Classifying Compactly generated t-structures on the derived category of a Noetherian ring

COMP A CTL Y GENERA TED t -STR UCTURES ON THE DERIVED CA TEGOR Y OF A NOETHERIAN RING LEOVIGILDO ALONSO T ARR ´ IO, ANA JEREM ´ IAS L ´ OPEZ, AND MAN UEL SAOR ´ IN Abstra ct. W e study t -structures on D ( R ) the derived category of mod u les ov er a commutativ e No etherian ring R generated by com- plexes in D − fg ( R ). W e pro ve that they are ex actly the compactly gen- erated t -structures on D ( R ) and describe them in terms of decreasing ﬁltrations by supp orts of Sp ec( R ). A decreasing ﬁltration by sup p orts φ : Z → S p ec( R ) satisﬁes the w eak Cousin condition if for any integer i , th e set φ ( i ) con tains all the immediate generalizations of eac h p oint in φ ( i + 1). If a compactly generated t - structure on D ( R ) restricts to a t -structure on D fg ( R ) then the corresponding ﬁ ltration satisﬁes the weak Cousin condition. If R has a p oint wise du alizing complex the convers e is tru e. If the ring R has dualizing complex then these are exactly all the t -structures on D b fg ( R ). Contents In tro du ction 2 1. Notatio n and preliminaries on t -structures 5 2. Aisles d etermined by ﬁltrations of su pp orts 11 3. Compactly generated aisles 14 4. The weak Cousin condition 20 5. Aisles d etermined by ﬁnite ﬁltrations by sup p orts 26 6. The classiﬁcation o v er rings w ith dualizing complex 31 References 40 Date : Nov em b er 2, 2021. 2000 Mathematics Subje ct Classiﬁc ation. 14B1 5 (primary); 18E30, 16D90 (secondary). L.A.T. and A .J.L. hav e b een partially supp orted by Spain’s MEC and E.U.’s FEDER researc h p ro jects MTM2 005-05754 and MTM2008-03465 together with Xunta de Galicia ’s gran t PGIDIT06PXIC20 7056PN and ac knowledge hospitality and supp ort from Purdu e Universit y and Universidad de Murcia. M.S. has b een su pp orted by the D.G.I. of the Spanish Ministry of Education and the F undaci´ on “S´ eneca” of Murcia, with a part of FEDER funds from the Europ ean Union. Manuel Saor ´ ın de di c a este art ´ ıculo a sus p adr es en el 50 aniversario de su b o da . 1 2 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN Introduction The concept of t -structur e on a triangulated categ ory arises as a categ or- ical fr amew ork for Goresky-MacPherson’s intersectio n homology . Through this construction Be ˘ ılinson, Bernstein, Deligne and Gabb er extended in ter- section cohomology to the ´ etale con text. A t -stru cture p ro vides a homo- logica l fun ctor with v alues in a certain ab elian category con tained in th e original triangulated categ ory d en ominated the h eart of the t -stru cture. In Grothendiec ks’s terms, this study accoun ts for the study of extraordinary cohomology theories for discr ete co eﬃcients. In tersection cohomology and its v ariants ha v e b een studied successfully w ith these metho d s o v er the last t w en t y y ears. Let us p oin t out [BBD] and [GN] and references therein. On the side of c ontinuous co eﬃcien ts the dev elopmen t has p ro ceeded at a slo w er pace. Deligne w as ﬁrs t to mak e con tributions to the problem of con- structing t -structures on the b ounded derive d catego ry of coheren t sheav es on a No etherian scheme u nder certain h yp othesis, n amely the existence of a d u alizing complex and of global lo cal ly f ree r esolutions. His work wa s not a v ailable un til the exp osito ry e-print by Bezruk avnik o v [Be]. Later Kashi- w ara [K a] constructs from a decreasing family of sup p orts satisfying certain condition a t -stru ctur e on the b ounded derive d category of coherent shea v es on a complex manifold, w hic h corresp onds in the algebraic case to a smo oth separated sc heme of ﬁnite t yp e o v er C . Also, Y ekutieli and Z hang [YZ] con- sidered the Grothendiec k dual t -str u cture of the canonical one on the deriv ed catego ry of ﬁnitely generated mo du les o v er a No etherian ring w ith dualiz- ing complex. This t -stru cture is calle d the Cohen-Mac aulay t -structur e in the presen t pap er and it is shown to exist in the whole un b ounded derive d catego ry . Deligne, Bezruk a vnik o v an d Kashiwa ra bu ilt, on the deriv ed category of b ound ed complexes with ﬁnitely generated homologies, a t -structure start- ing with a ﬁnite ﬁltration b y supp orts X = Z s ) Z s +1 ) · · · ) Z i ) · · · ) Z n − 1 ) Z n = ∅ in the corresp on d ing top olog ical space X . In Bezruk avnik ov’s pap er No etherian ind uction is used. Kashiwa ra constructs the triangle of the corresp onding t -structure usin g in an essent ial wa y that the complexes are b ounded. The ﬁ ltrations by supp orts used b y b oth au- thors are ﬁn ite and satisfy a condition that we call in this pap er the we ak Cousin c ondition (for any integ er i , the set Z i con tains all the immedi- ate generalizations of eac h p oin t in Z i +1 ). This name refers to a wea k en- ing and reformulat ion of the n otion of co dimension f unction in tro du ced b y Grothendiec k, see [H, V.7] an d § 4 b elo w. An equiv alen t n otion was used in [Be] un der the name c omono tone p erversit y . In the aforementio ned pap ers, the authors rely on ﬁnite step-by-ste p con- structions and do n ot take into account the p ossibilit y allo w ed by inﬁn ite constructions if one considers the unbou n ded derive d category . This ap- proac h mak es sense after [AJS2] where it is p ro v ed that for a collectio n of COMP ACTL Y GENERA TED t -STRUCTURES 3 ob jects in the unb ounded deriv ed category of a Grothendieck category there is a t -structure w hose aisle is generated b y this set of ob jects. A logical next step is to try to classify t -structures on D ( R ) by ﬁltrations of su bsets of Sp ec( R ) extending the fact, pro v ed in the key pap er [N1], that Bousﬁeld lo calizations (the class of triangulated t -structures) are classiﬁed b y s u bsets of Sp ec( R ). Ho we v er, a coun terexample b y Neeman and further dev elopmen ts by th e third author made clear that residu e ﬁelds of pr ime ideals w ere not the right ob jects to use in ord er to ac hiev e the classiﬁcati on of t -structur es (see remark on page 19). Stanley in his preprint [S ta] treated the prob lem of studying t -structur es on D b fg ( R ), the su b category of D ( R ) of b oun ded complexes with ﬁn itely gen- erated homologies, where R is a comm utativ e No etherian r ing. He sho we d that it is n ot p ossible to classify all t -stru ctures on D ( R ) b ecause the class of t -structures on D ( Z ) is not a set [Sta, Corollary 8.4]. Th en it is not p ossible to pu t t -str u ctures on D ( R ) in corresp ond en ce with collect ions of su bsets of Sp ec( R ). On the p ositiv e side, Stanley show ed that th er e is an order pr eserving bijection b et w een ﬁltrations of Sp ec( R ) by stable un der sp ecialization sub- sets an d nullit y classes in D b fg ( R ). Theorem B in [Sta] states th at the w eak Cousin condition is a necessary co ndition o v er a ﬁltration for the corre- sp ond ing nullit y class in D b fg ( R ) to b e an aisle, and he conjectured th at the con v erse is true. How ev er, the pro of of Theorem B in [Sta] do es not seem to b e complete. W e giv e here an alternate approac h to Stanley’s result en- compassing the unb ounded category and, in addition, w e answ er Stanley’s conjecture in the aﬃrmative . Sp eciﬁcally , th e cate gory D fg ( R ) is sk eletal ly small so an y t -structure on D fg ( R ) is the restriction of a t -str u cture on D ( R ) (see Lemma 1.3 and Prop osition 1.4). W e lo ok at t -structures on D ( R ) generated by complexes with ﬁnitely generate d homologies and charact erize those that restrict to t - structures on D fg ( R ) —and in general to an y of the sub cate gories D ♯ fg ( R ) for an y b oun dedness condition ♯ ∈ { + , − , b , “blank” } . W e obtain the classiﬁca- tion of all t -structures on D ( R ) generated b y complexes in D − fg ( R ) in terms of ﬁ ltrations by sup p orts of S p ec( R ). They are exactly all compactly gen- erated t -str u ctures (Th eorem 3.1 1). W e prov e that the ﬁltration asso ciated to a compactly generated t -stru cture on D ( R ) that restricts to a t -structur e on D fg ( R ) (or in an y of the ab ov e sub categ ories D ♯ fg ( R )) n ecessarily satis- ﬁes the w eak Cousin condition (Corollary 4.5). So we giv e a pro of of [Sta, Theorem B] b y d iﬀeren t means. The compact ob jects in D ( R ) are the p erfect complexes. T ranslated to our con text, Stanley’s qu estion asks whether t -structur es generat ed b y p er- fect complexes on D fg ( R ) (or, in general, in D ♯ fg ( R )) are th ose determined b y ﬁ ltrations satisfying the weak Cousin cond ition. I n the last section of our pap er we answer this question in the aﬃrmativ e for D fg ( R ) wh en the ring R p ossesses a p oin t wise dualizing complex. F or D b fg ( R ) w e get an aﬃrmativ e 4 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN answ er when the ring R has a du alizing complex, a mild hyp othesis already present —as w e h a v e recalled— in previous works. Let us describ e the con ten ts of this pap er. In the ﬁrst sectio n we start recalling the notations and deﬁnitions used in this pap er. W e introd uce the notion of total pr e - aisle and show that an y total pr e-aisle of D ♯ fg ( R ) is the restriction of an aisle of D ( R ), with ♯ as b efore. W e also study ho w total pre-aisles b eha v e un der change of rings. A ﬁltration by sup p orts of Sp ec ( R ) is a decreasing family · · · ⊃ φ ( i ) ⊃ φ ( i + 1) ⊃ . . . of s table un der sp ecialization subsets of S p ec( R ). Eac h ﬁltra- tion by supp orts φ : Z → P (Sp ec( R )) has an asso cia ted aisle U φ , generated b y { R/ p [ − i ] ; i ∈ Z and p ∈ φ ( i ) } . In § 2 we consider aisles generated by sus- p ensions of cyclic mo d ules. They are pr ecisely the aisles asso ciated to ﬁ ltra- tions by su pp orts. W e c haracterize in section 3 the aisles of D ( R ) generated b y complexes in D − fg ( R ) in terms of homological sup p orts (Prop ositio n 3.7). They corresp ond to compact ly generated aisle s of D ( R ) (Theorem 3.10). W e study in § 4 th e ﬁltrations b y supp orts of Sp ec( R ) that p ro vide aisles in D ♯ fg ( R ). W e note that all t -structur es on D − fg ( R ) and D b fg ( R ) are generated b y p er f ect complexes. Theorem 4.4 is the ﬁrst main result in this section. It sa ys th at the we ak Cousin c ondition is necessary on a ﬁltration by su pp orts on Sp ec( R ) in ord er to r estrict the corresp onding compactly generated t - structure on D ( R ) to a t -structure on D fg ( R ). This theorem corresp onds to Stanley’s Theorem 7.7 1 . W e deal with the unboun ded category D fg ( R ) and obtain f rom this the r esu lt 2 for D b fg ( R ). Next we d escrib e the ﬁltrations b y supp orts of Sp ec( R ) satisfying the w eak Cousin condition (Prop osition 4.7 and Corollary 4.8). As a consequ en ce, if Sp ec( R ) is connected an d R has ﬁnite Krull dimension then ﬁltrations that satisfy the w eak Cousin condition are ﬁn ite an d exhaustive . Therefore, there is a bijection b et w een t -structures on D − fg ( R ) and D b fg ( R ) (Corollary 4.11). In § 5 w e giv e a description of the tru ncation fu nctors asso ciated to a ﬁ- nite ﬁltration. Pr op osition 5.8 sh o ws that the aisle asso ciated to a t wo -step ﬁltration b y sup p orts that satisﬁes the weak Cousin condition restricts to a t -structure on D fg ( R ), f or an y No etherian ring R . With all these to ols at hand, in the last s ection we pr o v e the remaining m ain result (Theorem 6.9). Our strategy of pr o of is related to the one used in [Ka]. Na mely , th is The- orem asserts that if R p ossesses a dualizing complex, then th e aisles of D b fg ( R ) are exactly those in duced b y ﬁltrations by su pp orts satisfying the we ak Cousin c onditio n . As a consequence we obtain a bijection b etw een aisles of D fg ( R ) and ﬁ ltrations satisfying the w eak Cousin condition under the weak er hyp othesis that R p ossesses p ointwise du alizing complex. The existence of a dualizing complex on R is a ve ry mild condition. I t is satisﬁed 1 The weak Cousin condition here corresp onds to b eing comonotone in [Sta]. 2 W e should remark th at the statement of Theorem 4.4 is a va riation of [St a, Proposi- tion 7.4], th e key ingredient in [Sta, Theorem B]. COMP ACTL Y GENERA TED t -STRUCTURES 5 b y all rings of ﬁn ite Kr ull d imension that are quotien ts of a Gorenstein rin g. This is the case for all ﬁnitely generated algebras o v er a regular ring ( e.g. o v er a ﬁeld or o v er Z ). 1. Not a tion and prel iminaries on t -st ructures Notation and Conv entions. All rings in this pap er w ill b e comm utativ e and No etherian. Giv en a pr ime ideal p ∈ Sp ec( R ), k ( p ) stands for the residue ﬁeld of p and R p for the lo calization of R with resp ect to p . The sup p ort of an R -mo du le N is the set of prime ideals S upp( N ) = { p ∈ Sp ec( R ) / N p = N ⊗ R p 6 = 0 } . F or an ideal a ⊂ R w e denote by V( a ) := { p ∈ Sp ec( R ) ; a ⊂ p } . As usual Mo d ( R ) d enotes the catego ry of mo d ules o v er a ring R , C ( R ) th e catego ry of complexes of R -mo dules, K ( R ) its homotopy category and D ( R ) its deriv ed category . F or complexes we use the upw ard gradings. Let n and m b e inte gers, as usual D ♯ ( R ) ⊂ D ( R ) denotes the fu ll sub categ ory of those complexes whose homologies satisfy one of the standard b oundedness con- ditions ♯ ∈ { ≤ n, < n, ≥ n, > n, + , − } , D [ n,m ] ( R ) := D ≥ n ( R ) ∩ D ≤ m ( R ) and D b ( R ) = D − ( R ) ∩ D + ( R ). Let D fg ( R ) ⊂ D ( R ) b e the full sub cate gory of complexes with ﬁ nitely generated h omologies. T he sym b ol D ♯ fg ( R ) stands for D fg ( R ) ∩ D ♯ ( R ) for any sup erscript ♯ . Basics on t -structures. Let T b e any triangulated c ategory . W e will denote b y ( − )[1] the translation auto-equiv alence of T and its iterates by ( − )[ n ], w ith n ∈ Z . A t -stru ctur e on T in the sense of Be ˘ ılins on , Bern s tein, Deligne and Gabb er ([BBD, D ´ eﬁnition 1.3.1] ) is a couple of full sub categories ( U , F [1]) suc h that U [1] ⊂ U , F [1] ⊃ F , Hom T ( Z, Y ) = 0 for Z ∈ U and Y ∈ F ( i.e. U ⊂ ⊥ F , equiv alentl y F ⊂ U ⊥ ), and f or eac h X ∈ T there is a distin- guished triangle τ ≤ U X − → X − → τ > U X + − → (1.0.1) with τ ≤ U X ∈ U and τ > U X ∈ F . The sub categ ory U is called the aisle of the t -structure, and F is called the co-aisle . It follo ws fr om the d eﬁnition that U = ⊥ F and F = U ⊥ . It also follo ws that the inclusion U ֒ → T h as a righ t adjoin t τ ≤ U called the left truncation f unctor and, du ally , the inclusion F ֒ → T has a left adjoin t functor τ > U , the righ t truncation functor. In fact, X ∈ U if and only if τ > U X = 0, similarly X ∈ F if and only if τ ≤ U X = 0. T he t -structure can b e describ ed just in terms of its aisle U . This fact ju stiﬁes the notation for the truncation f u nctors. W e call ( D ≤ 0 ( R ) , D > 0 ( R )[1]) the c anonic al t -structure on D ( R ). With n ∈ Z , the t -structures ( D ≤ n ( R ) , D >n ( R )[1]) obtained b y translations of the canonical one are called standar d t -structures on D ( R ). As usual τ ≤ n = τ n denote th e left and right truncation functors asso- ciated to the n -th standard t -structure. F or eac h X ∈ D ( R ), τ ≤ 0 X − → X − → τ > 0 X + − → 6 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN denotes the distinguished triangle determined b y the canonical t -structure. 1.1. A cla ss U ⊂ T is a pr e-aisle of T if U endow ed with the class of distinguished triangles in T with vertices in U is a s usp ended category in the sense of K eller and V ossiec k [KeV], that is, U is a class closed f or extensions suc h that U [1] ⊂ U . A pre-aisle U of T is total if U = ⊥ ( U ⊥ ) (orthogonal alw a ys tak en in T ). If U ⊂ T is a class of ob jects such that U [1] ⊂ U then the class ⊥ ( U ⊥ ) is a total pre-aisle of T , it is the s m allest total pre-aisle of T con taining U . T he p rop erty U [1] ⊂ U implies U ⊥ ⊂ U ⊥ [1]. In general, giv en a class Y ⊂ T suc h that Y ⊂ Y [1] th en the class ⊥ Y ⊂ T is a total pr e-aisle of T . As a consequence, if T ′ is a triangulated sub category of T and V is a total pre-aisle of T ′ then V = U ∩ T ′ where U is a total p re-aisle of T . An aisle (or in general, a total p re-aisle) U ⊂ T is gener ate d by a set of ob jects W ⊂ T if U is the smallest aisle (total pre-aisle) of T conta ining W . W e will say that a t -structure on T is gener ate d b y the set of ob jects W if so is its aisle. 1.2. Amon g th e triangulated categories w e are concerned with, only the unboun ded cate gory D ( R ) h as copro du cts. Starting with a family of ob jects in D ( R ) it is p ossible to constru ct its asso ciated t -stru cture on D ( R ) as follo ws. Giv en a set o f ob jects M ⊂ D ( R ) , let M [ N ] := { M [ i ] ; M ∈ M and i ≥ 0 } ⊂ D ( R ) . By [AJS2, Prop osition 3.2] th e s m allest co complete pre-aisle contai ning the ob jects in M is an aisle, that w e w ill denoted here by aisle hMi . Note that aisle hMi = ⊥ ( M [ N ] ⊥ ). F urth eremore w e can alw a y s assume that aisle hMi is generated b y a single ob ject b ecause aisle hMi = aisle h M i with M := ⊕ M ∈M M . Let us ﬁx a su p erscrip t ♯ ∈ { “blank” , + , − , b } . W e kno w that an y total pre-aisle of D ♯ fg ( R ) is the restriction of a total p re-aisle of D ( R ); Prop osi- tion 1.4 b elo w pro vides a more usefu l resu lt. Lemma 1.3. The c ate gories D ♯ fg ( R ) ar e skeletal ly smal l. Pr o of. Let us treat ﬁ rst the case ♯ = − . By using step b y step free r esolutions and taking in to acco unt that R is no etherian, we s ee that ev ery ob ject in D − fg ( R ) is isomorphic to a b ounded ab o v e complex of ﬁnitely generated free mo dules and they form a set W that con tains a representa tiv e for ev ery complex in D − fg ( R ). The rest of the cases will b e settled if we sho w that D fg ( R ) is s keleta lly small i.e. , the case ♯ = “blank”. Let X ∈ D fg ( R ). Note that X ˜ → lim − → n ∈ N τ ≤ n X. No w ev ery τ ≤ n X is quasi-isomorphic to an ob ject W n in W . On the other hand [AJS1, Pro of of Lemma 3.5 ] there is a qu asi-isomorphism lim − → n ∈ N τ ≤ n X ˜ → h olim − → n ∈ N τ ≤ n X. COMP ACTL Y GENERA TED t -STRUCTURES 7 Summing up X is isomorphic to the cone of an endomorp hism of L n ∈ N W n . But it is clear th at the collecti on of endomorp hisms of coun table copro ducts of ob jects in W form a set M and this set conta ins a representa tiv e for eve ry complex in D fg ( R ).  Prop osition 1.4. L et T ♯ b e any of the c ate gories D ♯ fg ( R ) . L et V b e a total pr e-aisle of T ♯ , and let E b e its right ortho gonal in T ♯ . Then (1) U = ⊥ ( V ⊥ ) is an aisle of D ( R ) (we ar e using the symb ol ⊥ for the ortho gonal in D ( R ) ); (2) the c orr esp onding t -structur e ( U , F [1]) on D ( R ) satisﬁes that V = U ∩ T ♯ and E = F ∩ T ♯ ; (3) if ( V , E [1] ) is a t - structur e on T ♯ then for any X ∈ T ♯ the distin- guishe d triangle in T ♯ deﬁne d by the t -structur e ( V , E [1]) N − → X − → B + − → is the distinguishe d triangle in D ( R ) asso ciate d to ( U , F [1]) . Pr o of. By th e p revious lemma the category T ♯ is s keleta lly small, then w e can choose a set of ob jects W ⊂ V su c h that for eac h ob ject in V th ere is an isomorpic ob ject in W . Then E is the right orthogonal of W in T ♯ . Th e class U = ⊥ ( W ⊥ ) is the aisle of D ( R ) generated by W , and trivially V ⊂ U . Let F := U ⊥ , that is F = W ⊥ . In particular E = F ∩ T ♯ ⊂ F and th er efore U ∩ T ♯ = ⊥ F ∩ T ♯ ⊂ ⊥ E ∩ T ♯ = V . Whence U ∩ T ♯ = V . T h e last assertion in the prop osition is ob vious b ecause V ⊂ U and E ⊂ F .  1.5. Let T b e a tr iangulated category with copro ducts. An ob ject E of T is called c omp act if the functor Hom T ( E , − ) commute s with a rbitrary copro ducts. By R ick ard’s criterion th e compact ob jects of D ( R ) are the p erfect complexes, i.e. those complexes isomorphic to b ou n ded complexes of ﬁn ite-t yp e p ro jectiv e m o dules (see [R, Prop osition 6.3 and its pr o of ]) 3 . W e w ill say that an aisle (or in general, a total pre-aisle) U ⊂ T is c om- p actly gener ate d if there is a set E ⊂ U of compact ob jects in T suc h that E generates U . W e w ill sa y that a t -structur e is c omp actly gener ate d if its aisle is compactly generated. Example. D ≤ 0 ( R ) is a compactly generated aisle of D ( R ), it is generated b y the stalk complex R = R [0]. Compactly generated Bousﬁeld lo calizations on D ( R ). In general the aisle of a t -structure on a triangulated category T is n ot a triangulated sub ca tegory . In fact, an aisle U of T , U is a triangulated su b category of T if a nd only if U [ − 1] ⊂ U , equiv alen tly th e left truncation functor τ ≤ U (equiv alen tly , r igh t tr uncation fun ctor τ > U ) is a ∆-fu nctor. A t -structure whose aisle U is a triangulated sub categ ory of th e am bient triangulated categ ory T is called a Bousﬁeld lo c alization of T , an d th e class 3 In [A JS2, Lemma 4.3] the reader can ﬁn d a simpler pro of of this fact u sing results of Neeman [N2, Lemma 2.2]. 8 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN U is called a lo calizing sub catego ry of T . T he ob jects in U are called acyclic and the fun ctor τ ≤ U is the asso ciated acyclizat ion functor. The ob jects in U ⊥ are cal led lo cal ob jects an d th e fun ctor τ > U is calle d the Bousﬁeld localization functor. F or a reference on Bousﬁeld localizations in th is conte xt, see [AJS1]. F or its classiﬁcati on see [N1] f or rings and [AJS3] for sc hemes and formal sc hemes. 1.6. Prop ositio n 5.7 in [AJS3] sh o ws that compactly generated Bousﬁ eld lo calizati ons on D ( R ) corresp ond to stable und er sp ecia lization subsets of Sp ec( R ). Let u s recall in our cont ext some facts ab out compactly generated Bousﬁeld lo calizati ons from [AJS 3] that w ere ob tained follo wing the path initiated in [AJL]. A su bset Z ⊂ Sp ec( R ) is stable und er sp ecializati on if for any couple of prime ideals p ⊂ q with p ∈ Z , it holds that q ∈ Z , in other w ords, it is the union of a dir ected system of clo sed subsets of Sp ec( R ). F rom no w on, to abbr eviate, we will refer to this kind of subsets as sp-subsets . As usual, for eac h R -mo d u le N le t us denote b y Γ Z ( N ) th e biggest submo dule of N whose sup p ort is conta ined in Z . The functor Γ Z : Mod ( R ) → Mo d ( R ) is an idemp otent ke rnel fu n ctor, th us it is determined by its Gabr iel top ology 4 of ideals: the set of ideals a ⊂ R suc h that Supp( R/ a ) = V ( a ) ⊂ Z . Namely , Γ Z := lim − → V( a ) ⊂ Z Hom R ( R/ a , − ) . A basis of id eals of the Gabriel top ology suﬃces to compute Γ Z . Let Q Z : Mo d ( R ) → Mo d ( R ) b e the (ab elia n) lo calizatio n f u nctor asso ciated to Γ Z . T he canonical tr an s formations Γ Z → id and id → Q Z induce isomor- phisms Γ Z Γ Z ∼ = Γ Z and Q Z ∼ = Q Z Q Z . Using K-inj ectiv e resolutions these relations can b e extended to the de- riv ed category D ( R ). In su ch a w a y that R Γ Z E ρ ( E ) − − − → E − → R Q Z E + − → (1.6.1) is the Bousﬁeld localization triangle whose localization functor is R Q Z and its acycliza tion fu n ctor R Γ Z (see [AJS3, § 2 an d the example in page 16] for the results ment ioned in th is paragraph). Moreo ve r, for all E ∈ D ( R ) the natural map E ⊗ L R R Γ Z R → E factors th r ough R Γ Z E , pro viding a canonical isomorphism E ⊗ L R R Γ Z R ˜ → R Γ Z E [AJS3, § 2.3] in suc h a wa y that the ab o v e triangle (1.6.1) is canonically isomorphic to the triangle E ⊗ L R R Γ Z R E ⊗ L R ρ ( R ) − − − − − − → E − → E ⊗ L R R Q Z R + − → [AJS3, § 2.1]. Th ese p rop erties are summarized in [AJS3, § 5, p. 603] by say- ing that the corresp onding Bousﬁeld lo calization is ⊗ -c omp atible . In lo c. cit. it is prov ed that these are exactly the Bousﬁeld lo calizations on D ( R ) whose lo calizati on functors comm ute with copro ducts, i.e. the smashing lo calizing sub ca tegories of D ( R ) ( cf . [N1, § 3]) . 4 See [Ste, Ch. VI , § 5]. COMP ACTL Y GENERA TED t -STRUCTURES 9 1.7. As a d irect consequence of the ab o ve results giv en Z 1 , Z 2 ⊂ Sp ec ( R ) t w o s p -subsets, then: (1) Th e canonical transformation R Γ Z 1 ∩ Z 2 → R Γ Z 2 induces a natural isomorphism R Γ Z 1 ∩ Z 2 → R Γ Z 1 R Γ Z 2 . (2) Th e canonica l map of f u nctors R Q Z 1 R Γ Z 2 → R Q Z 1 induces a n atu- ral isomorphism R Q Z 1 R Γ Z 2 ˜ → R Γ Z 2 R Q Z 1 . F u r thermore R Q Z 1 R Q Z 2 and R Q Z 2 R Q Z 1 are canonically isomorph ic, and they are isomorphic to R Q Z 2 ∪ Z 1 . Theorem 1.8. L et Z ⊂ Sp ec( R ) b e a sp-subset and F ∈ D ( R ) . The c anon- ic al map R Γ Z F → F is an isomorp hism if and only if S upp(H j ( F )) ⊂ Z , for every j ∈ Z . Pr o of. It is [AJS3, Th eorem 5.6] translated into the presen t conte xt.  Corollary 1.9. L et Z ⊂ S p ec( R ) b e a sp-subset and i ∈ Z . The pr e-aisle U i Z := { N ∈ D ≤ i ( R ) ; Supp(H j ( N )) ⊂ Z f or al l j ≤ i } is an aisle of D ( R ) with τ ≤ i R Γ Z as its asso ciate d lef t trunc ation functor. Pr o of. By Th eorem 1.8, U i Z is the class of ob jects N ∈ D ( R ) suc h that τ ≤ i R Γ Z N ∼ = N . Then U i Z is an aisle of D ( R ) and the right adjoin t functor of the inclusion U i Z ֒ → D ( R ) is τ ≤ i R Γ Z .  T otal pre-aisles and base c hange. Let f : R → A b e a homomorphism of rings. T he exact f orgetful functor f ∗ : Mod ( A ) → Mo d ( R ) has adjoint s on b oth sides. Th e base c hange fu nctor f ∗ = A ⊗ R − is its left adjoin t, and its righ t adjoin t is Hom R ( A, − ) . The d eriv ed fu nctors L f ∗ : D ( R ) → D ( A ) , f × := R Hom · R ( A, − ) : D ( R ) → D ( A ) , deﬁned using K -pro jectiv e and K -injectiv e resolutions in K ( R ) (see [BN , Theorem 2.14] and [Sp]), satisfy the corresp ond ing natural ad j unction for- m ulas: Hom D ( R ) ( M , f ∗ N ) ∼ = Hom D ( A ) ( L f ∗ M , N ) , Hom D ( R ) ( f ∗ N , M ) ∼ = Hom D ( A ) ( N , R Hom · R ( A, M )) , for all M ∈ D ( R ) , and N ∈ D ( A ). As it is us ual, if there is n o am biguit y , w e will w rite N = f ∗ N for ev ery N ∈ D ( A ) . The fu n ctor f ∗ transforms acyclic complexes in to acyclic complexes, h ence Hom · R ( A, − ) : K ( R ) → K ( A ) transforms a K -injectiv e complex of (injectiv e) R -mo d ules in to a K -i njectiv e complex of (injective ) A -mod u les. As a consequence if g : A → B is another morphism of rings then ( g f ) × = f × g × . Let f : R → A b e a homomorphism of rings and let W ⊂ D ( R ) b e a class of ob jects. W e den ote b y W [ f ∗ ] = { N ∈ D ( A ) ; f ∗ N ∈ W } th e pre-image of W through f ∗ , and by L f ∗ W the image of W throu gh L f ∗ . W e use the same orthogonal sym b ols for classes in D ( R ) and D ( A ) in eac h case the am bien t catego ry will tell us where the orthogonals are tak en. The follo wing is a sligh tly more general reformulatio n of the statemen t in [AJS2, Corollary 5.2] that is us eful in the p resen t context . 10 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN Prop osition 1.10. L et U b e a c o c omplete pr e-aisle of D ( R ) , then X ⊗ L R M ∈ U for al l X ∈ U and M ∈ D ≤ 0 ( R ) . Pr o of. The class V = { M ∈ D ( R ) ; X ⊗ L R M ∈ U } is a co complete pre-aisle of D ( R ) that con tains R = R [0] , therefore D ≤ 0 ( R ) ⊂ V .  Corollary 1.11. L et f : R → A b e a mo rphism of rings an d let U b e a c o c omplete pr e-aisle of D ( R ) . Then: (1) f ∗ L f ∗ U ⊂ U ; (2) ( L f ∗ U ) ⊥ = ( U ⊥ ) [ f ∗ ] ; (3) f × ( U ⊥ ) ⊂ ( L f ∗ U ) ⊥ . If furthermo r e U is a total pr e -aisle and W := ⊥ (( L f ∗ U ) ⊥ ) , then f ∗ W ⊂ U . Pr o of. If M ∈ U th en Pr op osition 1.10 sho ws that f ∗ L f ∗ M = A ⊗ L R M ∈ U , from which assertion (1) follo ws. Assertion (2) follo ws immediately from the adjunction isomorp hism Hom D ( A ) ( L f ∗ M , N ) ∼ = Hom D ( R ) ( M , f ∗ N ) for all M ∈ D ( R ) and N ∈ D ( A ). Due to (1), for all V ∈ U ⊥ and U ∈ U w e ha v e th at Hom D ( A ) ( L f ∗ U, f × V ) ∼ = Hom D ( R ) ( f ∗ L f ∗ U, V ) = 0, that is (3) follo ws. In ord er to chec k the last assertion note that for any W ∈ W and V ∈ U ⊥ one has that 0 = Hom D ( A ) ( W , f × V ) b y (3); therefore, 0 = Hom D ( R ) ( f ∗ W , V ) . So f ∗ W ⊂ ⊥ ( U ⊥ ) = U .  1.12. Let S ⊂ R b e a m ultiplicativ e closed subset and f : R → S − 1 R b e the canonical ring homomorphism. The f u nctor f ∗ is exact so L f ∗ = f ∗ : D ( R ) → D ( S − 1 R ). As usu al we denote f ∗ X = S − 1 X for ev ery ob ject X ∈ D ( R ), and give n a class V in D ( R ), S − 1 V stands for f ∗ V . The forgetful functor identiﬁes D ( S − 1 R ) with a full su b category of D ( R ). Throughout the pap er w e identify Sp ec( S − 1 R ) with the s u bset { p ∈ Sp ec( R ) ; p ∩ S = ∅ } ⊂ Sp ec( R ). If S = R \ q where q is a prime ideal of R w e will write, as usual, S − 1 V = V q . Prop osition 1.13. L e t Z ⊂ Sp ec( R ) b e a sp-subset, and let u s ﬁx i an inte ger. L et us denote U = U i Z (se e Cor ol lary 1.9), and F = U ⊥ . F or any multiplic ative close d subset S ⊂ R the p air ( S − 1 U , S − 1 F [1]) is a t -structur e on D ( S − 1 R ) , furthermor e S − 1 U = U ∩ D ( S − 1 R ) and S − 1 F = F ∩ D ( S − 1 R ) . Pr o of. F or eve ry M ∈ D ( R ) th e canonical map S − 1 R Γ Z M → R Γ Z S − 1 M is an isomorphism (see § 1.7), therefore τ ≤ U S − 1 M ∼ = S − 1 τ ≤ U M . It follo ws that S − 1 U = U ∩ D ( S − 1 R ) and S − 1 F = F ∩ D ( S − 1 R ). Moreo ver, f or any M ∈ D ( S − 1 R ) th e tr iangle in D ( R ) τ ≤ U M − → M − → τ > U M + − → is also in D ( S − 1 R ), b ecause τ ≤ U M = τ ≤ U S − 1 M ∼ = S − 1 τ ≤ U M . As a result ( S − 1 U , S − 1 F [1]) is a t -structur e on D ( S − 1 R ).  COMP ACTL Y GENERA TED t -STRUCTURES 11 R emark. No te that in p articular w e ha v e that ( S − 1 U ) ⊥ = S − 1 ( U ⊥ ) (w h ere the ﬁr st orthogonal is tak en in D ( S − 1 R ) and the second in D ( R )). 2. Aisles deter mined by fil tra tions of suppor ts 2.1. Let u s denote by j a : R → R/ a the canonical morphism determined b y the ideal a ⊂ R . W e b egin this section describing the indu ced t -structures on D ( R ) by the standard t -str u ctures on D ( R / a ) thr ough the adju nction j a ∗ ⊣ j × a . Let X b e a complex of R -mo dules such that a X i = 0, for all i ∈ Z . T hen X can b e also viewe d as a complex of R / a -mo du les, in such a wa y that j a ∗ X = X. That b eing so, for an y Y ∈ D ( R ), there are isomorphisms Hom D ( R ) ( X, Y ) ∼ = Hom D ( R/ a ) ( X, j × a Y ) (2.1.1) Lemma 2.2. L et a ⊂ R b e an ide al, and k ∈ Z . F or a c omplex Y ∈ D ( R ) , the fol lowing statements ar e e quivalent: (1) Y ∈ aisle h R/ a [ − k ] i ⊥ ; (2) j × a Y ∈ D >k ( R/ a ) . Pr o of. As a consequence of 2.1 it holds that Hom D ( R ) ( R/ a [ i ] , Y ) ∼ = Hom D ( R/ a ) ( R/ a [ i ] , j × a Y ) = H − i ( R Hom · R/ a ( R/ a , j × a Y )) = H − i ( j × a Y ) for all i ∈ Z . S o the result follo ws.  Lemma 2.3. L et a ⊂ R b e an ide al. If X ∈ D ≤ 0 ( R/ a ) , then X = j a ∗ X ∈ aisle h R/ a i . Pr o of. Clear.  Prop osition 2.4. The fol lowing statements hold for any ide als a , b ⊂ R : (1) If a ⊂ b then R/ b ∈ ai sle h R/ a i . (2) We have that aisle h R / ab i = aisl e h R/ a , R / b i = aisle h R / a ∩ b i . (3) F or al l n ≥ 1 , aisl e h R/ a n i = ai sle h R/ a i . (4) If rad( a ) = rad( b ) then aisle h R/ a i = a isle h R/ b i . Pr o of. The statemen t (1) is th e particular case of Lemm a 2.3 in wh ic h X ∈ D ≤ 0 ( R/ a ) is the stalk complex R/ b . In order to pro v e (2) note that a / ab is also an R / b -mo dule then a / ab ∈ aisle h R/ b i b y Lemma 2.3. As a consequ en ce the middle p oint in the exact sequence 0 → a / ab → R/ ab → R / a → 0 b elongs to ais le h R/ a , R / b i b ecause the extreme p oin ts do. Th erefore aisl e h R/ ab i ⊂ ais le h R/ a , R / b i . W e ﬁnish the p r o of of (2) applying (1) to the c hains of ideals ab ⊂ a ∩ b ⊂ a and ab ⊂ a ∩ b ⊂ b . Statemen t (3) follo ws by in duction on n ≥ 1 from the ﬁrst equalit y in (2). The ring R is No etherian, so if rad( a ) = rad( b ) there exist s , t ∈ N such that a s ⊂ b and b t ⊂ a . Therefore (4) follo ws from (1) and (3).  12 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN Corollary 2.5. L et { p 1 , . . . , p s } b e the minimal prime ide als over the ide al a ⊂ R . Then aisle h R/ a i = aisle h R/ p 1 , . . . , R / p s i . Pr o of. Using that rad( a ) = p 1 ∩ · · · ∩ p s and the second identit y in Prop o- sition 2.4(2) w e easily pro ve b y induction on s that aisle h R/ rad( a ) i = aisle h R/ p 1 , . . . , R/ p s i . Finally apply (4) in Pr op osition 2.4.  Corollary 2.6. L et a ⊂ R b e a n ide al and Z := V( a ) ⊂ Sp ec( R ) . L et Y ∈ D ( R ) b e a c omplex such that Y ∈ aisl e h R/ a i ⊥ , then R Γ Z Y ∈ D > 0 ( R ) . Pr o of. Without lost of generalit y w e ma y assume that Y is K -injectiv e, so R Γ Z Y = Γ Z Y . By Prop osition 2.4 it holds that Y ∈ aisle h R/ a n i ⊥ for all n ≥ 1, that is, Hom D ( R ) ( R/ a n [ i ] , Y ) = 0 for all n ≥ 1 and all i ≥ 0. As a consequence f or all i ≥ 0 H − i Γ Z Y = H − i lim − → n ≥ 1 Hom · R ( R/ a n , Y ) = lim − → n ≥ 1 H − i Hom · R ( R/ a n , Y ) = lim − → n ≥ 1 Hom D ( R ) ( R/ a n [ i ] , Y ) = 0 .  Prop osition 2.7. L et Z ⊂ Sp ec( R ) b e a sp-su b set, i ∈ Z , and U i Z b e the aisle deﬁne d in Cor ol lary 1.9. Then U i Z = ai sle h R/ p [ − i ]; p ∈ Z i . Pr o of. T o pro v e this result it is enough to deal with the case i = 0. Let us re- call from C orollary 1.9 that τ ≤ 0 R Γ Z is the left trun cation functor asso ciated to the aisle U 0 Z . Let { a α } α ∈ I b e the Gabriel ﬁlter of ideals such that Z α := V ( a α ) ⊂ Z. By Corollary 2.5 it is enough to pro v e that U 0 Z = aisle h R/ a α ; α ∈ I i . T rivially U := ais le h R/ a α ; α ∈ I i ⊂ U 0 Z . T o p ro v e the equalit y let us chec k th at U ⊥ ⊂ U 0 Z ⊥ . Let j ≤ 0 an d Y ∈ U ⊥ , then Corollary 2.6 asserts that H j ( R Γ Z α Y ) = 0 for all α ∈ I , therefore H j ( R Γ Z Y ) = lim − → α ∈ I H j ( R Γ Z α Y ) = 0 That is, τ ≤ 0 R Γ Z Y = 0, equiv alen tly Y ∈ U 0 Z ⊥ .  2.8. A ﬁltr ation by supp orts of Sp ec( R ) is a d ecreasing map φ : Z − → P (S p ec( R )) suc h th at φ ( i ) ⊂ Sp ec ( R ) is a sp-su bset for eac h i ∈ Z . T o abb reviate, we will refer to a ﬁltration by supp orts of Sp ec( R ) simply by a sp-ﬁltr ation of Sp ec( R ). Let U b e an aisle of D ( R ). Having in min d th at U [1] ⊂ U and the s tate- men t (1) in Prop osition 2.4, the aisle U determines a s p-ﬁltration φ U : Z → COMP ACTL Y GENERA TED t -STRUCTURES 13 P (Sp ec( R )) b y s etting, for eac h i ∈ Z , φ U ( i ) := { p ∈ Sp ec( R ) ; R / p [ − i ] ∈ U } . The other wa y round a sp-ﬁ ltration φ : Z → P (Sp ec( R )) h as an asso ciate d aisle U φ := ai sle h R/ p [ − i ] ; i ∈ Z and p ∈ φ ( i ) i . Fix i an in teger and Z ⊂ Sp ec( R ) a sp -subset. Let φ : Z → P (Sp ec( R )) b e the sp-ﬁltration d eﬁ ned by φ ( j ) = Z for all j ≤ i , and φ ( j ) = ∅ if j > i . The previous pr op osition shows that U i Z = U φ . The f ollo wing sh o ws the compatibilit y of these aisles with resp ect to lo calization in a m ultiplicativ e closed su bset of R , generalizing Prop ositio n 1.13 : Prop osition 2.9. L et S ⊂ R b e a multiplic ative close d su b set. Given a sp- ﬁltr ation φ : Z → P (Sp ec( R )) let us denote by F φ the right ortho gonal of U φ in D ( R ) . Then ( S − 1 U φ , S − 1 F φ [1]) is a t -structur e on D ( S − 1 R ) , furthermor e S − 1 U φ = U φ ∩ D ( S − 1 R ) and S − 1 F φ = F φ ∩ D ( S − 1 R ) . B e sides S − 1 U φ ⊂ D ( S − 1 R ) is the asso ciate d aisle to the sp-ﬁltr ation φ S : Z → P (S p ec( S − 1 R )) deﬁne d by φ S ( i ) := φ ( i ) ∩ Sp ec( S − 1 R ) , for i ∈ Z . Pr o of. The aisle U := U φ is the sm allest con taining all the aisles in the set {U i := U i φ ( i ) ; i ∈ Z } , equiv alen tly F := F φ is obtained by in tersecting the classes {F i := U i ⊥ φ ( i ) ; i ∈ Z } . F or any M ∈ D ( S − 1 R ) the d istin gu ish ed triangle in D ( R ) associated to ( U , F [1]) N − → M − → Y + − → (2.9.1) b elongs to D ( S − 1 R ). Indeed, S − 1 N ∈ U b y Prop osition 1.10. Giv en a com- plex Y ∈ F we hav e that Y ∈ F i for any i ∈ Z , then by Prop ositio n 1.13, w e ha v e that S − 1 Y ∈ F i for all i ∈ Z , that is S − 1 Y ∈ F . Necessarily the distin- guished triangle S − 1 N − → M − → S − 1 Y + − → is canonically isomorph ic to (2.9.1) . As a consequence S − 1 U is an aisle of D ( S − 1 R ) with righ t orthogonal class S − 1 F . Moreo v er S − 1 U = U ∩ D ( S − 1 R ) and S − 1 F = F ∩ D ( S − 1 R ). Finally , note that E ∈ S − 1 F if and only if, for eac h i ∈ Z 0 = Hom D ( R ) ( R/ p [ − i ] , E [ j ]) ∼ = Hom D ( S − 1 R ) ( S − 1 ( R/ p )[ − i ] , E [ j ]) . for all p ∈ φ ( i ) and all j ≤ 0. Th is fact amounts to sa ying that 0 = Hom D ( S − 1 R ) ( S − 1 R/ q [ − i ] , E [ j ]) , for eac h i ∈ Z , all q ∈ φ ( i ) ∩ Sp ec( S − 1 R ) and all j ≤ 0. W e conclude that S − 1 U is the aisle of D ( S − 1 R ) asso cia ted to φ S : Z → P (S p ec( S − 1 R )).  R emark. Let u s consider the notation in Prop ositio n 2.9. Let W b e any of the classes U φ or F φ . F rom the previous results it follo w s that a complex X ∈ D ( R ) is in W if and only if X p b elongs to W for an y p ∈ Sp ec ( R ). 14 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN 3. Comp actl y gene ra ted aisles In Prop ositio n 3.7 w e study the ai sles of D ( R ) generated by b ounded ab o v e complexes with ﬁnitely generated h omologies. It is a k ey r esult in the pro of of Theorem 3.10 and Theorem 3.11, the main results in this section. W e b egin by pr o ving some useful lemmas. Let us adopt the conv ention that D ≤−∞ ( R ) = 0 and D > −∞ ( R ) = D ( R ). Lemma 3.1. L et a ⊂ R b e an ide al, j a : R → R/ a the c anonic al map, and Y a c omplex of R - mo dules. Assume that the fol low ing two c onditions hold for a ﬁxe d m ∈ Z : (1) j × a Y ∈ D >m ( R/ a ) , (2) Su pp(H i ( Y )) ⊂ V ( a ) for al l i ≤ m. Then Y ∈ D >m ( R ) . Pr o of. It is enough to d eal with the case m = 0. Le t Z := V ( a ). By Theorem 1.8 the hyp othesis (2) is equiv alent to assuming that the canonical map R Γ Z τ ≤ 0 Y → τ ≤ 0 Y is an isomorphism. F u rthermore, by Lemma 2.2 and Corollary 2.6, hyp othesis (1) implies that R Γ Z Y ∈ D > 0 ( R ). Bearing in mind that R Γ Z D ≥ 0 ( R ) ⊂ D ≥ 0 ( R ) , it follo ws that the canonical map τ ≤ 0 Y → Y induces isomorphims H − i ( R Γ Z τ ≤ 0 Y ) → H − i ( R Γ Z Y ) , for all i ≥ 0 ; and so H − i ( τ ≤ 0 Y ) ˜ ← H − i ( R Γ Z τ ≤ 0 Y ) ˜ → H − i ( R Γ Z Y ) = 0.  Lemma 3.2. Assume R is lo c al with maximal ide al m ⊂ R . If Y ∈ D ( R ) is a c omplex such that Sup p(H j ( Y )) ⊂ { m } , for al l j ∈ Z , then for any X ∈ D − fg ( R ) the fol lowing ar e e q uivalent: (1) F or al l i ≥ 0 , Hom D ( R ) ( X [ i ] , Y ) = 0 . (2) Ther e is n ∈ Z ∪ {−∞} , such that X ∈ D ≤ n ( R ) and Y ∈ D >n ( R ) . Pr o of. The imp lication (2) = ⇒ (1) is trivial. Assume (1) for a non acyclic complex X ∈ D − fg ( R ). Due to Prop osition 1.10 we ha v e that, for all i ≥ 0, Hom D ( R ) ( X ⊗ L R R/ m [ i ] , Y ) = 0 . Let n := m ax { j ∈ Z ; H j ( X ) 6 = 0 } , then H n ( X ⊗ L R R/ m ) ∼ = H n ( X ) ⊗ R R/ m 6 = 0 b y Nak ay ama’s lemma. Sin ce X ⊗ L R R/ m is a complex of R / m -v ector spaces w e get Hom D ( R/ m ) ( X ⊗ L R R/ m [ i ] , j × m Y ) ∼ = Hom D ( R ) ( X ⊗ L R R/ m [ i ] , Y ) = 0 , for all i ≥ 0. Notice that H n ( X ⊗ L R R/ m )[ − n ] is isomorph ic in D ( R/ m ) to a direct sum m and of X ⊗ L R R/ m , th erefore 0 = Hom D ( R/ m ) (H n ( X ⊗ L R R/ m )[ − n + i ] , j × m Y ) , for any i ≥ 0, whic h in turn imp lies th at 0 = Hom D ( R/ m ) ( R/ m [ i ] , j × m Y ) , for all i ≥ − n . That is j × m Y ∈ D >n ( R/ m ). No w from Lemma 3.1 we can conclude that Y ∈ D >n ( R ).  COMP ACTL Y GENERA TED t -STRUCTURES 15 3.3. Let us ﬁ x the con v en tion that m ax( ∅ ) = min( ∅ ) = − ∞ for the empty subset ∅ ⊂ Z . T hen for X ∈ D − fg ( R ) and p ∈ Sp ec( R ) the follo wing are w ell-deﬁned ele ments in the set Z ∪ {−∞} m p ( X ) := max { j ∈ Z ; p ∈ Su p p(H j ( X )) } h p ( X ) := max { j ∈ Z ; Supp(H j ( X ⊗ L R R/ p )) = Sp ec( R/ p ) } . Lemma 3.4. F or any X ∈ D − fg ( R ) and p ∈ Sp ec( R ) , it holds that m p ( X ) = h p ( X ) . Pr o of. Let us p ut m = m p ( X ) and h = h p ( X ). F rom the canonical isomor- phisms ( X ⊗ L R R p ) ⊗ L R p k ( p ) ∼ = X ⊗ L R k ( p ) ∼ = ( X ⊗ L R R/ p ) ⊗ L R/ p k ( p ) , it follo ws that for an y in teger j ∈ Z such that X p ∈ D ≤ j ( R p ) necessary ( X ⊗ L R R/ p ) ⊗ L R/ p k ( p ) ∈ D ≤ j ( k ( p )). Then ha ving in mind that m and h can b e computed as m = min { j ∈ Z ; X p ∈ D ≤ j ( R p ) } and h = min { j ∈ Z ; ( X ⊗ L R R/ p ) ⊗ L R/ p k ( p ) ∈ D ≤ j ( k ( p ) } w e get that h ≤ m . T rivially h = m = −∞ when p 6∈ S i ∈ Z Supp(H i ( X )). Ass u me that p ∈ S i ∈ Z Supp(H i ( X )). Th en X ⊗ L R R p ∈ D ≤ m ( R p ) and H m ( X ⊗ L R R p ) ∼ = H m ( X ) ⊗ R R p 6 = 0. Hence ( X ⊗ L R R p ) ⊗ L R p k ( p ) ∈ D ≤ m ( k ( p )) and H m (( X ⊗ L R R p ) ⊗ L R p k ( p )) ∼ = H m ( X ⊗ L R R p ) ⊗ R p k ( p ) By Nak a y ama’s lemma, th e mo d ule H m ( X ⊗ L R R p ) ⊗ R p k ( p ) is n onzero, so H m (( X ⊗ L R R/ p ) ⊗ L R/ p k ( p )) is nonzero, hence m = h .  Lemma 3.5. L et R b e a c ommutat ive No etherian inte gr al domain. L et X ∈ D − fg ( R ) b e a c omplex and 0 ∈ Sp ec ( R ) b e the generic p oint. With the notation i n 3.3, m 0 ( X ) = max { i ∈ Z ; Supp(H i ( X )) = Sp ec ( R ) } . If Y ∈ D ( R ) satisﬁes the fol lowing c onditions: (1) Hom D ( R ) ( X, Y [ i ]) = 0 , for al l i ≤ 0 , and (2) for every 0 6 = p ∈ Sp ec( R ) , R Hom · R ( R/ p , Y ) ∈ D >m 0 ( R ) , then Y ∈ D >m 0 ( R ) . Pr o of. Let K b e the ﬁeld of fractions of R and let us set Z := Sp ec ( R ) \ { 0 } and m := m 0 ( X ). L et us stud y the non trivial case, so assum e that m ∈ Z . The complex X b elongs to D ≤ n fg ( R ) for an intege r n ≥ m . By Prop osition 2.7, the hyp othesis (2) on Y is equiv alen t to sa ying R Γ Y ∈ D >m ( R ), w h ere Γ := Γ Z : M o d ( R ) → M o d ( R ) is the us u al torsion r ad ical. App lying the homologica l fu nctor Hom( X, − ) := Hom D ( R ) ( X, − ) to the canonical triangle R Γ Y − → Y − → Y ⊗ L R K + − → (3.5.1) w e get exact s equences Hom( X , Y [ i − 1]) → Hom( X, Y ⊗ L R K [ i − 1]) → Hom ( X, R Γ Y [ i ]) 16 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN for all i ∈ Z . Notice that X ∈ D ≤ n ( R ), R Γ Y [ m − n ] ∈ D >n ( R ) and Y [ − 1] ∈ aisle h X i ⊥ , then w e get 0 = Hom D ( R ) ( X, Y ⊗ R K [ i − 1]) for i ≤ m − n ≤ 0. As a consequence Hom D ( K ) ( X ⊗ R K, Y ⊗ R K [ i − 1]) ∼ = Hom D ( R ) ( X, Y ⊗ R K [ i − 1]) = 0 for all i ≤ m − n . Recall that X ⊗ R K ∈ D ≤ m fg ( K ) and H m ( X ⊗ R K ) 6 = 0, therefore Y ⊗ R K [ m − n ] ∈ D >m ( K ) since K is a ﬁeld and so Lemma 3.2 applies h ere. Th en we conclude that Y ⊗ R K ∈ D + ( R ). T herefore Y ∈ D + ( R ) by the existence of th e d istinguished triangle (3.5.1). F rom this fact w e are going to pr o v e a more p recise h omologic al b ound for Y ⊗ R K , namely Y ⊗ R K ∈ D >m ( K ). I ndeed, f or all i ∈ Z ther e is a canonical isomorphism Hom D ( K ) ( X ⊗ R K, Y ⊗ R K [ i ]) ∼ = Hom D ( R ) ( X, Y [ i ]) ⊗ R K, since Y ∈ D + ( R ) and X ∈ D − fg ( R ). T hen, by adjunction, hyp othesis (1) and Prop osition 1.10 Hom D ( K ) ( X ⊗ R K, Y ⊗ R K [ i ]) ∼ = Hom D ( R ) ( X, Y ⊗ R K [ i ]) = 0 , for all i ≤ 0. Hence Y ⊗ R K ∈ D >m ( K ) b y Lemm a 3.2. Using once again the distinguish ed triangle (3.5.1 ) we conclude Y ∈ D >m ( R ) as desired .  Lemma 3.6. L et R b e a c ommutative No etherian ring. L et X ∈ D − fg ( R ) and Y ∈ D ( R ) . If Hom D ( R ) ( X, Y [ i ]) = 0 for al l i ≤ 0 , then Hom D ( R ) ( R/ p [ − k ] , Y [ i ]) = 0 for al l i ≤ 0 , k ∈ Z and any p ∈ Supp(H k ( X )) . Pr o of. As a consequence of Corollary 1.11, it f ollo ws f r om the h yp othesis that Hom D ( R ) ( X ⊗ L R R/ p , Y [ i ]) = 0 f or any i ≤ 0, and Hom D ( R/ p ) ( X ⊗ L R R/ p , R Hom · R ( R/ p , Y )[ i ]) = 0 for all i ≤ 0 and ev ery p ∈ Sp ec ( R ). Let u s ﬁ x an in teger k ∈ Z such that H k ( X ) 6 = 0 and take any p ∈ Supp(H k ( X )). Then we h a v e k ≤ m p = h p , wh ere m p = m p ( X ) and h p = h p ( X ) (see L emma 3.4). W e pro ceed by r e ductio ad absur dum assum ing that the s et S := { p ∈ Sp ec ( R ) ; p ∈ Supp(H k ( X )) and Y / ∈ a isle h R/ p [ − k ] i ⊥ } is nonemp ty . The ring R is No etherian so w e can c ho ose a m aximal elemen t p 0 in S . Note that for X 0 := X ⊗ L R R/ p 0 ∈ D − fg ( R/ p 0 ) the complex Y 0 := R Hom · R ( R/ p 0 , Y ) ∈ D ( R/ p 0 ) satisﬁes the f ollo wing prop erties: (1) Hom D ( R/ p 0 ) ( X 0 , Y 0 [ i ]) = 0, for all i ≤ 0 (see the remark at the b eginning of the pro of ). (2) If q ∈ Sp ec( R ) is su c h th at p 0 ( q , then q ∈ Supp(H k ( X )) and q / ∈ S since p 0 is maximal in S . Th erefore Y ∈ aisle h R/ q [ − k ] i ⊥ . This fact amoun ts to sa ying that R Hom · R/ p 0 ( R/ q , Y 0 ) ∈ D >k ( R/ p 0 ), since Hom D ( R/ p 0 ) ( R/ q , Y 0 [ i ]) ∼ = Hom D ( R ) ( R/ q , Y [ i ]). COMP ACTL Y GENERA TED t -STRUCTURES 17 Then Lemma 3.5 sho ws that Y 0 ∈ D >k ( R/ p 0 ), so Hom D ( R ) ( R/ p 0 [ i ] , Y ) = 0 for all i ≥ − k . This fact con tradicts the assum p tion p 0 ∈ S , then necessarily S = ∅ , and the result follo ws.  W e are n o w ready to prov e a k ey result in this section. Prop osition 3.7. L et R b e a c omm utative No etherian ring. F or X ∈ D − fg ( R ) and Y ∈ D ( R ) , the fol lowing ar e e quivalent: (1) Hom D ( R ) ( X, Y [ i ]) = 0 , for al l i ≤ 0 ; (2) Hom D ( R ) (H j ( X )[ − j ] , Y [ i ]) = 0 , for any j ∈ Z and i ≤ 0 ; (3) Hom D ( R ) ( R/ p [ − j ] , Y [ i ]) = 0 , for al l j ∈ Z , i ≤ 0 and al l prime ide als p (minimal) in Supp(H j ( X )) ; (4) Hom D ( R ) ( R/ p [ − j ] , Y [ i ]) = 0 , for every j ∈ Z , i ≤ 0 and al l prime ide als p (minimal) in Ass(H j ( X )) . Pr o of. The equiv ale nce b et we en (3) and (4) follo ws directly from Prop osi- tion 2.4(1). Lemma 3.6 is just (1) ⇒ (3). T o s ho w th at (3) ⇒ (1), w e only need to assume here that X ∈ D − ( R ) . F or sim p licit y let u s supp ose that X ∈ D ≤ 0 ( R ). Let U = U φ where φ : Z → P (Sp ec( R )) is the sp-ﬁltration deﬁn ed, for eac h i ∈ Z , by setting φ ( i ) := ∪ j ≥ i Supp(H j ( X )) ( cf. 2.8). Item (3) says that Y ∈ U ⊥ . S o to prov e (1) is enough to c hec k that X ∈ U . Let us consider the canonical triangle τ ≤ φ X − → X − → τ > φ X + − → and d enote N = τ ≤ φ X and B = τ > φ X. W e claim that B = 0 , equiv alently the canonical m ap N → X is an isomorph ism wh ence we get the desired result X ∼ = N ∈ U . Note that B ∈ D ≤ 0 ( R ) b ecause N an d X b elong to D ≤ 0 ( R ) . If B 6 = 0 , let us c ho ose q ∈ Sp ec( R ) minimal in the set ∪ t ≤ 0 Supp(H t ( B )). By localizing from the ab o ve triangle we get the d istinguished triangle in D ( R q ) N q − → X q − → B q + − → (3.7.1) Recall fr om Prop ositio n 2.9 that N q ∈ U q , and B q ∈ ( U ⊥ ) q = ( U q ) ⊥ (nota- tion as in 1.12). Let b := max { j ≤ 0 / q ∈ Su pp(H j ( B )) } and Z := { q R q } ⊂ Sp ec( R q ). Th en R Γ Z ( B q ) ∼ = B q ∈ D ≤ b ( R q ), that is τ ≤ b R Γ Z ( B q ) ∼ = B q . Hence B q ∈ ais le h R q / q R p [ − b ] i as a consequence of Pr op osition 2.7. If X q = 0 then B q ∼ = N q [1] ∈ U q , so in this case B q = 0. Supp ose that X q 6 = 0 , and set m := max { j ≤ 0 ; q R q ∈ Supp(H j ( X q )) } = max { j ≤ 0 ; q ∈ Supp(H j ( X )) } . Notice that then X q ∈ D ≤ m ( R q ). T h e aisle U q ⊂ D ( R q ) is generated b y the set { R q / p R q [ − i ] ; p ∈ Su pp(H i ( X )) , i ∈ Z } = { R q / p R q [ − i ] ; p ∈ Supp(H i ( X q )) , i ∈ Z } (see Prop osition 2.9). Hence U q is con tained in D ≤ m ( R q ), and so N q ∈ D ≤ m ( R q ). Using the triangle (3.7.1) w e get that b ≤ m . Therefore aisle h R q / q R q [ − b ] i ⊂ ai sle h R q / q R q [ − m ] i ⊂ U q . Then B q ∈ U q , and again as a resu lt B q = 0 . 18 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN Finally the equiv ale nce b et we en (2) an d (3) is a consequence of (1) ⇔ (3) for the complex M j ∈ Z H j ( X )[ − j ] ∈ D − fg ( R ) .  Corollary 3.8. F or any X ∈ D − fg ( R ) , aisle h X i = aisle h H j ( X )[ − j ]; j ∈ Z i = ai sle h R/ p [ − j ]; j ∈ Z and p ∈ Sup p H j ( X ) i . Pr o of. Immediate from the ab o v e p rop osition.  Corollary 3.9. L et i b e an i nte ger and let Z b e a sp-subset of Sp ec( R ) . The aisle U i Z ⊂ D ( R ) is c omp actly ge ne r ate d. Pr o of. It is enough to discuss th e case i = 0 . F or eac h ideal a ⊂ R s u c h that V( a ) ⊂ Z , let us ﬁx a system of generato rs { a 1 , . . . , a r } of a . Let K · ( a 1 , . . . , a r ) b e the K oszul complex associated to the sequence { a 1 , . . . , a r } ([EGA, I I I , (1.1.1) ]). Recall that K · ( a 1 , . . . , a r ) is the complex of R -mo du les deﬁned by K · ( a 1 , . . . , a r ) := ⊗ r j =1 K · ( a j ) , where K · ( a j ) is th e complex (0) in all degrees ap art from d egrees − 1 and 0, and whose diﬀeren tial in degree − 1 is R a j − → R the map m ultiplying b y a j . The complex K · ( a 1 , . . . , a r ) is a complex of ﬁnitely generated free mo dules in degrees [ − r , 0] and 0 elsewh er e, and whose homologies are killed b y the ideal a . The complex K · ( a 1 , . . . , a r ) is compact ( cf. 1.5). F urthermore Supp(H i ( K · ( a 1 , . . . , a r ))) ⊂ V( a ) and H 0 ( K · ( a 1 , . . . , a r )) = R/ a . Therefore, Prop osition 3.7 sho ws that the aisle generated b y the family of complexes { K · ( a 1 , . . . , a r ) ; { a 1 , . . . , a r } ⊂ R and V( h a 1 , . . . , a r i ) ⊂ Z } agrees with U 0 Z .  W e are n o w ready to state and p ro v e th e m ain resu lts in this section. Theorem 3.10. L et R b e a c ommutative No etherian ring and ( U , F [1]) b e a t-structur e on D ( R ) . The fol lowing assertions ar e e quivalent: (1) U is c omp actly gener ate d; (2) U is gener ate d by stalk c omplexes of ﬁnitely gener ate d (r esp. c yclic) R -mo dules; (3) U is gener ate d by c omp lexes in D b fg ( R ) ; (4) U is gener ate d by c omp lexes in D − fg ( R ) ; (5) ther e exists a sp-ﬁltr ation φ : Z → P (Sp ec( R )) such that U = U φ . Pr o of. Using [R, Prop osition 6.3], the equ iv alence (1) ⇔ (2) follo ws from Prop osition 3.7 and Corollary 3.9. Again Prop osition 3.7 pr o vides (2) ⇔ (3) ⇔ (4). (5) ⇒ (2) is obvious. In ord er to pro ve (2) ⇒ (5) w e only s hould realize that U = U φ where φ is the sp-ﬁltration φ = φ U . T rivially U φ ⊂ U b eca use COMP ACTL Y GENERA TED t -STRUCTURES 19 φ = φ U . Moreo ver, un der hypothesis (2) for U , P r op osition 3.7 shows that U ⊥ φ = U ⊥ that imp lies U φ = U .  Let Ais ( R ) b e the class of aisles of D ( R ) and Fil sp ( R ) b e the s et of all sp- ﬁltrations of S p ec( R ). Let us denote by Ais cp ( R ) the compactly generated aisles of D ( R ). Let us consider on Ais ( R ) the usual inclusion relation. The order on Fil sp ( R ) is the induced order by the usual one on P (Sp ec( R )). W e deﬁne a couple of order pr eserving maps Ais ( R ) f ⇄ a Fil sp ( R ) b y setting a ( φ ) := U φ for any φ ∈ Fil sp ( R ), and f ( U ) := φ U for an y aisle U of D ( R ) (see 2.8). Theorem 3.11. The maps f and a establish a bi je ctiv e c orr esp ondenc e b e- twe en Ais cp ( R ) and Fil sp ( R ) . F urthermor e, given φ ∈ Fil sp ( R ) the c orr e- sp onding t-structur e ( U φ , U φ ⊥ [1]) is describ e d in terms of the sp-ﬁltr ation by: U φ = { X ∈ D ( R ) ; Sup p(H j ( X )) ⊂ φ ( j ) , for al l j ∈ Z } U φ ⊥ = { Y ∈ D ( R ) ; R Γ φ ( j ) Y ∈ D >j ( R ) , for al l j ∈ Z } Pr o of. In the pro of of (2) ⇒ (5) in Theorem 3.10 we ha v e sho wn that a ◦ f ( U ) = U for an y U ∈ Ais cp ( R ). Con v ersely , if φ ∈ Fil sp ( R ) let us p ro v e that φ = f ◦ a ( φ ). Let U := { X ∈ D ( R ) ; Su pp(H j ( X )) ⊂ φ ( j ) for all j ∈ Z } . The class U is a co complete p r e-aisle of D ( R ) wh ic h con tains { R/ p [ − j ] ; j ∈ Z and p ∈ φ ( j ) } and , h ence, it also con tains U φ = a ( φ ). If X ∈ U then the pro of of the equiv alence (3) ⇔ (1) in Theorem 3.7 shows that τ ≤ n X b elongs to U φ , for ev ery n ∈ Z . So in the canonica l distinguished triangle ⊕ n ≥ 0 τ ≤ n X − → ⊕ n ≥ 0 τ ≤ n X − → X + − → , the t wo left v ertices are ob jects in U φ , as a consequence X ∈ U φ . Th erefore U φ = U , and from this identiﬁcat ion it is easy to derive that φ = f ( U φ ). T o conclude let us r ecall from Prop osition 2.7 th at ( U i φ ( i ) ) ⊥ = { Y ∈ D ( R ) ; R Γ φ ( i ) Y ∈ D >i ( R ) } ( ∀ i ∈ Z ) . Hence the disp lay ed description for U ⊥ φ in the statemen t of the c urr en t theorem follo ws from the ob vious relation U ⊥ φ = T i ∈ Z ( U i φ ( i ) ) ⊥ .  R emark. The existence of aisles of D ( R ) whic h are n ot compactly generated is w ell-kno wn, as it can b e easily deriv ed from [N1, Theorem 3.3]. But unlike the situation in lo c. cit. residu e ﬁelds are not the r igh t ob jects to classify compactly generat ed t -structures. If R is an in tegral domain (comm utativ e and No etherian) and not a ﬁeld, then the aisle U of D ( R ) generated by { k ( p ) ; p ∈ Sp ec ( R ) } is n ot compactly ge nerated. In deed, otherwise it s 20 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN asso ciated sp-ﬁltration wo uld b e giv en by φ ( i ) = Sp ec ( R ), for i ≤ 0, and φ ( i ) = ∅ , for i > 0 (see Theorem 3.11). Th en we would hav e U = D ≤ 0 ( R ). But that is imp ossible b ecause R [0] ∈ U ⊥ since Hom R ( k ( p ) , R ) = 0, for ev ery p ∈ Sp ec( R ). Corollary 3.12. L et ♯ ∈ {− , + , b , “ blank” } . L et V b e an aisle (or mor e gen- er al ly, any total pr e-aisle) of D ♯ fg ( R ) gener ate d by b ounde d ab ove c omplexes, and let E b e its right ortho g onal in D ♯ fg ( R ) . Then ther e exists a unique φ ∈ Fil sp ( R ) such that V = U φ ∩ D ♯ fg ( R ) and E = F φ ∩ D ♯ fg ( R ) . Pr o of. It follo ws from Prop ositio n 1.4 and Prop osition 3.7. The u niqueness of φ follo ws from the fact V = U φ ∩ D ♯ fg ( R ) determines φ (b y Theorem 3.11 ab o v e).  4. The weak Cousin condition W e pro ceed to classify , und er suﬃcien tly general hypotheses, on R , all the compactly generated t -stru ctures on D ( R ) that r estrict to t -structures on D ♯ fg ( R ). Th e main result of this s ection is Theorem 4.4 —it pr o vides a necessary condition on a sp-ﬁltration φ in order to U φ ∩ D ♯ fg ( R ) b e an aisle of D ♯ fg ( R ). Note that the statemen t of Theorem 4.4 here and [Sta, Prop osition 7.4] are almost the same. Here we treat w ith the case of the unboun ded categ ory D fg ( R ) and obtain in particular the result for D b fg ( R ) (the f ramew ork in lo c. cit. ) . 4.1. Given a s p -ﬁltration φ : Z − → P (Sp ec( R )) we will denote by τ ≤ φ the left truncation fun ctor asso cia ted to the aisle U φ and by τ > φ the r igh t trun cation functor. So that for eac h M ∈ D ( R ) the diagram τ ≤ φ M − → M − → τ > φ M + − → denotes the natural distinguished triangle determined b y the t -structure ( U φ , U ⊥ φ [1]) for M ; we refer to this triangle as th e φ - triangle with c entr al vertex M ∈ D ( R ) or just a φ - trian gle . The assu mption that U φ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) is equiv ale nt to the fact that th e φ -triangle in D ( R ) with cen tral v ertex X b elongs to D ♯ fg ( R ) whenev er X ∈ D ♯ fg ( R ). In other wo rds, U φ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) if and only if τ ≤ φ X ∈ D ♯ fg ( R ) (or equiv alen tly τ > φ X ∈ D ♯ fg ( R )) for all X ∈ D ♯ fg ( R ). The f ollo wing lemmas are useful in the pro of of Theorem 4.4. Lemma 4.2. L et φ : Z → P (Sp ec( R )) b e a sp-ﬁltr ation. Then, for every j ∈ Z , we ge t τ ≤ φ D ≥ j ( R ) ⊂ D ≥ j ( R ) and τ > φ D ≥ j ( R ) ⊂ D ≥ j ( R ) . Pr o of. Without loss of generalit y , we ma y assume that j = 0. Let X ∈ D ≥ 0 ( R ) b e a ny complex and p ut T = τ ≤ φ X and Y = τ > φ X . F rom the COMP ACTL Y GENERA TED t -STRUCTURES 21 long exact sequence of homology associated to the canonical φ -triangle with cen tral v ertex X we obtain isomorphisms H i − 1 ( Y ) ∼ = H i ( T ) for all i < 0, and a monomorphism of R -mo dules H − 1 ( Y ) ֒ → H 0 ( T ). In particular Supp(H j ( Y )) ⊂ Su pp(H j +1 ( T )) ⊂ φ ( j + 1) ⊂ φ ( j ) for all j ≤ − 1. The explicit d escription of U φ giv en in Th eorem 3.11 shows that τ ≤− 1 Y ∈ U φ . So ha ving in mind that Y ∈ U ⊥ φ the canonical map τ ≤− 1 Y → Y is zero. Th us Y ∈ D ≥ 0 ( R ) and as a consequence T ∈ D ≥ 0 ( R ).  As usual, w e denote b y Ass( M ) the set of asso ciated p rime ideals of a mo dule M ∈ Mo d ( R ) (for the basic pr op erties of asso ciated prime ideals cf. [Mat, § 6, page 38]). Lemma 4.3. L et φ b e a sp-ﬁltr ation of Sp ec ( R ) . L et j b e an inte ger and M b e a ﬁnitely gene r ate d R - mo dule such that Ass( M ) ∩ φ ( j ) = ∅ ( e.g. M = R / p , with p 6∈ φ ( j ) ). L et T a − → M [ − j ] b − → Y + − → b e the c anonic al φ -triangle with c entr al vertex M [ − j ] ∈ D ( R ) . Then: (1) T ∈ D >j ( R ) ; (2) Y ∈ D ≥ j ( R ) and Γ φ ( j ) (H j ( Y )) = 0 ; and (3) the hom omorphism of R -mo dules H j ( b ) : M → H j ( Y ) is an essential extension. Pr o of. Assuming again that j = 0 and rewriting the p r o of of Lemma 4.2 for X = M [0] = M , we get that Y and T b elong to D ≥ 0 ( R ) and that the canonical m ap H 0 ( a ) : H 0 ( T ) → M is a monomorph ism of R -mo dules. Then Ass(H 0 ( T )) ⊂ Ass( M ). The hyp othesis on Ass( M ) implies that Ass(H 0 ( T )) = ∅ , then H 0 ( T ) = 0 and T ∈ D > 0 ( R ). Ha ving in mind that H 0 ( Y )[0] ∼ = τ ≤ 0 Y , we get that Hom R ( N , H 0 ( Y )) = Hom D ( R ) ( N [0] , τ ≤ 0 Y ) ∼ = Hom D ( R ) ( N [0] , Y ) = 0 for eve ry R -mo d ule N such that Supp( N ) ⊂ φ (0); therefore Γ φ (0) (H 0 ( Y )) = 0. Finally , let u s chec k that th e monomorphism ι := H 0 ( b ) : M ֒ → H 0 ( Y ) is essen tial. Consid er the exact sequence 0 → M ι − → H 0 ( Y ) ν − → H 1 ( T ) → 0 of R -mo d ules asso ciated to the φ -triangle with cen tral ve rtex M = M [0]. Let V ⊂ H 0 ( Y ) b e a ﬁnitely generated submo dule such that Im( ι ) ∩ V = 0 . Then Ker( ν ) ∩ V = Im( ι ) ∩ V = 0, so the comp osition V ֒ → H 0 ( Y ) ν − → H 1 ( T ) is also a monomorp hism, hence Sup p( V ) ⊂ Su pp(H 1 ( T )) ⊂ φ (1) ⊂ φ (0). Therefore V = Γ φ (0) ( V ) ⊂ Γ φ (0) (H 0 ( Y )) = 0.  Theorem 4.4. L et φ : Z → P (Sp ec( R )) b e a sp-ﬁltr ation. Supp ose that p ( q is a strict inclusion of prime ide als of R such that p is maximal under 22 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN q . L et T − → R / p [ − j + 1] − → Y + − → b e a φ -triangle in D ( R ) with c entr al vertex R / p [ − j + 1] . If q ∈ φ ( j ) and p 6∈ φ ( j − 1) , then neither T nor Y b elongs to D fg ( R ) . Pr o of. F or simplicit y assu me j = 1, and pu t U = U φ . Su p p ose that one of the ob jects T or Y b elongs to D fg ( R ), then the other b elongs as w ell. Hence T → R / p [0] → Y + → is a triangle in D fg ( R ) with T ∈ U and Y ∈ U ⊥ . By Prop osition 2.9 lo calizing at q , w e get a triangle T q − → R q / p R q [0] − → Y q + − → in D fg ( R q ) , su c h that T q ∈ U q and Y q ∈ U q ⊥ . Moreo v er, from Prop ositi on 2.9 w e kno w that the sp-ﬁltration φ q of Sp ec ( R q ) asso ciated to U q is giv en by φ q ( i ) = φ ( i ) ∩ S p ec( R q ). As a consequence q R q ∈ φ q (1) b u t p R q 6∈ φ q (0). Let us simplify the notation assumin g that R = R q is lo cal, q = m is the maximal ideal of R , and p is maximal u nder m . By Lemma 4.3, un der the presen t hyp othesis T ∈ D > 0 ( R ), Y ∈ D ≥ 0 ( R ), Γ m (H 0 ( Y )) = 0 and the in duced h omomorphism R/ p → H 0 ( Y ) is an es- sen tial extension. Then Su pp(H 0 ( Y )) = V ( p ), and hence S upp(H 1 ( T )) ⊂ V( p ) ∩ φ (1) = { m } . Therefore H 1 ( T ) is a ﬁn itely generated R -mo du le w ith Supp(H 1 ( T )) ⊂ { m } , so one can ﬁnd r ∈ N suc h that m r H 1 ( T ) = 0 . Let us ﬁx an in teger k > 0. Let j : R → R / p b e the canonical h omo- morphism of rings. The r ing A := R/ p is an in tegral lo cal domain of Krull dimension 1 with n := m / p as its m aximal ideal. Not ice th at the canonica l map Ext 1 A ( A/ n k , A ) α − → Ext 1 R ( R/ ( p + m k ) , R/ p ) is in jectiv e. Moreo v er, applying the homologi cal fu nctor Hom D ( R ) ( − , R/ p ) to the short exact sequence of R -mo d ules 0 → ( p + m k ) / m k − → R/ m k − → R/ ( p + m k ) → 0 , w e obtain an exac t sequence 0 − → Ext 1 R ( R/ ( p + m k ) , R/ p ) β − → Ext 1 R ( R/ m k , R/ p ) b ecause Hom R (( p + m k ) / m k , R/ p ) = 0. Therefore w e get a monomorph ism β α : Ext 1 A ( A/ n k , A ) ֒ → Ext 1 R ( R/ m k , R/ p ). No w Ext 1 A ( A/ n k , A ) ∼ = Hom A ( A/ n k , Q ( A ) / A ) , b ecause Q ( A ) = k ( p ), the ﬁ eld of quotien ts of A = R/ p , is the injectiv e hull of A in Mo d ( A ). No te that this hom can b e describ ed as the A -submo du le of Q ( A ) / A of th ose elements ¯ x = x + A ∈ Q ( A ) / A suc h that n k ¯ x = 0. Since for eac h k > 0 one can alw a ys ﬁnd elements ¯ x ∈ Q ( A ) / A such that n k − 1 ¯ x 6 = 0 = n k ¯ x , we conclud e from the existence of the m onomorphism β α that m k − 1 Ext 1 R ( R/ m k , R/ p ) 6 = 0 , ∀ k > 0 (4.4.1) COMP ACTL Y GENERA TED t -STRUCTURES 23 On the other hand, since m ∈ φ (1) (i.e. R/ m [ − 1] ∈ U ), b y Pr op osi- tion 2.4(3 ) we get that R/ m k [ − 1] ∈ U , for all k > 0. The p rop erties of the triangle in the hyp othesis of the theorem giv e us isomorphisms Ext 1 R ( R/ m k , R/ p ) ∼ = Hom D ( R ) ( R/ m k [ − 1] , R/ p ) ∼ = Hom D ( R ) ( R/ m k [ − 1] , T ) But, since T ∈ D > 0 ( R ), w e ha v e that Hom D ( R ) ( R/ m k [ − 1] , T ) ∼ = Hom R ( R/ m k , H 1 ( T )) , so Hom D ( R ) ( R/ m k [ − 1] , T ) is isomorph ic to a su bmo du le of H 1 ( T ). Hence m r Ext 1 R ( R/ m k , R/ p ) = 0, for all k > 0. T his fact con tradicts (4.4. 1).  R emark. A d ualizing complex can b e explicitly realize d as a r esidual complex and determines a co d imension fun ctor (see remark 6.2 f urther on). T he co dimension fu n ction pro vides a sp -ﬁ ltration φ cm : Z → P (Sp ec ( R )) that satisﬁes the follo wing condition: F or an y j ∈ Z , and any pair of prime ideals p ( q , with p maximal under q , then q ∈ φ cm ( j ) if and only if p ∈ φ cm ( j − 1) . W e call this prop erty the str ong Cousin c ondition . F or our purp oses it is con v enien t to consider sp -ﬁltrations under a w eak er version of the ab ov e condition ( cf . Theorem 4.4). This fact justiﬁes the follo wing. Deﬁnition. Let φ : Z → P (Sp ec( R )) be a sp-ﬁltration. W e say that φ satisﬁes the we ak Cousin c ondition if the follo wing prop erty holds: F or ev ery j ∈ Z , if p ( q are prime ideals, with p maximal under q , and q ∈ φ ( j ) then p ∈ φ ( j − 1) . Corollary 4.5. L et ♯ ∈ {− , + , b , “blank” } . If φ is a sp-ﬁltr ation of Sp ec ( R ) such that U φ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) , then φ satisﬁes the we ak Cousin c ondition. Pr o of. Straigh tforw ard consequence of Theorem 4.4.  Our next goal is to see whether the con ve rse of the statement in Corol- lary 4.5 is also true. F or that w e stud y the sp-ﬁltrations satisfying the w eak Cousin condition. 4.6. Recall that for t w o prime id eals p , q ∈ Sp ec ( R ) the relation p ⊂ q can b e expressed sayi ng that p is a generalizatio n of q or, equ iv alen tly , q is a sp ecialization of p . A su b set Y ⊂ Sp ec( R ) is stable u nder generalizat ion if p ∈ Y wh enev er p ⊂ q with q ∈ Y . F or instance if q ∈ Sp ec( R ) w e identify Sp ec( R q ) w ith the subset of all generalizations of q in S p ec( R ). Under the assump tion th at R is a No etherian r ing, a su bset Y ⊂ Sp ec ( R ) is stable und er sp ecialization (sp-su b set) and generalizatio n if and only if Y is op en and closed, equiv alen tly Y is the union of connecte d comp onents of Sp ec ( R ). In deed, let Y ⊂ Sp ec ( R ) b e stable un der sp ecialization and generalizat ion. If p ∈ Sp ec( R ) is a minimal prim e ideal su c h that V ( p ) ∩ Y 6 = 24 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN ∅ necessarily p ∈ Y b ecause Y is stable under generalizati on; thus V( p ) ⊂ Y since Y is also stable un der sp ecializatio n. Let Min( R ) = { p 1 , . . . , p s } b e the set of minimal prime ideals of R order in suc h a wa y that Min ( R ) ∩ Y = { p 1 , . . . , p r } , for an in teger r ≤ s . Then Y = ∪ r i =1 V( p i ) so it is closed, and it is op en b ecause Sp ec ( R ) \ Y = ∪ s i = r +1 V( p i ). Prop osition 4 .7. L et φ : Z → P (Sp ec( R )) b e a sp-ﬁltr ation that satisﬁes the we ak Cousin c ondition. Then ther e exists an inte g er j 0 ∈ Z such that φ ( j ) = φ ( j 0 ) for al l j ≤ j 0 , and the subset φ ( j 0 ) ⊂ Sp ec( R ) i s op en and close d. A lso, the set T i ∈ Z φ ( i ) is op en and close d. Pr o of. The class of sp-subsets of Sp ec( R ) is closed u nder taking arb itrary unions and inte rsections, f rom which we get that T i ∈ Z φ ( i ) and S i ∈ Z φ ( i ) are sp-su b sets. F urthermore the wea k Cousin condition implies that the sets T i ∈ Z φ ( i ) and S i ∈ Z φ ( i ) are b oth also stable under generalization, so they are at once op en and closed in S p ec( R ) (see 4.6). T h e set of minimal prime ideals of Y = S i ∈ Z φ ( i ) is ﬁnite, so w e can ﬁnd a small en ough intege r j 0 suc h that φ ( j 0 ) con tains all m inimal prime id eals of Y s ince the sp -ﬁltration is decreasing. Then φ ( j 0 ) = Y , and φ ( j 0 ) = φ ( j ) for all j ≤ j 0 .  Corollary 4.8. If Sp ec ( R ) is c onne cte d and φ is not one of the two trivial c onstant sp-ﬁltr ations, then the fol low ing assertions hold: (1) The sp-ﬁltr ation φ is separated , i.e. T i ∈ Z φ ( i ) = ∅ . (2) Ther e exists an inte ger j 0 ∈ Z such that φ ( j 0 ) = Sp ec( R ) . (3) If R has ﬁnite Krul l dimension, then ther e exists a lar ge enough k ∈ Z such that φ ( k ) = ∅ . Pr o of. Being Sp ec( R ) connected and φ not one of the tw o trivial constan t sp-ﬁltrations it follo ws that T i ∈ Z φ ( i ) = ∅ and S i ∈ Z φ ( i ) = φ ( j 0 ) = Sp ec( R ) with j 0 as in the pr evious Prop osition. T o p ro v e (3), d enote by d the Kru ll dimension of R and b y Min( R ) the set of min im al prime id eals of Sp ec( R ). By (1) w e can asso ciate to eac h maximal m ⊂ R an in teger i m = max { i ∈ Z ; m ∈ φ ( i ) } . Let us ﬁx a maximal ideal m ⊂ R and tak e a maximal c hain of pr ime id eals p 0 ( p 1 ( · · · ( p r = m (then r ≤ d ). The wea k Cousin condition for φ implies that the minimal prime ideal p 0 b elongs to φ ( i m − r ) ⊂ φ ( i m − d ). I n particular, w e hav e Min( R ) ∩ φ ( i m − d ) 6 = ∅ , whic h implies that i m − d ≤ m := max { i ∈ Z ; Min( R ) ∩ φ ( i ) 6 = ∅ } . Then i m ≤ d + m , for ev ery maximal m ⊂ R . So φ ( d + m + 1) do es not con tain an y maximal id eal, wh ich means that φ ( d + m + 1) = ∅ .  R emark. F or a general s p-ﬁltration φ the result in Lemma 4.2 sh o ws th at U φ ∩ D + ( R ) is an aisle of D + ( R ). I f fu rthermore there is an in tege r k suc h that φ ( k ) = ∅ then U φ ⊂ D φ D − ( R ) ⊂ D − ( R )). In this case U φ ∩ D ♯ ( R ) is an aisle of D ♯ ( R ) for all ♯ ∈ {− , + , b } . As a consequence of the ab o v e Pr op osition w e get the follo wing. COMP ACTL Y GENERA TED t -STRUCTURES 25 Corollary 4.9. Assume that S p ec( R ) is c onne cte d, R has ﬁnite Krul l di- mension, and that φ i s a non-c onsta nt sp-ﬁltr ation of Sp ec ( R ) satisfying the we ak Cousin c ondition. F or any sup erscript ♯ ∈ {− , + , b } , U φ ∩ D ♯ ( R ) is an aisle of D ♯ ( R ) . Mor e over, ther e e xist inte gers j ≤ k for which the c anonic al maps τ j X induc e isomorph isms τ ≤ φ τ φ X ˜ → τ > φ τ >j X , for al l X ∈ D ( R ) . Pr o of. By Corollary 4.8 th ere exist in tegers j ≤ k su c h that φ ( j ) = Sp ec( R ) and φ ( k ) = ∅ . So the r emark preceding this corollary sh o ws that U φ ∩ D ♯ ( R ) is an aisle of D ♯ ( R ). The canonical homomorphism X → τ >j X induces natural isomorphisms Hom D ( R ) ( τ >j X, Y ) ˜ → Hom D ( R ) ( X, Y ) for all Y ∈ U ⊥ φ b ecause D ≤ j ( R ) ⊂ U φ . As a consequence of the natural adjunction isomorphism s Hom D ( R ) ( N , Y ) ˜ → Hom D ( R ) ( τ > φ N , Y ) for an y N ∈ D ( R ) and Y ∈ U ⊥ φ , w e ge t that the natural map τ > φ X → τ > φ τ >j X is a n isomorphism. F ollo wing a d ual p ath, note that the canonical map τ φ X ∈ D ♯ fg ( R )) for all X ∈ D ♯ fg ( R ). First let us sho w that ( − ) ⇒ ( b ). If X ∈ D b fg ( R ) then the φ -triangle with cen tral term X is in D + ( R ) (see Lemma 4.2). By assumption ( − ) w e also ha v e that it la ys on D − fg ( R ), hence on D b fg ( R ). In the rest of th e pro of we supp ose in addition th at R has ﬁn ite Krull dimension. Then Corollary 4.9 applies here. Let j ≤ k b e the in tegers in 26 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN the statemen t of Corollary 4.9. In ord er to pro v e ( b ) ⇒ (+) let us tak e an y complex X ∈ D + fg ( R ); then w e get that τ ≤ φ X ∼ = τ ≤ φ τ φ X ∼ = τ > φ τ >j X by Corollary 4.9; then τ > φ X ∼ = τ > φ τ >j X ∈ D + fg ( R ) ⊂ D fg ( R ) b y (+). Finally (“blank”) ⇒ ( − ) is a consequence of the f act that U φ ⊂ D ≤ k ( R ).  Corollary 4.11. Over a c ommuta tive No etherian ring of ﬁnite K rul l di- mension the pr oblems of classifying t- structur es on D − fg ( R ) and D b fg ( R ) ar e e qui valent. They ar e also e qu ivalent to classifying on D fg ( R ) and D + fg ( R ) al l t-structur es gener ate d by p erfe ct c omplexes (or, by b ounde d ab ove c om- plexes). Pr o of. It follo ws from C orollary 3.12 and C orollary 4.10.  4.12. T he sp-ﬁ ltrations of Sp ec ( R ) corresp ondin g to Bousﬁ eld localizations on D ( R ) are exactly the constan t sp-ﬁltrations. If φ is a constant sp-ﬁltration inducing a Bousﬁeld lo calization on D ♯ fg ( R ) then φ satisﬁes wea k Cousin condition (for an y ♯ ∈ {− , + , b , “blank” } ). By Prop osition 4.7, there exists a subset Z ⊂ Sp ec( R ) op en and closed such that φ ( j ) = Z for all j ∈ Z . Hence Z = V( e ) for an idemp oten t elemen t e ∈ R . F or eac h complex X ∈ D ( R ) the Bousﬁeld triangle associated to Z is R Γ V( e ) X − → X − → X e + − → Note that X ∼ = R Γ V( e ) ( X ) ⊕ X e , b ecause the third m ap in the triangle is zero. So trivially the Bousﬁeld lo calization ( U φ , U ⊥ φ [1]) restricts to a Bous- ﬁeld lo cal ization on D ♯ fg ( R ). These are all the Bousﬁeld localizatio ns on D ♯ fg ( R ). In particular, if Sp ec( R ) is connected then th e Bousﬁeld lo caliza- tions generated by b oun ded ab o ve complexes (equiv ale ntl y , the B ousﬁeld lo calizati ons generated b y p erfect complexes) on D ♯ fg ( R ) are just the trivial ones. 5. Aisles d etermined by finite fil tra tions by sup por ts In this section we set the stage to sho w that, und er suﬃciently general h yp othesis, the con v erse of the statemen t in C orollary 4.5 is true. 5.1. If R h as ﬁn ite K rull dimens ion and Sp ec( R ) is connected, for eac h nonconstan t sp-ﬁltration φ : Z → P (Sp ec( R )) satisfying the wea k Cousin condition there exist intege rs j 0 ≤ k suc h that φ ( j 0 ) = Sp ec( R ) and φ ( k ) = ∅ ( cf. Corolla ry 4.8). F or a general c omm utativ e Noetherian ring R we in tro duce the follo wing deﬁnition. Deﬁnition. Let φ : Z → P (Sp ec( R )) b e a sp -ﬁltration. Let s ≤ n t w o in tegers. W e sa y th at φ is determine d in the interval [ s, n ] if φ ( j ) = φ ( s ) for all j ≤ s , φ ( s ) ) φ ( s + 1) and φ ( n ) ) φ ( n + 1) = ∅ . COMP ACTL Y GENERA TED t -STRUCTURES 27 If the sp-ﬁltration φ : Z → P (Sp ec( R )) is d etermined in the in terv al [ s, n ] ⊂ Z we sa y th at φ is ﬁnite of length L( φ ) := n − s + 1. F or any ﬁnite sp -ﬁltration φ w e h a v e that L( φ ) ≥ 1 . R emark. Although a constan t sp-ﬁ ltration is nev er determined in an interv al in the ab o v e sense, we adopt the conv entio n that a constan t sp-ﬁ ltration φ of Sp ec( R ) is ﬁnite of length L( φ ) := 0. The asso ciated aisle to a constant sp-ﬁltration is a lo calizing class corresp ond ing to a ⊗ -compatible Bousﬁeld lo calizati on ( cf. § 1.6). 5.2. Let φ : Z → P (Sp ec( R )) b e a ﬁnite sp -ﬁltration of length 1, that is a sp-ﬁltration su c h that U φ = U i Z for a ﬁxed i ∈ Z and Z = φ ( i ). Corollary 1.9 sho ws that its asso cia ted left truncation fu nctor τ ≤ φ is τ ≤ i R Γ φ ( i ) . Note that for eac h complex X ∈ D ( R ), τ > φ X is determined by the existence of a distinguished triangle in D ( R ) τ >i R Γ Z X − → τ > φ X − → R Q Z X + − → built as follo ws. The natural m ap π : τ ≤ φ X → X is th e comp ositi on of the canonical maps α : τ ≤ i R Γ Z X → R Γ Z X and ρ : R Γ Z X → X. App lying the o c- tahedron axiom to the comm utativ e diagram π = ρ ◦ α X R Q Z X τ ≤ i R Γ Z X τ >i R Γ Z X R Γ Z X τ > φ X α + ρ + + + π w e get the v ertex of the triangle with base π : τ ≤ φ X → X inserted in the triangle we are looking for. 5.3. Our next task is the d escrip tion of the tru ncation functors associated to an y ﬁn ite sp-ﬁltration, see Pr op osition 5.5 and Corollary 5.6. These results are used in the pr o of of Lemma 5.7 that let us establish an ind uctiv e wa y to ac hiev e the classiﬁcat ion of aisles of D ♯ fg ( R ) in the last section of the p ap er. In § 5.2 we hav e describ ed the tr uncation functors asso ciated to any ﬁn ite sp-ﬁltration of length 1. Let φ : Z → P (Sp ec( R )) b e any sp -ﬁltration. F or eac h inte ger i ∈ Z let us den ote by φ i : Z → P (Sp ec( R )) the sp-ﬁltration determined by U φ i := U i φ ( i ) . The family { φ i ; i ∈ Z } o f sp-ﬁltrations of length 1 determines φ . Starting from a ﬁnite sp -ﬁltration φ : Z → P (Sp ec( R )) determined in the in terv al [ s, n ] ⊂ Z , w e construct a ﬁn ite sp-ﬁltration φ ′ : Z → P (S p ec( R )) of length L( φ ′ ) ≤ L( φ ) b y setting φ ′ ( n ) = ∅ and φ ′ ( j ) = φ ( j ), for all j ≤ n − 1 28 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN (that is, φ ′ j = φ j for an y j ≤ n − 1). If L( φ ) > 1 , then φ ′ is a ﬁ nite sp- ﬁltration of length 1 ≤ L( φ ′ ) = L( φ ) − 1. Note that if φ satisﬁes the wea k Cousin condition then so do es φ ′ . Lemma 5.4. L et us ﬁx two inte gers i ≤ j and Z j , Z i two sp-subsets of Sp ec( R ) . L et φ i and φ j b e the sp-ﬁltr ation of Sp ec( R ) of length 1 determine d by U φ k := U k Z k with k ∈ { i, j } . If M ∈ U φ i ⊥ then τ > φ j M ∈ U φ i ⊥ . Pr o of. By 5.2 there is a distinguish ed triangle in D ( R ) τ >j R Γ Z j M − → τ > φ j M − → R Q Z j M + − → First note that R Γ Z i τ >j R Γ Z j M b elongs to D >j ( R ) and as a consequence τ ≤ φ i τ >j R Γ Z j M = τ ≤ i R Γ Z i τ >j R Γ Z j M = 0 , that is, τ >j R Γ Z j M ∈ U φ i ⊥ . Let us chec k that R Q Z j M ∈ U φ i ⊥ . This follo w s from th e canonical isomorph isms τ ≤ φ i R Q Z j M = τ ≤ i R Γ Z i R Q Z j M ∼ = τ ≤ i R Q Z j R Γ Z i M . Indeed, M ∈ U φ i ⊥ therefore R Γ Z i M ∈ D >i ( R ) , hence R Q Z j R Γ Z i M ∈ D >i ( R ), th us τ ≤ φ i R Q Z j M ∼ = τ ≤ i R Q Z j R Γ Z i M = 0  Prop osition 5 .5. L et φ : Z → P (Sp ec( R )) b e a ﬁnite sp-ﬁltr atio n deter- mine d in the interval [ s, n ] ⊂ Z . Then τ ≤ φ X ∈ D ≤ n ( R ) for al l X ∈ D ( R ) . F urthermor e, u sing the notation in 5.3: (1) The right trunc ation functor τ > φ is c ompute d as the c omp osition τ > φ n τ > φ n − 1 · · · τ > φ s . (2) F or al l X ∈ D ( R ) , τ ≤ φ n τ > φ ′ X ∈ D [ n,n ] ( R ) . Pr o of. F or simplicit y we assume in th e pro of that [ s , n ] is the in terv al [0 , n ]. Let us set Z i := φ ( i ) for eac h i ∈ [0 , n ]. T r ivially τ ≤ φ X ∈ D ≤ n ( R ). In order to p ro v e the statemen t (1) we pro ceed by induction on the length of the sp-ﬁltration φ . The case L( φ ) = n + 1 = 1 is trivial. Let n > 0, then 1 ≤ L( φ ′ ) < L( φ ). By in ductiv e hyp othesis the resu lt is tru e for φ ′ . Let us consider the comm utativ e diagram of distinguished triangles X τ > φ n τ > φ ′ X τ ≤ φ ′ X τ ≤ φ n τ > φ ′ X N τ > φ ′ X + w + v u + + COMP ACTL Y GENERA TED t -STRUCTURES 29 built up from the comm utativ e diagram w = v ◦ u and the o ctahedron axiom. The triangle τ ≤ φ ′ X − → N − → τ ≤ φ n τ > φ ′ X + − → pro v es that N ∈ U φ b ecause τ ≤ φ ′ X ∈ U φ ′ ⊂ U φ and τ ≤ φ n τ > φ ′ X ∈ U φ n ⊂ U φ . Note that U φ ′ ⊥ = ∩ n − 1 i =0 U φ i ⊥ and U φ ⊥ = U φ ′ ⊥ ∩ U φ n ⊥ , therefore Lemma 5.4 sho ws that τ > φ n τ > φ ′ X b elongs to U ⊥ φ . Hence the horizonta l triangle in the ab o v e diagram is the φ -triangle with central v ertex X , that is τ ≤ φ X = N and τ > φ X = τ > φ n τ > φ ′ X . The n ext qu estion w e addr ess is to sho w that τ ≤ φ n τ > φ ′ X = τ ≤ n R Γ φ ( n ) τ > φ ′ X b elongs to D [ n,n ] ( R ) or, equiv alen tly , that R Γ Z n τ > φ ′ X ∈ D >n − 1 ( R ). T o pro ve it w e b egin by recalling some usefu l r esu lts. Let Γ Z n − 1 /Z n : Mod ( R ) → Mo d ( R ) b e the f u nctor d etermined b y the short exact sequence of fu nctors in Mo d ( R ) 0 − → Γ Z n − → Γ Z n − 1 − → Γ Z n − 1 /Z n − → 0 . ( cf. [H, v ariatio n 2 on p . 219]). W rite Γ n − 1 /n = Γ Z n − 1 /Z n , Γ n = Γ Z n for eac h n ∈ Z . Deriving these functors on the righ t the ab o ve ab elian exact sequence pr o vide a distinguished triangle for an y Y ∈ D ( R ) R Γ n Y − → R Γ n − 1 Y − → R Γ n − 1 /n Y + − → (5.5.1) Applying the functor R Γ n to this tr iangle we get R Γ n R Γ n − 1 /n Y = 0 s ince R Γ n R Γ n − 1 Y → R Γ n R Γ n Y = R Γ n Y is an isomorphism ( cf. § 1.7). F urther- more the natural transformation R Γ n − 1 → 1 indu ces m orp hisms of distin- guished triangles R Γ n R Γ n − 1 Y R Γ n − 1 R Γ n − 1 Y R Γ n − 1 /n R Γ n − 1 Y R Γ n Y R Γ n − 1 Y R Γ n − 1 /n Y R Γ n − 1 R Γ n Y R Γ n − 1 R Γ n − 1 Y R Γ n − 1 R Γ n − 1 /n Y + + + ≀ ≀ ≀ ≀ that are in fact isomorphisms of triangles b ecause the t w o vertic al maps on the left are isomorph isms (see § 1.7). Therefore R Γ n − 1 R Γ n − 1 /n Y ∼ = R Γ n − 1 /n R Γ n − 1 Y ∼ = R Γ n − 1 /n Y for all Y ∈ D ( R ). Going bac k to our aim, set Y := τ > φ ′ X . Note that Y ∈ ( U φ n − 1 ) ⊥ , so that τ ≤ n − 1 R Γ n − 1 Y = 0, that is, R Γ n − 1 Y ∈ D >n − 1 ( R ). But then R Γ n − 1 /n Y ∼ = R Γ n − 1 /n R Γ n − 1 Y also b elo ngs to D >n − 1 ( R ). No w the existence of the tri- angle (5.5 .1) allo ws us to conclude that R Γ n τ > φ ′ X = R Γ n Y ∈ D >n − 1 ( R ), as desired.  30 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN Corollary 5.6. L et us c onsider the notation in the ab ove pr op osition. Then for e ach X ∈ D ( R ) it holds that τ ≤ φ n τ > φ ′ X ∼ = H n ( τ ≤ φ X )[ − n ] and ther e is a diagr am of distinguishe d triangles in D ( R ) X τ > φ X τ ≤ φ ′ X τ ≤ φ n τ > φ ′ X τ ≤ φ X τ > φ ′ X + w + v u + + in which: (1) the triangle τ ≤ φ n τ > φ ′ X − → τ > φ ′ X − → τ > φ X + − → i s c anonic al ly i somor- phic to τ ≤ φ τ > φ ′ X − → τ > φ ′ X − → τ > φ τ > φ ′ X + − → ; and (2) the triangle τ ≤ φ ′ X − → τ ≤ φ X − → τ ≤ φ n τ > φ ′ X + − → i s c anonic al ly i somor- phic to τ ≤ n − 1 τ ≤ φ X − → τ ≤ φ X − → τ >n − 1 τ ≤ φ X + − → . Pr o of. The diagram wh ose existence w e assert is the diagram of d istin- guished triangles at th e b eginning of the pro of of Prop ositi on 5.5. F rom the ve ry same pro of note that τ ≤ φ n τ > φ ′ X ∈ U φ n ⊂ U φ hence the triangle τ ≤ φ n τ > φ ′ X − → τ > φ ′ X − → τ > φ X + − → is the φ -triangle with cen tral v ertex τ > φ ′ X , so assertion (1) follo ws. W e also d eriv e fr om Prop osition 5.5 that (2) holds true, since τ ≤ φ ′ X ∈ D ≤ n − 1 ( R ) and τ ≤ φ n τ > φ ′ X ∈ D [ n,n ] ( R ) ⊂ D >n − 1 ( R ). And also as a consequence τ ≤ φ n τ > φ ′ X ∼ = H n ( τ ≤ φ X )[ − n ].  Lemma 5.7. L et ♯ ∈ {− , + , b , “blank” } and let φ b e a ﬁnite sp-ﬁltr ation of Sp ec( R ) determine d in the interval [ s, n ] ⊂ Z . The fol lowing statements ar e e qui valent: (1) U φ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) . (2) U φ ′ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) and H n ( R Γ φ ( n ) M ) is a ﬁnitely gener ate d R - mo dule, for every M ∈ U ⊥ φ ′ ∩ D ♯ fg ( R ) . (3) U φ ′ ∩ D ♯ fg ( R ) is an aisle of D ♯ fg ( R ) and τ ≤ n ( R Γ φ ( n ) M ) ∈ D ♯ fg ( R ) , for every M ∈ U ⊥ φ ′ ∩ D ♯ fg ( R ) . Pr o of. As a consequence of the remark after Corollary 4.8, it is enough to pro v e the current Lemma f or ♯ = “blank”. COMP ACTL Y GENERA TED t -STRUCTURES 31 T ak e M ∈ U ⊥ φ ′ ∩ D fg ( R ) so the canonical map M → τ > φ ′ M is an isomor- phism. Then τ ≤ φ M ∼ = τ ≤ φ n M by Corollary 5.6(1). F rom the initial state- men t of that same corollary we get that τ ≤ φ M ∼ = τ ≤ φ n M = τ ≤ n R Γ φ ( n ) M ∼ = H n ( R Γ φ ( n ) M )[ − n ] . This p r o v es (2) ⇔ (3). And also pr ov es that τ ≤ φ M ∈ D fg ( R ) (equiv alen tly τ > φ M ∈ D fg ( R )) if and only if H n ( R Γ φ ( n ) M ) is a ﬁnitely generated R -mo dule. This said, to pro v e (1) ⇒ (2) we just n eed to c hec k that U φ ′ ∩ D fg ( R ) is an aisle of D fg ( R ). But it follo ws from the f act that, for ev ery X ∈ D fg ( R ), τ ≤ φ ′ X ∼ = τ ≤ n − 1 τ ≤ φ X , see Corollary 5.6(2). Finally let us sho w (2) ⇒ (1). Let X ∈ D fg ( R ), assuming (2) w e ha v e that b oth M = τ > φ ′ X and τ ≤ φ n τ > φ ′ X = H n ( R Γ φ ( n ) M )[ − n ] (see Pr op osition 5.5(2)) b elong to D fg ( R ) and we conclude by the triangle in Corollary 5.6(1).  Prop osition 5.8. L et φ b e a sp-ﬁltr ation satisfying the we ak Cousin c ondi- tion and such that L( φ p ) ≤ 2 for al l prime ide al p ∈ Sp ec( R ) (e quivalently for al l maximal ide al p ∈ Sp ec ( R ) ), then U φ ∩ D fg ( R ) is an aisle of D fg ( R ) . Pr o of. The question is lo cal so w e can assume that R is lo cal (hence Sp ec ( R ) is connected and of ﬁn ite Krull dimen s ion). T hen φ is a ﬁnite s p -ﬁltration of length ≤ 2 w hic h, without loss of generalit y , we assume φ nonconstan t and concen trated in the in terv al [0 , n ]. If L( φ ) = 1, that is n = 0, then U φ ∩ D fg ( R ) is trivially an aisle since in the p resen t setting th e condition L( φ ) = 1 is equiv alen t to s aying that U φ = D ≤ 0 ( R ). If L( φ ) = 2, that is n = 1, then U φ ′ ∩ D fg ( R ) = D ≤ 0 fg ( R ) is the aisle of the canonica l t -stru cture on D fg ( R ). O n the other hand , if M ∈ U φ ′ ⊥ ∩ D fg ( R ) = D > 0 ( R ) ∩ D fg ( R ), w e ha v e that H 1 ( R Γ φ (1) M ) ∼ = Γ φ (1) (H 1 ( M )), whic h is ﬁnitely generated b eca use it is a su bmo du le of the ﬁnitely generated mo du le H 1 ( M ). Hence U φ ∩ D fg ( R ) is an aisle of D fg ( R ) by Lemma 5.7.  6. The c lassifica t ion over rings with dua l izing co m plex W e b egin this secti on by recalling brieﬂy some basic results on d ualizing complexes fr om [H, Chapter V § 2] in our con text. 6.1. A c omplex X ∈ D ( R ) is r eﬂexive with r esp e ct to D ∈ D ( R ) if the natural morp hism σ X : X − → R Hom · R ( R Hom · R ( X, D ) , D ) is an isomorphism in D ( R ). Let D ∈ D b fg ( R ) b e a complex quasi-isomorphic to a b ounded complex of injectiv e R -mo dules. Then follo wing assertions are equiv alen t [H, Ch apter V § 2 Prop osition 2.1 ]: (1) Th e co nt ra v arian t functor R Hom · R ( − , D ) : D fg ( R ) → D fg ( R ) is a triangulated dualit y qu asi-in v erse of itself. 32 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN (2) Th e co nt ra v arian t functor R Hom · R ( − , D ) : D b fg ( R ) → D b fg ( R ) is a triangulated dualit y qu asi-in v erse of itself. (3) Every ﬁn itely generated R -mo du le is r eﬂexiv e w ith resp ect to D . (4) Th e stalk complex R [0] is reﬂexiv e with resp ect to D . A complex D ∈ D b fg ( R ) qu asi-isomorphic to a b oun ded complex of in- jectiv e R -mo d ules that satisﬁes th e ab o v e equiv alent cond itions is called a dualizing c omplex for R . More generally , D ∈ D + fg ( R ) is called a p ointwise dualizing c omp lex for R in case D p is a dualizing complex o v er R p , f or ev ery p ∈ Sp ec( R ). If R p ossesses a dualizing co mplex then R h as ﬁnite Kru ll dimen s ion ( cf. [H, Chapter V, Corollary 7.2, p. 283 ]). F urtherm ore, D is a dualizing complex for R if and only if D is a p oint wise dualizing complex and the Krull dimen s ion of R is ﬁnite. 6.2. Let D ∈ D b fg ( R ) b e a complex. As w e easily derive from [H, Ch ap ter V, Prop osition 3.4, p. 269], D is a p oint wise du alizing complex if, and only if, for eac h p ∈ Sp ec( R ) there is a unique i p ∈ Z suc h that Hom D ( R p ) ( k ( p ) , D p [ j ]) = ( 0 , if j 6 = i p , k ( p ) , if j = i p . In that case we deﬁne a map d : S p ec( R ) → Z by setting d ( p ) = i p , for all p ∈ Sp ec( R ). Observe that the map d : Sp ec( R ) → Z ob eys the rule: d ( p ) = i ⇐ ⇒ [ Hom D ( R p ) ( k ( p ) , D p [ j ]) = 0, ∀ j ∈ Z suc h that j 6 = i ] Moreo v er d : Sp ec( R ) − → Z is a co d imension fun ction, that is, if p ( q and h t( q / p ) = 1 then d ( q ) = d ( p ) + 1 [H, Chapter V, § 7 Prop osition 7.1]. The follo wing Lemma give s a usefu l c haracteriza tion of d : Sp ec( R ) → Z . Lemma 6.3. If D ∈ D ( R ) is a dualizing c omplex, then d ( p ) = max { n ∈ Z ; R Γ V( p ) D ∈ D ≥ n ( R ) } for e ach p ∈ Sp ec( R ) . Pr o of. First note that for ev ery p ∈ Sp ec ( R ) and j ∈ Z the supp ort of the R -mo du le Hom D ( R ) ( R/ p , D [ j ]) is con tained in V( p ). Moreo v er Hom D ( R ) ( R/ p , D [ j ]) p ∼ = Hom D ( R p ) ( k ( p ) , D p [ j ]) b ecause D is b ounded b elo w. Th e r esu lt mentioned in 6.2 guarantie s that 0 6 = ( R Hom · R ( R/ p , D )) p ∈ D [ d ( p ) , d ( p ) ] ( R p ). As a consequence ( R Γ V( p ) D ) p ∼ = R Γ p R p D p b elongs to D ≥ d ( p ) ( R p ) and do es not b elong to D > d ( p ) ( R p ). Whence H d ( p ) ( R Γ V( p ) D ) 6 = 0 and therefore max { n ∈ Z ; R Γ V( p ) D ∈ D ≥ n ( R ) } ≤ d ( p ) , for all p ∈ Sp ec ( R ) . Let T b e the s et of prime ideals in Sp ec( R ) for which the desired equalit y do es not hold, i.e. T = { p ∈ S p ec( R ) ; R Γ V( p ) D / ∈ D ≥ d ( p ) ( R ) } . Assum e that T is non empt y and c ho ose a prim e ideal p ∈ T maximal among the pr ime COMP ACTL Y GENERA TED t -STRUCTURES 33 ideals in T (recall that R is No etherian). T h en f or all q ∈ S p ec( R ) such that p ( q it holds that d ( q ) = max { n ∈ Z ; R Γ V( q ) D ∈ D ≥ n ( R ) } Let W p := { q ; q ∈ S p ec( R ) and q * p } = Sp ec ( R ) \ Sp ec( R p ). Let us consider the canonical Bousﬁeld triangle determined by W p for R Γ V( p ) D R Γ W p R Γ V( p ) D − → R Γ V( p ) D u − → ( R Γ V( p ) D ) p + − → . By the remark in th e pr evious paragraph ( R Γ V( p ) D ) p is in D ≥ d ( p ) ( R ) and do es not b elong to D > d ( p ) ( R ). Let us prov e that th e left v ertex in the ab ov e triangle is in D ≥ d ( p )+1 ( R ). Let W ′ p b e the set of prim e ideals W ′ p := W p ∩ V( p ) = V( p ) \ { p } . Th en usin g the canonical isomorp h ism R Γ W p R Γ V( p ) ∼ = R Γ W ′ p ( cf. § 1.7) we deduce that R Γ W p R Γ V( p ) D ∼ = R Γ W ′ p D ∈ D ≥ d ( p )+1 ( R ), b ecause R Γ V( q ) D ∈ D ≥ d ( q ) ( R ) ⊂ D ≥ d ( p )+1 ( R ) , for all q ∈ W ′ p (see C orollary 2.5). F rom the ab o v e Bousﬁ eld triangle w e conclude that R Γ V( p ) D ∈ D ≥ d ( p ) ( R ) agai nst the fact that p ∈ T .  6.4. Let D ∈ D b fg ( R ) b e a dualizing complex for R and d : S p ec( R ) → Z its asso ciated co dim en sion function. The du alit y fun ctor R Hom · R ( − , D ) : D b fg ( R ) − → D b fg ( R ) transform s the canonical t -structure on D b fg ( R ) on to a t -structure on D b fg ( R ). W e call this t -structure the Cohen-M ac aulay t -structur e on D b fg ( R ) with resp ect to D , b ecause it can be prov ed that the ob jects in its heart are precisely th e Cohen-Macaula y complexes in th e s ense of [H, p p. 238-239] . By Corollary 3.12 there exists a un ique sp-ﬁltration of S p ec( R ) asso ciated to the C ohen-Macaula y t -structure on D b fg ( R ) (with resp ect to D ). W e denote th is ﬁltration b y φ cm : Z − → P (Sp ec( R )) and we name it the Coh en-Mac aulay ﬁltr ation (with resp ect to D ). T rivially the ﬁ ltration φ cm satisﬁes the w eak Cousin condition, actually as a consequence of Prop ositio n 6.5 right b elo w, the ﬁ ltration φ cm do es satisfy the strong Cousin condition ( cf. the remark after Theorem 4.4) b ecause d is a co dimension function. Prop osition 6.5. L e t us c onsider the hyp oth esis and notatio n in the ab ove p ar agr aph . The Cohen-Mac aulay ﬁltr ation φ cm attaches to e ach i ∈ Z the set φ cm ( i ) = { p ∈ S p ec( R ) ; d ( p ) > i } . Pr o of. Let ( V , E [1]) b e the t -structure on D b fg ( R ) image by the dualit y func- tor R Hom · D ( R ) ( − , D ) of the canonical t -structure on D b fg ( R ). Th e class V 34 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN consists of those complexes X ∈ D b fg ( R ) suc h that 0 = Hom D ( R ) ( X, R Hom · R ( N , D )) for all N ∈ D ≤ 0 ( R ) ∩ D b fg ( R ). The canonical aisle D ≤ 0 fg ( R ) is generated by the stalk complex R , therefore a complex X ∈ D b fg ( R ) is in V if and only if 0 = Hom D ( R ) ( X, R Hom · R ( R [ i ] , D )) ∼ = Hom D ( R ) ( X [ i ] , D ) (6.5.1) for all i ≥ 0. T hen the ﬁltration φ cm is deﬁned, for eac h i ∈ Z , by φ cm ( i ) = { p ∈ S p ec( R ) ; Hom D ( R ) ( R/ p [ j ] , D ) = 0 , for all j ≥ − i } , a formula that can b e r ewr itten as φ cm ( i ) = { p ∈ S p ec( R ) ; R Γ V( p ) D ∈ D >i ( R ) } . (see Corollary 1.9 and Prop osition 2.7). Then w e get from Lemma 6.3 that φ cm ( i ) = { p ∈ S p ec( R ) ; d ( p ) > i } , for all i ∈ Z .  6.6. Given a total pre-aisle V of D b fg ( R ) and E its righ t orthogonal in D b fg ( R ) there is a u nique sp-ﬁltration φ ∈ Fi l sp ( R ) such that V = U φ ∩ D b fg ( R ) and E = F φ ∩ D b fg ( R ), w h ere F φ is the r igh t o rthogonal of U φ in D ( R ) (see Corollary 3.12). Assum e that R admits a dualizing complex D with co dimension function d : Sp ec( R ) → Z . Then the image b y the dualit y functor R Hom · R ( − , D ) of the class E is a total pre-aisle of D b fg ( R ) that w e denote by E d . The right orthogonal of E d in D b fg ( R ) is the image of V by th e dualit y functor R Hom · R ( − , D ), that w e denote by V d . Th erefore th er e exist a u n ique sp -ﬁltration φ d ∈ Fi l sp ( R ), that w e call the dual of φ (with resp ect to D ), suc h that E d = U φ d ∩ D b fg ( R ) and V d = F φ d ∩ D b fg ( R ). Recall that X ∈ E = F φ ∩ D b fg ( R ) if and only if Hom D ( R ) ( R/ p [ − j ] , X ) = 0, for all j ∈ Z and p ∈ φ ( j ). Then it follo w s from dualit y that X ∈ E if and only if Hom D ( R ) ( R Hom · R ( X, D ) , R Hom · R ( R/ p [ − j ] , D )) = 0 , for any j ∈ Z and p ∈ φ ( j ). That is, E d = U φ d ∩ D b fg ( R ) is the left orthogonal in D b fg ( R ) to the set of ob jects Y = { R Hom · R ( R/ p [ − j ] , D ) ; j ∈ Z , p ∈ φ ( j ) } . Therefore the dual of φ is the sp-ﬁltration d eﬁned by φ d ( k ) = { q ∈ Sp ec( R ) ; Hom D ( R ) ( R/ q [ − k ] , Y ) = 0 for all Y ∈ Y } for eac h k ∈ Z . Lemma 6.7. L e t R b e a ring that admits a dualizing c omplex D with φ cm : Z → P (S p ec( R )) as its asso ciate d Cohen-Mac aulay ﬁltr ation, and let Z ⊂ Sp ec( R ) b e a sp-subset. F or a c omplex X ∈ D b fg ( R ) and n ∈ Z , the fol lowing assertion s ar e e quiv alent: (1) R Γ Z X b elongs to D >n ( R ) ; COMP ACTL Y GENERA TED t -STRUCTURES 35 (2) for e ach k ∈ Z and al l q ∈ Supp(Hom D ( R ) ( X, D [ k ])) , one has that Supp(T or R i ( R/ q , R/ p )) ⊂ φ cm ( k + n − i ) for al l i ≥ 0 and al l p ∈ Z ; (3) Z ∩ Supp(Hom D ( R ) ( X, D [ k ])) ⊂ φ cm ( k + n ) , for al l k ∈ Z . Pr o of. Let φ b e the sp-ﬁltration determined b y th e aisle U n Z ⊂ D ( R ). Recall from th e ab o v e paragraph that φ d ( k ) =  q ∈ Sp ec( R )  R/ q [ − k ] ∈ ⊥ Y  , where now Y := { R Hom · R ( R/ p [ − j ] , D )  p ∈ Z, j ≤ n } . Note th at R Γ Z X ∈ D >n ( R ) is equiv ale nt to (1 ′ ) Supp(H k ( R Hom · R ( X, D ))) = Su pp(Hom D ( R ) ( X, D [ k ])) ⊂ φ d ( k ) for all k ∈ Z . In order to pro v e the equiv alence b et ween (1) and (2) we will giv e an alternativ e description of the du al sp-ﬁ ltration φ d . Let us ﬁx k ∈ Z an arbitrary integ er. Notice that q ∈ φ d ( k ) if and only if 0 = Hom D ( R ) ( R/ q [ − k ] , R Hom · R ( R/ p [ − j ] , D )) (6.7.1) ∼ = Hom D ( R ) ( R/ q , R Hom · R ( R/ p , D )[ k + j ]) = H k + j ( R Hom · R ( R/ q , R Hom · R ( R/ p , D ))) for all j ≤ n and all p ∈ Z . Using ⊗ − hom adju nction, the latter fact is equiv alent to 0 = H k + j ( R Hom · R ( R/ q ⊗ L R R/ p , D ) ∼ = H j − n ( R Hom · R ( R/ q ⊗ L R R/ p [ − k − n ] , D ) , for all j ≤ n and all p ∈ Z . Making the c hange of v ariables i = j − n , w e conclude that q ∈ φ d ( k ) if and only if Hom D ( R ) ( R/ q ⊗ L R R/ p [ − k − n ] , D [ i ]) = 0 , (6.7.2) for all i ≤ 0 and all p ∈ Z . Prop ositio n 3.7 sho ws that (6.7.2) is equiv alent to saying that 0 = Hom D ( R ) (H s ( R/ q ⊗ L R R/ p [ − k − n ])[ − s ] , D [ i ]) ∼ = Hom D ( R ) (H s − k − n ( R/ q ⊗ L R R/ p )[ − s ] , D [ i ]) for all s ∈ Z , all i ≤ 0 and all p ∈ Z . The expression lab eled (6.5.1) in the pro of of Prop osition 6.5 tells us that this last condition for q amoun ts to sa ying that H s − k − n ( R/ q ⊗ L R R/ p )[ − s ] ∈ U φ cm ∩ D b fg ( R ) or, equiv alen tly , that Supp(H s − k − n ( R/ q ⊗ L R R/ p )) ⊂ φ cm ( s ), for all s ∈ Z and all p ∈ Z . Since the homology of that (derived) tensor pr o duct could b e nonzero only in case s − k − n ≤ 0, we mak e a c hange of v ariable − t = s − k − n , s o th at H − t ( R/ q ⊗ L R R/ p ) = T or R t ( R/ q , R/ p ) 36 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN and s = n + k − t , for all t ≥ 0. W e then conclude that q ∈ φ d ( k ) if and only if, for all t ≥ 0 and all p ∈ Z , Supp(T or R t ( R/ q , R/ p )) ⊂ φ cm ( k + n − t ) (6.7.3) T ogether with the p revious paragraph, this characte rization of φ d pro v es the lo ok ed-for equiv alence. Let us pr ov e no w the equiv ale nce b et we en (2) and (3). Let us in tro duce the sp-ﬁ ltration ξ : Z → P (Sp ec ( R )) deﬁned b y ξ ( k ) = { q ∈ Sp ec( R ) ; V( q ) ∩ Z ⊂ φ cm ( k + n ) } , for eac h k ∈ Z . An easy exercise sh o ws then that (3) is equ iv alent to Supp(Hom D ( R ) ( X, D [ k ])) ⊂ ξ ( k ) , for all k ∈ Z . Our goal will b e reac h ed once we sho w that, for eac h k ∈ Z , ξ ( k ) is the set of prime ideals q ∈ Sp ec ( R ) satisfying Supp(T or R i ( R/ q , R/ p )) ⊂ φ cm ( k + n − i ) for all i ≥ 0 and all p ∈ Z , otherwise said, ξ = φ d (see the description of φ d in (6.7.3)). F or that, let us ﬁx an intege r k . Let u s consider q ∈ ξ ( k ), let i ≥ 0 b e a natural n umb er and tak e an arbitrary p ′ ∈ S upp(T or R i ( R/ q , R/ p )). Then T or R p ′ i ( R p ′ / q R p ′ , R p ′ / p R p ′ ) 6 = 0 , and th is fact implies that p ′ con tains b oth q and p . Then p ′ ∈ V ( q ) ∩ V ( p ) ⊂ V( q ) ∩ Z . S ince q ∈ ξ ( k ), we conclude that p ′ ∈ φ cm ( k + n ), whic h implies th at p ′ ∈ φ cm ( k + n − i ) b ecause φ cm is decreasing. Th at p r o v es th at q ∈ φ d ( k ), so that w e get the inclusion ξ ( k ) ⊂ φ d ( k ). Con v ersely , assume that q ∈ φ d ( k ) that is (acco rdin g to (6.7.1)) 0 = H k + j ( R Hom · R ( R/ q , R Hom · R ( R/ p , D ))) for all j ≤ n and all p ∈ Z or, equiv alently , that 0 = H i ( R Hom · R ( R/ q , R Hom · R ( R/ p , D ))) , (6.7.4) for any i ≤ k + n and all p ∈ Z . W e need to prov e that V ( q ) ∩ Z ⊂ φ cm ( k + n ). Indeed, if p ′ ∈ V ( q ) ∩ Z then p ′ ∈ φ d ( k ) b ecause φ d ( k ) is a sp-sub set. So the equalit y in (6.7.4) is true for q = p = p ′ , that is 0 = H i ( R Hom · R ( R/ p ′ , R Hom · R ( R/ p ′ , D ))) = Hom D ( R ) ( R/ p ′ , R Hom · R ( R/ p ′ , D )[ i ]) , for all i ≤ k + n . But then, viewing R Hom · R ( R/ p ′ , D ) as an ob ject of D ( R/ p ′ ), w e get that Hom D ( R/ p ′ ) ( R/ p ′ , R Hom · R ( R/ p ′ , D )[ i ]) = 0 for all i ≤ k + n ( cf. 2.1). T he last is equ iv alent to saying that R Hom · R ( R/ p ′ , D ) b elongs to D >k + n ( R/ p ′ ). But R Hom · R ( R/ p ′ , D ) is a du alizing complex o v er R/ p ′ (cf. [H, C hapter V, Prop osition 2.4, p . 260]) and then the asso ciated co dimension fun ction, whic h we denote b y ¯ d , has the p rop erty th at ¯ d ( ¯ 0) > k + n , wh ere ¯ 0 ∈ Sp ec( R/ p ′ ) is th e generic p oin t. It will b e enough to c hec k that d ( p ′ ) ≥ ¯ d ( ¯ 0) or, equiv alen tly , to pro v e that if Hom D ( R p ′ ) ( k ( p ′ ) , D p ′ [ i ]) 6 = COMP ACTL Y GENERA TED t -STRUCTURES 37 0 then Hom D ( k ( p ′ )) ( k ( p ′ ) , R Hom · k ( p ′ ) ( k ( p ′ ) , D p ′ [ i ])) 6 = 0. But this last f act follo ws f r om remark 2.1.  Lemma 6.8. Under the hyp othesis of L emma 6.7 the fol lowing assertions ar e e quivalent for X ∈ D b fg ( R ) : (1) τ ≤ n R Γ Z X b elongs to D b fg ( R ); (2) for e ach k ∈ Z and al l q ∈ Sup p(Hom D ( R ) ( X, D [ k ])) , e ither q ∈ Z or Z ∩ V ( q ) ⊂ φ cm ( k + n ) . Pr o of. Con tinuing with the notation in the p ro of of Lemma 6.7 , w e h a v e φ d ( k ) = ξ ( k ) = { q ∈ Sp ec( R ) ; V( q ) ∩ Z ⊂ φ cm ( k + n ) } , for all k ∈ Z , so that condition (2) can b e rewritten as: (2 ′ ) Supp(Hom D ( R ) ( X, D [ k ]) ⊂ Z ∪ φ d ( k ), for all k ∈ Z . Let us c hec k that (1) implies (2 ′ ). Assum e that τ ≤ φ X = τ ≤ n R Γ Z X b elongs to D b fg ( R ), then the third ve rtex in the canonical triangle τ ≤ φ X − → X − → τ > φ X + − → , (6.8.1) is in D b fg ( R ). Observe that the complex R Γ Z τ > φ X b elongs to D >n ( R ) since τ > φ X ∈ U φ ⊥ = U n ⊥ Z . Th en, b y Lemma 6.7, w e get that Z ∩ Su pp(Hom D ( R ) ( τ > φ X, D [ k ])) ⊂ φ cm ( k + n ) , for all k ∈ Z or, equiv alen tly (see the pro of of the referr ed Lemma), that Supp(Hom D ( R ) ( τ > φ X, D [ k ])) ⊂ φ d ( k ) , for any k ∈ Z . F urtherm ore, n ote that (Hom D ( R ) ( τ ≤ φ X, D [ k ])) p ∼ = Hom D ( R ) (( τ ≤ φ X ) p , D p [ k ])) for any p ∈ Sp ec ( R ), therefore Supp(Hom D ( R ) ( τ ≤ φ X, D [ k ])) ⊂ Z b ecause Supp( τ ≤ φ X ) ⊂ Z . No w applying the homological fu nctor Hom ( − , D ) := Hom D ( R ) ( − , D ) to the triangle (6.8.1) w e get an exact sequence of R -mo d ules Hom( τ > φ X, D [ k ]) − → Hom( X, D [ k ]) − → Hom( τ ≤ φ X, D [ k ]) from w hic h S upp(Hom D ( R ) ( X, D [ k ])) ⊂ φ d ( k ) ∪ Z as desired . Let u s chec k that (2 ′ ) implies (1). Th e fun ctor R Hom · R ( − , D ) : D b fg ( R ) − → D b fg ( R ) is a dualit y of triangulated cate gories and the class of ob jects V = { Y ∈ D b fg ( R ) ; Supp(H k ( Y )) ⊂ φ d ( k ) ∪ Z for all k ∈ Z } can b e bu ilt up by a ﬁnite num b er of iterated extensions from those ob jects in the class w hic h are stalk complexes of the form M [ − k ], with M a ﬁ nitely generated R -mo du le su c h that Supp( M ) ⊂ φ d ( k ) ∪ Z . So it is enough to p ro v e that (2 ′ ) implies (1) for those complexes X ∈ D b fg ( R ) such that 38 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN R Hom · R ( X, D ) ∼ = M [ − k ], with Su pp( M ) ⊂ φ d ( k ) ∪ Z . Moreo v er every suc h M admits a ﬁltration 0 = M 0 ( M 1 ( · · · ( M n − 1 ( M n = M suc h that M i /M i − 1 ∼ = R/ p i , with p i ∈ Supp( M ) for i ∈ { 1 , . . . , n } (see [Mat, Theorem 6.4]). This imp lies that M [ − k ] c an be b uilt by iterated extensions from stalk complexes R / p [ − k ], with p ∈ Sup p( M ) ⊂ φ d ( k ) ∪ Z . Therefore it is enough to pro v e (2 ′ ) = ⇒ (1) in the particular case in w hic h R Hom R ( X, D ) ∼ = R/ p [ − k ], with p ∈ φ d ( k ) ∪ Z . So let X = R Hom · R ( R/ p [ − k ] , D ) with p ∈ φ d ( k ) ∪ Z . If p ∈ Z then Supp( X ) ⊂ S upp( R/ p [ − k ]) = V( p ) ⊂ Z , whence R Γ Z X ∼ = X (b y The- orem 1.8). Th en τ ≤ n R Γ Z X ∼ = τ ≤ n X b elongs to D b fg ( R ). In case that p ∈ φ d ( k ), w e hav e R Hom · R ( X, D ) ∼ = R/ p [ − k ] so Supp(Hom D ( R ) ( X, D [ j ])) = Su pp(H j ( R/ p [ − k ])) ⊂ φ d ( j ) , for all j ∈ Z . T hat is exactly what condition (1 ′ ) in the p r o of of Lemma 6.7 sa ys, h ence R Γ Z X ∈ D >n R , that is, τ ≤ n R Γ Z X = 0 and assertion (1) triv- ially holds in this case.  Theorem 6.9. L et R b e a c ommuta tive N o etherian ring that admits a du- alizing c omplex. Given a sp-ﬁltr ation φ : Z → P (Sp ec( R )) the f ol low ing ar e e qui valent: (1) U φ ∩ D b fg ( R ) is an aisle of D b fg ( R ) ; (2) φ satisﬁes the we ak Cousin c ondition . Pr o of. By C orollary 4.5 we only need to pr o v e (2) ⇒ (1). Without loss of generalit y w e ma y assum e that Sp ec ( R ) is conn ected and th at φ is a non- constan t sp-ﬁltration (if it is n ecessary , lo calize resp ect to an idemp oten t elemen t of R and use Prop osition 2.9). Let D b e a dualizing complex for R , with asso ciated co dimens ion f unction d : S p ec( R ) → Z . Under this hyp oth- esis R has ﬁnite Kru ll dimension ( cf. [H, Ch apter V, Corollary 7.2, p. 283]) and, h ence, w e know that the sp-ﬁ ltration φ is ﬁ nite of length ≥ 1. More precisely , there are integ ers t ≤ n such that φ is determined in the in terv al [ t, n ] ⊂ Z , with φ ( t ) = Sp ec( R ) (see Corollary 4.8). W e claim that the f ollo wing is true f or an in teger m ∈ Z and a p rime id eal q ∈ Sp ec( R ): If q has th e pr op erty that V( q ) ∩ φ ( i ) ⊂ φ cm ( k + i ) , for all i < m , then either q ∈ φ ( m ) or V ( q ) ∩ φ ( m ) ⊂ φ cm ( k + m ). Indeed, supp ose that q 6∈ φ ( m ). If V ( q ) ∩ φ ( m ) = ∅ we are done, so we assume that V ( q ) ∩ φ ( m ) is nonempt y . Ch o ose a minimal elemen t p of V( q ) ∩ φ ( m ) and consider a maximal c hain of prime ideals q = q 0 ( q 1 ( · · · ( q s = p . Then the wea k Cousin condition says th at q s − 1 ∈ φ ( m − 1), so that q s − 1 ∈ V( q ) ∩ φ ( m − 1) ⊂ φ cm ( k + m − 1). Therefore d ( q s − 1 ) > k + m − 1 and, since COMP ACTL Y GENERA TED t -STRUCTURES 39 d is a co d imension f u nction, we conclude th at d ( p ) > k + m or, equiv alen tly , that p ∈ φ cm ( k + m ). That pro v es our claim. F or the rest of the pro of assu me, w ithout loss of generalit y , that Sp ec( R ) = φ (0) ) φ (1) ) · · · ) φ ( n ) ) φ ( n + 1) = ∅ with n + 1 = L( φ ). The class U φ ∩ D b fg ( R ) is an aisle of D b fg ( R ) if L( φ ) = n + 1 ≤ 2 b y Prop osition 5.8. Sup p ose that L( φ ) = n + 1 > 2, th en φ ′ is a sp-ﬁltration satisfying the wea k Cous in condition su c h that L( φ ′ ) = n . By induction on the length of the sp-ﬁ ltrations we can assume that U φ ′ ∩ D b fg ( R ) is an aisle of D b fg ( R ). T h en, as a consequence of Lemma 5.7, c hec king that U φ ∩ D b fg ( R ) is an aisle of D b fg ( R ) turns out to b e equiv alen t to pr oving that τ ≤ n R Γ φ ( n ) M b elo ngs to D b fg ( R ) f or an y M ∈ U ⊥ φ ′ ∩ D b fg ( R ). So let M ∈ U ⊥ φ ′ ∩ D b fg ( R ), then we h a v e that R Γ φ ( i ) M ∈ D >i ( R ), for all i < n . F rom Lemma 6.7, we conclude th at φ ( i ) ∩ Supp(Hom D ( R ) ( M , D [ k ])) ⊂ φ cm ( k + i ), for all k ∈ Z and all i < n . T his, in particular, implies that if k ∈ Z and q ∈ Supp(Hom D ( R ) ( M , D [ k ])) then V ( q ) ∩ φ ( i ) ⊂ φ cm ( k + i ), for all i < n . No w, applying the claim ab ov e, we get that, for ev ery k ∈ Z and eve ry q ∈ Supp(Hom D ( R ) ( M , D [ k ])), either q ∈ φ ( n ) or V ( q ) ∩ φ ( n ) ⊂ φ cm ( k + n ). Then, by Lemma 6.8, w e get that τ ≤ n R Γ φ ( n ) M ∈ D b fg ( R ) as it is d esired.  Corollary 6.10. L e t R b e a c ommutative No etherian ring with a p oint- wise dualizing c omp lex. Then for any sp-ﬁltr ation φ : Z → P (Sp ec( R )) the fol lowing ar e e quivalent: (1) U φ ∩ D fg ( R ) is an aisle of D fg ( R ) ; (2) φ satisﬁes the we ak Cousin c ondition . Pr o of. Giv en a complex X ∈ D fg ( R ) let us co nsider the φ -triangle with cen tral vertex X U − → X − → V + − → (6.10.1 ) By lo calizing at an y pr ime ideal p , w e obtain a φ p -triangle in D ( R p ) with cen tral vertex X p U p − → X p − → V p + − → . (6.10.2 ) Note that the triangle (6 . 10 . 1 ) is in D fg ( R ) if and only if f or all p ∈ Sp ec ( R ) the triangle (6.10.2 ) b elongs to D fg ( R p ). F urth ermore, a sp -ﬁltration φ of Sp ec( R ) satisﬁes the we ak Cousin condition if and only for an y p ∈ Sp ec ( R ) the sp-ﬁltration φ p of Sp ec( R p ) satisﬁes the w eak Cous in condition. So we deriv e the tru th of this r esult from Theorem 6.9 and Corollary 4.10 b ecause for eac h p ∈ S p ec( R ) the r in g R p admits a dualizing complex and, hence, has ﬁ n ite Kru ll dimension.  Corollary 6.11. L et R b e a c ommutative No etherian ring with dualizing c omplex. F or any ♯ ∈ {− , + , b , “blank” } , the assignment φ U φ ∩ D ♯ fg ( R ) deﬁnes a one-to-one c orr esp ondenc e b etwe e n: (1) sp-ﬁltr atio ns of Sp ec( R ) satisfying the we ak Cousin c onditio n; 40 L. ALONSO, A. JEREM ´ IAS, AND M. SA OR ´ IN (2) aisles of D ♯ fg ( R ) gener ate d b y b ounde d c omp lexes; and (3) aisles of D ♯ fg ( R ) gener ate d b y p erfe ct c omplexes. In p articular, in c ase ♯ ∈ {− , b } , the assignment φ U φ ∩ D ♯ fg ( R ) deﬁnes a bije ction b etwe en the set of sp-ﬁltr ations of Sp ec( R ) and the set of aisles of D ♯ fg ( R ) . Pr o of. 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