aa r X i v : . [ m a t h . AG ] A p r Motivic intersection complex by J¨org Wildeshaus ∗ LAGAUMR 7539Institut Galil´eeUniversit´e Paris 13Avenue Jean-Baptiste Cl´ementF-93430 VilletaneuseFrance [email protected]
April 28, 2011
Abstract
In this article, we give an unconditional definition of the motivic ana-logue of the intersection complex, establish its basic properties, andprove its existence in certain cases.Keywords: Beilinson motives, motivic weight structure, Chow mo-tives, motivic intersection complex.
Math. Subj. Class. (2010) numbers: 19E15 (14C25, 14F42, 14J17). ∗ Partially supported by the
Agence Nationale de la Recherche , project no. ANR-07-BLAN-0142 “M´ethodes `a la Voevodsky, motifs mixtes et G´eom´etrie d’Arakelov”. ontents This paper contains largely extended notes of the talk the author gave duringthe conference
Regulators III , which took place at the University of Barcelonain July 2010. Its main purpose is to propose an unconditional definition ofthe motivic intersection complex.The use of the intersection complex, say in the context of (topological)sheaves on schemes over the complex numbers, or of ( ℓ -adic) sheaves onschemes over a field, can be motivated by purity . Let X be proper over k .Its singular cohomology (if k = C ) carries a pure Hodge structure, and its ℓ -adic cohomology (if k is finite or a number field) a pure Galois action, pro-vided that X is smooth. If this latter hypothesis is not met, then in order toget analogous purity statements, the constant sheaf on X has to be replacedby the intersection complex [BBD] (with respect to the inclusion of the regu-lar locus of X ). Its (hyper)cohomology is known as intersection cohomology of X .One of the main arithmetic applications to keep in mind concerns the Baily–Borel compactification of a smooth
Shimura variety : it is canonical,and even minimal in a precise sense, but rarely smooth. Its intersection co-homology contains valuable arithmetic information, e.g. , certain of its directfactors allow to realize Hodge structures and Galois representations associ-ated to automorphic forms.In order to construct motives inducing these Hodge structures and Galoisrepresentations via the respective realizations, one is thus led to try first toconstruct the intersection motive . One minimal requirement on this objectwould be that its realizations equal intersection cohomology.2his construction succeeded in a small number of cases. Let us cite vari-eties (over C ) admitting semismall resolutions [CM], which includes the caseof surfaces, and Baily–Borel compactifications of Hilbert–Blumenthal vari-eties [GHM] (over C , and with more general than just constant coefficients).A general program for the construction of the intersection motive, assumingGrothendieck’s standard conjectures, was developed (still over C ) in [CH].When the construction works unconditionally, then it does so for specificgeometric reasons. For example, such a reason would be that the relevantcycle classes are isomorphisms. The idea is basically to obtain an explicitformula for intersection cohomology sitting in the cohomology of a desingu-larization of X ; the specific geometric reasons in question then allow to givea motivic sense to the explicit formula. Unfortunately, some functorialityproperties valid for intersection cohomology are not a consequence of the ex-plicit formula, and hence do not obviously hold for the intersection motive.This concerns for example the action of the Hecke algebra (which is neededin order to cut out the motive of an individual automorphic form from theintersection motive).In [W1], we gave an unconditional construction of the intersection mo-tive of Baily–Borel compactifications of smooth Hilbert–Blumenthal varietieswith non-constant coefficients. It is a Chow motive over Q , and behaves wellunder Hecke correspondences. Again, the construction works for specific geo-metric reasons, which translate into saying that “the boundary avoids weights − weight .Assume first that our base scheme X equals the spectrum of a perfectfield k . According to Bondarko [Bo1], the category of geometrical motives [VSF] carries a weight structure , whose heart equals the category CHM ( k ) ofChow motives over k . The precise definitions of weight structures and heartswill be recalled in the present Section 1; for the moment, let us keep in mindthat the motivic weight structure allows for an intrinsic characterization ofthe full sub-category CHM ( k ) of the category of geometrical motives. Thisis the key for everything to follow. Roughly speaking, the construction from[W1] works since geometrical motives are flexible enough to preserve functo-riality; the problem of knowing whether the result of this construction is aChow motive is then reduced to a computation of weights.In general, the properties of intersection cohomology (functoriality, pu-rity,...) are consequences of properties of the intersection complex. A generalsolution to the problem of constructing the intersection motive therefore re-quires the construction of the motivic intersection complex. Here, one isconfronted with a foundational problem: the na¨ıve generalization of the de-finition via truncations [BBD] cannot work since it requires the existence of3 (perverse) t -structure . But even when the base is of the form Spec k , thenexcept for certain fields k , such a t -structure is not known to exist on thecategory of geometrical motives. Thus, the mere problem of giving an uncon-ditional definition of the motivic intersection complex is a priori non-trivial.The solution to this problem that we shall propose, is again based on thenotion of weight structure. In a way, our approach can be seen as “reading[BBD] backwards”, i.e. , starting from [BBD, Chap. 5] on weights. This con-cerns in particular the Decomposition Theorem [BBD, Thm. 5.4.5], whichimplies that every pure complex on X restricting to the structure sheaf onan open smooth sub-variety, contains the intersection complex as a directfactor. Let us indicate already here that the motivic analogue of this result(Theorem 3.1 (b)) is a rather elementary exercice in weight structures...Let us now give a detailed overview of the individual sections of thispaper. Section 1 starts with a review of Beilinson motives [CD], which con-veniently generalize geometrical motives from
Spec k to arbitrary bases X .We then recall the basic notions related to weight structures. We reviewthe main results from [H´e1] on the existence of the motivic weight struc-ture on Beilinson motives (generalizing [Bo1] from Spec k to X ), and on thebehaviour of weights under the six operations from [CD]. We then define the category CHM ( X ) Q of Chow motives over X as the heart of the mo-tivic weight structure, and establish two complements of the theory. First(Theorem 1.7), we show that for an open sub-scheme U of X , the inverseimage from CHM ( X ) Q to CHM ( U ) Q is both essentially surjective and full.Following the terminology introduced in [Bo2], this can be seen as a mo-tivic version of resolution of singularities . Theorem 1.7 strenghens [Bo2,Thm. 2.2.1 III 1], where essential surjectivity is proved up to pseudo-Abeliancompletion. Second (Theorem 1.12), we show that local duality respects theweights in a strict sense; in particular, the dual of a Chow motive is again aChow motive. This complements [H´e2, Cor. 2.2.5], where the same result isproved provided X is regular, and also [H´e1, Cor. 3.9], where left exactness(with respect to the weights) is established for any X .Having in mind the Decomposition Theorem [BBD, Thm. 5.4.5], an in-termediate extension of a Chow motive M U over a dense U should satisfya certain minimality condition among all possible extensions of M U to aChow motive over X . In Section 2, we make this precise for regular U , and M U = U , the structure motive on U . More precisely (Definition 2.1), themotivic intersection complex j ! ∗ U is a Chow motive on X restricting togive U , and admitting no non-trivial endomorphisms restricting trivially to U . We then establish independence of j ! ∗ U of U (Proposition 2.4). In itsessence, it results from the study of a basic, but important example: when X is regular, then j ! ∗ U = X (Example 2.3).4ection 3 contains our main results. According to Theorem 3.1 (a), themotivic intersection complex is unique up to unique isomorphism. As al-ready indicated, Theorem 3.1 (b) states that any extension of U to a Chowmotive over X contains j ! ∗ U as a direct factor — provided the latter exists.Under the same hypothesis, j ! ∗ U is auto-dual (Corollary 3.8), meaning thatthe motivic intersection pairing can be defined. Theorem 3.11 identifies thefew cases where we actually know the motivic intersection complex to exist.Section 4 contains the proof of Theorem 3.11.We choose to add a number of “Problems” in the text. While they con-cern properties that one might reasonably expect j ! ∗ U to satisfy, the authordoes not know to solve any of them. The paper also contains a number ofmiscellaneous results, which are not needed elsewhere in the text, but seemworth to be mentioned nonetheless. In particular, this concerns Corollar-ies 1.10 and 3.5. The first (Corollary 1.10) states that for an open immersion j : U ֒ → X and any Chow motive N U over U , the image under the inverseimage j ∗ of motivic cohomology of N in motivic cohomology of N U is in-dependent of the extension of N U to a Chow motive N over X . We relatethis to Scholl’s construction of “integral” sub-spaces of motivic cohomologyof Chow motives over number fields (Remark 1.11). According to the second(Corollary 3.5), a Beilinson motive which is Nisnevich-locally isomorphic to X , for a regular base X , is (globally) isomorphic to X . This allows to gen-eralize absolute purity [CD, Thm. 13.4.1] to arbitrary morphisms a : X → S ′ between regular schemes: as soon as a is of pure relative dimension d , thereis an isomorphism X ( d )[2 d ] ∼ = a ! S ′ (Corollary 3.7).Part of this work was done while I was enjoying a modulation de ser-vice pour les porteurs de projets de recherche , granted by the Universit´eParis 13 , and during a stay at the University of Tokyo. I am grateful toboth institutions. I wish to thank the organizers of
Regulators III for theinvitation to Barcelona, C. Soul´e for a stimulating question asked during mytalk, F. D´eglise and D. H´ebert for useful comments on a first draft of thispaper, G. Ancona for strengthening an earlier version of Proposition 2.5, andthe referee for her or his remarks and suggestions.
Conventions : Throughout the article, S denotes a fixed base scheme,which we assume to be of finite type over an excellent scheme of dimension atmost two. By definition, schemes are S -schemes which are separated and offinite type (in particular, they are excellent, and Noetherian of finite dimen-sion), morphisms between schemes are separated morphisms of S -schemes,and a scheme is regular if the underlying reduced scheme is regular in theusual sense. 5 Review of weights on Beilinson motives
We fix our base S , and work in the triangulated, Q -linear categories DM B ( X )of Beilinson motives over X [CD, Def. 13.2.1], indexed by schemes X (al-ways in the sense of the conventions fixed at the end of our Introduction).As in [CD], the symbol X is used to denote the unit for the tensor productin DM B ( X ). We shall employ the full formalism of six operations devel-oped in [loc. cit.]. Below, we shall list the principles (A)–(E) which willbe particularly important to us. The global assumptions made in [loc. cit.]to establish these principles are met since DM B ( • ) is a motivic category [CD, Cor. 13.2.11], which by definition [CD, Def. 2.4.2] implies that it is pregeometric . Therefore, it is Sm -fibred [CD, Def. 1.1.9], the localizationproperty ( Loc i ) from [CD, Def. 2.3.2] holds, and by [CD, Thm. 2.4.12] sodo the proper transversality property from [CD, Def. 1.1.16] and the supportproperty from [CD, Def. 2.2.5]. Furthermore, by [CD, Prop. 14.2.16], thecategory DM B ( • ) is separated in the sense of [CD, Def. 2.1.11]. By [CD,Ex. 14.3.20], it is pure in the sense of [CD, Def. 14.3.19]. (A) Absolute purity.Relation to K -theory: if i : Z ֒ → X is a closed immersion of pure codimension c between regular schemes, then there is a canonical isomorphism Z ( − c )[ − c ] ∼ −−→ i ! X in DM B ( Z ) [CD, Thm. 13.4.1]. For any regular scheme X , and any pair ofintegers ( p, q ), there is a canonical isomorphismHom DM B ( X ) ( X , X ( p )[ q ]) ∼ = Gr pγ K p − q ( X ) Q , where K • ( X ) Q denotes the tensor product of K -theory of X with the ratio-nals, and Gr γ the graded object with respect to the (Adams) gamma filtra-tion [CD, Cor. 13.2.14]. Furthermore, this isomorphism is contravariantlyfunctorial with respect to morphisms of regular schemes [CD, Cor. 13.2.11].(B) Base change: for any morphism f , there is a natural transformation α f : f ! −→ f ∗ , which is an isomorphism is f is proper [CD, Thm. 2.2.14 (1)]. If f is thebase of a cartesian diagram Y ′ g ′ (cid:15) (cid:15) f ′ / / X ′ g (cid:15) (cid:15) Y f / / X of schemes, then the exchange transformation g ∗ f ! −→ f ′ ! g ′∗ is an isomorphism [CD, Prop. 2.2.13 (b)]. Hence so is the adjoint exchangetransformation g ′∗ f ′ ! −→ f ! g ∗ . Constructibility: by definition [CD, Def. 1.4.7], the full thick triangulatedsub-category DM B ,c ( X ) of DM B ( X ) of constructible objects is generatedby the Tate twists M X ( T )( p ) of the motives M X ( T ) [CD, Sect. 1.1.33] ofsmooth X -schemes T . In particular, all twists X ( p ) belong to DM B ,c ( X ).By [CD, Ex. 14.1.3], an object of DM B ( X ) is constructible if and only if it iscompact. According to [CD, Thm. 14.1.31], the sub-categories DM B ,c ( • ) ⊂ DM B ( • ) are respected by the six functors. (D) Duality: fix a scheme X whose structure morphism to S factors over a regular scheme; this is of coursethe case if S is itself regular. According to [CD, Thm. 14.3.28], the category DM B ,c ( X ) then contains dualizing objects in the sense of [CD, Def. 14.3.10].Fix such a dualizing object R . Define the local duality functor (with respectto R ) as D X := Hom X ( • , R ) . It is right adjoint to itself [CD, Sect. 14.3.30]. It preserves constructibleobjects, and the adjunction id X → D X is an isomorphism on DM B ,c ( X ) [CD,Cor. 14.3.31 (a), (b)]. Furthermore, it exchanges f ∗ and f ! , as well as f ! and f ∗ in the following sense: for a morphism f : Y → X , put D Y := Hom Y ( • , f ! R ) ;note that according to [CD, Prop. 14.3.29 (ii)], the motive f ! R is dualizingon Y . Then there are natural isomorpisms of functors D Y f ∗ ∼ −−→ f ! D X and f ∗ D Y ∼ −−→ D X f ! on DM B ( • ) [CD, Cor. 14.3.31 (d) and its proof]. Therefore, f ∗ D X ∼ −−→ f ! D Y and D X f ∗ ∼ −−→ f ! D Y on DM B ,c ( • ). For the applications of duality that we have in mind, we needto make explicit choices of dualizing object R . Fix a pair of integers ( p, q ),and a morphism a : X → S ′ with regular target. Then R := a ! S ′ ( p )[ q ] ∈ DM B ,c ( X )is a dualizing object [CD, Prop. 14.3.29]. It will be necessary to identify R under the following additional hypotheses on X : the morphism a : X → S ′ is quasi-projective, and X is regular and connected of relative dimension e over S ′ . We claim that in this case, there is an isomorphism R ∼ −−→ X ( p + e )[ q + 2 e ] . Indeed, absolute purity (see point (A)) and the formula j ! = j ∗ for an openimmersion j [CD, Thm. 2.2.14 (2)] reduces us to the case when X is a pro-jective space over S ′ . Our claim then follows from [Ay, Scholie 1.4.2 3] ( via [CD, Cor. 2.4.9]). (E) Localization: if i : Z ֒ → X and j : U ֒ → X are comple-mentary closed, resp. open immersions of schemes, then there are canonicalexact triangles j ! j ∗ −→ id X −→ i ∗ i ∗ −→ j ! j ∗ [1] , ∗ i ! −→ id X −→ j ∗ j ∗ −→ i ∗ i ! [1]of exact endo-functors of DM B ( X ) [CD, Prop. 2.3.3 (2), (3), Thm. 2.2.14 (2)].The adjunctions id U → j ∗ j ! , j ∗ j ∗ → id U and i ∗ i ∗ → id Z are isomorphisms,and the compositions i ∗ j ! and j ∗ i ∗ are trivial [CD, Sect. 2.3.1]. From whatprecedes, it follows formally that the adjunction id Z → i ! i ∗ is an isomorphism,and that the composition i ! j ∗ is trivial. We also see, putting i equal to theimmersion of the reduced scheme structure X red on X , that i ∗ : DM B ( X red ) −→ DM B ( X )is an equivalence of categories, with canonical quasi-inverse i ! = i ∗ . Thisjusitifies a posteriori the abuse of language fixed in the conventions at theend of our Introduction.Now recall the following notions, due to Bondarko. Definition 1.1 ([Bo1, Def. 1.1.1]) . Let C be a triangulated category. A weight structure on C is a pair w = ( C w ≤ , C w ≥ ) of full sub-categories of C ,such that, putting C w ≤ n := C w ≤ [ n ] , C w ≥ n := C w ≥ [ n ] ∀ n ∈ Z , the following conditions are satisfied.(1) The categories C w ≤ and C w ≥ are Karoubi-closed: for any object M of C w ≤ or C w ≥ , any direct summand of M formed in C is an object of C w ≤ or C w ≥ , respectively.(2) (Semi-invariance with respect to shifts.) We have the inclusions C w ≤ ⊂ C w ≤ , C w ≥ ⊃ C w ≥ of full sub-categories of C .(3) (Orthogonality.) For any pair of objects A ∈ C w ≤ and B ∈ C w ≥ , wehave Hom C ( A, B ) = 0 . (4) (Weight filtration.) For any object M ∈ C , there exists an exact triangle A −→ M −→ B −→ A [1]in C , such that A ∈ C w ≤ and B ∈ C w ≥ .Slightly generalizing the above terminology, for n ∈ Z , we shall refer toany exact triangle A −→ M −→ B −→ A [1]in C , with A ∈ C w ≤ n and B ∈ C w ≥ n +1 , as a weight filtration of M .8 efinition 1.2 ([Bo1, Def. 1.2.1 1]) . Let w be a weight structure on C .The heart of w is the full additive sub-category C w =0 of C whose objects lieboth in C w ≤ and in C w ≥ .Beilinson motives can be endowed with weight structures, thanks to themain results from [H´e1]. More precisely, the following holds. Theorem 1.3 ([H´e1, Thm. 3.3, Thm. 3.8 (i)–(ii)]) . (a) There are cano-nical weight structures w on the categories DM B ,c ( • ) . They are uniquelycharacterized by the following properties.(a1) The objects X ( p )[2 p ] belong to the heart DM B ,c ( X ) w =0 , for all integers p , whenever X is regular.(a2) For a morphism of schemes f , left adjoint functors f ∗ , f ! and f ♯ (thelatter for smooth f ) are w -left exact , i.e. , they map DM B ,c ( • ) w ≤ to DM B ,c ( • ) w ≤ , and right adjoint functors f ∗ , f ! and f ∗ (the lat-ter for smooth f ) are w -right exact , i.e. , they map DM B ,c ( • ) w ≥ to DM B ,c ( • ) w ≥ .(b) There are canonical weight structures W on the categories DM B ( • ) .They induce the weight structures w on the categories DM B ,c ( • ) . They areuniquely characterized by the requirement that any small sum of objects of DM B ,c ( X ) w =0 lie in DM B ( X ) W =0 . Let us refer to the weight structure w on DM B ,c ( • ) as the motivic weightstructure . Theorem 1.3 generalizes an earlier result of Bondarko’s [Bo1,Prop. 6.5.3] concerning the case X = Spec k , for a perfect field k (use [CD,Rem. 10.1.5, Thm. 15.1.4] to get the equivalence between the triangulatedcategory of geometrical motives `a la Voevodsky and DM B ,c ( Spec k )). Remark 1.4.
Since the first appearance of [H´e1], a different proof ofexistence of the motivic weight structure was given in [Bo2, Thm. 2.1.1]. The w -exactness properties from [H´e1, Thm. 3.8] are shown in [Bo2, Thm. 2.2.1 II]for quasi-projective morphisms of schemes. The results of [Bo2] were obtainedindependently from [H´e1].Note that for perfect fields k , [Bo1, Sect. 6.6] allows to identify the heart ofthe motivic weight structure on DM B ,c ( Spec k ) with the category (oppositeto the category) of Chow motives over k . This motivates the following. Definition 1.5.
The Q -linear category CHM ( X ) Q of Chow motives over X is defined as the heart DM B ,c ( X ) w =0 of the motivic weight structure. Remark 1.6.
The categories DM B ,c ( • ) are pseudo-Abelian (see [H´e1,Sect. 2.10]). Hence so are their hearts CHM ( • ) Q . For a fixed scheme X , thecategory CHM ( X ) Q can be constructed as the pseudo-Abelian completionof the category of motives over X of the form f ! Y ( p )[2 p ] , f : Y → X with regular source Y , and integers p [H´e1,Thm. 3.3 (ii)]. Since by [CD, Cor. 14.3.9] these motives generate DM B ,c ( X ) asa thick triangulated category, we see in particular that the latter is generatedby the heart of its weight structure.Here is our first application of the formalism of motivic weight structures. Theorem 1.7.
Let j : U ֒ → X be an open immersion of schemes.(a) The inverse image j ∗ : CHM ( X ) Q −→ CHM ( U ) Q is essentially surjective.(b) The inverse image j ∗ is full. Note that by Theorem 1.3 (a2), the functor j ∗ is w -exact , meaning that itis both w -left and w -right exact ( j is smooth). In particular, it preserves thehearts of the weight structures on DM B ,c ( X ) and on DM B ,c ( U ). Note alsothat essential surjectivity of j ∗ on both DM B ,c ( • ) w ≤ and DM B ,c ( • ) w ≥ is aformal consequence of the existence of j ! and j ∗ , and the formulae id U ∼ = j ∗ j ! and j ∗ j ∗ ∼ = id U . (By contrast, j ∗ should not in general be expected to be fullon DM B ,c ( • ) w ≤ or on DM B ,c ( • ) w ≥ !) Theorem 1.7 (a) strenghens [Bo2,Thm. 2.2.1 III 1], where it is proved that j ∗ is essentially surjective up topseudo-Abelian completion. Proof of Theorem 1.7. (a) Let M U be an object of CHM ( U ) Q , andconsider the morphism m := α j ( M U ) : j ! M U −→ j ∗ M U of motives over X (see point (B) above). Applying j ∗ to m yields an iso-morphism. Therefore, by localization, any cone of m is of the form i ∗ C , fora motive C over the complement i : Z ֒ → X of U in X (with the reducedscheme structure, say). Choose and fix such a cone i ∗ C , as well as a weightfiltration C ≤ c − −→ C c + −→ C ≥ δ −→ C ≤ [1]of C ∈ DM B ,c ( Z ) (Theorem 1.3 (a)). Thus, C ≤ ∈ DM B ,c ( Z ) w ≤ and C ≥ ∈ DM B ,c ( Z ) w ≥ . According to axiom TR4’ of triangulated categories (see [BBD, Sect. 1.1.6]10or an equivalent formulation), the diagram of exact triangles0 (cid:15) (cid:15) / / i ∗ C ≥ [ − i ∗ C ≥ [ − i ∗ δ [ − (cid:15) (cid:15) / / (cid:15) (cid:15) j ! M U i ∗ C ≤ i ∗ c − (cid:15) (cid:15) / / j ! M U [1] j ! M U (cid:15) (cid:15) m / / j ∗ M U (cid:15) (cid:15) / / i ∗ C i ∗ c + (cid:15) (cid:15) / / j ! M U [1] (cid:15) (cid:15) / / i ∗ C ≥ i ∗ C ≥ / / DM B ,c ( X ) can be completed to give0 (cid:15) (cid:15) / / i ∗ C ≥ [ − (cid:15) (cid:15) i ∗ C ≥ [ − i ∗ δ [ − (cid:15) (cid:15) / / (cid:15) (cid:15) j ! M U / / M (cid:15) (cid:15) / / i ∗ C ≤ i ∗ c − (cid:15) (cid:15) / / j ! M U [1] j ! M U (cid:15) (cid:15) m / / j ∗ M U (cid:15) (cid:15) / / i ∗ C i ∗ c + (cid:15) (cid:15) / / j ! M U [1] (cid:15) (cid:15) / / i ∗ C ≥ i ∗ C ≥ / / M ∈ DM B ,c ( X ). Since the composition of functors j ∗ i ∗ is trivial, the in-verse image j ∗ M is isomorphic to M U . Now observe that by Theorem 1.3 (a2),the functors i ! = i ∗ and j ! are w -left exact, and i ∗ and j ∗ are w -right exact.Thus, by the above diagram, the motive M is simultaneously an extensionof motives of weights ≤
0, and an extension of motives of weights ≥
0. Itfollows easily (see [Bo1, Prop. 1.3.3 3]) that M is pure of weight zero.(b) Now let M and N be Chow motives over X , and assume that amorphism β U : j ∗ M −→ j ∗ N between their restrictions to U is given. Consider the localization trianglesfor M and for N . i ∗ i ∗ M [ − / / j ! j ∗ M j ! β U (cid:15) (cid:15) / / M / / i ∗ i ∗ Mi ∗ i ∗ N [ − / / j ! j ∗ N / / N / / i ∗ i ∗ N According to Theorem 1.3 (a2), they provide weight filtrations of j ! j ∗ M andof j ! j ∗ N , respectively. By orthogonality (condition (3) in Definition 1.1), anymorphism from i ∗ i ∗ M [ −
1] to N is zero. Therefore, the above diagram canbe completed to give a morphism of exact triangles. q.e.d.Remark 1.8. Following the lines of part (a) of the above proof, one canshow that there is in fact a canonical bijection between the isomorphismclasses of extensions of M U to X as Chow motives on the one hand, and11somorphism classes of weight filtrations of the restriction of a cone of j ! M U → j ∗ M U to the complement X − U on the other hand.Let us note a consequence of Theorem 1.7, which we think of as usefuleven though it will not be used in the rest of this paper. Corollary 1.9.
Let j : U ֒ → X be an open immersion of schemes. Let N U , N U ∈ CHM ( U ) Q and M , M ∈ DM B ( X ) . Then the image of theinverse image j ∗ : Hom X (cid:0) M ⊗ X N , M ⊗ X N (cid:1) −→ Hom U (cid:0) j ∗ M ⊗ U N U , j ∗ M ⊗ U N U (cid:1) is independent of the extensions of N nU to Chow motives N n over X , n = 1 , .Proof. Let N nr ∈ CHM ( X ) Q , r = 1 , N nU , n =1 ,
2. By Theorem 1.7 (b), there are morphisms β n : N n → N n and β n : N n → N n extending id N nU . But then, j ∗ : Hom X (cid:0) M ⊗ X N , M ⊗ X N (cid:1) −→ Hom U (cid:0) j ∗ M ⊗ U N U , j ∗ M ⊗ U N U (cid:1) factors through Hom X ( M ⊗ X N , M ⊗ X N ), and j ∗ : Hom X (cid:0) M ⊗ X N , M ⊗ X N (cid:1) −→ Hom U (cid:0) j ∗ M ⊗ U N U , j ∗ M ⊗ U N U (cid:1) factors through Hom X ( M ⊗ X N , M ⊗ X N ). q.e.d.Corollary 1.10. Let j : U ֒ → X be an open immersion of schemes. Let N U ∈ CHM ( U ) Q and ( p, q ) ∈ Z . Then the image of the inverse image j ∗ : Hom X (cid:0) X , N ( p )[ q ] (cid:1) −→ Hom U (cid:0) U , N U ( p )[ q ] (cid:1) is independent of the extension of N U to a Chow motive N over X . Remark 1.11.
Corollary 1.10 should be compared to Scholl’s construc-tion of “integral” sub-spaces of motivic cohomology for Chow motives overlocal and global fields [S, Sect. 1]. In fact, continuity [CD, Thm. 14.2.5]implies that both statements of Theorem 1.7 continue to hold when passingto the limit over all open sub-schemes of a given scheme X . In particu-lar, for any Dedekind domain A with fraction field K , the restriction from CHM ( Spec A ) Q to CHM ( Spec K ) Q is essentially surjective and full. Thisyields the categorial interpretation of [S, Sect. 1]. It also shows that Scholl’sconstruction generalizes to the inclusion of a generic point of any scheme X (always in the sense of our conventions), which may thus be chosen differentlyfrom the spectrum of a Dedekind domain.We finish this section with a discussion of the behaviour of weights underduality. Fix X , and suppose that the structure morphism of X factors overa morphism a : X → S ′ with regular target. Fix an integer d , put R := a ! S ′ ( − d )[ − d ] ∈ DM B ,c ( X ) , D X with respect to this choice of R (seepoint (D) above). Part (a) of the following is contained in [H´e1, Cor. 3.9];statements (a)–(c) are proved for regular X in [H´e2, Cor. 2.2.5]. Theorem 1.12.
Let n be an integer, and consider the functor D X : DM B ,c ( X ) opp −→ DM B ,c ( X ) . (a) D X maps DM B ,c ( X ) oppw ≤ n to DM B ,c ( X ) w ≥− n .(b) D X maps DM B ,c ( X ) oppw ≥ n to DM B ,c ( X ) w ≤− n .(c) D X maps CHM ( X ) Q opp to CHM ( X ) Q . Given that id X = D X on DM B ,c ( X ), we see that D X actually inducesequivalences of categories DM B ,c ( X ) oppw ≤ n ∼ = DM B ,c ( X ) w ≥− n etc.Proof of Theorem 1.12. The thick triangulated category DM B ,c ( X ) isgenerated by its heart CHM ( X ) Q , and D X inverts the sign of the shifts.Therefore, it suffices to prove part (c). By [H´e1, Thm. 3.3 (ii)] (see Re-mark 1.6), it is enough to prove that for any proper morphism f : Y → X with regular source Y , and any integer p , the constructible Beilinson motive D X (cid:0) f ! Y ( p )[2 p ] (cid:1) is actually a Chow motive. From the formulae recalled in point (D) above, D X (cid:0) f ! Y ( p )[2 p ] (cid:1) ∼ = f ! D Y (cid:0) Y ( p )[2 p ] (cid:1) (recall that f is proper), provided D Y is formed with respect to f ! R . But f ! R = ( a ◦ f ) ! S ′ ( − d )[ − d ], hence D Y (cid:0) Y ( p )[2 p ] (cid:1) ∼ = ( a ◦ f ) ! S ′ ( − ( d + p ))[ − d + p )] .Y has a finite Zariski covering by connected quasi-projective schemes Y i over S ′ . Therefore (still thanks to point (D) above), the restriction to any Y i ofthe motive ( a ◦ f ) ! S ′ ( − ( d + p ))[ − d + p )] is isomorphic to Y i ( m )[2 m ], forsome integer m . In particular, we see that D Y (cid:0) Y ( p )[2 p ] (cid:1) is Zariski-locallyof weight zero. The two localization triangles, together with the w -exactnessproperties from Theorem 1.3 (a2) then show that D Y (cid:0) Y ( p )[2 p ] (cid:1) is itself ofweight zero. Again by Theorem 1.3 (a2), the same is true for its image under f ! . q.e.d. Fix a scheme X . Since (by the conventions fixed in the beginning) X isexcellent, there is an open immersion j : U ֒ → X whose image U is dense in X , and regular. Recall that by Theorem 1.3 (a1), the Beilinson motive U CHM ( U ) Q , and that by Theorem 1.7 (a), it can be extended to CHM ( X ) Q . Definition 2.1.
A pair ( j ! ∗ U , α ) is called motivic intersection complex on X if the following conditions are satisfied.(1) The object j ! ∗ U belongs to CHM ( X ) Q , and α : j ∗ j ! ∗ U ∼ −−→ U is an isomorphism in CHM ( U ) Q .(2) The morphism induced by α , j ∗ : End CHM ( X ) Q (cid:0) j ! ∗ U (cid:1) −→ End
CHM ( U ) Q (cid:0) U (cid:1) is injective.Given that j ∗ is full (Theorem 1.7), axiom (2) is equivalent to requiringthe restriction from End CHM ( X ) Q ( j ! ∗ U ) to End CHM ( U ) Q ( U ) to be bijective.Denote by i the closed immersion of the complement Z (with the reducedstructure, say) into X . Remark 2.2.
When S = Spec k for a finite field k of characteristic p ,let us consider the formalism of weights on perverse ℓ -adic sheaves , for ℓ = p [BBD, Sect. 5].(a) One of the main results from [loc. cit.] states that j ! ∗ is a functor whichtransforms perverse sheaves which are pure of a given weight into perversesheaves which are pure of the same weight [BBD, Cor. 5.4.3]. In particular,the intersection complex j ! ∗ Q ℓ is indeed pure of weight 0.(b) Localization implies that the kernel of j ∗ : End X (cid:0) j ! ∗ Q ℓ (cid:1) −→ End U (cid:0) Q ℓ (cid:1) is a quotient of the group Hom Z (cid:0) i ∗ j ! ∗ Q ℓ , i ! j ! ∗ Q ℓ (cid:1) . But this group is zero since with respect to the perverse t -structure , theobject i ∗ j ! ∗ Q ℓ in concentrated in degrees ≤ −
1, while i ! j ! ∗ Q ℓ is in degrees ≥ t -structure for Beilinson motives is not known toexist in general, the na¨ıve generalization of the definition of the intersectioncomplex is not possible (but see [Sb, Sect. 3] for the case of Artin–Tatemotives over a number ring). Definition 2.1 circumvents this problem byreplacing the use of a t -structure by the use of the motivic weight structure!14 xample 2.3. If X is regular, then ( X , id) is a motivic intersectioncomplex, as follows from the relation to K -theory (see Section 1, point (A)),and from the invariance under passage from X to its reduced structure X red (see Section 1, point (E)). Indeed, the restriction j ∗ : End CHM ( X ) Q (cid:0) X (cid:1) −→ End
CHM ( U ) Q (cid:0) U (cid:1) then corresponds to j ∗ : Gr γ K ( X red ) Q −→ Gr γ K ( U red ) Q . The latter is an isomorphism since both sides are canonically isomorphic ( via the rank) to r copies of Q , where r is the number of connected componentsof X , which coincides with the number of connected components of U (recallthat U is dense in X ).The same argument shows the following. Proposition 2.4.
The motivic intersection complex does not depend onthe choice of dense open regular sub-scheme of X . More precisely, if V isa dense open regular sub-scheme of X contained in U , and if ( j ! ∗ U , α ) is amotivic intersection complex with respect to U , then (cid:0) j ! ∗ U , α | V (cid:1) is a motivic intersection complex with respect to V . The proof of the following requires more efforts.
Proposition 2.5.
The motivic intersection complex is compatible withrestriction to open sub-schemes W of X . More precisely, if ( j ! ∗ U , α ) is amotivic intersection complex on X , then (cid:0) ( j ! ∗ U ) | W , α | W ∩ U (cid:1) is a motivic intersection complex on W .Proof. Assume first that W is dense in X . By Proposition 2.4, we mayassume U to be contained in W . Let β W : ( j ! ∗ U ) | W −→ ( j ! ∗ U ) | W be an endomorphism restricting trivially to U . The inverse image from X to W is full (Theorem 1.7 (b)), therefore β W is the restriction to W of anendomorphism β of j ! ∗ U . By assumption, we have j ∗ β = 0. Condition (2)of Definition 2.1 implies that β = 0. Hence β W = 0.In the general case, we follow an argument due to G. Ancona [An]. First,choose an open sub-scheme W ′ of X , contained in the complement of W , andsuch that W ` W ′ is dense in X . By the above, the restriction to W ` W ′ of j ! ∗ U is a motivic intersection complex. We are thus reduced to the casewhere X = W ` W ′ . We leave it to the reader to show that the restrictionsto W and W ′ of j ! ∗ U are then motivic intersection complexes on W and W ′ ,respectively. q.e.d. Basic properties
We keep the previous setting. Thus, X is fixed scheme, and j : U ֒ → X the immersion of a dense open regular sub-scheme. The complementaryimmersion is denoted by i : Z ֒ → X . Theorem 3.1. (a) The motivic intersection complex is unique up to uni-que isomorphism.(b) If the motivic intersection complex ( j ! ∗ U , α ) exists, and if β : j ∗ M ∼ −−→ U is an isomorphism in CHM ( U ) Q , with M ∈ CHM ( X ) Q , then j ! ∗ U is (ingeneral non-canonically) a direct factor of M . More precisely, there is anisomorphism M ∼ −−→ j ! ∗ U ⊕ i ∗ L Z restricting to α − ◦ β on U , with L Z ∈ CHM ( Z ) Q .Proof. Recall that the inverse image j ∗ is full on CHM ( • ) (Theo-rem 1.7 (b)). Therefore, there exist morphisms of Chow motives ϕ : j ! ∗ U −→ M and ψ : M −→ j ! ∗ U extending β − ◦ α and α − ◦ β , respectively. Observe that the composition ψ ◦ ϕ restricts to the identity on U . Injectivity of j ∗ : End CHM ( X ) Q (cid:0) j ! ∗ U (cid:1) −→ End
CHM ( U ) Q (cid:0) U (cid:1) therefore implies that ψ ◦ ϕ = id j ! ∗ U .Similarly, ϕ ◦ ψ = id M if ( M, β ) is another choice of motivic intersectioncomplex; note that in this case, the relations ψ ◦ ϕ = id j ! ∗ U and ϕ ◦ ψ = id M hold for any choices of ϕ , ψ , meaning that they are actually unique.In the general case, ϕ ◦ ψ is an idempotent endomorphism of M . Since itsrestriction to U is the identity, localization (see Section 1, point (E)) showsthat its kernel is necessarily a Chow motive of the form i ∗ L Z . The Beilinsonmotive L Z ∈ DM B ,c ( Z ) equals both i ∗ i ∗ L Z and i ! i ∗ L Z . By Theorem 1.3 (a2),it is of weight zero, hence a Chow motive over Z . q.e.d.Remark 3.2. In the context of perverse ℓ -adic sheaves over schemes offinite type over a finite field, the analogue of Theorem 3.1 (b) (concerningpure complexes M of ℓ -adic sheaves on X ) is a consequence of the Decomposi-tion Theorem [BBD, Thm. 5.4.5]. As illustrated by our proof, the formalismof weight structures yields a structural reason for the non-canonicity of theisomorphism of [loc. cit.].According to Theorem 3.1 (b), the motivic intersection complex (providedit exists) is indeed minimal among all possible extensions of U to a Chowmotive over X . Furthermore, our result suggests a possible strategy for16ts construction: first, use Theorem 1.7 (a) to choose any extension M ∈ CHM ( X ) Q of U ; then, choose idempotent endomorphisms of M to splitoff direct factors of the shape i ∗ L Z , until no such factor is left. Note thatit is not clear that the result is independent of the choices (of M and ofthe splittings) made in this process. Nor is it clear that the result actuallysatisfies axiom (2) of Definition 2.1. We plan to elaborate on this elsewhere. Remark 3.3.
Let X = X ∪ X be a covering by two dense open sub-schemes, and assume that the motivic intersection complexes on X and on X exist. Using Proposition 2.5, Theorem 3.1 and Theorem 1.7, one can showthat they can be glued along X ∩ X to give ( M, α ), with M ∈ CHM ( X ) Q ,and α : j ∗ M ∼ −−→ U . Problem 3.4.
In the situation of Remark 3.3, show that (
M, α ) satisfiesaxiom (2) of Definition 2.1.There is one specific case where we know the solution to Problem 3.4. Itis worthwhile to spell it out.
Corollary 3.5.
Assume that X is regular, and that M ∈ DM B ( X ) isNisnevich-locally isomorphic to X , i.e. , there is a finite Nisnevich coveringof X by schemes U n such that M | U n ∼ = U n for all n . Then X ∼ = X .Proof. The separation property of DM B ( • ) [CD, Def. 2.1.11] and the w -exactness properties from Theorem 1.3 (a2) allow to control the weightsof M locally for the smooth topology; in particular, our assumptions implythat M ∈ CHM ( X ) Q . The given covering of X can be refined to constructa dense open sub-scheme j : U ֒ → X and an isomorphism β : j ∗ M ∼ −−→ U . By Example 2.3 and Theorem 3.1 (b), M ∼ = X ⊕ i ∗ L Z for some L Z ∈ CHM ( Z ) Q . On each U n , the restriction U n ⊕ ( i ∗ L Z ) | U n has the same endomorphisms as U n . Therefore, ( i ∗ L Z ) | U n = 0. Separationimplies that i ∗ L Z = 0. q.e.d. Note that if M is Zariski-locally isomorphic to X , then separation canbe replaced by an application of the two localization triangles. Remark 3.6.
Corollary 3.5 and [CD, Prop. 14.3.29 (i)] can be employedto show that on a regular scheme X , two ⊗ -invertible objects of DM B ,c ( X )are isomorphic as soon as they are Nisnevich-locally isomorphic.17 orollary 3.7 (Absolute purity) . Let a : X → S ′ be a morphism of purerelative dimension d : for any irreducible component of S ′ of dimension n , itspre-image under a is of pure dimension n + d . Assume that both X and S ′ are regular. Then there is an isomorphism a ! S ′ ( − d )[ − d ] ∼ = X . Proof.
Cover X by open sub-schemes which are quasi-projective over S ′ .The discussion from point (D) of Section 1 then shows that the assumptionof Corollary 3.5 is satisfied (even Zariski-locally) for M = a ! S ′ ( − d )[ − d ]. q.e.d. Let us come back to the general situaton, i.e. , drop the regularity assump-tion on X . We aim at a motivic analogue of [BBD, Prop. 2.1.17] which statesthat j ! ∗ F is auto-dual on X provided that F is auto-dual on U . In order tohave the motivic analogue of that assumption satisfied for U , we supposethat the structure morphism of X factors over a morphism a : X → S ′ withregular target. We also suppose that a is of pure relative dimension d . Put R := a ! S ′ ( − d )[ − d ] ∈ DM B ,c ( X ) , and form the local duality functor D X with respect to this choice of R . Byabsolute purity (Corollary 3.7), there is an isomorphism γ : U ∼ −−→ j ∗ R .
We thus have D X = Hom X ( • , R ) , and composition with γ is an isomorphism of functors Hom U ( • , U ) ∼ −−→ Hom U ( • , j ∗ R ) = D U . When evaluated on U , this gives γ ∗ : U = Hom U ( U , U ) ∼ −−→ D U ( U ) ;it is in this precise sense that U is auto-dual. Theorem 3.1 has the followingformal consequence. Corollary 3.8 (Auto-duality) . If the motivic intersection complex exists,then it is auto-dual. More precisely, there is a unique isomorphism j ! ∗ U ∼ −−→ D X ( j ! ∗ U ) compatible with α and γ ∗ in the sense that its restriction to U equals thecomposition D ( α ) ◦ γ ∗ ◦ α : j ∗ j ! ∗ U ∼ −−→ D U ( j ∗ j ! ∗ U ) . Proof.
By Theorem 1.12 (c), D X ( j ! ∗ U ) is a Chow motive over X .Define β := γ − ∗ ◦ D ( α − ) : j ∗ D X ( j ! ∗ U ) = D U ( j ∗ j ! ∗ U ) ∼ −−→ U . CHM ( X ) Q (cid:0) D X ( j ! ∗ U ) (cid:1) = Hom CHM ( X ) Q (cid:0) j ! ∗ U , D X ( j ! ∗ U ) (cid:1) , and id X = D X on CHM ( X ) Q . Our claim follows from Theorem 3.1 (a). q.e.d.Definition 3.9. Assume that the motivic intersection complex ( j ! ∗ U , α )exists. The pairing j ! ∗ U ⊗ X j ! ∗ U −→ a ! S ′ ( − d )[ − d ]obtained by adjunction from the auto-duality isomorphism is called the mo-tivic intersection pairing .By definition, the motivic intersection pairing is non-degenerate in thesense that its adjoint is an isomorphism. Applying a ! to the first componentof its source, and a ∗ to the second, we get a ! j ! ∗ U ⊗ S ′ a ∗ j ! ∗ U , which maps (isomorphically, by the projection formula [CD, Thm. 2.4.21 (v)])to a ! (cid:0) j ! ∗ U ⊗ X a ∗ a ∗ j ! ∗ U (cid:1) , and finally, via the adjunction ( a ∗ , a ∗ ), to a ! (cid:0) j ! ∗ U ⊗ X j ! ∗ U (cid:1) . Composition with a ! of the intersection pairing, and application of the ad-junction ( a ! , a ! ) yields the pairing a ! j ! ∗ U ⊗ S ′ a ∗ j ! ∗ U −→ S ′ ( − d )[ − d ] . It is non-degenerate since by construction, its adjoint is the isomorphism a ! j ! ∗ U ∼ −−→ D S ′ a ∗ j ! ∗ U obtained from a ! of auto-duality and the formula D S ′ a ∗ = a ! D X (see Section 1,point (D)). In particular, we get the motivic analogue of Poincar´e duality forintersection cohomology. Corollary 3.10.
Assume that the motivic intersection complex ( j ! ∗ U , α ) exists, and that the morphism a : X → S ′ is proper. Then a ! j ! ∗ U is auto-dual. Note that under the assumptions of Corollary 3.10, the object a ! j ! ∗ U isa Chow motive over S ′ (Theorem 1.3 (a2)).Here are the few cases where we actually know the hypotheses of Theo-rem 3.1 (b) and Corollaries 3.8 and 3.10 to be satisfied.19 heorem 3.11. The motivic intersection complex exists in the followingcases.(a) The normalization X norm of the reduced scheme underlying X is regular.(b) X is of dimension at most two, and the residue fields of the singularpoints of X norm are perfect. The proof of Theorem 3.11 will be given in the next section.
Remark 3.12.
Let us discuss the case S = Spec C . We consider theHodge theoretic realization R : DM B ,c ( Spec C ) −→ D b (cid:0) MHS Q (cid:1) ([Hu, Sect. 2.3 and Corrigendum]; see [DG, Sect. 1.5] for a simplification ofthis approach). Here, D b ( MHS Q ) is the bounded derived category of theAbelian category MHS Q of mixed graded-polarizable Q -Hodge structures.It is reasonable to expect the Hodge realization to extend to the relativesetting, yielding exact, monoidal functors R : DM B ,c ( X ) −→ D b (cid:0) MHM Q X (cid:1) for all schemes X over C . Here, D b ( MHM Q X ) is the bounded derived cate-gory of algebraic mixed Q -Hodge modules on X [Sa]. Let us assume such anextension R to exist, and to be compatible with the six operations from [CD]and from [Sa]. According to [Bo3, Prop. 2.7 I], the category D b ( MHM Q X )carries a weight structure, with D b ( MHM Q X ) w ≤ and D b ( MHM Q X ) w ≥ equal to the sub-categories of complexes of Hodge modules of weights ≤ ≥ w -exact: indeed, since DM B ,c ( X ) is generatedby its heart (Remark 1.6), it suffices to show that Chow motives over X aremapped to Hodge modules which are pure of weight zero. This in turn followsfrom the explicit description of CHM ( X ) Q , and from the w -exactness prop-erties of the six operations on algebraic Hodge modules: any Chow motiveover X is a direct factor of f ! Y ( p )[2 p ] , for a proper morphism f : Y → X with regular source Y , and an integer p (Remark 1.6). Its image under R is therefore a direct factor of f ! Q HY ( p )[2 p ] . The C -scheme Y is regular, hence smooth over C . By [Sa, Thm. 3.27], thevariation of Hodge structure Q HY ( p ) on Y is a complex of algebraic Hodgemodules; as such, it is pure of weight − p . Therefore, its shift Q HY ( p )[2 p ] ispure of weight zero. But by [Sa, Sect. (4.5.2)], f ! = f ∗ is w -exact.The canonical t -structure on D b ( MHM Q X ) allows to define the Hodgetheoretic intersection complex IC X Q H on X [Sa, Sect. 4.5]. Due to thenormalization we chose for the motivic intersection complex, we define j ! ∗ Q HU := IC X Q H [ − d ]20f X is of pure dimension d . (Thus, j ! ∗ Q HU is a complex of Hodge modulesconcentrated in degree d .) According to [Sa, Sect. 4.5], j ! ∗ Q HU is pure of weightzero, and extends Q HU . It satisfies the Hodge theoretic analogue of axiom (2)of Definition 2.1. From the Hodge theoretic analogue of Theorem 3.1 (b), weconclude that the realization R ( j ! ∗ U ) of the motivic intersection complexcontains j ! ∗ Q HU as a direct factor.When S = Spec k is the spectrum of a finite field k , similar remarksapply to perverse ℓ -adic sheaves over k -schemes. Problem 3.13.
In the situation of Remark 3.12, show the equality R ( j ! ∗ U ) = j ! ∗ Q HU . Note that it implies that the intersection motive of X realizes to give (thecomplex computing) intersection cohomology of X . We keep the situation considered before: X is a scheme, and j : U ֒ → X is the immersion of a dense open regular sub-scheme. The complementaryimmersion is denoted by i : Z ֒ → X . Proposition 4.1.
Let Y be a scheme, h : V ֒ → Y an open immersion,and k : T ֒ → Y the complement. Let L and N be objects of DM B ( Y ) . Assumethat Hom T ( k ∗ L, k ! N ) = 0 . Then the restriction h ∗ : Hom Y (cid:0) L, N (cid:1) −→ Hom V (cid:0) h ∗ L, h ∗ N (cid:1) is injective.Proof. Either one of the localization triangles implies that the kernelof h ∗ : Hom Y (cid:0) L, N (cid:1) −→ Hom V (cid:0) h ∗ L, h ∗ N (cid:1) is a quotient of Hom T ( k ∗ L, k ! N ). q.e.d. In the setting of interest for us, Proposition 4.1 implies the following.
Corollary 4.2.
Assume that M ∈ CHM ( X ) Q is given, together with anisomorphism α : j ∗ M ∼ −−→ U . If Hom Z (cid:0) i ∗ M, i ! M (cid:1) = 0 , then ( M, α ) equals the motivic intersection complex on X . Z (cid:0) i ∗ j ! ∗ U , i ! j ! ∗ U (cid:1) might possibly provide a “better” definition of the motivic intersection com-plex. At least, the proof of Theorem 3.11 will consist in showing this vanish-ing. In order to do so, the following principle will be frequently used. Corollary 4.3.
Let Y be a scheme, h : V ֒ → Y an open immersion, and k : T ֒ → Y the complement. Let L and N be objects of DM B ( Y ) . If Hom V (cid:0) h ∗ L, h ∗ N (cid:1) and Hom T (cid:0) k ∗ L, k ! N (cid:1) = 0 , then Hom Y (cid:0) L, N (cid:1) = 0 . Successive applications of this principle show that the vanishing assump-tion of Corollary 4.2 can be verified on a finite stratification.
Example 4.4.
We get another proof of the equality “ j ! ∗ U = X ” forregular X (Example 2.3): choose a stratification of Z by regular sub-schemes T . Then apply absolute purity and the relation to K -theory to see thatHom T (cid:0) i ∗ X , i ! X (cid:1) = 0for each T .Let us turn to the proof of Theorem 3.11. We may assume that X isreduced. Denote by p : X norm −→ X the normalization of X ; note that p is finite since X is excellent. Note alsothat j factors uniquely through an open immersion j norm into X norm , iden-tifying U with its pre-image under p . Part (a) of Theorem 3.11 is containedin the following. Proposition 4.5.
Assume that X norm is regular. Then ( p ! X norm , id) equals the motivic intersection complex on X .Proof. First, note that X norm being supposed regular, the Beilin-son motive X norm is indeed a Chow motive (Theorem 1.3 (a1)). Since p ! = p ∗ , the same is true for p ! X norm (Theorem 1.3 (a2)). Define X nn := X norm × X X norm , denote by p and p the projections of X nn to the twofactors X norm , and by P the projection to X . Base change (see Section 1,point (B)) and adjunction, first from Z to p − ( Z ) and then to P − ( Z ), showthatHom Z (cid:0) i ∗ p ! X norm , i ! p ! X norm (cid:1) = Hom P − ( Z ) (cid:0) i nn, ∗ p ∗ X norm , i nn, ! p !1 X norm (cid:1) , where we let i nn denote the immersion of P − ( Z ) into X nn . Let k : T ֒ → P − ( Z ) be a regular connected locally closed sub-scheme. It is necessarily22uasi-finite over Z . In particular, its relative dimension e over X norm via p is strictly negative. As recalled in Section 1, point (D), k ! i nn, ! p !1 X norm = T ( e )[2 e ] . Of course, k ∗ i nn, ∗ p ∗ X norm = T . The relation to K -theory shows thatHom T (cid:0) T , T ( e )[2 e ] (cid:1) = 0 . Indeed, the graded object Gr eγ K ( T ) is zero since the gamma filtration isconcentrated in non-negative degrees. Now apply Corollaries 4.3 and 4.2. q.e.d.Problem 4.6. Without the regularity assumption on X norm , show that j ! ∗ U = p ! j norm ! ∗ U , whenever the motivic intersection complex j norm ! ∗ U on X norm exists.In order to prove part (b) of Theorem 3.11, note first that for reducedschemes X of dimension at most one, the normalization X norm is regular. Forthe rest of this section, let us therefore assume that X is a reduced surface( i.e. , all irreductible components of X are integral and of dimension two),and that the residue fields of the singular points of X norm are perfect.Let us start by the construction of j norm ! ∗ U on X norm . It is a variant of theconstruction from [CM] for surfaces defined over a field. By Proposition 2.4,we may perform the computation after replacing U by the regular locus V of X norm . Since X norm is regular in codimension one, the complement Z ′ of V (with the reduced structure) is finite; in fact, by our assumption, Z ′ is the spectrum of a finite product of perfect fields. By Abhyankar’sresult on resolution of singularities in dimension two [L2, Theorem], X canbe desingularized. In addition (see the discussion in [L1, pp. 191–194]), byfurther blowing up possible singularities of (the components of) the pre-image D of Z ′ , it can be assumed to be a divisor with normal crossings,whose irreducible components are regular. Fix such a resolution, that is, fixthe following diagram, assumed to be cartesian: V (cid:31) (cid:127) / / e X o o ˜ ı ? _ π (cid:15) (cid:15) D π (cid:15) (cid:15) V (cid:31) (cid:127) / / X norm o o i ′ ? _ Z ′ where π is proper (and birational), e X is regular, and D is a divisor withnormal crossings, whose irreducible components D m are regular.23hus, the D m are regular curves over perfect fields (the points of Z ′ ).Therefore, they are smooth. In addition, they are proper. Denote by ˜ ı m the closed immersion of D m into e X , and by π m the restriction of π to D m .The classical theory of Chow motives yields canonical (split) sub-objects( π m, ! D m ) and (split) quotients ( π m, ! D m ) of π m, ! D m . The adjunctionsid e X → ˜ ı m, ∗ ˜ ı ∗ m and ˜ ı m, ∗ ˜ ı ! m → id e X , and absolute purity for ˜ ı m yield canonicalmorphisms ˜ ı ∗ : π ! e X −→ M m π m, ! D m −→→ M m (cid:0) π m, ! D m (cid:1) and ˜ ı ∗ : M m (cid:0) π m, ! D m (cid:1) ( − − ֒ −→ M m π m, ! D m ( − − −→ π ! e X of Chow motives over X norm . Proposition 4.7. (i) The composition α := ˜ ı ∗ ˜ ı ∗ is an isomorphism.(ii) The composition ε := ˜ ı ∗ α − ˜ ı ∗ is an idempotent on π ! e X . Hence so is thedifference id π ! e X − ε .(iii) The image im ε is canonically isomorphic to ⊕ m ( π m, ! D m ) .Proof. The proof is formally identical to the one of [W2, Thm. 2.2].Observe that the non-degeneracy of the intersection pairing on the compo-nents of D holds since the proof [M, p. 6] carries over to the general contextof normal surfaces. q.e.d. Note that the ( π m, ! D m ) restrict trivially to V . Therefore, the imageim(id π ! e X − ε ) restricts to give V . Part (b) of Theorem 3.11 is contained inthe following. Proposition 4.8.
Assume that X is a surface, and that the residue fieldsof the singular points of X norm are perfect.(a) The pair (im(id π ! e X − ε ) , id) equals the motivic intersection complex on X norm .(b) The pair ( p ! im(id π ! e X − ε ) , id) equals the motivic intersection complex on X . Proof. It suffices to prove part (b). Write M := im(id π ! e X − ε ). As inthe proof of Proposition 4.5, let X nn = X norm × X X norm , denote by p and p the projections of X nn to the two factors X norm , and by P the projectionto X . Base change and adjunction show thatHom Z (cid:0) i ∗ p ! M, i ! p ! M (cid:1) = Hom P − ( Z ) (cid:0) i nn, ∗ p ∗ M, i nn, ! p !1 M (cid:1) , where i nn denotes the immersion of P − ( Z ) into X nn . In order to apply Corol-lary 4.2, we need to show the vanishing of this group. We shall repeatedlyapply Corollary 4.3. In order to do so, stratify P − ( Z ) as follows: the open24tratum (possibly empty) is the intersection of the pre-images under p andunder p of V (which contains U ), the closed stratum is the complement,which is a finite set of points.If k : T ֒ → P − ( Z ) is a regular connected locally closed sub-scheme ofthe open stratum, then its relative dimension e over X norm via p is strictlynegative. Since p ( T ) and p ( T ) are contained in V , and im(id π ! e X − p )restricts to V , we argue as in the proof of Proposition 4.5 to see thatHom T (cid:0) k ∗ i nn, ∗ p ∗ M, k ! i nn, ! p !1 M (cid:1) = 0 . It remains to check the points k : T ֒ → P − ( Z ) of the closed stratum.Depending on whether p ( T ) is regular or not, we have k ∗ i nn, ∗ p ∗ M = T or k ∗ i nn, ∗ p ∗ M = (cid:0) π ! D p T ) (cid:1) ≤ , where D p ( T ) is the base change of the exceptional divisor D to p ( T ), andwhere the symbol ( π ! D p T ) (cid:1) ≤ denotes the kernel of the projection π ! D p T ) −→→ ( π ! D p T ) (cid:1) := M m ( π m, ! D m,p T ) ) induced by ε . Similarly (see Section 1, point (D)), k ! i nn, ! p !1 M = T ( − − k ! i nn, ! p !1 M = (cid:0) π ! (˜ ı | D p T ) ) ! e X (cid:1) ≤ . We thus need to distinguish four cases. As usual,Hom T (cid:0) T , T ( − − (cid:1) = 0 . In order to show thatHom T (cid:0) T , (cid:0) π ! (˜ ı | D p T ) ) ! e X (cid:1) ≤ (cid:1) = 0 , note that π ! = π ∗ , and that by adjunction,Hom T (cid:0) T , π ∗ (˜ ı | D p T ) ) ! e X (cid:1) = Hom D p T ) (cid:0) D p T ) , (˜ ı | D p T ) ) ! e X (cid:1) . Stratify D p ( T ) by regular sub-schemes, using that ˜ ı is of stricly positivecodimension, to see that the latter group is zero.Similarly, Hom T (cid:0)(cid:0) π ! D p T ) (cid:1) ≤ , T ( − − (cid:1) = 0 . It remains to considerHom T (cid:0)(cid:0) π ! D p T ) (cid:1) ≤ , (cid:0) π ! (˜ ı | D p T ) ) ! e X (cid:1) ≤ (cid:1) . T (cid:0) π ! D p T ) , π ! (˜ ı | D p T ) ) ! e X (cid:1) and Hom D p T ) × T D p T ) (cid:0) D p T ) × T D p T ) , pr ! D p T ) (˜ ı | D p T ) ) ! e X (cid:1) , where D p ( T ) × T D p ( T ) is the (singular) surface obtained by base change over T of the curves D p ( T ) and D p ( T ) , and pr D p T ) denotes the projection to D p ( T ) . On regular sub-schemes contained in the singular locus of D p ( T ) × T D p ( T ) , the same considerations as before show that there are no non-zeromorphisms of the required type. Hence the assumption of Proposition 4.1 issatisfied, andHom D p T ) × T D p T ) (cid:0) D p T ) × T D p T ) , pr ! D p T ) (˜ ı | D p T ) ) ! e X (cid:1) injects intoHom D p T ) reg × T D p T ) reg (cid:0) D p T ) reg × T D p T ) reg , pr ! D p T ) reg (˜ ı | D p T ) reg ) ! e X (cid:1) , where the subscripts reg denote the regular loci. Both D p ( T ) reg × T D p ( T ) reg and e X are regular surfaces, hence pr ! D p T ) reg (˜ ı | D p T ) reg ) ! e X = D p T ) reg × T D p T ) reg . Therefore,Hom D p T ) reg × T D p T ) reg (cid:0) D p T ) reg × T D p T ) reg , pr ! D p T ) reg (˜ ı | D p T ) reg ) ! e X (cid:1) equals r copies of Q , where r is the number of connected components of D p ( T ) reg × T D p ( T ) reg . But the same result, with compatible identificationsis obtained by computingHom T (cid:0)(cid:0) π ! D p T ) (cid:1) , (cid:0) π ! (˜ ı | D p T ) ) ! e X (cid:1) (cid:1) . q.e.d. References [An] G. Ancona, letter to the author dated Apr. 14, 2011.[Ay] J. Ayoub,
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