BIRATIONAL MOTIVES, I

BIRATIONAL MOTIVES, I BRUNO KAHN AND R. SUJATHA Preliminary version Contents Introduction 1 1. Places and morphisms 8 2. Calculus of fractions 13 3. Equivalences of categories 14 4. Birational finite correspondences 21 5. Triangulated category of irational motives 27 6. Birational motivic complexes 32 7. Main theorem 36 8. Equivalence of categories 41 9. Aelian schemes 43 10. Birational motives and locally aelian schemes 49 References 54 Introduction In this article, we try to conciliate two important ideas in algeraic geometry: motives and irational geometry. There are many reasons to do this; the one which motivated us originally was to understand unramified cohomology from a motivic point of view. Although this is not yet achieved in this paper (ut see (2) at the end of this introduction), it led us to rather large developments and surprising structure theorems. For an application of an elementary part of our theory to function fields over finite fields, see [20]. In order to give the reader a rief resumé of our results, we assume familiarity with Voevodsky s construction of his triangulated categories of motives and recall the naturally commutative diagram of categories Date: Octoer 22, 2002. 1

2 BRUNO KAHN AND R. SUJATHA [38, 40], where F is a perfect field Sm(F ) SmCor(F ) ff ff DMgm(F eff ) ff Sm proj (F ) SmCor proj (F ) Chow eff (F ) ff DM eff (F ) in which ff means fully faithful. Here Sm(F ) is the category of smooth connected F -varieties, Sm proj (F ) is its full sucategory consisting of smooth projective varieties, SmCor(F ) is the category of smooth varieties with morphisms finite correspondences, SmCor proj (F ) its full sucategory consisting of smooth projective varieties, Chow eff (F ) the category of effective Chow motives, DMgm(F eff ) the triangulated category of effective geometrical motives and DM eff (F ) the triangulated category of effective motivic complexes (for the Nisnevich topology). Then, for F of characteristic 0, we construct a diagram to which the former maps naturally S 1 S 1 r r Sm(F ) Sr 1 SmCor(F ) ff DMgm(F o ) ff Sm proj (F ) Sr 1 SmCor proj (F ) Chow o (F ) Al T 1 place(f ) op AS(F ) ff DM o (F ) Here means an equivalence of categories, denotes karouian envelope (or pseudo-aelian envelope, or idempotent completion), Sr 1 and T 1 denote localisation with respect to certain sets of morphisms S r and T, place(f ) is the category of function fields over F with morphisms given y F -places, Chow o (F ) is the category of irational Chow motives, DMgm(F o ) the triangulated category of irational geometrical motives, DM (F o ) the triangulated category of rationally invariant motivic complexes and AS(F ) the category of locally aelian schemes. As a very special case, this diagram shows that any function field has a irational Chow motive which is natural with respect to F -places. It also shows that the situation is strikingly simpler in the irational case than in the regular case. In particular, the equivalence etween a localisation of the category of finite correspondences on smooth projective varieties and that of irational Chow motives, although not one of the most difficult results of this paper, is crucial in our proof that the functor Chow o (F ) DMgm(F o ) is fully faithful.

BIRATIONAL MOTIVES, I 3 Let us now give a few more details on the various categories introduced aove and some intuition of the situation. First, if one wants to make sense of a irational category, one is confronted at the outset with two different ideas: use the notion of place of Zariski-Samuel [41] or use the geometric idea of a irational map. It turns out that oth ideas work ut give rather different answers. The first idea gives the category place(f ), that we like to call the coarse irational category. For the second idea, one has to e a little careful: the naïve attempt at taking as ojects smooth varieties and as morphisms irational maps does not work ecause, as was pointed out to us y Hélène Esnault, one cannot compose irational maps in general. On the other hand, one can certainly start from the category Sm(F ) and localise it with respect to the multiplicative set S of irational isomorphisms. We like to call the resulting category S 1 Sm(F ) the fine irational category. By hindsight, the prolem mentioned just aove can e understood as a prolem of calculus of fractions in S 1 Sm(F ). These definitions raise the following issues. First, in place(f ), the Hom sets are very ig. Second, in Sm(F ), the set S does not admit calculus of left or right fractions in the sense of Gariel-Zisman [14]. And finally, there is no ovious comparison functor etween the coarse and the fine irational categories at this stage. In order to answer these issues at least to an extent, we introduce an incidence category SmP(F ), whose ojects are smooth connected F - varieties and morphisms from X to Y are given y pairs (f, v), where f is a morphism X Y, v is a place F (Y ) F (X) and f, v are compatile in an ovious sense (see Definition 1.4 elow). This category maps to oth place(f ) op and Sm(F ) y ovious forgetful functors. Denote y Sm proj P(F ) the full sucategory of SmP(F ) consisting of smooth projective varieties. Note that the set S lifts naturally to SmP(F ) and restricts to Sm proj (F ) and Sm proj P(F ) (same notation). Then: Theorem 1 (cf. Theorem 3.8). Assume F of characteristic 0. Then localisation with respect to S yields a naturally commutative diagram of categories, in which vertical functors are equivalences of categories

4 BRUNO KAHN AND R. SUJATHA while horizontal functors are full and essentially surjective: S 1 Sm proj P(F ) S 1 SmP(F ) place(f ) op. S 1 Sm proj (F ) S 1 Sm(F ) In particular, we get a full and essentially surjective functor place(f ) op S 1 Sm(F ), justifying the terminology of coarse and fine irational categories. In the course of the proof, we get that, at least, the first axiom of calculus of right fractions ( Ore condition ) holds for S, which is y no means ovious a priori (see Proposition 3.2). Hence in S 1 Sm(F ) any morphism may e written as a single fraction fs 1, with f a regular map and s S. On the other hand, the second axiom definitely does not hold. If X, Y are two connected smooth projective varieties over F, we get in this way a surjective map from the set of F -places from F (Y ) to F (X) to the set of morphisms from X to Y in S 1 Sm proj (F ). This defines a natural equivalence relation on the first set, which is stale under composition of places. Unfortunately we don t know how to descrie this equivalence relation in concrete terms, and this appears to e a very challenging question. If we now define T to e the multiplicative set of (trivial) places in place(f ) given y purely transcendental extensions of function fields, and S r as the multiplicative set of rational morphisms in Sm(F ), i.e. dominant morphisms such that the corresponding extension of function fields is purely transcendental, then we may localise further the diagram of Theorem 1, getting part of the second diagram in this introduction (see Theorem 3.8). We call the corresponding categories the coarse and fine stale irational categories. Having the aove categories at hand is well and good ut, as for usual algeraic geometry, it is very difficult to compute with them. So we apply the classical idea to enlarge morphisms y adding algeraic correspondences and making the categories additive. In this programme, having Voevodsky s categories in mind, the first logical step (even if for us it came rather late in the development) is to extend part of Theorem 1 to finite correspondences, i.e. to study the functor S 1 SmCor proj (F ) S 1 SmCor(F )

BIRATIONAL MOTIVES, I 5 where S denotes the image of S under the natural functor Sm(F ) SmCor(F ), i.e. the multiplicative set of graphs of irational morphisms. In Proposition 4.1, we prove that this functor is an equivalence of categories if F has characteristic zero. (The issue is faithfulness.) Although the idea of the proof is simple (interpret finite correspondences as maps to symmetric powers as in [34]), details are not really straightforward and the proof takes several pages. It uses the category SmP(F ) at a crucial step. Although, in view of the aove, the most natural definition of irational Chow motives would e to localise effective Chow motives with respect to S r, this is not the way we proceed and we go through a more convoluted way, even though in the end our definition coincides with the one just outlined (see Corollary 7.3). We define the category Chow o (F ) as the pseudo-aelian envelope of the quotient of Chow eff (F ) y the ideal I characterised as follows: for X, Y two smooth projective varieties, I(h(X), h(y )) is the sugroup of CH dim Y (X Y ) formed of those correspondences whose restriction to U Y is 0 for some dense open suset U of X. The fact that I is an ideal, i.e. is stale under left and right composition y any correspondence, is not ovious and amounts to a generalisation of the argument in [13, Ex. 16.1.11]. In fact, I is even a monoidal ideal, see Lemma 5.3, which means that the tensor structure of Chow eff (F ) passes to Chow o (F ). In characteristic 0, I can e descried more concretely as the set of those morphisms factoring through an oject of the form M L, where L is the Lefschetz motive; in characteristic p, this remains true at least if one tensors all morphisms with Q (Lemma 5.4). Hence, to otain irational Chow motives, we do something orthogonal to what is done haitually: instead of inverting the Lefschetz motive, we kill it! This rather surprising picture ecomes a little more natural if we consider the parallel one for Voeovdsky s triangulated motives. To get DMgm(F o ), we simply invert open immersions (or, equivalently, the image of S in DMgm(F eff )) and add projectors. The resulting category is a tensor triangulated category. It is easy to see, only assuming F perfect, that DMgm(F o ) = DMgm(F eff )/DMgm(F eff )(1) (Proposition 5.2 )); this is made intuitive y thinking of the Gysin exact triangles. Then one easily sees that the functor Chow eff (F ) DMgm(F eff ) of Voevodsky induces a functor Chow o (F ) DMgm(F o ). The main result of this paper is the computation of Hom( M(X), M(Y )[i]) for two smooth projective varieties X, Y and i Z, where M(X) and M(Y ) denote their motives in DMgm(F o ). It may e stated as follows:

6 BRUNO KAHN AND R. SUJATHA Theorem 2 (cf. Cor. 7.9). a) The functor Chow o (F ) DMgm(F o ) is fully faithful. ) For X, Y, i as aove, we have { Hom( M(X), M(Y CH 0 (Y F (X) ) if i = 0 )[i]) = 0 if i 0. The proof of this theorem is rather intricate. We follow the method Voevodsky used to compute Hom groups in DMgm(F eff ): we introduce another category DM (F o ) of rationally invariant motivic complexes and construct a functor DMgm(F o ) DM (F o ) that we show to e fully faithful. In fact the definition of DM (F o ) is simple: it is the full sucategory of DM eff (F ) consisting of those ojects C such that H i Nis (X, C) H i Nis (U, C) for any dense open immersion of smooth schemes U X. So DM (F o ) is defined naturally as a sucategory of DM eff (F ), while DMgm(F o ) is defined as a quotient of DMgm(F eff ). This is not surprising, as DM eff (F ) is a sucategory of the category of functors from DMgm(F eff ) to aelian groups. However, we have to show that the emedding (1) i : DM o (F ) DM eff (F ) has a left adjoint ν 0. This is done fairly easily in Lemma 6.6. The fact that Voevodsky s full emedding DMgm(F eff ) DM eff (F ) descends to a full emedding DMgm(F o ) DM (F o ) is then formal (Theorem 6.7). At this point, it remains to compute the group Hom( M(X), M(Y )[i]) within DM (F o ). This turns out to e very delicate and we can only refer the reader to Section 7 for the proof. We end this paper y relating the previous constructions to more classical ojects. We define a tensor additive category AS(F ) of locally aelian schemes, whose ojects are those F -group schemes that are extensions of a lattice (i.e. locally isomorphic for the étale topology to a free finitely generated aelian group) y an aelian variety. We then show that the classical construction of the Alanese variety of a smooth projective variety extends to a tensor functor Al : Chow o (F ) AS(F ) which ecomes full and essentially surjective after tensoring morphisms y Q (Proposition 10.2). So, one could say that AS(F ) is the representale part of Chow o (F ). We also show that, after tensoring with Q, Al has a right adjoint-right inverse, which identifies AS(F ) Q

BIRATIONAL MOTIVES, I 7 with the thick sucategory of Chow o (F ) Q generated y motives of varieties of dimension 1. This work is only the eginning of our investigations on irational motives. Let us just mention for the moment future lines of research: (1) Structure of DMgm(F o ). In the course of the proof of theorem 2, we realise Chow o (F ) as the sucategory of compact ojects in the aelian category HI o (F ) of rationally invariant Nisnevich sheaves with transfers (Propositions 7.2 and 7.4). In particular, this descries Chow o (F ) as an exact sucategory of an aelian category. What is the relationship etween DMgm(F o ) and the ounded derived category of Chow o (F )? (2) Where does unramified cohomology enter this picture? Expected answer: it should e a right adjoint to the functor i of (1). (3) We hope to use this formalism to study the unramified cohomology of BG, where G is a linear algeraic group. (4) Is it true that i(dmgm(f o )) DMgm(F eff ) and that i(chow o (F )) Chow eff (F )? This question was asked y Luca Barieri-Viale and is expected to e difficult to answer: it is closely related to a conjecture of Voevodsky [37, Conj. 0.0.11]. We expect this to e true after tensoring with Q. At the very least, the second statement ecomes true (and easy) if one replaces Chow motives y numerical motives, cf. [20, Prop. 1]. (5) The localisation of the Morel-Voevodsky A 1 -homotopy category of schemes H(F ) [23] with respect to S r should e studied, as well as that of Voevodsky s effective stale A 1 -homotopy category of schemes. For the latter, this is very likely any chunk of the slice filtration of [39]; this analogy was pointed out to us y Chuck Weiel. (6) The relationship etween these ideas and the proofs that the Bloch-Kato conjecture implies the Beilinson-Lichtenaum conjecture [35, 15, 16, 19] should definitely e investigated: a start is given in [19, Def. 2.14 and Lemma 2.15]. The est context for this seems to e in the previous item. (7) Equally the relationship with Déglise s category of generic motives [8, 9]: at this point it is not completely clear what this relationship is. (8) Finally, the exact relationship etween the category Chow o (F ) and Beilinson s correspondences at the generic point [3] should e investigated as well.

8 BRUNO KAHN AND R. SUJATHA Acknowledgements. We would like to thank Shreeram Ayankhar, Luca Barieri-Viale, Jean-Louis Colliot-Thélène, Frédéric Déglise, Hélène Esnault, Eric Friedlander, Jens Hornostel, Annette Huer-Klawitter, Bernhard Keller, Georges Maltsiniotis, Faien Morel, Amnon Neeman, Michel Vaquié, Vladimir Voevodsky and Chuck Weiel for various discussions and exchanges while writing this paper. Notation. F is the ase field. All varieties are F -varieties and all morphisms are F -morphisms. If X is irreducile, η X denotes its generic point. 1. Places and morphisms 1.1. Definition. Let K/F and L/F e two extensions. An F -place from K to L is a pair v formed of a valuation ring O of K/F (i.e. F O K) and an F -homomorphism M L, where M is the residue field of O. We write O = O v and M = F (v). Composition of places defines the category place(f ) with ojects finitely generated extensions of F and morphisms F -places. 1.2. Remark. If v : K L is a morphism in place(f ), then its residue field F (v) is finitely generated over F, as a sufield of the finitely generated field L. On the other hand, given a finitely generated extension K/F, there exist valuation rings of K/F with infinitely generated residue fields as soon as trdeg(k/f ) > 1, cf. [41, Ch. VI, 15, Ex. 4]. 1.3. Lemma. Let f, g : X Y e two morphisms, with X integral and Y separated. Then f = g if and only if f(η X ) = g(η X ) =: y and f, g induce the same map F (y) F (X) on the residue fields. Proof. Let ϕ : X Y F Y e given y (f, g). Then the diagonal Y is closed, so Ker(f, g) = ϕ 1 ( Y ) is closed in X and contains η X (compare [EGA, Ch. I, Prop. 5.1.5 and Cor. 5.1.6]). 1.4. Definition. Let X, Y e two integral varieties, with Y separated, f : X Y a morphism and v : F (Y ) F (X) a place. We say that f and v are compatile if v is finite on Y (i.e. has a centre in Y ). The corresponding diagram η X X v Spec O v f Y

commutes. BIRATIONAL MOTIVES, I 9 1.5. Remark. Any morphism which is a irational isomorphism is compatile with the identical place. This applies in particular to open immersions. 1.6. Proposition. Let X, Y, v e as in Definition 1.4. Suppose that v is finite on Y, and let y Y e its centre. Then a morphism f : X Y is compatile with v if and only if y = f(η X ) and the diagram of fields F (v) v f F (y) F (X) commutes. In particular, there is at most one such f. Proof. Suppose v and f compatile. Then y = f(η X ) ecause v (η X ) is the closed point of Spec O v. The commutation of the diagram then follows from the one in Definition 1.4. Conversely, if f verifies the two conditions, then it is oviously compatile with v. The last assertion follows from Lemma 1.3. 1.7. Corollary. a) Let Y e an integral variety, and let O e a valuation ring of F (Y )/F with residue field K and centre y Y. Assume that F (y) K. Then, for any morphism f : X Y with X integral, such that f(η X ) = y, there exists a unique place v : F (Y ) F (X) with valuation ring O which is compatile with f. ) If f is an immersion, the condition F (y) K is also necessary for the existence of v. The following lemma generalises Remark 1.5: 1.8. Lemma. Let f : X Y e dominant. Then f is compatile with the trivial place F (Y ) F (X), and this place is the only one with which f is compatile. Proof. This follows immediately from Proposition 1.6. 1.9. Proposition. Let f : X Y, g : Y Z e two morphisms of integral separated varieties. Let v : F (Y ) F (X) and w : F (Z) F (Y ) e two places. Suppose that f and v are compatile and that g and w are compatile. Then g f and v w are compatile.

10 BRUNO KAHN AND R. SUJATHA Proof. diagram We first show that v w is finite on Z. By definition, the η Y w Spec O w Spec O v Spec O v w is cocartesian. Since the two compositions η Y w Spec O w Z and η Y Spec O v Y g Z coincide (y the compatiility of g and w), there is a unique induced (dominant) map Spec O v w Z. In the diagram η X v Spec O v f Spec O v w X Y Z the left square commutes y compatiility of f and v, and the right square commutes y construction. Therefore the ig rectangle commutes, which means that g f and v w are compatile. 1.10. Definition. We denote y SmP(F ) the following category: Ojects are smooth F -schemes of finite type. Let X, Y SmP(F ). A morphism ϕ SmP(X, Y ) is a pair (v, f) with f : X Y, v : F (Y ) F (X) and v, f compatile. The composition of morphisms is given y Proposition 1.9. We denote y Sm proj P(F ) the full sucategory of SmP(F ) consisting of smooth projective varieties. 1.11. Lemma. Let f : X Y e a morphism from an integral variety to a regular variety. Then there is a place v : F (Y ) F (X) compatile with f. Proof. Let y = f(η X ). The local ring A = O Y,y is regular: By Corollary 1.7 a), it is sufficient to produce a valuation ring O containing A and with the same residue field as A. The following construction is certainly classical. Let m e the maximal ideal of A and let (x 1,..., x d ) e a regular sequence generating m, with d = dim A = codim Y y. For 0 i < j d + 1, let A i,j = A[x 1 1,..., x 1 i ]/(x j,..., x d ) (for i = 0 we invert no x k, and for j = d+1 we mod out no x k ). Then, for any (i, j), A i,j is a regular local g

BIRATIONAL MOTIVES, I 11 ring of dimension j i 1. In particular, F i = A i,i+1 is the residue field of A i,j for any j i + 1. We have A 0,d+1 = A and there are ovious maps A i,j A i+1,j A i,j A i,j 1 (injective) (surjective). Consider the discrete valuation v i associated to the discrete valuation ring A i,i+2 : it defines a place, still denoted y v i, from F i+1 to F i. The composition of these places is a place v from F d = F (Y ) to F 0 = F (y), whose valuation ring dominates A and whose residue field is clearly F (y). 1.12. Lemma. Suppose F perfect. Let place(f ) e the category of function fields over F with morphisms the F -places, and let Sm(F ) e the category of integral separated F -schemes of finite type. There are forgetful essentially surjective functors SmP(F ) Φ 2 Φ 1 Sm(F ) place(f ) op with Φ 1 full and Φ 2 faithful. The restriction of Φ 2 to Sm proj P(F ) is essentially surjective when F is of characteristic 0. Proof. The definitions and essential surjectivity of Φ 1 and Φ 2 are ovious. The restricted case of essential surjectivity for Φ 2 is clear y Hironaka s resolution of singularities. The fullness of Φ 1 follows from Lemma 1.11 and the faithfulness of Φ 2 follows from Proposition 1.6. 1.13. Lemma. Let Z, Z e two models of a function field K, with Z separated, and v a place of K with centres z, z respectively on Z and Z. Assume that there is a morphism g : Z Z which is a irational isomorphism. Then g(z) = z. Proof. Let f : Spec O v Z e the dominant map determined y z. Then f = g f is a dominant map Spec O v Z. By the valuative criterion of separatedness, it must correspond to z.

12 BRUNO KAHN AND R. SUJATHA 1.14. Lemma. Consider a diagram Z f X g f with g a irational isomorphism. Let K = F (X), L = F (Z) = F (Z ) and suppose given a place v : L K compatile oth with f and f. Then f = g f. Proof. This follows from Proposition 1.6 and Lemma 1.13. 1.15. Lemma. Let X, Y, v, y e as in Proposition 1.6. Then there exists an open suset U X and a morphism f : U Y compatile with v. Proof. Let V = Spec R e an affine neighourhood of y in Y, so that R O v, and let S e the image of R in F (v). Choose a finitely generated F -sualgera T of F (X) containing S, with quotient field F (X). Then X = Spec T is an affine model of F (X)/F. The composition X Spec S V Y is then compatile with v. Its restriction to a common open suset U of X and X defines the desired map f. 1.16. Definition. A projective irational isomorphism f : X Y of smooth varieties will e called an astract low-up. The isomorphism locus of f is the largest open suset U of Y such that f : f 1 (U) U is an isomorphism. The complement Z of U is the centre of f: it is considered as a reduced suscheme of X. The following proposition strengthens Lemma 1.15. 1.17. Proposition. a) Let v : L K e a morphism in place(f ), and let Y e a projective model of L. Then there exists a normal projective model X of K and a morphism f : X Y compatile with v. If F is perfect, we may choose X smooth affine instead. If char F = 0, we may choose X smooth projective. ) Suppose that char F = 0, and let X 0 e a smooth projective model of K. Then there exists another smooth projective model X 1 of K, an astract low-up u : X 1 X 0 and a map f 1 : X 1 Y compatile with v. Proof. a) By the valuative criterion of properness, v has a centre y in Y. Let N = F (y) e the residue field of y: it maps into K y v. Z

BIRATIONAL MOTIVES, I 13 Let Z = {y}: this is a projective model of N. Choose a projective model X 0 of K/N and spread X 0 Spec N to a projective map X 1 Z. Then X 1 is projective. Let X e the normalisation of X 1. The composition X X 1 Z Y is clearly compatile with v. If F is perfect, we may replace X y a smooth dense open suset. Finally, if char F = 0, we may take X 0 projective and end y desingularising X 1 instead of normalising it. ) Let U e a common open suset to X and X 0 (where X has een constructed in a)). We may view U as diagonally emedded into X X 0. Let X 1 e a desingularisation of the closure of U in X X 0 : we have a diagram of astract low-ups X 1 u g X X 0. By Proposition 1.9, f 1 = f g is compatile with v. 2. Calculus of fractions Let C e a category and S a set of morphisms of C. Recall from [14, I.1] the category C[S 1 ] and the canonical functor P S : C C[S 1 ]: P S is universal among functors from C that render all arrows of S invertile. We shall denote y S the set of morphisms s of C such that P S (s) is invertile: this is the saturation of S. Recall (loc. cit., I.2) that S admits a calculus of left fractions when it verifies the following conditions: (1) The identities are in S; S is stale under composition. (2) Each diagram (2.1) X s u Y X where s S can e completed in a commutative diagram X s u Y s X u Y

14 BRUNO KAHN AND R. SUJATHA with s S. (3) If f, g : X Y are two morphisms and if s : X X is a morphism of S such that fs = gs, there exists a morphism t : Y Y of S such that tf = tg: X s X f g Y t Y. Under calculus of left fractions, the category C[S 1 ] has a very nice description S 1 C (loc. cit. I.2.3). In this article, we shall encounter situations where Condition 2 is verified only for certain pairs (u, s) and where Condition 3 is not verified. This leads us to give more careful definitions. 2.1. Definition. A pair (u, s) as in Condition 2 aove admits a calculus of left fractions within S if there exists a pair (u, s ) as in the said condition. It admits cocartesian calculus of left fractions within S if the pushout of (2.1) exists and provides such a pair (u, s ). We won t repeat the dual definitions for calculus of right fractions. 3. Equivalences of categories In this section, we assume that char F = 0. We shall work with several multiplicative susets of Sm(F ) and Sm proj (F ): S o = {open immersions}. S = {irational isomorphisms}. S p = {projective morphisms in S }. S h = the multiplicative suset generated y morphisms of the form X A 1 pr 1 X. S r = {rational isomorphisms} (a morphism is a rational isomorphism if the corresponding extension of function fields is purely transcendental). Sr p = {projective morphisms in S r }. For S of this form, we also write S for the multiplicative suset of SmP(F ) or Sm proj P(F ) formed y the pairs (v, f) with f S. We shall write S 1 C for the localisation of a category instead of the heavier notation C[S 1 ], even when there is no calculus of fractions. 3.1. Lemma. a) We have the inclusions S o S S o and S S r. In particular, So 1 Sm(F ) = S 1 Sm(F ) and So 1 SmP(F ) = S 1 SmP(F ). ) Suppose char F = 0. In SmP(F ), any morphism of Sr p can e covered y the composition of a morphism in S p and morphisms of the form X P 1 pr 1 X.

BIRATIONAL MOTIVES, I 15 Proof. a) is ovious. For ), we can forget aout places thanks to Lemma 1.8. Let s : X Y Sr p. By assumption, X is irationally equivalent to X = Y (P 1 ) n for some n 0. We proceed as usual: let U e a common open suset of X and X and X the closure of the diagonal emedding of U in X Y X. Then X is also a model of F (X). Since s and pr 1 : Y (P 1 ) n Y are projective, so is the composition X X Y X Y. It remains to resolve the singularities of X via a succession of low-ups. 3.2. Proposition. Let X u Y e a diagram in SmP(F ), with s S p (resp. s Sp r ). Then (u, s) admits calculus of right fractions within S p (resp. within Sr p ). The same holds in Sm(F ). Proof. Note that the statement for Sm(F ) follows from the case of SmP(F ), thanks to Lemma 1.11. Moreover, the case of Sr p follows from that of S p y Lemma 3.1 ), since calculus of fractions is ovious for a morphism of the form X P 1 X. Here is the proof for S p SmP(F ). Let v : F (Y ) F (X) e the place compatile with u which is implicit in the statement. By assumption v has a centre z on Y. Since s is proper, v therefore has also a centre z on Y. By Lemma 1.13, s(z ) = z. Hence we have inclusions of fields (3.1) F (z) F (z ) F (v) F (X). Let Z = {z} and Z = {z }. The map u factors through a map ū : X Z. The chain (3.1) shows that ū lifts to a rational map from X to Z. Blowing up X suitaly, we get a commutative diagram X ū Y Z Y s s ū X Z Y in which s is an astract low-up; moreover, the composition u : X Y is compatile with v (compare Corollary 1.7 )). 3.3. Lemma. Any morphism in S 1 SmP(F ) of the form fj 1 with j an open immersion is equal to a morphism of the form a 1 gp 1, where p is an astract low-up and a, are open immersions. s

16 BRUNO KAHN AND R. SUJATHA Proof. Consider a diagram j X U f where j is an open immersion. Choose open emeddings a : Y Ȳ and : X X, with X, Ȳ smooth projective. This reduces us to the case where X and Y are projective. By proposition 1.17, choose a smooth projective model X of F (X) and a morphism f : X Y compatile with the underlying place v. Let X e a third smooth projective model of F (X) mapping oth to X and X, and let U e a common open suset of X, X and X contained in U: U j f f X U Y. j p j X p X By Lemma 1.14, all paths in this diagram commute. Hence we find f fj 1 = f p p 1. 3.4. Lemma. In S 1 SmP(F ), any morphism q S r can e written in the form q = j 1 qp 1 j with j, j S o, p S p and q Sp. Same statement within S 1 r SmP(F ) for q S r, with q Sr p. Proof. We don t need to take care of the places, thanks to Lemma 1.8. Let us prove the first statement. By resolution of singularities, we can find a commutative diagram Y X j X j 1 p q j 2 Y U X1 j q Ȳ

BIRATIONAL MOTIVES, I 17 where j, j, j 1, j 2 are open immersions, X, X1 and Ȳ are smooth projective, p is an astract low-up and q Sr p. Lemma 3.4 now follows from a small computation. The proof of the second statement is exactly the same. 3.5. Proposition. Any morphism in S 1 SmP(F ) can e written as j 1 fq 1, with j S o and q S p. Same statement for S 1 r SmP(F ), with q Sr p. Proof. We first show that any morphism in either category can e written as a composition of morphisms of the form j 1 fq 1 as in the statement of Proposition 3.5. It is sufficient to prove this for a morphism of the form fq 1, with q S (resp. q S r ). Write q = j 1 qp 1 j as in Lemma 3.4. Then we have fq 1 = fj 1 p q 1 j with j, j S o, p S p and q Sp (resp. q Sp r ). Applying now Lemma 3.3 to fj 1, we get fq 1 = a 1 gp 1 p q 1 j = (a 1 gp 1 )(p q 1 )j with a, S o and p S p. It now suffices to prove that the composition j1 1 f 1 q1 1 j2 1 f 2 q2 1 of two morphisms of this form is still of this form. By Lemma 3.4, write j 2 q 1 = j 1 q 3 p 1 j 3, with j 3, j S o, p S p and q 3 S p, so that f 1 q 1 1 j 1 2 = f 1 j 1 3 pq 1 3 j. By Lemma 3.3, write f 1 j3 1 = a 1 gp 1 1, where p 1 S p and a, S o, so that f 1 j3 1 pq3 1 j = a 1 gp 1 1 pq3 1 j and j1 1 f 1 q1 1 j2 1 f 2 q2 1 = j1 1 a 1 gp 1 1 pq3 1 j f 2 q2 1. It now suffices to apply Proposition 3.2 twice. 3.6. Proposition. Consider a diagram in Sm proj P(F ) Z f p X Y f p Z

18 BRUNO KAHN AND R. SUJATHA where p and p are astract low-ups. Let K = F (Z) = F (Z ) = F (Y ), L = F (X) and suppose given a place v : L K compatile oth with f and f. Then (v, fp 1 ) = (v, f p 1 ) in S 1 Sm proj P(F ). Proof. Complete the diagram as follows: Z f p p 1 X Z Y p 1 f p Z where p 1 and p 1 are astract low-ups and Z is smooth. Then we have pp 1 = p p 1, fp 1 = f p 1 (the latter y Lemma 1.14), hence the claim. 3.7. Definition. We extend the functor Φ 2 of Lemma 1.12 to functors Φ 2 : S 1 SmP(F ) place(f ) op Φ r 2 : Sr 1 SmP(F ) T 1 place(f ) op via Remark 1.5, where T is the multiplicative set of morphisms in place(f ) given y rational extensions. Let also JP : Sm proj P(F ) SmP(F ) J : Sm proj (F ) Sm(F ) denote the inclusion functors. Finally, we recall the forgetful functor Φ 1 : SmP(F ) Sm(F ) and its restriction to Sm proj P(F ), denoted y the same letter. 3.8. Theorem. The functors Φ 2, S 1 JP, Φ 2 S 1 JP and S 1 J are equivalences of categories, while the two functors S 1 Φ 1 are full and

BIRATIONAL MOTIVES, I 19 essentially surjective: S 1 Sm proj P(F ) S S 1 JP S 1 1 1 Φ 1 SmP(F ) S Φ 1 Φ 2 place(f ) op. S 1 Sm proj (F ) S 1 J S 1 Sm(F ) The same holds y replacing Φ 2 y Φ r 2 and S y S r : S 1 r Sm proj P(F ) S 1 r Φ 1 S 1 r JP S 1 r SmP(F ) S 1 r Φ 1 Φ r 2 T 1 place(f ) op. S 1 r Sm proj (F ) S 1 r J S 1 r Sm(F ) Proof. It is enough to prove the first assertion: the second one follows y localising the first with respect to S r. A) We first prove that Φ 2 S 1 JP is an equivalence of categories. By Lemma 1.12, this functor is essentially surjective. Proposition 1.17 shows that it is full. It remains to see that it is faithful. Let us construct a functor Ψ : place(f ) S 1 Sm proj P(F ) o as follows: To a function field K we associate a smooth projective model Ψ(K), chosen once and for all. Let v : L K e a place. By Proposition 1.17, there exists a pair (f, p) with f : X 1 Ψ(L) compatile with v and p : X 1 Ψ(K) an astract low-up. We define Ψ(v) as (v, fp 1 ). The map v Ψ(v) is well-defined y Proposition 3.6. To prove that it respects composition, consider two composale places K v L w

20 BRUNO KAHN AND R. SUJATHA M and the following diagram X 3 h r Ψ(M) Ψ(L) Ψ(K) q p g f X 2 X 1 with (v, fp 1 ) = Ψ(v), (w, gq 1 ) = Ψ(w) and (wv, hr 1 ) = Ψ(wv). We may complete it into a commutative diagram X 3 h r q 1 Ψ(M) Ψ(L) Ψ(K) X 4 q p g f X 2 X 1 q f Y where q, q 1, r 1 S p (the existence of (f, q ) comes from Proposition 3.2). The equality Ψ(wv) = Ψ(v)Ψ(w) now easily follows from this diagram. The equality Φ 2Sr 1 JP Ψ = Id is ovious and Ψ is clearly full. Therefore Φ 2S 1 JP is faithful, hence an equivalence of categories (with quasi-inverse Ψ). B) We now prove that S 1 JP is an equivalence of categories. The aove shows that it is faithful, and it is also essentially surjective thanks to resolution of singularities. It remains to see that it is full. By Proposition 3.5, for two ojects X, Y S 1 SmP(F ), any morphism ϕ : X Y is of the form j 1 fp 1 with j an open immersion and p a projective rational map. If Y is projective, j is necessarily an isomorphism. If furthermore X is projective, the source of p is projective too. Hence ϕ comes from S 1 Sm proj P(F ), as desired. C) Now the functor Φ 1 : SmP(F ) Sm(F ) is full and essentially surjective y Lemma 1.12, and this is preserved y localisation. Similarly for its restriction to Sm proj P(F ). As a consequence, S 1 J is full (and essentially surjective). r 1

BIRATIONAL MOTIVES, I 21 D) It remains to show that S 1 J is faithful. For this, we proceed exactly as in A): define a functor Π : Sm(F ) S 1 Sm proj (F ) y sending a smooth connected variety X to a smooth compactification Π(X) chosen once and for all, and a map f : X Y to a map Π(f) = s 1 f, where s : X Π(X) is a suitale astract low-up with centre disjoint from X and f : X Π(Y ) restricts to f on s 1 (X) = X. By an analogous statement to Proposition 3.6, Π(f) is well-defined. That Π is indeed a functor now is proven exactly as in A), y using Proposition 3.2, and Π is oviously full. Then Π factors through S 1 Sm(F ) into a full functor, which is a quasi-inverse to S 1 J. 3.9. Definition. We call place(f ) the coarse irational category of F and S 1 Sm proj (F ) the fine irational category of F. Similarly, we call T 1 place(f ) the coarse stale irational category of F and Sr 1 Sm proj (F ) the fine stale irational category of F. By Theorem 3.8, we have a commutative diagram of categories and functors place(f ) op Φ1 S 1 Sm proj (F ) T 1 place(f ) op T 1 Φ 1 S 1 r Sm proj (F ) where the horizontal functors are full and essentially surjective. 4. Birational finite correspondences We shall need the following proposition, which is an additive analogue of part of Theorem 3.8, in Section 6: 4.1. Proposition. Let SmCor(F ) e Voevodsky s category of finite correspondences on smooth varieties [38], and let SmCor proj (F ) e its full sucategory consisting of smooth projective varieties. Let J denote the inclusion functor SmCor proj (F ) SmCor(F ). Let further S SmCor(F ) e the set of [graphs of] morphisms that are irational isomorphisms, as well as its restriction to SmCor proj (F ). Then the equivalence of categories S 1 J of Theorem 3.8 extends to an equivalence of categories S 1 J : S 1 SmCor proj (F ) S 1 SmCor(F ) via the canonical functors Sm(F ) SmCor(F ) and Sm proj (F ) SmCor proj (F ). The same holds when replacing S y S r.

22 BRUNO KAHN AND R. SUJATHA The proof will occupy the entire section. Proof. We limit ourselves to S, the proofs for S r eing exactly the same. The functor S 1 J is clearly full and essentially surjective. We prove its faithfulness y constructing a quasi-inverse. In order to do this, we proceed using the notation and arguments as in part D) of the proof of Theorem 3.8. Thus we first extend the functor Π of loc. cit.to an additive functor Π : SmCor(F ) S 1 SmCor proj (F ). Given two smooth varieties X, Y, we need to define a homomorphism Π : c(x, Y ) S 1 SmCor proj (F )(Π(X), Π(Y )). Since the left-hand side is y definition the free aelian group on the set (X, Y ) of closed integral suschemes of X Y which are finite and surjective over a connected component of X, it would e sufficient to define a map Π : (X, Y ) S 1 SmCor proj (F )(Π(X), Π(Y )). To check functoriality, however, it will e more convenient to define Π directly on the monoid of positive correspondences 1 c + (X, Y ) = N(X, Y ), and to check that it is additive. We shall use the idea of Suslin-Voevodsky [34]: a cycle Z (X, Y ) of generic degree d over X defines a map [Z] : X S d (Y ). This rule extends to a homomorphism of aelian monoids (4.1) c + (X, Y ) n 0 Map F (X, S n (Y )) which is an isomorphism y [34, Th. 6.8] (ecause F is of characteristic 0). Let then Z c + (X, Y ), and [Z] : X S d (Y ) the corresponding map. Composing with the open immersion Y Π(Y ), we get a map X S d (Π(Y )). Consider its graph Γ in X S d (Π(Y )) and its closure Γ in Π(X) S d (Π(Y )). Clearly Γ is projective and irational to X (since the projection Γ X is an isomorphism). Desingularising it, we get s : X Π(X) with s S and a map f : X S d (Π(Y )) 1 We prefer to use the term positive rather than effective, in order not to create a possile confusion with effective motives later in this text.

extending f on some open suset of X: BIRATIONAL MOTIVES, I 23 X f s Π(X) Γ S d (Π(Y )). f Γ X On the other hand, y the ijectivity of (4.1), f corresponds to a unique positive cycle c c + ( X, Π(Y )). We may now define Π(Z) = cγ 1 s. We now proceed to check successively various compatiilities: 1) If Z is the graph Γ f of a morphism f : X Y, then d = 1 and [Z] = f, so we get exactly the same construction as in part D) of the proof of Theorem 3.8. 2) We check that cγ 1 s is independent of the choice of X. This follows from an argument analogous to that in Proposition 3.6. Indeed, if X 1 and X 2 are two different astract low-ups which give maps f 1 and f 2 into S d (Π(Y )), then we have a diagram as follows where t 1, t 2 are astract low-ups and hence t 1, t 2 S : f 1 t 1 S d (Π(Y )) f 2 X 1 s 1 X3 Π(X) s 2 X 2 t 2 If c 1, c 2 are the positive cycles corresponding to f 1 and f 2, then we have c 1 Γ t1 = c 2 Γ t2 in SmCor(F ), and also s 1 t 1 = s 2 t 2, hence c 1 Γ 1 s 1 = c 2 Γ 1 s 2. 3) We now check that Π(Z+Z ) = Π(Z)+Π(Z ) for two positive finite correspondences Z and Z. Let f : X S d (Y ) and f : X S d (Y ) e the two corresponding maps. Then to Z + Z is associated the composite f : X (f,f ) S d (Y ) S d (Y ) S d+d (Y ).

24 BRUNO KAHN AND R. SUJATHA Let s : X Π(X) and s : X Π(X) e two irational isomorphisms chosen as aove respectively for f and for f, and let t : X X, t : X X e two astract low-ups ( X smooth projective), defining two equal compositions s = s t = s t : X Π(X). Then we get a diagram X (f,f ) S d (Y ) S d (Y ) S d+d (Y ) Π(X) S d (Π(Y )) S d (Π(Y )) S d+d (Π(Y )) (ft,f t ) s X Let f : X S d+d (Π(Y )) e the composition from this diagram. Then, y 2), Π(Z + Z ) corresponds to f s 1, and the diagram concludes the check of 3). 4) We check that Π is a functor. Let α c + (X, Y ) and β c + (Y, Z) e positive correspondences, with f : X S d (Y ) (where d = n i [F (U i ) : F (X)] if α = n i α i with α i irreducile with supports U i in X Y ), and g : Y S k (Z) the corresponding maps coming from the Suslin-Voevodsky isomorphism. 4.2. Lemma. The composition β α c + (X, Z) maps under (4.1) to the composite X f S d (Y ) S d (g) S d (S k (Z)) S dk (Z). Proof. (Friedlander and Voevodsky) We give it using that F is of characteristic 0, although this assumption is proaly not necessary (Faien Morel has indicated us that he has a proof ased on the results of [34], at least after inverting the exponential characteristic). We have to show that the image of β α in Map F (X, S dk (Z)) equals S d (g) f. We may assume X connected. Let η = Spec K e its generic point. Since Map F (X, S dk (Z/F )) Map F (η, S dk (Z/F )) = Map K (Spec K, S dk (Z K /K)), we may assume that X = Spec F. Then Map F (Spec F, S dk (Z)) is the set of positive zero-cycles of degree dk on Z. Since F is of characteristic 0, this set injects into Map F (Spec F, S dk (Z F )) and we further reduce to the case where F is algeraically closed and finally where α

BIRATIONAL MOTIVES, I 25 is irreducile. Then it corresponds to a rational point of Y, d = 1 and the claim is ovious. With notation as aove, we now have a (not very) commutative diagram: X Π(X) f S d (Y ) s f S d (Π(Y )) X S d (g) S d (S k (Z)) S d (t) S d (Ỹ ) S d ( g) S d (S k (Π(Z)) S dk (Z) S dk (Π(Z)) and the proof will e finished if we can complete it into a (rather more) commutative diagram X Π(X) f S d (Y ) S d (g) S d (S k (Z)) s f S d (Π(Y )) S d (t) S d (S k (Π(Z)) S d ( g) X u X h S d (Ỹ ) S dk (Z) S dk (Π(Z))

26 BRUNO KAHN AND R. SUJATHA where X is smooth and u is a irational isomorphism. (The commutativity means that S d (t)h = fu, or equivalently S d (t) 1 f = hu 1, which implies S d ( g) S d (t) 1 fs 1 = S d ( g)hu 1 s 1.) This means extending Proposition 3.2 to a case (with its notation) where X and Y are not smooth. Examining the proof of this proposition, still with its notation, smoothness is irrelevant everywhere provided we have a place v compatile with u. Referring to Lemma 1.11, we do use that Y is regular. However, all we use is that u(η X ) is a regular point of Y. Therefore, coming ack to the aove notation, the proof of Proposition 4.1 will e complete if we can prove that f(η X ) is a smooth point of S d (Π(Y )). But this point is the image of f(η X ) y the open immersion S d (Y ) S d (Π(Y )), so it suffices to know that f(η X ) is a smooth point of S d (Y ). For any scheme T (over some ase), let Ũd (T ) e the open suset of T d defined y the conditions that all coordinates are distinct, and let U d (T ) e the image of Ũd (T ) in S d (T ): the projection Ũd (T ) U d (T ) is finite étale. If T is smooth, U d (T ) is smooth as well. Note that if i : S T is an immersion, then the immersion S d (i) carries U d (S) into U d (T ). Therefore, to check 4) we are reduced to proving the following 4.3. Lemma. Let U X e a finite surjective morphism of schemes, with U integral and X normal connected. Let d = [κ(u) : κ(x)], and assume the extension κ(x)/κ(u) separale. Then the Suslin-Voevodsky map [34, p. 81] X S d (U) sends the generic point η X in U d (U). Note that the assumption char F = 0 is needed to e ale to apply Lemma 4.3! Proof. We have a commutative diagram η X S d (η U ) X S d (U). Since the vertical maps are immersions, we are reduced to the case where X, hence U, is the spectrum of a field, say X = Spec K, U = Spec L. We shall prove the lemma in the slightly more general case where L is an étale K-algera. Suppose first that L = K d. Let x 1,..., x d e the d rational points Spec K Spec L. Then the image of Spec K in S d (Spec L)

BIRATIONAL MOTIVES, I 27 is (x 1,..., x d ), which is clearly in U d (Spec L). In general, we reduce to this case y passing to a separale closure of K. S 1 5) Finally we need to check that Π induces a functor S 1 SmCor(F ) SmCor proj (F ), i.e. that Π(Γ s ) is invertile if s S. This follows immediately from 1). The fact that Π is a quasi-inverse to S 1 immediate. 5. Triangulated category of irational motives J is In this section, we construct a triangulated category of irational geometric motives and an additive category of irational Chow motives, and construct a functor from the second to the first. We assume that the reader is familiar with Voevodsky s triangulated categories of motives [38]; for a smooth variety X over F, we simplify his notation M gm (X) into M(X) for the motive of X in DM eff gm(f ). 5.1. Definition. We denote y DMgm(F o ) the pseudo-aelian envelope of the quotient of DMgm(F eff ) [38] y the thick triangulated sucategory C generated y the cones of maps M(U) j M(X) where X is a smooth variety and j is an open immersion. We denote y M(X) the image of M(X) in DM o gm(f ). By [2], DM o gm(f ) is still triangulated. 5.2. Proposition. a) The tensor structure of DMgm(F eff ) passes to DMgm(F o ). ) Suppose F perfect. Then C consists of those motives of the form M(1) and the functor DMgm(F eff ) C given y M M(1) is an equivalence of categories. Proof. a) Recall that the category DMgm(F eff ) is generated y the M(Y ) (Y smooth) and there is a canonical isomorphism M(X 1 X 2 ) M(X 1 ) M(X 2 ). Clearly the tensor structure is compatile on further localising with respect to morphisms M(U) M(X) where U X is an open immersion. ) Since Z Z(1)[2] = M(P 1 ) = M(A 1 ) = Z in DMgm(F o ), Z(1) = 0 in DMgm(F o ). By a), DMgm(F o ) is therefore a localisation of DMgm(F eff )/C. To see conversely that Ker(DMgm(F eff ) DMgm(F o )) C, we have to j prove that cone(m(u) M(X)) is in C for any open immersion j. We argue y Noetherian induction on the (reduced) closed complement Z in a standard way. If Z is smooth of pure codimension

28 BRUNO KAHN AND R. SUJATHA c, then the cone is isomorphic to M(Z)(c)[2c] [38, Prop. 3.5.4]. In general, let Z sing Z e the singular locus of Z. Then Z Z sing is smooth in X Z sing. Let C and C denote respectively cones of the maps M(U) M(X) and M(X Z sing ) M(X). By the axioms of triangulated categories, we may complete the diagram M(U) M(U) M(X Z sing ) M(X) C M(Z Z sing )(c)[2c] M(U)[1] C M(U)[1] into a commutative diagram of exact triangles M(U) M(U) 0 M(X Z sing ) M(X) C M(Z Z sing )(c)[2c] C C M(U)[1] M(U)[1] 0 where C is y definition a cone of the map M(Z Z sing )(c)[2c] C. Hence we get an exact triangle C [ 1] M(Z Z sing )(c)[2c] C C. Since F is perfect, Z sing is strictly smaller than Z and we have C C y Noetherian induction, hence also C C. To conclude, we have to show that, if M, N DM eff gm(f ) and f Hom(M(1), N(1)), then the cone of f is of the form P (1). This follows from the cancellation theorem [40], which is now valid over any perfect field. (We are indeted to Faien Morel for pointing out this issue.) The last assertion similarly follows from the cancellation theorem.

BIRATIONAL MOTIVES, I 29 Let Cor eff (F ) denote the category of effective correspondences, i.e. the category whose ojects are smooth projective varieties and morphisms are Chow correspondences. The category Chow eff (F ) of effective Chow motives is the pseudo-aelian envelope of Cor eff (F ). Recall the functor [38, Prop. 2.1.4] (5.1) Chow eff (F ) DM eff gm(f ) from the category of effective Chow motives (viewed as a homological category: the natural functor Sm proj (F ) Chow eff (F ) is covariant). We would like to descrie a similar functor to DM o gm(f ). To this effect we quotient Chow eff (F ) in a parallel way. In Chow eff (F ), we write h(x) for the Chow motive associated to a smooth projective variety X, 1 for h(spec F ) and L for the Lefschetz motive, defined y h(p 1 ) = 1 L. Then L Z(1)[2] under (5.1). For M Chow eff (F ), we write M(n) for M L n, rather than the traditional notation M( n). 5.3. Lemma. For two smooth projective varieties X, Y, let I(X, Y ) e the sugroup of CH dim X (X Y ) consisting of those classes vanishing in CH dim X (U Y ) for some open suset U of X. Then I is a monoidal ideal in Cor eff (F ) (i.e. is closed with respect to composition and tensor products on the left and right). In the factor category Cor o (F ), one has the formula Hom( h(x), h(y )) = CH 0 (Y F (X) ) where h h denotes the natural composite functor Sm proj (F ) Cor eff (F ) Cor o (F ). Proof. Let X, Y, Z e 3 smooth projective varieties. If U is an open suset of X, it is clear that the usual formula defines a composition of correspondences CH dim X (U Y ) CH dim Y (Y Z) CH dim X (U Z) and that this composition commutes with restriction to smaller and smaller open susets. Passing to the limit on U, we get a composition or CH dim Y (Y F (X) ) CH dim Y (Y Z) CH dim Z (Z F (X) ) CH 0 (Y F (X) ) CH dim Y (Y Z) CH 0 (Z F (X) ). This pairing is actually nothing else than the action of correspondences on Chow groups of 0-cycles (it extends to an action of CH dim Y (Y F (X) Z F (X) )). We now need to prove that this action factors through an action of CH dim Y (V Z) for any open suset V of

30 BRUNO KAHN AND R. SUJATHA Y. Without loss of generality, we may pass to F (X) and hence assume that X = Spec F. The proof is asically a generalisation of Fulton s proof of the Colliot- Thélène Coray theorem that CH 0 is a irational invariant of smooth projective varieties [7], [13, Ex. 16.1.11]. Let M e a proper closed suset of Y, and i : M Y e the corresponding closed immersion. We have to prove that for any α CH 0 (Y ) and β CH dim Y (M Z), (i 1 Z ) (β)(α) := (p 2 ) ((i 1 Z ) β p 1α) = 0 where p 1 and p 2 are respectively the first and second projections on Y Z. We shall actually prove that (i 1 Z ) β p 1α = 0. For this, we may assume that α is represented y a point y Y (0) and β y some integral variety W M Z. Then (i 1 Z ) β p 1α has support in (i 1 Z )(W ) ({y} Z) (M Z) ({y} Z). If y / M, this suset is empty and we are done. Otherwise, up to linear equivalence we may replace y y a 0-cycle disjoint from M (cf. [30]), and we are ack to the previous case. This shows that I is an ideal of Cor eff (F ). The fact that it is a monoidal ideal is essentially ovious. We extend the ideal I from the category of effective Chow correspondences to the category of effective Chow motives (its pseudo-aelian envelope) in the ovious way, keeping the same notation. Let us define Chow o (F ) = (Chow eff (F )/I) where denotes pseudo-aelian envelope. 5.4. Lemma. In Chow eff (F ), I contains the morphisms factoring through an oject of the form M(1). If char F = 0, any morphism in I is of this form. If char F > 0, this is true at least after tensoring morphisms with Q. Proof. Consider the Lefschetz motive L. As it is defined y the projector P 1 CH 1 (P 1 P 1 ), L 0 Chow o (F ). Since I is monoidal, the first statement is clear. For the converse, it is enough to handle morphisms f I(h(X), h(y )) for two smooth projective varieties X, Y. Assume that the cycle f CH dim X (X Y ) is of the form (i 1 Y ) g for some closed immersion i : Z X, where g CH dim X (Z Y ). If Z is smooth of codimension c, then f factors through h(z)(c). Without loss of generality, we may assume that Z is an integral divisor. We may further assume that g is the class of an integral variety W Z Y. Moreover, let Z e the singular locus of