality of an ideal P is defined in terms o f pro ducts of ideals, i.e., if I J ⊂ P for some ideals I and J , then I ⊂ P o r J ⊂ P . I f this property holds at the level of elements (i.e., if ab ∈ P for some elements a , b of the ring, then a ∈ P or b ∈ P ) then we say P is strong ly prime. There are o bvious gra ded co unterparts of these notions and the following pro p e rty holds. Lemma 4. 4 (Nasta sescu, V an Oystaeyen, [2 7]) . F or a Z -gr ade d ring, a homo ge- ne ous ide al is gr-str ongly prime if and only if it is stro ngly prime. Since R is co mm utative, the no tions of prime and strong ly prime co incide. Hence, P is prime. Note now that there is a gra ded isomorphism from R/ Ann ( m )( deg ( m )) to mR and th us a gr aded injecti on from R/ Ann ( m )( deg ( m )) to M . Since Ann ( m ) is contained in a homo g eneous prime ideal P , R / Ann ( m )( d eg ( m )) maps nontrivially to E g ( R/P )( deg ( m )) and th us s o do es M . Since R sa tisfies the h yp othesis o f lemma 2.13, one has that M maps no ntrivially to E g ( R/P ) and thus, b y the previous lemma, Hom T ails ( R ) ( π M , π E g ( R/P )) 6 = 0 .  Before sta ting the main theorem, we need to prov e the following useful lemma. Lemma 4.5. Supp ose R is a c ommutative lo c al ring with maximal ide al m . Given X • a b ounde d c omple x of fi nitely gener ate d fr e e R-mo dules, define Y • to b e t he c o mplex R /m ⊗ R X • . If, for some fixe d inte ger α , H j 0 ( Y • ) = 0 fo r al l j ≥ α , then H j 0 ( X • ) = 0 for al l j ≥ α . Pr o of. Without loss o f generality , let us assume tha t H j 0 ( Y • ) = 0 for all j ≥ 1 , i.e., α = 1 . Since X • is a b ounded co mplex, let p ∈ Z b e its righ t b ound, i.e., X k = 0 (and hence Y k = 0 ) for all k > p . If p < 1 then the res ult trivially follows. Supp ose that p ≥ 1 . W e will first show that the co homology H p 0 ( X • ) v anishes. Consider the exa ct sequence X p − 1 − → X p − → cok er ( d p − 1 X ) − → 0 and apply to it the functor R/m ⊗ R − , thus getting ano ther exact seq uence Y p − 1 − → Y p − → R /m ⊗ R cok er ( d p − 1 X ) − → 0 , since R/m ⊗ R − is right exact. By definition of Y • , the first map of the sequence is the differential d p − 1 Y . Since 1 ≤ p , H p 0 ( Y • ) = 0 , thus proving that d p − 1 Y is surjective ( Y p +1 = 0 by definition). Therefor e R/ m ⊗ R cok er ( d p − 1 X ) = 0 which, by Nak ayama’s lemma (since R is lo ca l and cok er ( d p − 1 X ) is a finitely generated R -mo dule), implies that cok er ( d p − 1 X ) = 0 . Hence d p − 1 X is sur jectiv e, thus proving that H p 0 ( X • ) = 0 . W e now prov e our result by induction on p ≥ 1 . If p = 1 , the pre v ious para graph shows that H 1 0 ( X • ) = 0 and the result follows. Supp ose now that the result is v alid for all complexes of free R -mo dules X • with right b ound p ≥ 1 and let X • be a 18 JORG E VITÓRIA complex of free R -mo dules with r ight b ound p + 1 . The pr evious parag raph shows that H p +1 0 ( X • ) = 0 . Since X p +2 = 0 , there is a s hort exact sequence 0 − → K er ( d p X ) − → X p − → X p +1 − → 0 which splits since X p +1 is free. Thus, K er ( d p − 1 X ) is a summand of the free mo d- ule X p − 1 , i.e., it is a pro jectiv e mo dule. Howev er, it is well-kno wn (Ka plansky’s theorem) that pro jective mo dules ov er lo ca l r ings a re free and, hence, the co mplex ˜ X • := ... / / X p − 2 d p − 2 X / / X p − 1 d p − 1 X / / K er ( d p X ) / / 0 / / ... . is a complex of free R -mo dules whic h is quasi-isomor phic to X • . Since X • is a complex of fr e e R -mo dules, i ts tensor pro duct with R /m can be reg arded in D ( M od ( R )) as the derived tensor pro duct Y • ∼ = R/m ⊗ L R X • ∼ = R/m ⊗ L R ˜ X • in D ( M od ( R )) , where the second isomo rphism holds s ince X • and ˜ X • are isomor - phic in D ( M od ( R )) . Since ˜ X • is also a complex of free R -mo dules , we hav e that Y • is isomor phic to R /m ⊗ R ˜ X • in D ( M od ( R )) , i.e., they are ar e quasi- isomorphic complexes and, thus H i 0 ( R/m ⊗ R ˜ X • ) = 0 for all i ≥ 1 . Now, the induction hypoth- esis holds for ˜ X • and, thus, H i 0 ( ˜ X • ) = 0 for all i ≥ 1 . Since ˜ X • is quasi- is omorphic to X • , we als o hav e tha t H i 0 ( X • ) = 0 for all i ≥ 1 , thus finishing the pro of.  Finally we prov e our main theorem. It gives us a description of the aisle of a per verse coher ent t-structure D p, ≤ 0 is ter ms of torsio n classes. Theorem 4.6. L et X b e a sm o oth pr oje ctive scheme over K , R = Γ ∗ ( X ) its homo- gene o us c o or dinate ring and p a p erversity on X . Supp ose that R is a c ommutative c o nne cte d, no etherian, p osi tively gr ade d K - algebr a gener ate d in de gr e e 1. L et T i denote the t orsion cla ss c o gener ate d in T ail s ( R ) by πE i , wher e E i = Y { x ∈ X top : p ( x ) ≤ i } E g ( R/I x ) , with I x standing for the de fining ide al of x ∈ X top in R . The n we have: D p, ≤ 0 =  F • ∈ D b ( T ai l s ( R )) : H i 0 ( F • ) ∈ T j , ∀ i > j  ∩ D b ( tail s ( R )) . Pr o of. Let S b e the set of tor sion clas s es T i . By remar k 3.15 and theorem 3 .13, D S, ≤ 0 is a n a isle in D b ( T ai l s ( R )) . W e will prove that the sub catego ries D p, ≤ 0 and D S, ≤ 0 ∩ D b ( tail s ( R )) coincide. W e denote by b T i the torsion theory co generated by E i in Gr ( R ) . W e start by rewriting the co nditions defining the aisle D S, ≤ 0 . By definition, we hav e D S, ≤ 0 = n F • ∈ D b ( T ai l s ( R )) : H j 0 ( F • ) ∈ T k , ∀ j > k o and, given that E k , for all k , is tor sion-free injectiv e, by lemma 4 .1 we hav e D S, ≤ 0 = n F • ∈ D b ( T ai l s ( R )) : Γ ∗ ( H j 0 ( F • )) ∈ c T k , ∀ j > k o = n F • ∈ D b ( Qcoh ( X )) : ∀ x ∈ X top , Hom Gr ( R ) (Γ ∗ ( H j 0 ( F • )) , E g ( R/I x )) = 0 , ∀ j > p ( x ) o . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 19 W e intersect with D b ( coh ( X )) to pa ss from a quasi-coherent to a coherent setting. F or simplicity , define D s, ≤ 0 := D S, ≤ 0 ∩ D b ( coh ( X )) . By cor ollary 2.5, we g et D s, ≤ 0 = n F • ∈ D b ( coh ( X )) : ∀ x ∈ X top , Γ ∗ ( H j 0 ( F • )) ( x ) = 0 , ∀ j > p ( x ) o where Γ ∗ ( H j 0 ( F • )) ( x ) is the degree zer o part of the loca lisation of Γ ∗ ( H j 0 ( F • )) at the pr ime ideal I x , which is the sa me as sta lk a t x o f the sheaf H j 0 ( F • ) . Since taking stalks is an exact functor (th us t-exact for the sta ndard t-structure and there fore commut ing with cohomolog y functors) we ge t D s, ≤ 0 = n F • ∈ D b ( coh ( X )) : ∀ x ∈ X top , H j 0 ( F • x ) = 0 , ∀ j > p ( x ) o . On the other hand, recall that D p, ≤ 0 = n F • ∈ D b ( coh ( X )) : ∀ x ∈ X top , Li ∗ x ( F • ) ∈ D ≤ p ( x ) 0 ( O { x } - mod ) o , which is clearly the same as n F • ∈ D b ( coh ( X )) : ∀ x ∈ X top , H j 0 ( Li ∗ x ( F • )) = 0 , ∀ j > p ( x ) o . Therefore, it s uffices to pr ov e that H j 0 ( F • x ) = 0 for all j > p ( x ) if and o nly if L j i ∗ x ( F • ) = H j 0 ( Li ∗ x ( F • )) = 0 for all j > p ( x ) . Let F • ∈ D b ( coh ( X )) such that H j 0 ( F • x ) = 0 for all j > p ( x ) . B y definition of the pullback functor ( i ∗ x ( V ) = V x ⊗ O X,x k ( x ) for any coherent sheaf V , where k ( x ) = O X,x /m X,x , with m X,x being the maximal ideal o f the lo cal ring O X,x , is the res idue field at the p oint x ), ther e is a s pec tr al sequence of Gro thendiec k type of the fo llowing form: E ab 2 = T or O X,x a ( k ( x ) , H b 0 ( F • x )) = ⇒ L a + b i ∗ x ( F • ) . Our hypothesis shows that E ab 2 = 0 for all b > p ( x ) (and, of cours e , by definition of T or , also for all a < 0 ). Thus E ab ∞ = 0 for all a < 0 or b > p ( x ) . Let F i denote the i -th par t of the decreasing filtra tio n assumed to exist (by definition of conv ergent sp e ctral sequence) on the limit ob ject Ω a + b := L a + b i ∗ x ( F • ) . Then, for q > p ( x ) , ... = F − 2 Ω − 2+( q +2) = F − 1 Ω − 1+( q +1) = F 0 Ω q = F 1 Ω q and th us they ar e a ll equa l to zero, pr oving that Ω q = L q i ∗ x ( F • ) = 0 for all q > p ( x ) . Conv ersely , supp o se we hav e F • such tha t L j i ∗ x ( F • ) = 0 for all j > p ( x ) . Since X is smo oth, let G • be a b ounded c o mplex o f lo cally free sheav es such that G • is qua s i-isomorphic to F • (th us isomorphic in the derived categ o ry) - check, for example, [20], Pr o p o sition 3 .26. Then L j i ∗ x ( F • ) = 0 means that H j 0 (( i ∗ x G ) • ) = 0 , where ( i ∗ x G ) • denotes the complex resulting from applying i ∗ x comp onent wise to G • . T ak e now X • = G • x and Y • = ( i ∗ x G ) • and recall that G • x is a complex of free mo dules over the lo cal ring O X,x . This leav es us in the context of 4.5, thus proving that H j 0 ( G • x ) = H j 0 ( G • ) x = 0 for all j > p ( x ) . Finally we have H j 0 ( F • x ) = H j 0 ( F • ) x = H j 0 ( G • ) x = 0 for a ll j > p ( x ) , hence finishing the pro of.  R emark 4 .7 . This theorem co mpares tw o sub catego ries of D b ( coh ( X )) , showing that they coincide. It contains no pro of that either of the sub categ o ries inv olv ed are aisles of t-s tructures. Howev er, b y pr oving that the intersection D S, ≤ 0 ∩ D b ( coh ( X )) , for S defined as ab ov e, coincides with the aisle D p, ≤ 0 constructed in [10], w e do show that, under the as sumptions o f the theorem, D S, ≤ 0 (whic h is an aisle by section 3) in D b ( Qcoh ( X )) res tricts w ell to an aisle in D b ( coh ( X )) . 20 JORG E VITÓRIA 5. Per verse quasi-coherent t-structures f or noncommut a tive pr ojective planes The aim o f this section is to us e the constr uction of section 3 to crea te an analo gue of p erverse coherent t-structures in the derived ca tegories of certain noncommuta- tiv e pro jectiv e planes, motiv ated by the co mpa rison established on section 4 in the commut ative case. This entails finding an injectiv e cog e nerator in T ails ( R ) for a suitable class of K - a lgebras R and set up a definition of p erversit y that generalises the commutativ e o ne. R emark 5.1 . tails ( R ) is not co co mplete and therefore, in this section, we can o nly do the co ns tr uction of section 3 in T ail s ( R ) (hence the word quasi-c oher ent rather than c oher ent in the title). How ev er, tak ing in to a c c o unt theorem 4.6 , we conjecture that indeed the constructions in this s e c tion restrict well to D b ( tail s ( R )) . W e shall fo cus on the case wher e R is a graded elliptic 3-dimensiona l Artin- Sch elter regula r algebra which is finite ov er its centre. These alg e br as are interesting for our purp oses s ince they a r e fully b ounded no etheria n (mo re than that, they ar e PI, as pro ved in [7 ]). Also, a graded no etherian algebra which is fully b ounded is graded fully bounded ([32]). This is imp ortant for the following result that allows us to parametrise a useful collection of injectiv e ob jects via prime ideals. In this sense, although these examples a re noncommutativ e, we are s till very clo s e to the commut ative setting ([25]). Recall that there is a map from the set o f indecompo sable injective gr aded mo d- ules to the s e t of ho mogeneous prime ideals given by assigning to an injective E its ho mo geneous assassina tor ideal, Ass ( E ) . The assa ssinator ideal of an indecom- po sable ob ject is the only prime ideal asso ciated to E , i.e., the only prime ideal which is maximal among the annihilators o f no nzero submodules of E (a nd ther e is a na tural gra ded version of this co ncept - see [26] and [32]). Prop ositio n 5. 2 (Natasescu, V an Oystaeyen, [26], Theorem C.I.3 .2) . L et R b e a p ositively gr ade d no etherian ring. Then R is gr ade d ful ly b ounde d if and only if t he map that assigns the c orr esp onding assassinator ide al to an inde c omp osab le inje c- tive gr ade d mo dule induc es a bije ctio n b etwe en inde c omp osable inje ctive mo dules in Gr ( R ) (up to isomorphism and gr a de d shift) and ho mo gene ous prime ide als of R . R emark 5.3 . In the context o f this prop osition, the indecomp osable injective a sso- ciated with a homo geneous prime P is the unique (up to isomo rphism a nd shifts) indecomp o sable direct summand of E g ( R/P ) ([26], Theorem C.I.3.2), thu s es ta b- lishing an inv erse map. This result bring s us closer to the desired cogenerating s et. Its significance in o ur context comes fro m the work of Matlis o n the decomposition of injectiv e mo dules ov er no etheria n rings . Ma tlis proved that R is (right) no etherian if and only if every injective (r ight) module is the direct sum of indecomp osable injective (right) mo dules ([25]). This shows in particula r that the set of indecompos able injective ob jects cogenerates the categor y of mo dules ov er a no etheria n ring. There is a graded analo gue of this result, as follows. Prop ositio n 5.4 (Samir Mahmoud, [28]) . L et M b e a fi nitely gener ate d gr ad e d mo d ule over a gr ade d no etherian ring R . Then E g ( M ) is a fin ite dir e ct sum of inde c omp osable inje ctive obj e cts in Gr ( R ) . PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 21 These tw o results yield a useful set of torsion class e s parametris e d by the homo- geneous prime idea ls of R . Indeed, for an indecompos able injective E , denote the corresp onding torsio n clas s coge ner ated b y E in Gr ( R ) by T E , i.e., T E :=  M ∈ Gr ( R ) : H om Gr ( R ) ( M , E ) = 0  . Let b Y denote the set of all such classes where E runs ov er indecomp osable injectiv e ob jects, up to isomorphism and gr a ded s hift, s uch that its assa ssinator ideal is not the irrelev ant ideal, i.e., Ass ( E ) 6 = R + . A nalogous ly define Y to b e the set of the torsion classes in T ail s ( R ) of the form T π E :=  K ∈ T ail s ( R ) : H om T ails ( R ) ( K, π E ) = 0  for all E indecompo sable injective gra ded mo dule with non-irr e le v ant assa s sinator. Corollary 5. 5 . L et R b e a p ositivel y gr ade d ful ly b ounde d c onne cte d no etheria n K -algebr a gener ate d in de gr e e 1. Then the interse ction (in T ail s ( R ) ) of t he torsion classes in Y is zer o. Pr o of. Supp ose π M lies in the intersection o f the tor sion class es of Y . Then by lemma 4 .1, M lies in the intersection of the torsion clas ses in ˆ Y . By prop ostion 5.4, the indecompos able injective ob jects cogenera te Gr ( R ) a nd th us E g ( M ) must be a finite direct sum of the indeco mp os able injective asso ciat ed with R + . This inde- comp osable is a dir e ct summand of E g ( R/R + ) (see remark 5.3), whose pro jection in T ails ( R ) is therefor e z e ro. Th us π E g ( M ) = 0 and so π M = 0 .  W e pro ceed now to the desired construction. Let R be a p ositively graded Artin-Schelt er regular algebra of dimension 3 gener ated in degree o ne whic h is finitely generated o ver its cen tre. As discussed b efore, it is gra ded fully b ounded no etherian. W e need to define a p erversit y in Y , wher e Y is as befor e. Definition 5.6. A p erversity is a map p : Y − → Z such that, given T π E 1 , T π E 2 in Y , if there is a nonzero homo morphism from π E 2 to π E 1 then p ( T π E 1 ) − ( GK dim ( R / Ass ( E 2 )) − GK dim ( R / Ass ( E 1 ))) ≤ p ( T π E 2 ) ≤ p ( T π E 1 ) . W e now prov e that this definition of p e rversit y coincides, when the alg ebra is commut ative, with the definition of p erversit y of the introduction. W e s tart by a suppo rting lemma. Lemma 5 .7. L et R b e a p ositively gr ade d c ommut ative no etherian K -algebr a and X = P r oj ( R ) . Th e fo l lowing ar e e quivalent. (1) F or x 1 , x 2 ∈ X top , x 1 ∈ ¯ x 2 ; (2) P 2 := Ann ( x 2 ) ⊂ Ann ( x 1 ) =: P 1 , wher e Ann ( x i ) denotes the homo gene ous ide al of functions vanishing in x i ; (3) Ther e is a nonzer o homomorphism fr om R/P 2 to R/P 1 ; (4) Ther e is a nonzer o homomorphism fr om E g ( R/P 2 ) to E g ( R/P 1 ) . Pr o of. It is clear that (1) ⇔ (2) ⇒ (3) ⇒ (4) . W e only need to pr ove (4) ⇒ (2) . Let f be a homomo rphism from E g ( R/P 2 ) to E g ( R/P 1 ) and a ∈ h ( P 2 ) \ h ( P 1 ) . Clearly N := R / P 1 ∩ im ( f ) 6 = 0 s ince R/P 1 is a graded ess ent ial submo dule of E g ( R/P 1 ) . Now, N ∩ ( a + P 1 ) R 6 = 0 since any homog eneous ideal of a commutativ e graded domain is gr aded essential (the pr o duct of tw o nonzero ideals is nonzero a nd it is contained in the intersection). Hence, 0 6 = ( a + P 1 ) R ∩ N ⊂ ( a + P 1 ) R ∩ im ( f ) . 22 JORG E VITÓRIA Let then b b e a nonzero elemen t in ( a + P 1 ) R ∩ im ( f ) and y ∈ E g ( R/P 2 ) such that b = ar + P 1 = f ( y ) . Note that y a ∈ P 2 E g ( R/P 2 ) and P 2 annihilates E g ( R/P 2 ) , th us 0 = f ( y a ) = a 2 r + P 1 and r ∈ P 1 , since P 1 is prime. Hence b = 0 in R/P 1 , reaching a contradiction and pr oving the r esult.  Prop ositio n 5.8 . If R is a p ositively gr ade d no etherian c onne cte d c ommutative K -algebr a gener a te d in de gr e e 1, the definition of p erversity ab ove is e quivalent to the c o mmutative definition of p erv ersity in e quation (1.1). Pr o of. Let X b e the pro jective sc heme asso ciated with R . Note that p oints x ∈ X top are in bijection with homogeneous prime ideals not equa l to the irr e le v ant ideal of R and these are in bijection with graded tors ion-free indecomp osable injectives in Gr ( R ) ( up to isomorphisms a nd shifts). Suppose x 1 , x 2 ∈ X top , P 1 , P 2 the asso ciated homogeneous prime idea ls and E 1 , E 2 the corres p o nding indecomp osable injectiv es. The condition x 1 ∈ ¯ x 2 translates in to the existence of a nonzero ma p from E 2 to E 1 b y lemma 5.7 (no te that, in this cas e, E i = E g ( R/P i ) since R/ P is indecomp o sable in Gr ( R ) and hence so is its injective env elop e) and b y lemma 4.1 this is equiv alent to the existence o f a map from π E 2 to π E 1 . Since R is finitely generated o ver K (as it is noetheria n), and hence are all its quotients, it is known the Krull dimension of R/ P i (whic h is the sa me as dim ( x i ) in the g eometric definition of p erversity - see in tro duction) c oincides with the Gelfand- Kirillov dimensio n o f R /P i ([23], Theorem 4 .5). The res ult then follows b y making the adequa te substitutions in equation (1.1).  Recall that 3 -dimensional Artin-Schelter reg ular algebras ar e no ether ia n domains - in par ticular, they a re prime ring s ([6], [7]). This allows us to pr ov e the following useful lemma. Lemma 5.9. L et R b e a p ositively gr ade d c onne cte d 3-dimensional Artin-Schelter r e gular algebr a gener ate d in de gr e e 1 which is finitely gener ate d over its c ent r e. Then the image of a p erversity p as define d ab ove is fin ite. Pr o of. Since R is prime, (0) is a prime ideal not equal to the irr elev an t ideal. Thu s it co rresp onds to an indecomp osable injective ob ject which we denote by E 0 . F urthermore, as a co ns e quence of remark 5.3, E g ( R ) is a finite direct s um of copies of E 0 . Similarly , E g ( R/P ) is a finite direct s um of copies o f E P , the indecompo sable injectiv e o b ject asso ciated to the homog eneous prime ideal P . W e observe that for any such P , there is a map from E g ( R ) to E g ( R/P ) induced by the canonical pro jection from R to R/P . Ther efore, there is a nontrivial map fro m E 0 to E P . The p erversit y c ondition then assures that: p ( T π E P ) − ( GK dim ( R ) − GK dim ( R/P )) ≤ p ( T π E 0 ) ≤ p ( T π E P ) . Since, by definition, the Gelfand-Kir illov dimension of R is finite (and so is the dimension of a n y of its quotients - [23], Lemma 3.1) we hav e that, for a fix e d v alue of p ( T π E 0 ) , p ( T π E P ) is an in teger that differs a t most GK dim ( R ) from it. Hence the image o f p is finite.  Thu s, for R Artin-Schelter re g ular algebra of dimension 3 a nd finite over its cen- tre, w e can form a finite chain of hereditary to r sion cla sses of T ails ( R ) , compactly PER VERSE COHERENT T-STR UCTURES THROUG H TORSION THEORIES 23 generated in D b ( T ai l s ( R )) : S :=    T i := \ T : p ( T ) ≤ i T , min ( p ) ≤ i ≤ max ( p )    . By coro lla ry 5.5, the last element of the chain, T max ( p ) , is zero. Finally , section 3 provides a w ay o f building a pe r verse quasi-c oherent t-structure with r esp ect to p b y defining its aisle to b e D S, ≤ 0 in D b ( T ai l s ( R )) . As mentioned in re ma rk 5.1, in light of sectio n 4, w e conjecture tha t these t-structures res trict to D b ( tail s ( R )) . References [1] Ajitabh, K., Mo dules over el liptic algebr as and quantum planes , Pro c. Lond. Math. Soc. 72, 567–587, 1996. 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