Boundary motive, relative motives and extensions of motives

Boundary motiv e, relativ e motiv es and extensions of moti v es by J¨ org Wildeshaus ∗ LA GA UMR 7539 Institut Galil´ ee Univ er sit´ e P ar is 13 Aven ue Jean-Baptiste Cl ´ ement F-9343 0 Villetaneuse F rance wildes h@mat h.univ-paris13.fr Ma y 5, 20 11 Abstract W e explain the role of the b ound ary motiv e in the construction of certain Cho w motiv es, and of extensions of Chow mo tives. Our t wo main examples concern prop er, singular surfaces and ﬁ b re pro du cts o f a unive rs al elliptic curv e. Keyw ords : w eigh t structures, b oundary motiv e, relativ e motiv es, in- tersection motiv e, interior motiv e, extensions of motiv es. Math. Sub j. Class. (20 10) num b ers: 19E15 (1 4F25, 1 4F42, 14G35). ∗ Partially suppo rted by the A genc e Nationale de la R e cher che , pro ject no. ANR-0 7- BLAN-0142 “M´ etho des ` a la V o evo ds ky , motifs mixtes et G´ eo m´ etrie d’Ara kelo v”. 1 Con tents 0 In tro duction 2 1 Motiv ation 6 2 Relativ e motiv es and functor ialit y of the b oundary motiv e 22 3 Motiv es asso ciated to Ab elian sc hemes 35 4 The intersection motive of a surface 37 5 The interior motiv e of a product of univ ersal elliptic curv es 42 0 In tro ducti o n This art icle con ta ins largely extended notes of a short series of lectures de- liv ered during the Ec ole d’´ et ´ e fr anc o-asiatique “Autour des m otifs” , whic h to ok pla ce at the IHES in July 2006 . The task whic h I was assigned w as to explain the role of the b oundary motive , and I hop e that the presen t article will mak e a mo dest con tributio n to this eﬀect. By deﬁnition [W1], the b oundary motiv e ∂ M ( X ) of a v ariet y X o v er a p erfect ﬁeld k ﬁts in to a canonical exact triangle ( ∗ ) ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1 ] in the category DM ef f g m ( k ) of eﬀectiv e ge o metric al mo tives . This triangle establishes the relation of the b oundary motive to M ( X ) and M c ( X ), the motive o f X and its motive with c omp act supp ort , respective ly [V1]. One w ay to explain its in terest is to start with the notion of extensions. Indeed, most of the existing attempts to pr ov e the Beilinson or Blo ch – K a to conjectures on sp ecial v alues of L -functions necessitate the construction of extensions of (Chow ) motiv es, and the explicit con trol o f their realizations (Betti, de Rham, ´ etale...). Often, the source of these extensions is lo c aliza- tion , whic h expresses the motiv e with compact supp ort of a non-compact v ar iety X as an extension o f the motiv e of a compactiﬁcation X ∗ b y the motiv e of the complemen t X ∗ − X . The realizations of these extensions then corresp ond to cohomo lo gy with compact supp ort of X . This approac h is clearly presen t e.g. in Harder’s w ork on sp ecial v alues [H ]. Th us, giv en t w o Chow motives , one ma y try to use lo calization to con- struct an extension of one b y the other. Here, w e base ourselv es on the principle that the giv en Chow motiv es are “basic”, and that the extension 2 is “diﬃcult” to obtain. But one may also in v ert the logic: giv en a “mixed” motiv e, try to use lo calization to construct the Chow motiv es used to build it up; let us refer to this problem as “resolution of extensions”. The purp ose of this article is to establish that the b oundary motive pla ys a role b oth for the construction and for the resolution of extensions via lo- calization. In Section 1, w e start by making precise the relation b et w een lo calization and the b o undary motive. In fact, the triangle ( ∗ ) turns out to b e obtained b y “splicing” the lo calization tria ng le a nd its dual. W e c hose to discuss this relation ﬁrst in t he Ho dge theoretic realization, a nd in the sp ecial case of a complemen t X of tw o p oints in an elliptic curv e o ve r C (Examples 1.1, 1.3 and 1.5), and deduce from that discussion the general picture in Ho dge theory ( Theorems 1.6 and 1.7), concerning compactiﬁca- tions of a ﬁxed v ariet y X ov er C . W e observ e in particular (Corollary 1.8) that when X is smo oth, then an y smo oth compactiﬁcation induces a weight ﬁltr a tion on the b oundary cohomology of X , i.e., on the Ho dge realization of t he b oundary motiv e. In order to formulate the motivic analogues of these results, w e need the righ t notion of w eights fo r motive s. It turns o ut that t his notion is give n by weight structur es , as recen tly intro duced and studied b y Bondarko [Bo2]. W e review the deﬁnition, and the basic prop erties of w eigh t structures, including their application to motiv es (Theorem 1.11): a ccording to Bondark o, there is a canonical suc h structure on t he triangulated category D M ef f g m ( k ), and its he art equals t he category C H M ef f ( k ) of eﬀectiv e Chow motiv es. The motivic a nalogue o f Corollary 1.8 holds: according to Corolla ry 1.16, any smo oth compactiﬁcation of a ﬁxed v ariet y X whic h is smo oth ov er k induces a w eigh t ﬁltration on ∂ M ( X ). Then w e try to inv ert this pro cess (hoping for this in ve rsion to allow us to resolv e extens ions). The precise statemen t is given in Theorem 1.18, whic h states that for ﬁxed X , there is a canonical bijectiv e corresp ondence (dis- cussed at length in Construction 1.17) b et wee n isomorphism classes of tw o t yp es of ob jects: (1) w eigh t ﬁltrations on ∂ M ( X ), and (2) certain eﬀectiv e Cho w motiv es M 0 through which the morphism M ( X ) → M c ( X ) factors. An analogous statemen t (V arian t 1.23) holds fo r direct factors of ∂ M ( X ), M ( X ), and M c ( X ), pro vided that they are imag es o f an idemp oten t endo- morphism of the whole exact triangle ( ∗ ). In this correspo ndence, the passage to isomorphism classes cannot b e a v o ided b ecause of the necessit y to cho ose cones of certain morphisms in the triangulated category D M ef f g m ( k ). This causes (at least) one imp orta n t pro blem, namely the la ck of functoriality of the represen tative s of the isomor phism classes. In order to obtain f unctori- alit y , Construction 1.17 th us needs to b e rigidiﬁed. In the rest of Section 1, w e describ e the approac h from [W3] to rigidi- 3 ﬁcation, hence functoriality . It is based on the notion of motives avoidin g c ertain weig hts . If a direct factor ∂ M ( X ) e of ∂ M ( X ) is without w eigh ts − 1 and 0, then an eﬀectiv e Chow motive M 0 is cano nically and functorially deﬁned (Complemen t 1.24). Giv en t he nature o f the realizat io ns of M 0 , it is natural to call it the e -p art of the interior motive of X . Its main prop erties are established in [W3, Sect. 4]. Note ho we ver (Problem 1.2 2) that t he ab ov e condition on absence of w eights is nev er satisﬁed for the whole of ∂ M ( X ) — unless ∂ M ( X ) = 0 . In order to make this approac h work, w e th us need an idemp oten t endomorphism e of the exact t r ia ngle ( ∗ ), giving rise to a direct factor ∂ M ( X ) e − → M ( X ) e − → M c ( X ) e − → ∂ M ( X ) e [1] . Section 2 shows ho w t he theory o f smo oth r elative Chow motives can b e emplo y ed to construct endomorphisms of the exact triangle ( ∗ ). Fix a base sc heme S , whic h is smo oth ov er k . Theorem 2.2 establishes t he existence of a functor f rom the category of smo ot h relative Chow motiv es ov er S to the category of exact triangles in D M ef f g m ( k ). On ob jects, it is giv en b y mapping a prop er, smo oth S -sc heme X to the exact triangle ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X ) [1 ] . W e should mention that as far as the M ( X )-comp onen t is concerned, the functorialit y statement from Theorem 2.2 is just a sp ecial feature of results b y D´ eglise [D´ e2], Cisinski–D ´ eglise [CiD´ e ] and Levine [L] (see Remarks 2.3 and 2 .13 for details). Ho we ver, the application of the results from [lo c. cit.] to the functor ∂ M is not ob vious. This is o ne of the reasons wh y w e follow an alternativ e approach. I t is based on a relative v ersion of moving c ycles [W1, Thm. 6 .1 4]. This also explains why w e are forced to supp ose the ba se ﬁeld k to admit a strict v ersion o f resolution of singularities. Theorem 2.5 and Corollary 2.15 then analyze the b ehav iour of the functor from Theo- rem 2.2 under change of the base S . Another reason for us t o ch o ose a cycle t heoretic approach w as that it b ecomes then easier to k eep tra c k of the corresp ondences on X × k X comm uting with our constructions. Our main application (Example 2.16) thus concerns corresp ondences “of Hec k e ty p e” yielding endomorphisms of the exact tria ngle ( ∗ ). In Section 3, w e apply these principles to Ab elian sche mes. M o r e pre- cisely , the main result of [DeMu] on the Cho w–K ¨ unneth decomp o sition of the relativ e motiv e of an Ab elian sc heme A o ve r S (recalled in Theorem 3.1) yields canonical pro jectors in t he r elat ive Cho w group. Giv en our analysis from Section 2, it follows that they act idemp oten tly on the exact triang le ∂ M ( A ) − → M ( A ) − → M c ( A ) − → ∂ M ( A )[1] . In Sections 4 and 5, w e discuss t w o examples. Section 4 concerns nor- mal, prop er surfaces X ∗ . W e ﬁrst recall the basic construction of t he in- terse ction m otive M ! ∗ ( X ∗ ) of X ∗ , follo wing previous work of Cataldo and 4 Migliorini [CatMi ], and r eview some of the material from [W5]. In particular (Prop osition 4.3), w e recall that M ! ∗ ( X ∗ ) is co- and con trav ariantly functo- rial under ﬁnite mor phisms of prop er surfaces. W e then analyze the precise relation to the weigh t ﬁltration o f the b oundary motive of a dense, op en sub- sc heme X ⊂ X ∗ , which is smo oth ov er k (Theorem 4.4) , fo llowing t he lines of Construction 1.17. W e ﬁnish the section with a discus sion of the case o f Baily–Borel compactiﬁcations of Hilb ert surfaces. W e recall, f ollo wing [W5, Sect. 6 a nd 7], that lo calizatio n allows to construct non-trivial extensions of a certain Artin motiv e b y a direct factor of M ! ∗ ( X ∗ ). Using Prop osition 4.3, w e then establish stabilit y of M ! ∗ ( X ∗ ) under the corresp ondences “of Hec k e t yp e” constructed in Example 2 .16. In Section 5, w e discuss ﬁbre pro ducts of the univ ersal elliptic curv e ov er the mo dular curv e of level n ≥ 3. W e review some of the material from [Sch] and [W3, Sect. 3 a nd 4]. Notably (Prop osition 5.3), w e recall that in this geometrical setting, the condition f r o m Complemen t 1.24 on the a bsence o f w eigh ts − 1 and 0 in the b oundary motive is satisﬁed. Th us, the interior mo- tiv e can b e deﬁned. The new ingredien t is Example 5.4, where w e use rigidity of our construction to giv e a pro of “a voiding compactiﬁcations” of equiv ari- ance of the interior motive under the corresp ondences “of Hec k e ty p e”. As men tioned ab o ve , this article is primarily in tended to b e a general in- tro duction to the construction and to the applications of b oundary mot iv es. F o r man y details of the pro ofs, w e shall refer to our earlier articles [W1] and [W3]. Let us ho w eve r indicate that v arious parts of this pap er discuss orig ina l constructions. This is true in particular for Section 2 (on relativ e motive s and functoriality), including our study o f Hec ke equiv a r ia nce. W e exp ect these constructions to b e of in terest in other conte xts than those discussed in Sections 4 and 5. F o r further dev elopmen ts of the theory of b oundary mo t ives and their applications to sp ecial classes of algebraic v arieties a nd to their asso ciated motiv es, in particular to the motiv es of Shim ura v arieties, we refer also to [W2, W4]. P art o f this w ork w as done while I w as enjoyin g a m o dulation de servic e p our les p orteurs de pr ojets de r e cher che , granted b y the Unive rs i t´ e Paris 13 , and during a sta y at t he Universit¨ at Z ¨ urich . I am g rateful t o b oth insti- tutions. I wish to thank the organizers of Au tour des motifs for the in vi- tation to Bures-sur-Yv ette, and J. Ay oub, F. D´ eglise, D. H´ eb ert, B. Kahn, F. Lecom te and M. Levine for useful dis cussions and commen ts. Sp ecial thanks go to J.-B. Bost for insisting on this article to b e written, and for his helpful suggestions to impro v e a n earlier v ersion. 5 Notation and c on v entions : k denotes a ﬁxed p erfect base ﬁeld, S ch/k the category of separated sc hemes of ﬁnite type ov er k , and S m/k ⊂ S ch/k the full sub-category of ob j ects whic h are smo ot h o v er k . When w e as- sume k to admit resolution of singularities, then it will b e in the sense of [FV, Def. 3.4]: (i) for an y X ∈ S ch/k , there exists an a bstract blow-up Y → X [FV, D ef. 3.1] whose source Y is in S m/k , (ii) for any X , Y ∈ S m/k , and an y abstract blo w-up q : Y → X , there exists a sequence of blo w-ups p : X n → . . . → X 1 = X with smo oth cen ters, such that p factors throug h q . W e say that k admits strict resolution of singularities, if in (i), for any given dense op en subset U of the smoo t h lo cus of X , the blow-up q : Y → X can b e c hosen to b e an isomorphism a b ov e U , and suc h that arbitrary in tersections of the irr educible comp onents of the complemen t Z of U in Y are smo oth (e.g., Z ⊂ Y a normal crossing divisor with smo oth irreducible comp onents ). As far as motives are concerned, the notation of this pap er follo ws that of [V1]. W e refer to Levine’s lecture notes (this volume) fo r 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 motive s o ve r k , and of the motiv e M ( X ) and the motiv e with compact supp ort M c ( X ) of X ∈ S ch/k . L et F b e a comm utativ e ﬂat Z -alg ebra, i.e., a comm utative unitary ring whose additive group is without torsion. The notation 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 ) a nd 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 motives , 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 a nd C H M ( k ) ⊗ Z F , resp ectiv ely . Using [V2, Cor. 2] ([V1, Cor. 4.2.6] if k admits resolution of singularities), w e canonically identify C H M ef f ( k ) F and C H M ( k ) F with a f ull additive sub-category of D M ef f g m ( k ) F and D M g m ( k ) F , resp ectiv ely . Note in particular that with t hese conv entions, C H M ( k ) Q is a ctually opp osite to the category denoted b y the same sym b ol in [W5]. 1 Motiv ation Let us start by r ecalling the geometrical in terpretation of (cup pro duct with) the Chern class in a v ery sp ecial contex t. Example 1.1. Let E b e an elliptic curve o ve r t he ﬁeld C of complex n um b ers, and P ∈ E ( C ) a p o in t unequal to zero. Put X := E − { 0 , P } , and consider the complemen tary inclusions X   j / / E o o i ? _ { 0 , P } . 6 Let us prepare the reader that t he follo wing asp ect o f this situatio n will b e generalized in the sequel: E is a smo oth compactiﬁcation of X . The asso ciated long exact lo c alization se quenc e for (singular) cohomology with co eﬃcien ts in Q reads as follow s. 0 / / H 0 c ( X ( C ) , Q ) / / H 0 ( E ( C ) , Q ) / / H 0 ( { 0 } , Q ) ⊕ H 0 ( { P } , Q ) / / H 1 c ( X ( C ) , Q ) / / H 1 ( E ( C ) , Q ) / / 0 / / H 2 c ( X ( C ) , Q ) / / H 2 ( E ( C ) , Q ) / / 0 It sho ws that H 0 c ( X ( C ) , Q ) = 0, that H 2 c ( X ( C ) , Q ) ∼ − − → H 2 ( E ( C ) , Q ) , and, most inte restingly , that H 1 c ( X ( C ) , Q ) is a Y oneda one-extension of H 1 ( E ( C ) , Q ) b y the cok ernel o f i ∗ : H 0 ( E ( C ) , Q ) − → H 0 ( { 0 } , Q ) ⊕ H 0 ( { P } , Q ) . The Ho dge structures on the three g roups H 0 ( E ( C ) , Q ), H 0 ( { 0 } , Q ) and H 0 ( { P } , Q ) all equal Q (0), and under these iden tiﬁcations, i ∗ corresp onds to the diagonal em b edding ∆ : Q (0) ֒ − → Q (0) ⊕ Q (0) . W e iden tify its cokernel with Q (0) by cho osing the class o f ( − 1 , 1 ) ∈ H 0 ( { 0 } , Q ) ⊕ H 0 ( { P } , Q ) as its g enerato r . The one-extension then t ak es the form 0 − → Q (0) − → H 1 c ( X ( C ) , Q ) − → H 1 ( E ( C ) , Q ) − → 0 . Let us denote it by E xt P . D eﬁning E xt 0 to b e the trivial extension, w e th us get a map E xt : E ( C ) − → Ext 1  H 1 ( E ( C ) , Q ) , Q (0)  , P 7− → E xt P (Ext 1 := the group of one-extensions in the category of mixed Q -Hodg e structures), whic h can b e c hec ked to b e a morphism of g r oups. An ana lo gous construction is p ossible for ℓ -adic cohomolog y and elliptic curv es o v er a ﬁeld of characteristic unequal t o ℓ . The one-extensions then tak e place in the category of mo dules ov er the absolute Galois group. Note that in the contex t considered in Example 1 .1, the morphism E xt induces an isomorphism E ( C ) ⊗ Z Q ∼ − − → Ext 1  H 1 ( E ( C ) , Q ) , Q (0)  . Here is what we w ould lik e the reader to recall f r om the ab o v e. 7 Principle 1.2. L o c alization p otential ly le ads to inter esting extensions of Ho d g e structur es or Galois mo dules. Actually , Principle 1.2 admits a more general v ersion, where w e replace “lo calization” by “the formalism of six op erations”. In the sequel of this article, w e shall ho we ver concen trate on lo calization a nd its dual. Example 1.3. W e contin ue to consider the situation from Example 1.1. (a) The lo ng exact sequence dual to the lo calization sequence asso ciated to X   j / / E o o i ? _ { 0 , P } ( X = E − { 0 , P } as b efore) will b e referred to as the c o-lo c alization se q uen c e . It sho ws that H 0 ( E ( C ) , Q ) ∼ − − → H 0 ( X ( C ) , Q ) , that H 2 ( X ( C ) , Q ) = 0 , and that H 1 ( X ( C ) , Q ) is a Y oneda one-extension 0 − → H 1 ( E ( C ) , Q ) − → H 1 ( X ( C ) , Q ) − → Q ( − 1) − → 0 (with the iden tiﬁcations dual to the one used in Example 1.1). (b) Let us now compare cohomology and cohomolo gy with compact supp ort of X . This comparison is expressed b y a third lo ng exact sequence, whic h w e shall refer to as t he b oundary se quenc e . H 0 c ( X ( C ) , Q ) / / H 0 ( X ( C ) , Q ) / / ∂ H 0 ( X ( C ) , Q ) / / H 1 c ( X ( C ) , Q ) / / H 1 ( X ( C ) , Q ) / / ∂ H 1 ( X ( C ) , Q ) / / H 2 c ( X ( C ) , Q ) / / H 2 ( X ( C ) , Q ) The third term in this sequence is b oundary c ohomolo gy o f X , deﬁned as co- homology of E ( C ) with co eﬃcien ts in the complex i ∗ j ∗ Q X . Here, the sym b ol j ∗ Q X denotes the to t a l direct image of Q X under j ; follo wing the con v en tio n used in [BBD ], we drop the letter “ R ” f rom our nota t ion. The mo r phism from H 1 c ( X ( C ) , Q ) to H 1 ( X ( C ) , Q ) factors o v er H 1 ( E ( C ) , Q ). Therefore, b y what was said b efore, w e see that the b oundary sequence is obtained b y splicing the t w o sequences 0 H 0 c ( X ( C ) , Q ) / / H 0 ( X ( C ) , Q ) / / ∂ H 0 ( X ( C ) , Q ) / / H 1 c ( X ( C ) , Q ) / / H 1 ( E ( C ) , Q ) / / 0 and 0 / / H 1 ( E ( C ) , Q ) / / H 1 ( X ( C ) , Q ) / / ∂ H 1 ( X ( C ) , Q ) / / H 2 c ( X ( C ) , Q ) / / H 2 ( X ( C ) , Q ) 0 . 8 This allo ws us in particular to iden tify b oundary cohomology: ∂ H 0 ( X ( C ) , Q ) ∼ = Q (0) 2 , ∂ H 1 ( X ( C ) , Q ) ∼ = Q ( − 1) 2 . (c) W e claim that the ab ov e “ﬁrst half ” of the b oundary sequence 0 H 0 c ( X ( C ) , Q ) / / H 0 ( X ( C ) , Q ) / / ∂ H 0 ( X ( C ) , Q ) / / H 1 c ( X ( C ) , Q ) / / H 1 ( E ( C ) , Q ) / / 0 equals the lo calization sequence 0 H 0 c ( X ( C ) , Q ) / / H 0 ( E ( C ) , Q ) / / H 0 ( { 0 } , Q ) ⊕ H 0 ( { P } , Q ) / / H 1 c ( X ( C ) , Q ) / / H 1 ( E ( C ) , Q ) / / 0 from Example 1.1. Indeed, the iden tiﬁcatio n is achie ve d by the canonical isomorphism H 0 ( { 0 } , Q ) ⊕ H 0 ( { P } , Q ) ∼ − − → ∂ H 0 ( X ( C ) , Q ) , induced by the adjunction i ∗ Q E → i ∗ j ∗ Q X . A dual statemen t relates the co-lo calization seque nce fro m (a) to the “second half ” of the b oundary se- quence. (d) Altogether, we see that the long exact b oundary sequence allow s us to reco v er cohomology of E , to gether with the lo calization and co-lo calization sequence s. One “half ” of b oundary cohomology (namely ∂ H 0 ) con tributes to the lo calization sequence , the other “half ” (namely ∂ H 1 ) to the c o- lo calization sequence . Here is what w e w ould the lik e the reader to recall f r om the ab o ve. Principle 1.4. Th e b oundary se quenc e al lows to r e c over c ohomolo gy of a smo oth c omp actiﬁc ation of X , to gether with the lo c alization and c o-lo c alization se quenc es . A few precisions are necessary . First, giv en that X is a curve , there is only one p ossible c hoice of smo oth compactiﬁcation ( na mely E ). But this c hanges of course in higher dimensions. Second, the “recov ery” of the lo - calization and co-lo calization sequ ences from the b oundary sequence see ms to require a c hoice of additional data, namely a division of b oundary coho- mology into t wo “halfs”. In order to address b oth p oints in a satisfactory manner (see Theorem 1 .6 b elo w), we need to formalize the pro blem. Since w e wish the discuss io n to apply to the triangulated category of mo- tiv es, for which no t -structure is av ailable at presen t, it is b est pla ced in the con text of triangulated categories. In the con text w e chos e t o discuss, namely that of Ho dge theory , the a ppropriate triangulated category is the category of algebr aic Q -Ho dge mo dules [Sa]. W e should immediately reassure readers 9 not fa miliar with t his t heory: for our purp oses, only its fo rmal prop erties (lo- calization, purit y , prop er base c hange,...) will b e needed. Therefore, in order to motiv ate what is to follo w, w e migh t just as w ell hav e placed ourselv es in the con text of ℓ -adic shea v es, which w ould allow to arg ue in a completely analogous fashion. Readers wishing nonetheless to hav e a surv ey on Ho dge theory at their dispo sal migh t ﬁnd it useful to consult [St ]. Example 1.5. The relation to the geometric situation X   j / / E o o i ? _ { 0 , P } studied b efore is a s follo ws. The lo calization sequence from Example 1 .1 is the result of application of the cohomological functor H ∗ ( E ( C ) , • ) to the exact lo c ali z ation triangle j ! Q X (0) − → Q E (0) − → i ∗ i ∗ Q E (0) − → j ! Q X (0)[1] of algebraic Q -Ho dge mo dules on E [Sa, (4.4.1)]. In the same wa y , the co-lo calization sequence fro m Example 1 .3 (a) is induced b y the exact c o- lo c aliza tion triangle i ∗ i ! Q E (0) − → Q E (0) − → j ∗ Q X (0) − → i ∗ i ! Q E (0)[1] of Ho dge mo dules on E [Sa, (4.4.1) ], using in a dditio n that thanks to p urity , w e hav e a cano nical identiﬁcation i ∗ i ! Q E (0) ∼ = Q E ( − 1)[ − 2] . Applying localizatio n to the Ho dge mo dule j ∗ Q X (0) (or equiv alen tly , co- lo calization to j ! Q X (0)), w e obtain the exact b oundary triangle j ! Q X (0) − → j ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → j ! Q X (0)[1] , whic h induces the b oundar y sequence from Example 1.3 (b). Note that the three triangles (lo calization, co-lo calizatio n and b oundary) exist for any pair of complemen tary immersions. The fo llowing results from Saito’s formalism of six o p erations on algebraic Ho dge mo dules [Sa]. Theorem 1.6. L et X   j / / X o o i ? _ D b e c omplementary immersions ( j op en, i clos e d) o f sep ar ate d scheme s of ﬁnite typ e over C . (a) If X is pr op er, then the r esult of applying H ∗ ( X ( C ) , • ) to the b oundary triangle j ! Q X (0) − → j ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → j ! Q X (0)[1] , do es only dep end on X . 10 (b) The mo rp h isms i ∗ i ! Q X (0) − → i ∗ i ∗ Q X (0) and i ∗ i ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) induc e d by the r esp e ctive adjunction s ﬁt into a c anonic al exac t triangle i ∗ i ! Q X (0) − → i ∗ i ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → i ∗ i ! Q X (0)[1] . It i s the thir d c olumn of a dia g r am of exac t triangles 0   / / i ∗ i ! Q X (0)   i ∗ i ! Q X (0)   / / 0   j ! Q X (0) / / Q X (0)   / / i ∗ i ∗ Q X (0)   / / j ! Q X (0)[1] j ! Q X (0)   / / j ∗ Q X (0)   / / i ∗ i ∗ j ∗ Q X (0)   / / j ! Q X (0)[1]   0 / / i ∗ i ! Q X (0)[1] i ∗ i ! Q X (0)[1] / / 0 whose se c ond and thir d r ows ar e the lo c alization and b oundary triangles, a n d whose se c ond c olumn is the c o-lo c alization triangle. (c) If X is smo oth of c onstant dimension d , then ther e is a c anonic al i s o - morphism i ∗ i ! Q X (0) ∼ = D X  i ∗ i ∗ Q X ( d )[2 d ]  ( D X := duality for Ho dge mo dules on X [Sa, (4.1.5)] ) . Pr o of. (a) is a consequenc e of prop er base c hange [Sa, (4.4.3)]. As for (b), let us deﬁne the morphism i ∗ i ∗ j ∗ Q X (0) − → i ∗ i ! Q X (0)[1] as i ∗ i ∗ of t he morphism j ∗ Q X (0) − → i ∗ i ! Q X (0)[1] o ccurring in the co-lo calizatio n triangle. T ogether with the morphisms de- ﬁned b efore, it yields the diagram of the statemen t. Exactness of it s thir d column is then a consequence of exactness of the ﬁrst and second column. Finally , part ( c) results from duality [Sa, (4.3.5)]. q.e.d. Fix X ∈ S ch/ C . In the sequel, the b oundary triang le j ! Q X (0) − → j ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → j ! Q X (0)[1] , will alw ay s b e assumed to b e formed using a compactiﬁcation j : X ֒ → X of X , with complemen t i : D ֒ → X . Theorem 1.6 gives the precise r elat io n b et we en the b oundary t r ia ngle on the one hand, and the lo calization and co- lo calization triangles on the o t her hand. W hile b oundary cohomology , i.e., 11 cohomology of i ∗ i ∗ j ∗ Q X (0) do es not dep end on X , cohomology of the t w o other terms of the tria ngle i ∗ i ! Q X (0) − → i ∗ i ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → i ∗ i ! Q X (0)[1] . from Theorem 1.6 (b) in general do es (unless X is itself prop er). Saito’s formalism allows to put restrictions o n the Ho dge structures p otentially o c- curring as suc h cohomology groups. Theorem 1.7. L et n b e an inte ger. Assume X to b e pr op er and sm o oth (henc e X is smo oth). (a) The Ho dge structur e H n ( X , i ∗ i ∗ Q X (0)) is of weights at most n . (b) The Ho dge structur e H n ( X , i ∗ i ! Q X (0)[1]) is of we i g hts at le ast n + 1 . Pr o of. The sc heme X b eing prop er, H n maps complexes of Ho dg e mo dules of weigh ts ≤ 0 to Ho dge structures of weigh ts ≤ n , a nd complexes of Ho dge mo dules of w eights ≥ 1 to Ho dge structures of w eigh ts ≥ n + 1 . W e th us need to show that i ∗ i ∗ Q X (0) is of weigh ts ≤ 0, and i ∗ i ! Q X (0) o f w eights ≥ 0. The sche me X b eing smo oth, Q X (0) is pure of w eight 0. Therefore [Sa, (4.5.2)], i ∗ Q X (0) is of w eights ≤ 0, and i ! Q X (0) of weigh ts ≥ 0, and the same remains true after a pplicatio n of the functor i ∗ . q.e.d. In “triangulated” language, Theorem 1.7 sa ys that the ob jects ¯ π ∗  i ∗ i ∗ Q X (0)  and ¯ π ∗  i ∗ i ! Q X (0)[1]  ( ¯ π := the structure morphism of X ) of the deriv ed category of Ho dge struc- tures a r e of w eights ≤ 0 a nd ≥ 1, resp ectiv ely , when the compactiﬁcation X is smo oth. Corollary 1.8. L et X ∈ S m/ C , and . . . − → A n − → ∂ H n ( X ( C ) , Q ) − → B n − → A n +1 − → . . . a long exa ct se quenc e of mixe d Q -Ho dge s tructur es. This se quenc e is the r esult of applying H ∗ ( X ( C ) , • ) to the triang le i ∗ i ! Q X (0) − → i ∗ i ∗ Q X (0) − → i ∗ i ∗ j ∗ Q X (0) − → i ∗ i ! Q X (0)[1] , for a suitable smo oth c o m p actiﬁc ation j : X ֒ → X , only if A n is of weights at mo st n , and B n is of weigths at le ast n + 1 , for al l n ∈ Z . Theorems 1 .6 and 1.7, and Corollary 1.8 admit motivic analogues (Theo- rems 1.13 and 1.15, and Corolla r y 1.16 b elo w), whic h we shall dev elop now . T o do so, it is necess a r y to use the right notion of w eights on triangulated categories. Let us recall the follow ing deﬁnitions a nd results of Bondark o [Bo2]. 12 Deﬁnition 1.9. Let C b e a triangulated category . A weight structur e on C is a pair w = ( C w ≤ 0 , C w ≥ 0 ) of full sub-categories of C , such that, putting C w ≤ n := C w ≤ 0 [ n ] , C w ≥ n := C w ≥ 0 [ n ] ∀ n ∈ Z , the follo wing conditions are satisﬁed. (1) The categories C w ≤ 0 and C w ≥ 0 are Karoubi-closed: for any ob ject M of C w ≤ 0 or C w ≥ 0 , an y direct summand o f M formed in C is an ob ject of C w ≤ 0 or C w ≥ 0 , resp ective ly . (2) (Semi-in v ariance with resp ect to shifts.) W e hav e the inclusions C w ≤ 0 ⊂ C w ≤ 1 , C w ≥ 0 ⊃ C w ≥ 1 of f ull sub-categor ies of C . (3) (Orthogonalit y .) F or an y pair o f ob j ects M ∈ C w ≤ 0 and N ∈ C w ≥ 1 , w e ha v e Hom C ( M , N ) = 0 . (4) (W eigh t ﬁltra t io n.) F or an y ob j ect 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 conditio n 1.9 (2 ), C w ≤ n ⊂ C w ≤ 0 for negativ e n , and C w ≥ n ⊂ C w ≥ 0 for p ositiv e n . Th ere a r e obv ious analog ues of the other conditions for all the categories C w ≤ n and C w ≥ n . In par ticular, they are a ll Kar o ubi-closed, and an y ob ject M ∈ C is part of an 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 sligh t generalization of the terminology introduced in condition 1 .9 (4), w e shall refer to an y suc h exact triangle as a weigh t ﬁltration of M . Remark 1.10. (a) Our con v en tion concerning the sign of the w eigh t is actually opp osite to the one from [Bo 2 , Def. 1.1.1], i.e., w e exc hanged the roles of C w ≤ 0 and C w ≥ 0 . (b) Note t ha t in condition 1.9 (4), “the” w eigh t ﬁltratio n is not assumed to b e unique. As observ ed b y Bondark o, w eight structures are rele v an t to m o t ives thanks to the follo wing result. 13 Theorem 1.11. L et F b e a c ommutative ﬂat Z -algebr a, and a s sume k to admit r esolution of singularities. (a) Ther e is a c anonic al weig ht structur e on the c ate gory D M ef f g m ( k ) F . It is uniquely cha r acterize d by the r e quir em ent that its he art e qual C H M ef f ( k ) F . (b) Ther e is a c anonic al weight structur e on the c ate go ry D M g m ( k ) F , ex- tending the weight structur e fr om (a). I t i s uniquely char acterize d by the r e quir ement that its he art e qual C H M ( k ) F . (c) Statements (a) and (b) hold without assuming r esolution of sin gularities pr ovide d F is a Q -algebr a. Pr o of. F or F = Z and k of characteristic zero, this is the con tent of [Bo2, Sect. 6.5 a nd 6.6]. F or the mo diﬁcations of the pro of in the remaining cases, see [W3, Thm. 1.13]. q.e.d. The following result is formally implied by Theorem 1.11, and t he funda- men tal prop erties of the category D M g m ( k ) F , notably lo c ali z ation and duality [V1, Prop. 4.1.5 and Thm. 4.3.7]. F or details of the pro of, w e refer to [W3, Cor. 1.14] ([Bo1, Thm. 6 .2.1 ( 1 ) and (2 ) ] if k is of c haracteristic zero). Corollary 1.12. L et X ∈ S ch/k b e of (Krul l) dimension d . Assume k to ad m it r esolution o f sin gularities. (a) The mo tive with c omp act supp ort M c ( X ) li es in D M ef f g m ( k ) w ≥ 0 ∩ D M ef f g m ( k ) w ≤ d . (b) If X ∈ S m/k , then the motive M ( X ) lies in D M ef f g m ( k ) w ≥− d ∩ D M ef f g m ( k ) w ≤ 0 . Fix X ∈ S ch/k . The motivic analogue of (the complex computing) b oundary cohomology (for k = C ) is giv en b y the b oundary motive ∂ M ( X ) of X [W1, Def. 2.1]. Th e analog ue of Theorem 1.6 reads as follows ; there as in the sequel, w e shall denote by M ∗ the dual of a geometrical motiv e M [V1, Thm. 4.3.7]. Theorem 1.13. (a) Ther e is a c anonic al exact b oundary triangle ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1 ] in D M ef f g m ( k ) . (b) Assume k to admit r esolution of singularities. L et X   j / / X o o i ? _ D b e c omplem entary immersions ( j op en, i close d) of schemes in S ch/k . As- sume X to b e pr op er. T h er e is a c anonic al morphi s m α : ∂ M ( X ) → M ( D ) . Deﬁne M ( X /X ) as the r elative motive of X mo dulo X [W1, Conv. 1.2], and 14 let β : M ( D ) → M ( X /X ) b e induc e d by the morp h ism i ∗ : M ( D ) → M ( X ) . Then the morph i s ms α and β ﬁt into a c anonic al exact triangle M ( X /X )[ − 1] − → ∂ M ( X ) α − → M ( D ) β − → M ( X /X ) . It i s the thir d c olumn of a c anonic al diagr am o f ex a ct triangles 0 M ( X /X ) o o M ( X /X ) 0 o o M c ( X ) O O M ( X ) O O o o M ( D ) β O O o o M c ( X )[ − 1] O O o o M c ( X ) M ( X ) O O o o ∂ M ( X ) α O O o o M c ( X )[ − 1] o o 0 O O M ( X /X )[ − 1] O O o o M ( X /X )[ − 1] O O 0 o o O O whose se c ond and thir d r ow ar e the lo c alization [V1, Pr op . 4.1.5] and b ound- ary triangles, and w h ose s e c ond c o lumn is c anonic al ly asso ciate d to the r ela- tive m otive M ( X / X ) . (c) In the situation of (b ) , assume X to b e pr op er a nd smo oth of c on - stant dimension d (henc e X is smo oth). Ther e is a c anonic al morphism α ∗ : M ( D ) ∗ ( d )[2 d − 1] → ∂ M ( X ) . L et γ := i ∗ i ∗ : M ( D ) → M ( D ) ∗ ( d )[2 d ] b e the c omp osition of i ∗ : M ( D ) → M ( X ) and its dual. Then the morp hisms α ∗ , α (fr om (b)) and γ form a c anonic al exact triangle M ( D ) ∗ ( d )[2 d − 1] α ∗ − → ∂ M ( X ) α − → M ( D ) γ − → M ( D ) ∗ ( d )[2 d ] . It i s the thir d c olumn of a se c ond c anonic al diagr am of ex a ct triangles 0 M ( D ) ∗ ( d )[2 d ] o o M ( D ) ∗ ( d )[2 d ] 0 o o M c ( X ) O O M ( X ) O O o o M ( D ) γ O O o o M c ( X )[ − 1] O O o o M c ( X ) M ( X ) O O o o ∂ M ( X ) α O O o o M c ( X )[ − 1] o o 0 O O M ( D ) ∗ ( d )[2 d − 1] O O o o M ( D ) ∗ ( d )[2 d − 1] α ∗ O O 0 o o O O whose se c ond and thir d r ow ar e the lo c alization and b oundary triangle s, and whose s e c ond c olumn is dual, up to a twist by ( d ) and a shift by [2 d ] , to the lo c alization triangle (it wil l b e r ef err e d to as the c o-lo c alization triangle). This diagr am is isomorphic to the diagr am fr om (b). Pr o of. F o r (a ) , let us brieﬂy recall the deﬁnition of ∂ M ( X ). First, according t o [V1, pp. 223, 224], a mono mo r phism o f Nisnevich she aves with tr an s f e rs ι X : L ( X ) ֒ → L c ( X ) is asso ciated to X : the sheaf L ( X ) is formed 15 using ﬁnite c o rr esp ondenc es , and L c ( X ) is formed using quasi-ﬁnite c orr e- sp ondenc es . Next [V1, pp. 207, 208], there is a functor R C asso ciating to a Nisnevic h sh eaf with tra nsfers its si n gular simplic ial c omplex . V o ev o d- sky go es on to deﬁne the motiv e M ( X ) a s R C ( L ( X )), and the motiv e with compact supp ort M c ( X ) as R C ( L c ( X )). Set ∂ M ( X ) := R C (cok er ι X )[ − 1] [W1, D ef. 2.1]. Claim (a ) is t hen a direct consequence of this deﬁnition. As fo r (b), w e refer to [W1, Prop. 2.4]. It remains to sho w part (c). The morphism α ∗ is deﬁned as the dual of α , tw isted by ( d ) and shifted by [2 d − 1] M ( D ) ∗ ( d )[2 d − 1] − → ∂ M ( X ) ∗ ( d )[2 d − 1] , follo wed by the auto-dua lity isomorphism ∂ M ( X ) ∗ ( d )[2 d − 1] ∼ − − → ∂ M ( X ) [W1, Thm. 6.1]. Note that by dualit y [V1, Thm. 4.3 .7 3 ], M ( X ) ∗ ( d )[2 d ] ∼ − − → M ( X ) M ( X ) ∗ ( d )[2 d ] ∼ − − → M c ( X ) canonically , and under these iden tiﬁcations, the dual of the canonical mor- phism M ( X ) → M c ( X ) o ccurring in the lo calization triangle equals the canonical morphism M ( X ) → M ( X ) o ccurring in the co-lo calization tria n- gle. It remains to show that the comp osition M ( D ) ∗ ( d )[2 d − 1] α ∗ − → ∂ M ( X ) − → M ( X ) equals the mo r phism M ( D ) ∗ ( d )[2 d − 1] − → M ( X ) in the co-lo calization sequenc e. But this iden tit y can b e c heck ed after apply- ing dualit y . Note that the b oundary tria ngle is auto- dual [W1, Thm. 6.1]. Therefore, the dual of the ab o v e comp osition equals M c ( X ) − → ∂ M ( X ) α − → M ( D ) , whic h in turn equals the morphism M c ( X ) − → M ( D ) in the lo calization sequence. q.e.d. Recall that motive s ` a la V o ev o dsky b ehav e homologically; this is wh y the sense of the ar ro ws is in v ersed when compar ed to cohomolo gy . Remark 1.14. If X is prop er and smo oth of constant dimension, there should b e a c anoni c al choice of isomorphism b etw een the tw o canonical di- agrams f rom Theorem 1.13 (b) and (c). If D is (prop er and) smo oth, then suc h a c hoice is induced by purity [V1, Prop. 3.5.4]. Here is the motivic analogue of T heorem 1.7; it follows directly from Corollary 1.12. 16 Theorem 1.15. Assume k to ad m it r esolution of si n gularities. L et X   j / / X o o i ? _ D b e c omplem entary immersions ( j op en, i close d) of schemes in S ch/k . As- sume X to b e pr op er an d smo oth (hen c e X is smo oth). (a) The mo tive M ( D ) lies in D M ef f g m ( k ) w ≥ 0 . (b) The mo tive M ( D ) ∗ ( d )[2 d − 1] lies in D M ef f g m ( k ) w ≤− 1 . In particular, the exact triang le M ( D ) ∗ ( d )[2 d − 1] α ∗ − → ∂ M ( X ) α − → M ( D ) γ − → M ( D ) ∗ ( d )[2 d ] . from Theorem 1.13 (c) is then a w eight ﬁltrat io n of ∂ M ( X ). Corollary 1.16. Assume k to a d mit r esolution of singularities. L e t A − → ∂ M ( X ) − → B − → A [1] b e an ex a ct triangle in D M ef f g m ( k ) , for X ∈ S m/k . This triangle is isomorp h ic to the triangle M ( D ) ∗ ( d )[2 d − 1] α ∗ − → ∂ M ( X ) α − → M ( D ) γ − → M ( D ) ∗ ( d )[2 d ] . for a suitable sm o oth c omp actiﬁc ation j : X ֒ → X , only if it is a wei g ht ﬁltr a tion o f ∂ M ( X ) : A ∈ D M ef f g m ( k ) w ≤− 1 and B ∈ D M ef f g m ( k ) w ≥ 0 . Altogether, for ﬁxed X ∈ S m/k , w e get a functor from the category of smo o t h compactiﬁcations of X to the category of w eight ﬁltrat io ns of ∂ M ( X ). It t urns out to b e v ery instructiv e to see what one gets when trying to in v ert this functor. Construction 1.17. Assume k to admit resolution of singularities. F ix a w eigh t ﬁltration ∂ M ( X ) ≤− 1 c − − → ∂ M ( X ) c + − → ∂ M ( X ) ≥ 0 δ − → ∂ M ( X ) ≤− 1 [1] of ∂ M ( X ), fo r X ∈ S m/k : ∂ M ( X ) ≤− 1 ∈ D M ef f g m ( k ) w ≤− 1 and ∂ M ( X ) ≥ 0 ∈ D M ef f g m ( k ) w ≥ 0 . Consider the b oundary triangle ∂ M ( X ) v − − → M ( X ) u − → M c ( X ) v + − → ∂ M ( X )[1] . According to axiom TR4’ of triangulated categories (see [BBD, Sect. 1.1.6] 17 for an equiv alen t form ulatio n), t he diagra m of exact triangles 0 ∂ M ( X ) ≤− 1 [1] o o ∂ M ( X ) ≤− 1 [1] 0 o o M c ( X ) O O ∂ M ( X ) ≥ 0 δ O O M c ( X )[ − 1] O O c + ( v + [ − 1]) o o M c ( X ) M ( X ) u o o ∂ M ( X ) c + O O v − o o M c ( X )[ − 1] v + [ − 1] o o 0 O O ∂ M ( X ) ≤− 1 v − c − O O o o ∂ M ( X ) ≤− 1 c − O O 0 o o O O can b e completed to g ive 0 ∂ M ( X ) ≤− 1 [1] o o ∂ M ( X ) ≤− 1 [1] 0 o o M c ( X ) O O M 0 δ − O O i 0 o o ∂ M ( X ) ≥ 0 δ O O δ + o o M c ( X )[ − 1] O O c + ( v + [ − 1]) o o M c ( X ) M ( X ) π 0 O O u o o ∂ M ( X ) c + O O v − o o M c ( X )[ − 1] v + [ − 1] o o 0 O O ∂ M ( X ) ≤− 1 v − c − O O o o ∂ M ( X ) ≤− 1 c − O O 0 o o O O Note that this completion necessitates choic es of M 0 and of factorizations u = i 0 π 0 and δ = δ − δ + . In general, the o b ject M 0 is unique up to p ossibly non-unique isomorphism; it is this problem that will b e addresse d in the last part of this section. F o r t he momen t, note that whatev er c hoice we mak e, M 0 will be in the heart of our weigh t structure: indeed, the second row of the diagram, together with Corollary 1.12 (a ) sho ws that M 0 ∈ D M ef f g m ( k ) w ≥ 0 , and the second column, together with Corolla ry 1.12 (b) sho ws t ha t M 0 ∈ D M ef f g m ( k ) w ≤ 0 . According to Theorem 1.11 ( a ), it is therefore an eﬀectiv e Cho w motive. Note that it comes equipp ed with a fa cto r izat io n M ( X ) π 0 − → M 0 i 0 − → M c ( X ) of t he canonical morphism u : M ( X ) → M c ( X ), and that the triangles ∂ M ( X ) ≥ 0 δ + − → M 0 i 0 − → M c ( X ) ( c + [1]) v + − → ∂ M ( X ) ≥ 0 [1] and ∂ M ( X ) ≤− 1 v − c − − → M ( X ) π 0 − → M 0 δ − − → ∂ M ( X ) ≤− 1 [1] are w eigh t ﬁltrations of M c ( X ) and of M ( X ), resp ectiv ely . Let us summarize the discussion. 18 Theorem 1.18. Assume k to a dmit r es olution of singularities, a nd ﬁx X ∈ S m/k . T h e map  ∂ M ( X ) ≤− 1 , ∂ M ( X ) ≥ 0  / ∼ = − →  M 0 , π 0 , i 0  / ∼ = fr om the pr e c e ding c onstruction is a bije ction b etwe en (1) the isom orphism classes of weight ﬁltr ations of the b ounda ry mo tive ∂ M ( X ) , (2) the isomorphis m classes of eﬀe ctive Cho w motives M 0 , to gether with a factorization M ( X ) π 0 − → M 0 i 0 − → M c ( X ) of the c anon ic al morphism u : M ( X ) → M c ( X ) , such that b oth i 0 and π 0 c an b e c omplete d to give w eight ﬁltr ations of M c ( X ) and of M ( X ) , r esp e ctively. There a r e ob vious F -linear v ersions of Theorem 1.18, for an y commuta- tiv e ﬂat Z -algebra F . Recall that w e start ed oﬀ with sp ecial choices of Cho w motiv es factorizing u , namely the motiv es of smo oth compactiﬁcations of X . But Theorem 1.18 should yield more general Cho w motives M 0 . F or exam- ple, one might hop e f o r the motivic v ersion o f in tersection cohomolo g y of a singular compactiﬁcation o f X to o ccur. F or surfaces, this will b e sp elled out in Section 4. Note that w e are forced to pass to the lev el of isomorphism classes b ecause of the c hoices made in Construction 1.17. One impo rtan t problem caused b y this is the lac k o f functoriality . Th us, an endomorphism of a given w eigh t ﬁltration ∂ M ( X ) ≤− 1 − → ∂ M ( X ) − → ∂ M ( X ) ≥ 0 − → ∂ M ( X ) ≤− 1 [1] will in general not yield an endomorphism of an y of the Cho w motiv es M 0 represen tating the asso ciated isomorphism class. Principle 1.19. In or der to obtain functoriality, Construction 1.17 ne e ds to b e rigidiﬁed . It turns out that an ad ho c geometrical metho d suﬃces to ac hiev e rig id- iﬁcation in the setting of surfaces (see Section 4). Let us ﬁnish this section b y describing another metho d (namely , that of [W3]), based again on the formalism of w eigh ts. It will b e illustrat ed in the setting of self-pro ducts of the univ ersal elliptic curve ov er a mo dular curve (see Section 5). Remark 1.20. G etting back to Construction 1.17, and start ing ag ain with a w eigh t ﬁlt r a tion ∂ M ( X ) ≤− 1 c − − → ∂ M ( X ) c + − → ∂ M ( X ) ≥ 0 δ − → ∂ M ( X ) ≤− 1 [1] , 19 let us see what obstacles there are for the triple ( M 0 , π 0 , i 0 ) to b e unique up to unique isomorphism. No te that ( M 0 , π 0 ) is a cone of v − c − : ∂ M ( X ) ≤− 1 − → ∂ M ( X ) − → M ( X ) . An y other choice of cone w ould map isomorphically t o ( M 0 , π 0 ), the isomor- phism in question b eing unique up to the image of an elemen t in Hom D M ef f gm ( k )  ∂ M ( X ) ≤− 1 [1] , M 0  . In general, t he ob ject ∂ M ( X ) ≤− 1 b elonging to D M ef f g m ( k ) w ≤− 1 , hence ∂ M ( X ) ≤− 1 [1] ∈ D M ef f g m ( k ) w ≤ 0 , there is no w ay of prev en ting suc h elemen ts from b eing non-zero. How ev er, if ∂ M ( X ) ≤− 1 ∈ D M ef f g m ( k ) w ≤− 2 ⊂ D M ef f g m ( k ) w ≤− 1 , then Hom D M ef f gm ( k )  ∂ M ( X ) ≤− 1 [1] , M 0  = 0 b y ortho g onalit y 1.9 (3) (recall that M 0 is o f weigh t zero). Th us, under this h yp othesis, the pair ( M 0 , π 0 ) is rigid. As fo r i 0 , the same type of r easoning sho ws unicity provide d that ∂ M ( X ) ≥ 0 ∈ D M ef f g m ( k ) w ≥ 1 ⊂ D M ef f g m ( k ) w ≥ 0 . W e a re th us led naturally to mak e the following deﬁnition [W3, Def. 1.6 and 1.10]. Deﬁnition 1.21. Let M ∈ D M ef f g m ( k ), and m ≤ n t w o inte g ers (which ma y b e iden tical). A weight ﬁltr ation of M avoiding weights m, m + 1 , . . . , n − 1 , n is an exact triangle M ≤ m − 1 − → M − → M ≥ n +1 − → M ≤ m − 1 [1] , with M ≤ m − 1 ∈ D M ef f g m ( k ) w ≤ m − 1 and M ≥ n +1 ∈ D M ef f g m ( k ) w ≥ n +1 . If suc h a w eigh t ﬁltration exists, then w e sa y t hat M is without w e i ghts m, . . . , n . W eight ﬁltrations av o iding w eights m, . . . , n b eha v e functoria lly [W3, Prop. 1.7]. In particular, if M ∈ D M ef f g m ( k ) is without w eights m, . . . , n , then its weigh t ﬁltra tion av o iding we ig hts m, . . . , n is unique up to unique ismorphism. Remark 1 .20 therefore sho ws that w e can rigidify Construc- tion 1.17 pro vided that the b oundary motiv e ∂ M ( X ) is without w eigh ts − 1 and 0. Problem 1.22. The b oundary motiv e ∂ M ( X ) of S m/k is nev er without w eigh ts − 1 and 0 — unless it is altogether trivial. 20 Here is a heuristic reason, using the weigh ts o ccurring in b oundary coho- mology o v er k = C : to say that ∂ M ( X ) is not trivial implies that X is not prop er. On the one hand, the cok ernel of H 0 c ( X ( C ) , Q ) − → H 0 ( X ( C ) , Q ) ∼ = Q (0) r is t hen non-trivial. On the other hand, it injects into ∂ H 0 ( X ( C ) , Q ). Therefore, the a ppro ac h of “a v oiding w eights” cannot work on the whole of the b oundar y mo t ive. W e need to restrict to dir ect factors. F ix a comm u- tativ e ﬂat Z -algebra F , and let ( ∗ ) ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1 ] b e the b oundary triangle asso ciated to a ﬁxed ob ject X of S m/k , view ed as a triangle in D M ef f g m ( k ) F . Fix a n idemp oten t endomorphism e of the triangle ( ∗ ), that is, ﬁx idemp oten t endomorphisms of eac h of M ( X ), M c ( X ) and ∂ M ( X ), view ed as ob jects of D M ef f g m ( k ) F , whic h yield an endomorphism of the triangle. Denote b y M ( X ) e , M c ( X ) e and ∂ M ( X ) e the images of e on M ( X ), M c ( X ) and ∂ M ( X ), resp ectiv ely , and consider the canonical mor- phism u : M ( X ) e → M c ( X ) e . By Corollary 1.12 and condition 1.9 (1), the ob ject M ( X ) e b elongs t o D M ef f g m ( k ) F ,w ≤ 0 , and M c ( X ) e to D M ef f g m ( k ) F ,w ≥ 0 . In this situation, Construction 1.17 yields the follo wing. V arian t 1.23. The m a p  ∂ M ( X ) e ≤− 1 , ∂ M ( X ) e ≥ 0  / ∼ = − →  M 0 , π 0 , i 0  / ∼ = is a b i j e ction b etwe en (1) the i s o morphism classe s of weight ﬁltr ations of ∂ M ( X ) e , (2) the isomorphism classes o f eﬀe ctive Chow motives M 0 ∈ C H M ef f ( k ) F , to g e ther with a factorization M ( X ) e π 0 − → M 0 i 0 − → M c ( X ) e of the c anonic al morphism u : M ( X ) e → M c ( X ) e , s uch that b oth i 0 and π 0 c an b e c omplete d to give weight ﬁltr ations of M c ( X ) e and of M ( X ) e , r esp e ctively. Complemen t 1.24. L et e denote an idemp otent endomo rphism of the b oundary triangle ( ∗ ) as ab ove, and assume that ∂ M ( X ) e is without weights − 1 and 0 . Then the isom o rphism clas s ( M 0 , π 0 , i 0 ) asso ciate d to the we ight ﬁltr a tion of ∂ M ( X ) e avoiding weights − 1 and 0 essential ly c ontains one sin- gle obje ct, which is unique up to unique iso morphism. This is the principle exploited in [W3 , Sect. 4]. D ue to the b eha viour of its realizations, the ob ject M 0 is r eferred to as the e -p art of the interior motive of X . It has very strong functoriality prop erties. They will b e illustrated in our Section 5, where w e shall establish equiv ariance under the Hec ke algebra of M 0 in a sp ecial geometrical contex t. Before that, w e need to a ddress t w o 21 v ery concrete questions: (I) Ho w do es one get endomorphisms of t he b oundary triangle ( ∗ ) ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1] ? (I I) Ho w can one sho w that a giv en suc h endomorphism is idemp o ten t? The following tw o sections attempt to answ er these questions, at least pa r- tially . 2 Relativ e motiv e s and functorialit y of the b oundary mot iv e In this and the next section, the base ﬁeld k is assumed to admit strict reso- lution of singularities. F or X ∈ S m/k , the algebra of ﬁnite c orr esp ondenc es c ( X , X ) acts on M ( X ) [V1, p. 190]. In order to apply the constructions from Section 1 , w e need to construct endomorphisms o f the whole b oundary triangle ( ∗ ) ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1] . One of the aims of this section is to show that the theory o f r elative m otives pro vides a source of suc h endomorphisms. This result is a sp ecial f eatur e of an analysis of the functorial b eha vior of the exact tria ng le ( ∗ ) under mor- phisms of relativ e motiv es (Theorems 2.2 and 2.5, Corollary 2.15). The main application (Example 2.16 ) concerns endomorphisms of ( ∗ ) “of Hec ke t yp e”. Let us ﬁx a base sc heme S ∈ S m/k . Recall that by deﬁnition, ob jects of S m/k ar e separated o ve r k . Th us, for an y tw o sche mes X and Y o v er S , the natural morphism X × S Y − → X × k Y is a closed immersion. Therefore, cycles on X × S Y can and will b e considered as cycles on X × k Y . D enote by S m/S the category of separated smo oth sc hemes of ﬁnite ty p e ov er S , by P r opS m/S ⊂ S m/S the full sub-category of o b jects whic h are pro p er and smo o th o v er S , a nd by P r oj S m/S ⊂ S m/S the full sub-category o f pro jectiv e, smo o th S -sche mes. Deﬁnition 2.1. Let X , Y ∈ S m/S . D enote by c S ( X , Y ) the subgroup of c ( X , Y ) of corresp ondences whose supp ort is con tained in X × S Y . The group c S ( X , Y ) is at the base of the theory of (eﬀe ctive) ge ometric al motives over S , as deﬁned and dev elop ed (for arbitrary regular No etherian bases S ) in [D ´ e1, D´ e2]. Note that a ny cycle Z in c S ( X , Y ) g iv es rise to a morphism f r o m M ( X ) to M ( Y ), whic h w e shall denote by M ( Z ). Re- call from [DeMu, Sect. 1.3, 1.6] the deﬁnition of the categories of smo oth 22 (eﬀe ctive) Chow motives over S ; note that the approac h of [lo c. cit.] do es not necessitate passage to Q -co eﬃcien ts, and that one ma y choose to p er- form t he construction using sc hemes in P r opS m/S instead of just sc hemes in P r oj S m/S . Denote b y C H M s,ef f ( S ) and C H M s ( S ) the resp ectiv e opp o- sites of these categories. Not e that f or X , Y ∈ P r opS m/S and Z ∈ c S ( X , Y ), the class of Z in the Cho w group CH ∗ ( X × S Y ) of cycles mo dulo rat ional equiv alence lies in the righ t degree, and therefore deﬁnes a morphism fro m the relativ e Cho w motive h ( X/ S ) of X to the relative Chow motive h ( Y /S ). Our aim is to prov e the f ollo wing. Theorem 2.2. (a) Ther e is a c anonic al additive c ovariant functor, de- note d ( ∂ M , M , M c ) = ( ∂ M , M , M c ) S , fr om C H M s ( S ) to the c ategory of exact triangles in D M g m ( k ) . On obje cts, it is char acterize d by the fol lowing pr op erties: (a1) for X ∈ P r opS m/S , the functor ( ∂ M , M , M c ) maps h ( X/S ) to the b oundary triangle ( ∗ ) X ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1 ] , (a2) the functor ( ∂ M , M , M c ) i s c omp atible with T ate twists. On morphisms, the functor ( ∂ M , M , M c ) maps the class o f a cycle Z ∈ c S ( X , Y ) in CH ∗ ( X × S Y ) , fo r X , Y ∈ P r opS m/S , to a morphism ( ∗ ) X → ( ∗ ) Y whose M -c omp onent M ( X ) → M ( Y ) c oincides w ith M ( Z ) . (b) Ther e is a c anonic al additive c ontr avaria nt functor ( ∂ M , M , M c ) ∗ = ( ∂ M , M , M c ) ∗ S fr om C H M s ( S ) to the c ate gory of exact triangles in D M g m ( k ) . On obje cts, it is char acterize d by the fol lowing pr op erties: (b1) for an obje ct X ∈ P r opS m/S which is of pur e absolute dime n sion d X , the functor ( ∂ M , M , M c ) ∗ maps h ( X/ S ) to the triangle ( ∗ ) ∗ X := ( ∗ ) X ( − d X )[ − 2 d X ] , (b2) the functor ( ∂ M , M , M c ) ∗ is anti-c omp atible with T a te twists. On morph isms, the functor ( ∂ M , M , M c ) ∗ maps the class of a cycle Z ∈ c S ( X , Y ) in CH ∗ ( X × S Y ) , for X , Y ∈ P r opS m/S of pur e ab solute d i- mensions d X and d Y , r esp e ctively, to a morphism ( ∗ ) ∗ Y → ( ∗ ) ∗ X whose M c - c omp o n ent c oincides with the dual of M ( Z ) . (c) The functor ( ∂ M , M , M c ) ∗ is c anonic al ly identiﬁe d with the c omp o s i tion of ( ∂ M , M , M c ) a nd duality in D M g m ( k ) . Recall that by [V1, Thm. 4.3.7 3], the ob ject M c ( X ) is indeed dual to M ( X )( − d X )[ − 2 d X ]. Note also [V1, Cor. 4.1.6 ] that the f unctor from Theo- rem 2.2 (a) ma ps the full sub-category C H M s,ef f ( S ) to the full sub-categor y [V1, Thm. 4.3.1] of exact triangles in D M ef f g m ( k ). Also recall that by con- v en tion, the T ate t wist ( n ) in C H M s ( S ) corresp onds to the (comp onent- wise) op eration M 7→ M ( n )[2 n ] in D M g m ( k ). Th us, a nti-compatibilit y of 23 the functor ( ∂ M , M , M c ) ∗ with T ate tw ists means that for any ob ject X of C H M s ( S ), there is a functorial isomorphism  ∂ M , M , M c  ∗ ( X ( n )) ∼ − − →  ∂ M , M , M c  ∗ ( X )  ( − n )[ − 2 n ] . Remark 2.3. As far as t he M - a nd M c -comp onen ts are concerned, The- orem 2.2, or at least its restriction to the full sub-category C H M s ( S ) pr oj of C H M s ( S ) generated b y the motiv es of pro jectiv e smo oth S -sc hemes, is a consequence o f the main results of [D ´ e2], especially [D ´ e2, Thm. 5.23], to - gether with the existence of an adjoin t pair ( La S,♯ , a ∗ S ) of exact functors [CiD ´ e, Ex. 4.12, Ex. 7.15] linking the category D M g m ( S ) of geometrical mot ives ov er S to D M g m ( k ) (here we let a S : S → Sp ec k denote the structure morphism of S ). W e should also men tion that this approach w ould allow to av oid the h yp othesis on strict resolution of singularities. Ho w ev er, the application of the results of [lo c. cit.] to the functor ∂ M is not obv ious. W e are therefore forced to follo w an alternativ e a pproac h. Remark 2.4. The follo wing sheaf-theoretical phenomenon explains wh y one should exp ect a statemen t lik e Theorem 2.2. W riting a = a X for the structure morphism X → Sp ec k , fo r X ∈ S ch/k , there is a n exact triangle of exact functors (+) X a ! − → a ∗ − → a ∗ /a ! − → a ! [1] from the deriv ed categor y D + ( X ) of complexes of ´ etale shea v es on X (sa y), b ounded from b elow , to D + ( Sp ec k ). Here, a ∗ denotes the deriv ed functor of the dir ect image, a ! is its a nalogue “ with compact supp ort” , and a ∗ /a ! is a canonical choic e of cone ( which exists since the category of compactiﬁcations of X is ﬁltered). The triangle (+) X is con trav ariantly f unctor ia l with resp ect to prop er morphisms. Up to a t wist and a shift, it is cov ariantly f unctorial with respect to prop er smo oth morphisms. This show s that a suitable v er- sion of Theorem 2.2 (a) is lik ely to extend to the sub-category of D M g m ( S ) generated by the relativ e motiv es of sc hemes whic h are (only) prop er ov er S . F o r an y prop er smo oth morphism f : T → S in the category S m/k , denote by f ♯ : C H M s ( T ) → C H M s ( S ) the canonical functor induced b y h ( X/T ) 7→ h ( X/S ), for a n y pro p er smo oth sc heme X ov er T (hence, ov er S ). F or any morphism g : U → S in S m/k , denote b y g ∗ : C H M s ( S ) → C H M s ( U ) the canonical tensor functor induced b y h ( Y /S ) 7→ h ( Y × S U /U ), for any prop er smo o th sc heme Y ov er S . When g is prop er and smo oth, the functor g ♯ is left adjo in t to g ∗ . The fo llo wing summarizes the behaviour of ( ∂ M , M , M c ) and ( ∂ M , M , M c ) ∗ under c hange of the base S . Theorem 2.5. (a) L et f : T → S b e a pr op er smo oth morphism in S m/k . Ther e ar e c anonic al isomorphisms of additive functors α f ♯ : ( ∂ M , M , M c ) S ◦ f ♯ ∼ − − → ( ∂ M , M , M c ) T 24 and α ∗ f ♯ : ( ∂ M , M , M c ) ∗ T ∼ − − → ( ∂ M , M , M c ) ∗ S ◦ f ♯ on C H M s ( T ) . The formation of b oth α f ♯ and α ∗ f ♯ is c omp atible with c om - p osition of pr op er smo oth morphisms in S m/k . Under the id e ntiﬁc ation of The or em 2.2 (c), the e quivalenc e α ∗ f ♯ c orr esp onds to the dual of the e quiva- lenc e α f ♯ . (b) L et g : U → S b e a pr op er smo oth morphism in S m/k . Then ther e ex i s ts a c anonic al tr ansformation of additive functors β g ∗ , id S : ( ∂ M , M , M c ) U ◦ g ∗ − → ( ∂ M , M , M c ) S . The formation of β g ∗ , id S is c om p atible with c omp osition of pr op er smo oth morphisms in S m/k . (c) The tr an s formations α f ♯ and β g ∗ , id S c ommute in the fol lowing se n se: let f : T → S and g : U → S b e p r op er smo oth morphi s ms in S m/k . Consider the c artesian diagr am V = T × S U g ′   f ′ / / U g   T f / / S and the c anonic al ide n tiﬁc ation of natur al tr an sformations f ′ ♯ ◦ g ′∗ = g ∗ ◦ f ♯ of functors fr om C H M s ( T ) to C H M s ( U ) . The n the tr ansformations β g ′∗ , id T ◦ ( α f ′ ♯ ◦ g ′∗ ) , α f ♯ ◦ ( β g ∗ , id S ◦ f ♯ ) of functors o n C H M s ( T ) ( ∂ M , M , M c ) U ◦ g ∗ ◦ f ♯ − → ( ∂ M , M , M c ) T c oincide. (d) L et g : U → S b e a pr op er smo oth m orphism in S m/k . Then ther e exi s ts a c anonic al tr ansformation of additive functors γ id S ,g ∗ : ( ∂ M , M , M c ) ∗ S − → ( ∂ M , M , M c ) ∗ U ◦ g ∗ . The formation of γ id S ,g ∗ is c om p atible with c omp os i tion of pr op er s mo oth morphisms in S m/k . Under the identiﬁc ation of The or em 2.2 (c), the tr ans- formation γ id S ,g ∗ c orr esp onds to the dual of the tr ans f o rmation β g ∗ , id S . (e) The tr an s formations α ∗ f ♯ and γ id S ,g ∗ c ommute in the fol lowing sense: let f : T → S and g : U → S b e p r op er smo oth morphi s ms in S m/k . Consider 25 the c artesian diagr am V = T × S U g ′   f ′ / / U g   T f / / S Then the tr ansformations ( γ id S ,g ∗ ◦ f ♯ ) ◦ α ∗ f ♯ , ( α ∗ f ′ ♯ ◦ g ′∗ ) ◦ γ id T ,g ′∗ of functors o n C H M s ( T ) ( ∂ M , M , M c ) ∗ T − → ( ∂ M , M , M c ) ∗ U ◦ g ∗ ◦ f ♯ c oincide. Remark 2.6. Sheaf-theoretical considerations sho w that parts (b)–(e) o f Theorem 2.5 should hold mor e generally for morphisms g whic h are (only) prop er. While this could b e sho wn to b e indeed the case, w e c hose to prov e the statemen ts only under the more r estrictiv e assumption on g : the pro of then simpliﬁes considerably since it is p ossible to mak e use of the functor g ♯ , whic h only exists when g is (prop er and) smo ot h. Let us prepare the pro ofs of Theorems 2.2 and 2.5. They are based on the follo wing result. Theorem 2.7 ([W1, Thm. 6.14 , Rem. 6 .1 5]) . L et W ∈ S m/k b e of pur e dimen s ion m , and Z ⊂ W a close d sub-scheme such that arbitr ary interse ctions of the irr e ducible c omp onents of Z ar e sm o oth. F ix n ∈ Z . (a) Ther e is a c anonic al m orphism cy c : h 0  z eq ui ( W , m − n ) Z  ( Sp ec k ) − → Hom D M ef f gm ( k ) ( M ( W / Z ) , Z ( n )[2 n ]) . (b) The morphism cy c is c omp atible with p assage fr om the p air Z ⊂ W to Z ′ ⊂ U , for op en sub-schemes U of W , and close d sub-schemes Z ′ of Z ∩ U such that arbitr ary interse ctions of the irr e ducible c omp onents of Z ′ ar e smo oth. (c) When Z is em pty, then cy c : h 0  z eq ui ( W , m − n )  ( Sp ec k ) − → Hom D M ef f gm ( k ) ( M ( W ) , Z ( n )[2 n ]) c oincides with the morp h i sm fr om [V1, Cor. 4 .2.5]. I n p articular, it is then an isomorphism. Some explanations are necessary . First, b y deﬁnition [W1, Def. 6.13], the Nisnevic h sheaf with transfers z eq ui ( W , m − n ) Z asso ciates to T ∈ S m/k the g roup of those cycles in z eq ui ( W , m − n )( T ) [V1, p. 228] hav ing empt y in tersection with T × k Z . In part icular, the group z eq ui ( W , m − n ) Z ( Sp ec k ) 26 equals the group of cycles on W of dimension m − n , whose supp ort is disjoin t from Z . R ecall then [V1, p. 207] that the group h 0  z eq ui ( W , m − n ) Z  ( Sp ec k ) is the quotien t of z eq ui ( W , m − n ) Z ( Sp ec k ) by the image under the diﬀeren tial “pull-bac k via 1 min us pull-ba ck via 0” of z eq ui ( W , m − n ) Z ( A 1 k ). Finally the ob ject M ( W / Z ) denotes the relat ive motive asso ciat ed to the immersion of Z into W [W1, Def. 6.4]. Remark 2.8. One ma y sp eculate ab out the v alidity of Theorem 2 .7 for arbitrary closed sub-sc hemes Z of W ∈ S m/S . While the author is optimistic ab out this p ossibilit y , he notes that the to o ls dev elop ed in [W1] to pro v e Theorem 2.7 require Z to satisfy our more restrictiv e h yp otheses. It is for that reason that w e are forced to supp ose k to admit strict r esolution of singularities. No w note the follo wing. Prop osition 2.9. In the ab ove situation, let in addition V ⊂ W b e a close d sub-scheme in S m/S , which is dis joint fr om Z . Then the na tur al map z eq ui ( V , m − n ) − → z eq ui ( W , m − n ) Z induc es a morphism CH m − n ( V ) − → h 0  z eq ui ( W , m − n ) Z  ( Sp ec k ) . Corollary 2.10. L et W ∈ S m/k b e of pur e d i m ension m , V , Z ⊂ W close d sub-schemes, and n ∈ Z . Supp ose that arbitr ary interse ctions of the irr e ducible c omp onents of Z ar e sm o oth, and that V ∩ Z = ∅ . T h en ther e is a c anonic al mo rphism cy c : CH m − n ( V ) − → Hom D M ef f gm ( k ) ( M ( W / Z ) , Z ( n )[2 n ]) . Given a n op en imm ersion j : U ֒ → W a nd a close d sub-scheme Z ′ of the interse ction Z ∩ U such that arbitr ary interse ctions of the irr e ducible c omp o- nents of Z ′ ar e smo oth, the diagr am CH m − n ( V ) j ∗   cy c / / Hom D M ef f gm ( k ) ( M ( W / Z ) , Z ( n )[2 n ]) j ∗   CH m − n ( V ∩ U ) cy c / / Hom D M ef f gm ( k ) ( M ( U / Z ′ ) , Z ( n )[2 n ]) c ommutes. No w ﬁx X , Y ∈ P r opS m/S . C ho ose a compactiﬁcation (o ver k ) S of S , and compactiﬁcations X of X , and Y of Y together with cartesian diagrams X     / / X   S   / / S 27 and Y     / / Y   S   / / S (this is possible since X and Y are proper o ver S ). The h yp othesis on k ensures that arbitra r y in tersections of the irreducible comp onen ts of the complemen ts ∂ X of X in X and ∂ Y of Y in Y can b e supposed to be smo oth. Eac h of the three constituen ts M , M c , ∂ M of the exact triangle ( ∗ ) will corresp ond to an application o f Corollary 2.10, with diﬀeren t choices of ( W , Z ). (1) for M , we deﬁne W := X × k Y , (2) for M c , we deﬁne W := X × k Y , (3) for ∂ M , w e deﬁne W := X × k Y − ∂ X × k ∂ Y . In all three cases, we put Z := W − X × k Y . That is, (1) Z = X × k ∂ Y , (2) Z = ∂ X × k Y , (3) Z = X × k ∂ Y ∪ ∂ X × k Y . W e also let V := X × S Y ⊂ X × k Y in all t hr ee cases. These c hoices satisfy the h yp otheses of Corollary 2.10 thanks to the follo wing. Lemma 2.11. The scheme X × S Y is close d in X × k Y − ∂ X × k ∂ Y . Pr o of. Indeed, the dia g ram X × S Y     / / X × k Y − ∂ X × k ∂ Y   S   ∆ / / S × k S is car t esian. q.e.d. Pr o of of The or em 2.2. W e may clearly assume S , X and Y to b e of pure absolute dimension d S , d X and d Y , resp ectiv ely . Let us treat M ﬁrst. Note that b y [V1, T hm. 4.3.7 3], the group o f morphisms in D M g m ( k ) from M ( X ) to M ( Y ) is canonically isomorphic to Hom D M gm ( k )  M ( X ) ⊗ M c ( Y ) , Z ( d Y )[2 d Y ]  . Lo calization f or the motiv e with compact supp ort [V1, Prop. 4.1.5] show s that M c ( Y ) = M ( Y / ∂ Y ). Giv en the deﬁnition of the tensor structure o n D M g m ( k ), the ab ov e therefore equals Hom D M gm ( k )  M ( X × k Y / X × k ∂ Y ) , Z ( d Y )[2 d Y ]  . 28 By Corollar y 2.10, a pplied to the setting (1), this group is the targ et of the morphism cy c 1 on CH d X ( X × S Y ) = CH d Y − d S ( X × S Y ). Note that on a class whic h comes from Z ∈ c S ( X , Y ), the map cy c 1 tak es indeed the v alue M ( Z ). The case of M c is similar. F ir st, by dualit y , the g r o up of morphisms in D M g m ( k ) from M c ( X ) to M c ( Y ) is canonically isomorphic to Hom D M gm ( k )  M c ( X ) ⊗ M ( Y ) , Z ( d Y )[2 d Y ]  . By lo calization, this group then equals Hom D M gm ( k )  M ( X × k Y / ∂ X × k Y ) , Z ( d Y )[2 d Y ]  . By Corollar y 2.10, a pplied to the setting (2), this group is the targ et of the morphism cy c 2 on CH d Y − d S ( X × S Y ). In order to sho w that for a cycle class z in CH d Y − d S ( X × S Y ), the diagram M ( X ) cy c 1 ( z )   / / M c ( X ) cy c 2 ( z )   M ( Y ) / / M c ( Y ) comm utes, w e need to study the group of morphisms in D M g m ( k ) f r om M ( X ) to M c ( Y ). Aga in by duality , it is canonically isomorphic t o Hom D M gm ( k )  M ( X × k Y ) , Z ( d Y )[2 d Y ]  . The ab o ve comm utativity then follow s from the compatibility of cy c under restriction from X × k Y , resp. X × k Y , to X × k Y (Corolla ry 2.10) . No w let us treat ∂ M . Note that b y [W1, Thm. 6.1], the group of mor- phisms in D M g m ( k ) from ∂ M ( X ) to ∂ M ( Y ) is canonically isomorphic to Hom D M gm ( k )  ∂ M ( X ) ⊗ ∂ M ( Y )[1] , Z ( d Y )[2 d Y ]  . As in [W1, pp. 65 0 –651], one shows that ∂ M ( X ) ⊗ ∂ M ( Y )[1] maps canoni- cally to the relat ive motive M  ( X × k Y − ∂ X × k ∂ Y ) / ( X × k Y − ∂ X × k ∂ Y − X × k Y )  . Hence the group of morphisms Hom D M ef f gm ( k ) F ( ∂ M ( X ) , ∂ M ( Y )) receiv es an arro w, say α , from the group of morphisms from M  ( X × k Y − ∂ X × k ∂ Y ) / ( X × k Y − ∂ X × k ∂ Y − X × k Y )  to Z ( d Y )[2 d Y ]). By Corolla r y 2.10, applied to the setting (3), this group is the target of the morphism cy c 3 on CH d Y − d S ( X × S Y ). In order to sho w that for a cycle class z in CH d Y − d S ( X × S Y ), the diagram M c ( X ) cy c 2 ( z )   / / ∂ M ( X )[1] cy c 3 ( z )[1]   M c ( Y ) / / ∂ M ( Y )[1] 29 comm utes, w e need to study the group of mor phisms in D M g m ( k ) from M c ( X ) to ∂ M ( Y )[1]. Again b y [W1, Thm. 6.1], it is canonically isomor- phic to Hom D M gm ( k )  M c ( X ) ⊗ ∂ M ( Y ) , Z ( d Y )[2 d Y ]  . But M c ( X ) ⊗ ∂ M ( Y ) maps canonically to M c ( X ) ⊗ M ( Y ), whic h w as already iden tiﬁed with the relativ e motiv e M ( X × k Y / ∂ X × k Y ) . Hence the group of morphisms Hom D M gm ( k ) ( M c ( X ) , ∂ M ( Y )[1]) receiv es an arro w, say β , from Hom D M gm ( k )  M ( X × k Y / ∂ X × k Y ) , Z ( d Y )[2 d Y ]  . The desired comm utativity then follo ws from the compatibilit y of cy c under restriction f rom X × k Y − ∂ X × k ∂ Y to X × k Y (C or o llary 2.10 ), and from the compatibilit y of β a nd the map α fr o m ab o ve . The latter is a consequence of t he compatibility of t he isomorphism ∂ M ( Y )[1] ∼ − − → ∂ M ( Y ) ∗ ( d Y )[2 d Y ] with dualit y M ( Y ) ∼ = M c ( Y ) ∗ ( d Y )[2 d Y ] [W1, Thm. 6.1]. The pro of of the commu ta tivit y of ∂ M ( X ) cy c 3 ( z )   / / M ( X ) cy c 1 ( z )   ∂ M ( Y ) / / M ( Y ) is similar. Altogether, this pro v es pa rt (a) of the statemen t. As for parts (b) and (c), simply comp ose the functor from (a) with duality in D M g m ( k ), using [V1, Thm. 4.3.7 3] and [W1, Thm. 6.1]. By [W1, Rem. 6.1 5], our construction is indep enden t of the compactiﬁ- cations S , X , Y . q.e.d. Pr o of o f The or em 2.5. W e ke ep t he notations of the previous pro of. Cho ose compactiﬁcations T of T , and U of U together with cartesian dia- grams T f     / / T   S   / / S and U g     / / U   S   / / S 30 ( f and g are pr o p er). (a) Chec king t he deﬁnitions, the transformation α f ♯ is in fact giv en b y the identit y . Indeed, b oth ( ∂ M , M , M c ) S ◦ f ♯ and ( ∂ M , M , M c ) T map the ob ject h ( X/T ), f or X ∈ P r opS m/T , to the exact triang le ( ∗ ) X ∂ M ( X ) − → M ( X ) − → M c ( X ) − → ∂ M ( X )[1 ] . Note that on mor phisms, the functor f ♯ corresp onds to the push-forw ard CH ∗ ( X × T Y ) − → CH ∗ ( X × S Y ) along the closed immersion X × T Y ֒ → X × S Y . The latter factors the closed immersion X × T Y ֒ − → X × k Y − ∂ X × k ∂ Y . The construction (see the preceding pro o f ) sho ws then that the eﬀects of t he functors ( ∂ M , M , M c ) S ◦ f ♯ and of ( ∂ M , M , M c ) T coincide also on CH ∗ ( X × T Y ). This sho ws the ﬁrst half of statemen t (a ). The second is implied formally b y Theorem 2.2 (c). (b) W e ﬁrst consider an auxiliary functor. The morphism g b eing prop er and smo o th, w e ma y consider the comp osition g ♯ ◦ g ∗ on C H M s ( S ), whic h on ob jects is g iv en b y h ( Y /S ) 7→ h ( Y × S U /S ) , for an y prop er smo oth sc heme Y o v er S . Pro j ection on to the ﬁrst comp onent then yields a transformation of f unctor s, namely the adjunction b g ∗ , id S : g ♯ ◦ g ∗ − → id C H M s ( S ) . Then deﬁne β g ∗ , id S to b e the comp osition of transformations β g ∗ , id S :=  ( ∂ M , M , M c ) S ◦ b g ∗ , id S  ◦  α g ♯ ◦ g ∗  − 1 , observing the equiv alence α g ♯ ◦ g ∗ : ( ∂ M , M , M c ) S ◦ g ♯ ◦ g ∗ ∼ − − → ( ∂ M , M , M c ) U ◦ g ∗ from part (a). W e lea v e it to the reader to chec k the compatibility of this construction with comp osition of prop er smo oth morphisms in S m/k . (c) Similarly , this comm utativity statemen t is left as an exercice. (d), (e) G iv en Theorem 2.2 (c), these statements follow formally from (b) and (c), resp ective ly . q.e.d. F o r X , Y ∈ P r opS m/S , denote b y ¯ c S ( X , Y ) the quotien t of c S ( X , Y ) by the group of cycles Z satisfying M ( Z ) = 0 , M c ( Z ) = 0 , ∂ M ( Z ) = 0 . Note that comp o sition of corresp ondences induces a w ell-deﬁned comp osition on ¯ c S . In particular, for a n y X ∈ P r opS m/S , the group ¯ c S ( X , X ) carries the structure of an alg ebra. Corollary 2.12. L et X and Y b e in P r opS m/S . Then the pr oje c tion c S ( X , Y ) − → → ¯ c S ( X , Y ) 31 factors thr ough the ima g e of c S ( X , Y ) in CH ∗ ( X × S Y ) . In other wor ds, two cycles Z 1 , Z 2 ∈ c S ( X , Y ) induc e the same morphisms M ( Z i ) , r esp. M c ( Z i ) , r esp. ∂ M ( Z i ) , if they ar e r ational ly e quivalent (on X × S Y ). Remark 2.13. (a) Let X , Y ∈ S m/S . As shown in [L, Lemma 5.1 8], the map c S ( X , Y ) − → CH d X ( X × S Y ) is surjectiv e, whenev er Y is pro jectiv e, and X of pure a bsolute dimension d X . Therefore, b y Corollary 2.12, the group ¯ c S ( X , Y ) is canonically a quotien t of CH d X ( X × S Y ) if X ∈ P r opS m/S and Y ∈ P r oj S m/S . (b) The observ ation from (a) ﬁts in the functorial picture sk etc hed in Re- mark 2.3. Indeed, [D´ e2, Thm. 5.23] implies that the restriction of the functor M , M : C H M s ( S ) pr oj − → D M g m ( k ) factors canonically through a fully faithful em b edding C H M s ( S ) pr oj ֒ − → D M g m ( S ) . (c) In [L, Prop. 5.19], an em b edding result analogous t o ( b) is pro ve n fo r a dg-v ersion of D M g m ( S ), denoted S mM ot ( S ) in [lo c. cit.], from whic h the em b edding (b) can b e deduced [L, Cor. 7.1 3]. (d) Using [D ´ e2, Thm. 5 .23], one can show that Hom D M gm ( S ) ( M 1 , M 2 [ i ]) = 0 for a n y tw o smo o th relativ e Cho w motiv es M 1 , M 2 ∈ C H M s ( S ) pr oj , and a n y in teger i > 0. (e) When S = Sp ec k , results (b) a nd (d) are contained in [V1, Cor. 4.2.6]. Remark 2.14. Fix a non-negativ e in teger d , and consider the full sub- category C H M s ( S ) d of C H M s ( S ) of smo ot h relativ e Cho w mo t ives gener- ated b y the T ate t wists of h ( X/ S ), for X ∈ P r opS m/S of pure absolute di- mension d . The construction o f the dua lity isomorphisms [V1, Thm. 4.3 .7 3], [W1, Thm. 6.1] sho ws that the identiﬁcation  ∂ M , M , M c  ∗ =  ∂ M , M , M c  ( − d )[ − 2 d ] of the restriction of the functors from Theorem 2.2 to C H M s ( S ) d admits an alternative description, when S is of pure a bsolute dimension, say s : on C H M s ( S ) d , the functor ( ∂ M , M , M c ) ∗ equals then comp o sition of dualit y in the category C H M s ( S ) d with ( ∂ M , M , M c ), follow ed b y the functor M 7→ M ( − s )[ − 2 s ]. Note that on morphisms, duality in C H M s ( S ) d corresp onds to the transp osition CH ∗ ( X × S Y ) → CH ∗ ( Y × S X ). This observ ation allows to deduce the following statemen ts from Theo- rem 2 .5. 32 Corollary 2.15. ( a) L et g : U → S b e a pr op er s m o oth morphism in S m/k of pur e r elative dimen sion d g . Then ther e e x i s ts a c anonic al tr ans f o r- mation of addi tive functors δ id S ,g ∗ : ( ∂ M , M , M c ) S − → ( ∂ M , M , M c ) U ( − d g )[ − 2 d g ] ◦ g ∗ . The formation of δ id S ,g ∗ is c omp atible with c omp osition of pr op er smo oth mor- phisms i n S m/k of pur e r elative dime n sion. (b) L et g : U → S b e a ﬁnite ´ etale morphism in S m/k of c onstant (ﬁbr ewise) de gr e e u . Then the endomorphis m β g ∗ , id S ◦ δ id S ,g ∗ of the functor ( ∂ M , M , M c ) S e quals multiplic ation by u . Pr o of. (a) W e may a ssume S to b e of pure a bsolute dimension, sa y s . Consider the t ransformation γ id S ,g ∗ : ( ∂ M , M , M c ) ∗ S − → ( ∂ M , M , M c ) ∗ U ◦ g ∗ from Theorem 2.5 (d). Comp o sition with dualit y D S in C H M s ( S ) gives γ id S ,g ∗ ◦ D S : ( ∂ M , M , M c ) ∗ S ◦ D S − → ( ∂ M , M , M c ) ∗ U ◦ g ∗ ◦ D S . No w observ e the form ula D U ◦ g ∗ = g ∗ ◦ D S ( D U := dualit y in C H M s ( U )). Deﬁne δ id S ,g ∗ as the comp osition of γ id S ,g ∗ ◦ D S and M 7→ M ( s )[2 s ], observing that source and target o f δ id S ,g ∗ are iden tiﬁed with ( ∂ M , M , M c ) S and ( ∂ M , M , M c ) U ( − d g )[ − 2 d g ] ◦ g ∗ , resp ective ly . (b) The morphism g b eing ﬁnite and ´ etale, we hav e D S ◦ g ♯ = g ♯ ◦ D U . This sho ws that g ♯ is also right adjoint to g ∗ . Chec king the deﬁnitions, the comp osition β g ∗ , id S ◦ δ id S ,g ∗ equals up to twis t and shift the comp osition of the t w o adjunctions ξ : id C H M s ( S ) − → g ♯ ◦ g ∗ − → id C H M s ( S ) , preceded b y dualit y , and follo we d b y ( ∂ M , M , M c ) ∗ S . These functors b eing additiv e, it suﬃc es to sh ow that ξ equals m ultiplication bu u . But t his iden tit y on morphisms of smo oth relativ e Chow motiv es is classical. q.e.d. The main results of this section ha v e obvious F -linear v ersions, for an y comm utativ e Q -algebra F . Let us now describ e ho w o ur analysis of the functor ( ∂ M , M , M c ) will b e used in the sequel. Example 2.16. L et g 1 , g 2 : U → S b e t wo ﬁnite ´ etale morphisms in S m/k . Fix an o b ject X ∈ P r opS m/S , an idemp o ten t e on h ( X/S ) (p ossibly b elonging to CH ∗ ( X × S X ) ⊗ Z F , for some comm utative Q -algebra F ), and a morphism ϕ : g ∗ 1  h ( X/S ) e  − → g ∗ 2  h ( X/S ) e  33 in C H M s ( U ) (or C H M s ( U ) F ). (a) Let us deﬁne an endomorphism of ( ∂ M , M , M c )( h ( X/S ) e ) “of Hec k e t yp e”, denoted ϕ ( g 1 , g 2 ), b y comp osing δ id S ,g ∗ 1 : ( ∂ M , M , M c )  h ( X/S ) e  − → ( ∂ M , M , M c )  g ∗ 1  h ( X/S ) e  ﬁrst with ( ∂ M , M , M c ) ◦ ϕ , and then with β g ∗ 2 , id S : ( ∂ M , M , M c )  g ∗ 2  h ( X/S ) e  → ( ∂ M , M , M c )  h ( X/S ) e  . (b) Note that unless g 1 = g 2 , the endomorphism ϕ ( g 1 , g 2 ) is in g eneral not the image of an endomorphism o n the smo oth relat ive Cho w motiv e h ( X/S ) e under the functor ( ∂ M , M , M c ). (c) If ϕ is a n isomorphism, with inv erse ψ , t hen using the construction from (a), the endomorphism ψ ( g 2 , g 1 ) on ( ∂ M , M , M c )( h ( X/S ) e ) can b e deﬁned. If X is of pure absolute dimension d X , then ψ ( g 2 , g 1 ) equals t he dual of ϕ ( g 1 , g 2 ), t wisted b y d X and shifted by 2 d X , under the identiﬁc atio n ( ∂ M , M , M c ) ∗  h ( X/S )  = ( ∂ M , M , M c )  h ( X/S )  ( − d X )[ − 2 d X ] from Theorem 2.2 (b1). W e leav e the details of the v eriﬁcation to the reader. (d) In practice, the morphism ϕ : g ∗ 1 ( h ( X/S ) e ) → g ∗ 2 ( h ( X/S ) e ) will b e ob- tained from a mor phism o f smo oth relativ e Cho w motives ov er U ϕ : h ( X × S,g 1 U /U ) = g ∗ 1 ( h ( X/S )) − → g ∗ 2 ( h ( X/S )) = h ( X × S,g 2 U /U ) satisfying the equation ϕ ◦ g ∗ 1 ( e ) = g ∗ 2 ( e ) ◦ ϕ in CH ∗ (( X × S,g 1 U ) × U ( X × S,g 2 U )) (or CH ∗ (( X × S,g 1 U ) × U ( X × S,g 2 U )) ⊗ Z F ). In that cas e, ϕ ( g 1 , g 2 ) can b e seen a s an endomorphism of the whole of ( ∂ M , M , M c )( h ( X/S )) comm uting with e . (e) In t he setting of (d), assume that the morphism ϕ : h ( X × S,g 1 U /U ) − → h ( X × S,g 2 U /U ) is r epresen ted by the cycle Z in c U ( X × S,g 1 U, X × S,g 2 U ) (or in c U ( X × S,g 1 U, X × S,g 2 U ) ⊗ Z F ). Chec king the deﬁnitions, o ne sees that the M -comp onen t of ϕ ( g 1 , g 2 ) is then represen ted b y the image o f Z under the direct image  g 1 × k g 2  ∗ : c U ( X × S,g 1 U, X × S,g 2 U ) − → c ( X , X ) (or under ( g 1 × k g 2 ) ∗ ⊗ F ). 34 3 Motiv es ass o c iated to Ab e lian sc hemes Fix a ﬁeld k admitting strict resolution o f singularities, a nd a base S ∈ S m/S . In this section, w e combine the ma in result f rom [DeMu] with t he theory dev elop ed in Section 2. Recall the followin g. Theorem 3.1 ([DeMu, Thm. 3.1, Prop. 3.3]) . (a) L et A/S b e an A beli a n scheme of r elative dimension g . Then ther e is a unique de c omp osition of the cla ss of the d iagonal (∆) ∈ CH g ( A × S A ) ⊗ Z Q , (∆) = 2 g X i =0 p A,i such that p A,i ◦ (Γ [ n ] A ) = n i · p A,i for al l i , and al l inte ge rs n . The p A,i ar e mutual ly ortho gonal idem p otents, and (Γ [ n ] A ) ◦ p A,i = n i · p A,i for al l i . (b) F or an y morphism f : A → B of Ab elian schemes o v er S , an d a ny i , p B ,i ◦ (Γ f ) = (Γ f ) ◦ p A,i ∈ CH ∗ ( A × S B ) ⊗ Z Q . In other wor ds, the de c omp osition in (a) is c ovariantly func torial in A . (c) F or an y iso geny g : B → A of A b elian schemes over S , and any i , p B ,i ◦ ( t Γ g ) = ( t Γ g ) ◦ p A,i ∈ CH ∗ ( A × S B ) ⊗ Z Q . In other wor ds, the de c omp osition in (a) is c ontr avariantly functorial und e r iso genies. W e use the notation Γ h for the graph of a morphism h of S -sche mes, [ n ] A for the m ultiplication b y n on the Ab elian sche me A , ( Z ) fo r the class of a cycle Z , a nd t Z for it s transp osition. Let h ( A/S ) = M i h i ( A/S ) b e the decomp osition of the relativ e mot iv e of A corresp onding to the decom- p osition (∆) = P i p A,i . Th us, on the t erm h i ( A/S ), the cycle class ( Γ [ n ] A ) acts via multiplic at io n b y n i . No w recall the exact tria ngle ( ∗ ) A ∂ M ( A ) − → M ( A ) − → M c ( A ) − → ∂ M ( A )[1] . By Theorem 2.2 (a), the cycle classes p A,i induce endomorphisms o f ( ∗ ) A , when considered as an exact triangle in D M ef f g m ( k ) Q . Theorem 3.2. (a) L et A/ S b e an Ab eli a n scheme o f r elative dim ension g . F or 0 ≤ i ≤ 2 g , den ote by M ( A ) i , M c ( A ) i and ∂ M ( A ) i the images of the idem p otent p A,i on M ( A ) , M c ( A ) and ∂ M ( A ) , r esp e ctively, c onsider e d as obje cts of the c ate gory D M ef f g m ( k ) Q . Then for any i , the triangle ( ∗ ) A,i ∂ M ( A ) i − → M ( A ) i − → M c ( A ) i − → ∂ M ( A ) i [1] 35 in D M ef f g m ( k ) Q is exact. (b) The dir e ct sum of the triangles ( ∗ ) A,i yields a de c omp osition ( ∗ ) A = 2 g M i =0 ( ∗ ) A,i . It h as the fol low ing pr op erties: (b1) for any inte ger n , the de c om p osition is r esp e cte d by [ n ] A . (b2) for e ac h i and n , the induc e d m orphisms [ n ] A,i on the thr e e terms of ( ∗ ) A,i e qual multiplic ation by n i . (c) As a de c omp osition of ( ∗ ) A into some ﬁn i te dir e ct sum of e xact triangles in D M ef f g m ( k ) Q , ( ∗ ) A = M i ( ∗ ) A,i is uniquely determine d by pr op erties (b1) and (b2). Mor e pr e cisely, it is uniquely determ i n e d by the fol lowing pr op erties: (c1) for some inte ger n 6 = − 1 , 0 , 1 , the de c omp osition is r esp e cte d by [ n ] A . (c2) for the choic e of n made in (c1) and e ach i , the induc e d morph ism [ n ] A,i on the thr e e terms of ( ∗ ) A,i e quals multiplic ation by n i . (d) The de c omp osition ( ∗ ) A = M i ( ∗ ) A,i is c ovariantly functorial under morphi s ms, and c ontr avarian tly functorial un- der iso genies of A b elian schemes over S . Pr o of. P art ( a ) is a formal consequence of the f a ct that the p A,i are idemp oten t. P arts (b) and (d) follow from Theorem 3.1 and the functoriality statemen t from Theorem 2.2 (a). P art (c) is left to the reader. q.e.d. The following seems worth while to note explicitly . Corollary 3.3. L et A/S b e an A b elian scheme of r elative dimension g . Then the b o unda ry motive ∂ M ( A ) de c omp oses functorial ly into a dir e ct sum ∂ M ( A ) = 2 g M i =0 ∂ M ( A ) i . On ∂ M ( A ) i , the endomorphism [ n ] A acts via multiplic ation by n i , for any inte ger n , and any 0 ≤ i ≤ 2 g . 36 Here is an illustration of the surjectivit y pr ov ed in [L, Lemma 5.18 ]. Prop osition 3.4. L et A/S b e a n Abelian scheme. The ele m ents p A,i of CH ∗ ( A × S A ) ⊗ Z Q lie in the image o f c S ( A, A ) ⊗ Z Q − → CH ∗ ( A × S A ) ⊗ Z Q . Mor e pr e cisely, for any inte ger n 6 = − 1 , 0 , 1 , π A,i,n := Y j 6 = i Γ [ n ] A − n j n i − n j is a p r e-image of p A,i in c S ( A, A ) ⊗ Z Q . Pr o of. On eac h of the direct facto r s h j ( A/S ) ⊂ h ( A/S ), the pro jector p A,i acts via m ultiplication b y the Kr o nec k er sym b ol δ ij , while (Γ [ n ] A ) acts via multiplication b y n j . Therefore, p A,i = Y j 6 = i (Γ [ n ] A ) − n j n i − n j ∈ CH ∗ ( A × S A ) ⊗ Z Q , for any integer n 6 = − 1 , 0 , 1. Therefore, the elemen t π A,i,n ∈ c S ( A, A ) ⊗ Z Q is indeed a pre-image of p A,i . q.e.d. 4 The intersection motive of a surface Fix a normal, prop er surface X ∗ o v er k . Let us ﬁr st recall, follow ing [CatMi], the construction and the basic prop erties of the interse ction motive o f X ∗ . Cho ose X   / / X ∗ o o ? _ Z where Z is a closed sub-sc heme of X ∗ whic h is ﬁnite o ver k , and whose com- plemen t X is smo oth. Choo se a resolution of singularities. More precise ly , consider in addition the follo wing diagram, assumed to b e cartesian: X   / / e X o o i ? _ π   D π   X   / / X ∗ o o ? _ Z where π is prop er (and birational), e X is smo oth (and prop er), and D is a divisor with normal crossings, whose irreducible comp onents D m are smo o th (and prop er). Recall [Sch, Sect. 1.13] t ha t the “degree 2 parts” M 2 ( D m ) ar e canoni- cally deﬁned as sub-ob jects of the motiv es M ( D m ) (w e remind the reader 37 that througho ut t he article, w e use homolo g ical no t a tion). Henc e t here is a canonical morphism i ∗ , 2 : M 2 ( D ) := M m M 2 ( D m ) ֒ − → M m M ( D m ) − → M ( e X ) of Chow motives . Similarly [Sc h , Sect. 1.11], t here is a canonical morphism i ∗ 0 : M ( e X ) − → M m M ( D m )(1)[2] − → → M m M 0 ( D m )(1)[2] , where M 0 ( D m ) denote the “degree 0 parts”, canonically deﬁned a s quotien ts of M ( D m ). The following is a sp ecial case of [CatMi, Sect. 2 .5] (see also [W5, Thm. 2.2]). Theorem 4.1. (i) The c omp os i tion α := i ∗ 0 i ∗ , 2 is an isomo rp h ism in the Q -line ar c ate gory D M ef f g m ( k ) Q . (ii) The c omp o sition p := i ∗ , 2 α − 1 i ∗ 0 is an idemp otent on M ( e X ) ∈ D M ef f g m ( k ) Q . Henc e so is the d i ﬀer enc e id e X − p . (iii) The im age im p ∈ D M ef f g m ( k ) Q is c anonic al ly i s omorphic to M 2 ( D ) . The pro of relies on the non-degeneracy of the inte rsection pairing on the comp onen ts of D . Deﬁnition 4.2 ([CatMi, p. 158], [W5, Def. 2.3 ]) . The interse ction motiv e of X ∗ is deﬁned a s M ! ∗ ( X ∗ ) := im(id e X − p ) ∈ D M ef f g m ( k ) Q . The name is mo t iv ated by the b ehav iour of t he realizations o f the inter- section motive. Its functorialit y prop erties are give n in [W5, Prop. 2.5]. It will b e useful to recall in particular the b ehaviour under ﬁnite morphisms f : Y ∗ → X ∗ b et we en normal, prop er surfaces ov er k . Assume that Z is suc h that the pre-image under f of X = X ∗ − Z is dense, and smo oth (this can b e achie ved b y enlarging Z , if necessary). The closed sub-sc heme f − 1 ( Z ) of Y con tains the singularities of Y ∗ . W e t hus can ﬁnd a cartesian diag ram of desingularizations of X ∗ and Y ∗ of the t yp e considered b efore: e Y o o ? _ F   C F   e X o o ? _ D The follo wing is the con tent of [W5, Prop. 2.5 (iii) a nd (iv), Prop. 2 .4]. Prop osition 4.3. (a) Both F ∗ and F ∗ r esp e ct the de c omp ositions M ( e X ) = M ! ∗ ( X ∗ ) ⊕ M 2 ( D ) and M ( e Y ) = M ! ∗ ( Y ∗ ) ⊕ M 2 ( C ) 38 of M ( e X ) and of M ( e Y ) , r esp e ctively. The c omp osition F ∗ F ∗ e quals multipli- c ation with the d e gr e e of f . (b) The deﬁnition of M ! ∗ ( X ∗ ) is ind ep endent of the choic es of the ﬁnite sub- scheme Z c ontaining the singularities, and of the desingularization e X of X ∗ . Next, let us establish t he connection to the b oundary motiv e of X , and to the constructions of Section 1. T o do so, assume k to admit resolution of singularities, ﬁx a dense op en sub-sc heme X ⊂ X ∗ whic h is smo oth, and cho ose X   / / e X o o i ? _ π   D π   X   / / X ∗ o o ? _ Z as ab ov e. Recall the diagram of exact triangles 0 M ( D ) ∗ (2)[4] o o M ( D ) ∗ (2)[4] 0 o o M c ( X ) O O M ( e X ) i ∗ O O o o M ( D ) O O i ∗ o o M c ( X )[ − 1] O O o o M c ( X ) M ( X ) O O o o ∂ M ( X ) O O o o M c ( X )[ − 1] o o 0 O O M ( D ) ∗ (2)[3] O O o o M ( D ) ∗ (2)[3] O O 0 o o O O from Theorem 1.13 (c), and let us refer to it using the sym b ol ( A ). It turns out tha t the three comp onen ts of the idemp ot ent p = i ∗ , 2 α − 1 i ∗ 0 on M ( e X ) all extend to giv e morphisms o f diagrams of exact triangles: the ﬁrst, denoted i ∗ 0 , maps ( A ) to 0 L m M 0 ( D m )(1)[2] o o L m M 0 ( D m )(1)[2] 0 o o 0 L m M 0 ( D m )(1)[2] o o L m M 0 ( D m )(1)[2] 0 o o 0 0 O O 0 O O 0 0 L m M 0 ( D m )(1)[1] O O o o L m M 0 ( D m )(1)[1] O O 0 o o The second comp onen t α − 1 maps this diagram isomorphically t o the follow- 39 ing, whic h w e shall denote by ( B ). 0 M 2 ( D ) o o M 2 ( D ) 0 o o 0 M 2 ( D ) o o M 2 ( D ) 0 o o 0 0 O O 0 O O 0 0 M 2 ( D )[ − 1] O O o o M 2 ( D )[ − 1] O O 0 o o Finally the third comp onen t i ∗ , 2 maps ( B ) bac k to ( A ). The comp osition of the three morphisms, denoted b y p : ( A ) − → ( A ) , is idempo ten t. Its image is diagram ( B ). Denote the image of id − p on M ( D ) b y M ≤ 1 ( D ). Then the image of id − p on the whole diagra m equals 0 M ≤ 1 ( D ) ∗ (2)[4] o o M ≤ 1 ( D ) ∗ (2)[4] 0 o o M c ( X ) O O M ! ∗ ( X ∗ ) i ∗ O O o o M ≤ 1 ( D ) O O i ∗ o o M c ( X )[ − 1] O O o o M c ( X ) M ( X ) O O o o ∂ M ( X ) O O o o M c ( X )[ − 1] o o 0 O O M ≤ 1 ( D ) ∗ (2)[3] O O o o M ≤ 1 ( D ) ∗ (2)[3] O O 0 o o O O Theorem 4.4. Assume k to ad m it r esolution o f sin gularities. (a) F or ﬁxe d X ∗ and smo oth X ⊂ X ∗ , the ab ove diag r a m o f exact triangles is indep endent of the c h oic e of e X . (b) The diagr am is c ovariantly a n d c ontr avaria ntly functorial under ﬁnite morphisms f : Y ∗ → X ∗ of normal, pr op er surfac es, which ar e c omp atible with the cho ic es of sm o oth s ub-schemes X ⊂ X ∗ and Y ⊂ Y ∗ : f − 1 ( X ) = Y . (c) The thir d c olumn of the diagr am M ≤ 1 ( D ) ∗ (2)[3] − → ∂ M ( X ) − → M ≤ 1 ( D ) − → M ≤ 1 ( D ) ∗ (2)[4] is a w eight ﬁltr ation of ∂ M ( X ) : M ≤ 1 ( D ) ∗ (2)[3] ∈ D M ef f g m ( k ) Q ,w ≤− 1 and M ≤ 1 ( D ) ∈ D M ef f g m ( k ) Q ,w ≥ 0 . (d) The is omorphism classes of the weight ﬁltr ation fr om (c) and of the Chow motive M ! ∗ ( X ∗ ) c orr e s p ond under the bije ction fr om The or em 1.18 ( Q -line ar version). Pr o of. P arts (a) and (b) follow from Prop osition 4.3. Part (c) fo llo ws from stabilit y under passage to direct factors 1.9 (1), a nd the f act that M ( D ) ∗ (2)[3] − → ∂ M ( X ) − → M ( D ) − → M ( D ) ∗ (2)[4] 40 is a we ig ht ﬁltration (Theorem 1.15). Finally , (d) is a direct consequence o f Construction 1.17, and of t he shap e of the diagra m im(id − p ). q.e.d. Remark 4.5. Let us discuss Construction 1.17 in the presen t geometrical setting. The w eigh t ﬁltration of ∂ M ( X ) is M ≤ 1 ( D ) ∗ (2)[3] c − − → ∂ M ( X ) c + − → M ≤ 1 ( D ) δ − → M ≤ 1 ( D ) ∗ (2)[4] ; according to Theorem 4.4 (a), it is indep enden t of the c hoice of e X (hence of D ). It ﬁts into the diagra m of exact triangles 0 M ≤ 1 ( D ) ∗ (2)[4] o o M ≤ 1 ( D ) ∗ (2)[4] 0 o o M c ( X ) O O M ≤ 1 ( D ) δ O O M c ( X )[ − 1] O O c + ( v + [ − 1]) o o M c ( X ) M ( X ) u o o ∂ M ( X ) c + O O v − o o M c ( X )[ − 1] v + [ − 1] o o 0 O O M ≤ 1 ( D ) ∗ (2)[3] v − c − O O o o M ≤ 1 ( D ) ∗ (2)[3] c − O O 0 o o O O In the general situation of Construction 1.17, what w e did next was to cho ose some ob ject M 0 completing the diagram. In the sp eciﬁc situation w e a re con- sidering at presen t, there is only one c hoice, up to unique isomorphism, whic h in addition is compatible with any of the diagrams o f t yp e ( A ) asso ciated to desingularizations e X of X ∗ . This c hoice is M ! ∗ ( X ∗ ). W e thus obta in rigidiﬁ- cation of the in tersection motive , while the condition from Complemen t 1.24 on the absence of we ig hts − 1 and 0 in the b oundary motiv e is clearly not satisﬁed — unless X ∗ = X is itself (prop er and) smo oth (cmp. Problem 1.22). Let us ﬁnish this section by an example, whic h will allo w us to illustrate b oth Principle 1.2 (on extensions) and Principle 1.19 (on functorialit y). Example 4.6. Our base ﬁeld k equals the ﬁeld of ratio nal n umbers Q . Fix a real quadratic n um b er ﬁeld L , and let X b e a Hilb ert mo dular surfac e asso ciated to L and some lev el K . W e view K a s an op en compact subgroup of the g roup G ( A f ) of (ﬁnite) adelic p oints of the group sc heme G from [R, Sect. 1.27 ]. The subgroup K ⊂ G ( A f ) is assumed t o b e suﬃcien tly small, a condition which ensures that X is smo oth o v er Q . Denote b y X ∗ its Baily– Bor el c omp actiﬁc a tion ; it is normal and pro jectiv e ov er Q . (a) The mo r phism i ∗ : M ≤ 1 ( D ) − → M ! ∗ ( X ∗ ) o ccurring in the w eigh t ﬁltration M ≤ 1 ( D ) i ∗ − → M ! ∗ ( X ∗ ) − → M c ( X ) − → M ≤ 1 ( D )[1] 41 of M c ( X ) can b e used to construct a morphism of certain sub-quotien ts of its source and target, M 1 ( D ) − → M ! ∗ 2 ( X ∗ ) [W5, Thm. 6.6]. This morphism can b e interpreted as an elemen t o f Ext 1 D M ef f gm ( Q ) Q  M 1 ( D )[ − 1] , M ! ∗ 2 ( X ∗ )[ − 2]  , i.e., a one-extension in the triangula t ed category D M ef f g m ( Q ) Q (note that ac- cording t o [W5, Prop. 6.5], M 1 ( D )[ − 1] is an Artin motiv e). F ollowing [W5, Ex. 7.4 ], it can b e related to the Kummer–Chern–Eisenstein extensions con- sidered in [Cas]. In particular [W5, Ex. 7.4 (6)], the extension is non-trivial. (b) The in tersection motive M ! ∗ ( X ∗ ) carries a natural action o f the Hec k e al- gebra R ( K , G ( A f )) asso ciated to K ⊂ G ( A f ). More precisely , let x ∈ G ( A f ). The Hilb ert surface X is the target of tw o ﬁnite ´ etale morphisms g 1 , g 2 : Y → X , where Y denotes the Hilb ert surface of lev el K ′ := K ∩ x − 1 K x . In standard notation from the theory of Shimura v arieties, the morphism g 1 corresp onds to [ · 1 ], and the morphism g 2 to [ · x − 1 ]. Both morphisms can b e extended to ﬁnite morphisms b et wee n the Baily–Borel compactiﬁcations g i : Y ∗ → X ∗ , satisfying the formulae g − 1 i ( X ) = Y , i = 1 , 2. According to Theorem 4.4 (b), the diagram 0 M ≤ 1 ( D ) ∗ (2)[4] o o M ≤ 1 ( D ) ∗ (2)[4] 0 o o M c ( X ) O O M ! ∗ ( X ∗ ) i ∗ O O o o M ≤ 1 ( D ) O O i ∗ o o M c ( X )[ − 1] O O o o M c ( X ) M ( X ) O O o o ∂ M ( X ) O O o o M c ( X )[ − 1] o o 0 O O M ≤ 1 ( D ) ∗ (2)[3] O O o o M ≤ 1 ( D ) ∗ (2)[3] O O 0 o o O O is therefore stable under the comp osition g 2 , ∗ g ∗ 1 . By deﬁnition, this comp o - sition equals the action of t he class K xK ∈ R ( K , G ( A f )). In particular, the Hec k e alg ebra acts on the whole of the ab ov e diag r a m. It is useful to note that it s eﬀect o n the third r o w, i.e., on the b o undar y triangle, is the one described in Example 2.16 (d), f or X = S , ϕ = id Y , and e = id X . 5 The interior motiv e of a pro d uct of univer- sal ell iptic curv es In this section, our base ﬁeld k equals the ﬁeld of r ational n umbers Q . Fix in tegers n ≥ 3 and r ≥ 0, and let S ∈ S m/ Q denote the mo dular curv e 42 parametrizing elliptic curv es with lev el n structure. W rite X → S fo r the univ ersal elliptic curv e, and X r := X × S × . . . × S X for the r -fold ﬁbre pro duct of X ov er S . Recall the decompo sition h ( X/S ) = 2 M i =0 h i ( X/S ) of t he relative motive of X from Theorem 3.1. Deﬁnition 5.1. D eﬁne r V ∈ C H M s ( S ) Q as r V := Sym r h 1 ( X/S ) . The tensor pro duct is in C H M s ( S ) Q , and the symm etric p ow ers are formed with the usual con v ention concerning the (t wist of ) the natural action of the symmetric g roup on a p ow er of X ov er S (see e.g. [Sc h , Sect. 1.1.2]). Th us, r V is a direct factor of h ( X r /S ) ∈ C H M s ( S ) Q . That is, it is asso ciated to an idemp o ten t e ∈ CH r ( X r × S X r ) ⊗ Z Q . F ro m Theorem 2.2 (a ), we get a natural action of e on the b o undary triangle of X r . In particular, we hav e the fo llo wing result. Prop osition 5.2. The triangle ( ∗ ) e X r ∂ M ( X r ) e − → M ( X r ) e − → M c ( X r ) e − → ∂ M ( X r ) e [1] in D M ef f g m ( k ) Q is exact. This triangle equals the image ( ∂ M , M , M c )( h ( X r /S ) e ) of the smo oth relativ e Cho w mo t iv e r V = h ( X r /S ) e under the functor ( ∂ M , M , M c ) S from Theorem 2.2. It is not v ery diﬃcult to che ck that the idempotent e coincides with the one used in [Sc h , Sect. 1] and [W3, Sect. 3 and 4]. Prop osition 5.3 ([W3, Ex. 4.16 (d)]) . The dir e ct factor ∂ M ( X r ) e of the b oundary motive of X r is without weights − 1 and 0 wh e n ever r ≥ 1 . By Complemen t 1.24, the e -part of the in terior motiv e of X r can b e constructed, and is unique up to unique isomorphism. It is sho wn in [W3, Thm. 3.3 (b) and Cor. 3.4 (b)] that it is canonically isomorphic to the Chow motiv e r n W constructed in [Sc h ] out of a compactiﬁcation of X r . In that article, the action of the Hec k e algebra on that compactiﬁcation, hence on r n W is then used to construc t the G r o thendiec k motiv e M ( f ) for elliptic normalized newforms f of lev el n ≥ 3 and weigh t w = r + 2 ≥ 3. Let us ﬁnish this section by giving an alternat ive description of the action of the Hec k e algebra on r n W , whic h av oids compactiﬁcations. 43 Example 5.4. Assume that r ≥ 1, and consider the diagram 0 ∂ M ( X r ) e ≤− 2 [1] o o ∂ M ( X r ) e ≤− 2 [1] 0 o o M c ( X r ) e O O r n W O O o o ∂ M ( X r ) e ≥ 1 O O o o M c ( X r ) e [ − 1] O O o o M c ( X r ) e M ( X r ) e O O o o ∂ M ( X r ) e O O o o M c ( X r ) e [ − 1] o o 0 O O ∂ M ( X r ) e ≤− 2 O O o o ∂ M ( X r ) e ≤− 2 O O 0 o o O O from Complemen t 1 .24 asso ciated to the we ight ﬁltrat io n of ∂ M ( X r ) e a v oid- ing w eigh ts − 1 and 0. W e shall sho w that this dia g ram, and hence r n W in particular, carries a natural action of the Hec ke algebra R ( K n , G L 2 ( A f )) asso- ciated to t he pr incipal subgroup K n ⊂ GL 2 ( A f ) of lev el n . Let x ∈ GL 2 ( A f ). The curv e S is the target of tw o ﬁnite ´ etale morphisms g 1 , g 2 : U → S , where U denotes the mo dular curve of leve l K ′ := K n ∩ x − 1 K n x . In standard no- tation from the theory of Shim ura v arieties, the morphism g 1 corresp onds to [ · 1], and the morphism g 2 to [ · x − 1 ]. Denote by X 1 , X 2 the base changes of the univ ersal elliptic curve X to U via g 1 and g 2 , resp ectiv ely . T o the data K n and x , the follow ing are cano nically asso ciated: a third elliptic curv e Y o v er U , and isogenies f 1 : Y → X 1 and f 2 : Y → X 2 . No w note that ϕ := Γ f r 2 ◦ t Γ f r 1 deﬁnes a morphism of smo o th relativ e Cho w motiv es o ver U , ϕ : h ( X r 1 /U ) = g ∗ 1 ( h ( X r /S )) − → g ∗ 2 ( h ( X r /S )) = h ( X r 2 /U ) . Since b oth f 1 and f 2 are isogenies, this morphism is compatible with the external pro ducts of the idemp ot ents p X i , 1 pro jecting on to the h 1 (Theo- rem 3.1 (b) and (c)). The mo r phism ϕ is also compatible with the ac- tion of the symmetric g roup; hence it is compatible with the cycle classes g ∗ i ( e r ) ∈ CH r ( X r i × U X r i ) ⊗ Z Q . This means that w e hav e the relation ϕ ◦ g ∗ 1 ( e r ) = g ∗ 2 ( e r ) ◦ ϕ of morphisms of smo oth relativ e Cho w motives o ve r U . 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Boundary motive, relative motives and extensions of motives

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