Chow motives without projectivity

Cho w motiv es without pro jectivit y by J¨ org Wildeshaus ∗ LA GA UMR 7539 Institut Galil´ ee Univ er s it ´ e P aris 13 Aven ue Jean-Baptiste Cl ´ emen t F-9343 0 Villetaneuse F rance wildes h@mat h.univ-paris13.fr Jan uar y 16, 2009 Abstract In [Bo2], Bondark o recentl y d eﬁned the notion of w eigh t stru cture, and prov ed th at th e catego ry D M g m ( k ) of geometrical motive s o v er a p erfect ﬁeld k , as deﬁned and studied by V oevodsky , Su s lin and F ried- lander [VSF], is canonically equipp ed w ith suc h a structure. Building on this result, and under a condition on the w eigh ts av oided b y the b ound ary mo tive [W1], w e describ e a me tho d to construct in trinsically in D M g m ( k ) a motivic v ersion of in terior cohomolog y o f smo oth, but p ossibly non-pro jectiv e sc hemes. I n a sequel to t his wo rk [W2], this metho d will be applied to Sh im ura v arieties. Keyw ords: w eigh t structures, w eigh t ﬁltrations, C ho w motiv es, geo- metrical motiv es, motiv es for mo du lar forms, b oundary motiv e, inte- rior motiv e. Math. Sub j. Class. (2000) num b ers: 14F42 (14F20 , 14F2 5 , 14F3 0, 14G35 , 18E 30, 19E 15). ∗ Partially supp or ted by the A genc e Nationale de la Re cher che , pro ject no. ANR-07 - BLAN-0142 “M´ etho des ` a la V o evods ky , motifs mixtes et G´ eom´ etrie d’Arakelov”. 1 Con ten ts 0 In tro duct ion 2 1 W eigh t structures 7 2 W eigh t zero 14 3 Example: motiv es for mo dular forms 19 4 W eigh t s, b oundary motiv e and interior motiv e 25 0 In tro duction The full t itle of this w ork w ould b e “Approximation of motive s of v arieties whic h are sm o oth, but not necessarily pro jectiv e, by Chow motiv es, using homological rather than purely geometrical metho ds”. This might giv e a b etter idea of our progra m — but as a tit le, it app eared t o o long. So there. One w a y t o pla ce t he problem historically is to start with the question ask ed b y Serre [Se, p. 341], whether the “virtual motiv e” χ c ( X ) of an ar- bitrary v ar iet y X o v er a ﬁxed base ﬁeld k can b e (w ell) deﬁned in the Grothendiec k gr oup of the category C H M ef f ( k ) of eﬀectiv e Chow mo tives . When k admits resolution of singularities, Gillet and Soul´ e [GiSo] (see also [GuNA] when char ( k ) = 0) provided an aﬃrmativ e answ er. In fact, their s o- lution yields muc h more information: they deﬁne the weight c omplex W ( X ) ( h c ( X ) in [GuNA]) in the category of complexes ov er C H M ef f ( k ), w ell de- ﬁned up to canonical homotop y equiv alence . By deﬁnition, χ c ( X ) equals the class of W ( X ) in K 0  C H M ef f ( k )  . Th us, give n an y represen tativ e M • : . . . − → M n − → M n − 1 − → . . . of W ( X ), w e hav e the fom ula χ c ( X ) = P n ( − 1) n [ M n ]. Consider the fully faithful em bedding ι of C H M ef f ( k ) in to D M ef f g m ( k ), the category of eﬀe ctiv e ge ometric al motives , as deﬁned and s tudied by V o e- v o dsky , Suslin and F riedlande r [VSF]. As observ ed in [GiSo], lo calization for b oth χ c ( X ) and the mo tive with c omp act supp ort M c g m ( X ) of X sho ws that the elemen t χ c ( X ) is mapped to the c lass of M c g m ( X ) under K 0 ( ι ). Gillet and Soul´ e we nt on and ask ed [GiSo, p. 153 ] whether K 0 ( ι ) : K 0  C H M ef f ( k )  − → K 0  D M ef f g m ( k )  is an isomorphism. 2 Just as Serre, Gillet and Soul ´ e pro v ok ed muc h more than just an aﬃrma- tiv e answ er to their question. Bo ndark o deﬁned in [Bo2] the notion of we ight structur e on a triangulat ed category C . He prov ed that the inclusion of the he art of the w eigh t structure into C induces an isomorphism on the lev el of Grothendiec k g roups, whenev er the w eigh t structure is b ounded, and the heart is pseudo-Ab elian. The deﬁnitions will b e recalled in our Section 1 ; let us just note that shift by [ m ], f or m ∈ Z , adds m to the we ight of an ob ject of C . According to one of the main results of [lo c. cit.] (recalled in Theo- rem 1 .13), D M ef f g m ( k ) carries a canonical w eigh t structure, whic h is indeed b ounded, and a dmits C H M ef f ( k ) as it s heart. By deﬁnition, this means that a mong all geometrical motives , Cho w motiv es distinguish themselv es as b eing the motiv es whic h are pure of we ight zero. In particular, this g iv es an in trinsic c haracterization of o b jects of the category DM ef f g m ( k ) belonging to C H M ef f ( k ). W e think of this insigh t a s nothing less than rev olutionary . T o come back t o t he b eginning, the comp onent M n of the w eigh t com- plex W ( X ) can b e considered as a “Gr n W ( X )” with r espect to the weigh t structure. In the con text of Bondarko’s theory , the formula for the “virtual motiv e” of X th us reads χ c ( X ) = X n ( − 1) n Gr n W ( X ) . Basically , our approac h to construct a Cho w motiv e o ut of X , when X is smo oth, is v ery simple: according to Bondark o (see Coro llary 1.14), the mo- tive M g m ( X ) of X is of w eigh ts ≤ 0. W e w ould lik e to c onsider Gr 0 M g m ( X ), the “quotien t” o f M g m ( X ) of maximal w eight zero. Ho w ev er, one of the main subtleties of the notion of w eigh t structure is that “t he” w eight ﬁltr ation of an ob ject is almost nev er unique. (In the ex- ample o f the w eigh t complex , this corres p o nds to the fact that W ( X ) is only w ell-deﬁned up to homotop y .) Hence “G r n M g m ( X )” is not w ell-deﬁned, and the ab o ve approa c h cannot w ork as s tated. In fact, for an y smo oth compacti- ﬁcation e X of X , the Chow motiv e M g m ( e X ) o ccurs as “Gr 0 M g m ( X )” for a suitable weigh t ﬁltratio n! Our main con tribution is to iden tify a criterion assuring existenc e and unicit y of a “b est c hoice” of Gr 0 M g m ( X ). It is b est understo o d in the ab- stract setting created b y Bondark o, that is, in the con text o f a w eight struc- ture on a triangulated category C . Let M b e an ob ject of C of non-p ositiv e w eigh ts. Then Bondarko’s axioms imply that if M admits a w eigh t ﬁltr ation in whic h t he a djacen t we ight − 1 do es not o ccur, then the asso ciated Gr 0 M is unique up to unique isomorphism. In the motivic contex t, this observ atio n leads to our criterion on w eigh ts av oided b y the b oundary motive ∂ M g m ( X ) 3 of X , in tro duced and studie d in [W1]. T he b eha vior of the realizations o f the Chow motiv e Gr 0 M g m ( X ) mot iv ates its name: w e c hose to call it the interior motive of X . Let us no w give a mor e detailed des cription o f the conten t of this article. Section 1 claims no orig inalit y whatso ev er. W e giv e Bondark o’s deﬁnition of w eigh t structures (Deﬁnition 1.1) and review the results from [Bo2] needed in the seque l. W e treat particularly carefully the phenomenon whic h will turn out to b e the main theme of this article, namely the abse n c e of c ertain weights . Th us, w e intro duce the cen tral notion o f weight ﬁltr a tion avoiding weights m, m + 1 , . . . , n − 1 , n , for ﬁxed in tegers m ≤ n . Bondarko’s axioms imply that whenev er suc h a ﬁltration exists, it b eha v es functorially (Prop o- sition 1.7). In particular, fo r an y ﬁxed ob ject, it is unique up to unique isomorphism (Corollary 1.9 ). W e conclude Section 1 with Bondark o’s ap- plication of his theory to geometrical motiv es, whic h w e alr eady men tioned b efore (Theorem 1.13, Corollary 1 .14). Section 2 is the tec hnical cen ter of t his ar ticle. It is devoted to a further study of functorialit y of w eigh t ﬁltrations av oiding certain w eigh ts. W e work in the con text of an abstract w eigh t structure w = ( C w ≤ 0 , C w ≥ 0 ) on a tria n- gulated category C . W e ﬁrst sho w (Prop o sition 2 .2) that the inclusion ι − of the heart C w =0 in to the full sub-category C w ≤ 0 , 6 = − 1 of ob jects of no n-p ositiv e w eigh ts 6 = − 1 admits a left adjo in t Gr 0 : C w ≤ 0 , 6 = − 1 − → C w =0 . There is a dual v ersion of this statemen t for non-negative weigh ts 6 = 1. W e then consider the situation whic h will b e of in terest in our application to motiv es. W e thus ﬁx a morphism u : M − → M + b et w een ob jects M − ∈ C w ≤ 0 and M + ∈ C w ≥ 0 , and a cone C [1] of u . While the axioms c haracterizing w eigh t structures easily show that u can b e f actored through some ob ject of C w =0 , our aim is to do so in a c anonic al way . W e therefore form ulate Assumption 2.3: the ob ject C is without w eights − 1 and 0. Theorem 2.4 states that Assumption 2.3 not only allow s to factor u as desired; in a ddi- tion this factorization is t hrough an ob ject w hic h is sim ultaneously iden ti- ﬁed with Gr 0 M − and with Gr 0 M + . As a formal consequence of this, and of the f unctorialit y prop erties of Gr 0 from Prop osition 2 .2, w e get a state- men t o n a bstract factorization of u (Corollary 2.5), whose rigidit y ma y ap- p ear surprising at ﬁrst sight, giv en tha t we w ork in a triangula ted category: whenev er u : M − → N → M + factors through an ob ject N of C w =0 , then Gr 0 M − = Gr 0 M + is canonically identiﬁed with a direct fa ctor of N , admit- ting in addition a canonical direct complemen t. The reader willing to t urn directly to the a pplication of these results to geometrical motiv es may choo se to skip Section 3, in whic h Sc holl’s con- struction of motiv es for mo dular fo rms [Scho1] is discussed at length. As in 4 [lo c. cit.], w e consider a self-product X r n of the univ ersal elliptic curv e X n o v er a mo dular curv e. W e sho w (Theorem 3.3, Corollary 3.4, Corollary 3.6) that certain direct f actors M g m ( X r n ) e of the mot iv e M g m ( X r n ) and M c g m ( X r n ) e of the motiv e with compact suppo rt M c g m ( X r n ), together with the canonical morphism u : M g m ( X r n ) e → M c g m ( X r n ) e , satisfy the conclusions of Theo- rem 2 .4. In particular, the Cho w motiv es Gr 0 M g m ( X r n ) e and Gr 0 M c g m ( X r n ) e are deﬁned, and canonically isomorphic. In fa ct, they are b o th canonically isomorphic to the motive denoted r n W in [Sc ho1]. W e insist on giving a pro of o f the ab o v e statemen ts whic h is indep enden t of the theory dev elop ed in Section 2, that is, w e prov e the conclusions of Theorem 2.4 without ﬁrst c hec king Assumption 2.3. Instead, we use the detailed analysis from [Sc ho1] of the geometry of the b oundary of a “go o d c hoice” of smo oth compactiﬁ- cation of X r n . If o ne forgets ab out the language of w eigh t structures, whose use could not b e completely av oided, Section 3 is th us tech nically indep en- den t of the material preceding it. The reader may ﬁnd an interes t in the re-in terpretation of Sc holl’s construction in the con text of V o ev o dsky ’s g eo- metrical motiv es, whic h were not y et deﬁned a t the time when [Scho1] w as written. This concerns in particular the exact triangle M g m  S ∞ n  ( r + 1)[ r + 1] − → M g m ( X r n ) e − → r n W − → M g m  S ∞ n  ( r + 1)[ r + 2] from Corollary 3.4 ( S ∞ n := the cuspidal lo cus of the mo dular curv e). As a b y-pro duct of Sc holl’s analysis , we pro v e that the triangle is deﬁned usin g Z [1 / (2 n · r !)]- co eﬃcien ts. An in terpretation of Beilinson’s Eisenstein symb ol [B] in this con text (namely , as a splitting of this triangle) is clearly de sirable. Section 4 is dev o ted to the application of the res ults from Section 2 to geometrical motiv es. As u : M − → M + , w e tak e the canonical morphism M g m ( X ) e → M c g m ( X ) e , for a ﬁxed smo o th v ariet y X ov er k , and a ﬁxed idemp oten t e . Then the role o f the o b ject C fro m Section 2 is canonically pla y ed b y ∂ M g m ( X ) e , the e - part o f the b oundary motiv e. In this con text, Assumption 2.3 reads as follows : the ob ject ∂ M g m ( X ) e is without w eights − 1 and 0 (Ass umption 4.2). Our main res ult Theorem 4.3 is then simply the translation of the sum of the results from Sec tion 2 into this particu- lar motivic con text. Th us, the Chow mo tiv e G r 0 M g m ( X ) e = Gr 0 M c g m ( X ) e is deﬁned. F urthermore (Coro llary 4.6), w e get the motivic v ersion of ab- stract factorization: whenev er e X is a smo oth compactiﬁcation of X , then Gr 0 M g m ( X ) e is canonically a direct fa ctor of the Cho w motiv e M g m ( e X ), with a canonical direct complemen t. W e then study the implications o f these results f or the Hodge theoretic and ℓ -adic realizatio ns. Theorems 4.7 and 4.8 state that they are equal to the resp ectiv e e -parts of in terior cohomology o f X . Abstract f actorization a llo ws to say more ab out the qualit y of the Galois represen tation o n the ℓ -adic realization of G r 0 M g m ( X ) e (Theorem 4.14). F or example, simple semi-stable redu ction of som e smo ot h compactiﬁcation of X implie s that t he represen tation is semi-stable. 5 T o conclude, w e get bac k to Scholl’s construction. W e sho w (Remark 4.17) that essen tially all results of Section 3 can b e deduced (just) from Assump- tion 4.2, and t he theory dev elop ed in Section 4. W e consider that in spite of the tec hnical indep endence of Section 3, there is a go o d reason to include that mat erial in this ar ticle: for higher dimens ional Shim ura v arieties, “go o d c hoices” of smo oth compactiﬁcations as the one used in [Scho1] may simply not b e a v ailable. Therefore, t he purely geometrical strategy of pro of of the results of Section 3 can safely b e expected not to b e g eneralizable. W e think that a promising wa y to g eneralize is v ia a v eriﬁcation of Ass umption 4.2 b y other than purely geometrical means. W e refer to [W2] for the dev elopmen t of suc h an alternative in a con text including that of (p o w ers of univ ersal elliptic curve s ov er) mo dular curv es. P art of this w ork was done while I w as enjoy ing a mo dulation de servic e p our le s p orteurs de pr ojets de r e cher c h e , gran ted by the Universit´ e Paris 13 . I wish to thank M.V. Bondark o, F. D´ eglise, A. Deitmar, O. Ga bb er, B. Kahn, A. Mokrane, C. So ul ´ e and J. Tilouine for useful d iscussions, and the re feree(s) for helpful suggestions. Notation and con v en tions : k denotes a ﬁxe d perfect base ﬁeld, S ch/k the cat egory of separated sc hemes of ﬁnite ty p e ov er k , and S m/k ⊂ S ch/k the full sub-category of ob jects which are smo oth ov er k . Whe n we as- sume k to admit res olution of singularities, then it will b e in the se nse of [FV, Def. 3.4]: (i) fo r an y X ∈ S ch/k , there exists an abstract blo w-up Y → X [FV, Def. 3.1] whose source Y is in S m/k , (ii) for an y X , Y ∈ S m/k , and an y abstract blow-up q : Y → X , there exists a s equence of blow-ups p : X n → . . . → X 1 = X with smo oth cen ters, suc h that p factors thr ough q . Let us note that the main reason for us to supp ose k to admit resolution of singularities is to ha v e the motiv e with compact supp ort satisfy lo calization [V1, Prop. 4.1.5]. As fa r as mot iv es are concerned, the notation of this pap er follo ws that of [V1]. W e refer to [W1, Sect. 1 ] for a review of this notation, and in particular, of the deﬁnition of the categories D M ef f g m ( k ) and D M g m ( k ) of (eﬀectiv e) geometrical motiv es ov er k , and of the mot iv e M g m ( X ) a nd the motiv e with compact suppor t M c g m ( X ) of X ∈ S ch/k . Let F b e a comm u- tativ e ﬂat Z -algebra, i.e., a commutativ e unitary ring whose additiv e group is without torsion. The notat ion D M ef f g m ( k ) F and D M g m ( k ) F stands for the F -linear analogues of D M ef f g m ( k ) and D M g m ( k ) deﬁned in [A, Sect. 16.2.4 and Sect. 17.1.3]. Similarly , let us denote b y C H M ef f ( k ) and C H M ( k ) the categories opp osite to the categories of (eﬀectiv e) Cho w motiv es, and b y C H M ef f ( k ) F and C H M ( k ) F the pseudo-Ab elian completion of the category C H M ef f ( k ) ⊗ Z F and C H M ( k ) ⊗ Z F , resp ective ly . Using [V2, Cor. 2] ([V1, Cor. 4.2.6] if k admits r esolution of singularities), we canonically iden tify C H M ef f ( k ) F and C H M ( k ) F with a full additive sub-category of D M ef f g m ( k ) F 6 and D M g m ( k ) F , resp ectiv ely . 1 W e igh t s tructure s In this section, we review deﬁnitions and r esults of Bonda rk o’s recen t pap er [Bo2]. Deﬁnition 1.1. L et C b e a triangulated category . A weight structur e o n C is a pair w = ( C w ≤ 0 , C w ≥ 0 ) of full sub-categories of C , suc h that, putting C w ≤ n := C w ≤ 0 [ n ] , C w ≥ n := C w ≥ 0 [ n ] ∀ n ∈ Z , the following conditions a re satisﬁed. (1) The categories C w ≤ 0 and C w ≥ 0 are Karoubi-closed: for an y o b ject M of C w ≤ 0 or C w ≥ 0 , an y direct sum mand of M formed in C is an ob ject of C w ≤ 0 or C w ≥ 0 , resp ectiv ely . (2) (Semi-in v ariance with r espect to shifts.) W e hav e the inclusions C w ≤ 0 ⊂ C w ≤ 1 , C w ≥ 0 ⊃ C w ≥ 1 of full sub-categories of C . (3) (Orthogonality .) F or any pair of ob jects M ∈ C w ≤ 0 and N ∈ C w ≥ 1 , w e ha v e Hom C ( M , N ) = 0 . (4) (W eight ﬁltration.) F or any ob ject M ∈ C , there exists an exact triangle A − → M − → B − → A [1] in C , suc h that A ∈ C w ≤ 0 and B ∈ C w ≥ 1 . By condition 1.1 (2), C w ≤ n ⊂ C w ≤ 0 for negativ e n , and C w ≥ n ⊂ C w ≥ 0 for p ositiv e n . There are ob vious analogues of the o ther c onditions for all the categories C w ≤ n and C w ≥ n . In particular, they are all Karo ubi-closed, and an y ob ject M ∈ C is part of a n exact tria ngle A − → M − → B − → A [1] in C , suc h that A ∈ C w ≤ n and B ∈ C w ≥ n +1 . By a slight generalization of the terminology in tro duced in condition 1.1 ( 4), w e shall refer to an y suc h exact triangle as a we ight ﬁltration o f M . 7 Remark 1.2. ( a) Our con v en tion concerning the sign o f the we ight is actually opposite to the one fro m [Bo2, Def. 1.1.1], i.e., w e exc hanged the roles o f C w ≤ 0 and C w ≥ 0 . (b) Note that in condition 1.1 (4) , “the” weigh t ﬁltration is not assumed to b e unique. (c) Recall the notion of t -structur e on a triang ulated category C [BBD, D ´ ef. 1.3.1]. It consists of a pair t = ( C t ≤ 0 , C t ≥ 0 ) of full sub-categories sa- tisfying f ormal analogues of conditions 1.1 (2 )–(4), but putting C t ≤ n := C t ≤ 0 [ − n ] , C t ≥ n := C t ≥ 0 [ − n ] ∀ n ∈ Z . Note that in the con text of t -structures, t he analogues of the exact triangles in 1.1 (4) are then unique up to unique isomorphism, and that the ana logue of condition 1.1 (1) is fo rmally implied b y t he others. The follow ing is contained in [Bo 2, Def. 1.2.1 ]. Deﬁnition 1.3. Let w = ( C w ≤ 0 , C w ≥ 0 ) b e a w eigh t structure on C . The he art o f w is the full additiv e sub-catego ry C w =0 of C whose ob jects lie b o th in C w ≤ 0 and in C w ≥ 0 . Among the basic prop erties dev elop ed in [Bo2], let us note the follo wing. Prop osition 1.4. L et w = ( C w ≤ 0 , C w ≥ 0 ) b e a w e ight structur e on C , L − → M − → N − → L [1] an exa c t triang l e in C . (a) I f b oth L and N b elong to C w ≤ 0 , then so do es M . (b) I f b oth L and N b elong to C w ≥ 0 , then so do es M . Pr o of. This is the conte nt of [Bo2 , Prop. 1.3 .3 3]. q.e.d. The reader ma y w onder whether there is an easy criterion on a giv en sub-category of a triangulated catego ry t o b e the heart of a suitable w eigh t structure. Bondark o has results [Bo2, Thm. 4.3.2 ] a nsw ering this question. F or our purp oses, the result with the most restrictiv e ﬁniteness condition will b e suﬃcien t. Prop osition 1.5. L et H b e a ful l a dditive sub-c ate gory of a triangulate d c ate gory C . Supp ose that H gener ates C , i.e., C is the smal lest ful l trian g u- late d sub-c ate gory c ontaining H . (a) I f ther e is a w e ight structur e on C whose he art c ontains H , then it is unique. In this c ase, the he art is e qual t o the Karoubi env elop e of H , i.e., the c ate g ory of r etr acts of H in C . (b) Th e fol lowing c onditions ar e e quivalent. (i) Ther e is a weight structur e on C whose he art c ontains H . 8 (ii) H is negativ e , i.e., Hom C  A, B [ i ]  = 0 for any two ob j e cts A , B of H , and any inte ger i > 0 . Pr o of. Condition (ii) on H is clearly necessary for H to b elong t o the heart, giv en orthogonality 1.1 (3). As for (a), note that by Prop osition 1.4, there is only o ne p ossible deﬁnition of the category C w ≤ 0 (resp. C w ≥ 0 ): it is necessarily t he full sub-category of successiv e extensions of o b jects of the form A [ n ], for A ∈ H and n ≤ 0 (resp. n ≥ 0). The main p oint is to sho w that under condition (ii), the ab ov e construc- tion indeed yields a w eight structure on C . W e refer to [Bo 2, Thm. 4.3.2 I I] for details. q.e.d. F or the rest of this section, we consider a ﬁxed weigh t structure w on a triangulated cat egory C . Deﬁnition 1.6. Let M ∈ C , and m ≤ n tw o in tegers (whic h may be iden tical). A weight ﬁltr ation of M avoid ing weigh ts m, m + 1 , . . . , n − 1 , n is an exact triangle M ≤ m − 1 − → M − → M ≥ n +1 − → M ≤ m − 1 [1] in C , with M ≤ m − 1 ∈ C w ≤ m − 1 and M ≥ n +1 ∈ C w ≥ n +1 . The follow ing observ a tion is vital. Prop osition 1.7. Assume that m ≤ n , and that M , N ∈ C admit weight ﬁltr a tions M ≤ m − 1 x − − → M x + − → M ≥ n +1 − → M ≤ m − 1 [1] and N ≤ m − 1 y − − → N y + − → N ≥ n +1 − → N ≤ m − 1 [1] avoiding weights m, . . . , n . Then any morphism M → N in C extends uniquely to a morphi s m of exact triangles M ≤ m − 1 / /   M / /   M ≥ n +1 / /   M ≤ m − 1 [1]   N ≤ m − 1 / / N / / N ≥ n +1 / / N ≤ m − 1 [1] Pr o of. This follo ws from [Bo2, Lemma 1.5.1 2]. F or the con v enienc e o f the reader, let us recall the pro of. Let α ∈ Hom C ( M , N ). The comp osition y + ◦ α ◦ x − : M ≤ m − 1 → N ≥ n +1 is zero b y orthogonality 1.1 (3): m − 1 is strictly smaller than n + 1 . Hence α ◦ x − factors through N ≤ m − 1 . W e claim that this factorization is unique. Indeed, the error term comes fr om 9 Hom C ( M ≤ m − 1 , N ≥ n +1 [ − 1]). But this group is trivial, thanks to orthogonality , and our assumption on the w eigh ts: the ob ject N ≥ n +1 [ − 1] lies in C w ≥ n +1 [ − 1] = C w ≥ n , and m − 1 is still stricly smaller than n . Similarly , the comp osition y + ◦ α factors uniquely through M ≥ n +1 . q.e.d. Remark 1.8. Note tha t the h yp othesis of Prop osition 1.7 do es not im- ply unicit y o f weigh t ﬁltrations in the (more general) sense of 1.1 (4). F or example, assume that m = n = − 1, and let ( ∗ ) M ≤− 2 − → M − → M ≥ 0 − → M ≤− 2 [1] b e a w eight ﬁltra tion a v oiding w eigh t − 1. Cho o se an y ob ject M 0 in C w =0 and replace M ≥ 0 b y M 0 ⊕ M ≥ 0 , and M ≤− 2 b y M 0 [ − 1] ⊕ M ≤− 2 . Arguing as in the pro of of Prop osition 1.7, o ne sho ws that any w eigh t ﬁltratio n of M is isomorphic to one obtained in this w ay . Th us, the ex act triangle ( ∗ ) satisﬁes a minimality prop erty among all weigh t ﬁltratio ns of M . Corollary 1.9. Assume that m ≤ n . Then if M ∈ C admits a weight ﬁltr a tion avoid i n g weights m, . . . , n , it i s unique up to unique isomorphism. Deﬁnition 1.10. Assume that m ≤ n . W e sa y that M ∈ C do es not have w eights m, . . . , n , or that M is without weights m, . . . , n , if it admits a w eigh t ﬁltration av o iding w eights m, . . . , n . Let us no w state what we consider as one of t he main results of [Bo2]. Theorem 1.11. Assume that the triangulate d c ate gory C is gener ate d by its he art C w =0 . (a) The pse udo - Ab e l i a n c ompletion C ′ w =0 of C w =0 gener ates the p seudo-Ab elian c ompletion C ′ of C . (b) Ther e is a wei ght structur e w ′ on C ′ , uniquely char acterize d by any of the fol lowing c onditions. (i) The weight structur e w ′ extends w . (ii) The he art of w ′ e quals C ′ w =0 . (iii) The he art of w ′ c ontains C ′ w =0 . Pr o of. This is [Bo2, Prop. 5.2.2]. Let us describe the main steps of the pro of. Recall that by [BaSc hl, Thm. 1.5 ], the category C ′ is indeed triangulated. The criterion from Prop osition 1.5 implies the existence of a w eigh t structure w ′ on the full triangulat ed sub-category D o f C ′ generated b y C ′ w =0 (hence containing C ), and uniquely characterized by condition (iii), hence also b y (i) or (ii). The claim then follows from [Bo2, Lem ma 5.2.1], whic h states that D is pseudo-Ab elian, and hence equal t o C ′ . q.e.d. 10 Remark 1.12. Note tha t giv en Prop osition 1.5, part (b) of T heorem 1.11 follo ws fo rmally from its part (a). One may see Theorem 1.11 (a) as a gene- ralization o f [BaSchl, Cor. 2 .12], which states that the pseudo-Ab elian com- pletion of the bounded deriv ed category D b ( A ) of an exact category A equals the b ounded derive d categor y D b ( A ′ ) of the pseudo-Ab elian completion A ′ of A . F or our purp oses, the main application o f the preceding is the follo wing (cmp. [Bo 2, Sect. 6]). Theorem 1.13. L et F b e a c ommutative ﬂat Z -alge b r a, and assume k to admit r esolution of singularities. (a) Ther e is a c a nonic al we ight structur e on the c ate gory D M ef f g m ( k ) F . It i s uniquely char acterize d by the r e quir ement that its he art e qual C H M ef f ( k ) F . (b) Ther e is a c anonic al w eight structur e on the c ate go ry D M g m ( k ) F , ex- tending the weight structur e fr om (a). It is uniquely char acterize d b y the r e quir ement that its he art e qual C H M ( k ) F . (c) Statemen ts (a) and (b) hold without a s suming r esolution of singularities pr ovide d F is a Q -alge b r a. Pr o of. F or F = Z and k of characteristic zero, this is the conten t of [Bo2, Sect. 6.5 and 6.6]: (1) As in [Bo2], denote b y D M s the full triangulated sub-category of D M ef f g m ( k ) generated b y the motives M g m ( X ) [V1, Def. 2.1.1] o f ob jects X of S m/k , b y J 0 the full additiv e sub -catego ry of D M s generated by M g m ( X ) for X smoo th a nd pro jectiv e, and b y J ′ 0 the K aroubi env elop e of J 0 . Th us, D M ef f g m ( k ) is the pseudo-Ab elian completion of D M s , and C H M ef f ( k ) is the pseudo-Ab elian completion of b oth J 0 and J ′ 0 . (2) W e need tw o of the main results from [V1]. First, by [lo c. cit.], Cor. 3.5.5, the additiv e category J 0 generates the t riangulated category D M s . Next, by [lo c. cit.], Cor. 4.2.6 , the category J 0 is negativ e: Hom D M s  A, B [ i ]  = 0 for any t w o ob jects A , B of J 0 , and any in teger i > 0. (3) By Prop osition 1.5, there is a w eigh t structure on D M s , uniquely c haracterized b y the f act that J 0 is con tained in the heart. F urthermore, the heart equals the Ka roubi en v elop e J ′ 0 . (4) By Theorem 1.11, the pseudo-Ab elian completion C H M ef f ( k ) of J 0 generates the pseudo-Ab elian completion D M ef f g m ( k ) of D M s (let us r emark that this is stated, but not pro v ed in [V1, Cor. 3.5.5]). Th us, part (a) of our claim holds for F = Z . (5) Recall that C H M ( k ) and D M g m ( k ) are obtained from C H M ef f ( k ) and D M ef f g m ( k ) by inv erting an ob ject, namely the T a te o b ject T , with resp ect to the tensor structures. Henc e C H M ( k ) generates the triangula ted category D M g m ( k ). Its negativity follow s f ormally from that of C H M ef f ( k ): indeed, 11 for t w o ob jects A and B of C H M ( k ), and an y in teger i > 0, the group Hom D M gm ( k ) ( A, B [ i ]) is b y deﬁnition the direct limit o v er large inte gers r of the groups Hom D M ef f gm ( k )  A ⊗ T ⊗ r , B ⊗ T ⊗ r [ i ]  , whic h are a ll zero b y part (a). Th us, w e ma y aga in apply Prop osition 1.5. The resulting w eigh t structure extends the one o n D M ef f g m ( k ): in fact, its re- striction to D M ef f g m ( k ) is a w eight structure, whose heart equals C H M ef f ( k ). This prov es part (b) of our claim for F = Z . If F is ﬂat o v er Z , then t he same pro of w orks. (1’) Replace D M s b y the full F -linear triangulated sub-category DM s F of DM ef f g m ( k ) F generated b y the motiv es M g m ( X ) of ob jects X of S m/k , and J 0 b y J 0 ⊗ Z F . (2’) The tw o results cited in (2) fo rmally imply that J 0 ⊗ Z F generates DM s F , and that J 0 ⊗ Z F is negativ e. Steps (3’) and (4’) are f ormally iden t ical to (3) and ( 4), pro ving part (a ) of the claim. Step (5’) sho ws pa rt (b), once we observ e that C H M ( k ) F and D M g m ( k ) F are obtained fro m C H M ef f ( k ) F and D M ef f g m ( k ) F b y in v erting the T ate ob ject. As for part (c) of our claim, ev erything reduces to showing analogues o f the t w o statemen ts made in step (2). By [A, Cor. 18.1.1.2], the additive cate- gory J 0 ⊗ Z F generates the triangulated categor y D M s ⊗ Z F . The argumen t uses a lterations ` a la de Jong; since this in v olv es ﬁnite extensions of ﬁe lds, whose degrees need to b e in v erted, one requires F to b e a Q -algebra. The generalization of [V1, Cor. 4.2.6 ] to arbitr ary ﬁelds [V2 , Cor. 2] sho ws that the category J 0 is negativ e. Hence so is J 0 ⊗ Z F . q.e.d. The follow ing is the con ten t of [Bo1, Thm. 6.2.1 1 and 2 ]. Corollary 1.14. Assume k to admit r esolution of singularities. L et X in S ch/k b e of (Krul l) dimen sion d . (a) Th e motive with c omp act supp ort M c g m ( X ) lies in D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d . (b) I f X ∈ S m/k , then the motive M g m ( X ) lies in D M ef f g m ( k ) w ≥− d ∩ D M ef f g m ( k ) w ≤ 0 . Pr o of. (a) W e pro ceed b y induction o n d . If d = 0, then M c g m ( X ) is an eﬀectiv e Cho w motiv e, hence of w eigh t 0 by Theorem 1.13 (a). F or d ≥ 1, Nagata ’s theorem on the exis tence of a compactiﬁcation of X , and resolution of singularities imply that there is an op en dense sub- sc heme U of X admitting a smo oth compactiﬁcation e X . Denote b y Z the complemen t of U in X , and b y Y the complemen t of U in e X (b oth with the reduced sc heme structure). By lo calizatio n for the motiv e with compact supp ort [V1, Prop. 4.1.5], there are exact triangles M c g m ( Z ) − → M c g m ( X ) − → M c g m ( U ) − → M c g m ( Z ) [1] . 12 and M c g m ( Y ) − → M c g m ( e X ) − → M c g m ( U ) − → M c g m ( Y )[1] . By induction, M c g m ( Y ) , M c g m ( Z ) ∈ D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d − 1 . Therefore, M c g m ( Y )[1] , M c g m ( Z ) ∈ D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d . Giv en that M c g m ( e X ) is of w eight 0, Pro p osition 1 .4 sho ws ﬁrst that M c g m ( U ) ∈ D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d , and then that M c g m ( X ) ∈ D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d . (b) By [V1, Thm. 4.3 .7 1 and 2], the category D M g m ( k ) is rigid tensor triangulated. The claim th us follows formally from (a ), and f rom the fol- lo wing observ ations: (i) assuming (a s w e may) X to b e of pure dimension d , the motiv e M g m ( X ) is dua l t o M c g m ( X )( − d )[ − 2 d ] [V1, Thm. 4.3.7 3], (ii) the ob ject Z ( − d )[ − 2 d ] is a Cho w motiv e, (iii) the heart of the w eight structure on D M g m ( k ) is stable under dualit y , henc e for a n y natural num - b er n , induction on n sho ws that the dual of a n ob ject of the inte rsec- tion D M g m ( k ) w ≥ 0 ∩ D M g m ( k ) w ≤ n b elongs to D M g m ( k ) w ≥− n ∩ D M g m ( k ) w ≤ 0 , (iv) the w eigh t structure on D M ef f g m ( k ) is induced fro m the w eigh t structure on D M g m ( k ) (Theorem 1 .13 (b)). q.e.d. Remark 1.15. (a) Corollary 1 .14 (a) and its pro o f should b e compared to the construction of the w eigh t complex W ( X ) from [GiSo, Sect. 2.1]. Let e j : e Y • → e X • b e a smoo th h yper-env elop e (in the sens e of [loc. cit.]) of a closed immersion Y ֒ → X of prop er sc hemes whose comple men t eq uals X . Both e Y • and e X • giv e rise to complexes Z e Y • and Z e X • in t he category denoted Z V in [lo c. cit.], i.e., the Z -linearized category asso ciated to the category V of smo oth prop er sch emes ov er k . Hence w e may form the complex Cone( e j ) in Z V . Applying the functor C ◦ L [V1, pp. 207, 223–224], w e get a comple x of Nisnevic h shea v es with transfers w hose cohomology shea v es are homotop y in v arian t. On the one hand, it should be possible to emplo y [V1, Thm. 4.1.2] in order t o sho w t hat the complex C ( L (Cone( e j ))) represen ts the mo tiv e with compact supp ort M c g m ( X ) in DM ef f g m ( k ). On the other hand, b y deﬁnition [GiSo, Sect. 2.1], the (opp osite of the) complex M g m (Cone( e j )) represen ts W ( X ) in the homoto p y category o v er C H M ef f ( k ). The statemen t M c g m ( X ) ∈ D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d from Corolla ry 1.14 (a) should b e compared to [GiSo, Thm. 2 (i)]. 13 (b) Similarly , the construction from [GuNA, Thm. (5.10) (3 )] of the ob ject h ( X ) should b e compared to Corollary 1 .14 (b). Corollary 1.16. A ssume k to admit r esolution of s i n gularities. Supp ose given a dir e ct factor M of M g m ( X ) , for X ∈ S m/k , which is abstr ac tly isomorphic to a dir e ct factor o f M c g m ( Y ) , for som e Y ∈ S ch/ k . Then M is an eﬀe ctive Chow motive. 2 W e igh t ze ro Throughout this section, we ﬁx a w eigh t structure w = ( C w ≤ 0 , C w ≥ 0 ) on a triangulated cat egory C . An ticipating the situation which will b e of in terest in our applications, w e f orm ulate Assumption 2.3 on the cone of a morphism u in C . As w e shall see (Theorem 2.4), this h yp othesis ensures in part icular unique factorizat ion of u through a n ob ject o f the heart C w =0 . Deﬁnition 2.1. Denote by C w ≤ 0 , 6 = − 1 the full sub-category of C w ≤ 0 of ob jects without weigh t − 1, and b y C w ≥ 0 , 6 =1 the f ull sub-category of C w ≥ 0 of ob jects without w eigh t 1. Prop osition 2.2. (a ) The inclusion of t he h e art ι − : C w =0 ֒ → C w ≤ 0 , 6 = − 1 admits a left adjoin t Gr 0 : C w ≤ 0 , 6 = − 1 − → C w =0 . On obj e cts, it is given by sending M to the term M ≥ 0 of a w eight ﬁltr ation M ≤− 2 − → M − → M ≥ 0 − → M ≤− 2 [1] avoiding wei g h t − 1 . The c omp osition Gr 0 ◦ ι − e quals the identity on C w =0 . (b) Th e inclusion of the he art ι + : C w =0 ֒ → C w ≥ 0 , 6 =1 admits a right adjoin t Gr 0 : C w ≥ 0 , 6 =1 − → C w =0 . On obj e cts, it is given by sending M to the term M ≤ 0 of a w eight ﬁltr ation M ≤ 0 − → M − → M ≥ 2 − → M ≤ 0 [1] avoiding wei g h t 1 . The c omp osition Gr 0 ◦ ι + e quals the iden tity on C w =0 . Pr o of. Giv en Prop osition 1.7 and Corollary 1.9 , all that remains to b e pro v ed is that t he o b jects M ≥ 0 (in (a)) resp. M ≤ 0 (in (b)) actually do lie in C w =0 . But this follows from Prop osition 1 .4. q.e.d. Let us no w ﬁx the following data. (1) A morphism u : M − → M + in C b etw een M − ∈ C w ≤ 0 and M + ∈ C w ≥ 0 . 14 (2) An exact triangle C v − − → M − u − → M + v + − → C [1] in C . Th us, the ob ject C [1] is a ﬁxed choice of cone of u . W e mak e the follow ing rather restrictiv e h yp othesis. Assumption 2.3. The ob ject C is without w eigh ts − 1 and 0, i.e., it admits a w eigh t ﬁltratio n C ≤− 2 c − − → C c + − → C ≥ 1 δ C − → C ≤− 2 [1] a v oiding we ights − 1 and 0. The v alidity o f this a ssumption is indep enden t o f the choice of C . Here is our main technic al to ol. Theorem 2.4. Fix the data (1), (2), and supp ose Assumption 2.3. (a) Th e obje ct M − is without weight − 1 , a nd M + is without weig h t 1 . (b) T h e morphisms v − ◦ c − : C ≤− 2 → M − and π 0 : M − → Gr 0 M − r esp. i 0 : G r 0 M + → M + and ( c + [1]) ◦ v + : M + → C ≥ 1 [1] c an b e c anonic al ly extende d to exact trian g l e s (3) C ≤− 2 v − c − − → M − π 0 − → Gr 0 M − δ − − → C ≤− 2 [1] and (4) C ≥ 1 δ + − → Gr 0 M + i 0 − → M + ( c + [1]) v + − → C ≥ 1 [1] . Thus, (3) is a w eight ﬁltr ation of M − avoiding we i g ht − 1 , and (4) is a weigh t ﬁltr a tion of M + avoiding wei g h t 1 . (c) T h er e is a c ano nic al isomorph i s m Gr 0 M − ∼ − − → Gr 0 M + . As a morphism, it i s uniquely determine d by the pr op erty of making the d iagr am M − u / / π 0   M + Gr 0 M − / / Gr 0 M + i 0 O O c ommute. Its i nverse makes the diagr am C ≥ 1 δ C / / δ +   C ≤− 2 [1] Gr 0 M + / / Gr 0 M − δ − O O c ommute. Pr o of. W e start by c ho osing a nd ﬁxing exact tria ngles (3 ′ ) C ≤− 2 v − c − − → M − π ′ 0 − → G − δ − − → C ≤− 2 [1] 15 and (4 ′ ) C ≥ 1 δ − − → G + i ′ 0 − → M + ( c + [1]) v + − → C ≥ 1 [1] . Th us, G − is a cone of v − ◦ c − , and G + a cone of c + ◦ v + [ − 1]. Observ e ﬁrst that by Prop osition 1 .4, G − ∈ C w ≤ 0 and G + ∈ C w ≥ 0 . Giv en this, the existence of so m e isomorphism G − ∼ = G + clearly implie s parts (a) a nd (b) of our statemen t. Let us now sho w that there is an isomorphism α : G − ∼ − − → G + making the diagrams M − u / / π ′ 0   M + G − α / / G + i ′ 0 O O and C ≥ 1 δ C / / δ +   C ≤− 2 [1] G + α − 1 / / G − δ − O O comm ute. T o see this, consider the f ollo wing. M − u [1]   C ≤− 2 v − ◦ c − o o x x p p p p p p p p C f f N N N N N N N N [1] & & N N N N N N N N M + [ − 1] 8 8 q q q q q q q q [1] c + ◦ v + [ − 1] / / C ≥ 1 [ − 1] δ C [ − 1] O O The three arrow s mark ed [1] link the source to the shift by [1] of the tar- get, the upp er and lo w er tria ngles are comm utativ e, and the left and righ t triangles are exact. In the terminology of [BBD , Sect. 1], this is a c alotte inf´ erie ur e , whic h thanks to the axiom TR4’ of triang ulated categories in the form ulation of [BBD, 1.1.6] can b e completed to an o ctahedron. In particu- lar, its con tour M − u [1]   C ≤− 2 v − ◦ c − o o M + [ − 1] [1] c + ◦ v + [ − 1] / / C ≥ 1 [ − 1] δ C [ − 1] O O 16 is part of a c alotte sup´ erieur e : M − u [1]   [1] & & N N N N N N N N C ≤− 2 v − ◦ c − o o G 8 8 p p p p p p p p x x q q q q q q q q M + [ − 1] [1] c + ◦ v + [ − 1] / / C ≥ 1 [ − 1] δ C [ − 1] O O f f N N N N N N N N Here, the upp er and lo w er triangles are exact, and the left a nd righ t triangles are c ommutativ e. Hence the same ob ject G [1] can be chose n as c one of v − ◦ c − and of c + ◦ v + [ − 1], and in addition, suc h that the morphisms u a nd δ C factor through G [1]. F or our ﬁxed choices of cones, this means precisely that there is an isomorphism G − ∼ = G + factorizing b oth u and δ C . As observ ed b efore, this implies (a) and (b). It remains to show the unicit y statemen t from (c). F or this, use the exact triangle (3) a nd apply Prop osition 2.2 (b) to see that Hom C  M − , M +  = Hom C  Gr 0 M − , M +  = Hom C w =0  Gr 0 M − , Gr 0 M +  . Under this iden tiﬁcation, a morphism M − → M + is sen t to its unique fac- torization Gr 0 M − → Gr 0 M + . q.e.d. Giv en Theorem 2.4 (c), w e may and do iden tify Gr 0 M − and Gr 0 M + . Corollary 2.5. F i x the da ta (1 ) , ( 2), and s upp ose Assumption 2.3. L et M − → N → M + b e a factorization of u thr ough an obje ct N o f C w =0 . Then Gr 0 M − = Gr 0 M + is c anonic al ly identiﬁe d with a dir e c t factor of N , admit- ting a c anonic al dir e ct c omplement. Pr o of. By Proposition 2.2, the mor phism M − → N factors uniquely through Gr 0 M − , and N → M + factors uniquely through Gr 0 M + . The com- p osition Gr 0 M − → N → G r 0 M + is therefore a factorization o f u . By Theo- rem 2.4 ( c), it th us equals the canonical iden tiﬁcation Gr 0 M − = Gr 0 M + . Hence Gr 0 M − = Gr 0 M + is a retract of N . Consider a cone o f Gr 0 M − → N in C : Gr 0 M − − → N − → P − → Gr 0 M − [1] The e xact triangle is split in the s ense that P → Gr 0 M − [1] is zero. Hence the morphism N → P admits a righ t in v erse P → N , unique up to morphisms P → G r 0 M − . There is therefore a unique right in v erse i : P → N suc h tha t its comp osition with the pro jection p : N → Gr 0 M + = Gr 0 M − is zero. The image of i is then a kerne l of p , whose existence is t h us established. This is the canonical complemen t of G r 0 M − = Gr 0 M + in N . As a retract of N , the o b ject P b elongs to C w =0 (condition 1.1 (1)) . q.e.d. 17 Remark 2.6. Our pro of of Corollary 2.5 uses the triangulated structure of the category C . This can b e a v oided when the heart C w =0 is pseudo- Ab elian. Namely , consider the comp osition p : N − → Gr 0 M + = Gr 0 M − − → N . It is an idemp oten t whose image is iden tiﬁed with G r 0 M − = Gr 0 M + . Since C w =0 is pseudo-Ab elian, the morphism p also admits a k ernel. F or future use, w e c hec k the compat ibilit y of Assumption 2.3 with t ensor pro ducts. Assume the refore that our category C is tens or triangulated. Th us, a bilinear bifunctor ⊗ : C × C − → C is giv en, and it is assumed to b e triangula ted in b ot h argumen ts. Assume also that the w eight structure w is compatible with ⊗ , i.e., that C w ≤ 0 ⊗ C w ≤ 0 ⊂ C w ≤ 0 and C w ≥ 0 ⊗ C w ≥ 0 ⊂ C w ≥ 0 . It follows that the heart C w =0 is a tensor category . Now ﬁx a second set of data a s ab ov e. (1’) A mo rphism u ′ : M ′ − → M ′ + in C b et w een M ′ − ∈ C w ≤ 0 and M ′ + ∈ C w ≥ 0 . (2’) An exact triangle C ′ v ′ − − → M ′ − u ′ − → M ′ + v ′ + − → C ′ [1] . Fix an exact triang le D − → M − ⊗ M ′ − u ⊗ u ′ − → M + ⊗ M ′ + − → D [1] . Prop osition 2.7. If C and C ′ ar e without weights − 1 and 0 , then so is D . I n other wor ds, the vali dity of Assumption 2.3 for u and u ′ implies the validity of Assumption 2.3 for u ⊗ u ′ . Pr o of. W e lea v e it t o the reader to ﬁrst construct an exact triangle D − → M + ⊗ C ′ v + ⊗ v ′ − − → C ⊗ M ′ − [1] − → D [1] , i.e., to show tha t D [1] is isomorphic to the cone of the morphism v + ⊗ v ′ − . Then, consider the morphisms δ − ⊗ v ′ − c ′ − : Gr 0 M − ⊗ C ′ ≤− 2 − → C ≤− 2 ⊗ M ′ − [1] and ( c + [1]) v + ⊗ δ ′ + : M + ⊗ C ′ ≥ 1 − → C ≥ 1 ⊗ G r 0 M ′ + [1] (notation as in Theorem 2.4). They are completed to give exact triangles D ≤− 2 − → Gr 0 M − ⊗ C ′ ≤− 2 − → C ≤− 2 ⊗ M ′ − [1] − → D ≤− 2 [1] 18 and D ≥ 1 − → M + ⊗ C ′ ≥ 1 − → C ≥ 1 ⊗ Gr 0 M ′ + [1] − → D ≤ 1 [1] . By compatibility o f w and ⊗ , a nd by Prop osition 1.4, the ob ject D ≤− 2 is o f w eigh ts ≤ − 2, and D ≥ 1 is o f w eights ≥ 1. Finally , it remains to construct an exact tria ngle D ≤− 2 − → D − → D ≥ 1 − → D ≤− 2 [1] . W e lea v e this to the reader (hint: use Theorem 2.4 (c)). q.e.d. Corollary 2.8. Under the hyp otheses of Pr op osition 2.7, the c anoni c al morphisms Gr 0  M − ⊗ M ′ −  − → Gr 0 M − ⊗ G r 0 M ′ − and Gr 0 M + ⊗ Gr 0 M ′ + − → Gr 0  M + ⊗ M ′ +  ar e isomorphisms. 3 Example: motiv es for mo dular forms In his art icle [Scho1], Scholl constructs the Grothendiec k motiv e M ( f ) for elliptic normalized newforms f of ﬁxe d lev el n and w eigh t w = r + 2, for p ositiv e in tegers n ≥ 3 and r ≥ 1 . I t is a direct factor of a Grothendiec k motiv e, whic h underlies a Cho w motiv e denoted r n W in [lo c. cit.] (this Cho w motiv e dep ends only on n and r ). In order to establish the relation of r n W to the theory of weigh ts, let us b egin b y setting up the notation. It is iden tical to the one in tro duced in [Sc ho1], up to one exception: the letter M used in [lo c. cit.] to denote certain sub-sc hemes of the mo dular curv e will c hange to S in order to a v oid confusion with the motivic no tation used earlier in the presen t pap er. Th us, for o ur ﬁxed n ≥ 3 and r ≥ 1, let S n ∈ S m/ Q denote the mo dular curv e parametrizing elliptic curves with lev el n structure, j : S n ֒ → S n its smo oth compactiﬁcation, and S ∞ n the complemen t of S n in S n . Th us, S ∞ n is of dimension zero. W rite X n → S n for the univ ersal elliptic curv e, and X n → S n for the univ ersal generalized elliptic curv e. Th us, X n is smo oth and pro p er o v er Q . The r -fo ld ﬁbre pro duct X r n := X n × S n × . . . × S n X n of X n o v er S n is singular fo r r ≥ 2, a nd can b e desingularized canonically [D1, Lemmas 5.4, 5.5] (see also [Scho1, Sect. 3]). D enote by X r n this desingularization ( X r n = X r n for r = 1). W rite X r n for the r -fold ﬁbre pro duct X n o v er S n . The symmetric gro up S r acts on X r n b y p ermutations, the r - th p ow er of the 19 group Z /n Z b y tra nslations, a nd the r - th p ow er of the group µ 2 b y in v ersion in the ﬁbres. Altogether [Sc ho1 , Sect. 1.1.1 ], this g iv es a canonical action of the semi-direct pro duct Γ r :=  ( Z /n Z ) 2 ⋊ µ 2  r ⋊ S r b y automor phisms on X r n . By the canonical nature of the desingularisation, this extends to an action of Γ r b y automorphisms on X r n . Of course, this action resp ects the o p en sub-sc heme X r n of X r n . As in [Sc ho1, Sect. 1.1.2], let ε : Γ r → {± 1 } b e the morphism whic h is trivial on ( Z /n Z ) 2 r , is the pro duct map on µ r 2 , and is t he sign c haracter on S r . Deﬁnition 3.1. (a) Let F denote the Z -algebra Z [1 / (2 n · r !)]. (b) Let e denote the idemp ot en t in the gr oup ring F [Γ r ] asso ciated to ε : e := 1 (2 n 2 ) r · r ! X γ ∈ Γ r ε ( γ ) − 1 · γ = 1 (2 n 2 ) r · r ! X γ ∈ Γ r ε ( γ ) · γ (observ e that ε − 1 = ε ). Let M b e an ob ject of an F -linear pseudo-Ab elian catego ry . If M comes equipped with an action of the group Γ r , let us a gree to denote by M e the direct factor o f M on whic h Γ r acts via ε (in other w ords, the image o f e ). Let us deﬁne the ob ject r n W as in [Sc ho1 , 1.2.2]. Deﬁnition 3.2. Denote b y r n W := M g m  X r n  e ∈ DM ef f g m ( Q ) F the image of the idemp o ten t e on M g m  X r n  . Giv en that X r n is sm o oth and prop er, w e see that r n W is an eﬀec tive Cho w motiv e ov er Q . As ab ov e, denote b y M g m  X r n  e and M c g m  X r n  e the imag es of e o n M g m  X r n  and M c g m  X r n  , resp ectiv ely . The follow ing can b e seen a s a translation into the lang uage of geometrical motive s of t he detailed analysis from [Scho1, Sect. 2, 3] of the geometry of the b oundary of X r n . Theorem 3.3. (a) The motive M c g m  X r n  e is without wei g ht 1 . In p ar- ticular, the obje ct Gr 0 M c g m  X r n  e is deﬁn e d. (b) Th e r estriction j r, ∗ n : r n W = M g m  X r n  e − → M c g m  X r n  e induc e d by the op en immersion j r n of X r n into X r n factors c anonic al ly thr ough an isom orphism Gr 0 j r, ∗ n : r n W ∼ − − → Gr 0 M c g m  X r n  e . 20 (c) Th e r e is an exact triangle in D M ef f g m ( Q ) F C r δ + − → G r 0 M c g m ( X r n ) e i 0 − → M c g m ( X r n ) e p + − → C r [1] , wher e C r = M g m  S ∞ n  [ r ] is pur e of weight r . The exact triangle is c anonic al up to a r ep l a c ement o f the triple of m o rphisms ( δ + , i 0 , p + ) by (( − 1) r δ + , i 0 , ( − 1) r p + ) . Before giving the pro o f of Theorem 3.3, let us list some of its consequences. First, duality for smo o th sc hemes [V1, Thm. 4.3.7 3] implies t he following. Corollary 3.4. (a) The m o tive M g m  X r n  e is without weight − 1 . In p articular, the obje ct Gr 0 M g m  X r n  e is deﬁne d. (b) Th e morphism j r n : M g m  X r n  e − → M g m  X r n  e = r n W factors c anonic al ly thr ough an isomorphism Gr 0 j r n : Gr 0 M g m  X r n  e ∼ − − → r n W . (c) Th e r e is an exact triangle in D M ef f g m ( Q ) F C − ( r +1) ι − − → M g m ( X r n ) e π 0 − → Gr 0 M g m ( X r n ) e δ − − → C − ( r +1) [1] , wher e C − ( r +1) = M g m  S ∞ n  ( r + 1)[ r + 1] is p ur e of weight − ( r + 1) . The exa c t triangle is c anonic al up to a r eplac ement of the triple of morphisms ( ι − , π 0 , δ − ) by (( − 1) r ι − , π 0 , ( − 1) r δ − ) . Remark 3.5. (a) It is w ell know n that the motive M g m  S ∞ n  is isomor- phic to a ﬁnite sum of copies of M g m ( Sp ec Q ( µ n )). (b) As the pro of will sho w, Theorem 3.3 (c) and Coro llary 3.4 (c) remain true for r = 0 if one replaces Gr 0 M c g m ( X r n ) e and Gr 0 M g m ( X r n ) e b y M g m ( S n ). Next, note that Theorem 3.3 (b) and Corollary 3 .4 (b) together imply the follo wing. Corollary 3.6. The c anonic al morphism M g m  X r n  e → M c g m  X r n  e fac- tors c anonic al ly thr ough an iso morphism Gr 0 M g m  X r n  e ∼ − − → Gr 0 M c g m  X r n  e . F or any ob ject M of D M ef f g m ( Q ) F , deﬁne motivic cohomolog y H p M  M , F ( q )  := Hom D M ef f gm ( Q ) F  M , Z ( q )[ p ]  . 21 When M = M g m ( Y ) fo r a sc heme Y ∈ S m/ Q , this giv es motivic cohomolo gy H p M  Y , Z ( q )  of Y , tensored with F = Z [1 / (2 n · r !)]. Th us for example, H r +1 M  C − ( r +1) , F ( r + 1)  = H 0 M  S ∞ n , Z (0)  ⊗ Z F . Similarly , H r +2 M  C − ( r +1) , F ( r + ℓ + 2)  = H 1 M  S ∞ n , Z ( ℓ + 1)  ⊗ Z F for any in teger ℓ . W e get the fo llo wing reﬁnemen t of [Sc ho1, Cor. 1 .4.1]. Corollary 3.7. L et ℓ ≥ 0 b e a se c ond inte ger. Then the ke rnel of the morphism H M ( ι − ) :  H r +2 M  X r n , Z ( r + ℓ + 2)  ⊗ Z F  e − → H 1 M  S ∞ n , Z ( ℓ + 1)  ⊗ Z F e quals H r +2 M  Gr 0 M g m ( X r n ) e , F ( r + ℓ + 2)  . Pr o of. This follo ws from the exact triangle of Corollary 3.4 (c), and from the v anishing of H 0 M  S ∞ n , Z ( ℓ + 1)  (since ℓ + 1 ≥ 1). q.e.d. Recall that following ideas of Beilinson [B], this result can b e employ ed as follows : using the Eisenstein symb ol deﬁned in [lo c. cit.] o ne constructs elemen ts in  H r +2 M  X r n , Z ( r + ℓ + 2)  ⊗ Z Q  e . By Corolla ry 3.7, linear com- binations o f suc h elemen ts v anishing under H M ( ι − ) lie in the sub- Q -ve ctor space H r +2 M  Gr 0 M g m ( X r n ) e , Q ( r + ℓ + 2)  . It can then b e sho wn that there are suﬃ cien tly man y suc h linear com binations, in the sense that their im- ages under the regulato r generate Deligne c ohomolo gy [Sc hn, Sect. 2 and 4] H r +2 D  Gr 0 M g m ( X r n ) e / R , R ( r + ℓ + 2)  . F urthermore, the Q -span of these im- ages has the relation to the leading co eﬃcien t at s = − ℓ of the L -function of Gr 0 M g m ( X r n ) e , predicted b y Beilinson’s conjecture. F or details, see forth- coming w ork of Sc holl [Scho2]. Pr o of of The or em 3.3. Denote b y X r, ∞ n the complemen t of the smo o th sc heme X r n in the smo oth and proper sc heme X r n . Localizatio n fo r the motiv e with compact support [V1, Prop. 4.1.5] shows that there is a canonical exact triangle M g m  X r n  e j r, ∗ n − → M c g m ( X r n ) e − → M g m  X r, ∞ n  e [1] − → M g m  X r n  e [1] . F ollowing the strategy from [Sc ho1 ], we shall show the follow ing claim. ( C ) M g m  X r, ∞ n  e = M c g m  X r, ∞ n  e ∼ − − → M c g m  S ∞ n  [ r ] = M g m  S ∞ n  [ r ] canonically up to a sign ( − 1) r . In particular, the motiv e M g m  X r, ∞ n  e is pure of w eigh t r ≥ 1. Claim (C) implies that t he ab ov e exact triangle is a w eigh t ﬁltration of M c g m ( X r n ) e a v oiding we ight 1. F urthermore, the restriction j r, ∗ n iden tiﬁes M g m  X r n  e with G r 0 M c g m ( X r n ) e . 22 T o sho w claim ( C ), o bserv e ﬁrst that the motivic v ersion o f [Sc ho1 , Sta te- men t 1.3.0] remains v a lid: for a n y S ∈ S m/ Q , there is a decomp osition in D M ef f g m ( Q ) F (in fact, a lready in D M ef f g m ( Q ) Z [1 / 2] ) M c g m ( G m × Q S ) ∼ = M c g m ( S )(1)[2] ⊕ M c g m ( S )[1] , suc h that in v ersion x 7→ x − 1 on G m acts on the ﬁrst factor by +1, a nd on the second b y − 1. The pro jection on to the ﬁrst factor is canonical, and the pro jection on to the s econd fa ctor is canonical up to a sign. Proof: lo calization [V1, Prop. 4.1.5] for the inclusion of G m in to the pro jectiv e line; the choice of the second pro jection is equiv alen t to the choice of o ne of the residue morphisms to 0 or ∞ . Fix one of the tw o choice s of pro jection π − : M c g m ( G m ) − → → M c g m ( Sp ec Q )[1] (and use the same notation fo r M c g m ( G m × Q S ) − → → M c g m ( S )[1] obtained b y base change via S ). Then to deﬁne the morphism j 0 , ∗ : M c g m  X r, ∞ n  e − → M c g m  S ∞ n  [ r ] , consider the follow ing. (i) The intersec tion X r, ∞ , reg n of X r, ∞ n with the non-singular part X r, reg n of X r n . Explanation: b y [Sc ho1, Thm. 3 .1.0 ii)], the desingularization X r n − → → X r n is an isomorphism o v er X r, reg n . Thus , X r, ∞ , reg n is an op en sub-sc heme o f X r, ∞ n , (ii) the neutral comp onent X r, ∞ , 0 n of X r, ∞ , reg n , i.e., its inters ection with the N ´ eron mo del of X r n . Th us, X r, ∞ , 0 n is an op en sub-sc heme o f X r, ∞ , reg n , (iii) the identiﬁcation o f X r, ∞ , 0 n with the r -th p ow er (ov er the ba se S ∞ n ) of the neutral comp onen t X 1 , ∞ , 0 n for r = 1. The latter can b e iden tiﬁed with G m × Q S ∞ n , canonically up to a n automorphism x 7→ x − 1 . Hence X r, ∞ , 0 n ∼ = G r m × Q S ∞ n , canonically up to an auto morphism ( x 1 , . . . , x r ) 7− → ( x 1 , . . . , x r ) − 1 . The three steps (i)–( iii) give an op en immersion j 0 : G r m × Q S ∞ n ֒ − → X r, ∞ n , whic h by con trav ariance of M c g m induces a morphism j 0 , ∗ : M c g m  X r, ∞ n  e − → M c g m  G r m × Q S ∞ n  . Its comp osition with the r -th p ow er of π − giv es t he desired mor phism M c g m  X r, ∞ n  e − → M c g m  S ∞ n  [ r ] , 23 equally denoted j 0 , ∗ , and canonical up to a sign ( − 1) r . In the con text of t wisted P oincar ´ e dualit y theories ( H ∗ , H ∗ ), Scholl’s main techn ical result [Sc ho1, Thm. 1.3.3] is equiv alent to stating that the mor phism induced b y j 0 , ∗ on the lev el of the theory H ∗ is an isomorphism. Our observ ation is simply that the same pro of as the one given in [lo c. cit.] runs thro ugh, with ( H ∗ , H ∗ ) replaced b y ( M g m , M c g m ). More pre cisely , [Scho1, Lemma 1.3.1] holds for M c g m , and hence the pro of of [Sc ho1, Prop. 2.4.1] runs through for M c g m . The latter result implies that the sc hemes o ccurring in a suitable stratiﬁcation of the complemen t of X r, ∞ , reg n in X r, ∞ n all ha v e trivial M c,e g m (cmp. [Sc ho1, pro of of Thm. 3.1.0 ii)]). This show s that step (i) induces an isomorphism M c g m  X r, ∞ n  e ∼ − − → M c g m  X r, ∞ , reg n  e . T o deal with step (ii) , one sho ws that the group Γ r acts tr ansitiv ely on the set of comp onen ts of the singular part of X r, ∞ , reg n , and that the stabilizer of eac h componen t admits a subgroup of order t w o acting trivially on the comp onen t, but ha ving trivial in tersection w ith the ke rnel of ε (cmp. [Sc ho1 , pro of of Thm. 3.1.0 iii)]). This sho ws ﬁrst that the singular par t of X r, ∞ , reg n do es not con tribute to M c,e g m , and then that M c g m  X r, ∞ , reg n  e − → M c g m  X r, ∞ , 0 n  e ′ is an isomorphism, whe re e ′ denotes t he pro jection onto the eigenspace for the restriction of ε t o the s ubgroup µ r 2 ⋊ S r of Γ r . T o conclude, w e apply the motivic v ersion of [Sc ho1, Lemma 1.3.1] to see that π ⊗ r − induces a n isomor- phism M c g m  G r m × Q S ∞ n  e ′ ∼ − − → M c g m  S ∞ n  [ r ]. q.e.d. Remark 3.8. (a) T he pro of of Theorem 3.3 sho ws that the op en immer- sion of the non- singular part X r, reg n of X r n in to X r n induces a n isomorphism M g m  X r n  e ∼ − − → M c g m  X r, reg n  e . In particular, M c g m  X r, reg n  e is a Cho w motiv e. By duality for smo o th sc hemes [V1, Thm. 4.3.7 3], M g m  X r, reg n  e − → M g m  X r n  e is an isomorphism, to o, and hence so is t he canonical morphism M g m  X r, reg n  e − → M c g m  X r, reg n  e . The construction of the motive s M ( f ) can therefore also b e done using the smo oth no n-prop er sc heme X r, reg n instead of X r n . (b) A slightly closer lo o k at the pro of of [Sc ho1, Thm. 3.1.0 ii)] r ev eals that the op en immersion of X r, reg n in to X r n also induces an isomorphism M g m  X r n  e ∼ − − → M c g m  X r, reg n  e . 24 By (a), M g m  X r n  e = M c g m  X r n  e is a Chow motiv e. The construction o f the motiv es M ( f ) can therefore also b e done using the non-smo oth (f or r ≥ 2) prop er sche me X r n instead of X r n . 4 W e igh ts, b o undary motiv e and in ter i or mo- tiv e This section con tains our main result (Theorem 4.3). W e list its main con- sequence s, a nd deﬁne in particular the motivic analogue of (certain direct factors of ) in terior cohomology (Deﬁnition 4.9). Throughout, w e assume k to admit resolution of singularities. Let us ﬁx X ∈ S m/k . The b oundary motive ∂ M g m ( X ) of X [W1, Def. 2.1 ] ﬁts in to a canonical exact triangle ( ∗ ) ∂ M g m ( X ) − → M g m ( X ) − → M c g m ( X ) − → ∂ M g m ( X )[1] in DM ef f g m ( k ). The alg ebra of ﬁni te c orr esp on d enc es c ( X, X ) acts on M g m ( X ) [V1, p. 190]. Denote by t c ( X , X ) the transposed algebra: a c ycle Z on X × k X lies in t c ( X , X ) if and only if t Z ∈ c ( X, X ). The deﬁnition o f comp osition of corresp ondences [lo c. cit.] show s that the in tersection c ( X , X ) ∩ t c ( X , X ) acts on M c g m ( X ). Deﬁnition 4.1. (a) D eﬁne the algebra c 1 , 2 ( X , X ) as the inters ection of the a lgebras c ( X , X ) and t c ( X , X ). As an Ab elian g roup, c 1 , 2 ( X , X ) is t h us free on the sym bo ls ( Z ), where Z runs through the integral closed sub- sc hemes of X × k X , suc h that bo th pro jections to t he components X are ﬁnite on Z , and map Z surjectiv ely to a connected comp onen t of X . Mul- tiplication in c 1 , 2 ( X , X ) is deﬁned by comp osition of corresp ondences as in [V1, p. 190]. (b) D enote b y t the canonical anti-in v olution on c 1 , 2 ( X , X ) mapping a cycle Z to t Z . It results directly from the deﬁnitions t hat the algebra c 1 , 2 ( X , X ) acts on the triangle ( ∗ ) in the sense that it acts on the three o b jects, and the morphisms are c 1 , 2 ( X , X )-equiv arian t. Denote by ¯ c 1 , 2 ( X , X ) the quotien t of c 1 , 2 ( X , X ) b y the ke rnel of this action. Fix a comm utativ e ﬂa t Z -algebra F , and an ide mp otent e in ¯ c 1 , 2 ( X , X ) ⊗ Z F . Denote b y M g m ( X ) e , M c g m ( X ) e and ∂ M g m ( X ) e the images of e on M g m ( X ), M c g m ( X ) and ∂ M g m ( X ), respectiv ely , considered as ob jects of the category DM ef f g m ( k ) F . W e ar e ready to set up the data (1), (2 ) considered in Section 2, for C := DM ef f g m ( k ) F . 25 (1) The morphism u is the mor phism M g m ( X ) e → M c g m ( X ) e . By Corol- lary 1.14 and condition 1.1 (1), the ob ject M g m ( X ) e b elongs indeed t o D M ef f g m ( k ) F ,w ≤ 0 , and M c g m ( X ) e to D M ef f g m ( k ) F ,w ≥ 0 . (2) Our ch oice of cone of u is ∂ M g m ( X ) e [1], together with the exact triangle ∂ M g m ( X ) e v − − → M g m ( X ) e u − → M c g m ( X ) e v + − → ∂ M g m ( X ) e [1] in D M ef f g m ( k ) F induced by ( ∗ ). Observ e that our da ta (1), (2) are stable under the natural action o f GC en ¯ c 1 , 2 ( X,X ) ( e ) :=  z ∈ ¯ c 1 , 2 ( X , X ) ⊗ Z F , z e = eze  . In particular, they are stable under the a ction of the cen tralizer C en ¯ c 1 , 2 ( X,X ) ( e ) of e in ¯ c 1 , 2 ( X , X ) ⊗ Z F . In this con text, Assumption 2.3 reads as follow s. Assumption 4.2. The direct factor ∂ M g m ( X ) e of the b oundary motive of X is without we ights − 1 and 0. Th us, we ma y a nd do ﬁx a weigh t ﬁltration C ≤− 2 c − − → ∂ M g m ( X ) e c + − → C ≥ 1 δ C − → C ≤− 2 [1] a v oiding w eigh ts − 1 and 0. Theorem 2.4, Corollary 2.5 and the adjunction prop ert y from Pro p osition 2 .2 then give the following. Theorem 4.3. Fix the data (1), (2), and supp ose Assumption 4.2. (a) The motive M g m ( X ) e is without w eight − 1 , and the motive M c g m ( X ) e is without weight 1 . In p articular, the eﬀe c tive Chow motives G r 0 M g m ( X ) e and Gr 0 M c g m ( X ) e ar e deﬁne d, and they c arry a n atur al action of GC en ¯ c 1 , 2 ( X,X ) ( e ) . (b) Th e r e ar e c anonic a l exact triangles (3) C ≤− 2 v − c − − → M g m ( X ) e π 0 − → Gr 0 M g m ( X ) e δ − − → C ≤− 2 [1] and (4) C ≥ 1 δ + − → Gr 0 M c g m ( X ) e i 0 − → M c g m ( X ) e ( c + [1]) v + − → C ≥ 1 [1] , which ar e stable unde r the natur al action of GC en ¯ c 1 , 2 ( X,X ) ( e ) . (c) T h er e is a c ano nic al isomorphism Gr 0 M g m ( X ) e ∼ − − → Gr 0 M c g m ( X ) e in C H M ef f ( k ) F . As a m o rp hism, it is uniquely determi n e d by the pr op erty of making the diagr am M g m ( X ) e u / / π 0   M c g m ( X ) e Gr 0 M g m ( X ) e / / Gr 0 M c g m ( X ) e i 0 O O 26 c ommute; in p articular, it is GC en ¯ c 1 , 2 ( X,X ) ( e ) -e quivariant. Its inv e rs e makes the diagr am C ≥ 1 δ C / / δ +   C ≤− 2 [1] Gr 0 M c g m ( X ) e / / Gr 0 M g m ( X ) e δ − O O c ommute. (d) L et N ∈ C H M ( k ) F b e a Chow motive. Then π 0 and i 0 induc e isomor- phisms Hom C H M ( k ) F  Gr 0 M g m ( X ) e , N  ∼ − − → Hom D M gm ( k ) F  M g m ( X ) e , N  and Hom C H M ( k ) F  N , Gr 0 M c g m ( X ) e  ∼ − − → Hom D M gm ( k ) F  N , M c g m ( X ) e  . (e) L et M g m ( X ) e → N → M c g m ( X ) e b e a factorization of u thr ough a Cho w motive N ∈ C H M ( k ) F . Then Gr 0 M g m ( X ) e = Gr 0 M c g m ( X ) e is c anonic al ly a dir e ct f a ctor of N , with a c anonic al dir e ct c ompleme n t. W e explicitly men tion the fo llo wing immediate consequence of Theo- rem 4.3. Corollary 4.4. Fix X and e , and supp ose that ∂ M g m ( X ) e = 0 , i. e ., that u : M g m ( X ) e ∼ − − → M c g m ( X ) e . Then M g m ( X ) e ∼ = M c g m ( X ) e ar e eﬀe ctive Ch ow motives. Of course, this also follo ws from Corollary 1.16. The a uthor kno ws of no proo f of Corollary 4.4 “a v oiding w eigh ts” when e 6 = 1. (F o r e = 1, we lea v e it to t he reader to show (using for example [V1, Cor. 4.2 .5]) that the assumption ∂ M g m ( X ) = 0 is equiv alen t to X being prop er.) Remark 4.5. It is not diﬃcult to see t hat Assumption 4.2 is actually implied by parts (a ) and (c) of Theorem 4.3. Henceforth, w e iden tify Gr 0 M g m ( X ) e and Gr 0 M c g m ( X ) e via the canonical isomorphism of Theorem 4.3 (c). Corollary 4.6. In the si tuation c onsider e d in The or em 4.3, let e X b e any smo oth c om p actiﬁc ation of X . Then G r 0 M g m ( X ) e is c anonic al ly a dir e ct factor of the C h ow motive M g m ( e X ) , with a c anonic al dir e ct c omplemen t. Pr o of. Indeed, the morphism u factors canonically through M g m ( e X ): M g m ( X ) e ֒ − → M g m ( X ) − → M g m ( e X ) − → M c g m ( X ) − → → M c g m ( X ) e . Hence w e may apply Theorem 4.3 (e). q.e.d. 27 Recall that the category C H M ( k ) F is pseudo-Ab elian. Thus , the con- struction of a sub-motiv e of M g m ( e X ) do es not a prio ri neces sitate the iden- tiﬁc ation , but only the existenc e of a complemen t. In our situation, Corol- lary 4.6 states that the compleme nt of Gr 0 M g m ( X ) e is canonical. This sho ws that Assumption 4.2 is indeed rather restrictiv e, an observ atio n conﬁrmed b y part (c) of the follo wing results on the Hodg e theoretic and ℓ -adic realiza- tions ([H, Sect. 2 a nd Corrigendum]; see [DGo, Sect. 1 .5] for a s impliﬁcation of this approac h). They can b e seen as applications of the c ohomo lo gic al weight sp e c tr al se quenc e [Bo2, Thm. 2.4.1, R em. 2.4 .2] in a very sp ecial case. Theorem 4.7. Ke ep the situation c onsider e d in The or em 4.3. Assume that k c an b e emb e dde d into the ﬁeld C of c omplex numb ers. Fix one such emb e ddin g . L et H ∗ b e the Ho dge the or etic r e alization [H, Cor. 2 .3.5 an d Corrigendum], fo l low e d by the c anonic al c ohomolo gy functor, i.e., the functor on DM ef f g m ( k ) F given by Betti c ohomolo gy of the top olo gic al sp ac e of C -value d p oints, tensor e d with Q ⊗ Z F , and with its natur al m i x e d Ho dge structur e. L e t n ∈ N . (a) Th e morphisms π 0 and i 0 induc e isom o rphisms H n  Gr 0 M g m ( X ) e  ∼ − − → W n H n  M g m ( X ) e  =  W n H n ( X ( C ) , Q ) ⊗ Z F  e and  H n c ( X ( C ) , Q ) ⊗ Z F  e  W n − 1 H n c ( X ( C ) , Q ) ⊗ Z F  e = H n ( M c g m ( X ) e ) W n − 1 H n ( M c g m ( X ) e ) ∼ − − → H n  Gr 0 M g m ( X ) e  . Her e , W r denotes the r -th ﬁltr ation step of the weight ﬁltr ation of a mixe d Ho d g e structur e (thus, the weights of H n ( X ( C ) , Q ) ar e ≥ n , and those of H n c ( X ( C ) , Q ) ar e ≤ n ). (b) The isomorph isms of (a) identify H n (Gr 0 M g m ( X ) e ) with the ima ge of the natur al morphism  H n c ( X ( C ) , Q ) ⊗ Z F  e − →  H n ( X ( C ) , Q ) ⊗ Z F  e . (c) The im age of ( H n c ( X ( C ) , Q ) ⊗ Z F ) e in ( H n ( X ( C ) , Q ) ⊗ Z F ) e e quals the lowest we i g ht ﬁ ltr ation step W n of ( H n ( X ( C ) , Q ) ⊗ Z F ) e . The reader should b e a w are t hat the a lgebra ¯ c 1 , 2 ( X , X ) acts con trav ari- an tly on Betti coho mology H n ( X ( C ) , Q ). The same remark applies o f course to the ℓ -adic realization, whic h w e consider no w. Theorem 4.8. Ke ep the situation c onsider e d in The or em 4.3 , an d ﬁx a prime ℓ > 0 . Assume that k is ﬁnitely gener ate d over its prime ﬁeld, and of char acteristic zer o. L et H ∗ b e the ℓ -adic r e alization [H, Cor. 2.3.4 and C orrigendum], fol lowe d by the c anonic al c ohomolo gy functor, i.e., the functor on D M ef f g m ( k ) F given by ℓ -adic c ohomo lo gy of the b ase c hange t o a ﬁxe d algebr aic closur e ¯ k of k , tensor e d with Q ℓ ⊗ Z F , and with its natur al action of the absolute Galois gr oup G k of k . L et n ∈ N . 28 (a) Th e morphisms π 0 and i 0 induc e isom o rphisms H n  Gr 0 M g m ( X ) e  ∼ − − → W n H n  M g m ( X ) e  =  W n H n ( X ¯ k , Q ℓ ) ⊗ Z F  e and  H n c ( X ¯ k , Q ℓ ) ⊗ Z F  e  W n − 1 H n c ( X ¯ k , Q ℓ ) ⊗ Z F  e = H n ( M c g m ( X ) e ) W n − 1 H n ( M c g m ( X ) e ) ∼ − − → H n  Gr 0 M g m ( X ) e  . Her e , W r denotes the r -th ﬁltr ation step of the weight ﬁltr ation of a G k - mo dule, and X ¯ k denotes the b ase change X ⊗ k ¯ k of X to ¯ k (thus, the weights of H n ( X ¯ k , Q ℓ ) ar e ≥ n , and those of H n c ( X ¯ k , Q ℓ ) ar e ≤ n ). (b) The isomorph isms of (a) identify H n (Gr 0 M g m ( X ) e ) with the ima ge of the natur al morphism  H n c ( X ¯ k , Q ℓ ) ⊗ Z F  e − →  H n ( X ¯ k , Q ℓ ) ⊗ Z F  e . (c) T h e image of ( H n c ( X ¯ k , Q ℓ ) ⊗ Z F ) e in ( H n ( X ¯ k , Q ℓ ) ⊗ Z F ) e e quals the lowest weight ﬁltr ation step W n of ( H n ( X ¯ k , Q ℓ ) ⊗ Z F ) e . Deﬁnition 4.9. Fix the data (1), (2), and supp ose Assumption 4.2. W e call Gr 0 M g m ( X ) e the e -p art of the interior mo tive of X . This terminology is motiv ated by parts (b) of Theorems 4 .7 and 4.8, whic h show that after passage to ra tional co eﬃcien ts, the r ealizations of Gr 0 M g m ( X ) e are classes of complex es, whose cohomology equals the part of the inte rior cohomology of X ﬁxed b y e . Pr o of of The or em s 4.7 and 4. 8 . Consider the exact triangle (3) C ≤− 2 − → M g m ( X ) e π 0 − → Gr 0 M g m ( X ) e − → C ≤− 2 [1] from Theorem 4.3 (b). Recall that as suggested b y the notatio n, the motive C ≤− 2 is of w eigh ts ≤ − 2. The cohomological functor H ∗ transforms it in to a long exact sequence H n − 1  C ≤− 2  − → H n  Gr 0 M g m ( X ) e  H n ( π 0 ) − → H n  M g m ( X ) e  − → H n  C ≤− 2  . The essen tial information w e need to us e is that H n transforms Cho w motiv es in to ob jects whic h are pure of w eigh t n , for any n ∈ N . Since C ≤− 2 admits a ﬁltra tion whose cones are Cho w motiv es sitting in degrees ≥ 2 , the ob ject H n ( C ≤− 2 ) admits a ﬁltration whos e graded pieces are of w eights ≥ n + 2. Since our coeﬃcien ts ar e Q -v ector space s, the re are no non-trivial morphis ms b et w een ob jects of disjoin t w eigh ts. The a b o ve long exact sequence then sho ws that H n ( π 0 ) : H n  Gr 0 M g m ( X ) e  − → H n  M g m ( X ) e  is injectiv e, and it s image is iden tical to the part of w eigh t n of H n ( M g m ( X ) e ). This show s the pa rt of claim (a) concerning π 0 . No w recall t hat the realization functor is compatible with the tensor struc- tures [H, Cor. 2.3.5, Cor. 2.3 .4], and sends t he T at e motiv e Z (1) to the dual 29 of the T ate ob ject [H, Thm. 2.3.3]. It follows that it is compatible with dualit y . Since on the o ne hand, M g m ( X ) a nd M c g m ( X ) a re in dualit y [V1, Thm. 4.3.7 3 ], and on the other hand, the same is true for Be tti, resp. ℓ -adic cohomology and Betti, resp. ℓ -adic cohomology with compact supp ort, w e see that H n sends t he motive with compact supp o rt M c g m ( X ) to cohomology with compact supp ort H n c of X . No w rep eat the ab ov e argumen t for the exact triangle (4) C ≥ 1 − → Gr 0 M c g m ( X ) e i 0 − → M c g m ( X ) e − → C ≥ 1 [1] from Theorem 4.3 ( b). This shows the remaining part o f claim (a) . Claims (b) and (c) follo w, once w e observ e that the comp osition of H n ( i 0 ) and H n ( π 0 ) equals the canonical morphism from cohomology with compact supp ort to cohomolog y without supp ort. q.e.d. Remark 4.10. (a) The pro of of Theorems 4 .7 and 4.8 uses the fa ct that in the resp ectiv e target categories (mixed Ho dge structures in Theorem 4.7, Galois represen tations in Theorems 4.8), there are no non-trivial morphisms b et w een ob jects of disjoin t w eigh ts. This is true a s long as w e w ork with co eﬃcien ts which are Q -v ector spaces. Note that rece nt w ork of Lecom te [L] establishes the existenc e of a Betti realization which do es not require the passage to Q -co eﬃcien t s. I n pa rticular [L, Thm. 1.1], for a smo oth quasi-pro jectiv e v ariet y X , it yields the classical singular cohomology of the top ological space X ( C ). W e ha v e no statemen t (and not ev en a gues s) to oﬀer on the image of Gr 0 M g m ( X ) e under the realizat ion of [lo c. cit.]. (b) The author do es not k now whether for general Y ∈ S m/k it is possible (o r ev en reasonable to exp ect) to ﬁnd a complex computing in terior (Betti or ℓ - adic) coho mology of Y , and through whic h the natural morphism R Γ c ( Y ) → R Γ( Y ) factors. Remark 4.11. When k is a nu mber ﬁeld, Theorems 4.7 (b) and 4.8 (b) tell us in particular that the L - function of the C how motiv e Gr 0 M g m ( X ) e is computed via (the e - part of ) in terior cohomolo gy of X . Example 4.12. Let C b e a smo oth pro jective curv e, and P ∈ C a k - rational p oin t. Put X := C − P . L o calization [V1, Prop. 4.1.5] shows that there is an exact triangle M g m ( P ) − → M g m ( C ) − → M c g m ( X ) − → M g m ( P )[1] . The morphism M g m ( P ) → M g m ( C ) is split; hence M c g m ( X ) is a direct factor of M g m ( C ) . It is therefore a Cho w motiv e. By dua lit y [V1, Thm. 4.3.7 3], the same is t hen true for M g m ( X ). In particular, b oth M g m ( X ) and M c g m ( X ) are pure of we ight zero. But the morphism u : M g m ( X ) → M c g m ( X ) is not an isomorphism (lo ok at degree 0 o r − 2, or c hec k that the conclusions of Theorems 4.7 (c) and 4.8 (c) do not hold). Therefore, Assumption 4.2 is not 30 fulﬁlled for e = 1. Of course, this can b e seen direc tly: the boundary motiv e ∂ M g m ( X ) has a w eigh t ﬁltration Z (1)[1] − → ∂ M g m ( X ) − → Z (0) − → Z (1)[2] (whic h is necessarily split since t here are no non-trivial morphisms from Z (0) to Z (1)[2]). Orthogonalit y 1.1 (3) then show s that there are no non-trivial morphisms from an ob ject of w eights ≤ − 2 to ∂ M g m ( X ), and no non-trivial morphisms from ∂ M g m ( X ) to an ob ject of w eights ≥ 1. The ob ject ∂ M g m ( X ) b eing non-trivial, w e conclude that it do es not a dmit a weigh t ﬁltration a v oiding we ights − 1 and 0. More generally , w e hav e the fo llo wing, whic h again illustrates just ho w restrictiv e Assumption 4.2 is. Prop osition 4.13. F ix data ( 1 ) a n d ( 2 ) a s b efor e. Assume that X admits a sm o oth c omp a ctiﬁc ation e X such that the c omplemen t Y = e X − X is smo oth. Then the fol lowing statements ar e e quivalent. (i) Assumption 4.2 is valid, i.e., the obje ct ∂ M g m ( X ) e is without weights − 1 and 0 . (ii) The obje ct ∂ M g m ( X ) e is trivial (he n c e the c onclusion of Cor ol lary 4 .4 holds). Pr o of. Statemen t (ii) clearly implies (i). In order to sho w that it is implied by (i), let us show that the hy p ot hesis on e X and Y for ces the b oundary mot iv e ∂ M g m ( X ) to lie in the inters ection D M ef f g m ( k ) w ≥− 1 ∩ D M ef f g m ( k ) w ≤ 0 . By orthogonality 1.1 (3), the same is then true for its direct factor ∂ M g m ( X ) e . Th us, the o nly w a y for ∂ M g m ( X ) e to a v oid w eights − 1 and 0 is to b e trivial (again by orthogonality). In order to show our claim, apply [W1, Prop. 2.4 ] to see that ∂ M g m ( X ) is isomorphic t o the s hift b y [ − 1] of a c hoice of cone o f the c anonical morphism M g m ( Y ) ⊕ M g m ( X ) − → M g m ( e X ) . In pa rticular, there is a morphism c + : ∂ M g m ( X ) → M g m ( Y ), and an exact triangle ( W ) C − − → ∂ M g m ( X ) c + − → M g m ( Y ) − → C − [1] , where C − equals the shift b y [ − 1] o f a cone of M g m ( X ) − → M g m ( e X ) . By assumption, Y is smo oth and pro p er, hence M g m ( Y ), as a Cho w motiv e, is pure o f weigh t 0. Duality fo r smo oth sc hemes [V1, Thm. 4.3.7 3] shows that C − is pure o f w eigh t − 1. Hence ( W ) is a w eigh t ﬁltration of ∂ M g m ( X ) 31 b y ob jects of weigh ts − 1 a nd 0. Prop osition 1.4 then sho ws that ∂ M g m ( X ) b elongs indeed to D M ef f g m ( k ) w ≥− 1 ∩ D M ef f g m ( k ) w ≤ 0 . q.e.d. Corollary 4.6 allow s t o sa y more a b out the ´ etale realizations. Theorem 4.14. Ke ep the situation c onsider e d in The or em 4.3, and ﬁx a prime ℓ > 0 . Assume that k i s the quotient ﬁeld of a De dekind domain A , and that k is of char acteristic zer o. Fix a non-z er o prime ide al p of A , and let p denote its r esidue c h ar acteristic. (a) Assume that p 6 = ℓ . Then a suﬃcient c o n dition for the ℓ -adic r e aliza- tion H ∗ (Gr 0 M g m ( X ) e ) to b e unr amiﬁe d at p is the e x i stenc e of some smo oth c omp actiﬁc ation o f X having go o d r e duction at p . A s uﬃci e nt c ondition for H ∗ (Gr 0 M g m ( X ) e ) to b e se mi-stable at p is the existenc e of s o me s m o oth c om- p actiﬁc ation of X hav i n g simp le semi-stable r e duction at p . (b) Assume that p = ℓ , and that the r esidue ﬁeld A/ p is p erfe ct. Then a suf- ﬁcient c ondition fo r the p -a d ic r e aliza tion H ∗ (Gr 0 M g m ( X ) e ) to b e crystal l i ne at p is the existenc e of some s m o oth c omp actiﬁc ation of X having go o d r e- duction at p . A s uﬃci e nt c ondition for H ∗ (Gr 0 M g m ( X ) e ) to b e semi- s table at p is the existenc e o f some smo o th c omp actiﬁc ation of X having simple semi-stable r e duction at p . Pr o of. By Coro llary 4.6, G r 0 M g m ( X ) e is a direct fa ctor of the motiv e M g m ( e X ) of a n y smo oth compactiﬁcation e X of X . Hence H ∗ (Gr 0 M g m ( X ) e ) is a direct factor of the cohomology of an y suc h e X . P art (a) uses the spectral seque nce [D2, Sc holie 2.5] relating cohomo- logy with co eﬃcien ts in t he v anishing cycle shea v es ψ q to cohomology of the generic ﬁbre e X . By [RZ, Kor. 2.25] ([D2, Th m. 3.3 ] when p = 0), o ur assumption on the red uction of e X implies that the inertia group acts trivially on the ψ q . It therefore acts unip o ten tly o n the cohomolog y of e X . P art (b) follo ws from the C st -conjecture, prov ed by Tsuji [T, Thm. 0.2] (see also [N] for a pro of via K - theory). q.e.d. By [V1, Thm . 4.3.7], the category D M g m ( k ) F is a rigid tensor triangu- lated category . F urthermore, if X is smo oth of pure dimension n , then the ob jects M g m ( X ) and M c g m ( X )( − n )[ − 2 n ] are canonically dua l to each other. By [W1, Thm. 6.1], the b oundary motiv e ∂ M g m ( X ) is canonically dual to ∂ M g m ( X )( − n )[ − (2 n − 1)]. F urthermore, these dualities ﬁt together t o giv e an identiﬁc ation of the dual of the exact triang le ( ∗ ) ∂ M g m ( X ) − → M g m ( X ) − → M c g m ( X ) − → ∂ M g m ( X )[1] and the exact triangle ( ∗ )( − n )[ − 2 n ]. The construction of the dualit y isomor- phism [lo c. cit.] show s t hat the dual of the action of the algebra ¯ c 1 , 2 ( X , X ) on 32 ( ∗ ) equals the nat ural (an ti-) action giv en by the comp osition of the canoni- cal action on ( ∗ )( − n )[ − 2 n ], preceded by the anti-in v olution t . Consider the idemp oten t t e (i.e., the transp o sition of e ). Prop osition 4.15. (a) Assumption 4.2 is e quivalent to any of the fo l- lowing statements. (i) Both ∂ M g m ( X ) e and ∂ M g m ( X ) t e ar e without weight − 1 . (ii) Both ∂ M g m ( X ) e and ∂ M g m ( X ) t e ar e without weight 0 . In p articular, Assumption 4.2 is satisﬁe d for e if and onl y if it is satisﬁe d for the tr ansp osition t e . (b) Assume that e is symm etric, i.e., that e = t e . T hen Assumption 4.2 is e quivalent to any of the fol lowing statements. (i) ∂ M g m ( X ) e is without weight − 1 . (ii) ∂ M g m ( X ) e is without weight 0 . Pr o of. Indeed, ∂ M g m ( X ) e is dual to ∂ M g m ( X ) t e ( − n )[ − (2 n − 1)]. No w observ e that Z ( − n )[ − (2 n − 1)] is pure of w eigh t 1. The n use unicit y of w eight ﬁltrations av oiding weigh t − 1 resp. w eigh t 0 (Corolla ry 1.9). q.e.d. Example 4.16. Assume that a n abstract group G acts b y automorphisms on X . (a) The actio n of G translates in to a morphism of algebras Z [ G ] → c 1 , 2 ( X , X ). This morphism transforms the na tural anti-in v olution ∗ of Z [ G ] induced by g 7→ g − 1 in to the anti-in v olution t of c 1 , 2 ( X , X ). (b) Let F b e a ﬂat Z -alg ebra, and e a n id emp otent in F [ G ]. The m orphism of (a) then allows to consider the image of e (equally denoted b y e ) in c 1 , 2 ( X , X ), then in ¯ c 1 , 2 ( X , X ), and to ask whether Assumption 4.2 is v alid for e . (c) As a sp ecial case o f (b), consider the case when G is ﬁnite, it s or der r is in v ertible in F , and e is the idemp oten t in F [ G ] associated to a c haracter ε on G with v alues in t he m ultiplicativ e group F ∗ : e = 1 r X g ∈ G ε ( g ) − 1 · g . Observ e that the idemp oten t e ∈ ¯ c 1 , 2 ( X , X ) is symmetric if ε − 1 = ε . (d) Let us consider the situation from Section 3, and sho w that Assump- tion 4.2 is satisﬁed for X = X r n ∈ S m/ Q and e = 1 (2 n 2 ) r · r ! X γ ∈ Γ r ε ( γ ) − 1 · γ . 33 As in Section 3, denote by X r, ∞ n the complemen t of X r n in X r n . By [W1, Prop. 2.4], the ob ject ∂ M g m ( X r n ) e is canonically isomorphic to the shift b y [ − 1] of a canonical c hoice of cone of the canonical morphism M g m  X r, ∞ n  e ⊕ M g m ( X r n ) e − → M g m  X r n  e . In par ticular, there is a canonical morphism c + : ∂ M g m ( X r n ) e → M g m  X r, ∞ n  e , and an exact tria ngle C − − → ∂ M g m ( X r n ) e c + − → M g m  X r, ∞ n  e − → C − [1] , where C − equals the shift b y [ − 1] o f a cone of j r n : M g m ( X r n ) e − → M g m  X r n  e . By Corollary 3.4 (b) and (c), C − ∼ = M g m  S ∞ n  ( r + 1)[ r + 1] is pure of w eigh t − ( r + 1). It f ollo ws from t his and f rom Corollary 1 .14 (a) that the exact triang le C − − → ∂ M g m ( X r n ) e c + − → M g m  X r, ∞ n  e = M c g m  X r, ∞ n  e − → C − [1] is a w eight ﬁltration of ∂ M g m ( X r n ) e a v oiding w eights − r , . . . , − 1, and hence in particular, a v oiding w eigh t − 1 (since r ≥ 1) . Our claim then follo ws from Prop osition 4.15 (b) (observ e that e is symmetric). (Alternativ ely , use Claim (C) of the pro of o f The orem 3.3, to s ee directly that the last term of the w eigh t ﬁltra tion of ∂ M g m ( X r n ) e , M g m  X r, ∞ n  e ∼ = M g m  S ∞ n  [ r ] is pure of we ight r ≥ 1.) (e) As a b y-pro duct of the ab ov e iden tiﬁcation of the w eigh t ﬁltration of ∂ M g m ( X r n ) e , we see that the cohomolo gical Betti r ealization H i ( ∂ M g m ( X r n ) e ) [L], tensored with Z [1 / (2 n · r !)], is without tor sion for all integers i . W e th us reco v er a result of Hida’s (see [Gh, Prop. 3]): the o dd primes p dividing the torsion of the ( e -part of the) b oundary c ohomolo gy of X r n [Gh, Sect. 3.2] satisfy p ≤ r or p | n . Remark 4.17. Let us agree to forget the results f rom Section 3, a nd see what the theory dev elop ed in the presen t section implies in the situation studied in Example 4 .16 (d). W e only use the v alidit y of Assumption 4.2. (a) It follows formally from Theorem 4.3 (a)–(c) that Gr 0 M g m ( X r n ) e and Gr 0 M c g m ( X r n ) e are deﬁned, a nd canonically isomorphic, and that there are exact tria ngles C − − → M g m ( X r n ) e π 0 − → Gr 0 M g m ( X r n ) e δ − − → C − [1] 34 and M g m  X r, ∞ n  e δ + − → G r 0 M c g m ( X r n ) e i 0 − → M c g m ( X r n ) e − → M g m  X r, ∞ n  e [1] . (b) It follo ws fo rmally from Corollary 4.6 that Gr 0 M g m ( X r n ) e is a direct factor of r n W = M g m ( X r n ) e , with a canonical complemen t. Call this complemen t N . It f ollo ws f ormally from Theorems 4.7 (b) and 4.8 (b) tha t the re alizations of the motiv e Gr 0 M g m  X r n  e equal in terior cohomology , i.e., the image of the morphism H n c  X r n  e − → H n  X r n  e . By [Scho1, Sect. 1.2.0, Thm. 1.2.1 , Sect. 1.3.4], the same is true for r n W . Therefore, the complemen t N has trivial r ealizations. Th us, its underly- ing Gro thendiec k motiv e is trivial. This means that the construction of the Grothendiec k motiv es for mo dular fo rms M ( f ) can b e done r eplacing the Cho w motiv e r n W by Gr 0 M g m  X r n  e . (c) As recalled in [A, Sect. 11.5.2], V o evodsky’s nilp otenc e c onje ctur e implies that homological equiv alence equals smash-nilp oten t equiv alence. Therefore [A, Cor. 11.5.1 .2], it implies that the forgetful f unctor from Cho w motiv es to Grothendiec k motive s is conserv ative. Th us, the nilp ot ence conjecture gives a h ypothetical abstract reason f or the complemen t N o f Gr 0 M g m  X r n  e in r n W t o b e zero, whic h would mean that Gr 0 M g m  X r n  e = r n W . (d) Rec all that Sc holl’s motiv es M ( f ) are constructed out of r n W using cy cles, whic h are only know n to b e idemp otent after passage to the Grothendiec k motiv e underlying r n W : indee d, the Eich ler–Shimura isomorphism allo ws fo r a con trol of the action of the Hec k e algebra on the rele v an t cohomology gro up. In particular, there are only ﬁnitely man y eigen v alues, a fa ct whic h is need ed for the construction of the pro jector to the eigenspace corresp onding to f . The nilp o tence conj ecture implies [A, Cor. 11.5.1.2] that idempo ten ts can b e lifted from Grot hendiec k to Chow mot iv es. Hence its v alidit y would mean that Scholl’s cycles can b e mo diﬁed b y terms homologically equiv alen t to zero, t o give idemp oten t endomorphisms of r n W . This w ould pro duce Chow motiv es, whose underlying G rothendiec k motives are the M ( f ). (e) By [K, Cor. 7.8], the conclusion fro m (d) holds under an assumption, whic h is a priori w eak er than the nilp otence conjecture: the ﬁnite dimen- sionality [K, Def. 3.7] of the Chow motiv e r n W . By [K, T hm. 4.2, Cor. 5.11], ﬁnite dimensionality is known for motiv es of curv es and motive s of Ab elian v ar ieties. Unfortunately , the metho ds o f [lo c. cit.] do not seem to admit an immediate generalization to families of Ab elian v ar ieties ov er curve s, or degenerations of suc h families (lik e X r n ). Remark 4.18. (a) O f course, none of the implications listed in parts (a)–(c) of the pr eceding remark are new: they are all consequences of Theo- rem 3.3, Coro llary 3.4 and Corollary 3.6, whose pro of in v olv es the geometry of the b oundar y of the smo oth compactiﬁcation X r n of X r n . Observ e that 35 some of these results w ere ev en used in our pro of 4.1 6 (d) o f the v alidity of Assumption 4.2. In ot her w ords, w e applied the strategy f rom Remark 4.5, and prov ed Assumption 4 .2 via pa rts (a) a nd (c) of Theorem 4.3. (b) F or Shim ura v arieties of higher dimension, Heck e-equiv arian t smo oth compactiﬁcations (lik e X r n in the case of p ow ers o f the u niv ersal elliptic curv e o v er a mo dular curve) are not know n (and may b e not reasonable to exp ect) to exist. 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