VOEVODSKY S MOTIVES AND WEIL RECIPROCITY

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1 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY BRUNO KAHN AND TAKAO YAMAZAKI Abstract. We describe Somekawa s K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky s category of motives. While Somekawa s definition is based on Weil reciprocity, Voevodsky s category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles. Contents 1. Introduction 2 2. Mackey functors and presheaves with transfers 4 3. Presheaves with transfers and motives 9 4. Presheaves with transfers and local symbols K-groups of Somekawa type K-groups of geometric type Milnor K-theory K-groups of Milnor type Reduction to the representable case Proper sheaves Main theorem Application to algebraic cycles 32 Appendix A. Extending monoidal structures 33 References 38 Date: March 19, Mathematics Subject Classification. 19E15 (14F42, 19D45, 19F15). Key words and phrases. Motives, homotopy invariance, Milnor K-groups, Weil reciprocity. The second author is supported by Grant-in-Aid for Challenging Exploratory Research ( ), Grand-in-Aid for Young Scientists (A) ( ), and Inamori Foundation. 1

2 2 BRUNO KAHN AND TAKAO YAMAZAKI 1. Introduction 1.1. In this article, we construct an isomorphism (1.1) K(k; F 1,..., F n ) Hom DM eff (Z, F 1 [0] F n [0]). Here k is a perfect field, and F 1,..., F n are homotopy invariant Nisnevich sheaves with transfers in the sense of [32]. On the right hand side, the tensor product F 1 [0] F n [0] is computed in Voevodsky s triangulated category DM eff of effective motivic complexes. The group K(k; F 1,..., F n ) will be defined in Definition 5.1 by an explicit set of generators and relations: it is a generalization of the group which was defined by K. Kato and studied by M. Somekawa in [24] when F 1,..., F n are semi-abelian varieties In the introduction of [24], Somekawa wrote that he expected an isomorphism of the form K(k; G 1,..., G n ) Ext n MM(Z, G 1 [1] G n [1]) where MM is a conjectural abelian category of mixed motives over k, G 1,..., G n are semi-abelian varieties over k, and G 1 [1],..., G n [1] are the corresponding 1-motives. Since we do not have such a category M M at hand, (1.1) provides the closest approximation to Somekawa s expectation The most basic case of (1.1) is F 1 = = F n = G m. By [24, Theorem 1.4], the left hand side is isomorphic to the usual Milnor K- group K M n (k). The right hand side is almost by definition the motivic cohomology group H n (k, Z(n)). Thus, when k is perfect, we get a new and less combinatorial proof of the Suslin-Voevodsky isomorphism [28, Thm. 3.4], [16, Thm. 5.1] (1.2) K M n (k) H n (k, Z(n)) The isomorphism (1.1) also has the following application to algebraic cycles. Let X be a k-scheme of finite type. Write CH 0 (X) for the homotopy invariant Nisnevich sheaf with transfers U CH 0 (X k k(u)) (U smooth connected) see [7, Th. 2.2]. Let i, j Z. We write CH i (X, j) for Bloch s homological higher Chow group [14, 1.1]: if X is equidimensional of dimension d, it agrees with the group CH di (X, j) of [4] Theorem. Suppose char k = 0. Let X 1,..., X n be quasi-projective k-schemes. Put X = X 1 X n. For any r 0, we have an

3 isomorphism (1.3) VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 3 K(k; CH 0 (X 1 ),..., CH 0 (X n ), G m,..., G m ) where we put r copies of G m on the left hand side. 1 CH r (X, r), 1.6. When X 1,..., X n are smooth projective, 2 special cases of (1.3) were previously known. The case r = 0 was proved by Raskind and Spieß [20, Corollary 4.2.6], and the case n = 1 was proved by Akhtar [1, Theorem 6.1] (without assuming k to be perfect). The extension to non smooth projective varieties is new and nontrivial Theorem 1.5 is proven using the Borel-Moore motivic homology introduced in [6, 9]. We also have a variant which involves motivic homology, see Theorem Here is an application. Let C 1, C 2 be two smooth connected curves over our perfect field k, and put S = C 1 C 2. Assume that C 1 and C 2 both have a 0-cycle of degree 1. Then the special case n = 2, r = 0 of Theorem 12.3 gives an isomorphism Z Alb S (k) K(k; Alb C1, Alb C2 ) H 0 (S, Z). Here, for a smooth variety X, we denote by Alb X the Albanese variety of X in the sense of Serre [22]; it is a semi-abelian variety universal for morphisms from X to semi-abelian varieties (compare [32, Th ]). The right hand side in this case is Suslin homology [29], see Since Somekawa s groups are defined in an explicit manner, one can sometimes determine the structure of K(k; Alb C1, Alb C2 ) completely. For instance, when k is finite, we have K(k; Alb C1, Alb C2 ) = 0 by [9]. This immediately implies the bijectivity of the generalized Albanese map a S : H 0 (S, Z) deg=0 Alb S (k) of Ramachandran and Spieß-Szamuely [25]. Note that a S is not bijective for a smooth projective surface S in general, see [12, Prop. 9] We conclude this introduction by pointing out the main difficulty and main ideas in the proof of (1.1). The definitions of the two sides of (1.1) are quite different: the left hand side is based on Weil reciprocity, while the right hand side is based on homotopy invariance. Thus it is not even obvious how to define a map (1.1) to start with. Our solution is to write both sides as quotients of a common larger group, and to prove that one quotient factors 1 Using recent results of Shane Kelly [13], one may remove the characteristic zero hypothesis if we invert the exponential characteristic of k. 2 In this case, Theorem 1.5 is valid in any characteristic, see Remark 12.4.

4 4 BRUNO KAHN AND TAKAO YAMAZAKI through the other. This provides a map (1.1) which is automatically surjective (Theorem 5.3). The proof of its injectivity turns out to be much more difficult. We need to find many relations coming from Weil reciprocity. Our main idea, inspired by [24, Theorem 1.4] (recalled in 1.3), is to use the Steinberg relation to create Weil reciprocity relations. To show that this provides us with enough such relations, we need to carry out a heavy computation of symbols in 11. Acknowledgements. Work in this direction had been done previously by Mochizuki [17]. The surjective map (1.1) was announced in [26, Remark 10 (b)]. This research was started by the first author, who wrote the first part of this paper [11]. The collaboration began when the second author visited the Institute of Mathematics of Jussieu in October Somehow, the research accelerated after the earthquake on March 11, 2011 in Japan. We wish to acknowledge the pleasure of such a fruitful collaboration, along these circumstances. We also thank the referees for a very careful reading of this manuscript. We acknowledge the depth of the ideas of Milnor, Kato, Somekawa, Suslin and Voevodsky. Especially we are impressed by the relevance of the Steinberg relation in this story. 2. Mackey functors and presheaves with transfers 2.1. A Mackey functor over k is a contravariant additive (i.e., commuting with coproducts) functor A from the category of étale k-schemes to the category of abelian groups, provided with a covariant structure verifying the following exchange condition: if Y f g Y g X f X is a cartesian square of étale k-schemes, then the diagram A(Y ) g f A(Y ) g A(X f ) A(X) commutes. Here, denotes the contravariant structure while denotes the covariant structure. The Mackey functor A is cohomological if we further have f f = deg(f)

5 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 5 for any f : X X, with X connected. We denote by Mack the abelian category of Mackey functors, and by Mack c its full subcategory of cohomological Mackey functors Classically [30, (1.4)], a Mackey functor may be viewed as a contravariant additive functor on the category Span of spans on étale k-schemes, defined as follows: objects are étale k-schemes. A morphism from X to Y is an equivalence class of diagram (span) (2.1) X g Z f Y where, as usual, two spans (Z, f, g) and (Z, f, g ) are equivalent if there exists an isomorphism Z Z making the obvious diagram commute. Composition of spans is defined via fibre product in an obvious manner (compare Quillen s Q-construction in [18, 2]). If A is a Mackey functor, the corresponding functor on Span has the same value on objects, while its value on a span (2.1) is given by g f. Note that Span is a preadditive category: one may add (but not subtract) two morphisms with same source and target. We may as well view a Mackey functor as a contravariant additive functor on the associated additive category Z Span. In the notation of the appendix, we thus have Mack = Mod Z Span Let Cor be Voevodsky s category of finite correspondences on smooth k-schemes, denoted by SmCor(k) in [32, 2.1]. Recall that the objects of Cor are the same as the category Sm/k of smooth varieties over k. Following loc. cit., we denote by c(x, Y ) the group of morphisms in Cor from X to Y, which is, by definition, the free abelian group on the set of closed integral subschemes of X Y which are finite and surjective over some irreducible component of X. Let PST be the category of presheaves with transfers (i.e. contravariant additive functors from Cor to abelian groups), denoted by P reshv(smcor(k)) in [32, 3.1]. In the same style as 2.2, we have, by definition, PST = Mod Cor The category Z Span is isomorphic to the full subcategory of Cor consisting of smooth k-schemes of dimension 0 (= étale k-schemes). In particular, any presheaf with transfers in the sense of Voevodsky [32, Def ] restricts to a Mackey functor over k. By [31, Cor. 3.15], the restriction of a homotopy invariant presheaf with transfers yields a cohomological Mackey functor. In other words, we have exact functors (2.2) (2.3) ρ : PST Mack ρ : HI Mack c,

6 6 BRUNO KAHN AND TAKAO YAMAZAKI where HI is the full subcategory of PST consisting of homotopy invariant presheaves with transfers By definition, the functor (2.2) equals i, where i is the inclusion Z Span Cor. This inclusion has a left adjoint π 0 (scheme of constants). Both functors i and π 0 are symmetric monoidal: for π 0, reduce to the case where k is algebraically closed Let A be an additive symmetric monoidal category. In the appendix, we show that the symmetric monoidal structure of A extends canonically to the category Mod A of (right) A-modules ( A.8). Given a symmetric monoidal functor f : A B, its extension f! to right modules is symmetric monoidal ( A.12) If A = Cor in 2.6, we get a tensor structure in PST: we show in Example A.11 that it agrees with the one defined by Voevodsky in [32, p. 206]. For the reader s convenience, we recall how to compute this tensor product (see A.11 and [32, Sect. 3.2]) For a smooth variety X over k, denote as usual by L(X) the Nisnevich sheaf with transfers represented by X. Let F and F be presheaves with transfers. There are exact sequences of the form L(Y j ) L(X i ) F 0, j i j L(Y j ) i L(X i ) F 0. Then the tensor product F PST F of F and F is given by ( Coker L(Y j X i ) L(X i Y j ) L(X i X i ). ) j,i i,j i,j 2.8. If A = Z Span in 2.6, we get a tensor structure on Mack. Another tensor product of Mackey functors M was originally defined by L. G. Lewis (unpublished); it was used in [8, 5] and [9]. If either A or B is cohomological, A M B is cohomological. In Example A.18, we show that M agrees with the tensor structure from 2.6. For the reader s convenience, we recall the definition of. M Let A 1,..., A n be Mackey functors. For any étale k-scheme X, we define (A 1 M... M An )(X) := [ Y X A 1 (Y ) A n (Y )]/R,

7 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 7 where Y X runs through all finite étale morphisms, and R is the subgroup generated by all elements of the form a 1 f (a i )... a n f (a 1 ) a i... f (a n ), where Y 1 f Y2 Y is a tower of étale morphisms, 1 i n, a i A i (Y 1 ) and a j A j (Y 2 ) (j = 1,..., i 1, i + 1,..., n) As an upshot of 2.6, 2.7 and 2.8, (2.2) is symmetric monoidal with respect to Voevodsky s tensor structure PST on PST and the tensor structure M on Mack. In other words, if F and G are presheaves with transfers, then (2.4) ρf M ρg ρ(f PST G). The left hand side is sometimes abbreviated to F M G Let F PST. We define C 1 (F) PST by C 1 (F)(X) = F(X A 1 ) for all smooth X. For a k = A 1 (k), the morphism X X A 1, x (x, a) defines a morphism i a : C 1 (F) F in PST. The inclusion functor HI PST has a left adjoint h 0 given by h 0 (F) = Coker(i 0 i 1 : C 1 (F) F), and the symmetric monoidal structure of PST induces one on HI via h 0. In other words, if F, G HI, we define (2.5) F HI G = h 0 (F PST G). Note that (2.3) is not symmetric monoidal (since it is the restriction of (2.2)) For any F PST, the unit morphism F h 0 (F) induces a surjection (2.6) F(k) h 0 (F)(k). This is obvious from the formula h 0 (F) = Coker(C 1 (F) F) We shall also need to work with Nisnevich sheaves with transfers. We denote by NST the category of Nisnevich sheaves with transfers (objects of PST which are sheaves in the Nisnevich topology). By [32, Theorem 3.1.4], the inclusion functor NST PST has an exact left adjoint F F Nis (sheafification). The category NST then inherits a tensor product by the formula F NST G = (F PST G) Nis. Similarly, we define HI Nis = HI NST. The sheafification functor restricts to an exact functor HI HI Nis [32, Theorem ], and

8 8 BRUNO KAHN AND TAKAO YAMAZAKI HI Nis gets a tensor product by the formula F HINis G = (F HI G) Nis. To summarize, all functors in the following naturally commutative diagram are symmetric monoidal: (2.7) Nis PST NST h 0 h Nis 0 Nis HI HI Nis. where each functor is left adjoint to the corresponding inclusion Let F be a presheaf on Sm/k, and let F Nis be the associated Nisnevich sheaf. Then we have an isomorphism (2.8) F(k) F Nis (k). Indeed, any covering of Spec k for the Nisnevich topology refines to a trivial covering. In particular, the functor F F Nis (k) is exact. This applies in particular to a presheaf with transfers and the associated Nisnevich sheaf with transfers Let F 1,..., F n HI Nis. Then (2.4) yields a canonical isomorphism (2.9) (F 1 M... M Fn )(k) (F 1 PST PST F n )(k). Composing (2.9) with the unit morphism Id h Nis 0 from (2.7) and taking (2.5) into account, we get a canonical morphism (2.10) (F 1 M... M Fn )(k) (F 1 HINis HINis F n )(k). which is surjective by 2.11 and If G is a commutative k-group scheme whose identity component is a quasi-projective variety, then G has a canonical structure of Nisnevich sheaf with transfers ([25, proof of Lemma 3.2] completed by [2, Lemma 1.3.2]). This applies in particular to semi-abelian varieties and also to the full Albanese scheme [19] of a smooth variety (which is an extension of a lattice by a a semi-abelian variety). In particular, if G 1,..., G n are such k-group schemes, (2.10) yields a canonical surjection (2.11) (G 1 M... M Gn )(k) (G 1 HINis HINis G n )(k), where the G i are considered on the left as Mackey functors, and on the right as homotopy invariant Nisnevich sheaves with transfers.

9 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 9 3. Presheaves with transfers and motives 3.1. The left adjoint h Nis 0 in (2.7) extends to a left adjoint C of the inclusion DM eff D (NST) where the left hand side is Voevodsky s triangulated category of effective motivic complexes [32, 3, esp. Prop ]. More precisely, DM eff is defined as the full subcategory of objects of D (NST) whose cohomology sheaves are homotopy invariant. The canonical t-structure of D (NST) induces a t-structure on DM eff, with heart HI Nis. The functor C is right exact with respect to these t- structures, and if F NST, then H 0 (C (F)) = h Nis 0 (F) The tensor structure of 2.12 on NST extends to one on D (NST) [32, p. 206]. Via C, this tensor structure descends to a tensor structure on DM eff [32, p. 210], which will simply be denoted by. The relationship between this tensor structure and the one of 2.12 is as follows: if F, G HI Nis, then (3.1) F HINis G = H 0 (F[0] G[0]) where F[0], G[0] are viewed as complexes of Nisnevich sheaves with transfers concentrated in degree 0. We shall need the following lemma, which is not explicit in [32]: 3.3. Lemma. The tensor product of DM eff to the homotopy t-structure. Proof. By definition, C D = C (C L D) is right exact with respect for C, D DM eff, where L is the tensor product of D (NST) defined in [32, p. 206]. We want to show that, if C and D are concentrated in degrees 0, then so is C D. Using the canonical left resolutions of loc. cit., it is enough to do it for C and D of the form C (L(X)) and C (L(Y )) for two smooth schemes X, Y. Since C is symmetric monoidal, we have C (L(X)) C (L(Y )) C (L(X) L L(Y )) = C (L(X Y )) and the claim is obvious in view of the formula for C [32, p. 207] Let C DM eff. For any X Sm/k and any i Z, we have H i Nis(X, C) Hom DM eff (M(X), C[i])

10 10 BRUNO KAHN AND TAKAO YAMAZAKI where M(X) = C (L(X)) is the motive of X computed in DM eff (cf. [32, Prop ]). Specializing to the case X = Spec k (M(X) = Z) and taking 2.13 into account, we get (3.2) Hom DM eff (Z, C[i]) H i (C)(k). Combining (3.1), (2.8) and (3.2), we get: 3.5. Lemma. Let F 1,..., F n be homotopy invariant Nisnevich sheaves with transfers. Then we have a canonical isomorphism (3.3) (F 1 HINis HINis F n )(k) Hom DM eff (Z, F 1 [0] F n [0]) Summarizing, for any F 1,..., F n HI Nis we get a surjective homomorphism (3.4) (F 1 M... M Fn )(k) Hom DM eff (Z, F 1 [0] F n [0]). by composing (2.9), (2.6), (2.5), (2.8) and (3.3). 4. Presheaves with transfers and local symbols 4.1. Given a presheaf with transfers G, recall from [31, p. 96] the presheaf with transfers G 1 defined by the formula (4.1) G 1 (U) = Coker ( G(U A 1 ) G(U (A 1 {0})) ). If G HI Nis, then G(U (A 1 {0})) G(U) G 1 (U) for all smooth U. Thus G 1 HI Nis and G G 1 is an exact endofunctor of HI Nis. Suppose that G HI Nis. Let X Sm/k (connected), K = k(x) and x X be a point of codimension 1. By [31, Lemma 4.36], there is a canonical isomorphism (4.2) G 1 (k(x)) H 1 Zar,x(X, G) yielding a canonical map (4.3) x : G(K) G 1 (k(x)). The following lemma follows from the construction of the isomorphisms (4.2). It is part of the general fact that G defines a cycle module in the sense of Rost (cf. [5, Prop ]) Lemma. a) Let f : Y X be a dominant morphism, with Y smooth and connected. Let L = k(y ), and let y Y (1) be such that

11 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 11 f(y) = x. Then the diagram G(L) f ( y) G 1 (k(y)) ef G(K) x G1 (k(x)) commutes, where e is the ramification index of v y relative to v x. b) If f is finite surjective, the diagram ( y) G(L) G 1 (k(y)) commutes. f G(K) x y f 1 (x) f G1 (k(x)) 4.3. Proposition. Let G HI Nis. There is a canonical isomorphism G 1 = Hom(G m, G). Proof. The statement means that G 1 represents the functor H Hom HINis (H HINis G m, G). Sheafifying (4.1) for the Nisnevich topology and using homotopy invariance, we obtain an isomorphism Coker(G p p G) G 1 where p : A 1 {0} Spec k is the structural morphism. Moreover, [31, Prop. 5.4] shows R i p p G(K) = HNis i (A1 K {0}, p G) = 0 for any field K/k and i > 0, hence by [31, Prop. 4.20] one has R i p p G = 0 for any i > 0. It follows that p p G[0] Rp p G[0]. By [32, Prop ], we have Rp p G[0] = Hom(M(A 1 {0}), G[0]) where Hom is the (partially defined) internal Hom of DM eff. By [32, Prop ] (Gysin triangle) and homotopy invariance, we have an exact triangle, split by any rational point of A 1 {0}: Z(1)[1] M(A 1 {0}) Z +1 To get a canonical splitting, we may choose the rational point 1 A 1 {0}.

12 12 BRUNO KAHN AND TAKAO YAMAZAKI By [32, Cor ], we have an isomorphism Z(1)[1] G m [0]. Hence, in DM eff, we have an isomorphism (4.4) G 1 [0] Hom DM eff (G m [0], G[0]). Let H HI Nis. We get: as desired. Hom DM eff (H[0], G 1 [0]) Hom DM eff (H[0] G m [0], G[0]) by (4.4) Hom HINis (H 0 (H[0] G m [0]), G) by Lemma 3.3 = Hom HINis (H HINis G m, G) by (3.1) 4.4. Remark. The proof of Proposition 4.3 also shows that, in DM eff, we have an isomorphism Hom(G m [0], G[0]) Hom(G m, G)[0] where the left Hom is computed in DM eff and the right Hom is computed in HI Nis. In particular, Hom(G m [0], ) : DM eff DM eff is t-exact Proposition. Let C be a smooth, proper, connected curve over k, with function field K. For any G HI Nis, there exists a canonical homomorphism Tr C/k : H 1 Zar(C, G) G 1 (k) such that, for any x C, the composition G 1 (k(x)) Hx(C, 1 G) HZar(C, 1 G) Tr C G 1 (k) equals the transfer map Tr k(x)/k associated to the finite surjective morphism Spec k(x) Spec k. Proof. By [32, Prop ], we have H 1 Zar(C, G) H 1 Nis(C, G) Hom DM eff (M(C), G[1]). The structural morphism C Spec k yields a morphism of motives M(C) Z which, by Poincaré duality [32, Th ], yields a canonical morphism G m [1] Z(1)[2] M(C). (One may view this morphism as the image of the canonical morphism L h(c) in the category of Chow motives.) Therefore, by Proposition 4.3 and Remark 4.4, we get a map Tr C/k : H 1 Zar(X, G) Hom DM eff (G m [1], G[1]) = G 1 (k).

13 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 13 It remains to prove the claimed compatibility. Let M x (C) be the motive of C with supports in x, defined as C (Coker(L(C {x}) L(C)). Let Z k(x) = M(Spec k(x)). By [32, proof of Prop ], we have an isomorphism M x (C) Z k(x) (1)[2], and we have to show that the composition Z(1)[2] M(C) gx Z k(x) (1)[2] induces Tr k(x)/k, up to twisting and shifting. To see this, we observe that g x is the image of the morphism of Chow motives h(c) h(spec k(x))(1) dual to the morphism h(spec k(x)) h(c) induced by the inclusion Spec k(x) C: this is easy to check from the definition of g x in [32] (observe that in this special case, Bl x (C) = C and that we may use a variant of the said construction replacing C A 1 by C P 1 to stay within smooth projective varieties). The conclusion now follows from the fact that the composition Spec k(x) C Spec k is the structural morphism of Spec k(x) Proposition (Reciprocity). Let C be a smooth, proper, connected curve over k, with function field K. Then the sequence ( x) G(K) x C G x 1(k(x)) Tr k(x)/k G 1 (k) is a complex. Proof. This follows from Proposition 4.5, since the composition is 0. G(K) x C H 1 x(c, G) (gx) H 1 (C, G) 4.7. Theorem. Suppose F HI Nis. Then there exists a canonical isomorphism F (F HINis G m ) 1. Proof. We compute again in DM eff. As recalled in the proof of Proposition 4.3, we have G m [0] = Z(1)[1]. Hence the cancellation theorem [34] yields a canonical isomorphism By taking H 0, we obtain F[0] Hom DM eff (G m [0], F[0] G m [0]) F Hom HINis (G m, F HINis G m ).

14 14 BRUNO KAHN AND TAKAO YAMAZAKI The right hand side is isomorphic to (F HINis G m ) 1 by Proposition If F, G are presheaves with transfers, there is a bilinear morphism of presheaves with transfers (i.e. a natural transformation over PST PST): F(U) G 1 (V ) = Coker ( F(U) G(V A 1 ) F(U) G(V (A 1 {0})) ) Coker ( (F PST G)(U V A 1 ) (F PST G)(U V (A 1 {0})) ) which induces a morphism (4.5) F PST G 1 (F PST G) 1. = (F PST G) 1 (U V ) 4.9. Notation. Let F, G HI Nis and H = F HINis G. Let X, K, x be as in 4.1. For (a, b) F(K) G(K), we denote by a b the image of a b in H(K) by the map (4.6) F(K) G(K) H(K). We define the local symbol on F to be the composition F(K) K F(k(x)) F(K) K (F HINis G m )(K) x (F HINis G m ) 1 (k(x)) F(k(x)) where the first map is given by (4.6) with G = G m, and the last isomorphism is given by Theorem 4.7. The image of (a, b) F(K) K by the local symbol is denoted by x (a, b) F(k(x)) Proposition (cf. [5, Prop ]). Let F, G HI Nis, and consider the morphism induced by (4.5) F HINis G 1 Φ (F HINis G) 1. Let X, K, x be as in 4.1. Then the diagram F(O X,x ) G(K) (F HINis G)(K) i x x F(k(x)) G 1 (k(x)) x (F HINis G 1 )(k(x)) Φ (F HINis G) 1 (k(x))

15 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 15 commutes, where i x is induced by the reduction map O X,x k(x). In other words, with Notation 4.9 we have the identity (4.7) x (a b) = Φ(i xa x b) for (a, b) F(O X,x ) G(K) Corollary. Let F HI Nis ; let X, K, x be as in 4.1 and let (a, f) F(K) K. a) Suppose that there is ã F(O X,x ) whose image in F(K) is a. 3 Then we have x (a, f) = v x (f)a(x) where a(x) is the image of ã in F(k(x)). b) Suppose that v x (f 1) > 0. Then x (a, f) = 0. Proof. a) Since (4.3) is given by v x when G = G m, this follows from Proposition 4.10 (applied with G = G m ) and Theorem 4.7. b) This follows again from Proposition 4.10, after switching the rôles of F and G Proposition. Let G be a semi-abelian variety. The local symbol on G defined in Notation 4.9 agrees with Somekawa s local symbol [24, (1.1)] (generalising the Rosenlicht-Serre local symbol) on G. Proof. Up to base-changing from k to k (see Lemma 4.2 a)), we may assume k algebraically closed. By [23, Ch. III, Prop. 1], it suffices to show that the local symbol in Notation 4.9 satisfies the properties in [23, Ch. III, Def. 2] which characterize the Rosenlicht-Serre local symbol. In this definition, Condition i) is obvious, Condition ii) is Corollary 4.11 b), Condition iii) is Corollary 4.11 a) and Condition iv) is Proposition K-groups of Somekawa type 5.1. Definition. Let F 1,..., F n HI Nis. a) A relation datum of Somekawa type for F 1,..., F n is a collection (C, h, (g i ) i=1,...,n ) of the following objects: (i) a smooth proper connected curve C over k, (ii) h k(c), and (iii) g i F i (k(c)) for each i {1,..., n}; which satisfies the condition (5.1) for any c C, there is i(c) such that c R i for all i i(c), where R i := {c C g i Im(F i (O C,c ) F i (k(c)))}. b) We define the K-group of Somekawa type K(k; F 1,..., F n ) to be the 3 By [31, Cor. 4.19], ã is unique.

16 16 BRUNO KAHN AND TAKAO YAMAZAKI M M quotient of (F 1... Fn )(k) by its subgroup generated by elements of the form (5.2) Tr k(c)/k (g 1 (c) c (g i(c), h) g n (c)) c C where (C, h, (g i ) i=1,...,n ) runs through all relation data of Somekawa type Remark. In view of Proposition 4.12, our group K(k; F 1,... F n ) coincides with the Milnor K-group defined in [24] when F 1,..., F n are semi-abelian varieties over k. 4 (Note that Somekawa works with all finite extensions but over an arbitrary field; we work with finite separable extensions but are assuming k perfect.) 5.3. Theorem. Let F 1,..., F n HI Nis. The homomorphism (2.10) factors through K(k; F 1,..., F n ). Consequently, we get a surjective homomorphism (1.1). Proof. Put F := F 1 HINis HINis F n. Let (C, h, (g i ) i=1,...,n ) be a relation datum of Somekawa type. We must show that the element (5.2) goes to 0 in F(k) via (2.10). Consider the element g = g 1 g n F(K). It follows from (4.7) that, for any c C, we have g 1 (c) c (g i(c), h) g n (c) = g 1 (c) c (g i(c) {h}) g n (c) = c (g {h}). The claim now follows from Proposition K-groups of geometric type 6.1. Definition. Let F 1,..., F n PST. a) A relation datum of geometric type for F 1,..., F n is a collection (C, f, (g i ) i=1,...,n ) of the following objects: (i) a smooth projective connected curve C over k, (ii) a surjective morphism f : C P 1, (iii) g i F i (C ) for each i {1,..., n}, where C = f 1 (P 1 \ {1}). b) We define the K-group of geometric type K (k; F 1,..., F n ) to be the M M quotient of (F 1... Fn )(k) by its subgroup generated by elements of the form (6.1) v c (f) Tr k(c)/k (g 1 (c) g n (c)) c C 4 As was observed by W. Raskind, the signs appearing in [24, (1.2.2)] should not be there (cf. [20, p. 10, footnote]).

17 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 17 where (C, f, (g i ) i=1,...,n ) runs through all relation data of geometric type. (Here we used the notation g i (c) = ι c(g i ) F(k(c)), where ι c : c = Spec k(c) C is the closed immersion.) The rest of this section is devoted to a proof of the following theorem: 6.2. Theorem. Let F 1,..., F n HI Nis. The homomorphism (2.10) induces an isomorphism (6.2) K (k; F 1,..., F n ) Hom DM eff (Z, F 1 [0] F n [0]) Remark. By combining Theorems 5.3 and 6.2, we obtain a surjective homomorphism (6.3) K(k; F 1,..., F n ) K (k; F 1,..., F n ) for any F 1,..., F n HI Nis. The existence of this surjection is not clear from the definition Let F PST. Suppose that we are given the following data: (i) a smooth projective connected curve C over k, (ii) a surjective morphism f : C P 1, (iii) a map α : L(C ) F in PST, where C = f 1 ( ) and = P 1 \ {1}( A 1 ). To such a triple (C, f, α), we associate an element (6.4) α(div(f)) F(k), where we regard div(f) as an element of Z 0 (C ) = c(spec k, C ) = L(C )(k). One can rewrite the element (6.4) as follows. The map α : L(C ) F can be regarded as a section α F(C ). To each closed point c C, we write α(c) for the image of α in F(k(c)) by the map induced by c = Spec k(c) C. Now (6.4) is rewritten as (6.5) v c (f) Tr k(c)/k α(c). c C 6.5. Proposition. Let F PST. We define F(k) rat to be the subgroup of F(k) generated by elements (6.4) for all triples (C, f, α) as in 6.4. Then we have h 0 (F)(k) = F(k)/F(k) rat. Proof. By definition we have (6.6) h 0 (F)(k) = Coker(i 0 i : F( ) F(k)), where = P 1 \ {1}( A 1 ) and i a is the pull-back by the inclusion i a : {a} for a {0, }.

18 18 BRUNO KAHN AND TAKAO YAMAZAKI Suppose we are given a triple (C, f, α) as in 6.4, and set C = f 1 ( ). The graph γ f C of f C defines an element of c(, C ) = L(C )( ). In the commutative diagram L(C )( ) i 0 i L(C )(k) α α F( ) i 0 i F(k), the image of γ f C in L(C )(k) = Z 0 (C ) is div(f), which shows the vanishing of α(div(f)) in h 0 (F)(k). Conversely, given α F( ), (6.4) for the triple (P 1, id P 1, α) coincides with (i 0 i )(α). This completes the proof Lemma. Let F 1,..., F n PST. Put F := F 1 PST PST F n. Let (C, f, α) be a triple considered in 6.4. Then α F(C ) is the sum of a finite number of elements of the form (6.7) Tr h (g 1 g n ), where D is a smooth projective curve, h : D C is a surjective morphism, g i F i (h 1 (C )) for i = 1,..., n, and Tr h : F(h 1 (C )) F(C ) is the transfer with respect to h h 1 (C ). Proof. By the facts recalled in 2.7, we are reduced to the case F i = L(X i ) where X i is a smooth variety over k for each i = 1,..., n. Then we have F = L(X) with X = X 1 X n. Let Z be an integral closed subscheme of C X which is finite and surjective over C. It suffices to show that Z c(c, X) = L(X)(C ) can be written as (6.7). Let q : D Z be the normalization, and let h : D C be the composition D Z C, so that h is a finite surjective morphism. For i = 1,..., n, we define g i c(d, X i ) = L(X i )(D ) to be the graph of D X X i. If we set g = g 1 g n L(X)(D ), then by definition we have Tr h (g) = Z in L(X)(C ). The assertion is proved Now it follows from Definition 6.1 b), Proposition 6.5, Lemma 6.6 and (6.5) that (2.9) and (2.6) induce an isomorphism K (k; F 1,..., F n ) h 0 (F 1 PST PST F n )(k) for any F 1,..., F n PST. If F 1,..., F n HI Nis, the right hand side is canonically isomorphic to Hom DM eff (Z, F 1 [0] F n [0]) by (3.3) and (2.8). This completes the proof of Theorem 6.2.

19 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY Milnor K-theory 7.1. Our aim is to show that the map (6.3) is bijective. The first step is the special case of the multiplicative groups. Recall that, if C is a smooth projective connected curve over k, then the composition Kr+1(k(C)) M c Kr M c C (k(c)) N k(c)/k Kr M (k) is the zero map by Weil reciprocity [3, Ch. I, (5.4)]. Here, for each closed point c C, we write c : K M r+1(k(c)) K M r (k(c)) and N k(c)/k : K M r (k(c)) K M r (k) for the tame symbol and the norm map. The tame symbol satisfies (and is characterized by) the property c ({a 1,..., a n, f}) = v c (f){a 1 (c),..., a n (c)} for any a 1,..., a n O C,c and f k(c) Proposition. When F 1 = = F n = G m, the map (6.3) is bijective. Proof. It suffices to show that relations (6.1) vanish in K(k; G m,..., G m ). Because of Somekawa s isomorphism [24, Theorem 1.4] (7.1) K(k; G m,..., G m ) K M n (k) given by {x 1,..., x n } E/k N E/k ({x 1,..., x n }), it suffices to show this vanishing in the usual Milnor K-group Kn M (k), which follows from Weil reciprocity recalled above. The following lemmas appear to be crucial in the proof of the main theorem Lemma. Let C be a smooth projective connected curve over k, and let Z = {p 1,..., p s } be a finite set of closed points of C. If k is infinite, then we have K M 2 (k(c)) = {k(c), O C,Z }. Proof. Let p i be the maximal ideal of A = O C,Z corresponding to p i. Since A is a semi-local PID, we can choose generators π 1,..., π s of p 1,..., p s. Since k is infinite, we can change π i into µ i π i for suitable µ 1,..., µ s k to achieve π i + π j 0 (mod p k ) for i, j, k all distinct (indeed, the set of bad (µ 1,..., µ s ) is contained in a finite union of hyperplanes in k s ). It follows that π i + π j A for all i j. By the relation {f, f} = 0 (f k(c) ), we have K M 2 (k(c)) = {A, A } + i<j {π i, π j }. Now the identity {π i, π j } = {π i, π j } {π j, π j } = {π i /π j, π j } = {π i /π j, π j } + {π i /π j, 1 + (π i /π j )} = {π i /π j, π i + π j }

20 20 BRUNO KAHN AND TAKAO YAMAZAKI proves the lemma Lemma. Let C be a smooth projective connected curve over k, Z C a proper closed subset, and r > 0. If k is an infinite field, then K M r+1k(c) is generated by elements of the form {a 1,..., a r+1 } where the a i k(c) satisfy Supp(div(a i )) Z = for all 1 i r and Supp(div(a i )) Supp(div(a j )) = for all 1 i < j r. Proof. We proceed by induction on r. The assertion follows from Lemma 7.3 when r = 1. Suppose r > 1. Take a 1,..., a r+1 k(c). By induction, there exist b m,i k(c) such that Supp(div(b m,i )) Z = for all i < r and m, Supp(div(b m,i )) Supp(div(b m,j )) = for all i < j < r and m, and {a 1,..., a r } = {b m,1,..., b m,r } m holds in Kr M k(c). For each m, Lemma 7.3 shows that there exist c m,i, d m,i k(c) such that ( ) and that holds in K M 2 Supp(div(c m,i )) Z r1 j=1 {b m,r, a r+1 } = i k(c). Then we have Supp(div(b m,j )) {c m,i, d m,i } = {a 1,..., a r+1 } = m,i {b m,1,..., b m,r1, c m,i, d m,i } in K M r+1k(c), and we are done. 8. K-groups of Milnor type We now generalize the notion of Milnor K-groups to arbitrary homotopy invariant Nisnevich sheaves with transfers, although we shall seriously use this generalization only for special, representable, sheaves Let F HI Nis. We shall call a homomorphism G m F a cocharacter of F. (By Proposition 4.3, the group Hom HINis (G m, F) is canonically isomorphic to F 1 (k).) Let F 1,..., F n HI Nis. Denote by St(k; F 1,..., F n ) the subgroup of (F 1 PST PST F n )(k) generated by the elements (8.1) a 1 χ i (a) χ j (1 a) a n where χ i : G m F i, χ j : G m F j are 2 cocharacters with i < j, a k \ {1}, and a m F m (k) (m i, j).

21 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY Definition. For F 1,..., F n HI Nis, we write K(k; F 1,..., F n ) for the quotient of (F 1 M... M Fn )(k) by the subgroup generated by Tr E/k St(E; F 1,..., F n ), where E runs through all finite extensions of k. This is the K-group of Milnor type associated to F 1,..., F n Let F 1,..., F n HI Nis. We have a canonical isomorphism (F 1 M M Fn )(k) (Z M M Z M F 1 M M Fn )(k) because Z is the unit object for the tensor structure of Mackey functors. Since there is no non-trivial cocharacter of Z, it induces an isomorphism K(k; F 1,..., F n ) K(k; Z,..., Z, F r+1,..., F n ) The assignment k K(k; F 1,..., F n ) inherits the structure of a cohomological Mackey functor, which is natural in (F 1,..., F n ). In particular, the choice of elements f i F i (k) = Hom HINis (Z, F i ) for i = 1,..., r induces a homomorphism (8.2) K(k; Fr+1,..., F n ) K(k; Z,..., Z, F r+1,..., F n ) K(k; F 1,..., F n ) Lemma. Let F 1,..., F n HI Nis. The image of St(k; F 1,..., F n ) vanishes in K(k; F 1,..., F n ). Consequently, we have a surjective homomorphism K(k; F 1,..., F n ) K(k; F 1,..., F n ) and a composite surjective homomorphism (8.3) K(k; F1,..., F n ) K (k; F 1,..., F n ) Proof. This is a straightforward generalization of Somekawa s proof of [24, Th. 1.4]. We need to show the image of (8.1) vanishes in K(k; F 1,..., F n ). By functoriality, we may assume that F i = F j = G m for some i < j and χ i, χ j are the identity cocharacters. Given a m F m (k) (m i, j) and a k \ {1}, we put a i = 1 at 1, a j = 1 t G m (k(p 1 )) = k(t). Then (P 1, t, (a 1,..., a n )) is a relation datum of Somekawa type. Note that a i G m (P 1 \ {0, a}) and a j G m (P 1 \ {1, }). Direct computation shows a j (0) = (a j, t) = 1 (a j, t) = 1, a (a i, t) = a 1, a j (a) = 1 a. Thus this relation datum yields the vanishing of {a 1,..., a i1, a 1, a i+1,..., a j1, 1 a, a j+1,..., a n } k/k, which is the negative of the image of (8.1) in K(k; F 1,..., F n ) Lemma. Let F 1,..., F n HI Nis and let G G be an epimorphism in HI Nis. If (8.3) is bijective for (G, F 1,..., F n ), it is bijective for (G, F 1,..., F n ).

22 22 BRUNO KAHN AND TAKAO YAMAZAKI Proof. Let G = Ker(G G ). For H {G, G, G }, we put K H = K(k; H, F 1,..., F n ), K H = K (k; H, F 1,..., F n ). In the commutative diagram K G K G f K G 0 K G K G K G 0 the upper row is a complex and f is surjective. The lower row is exact because of Theorem 6.2 and Lemma 3.3, and all vertical arrows are surjective. The assertion now follows from a diagram chase Let E/k be an étale k-algebra and let F HI Nis. We define the Weil restriction R E/k F HI Nis of F by the formula R E/k F(U) = F(U k E) for all smooth variety U. If F is a semi-abelian variety, then R E/k F is the (usual) Weil restriction of F Lemma. Let E/k be a finite extension. Let F 1,..., F n1 HI Nis, and let F n be a Nisnevich sheaf with transfers over E. We have canonical isomorphisms K(k; F 1,..., F n1, R E/k F n ) K(E; F 1,..., F n ), K (k; F 1,..., F n1, R E/k F n ) K (E; F 1,..., F n ), K(k; F 1,..., F n1, R E/k F n ) K(E; F 1,..., F n ). Proof. The first isomorphism was constructed in [26, Lemma 4] when F 1,..., F n are semi-abelian varieties. The same construction works for arbitrary F 1,..., F n and also for K and K If F 1 = = F n = G m, (8.3) is bijective by Proposition 7.2. This is false in general, e.g. if all the F i are proper (Definition 10.1) and n > 1. However, we have: Proposition. a) Let F 1 = F 1 F 1. Then the natural map K(k; F 1,..., F n ) K(k; F 1,..., F n ) K(k; F 1,..., F n ) is bijective. b) Let T 1,..., T n be tori. Assume that, for each i, there exists an exact sequence of tori 0 P 1 i P 0 i T i 0 where P 0 i and P 1 i are invertible tori (i.e. direct summands of permutation tori). Then (8.3) is bijective for F i = T i.

23 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 23 Proof. a) This is formal, as K(k; F 1,..., F n ) is a quotient of the mul- M M tiadditive multifunctor (F 1... Fn )(k) (see 8.4). b) Note that, by Hilbert s theorem 90, the sequences 0 Pi 1 T i 0 are exact in HI Nis. Lemma 8.6 reduces us to the case P 0 i where all T i are permutation tori. Then Lemma 8.8 reduces us to the case where all T i are split tori. Finally, we reduce to F 1 = = F n = G m by a) Question. Is proposition 8.10 b) true for general tori? Let T 1,..., T n be as in Proposition 8.10 b). Let C/k be a smooth projective connected curve with function field K and v C a closed point. Put F = T 1 HINis HINis T n. By Theorem 4.7 and (4.3), we get a residue map v : F HINis G m (K) F(k(v)). From Lemma 3.5, Theorem 6.2 and Proposition 8.10 b), this can be reinterpreted as (8.4) v : K(K; T 1,..., T n, G m ) K(k(v); T 1,..., T n ). As v varies, these maps satisfy the reciprocity law of Proposition 4.6 and the compatibility of Lemma Reduction to the representable case Following [32, p. 207], we write h Nis 0 (X) := h Nis 0 (L(X)) for a smooth variety X over k Proposition. The following statements are equivalent: a) The homomorphism (6.3) is bijective for any F 1,..., F n HI Nis. b) Let F 1 = = F n = h Nis 0 (C ) for a smooth connected curve C /k. Then (6.3) is bijective. c) Let C be a smooth projective connected curve over k, and let f : C P 1 be a surjective morphism. Let C = f 1 (P 1 \ {1}). Let ι : L(C ) h Nis 0 (C ) =: A be the canonical surjection, which we regard as an element of A(C ). These data define a relation datum of geometric type (C, f, (ι,..., ι)) for F 1 = = F n = A, and its associated element (6.1) is (9.1) v c (f) Tr k(c)/k (ι(c) ι(c)) A M... M A(k). c C Then the image of (9.1) in K(k; A,..., A) vanishes. Proof. Only the implication c) a) requires a proof. Let (C, f, (g i )) be a relation datum of geometric type for F 1,..., F n. We need to show the vanishing of (9.2) v c (f){g 1 (c),..., g n (c)} k(c)/k in K(k; F 1,..., F n ). c C

24 24 BRUNO KAHN AND TAKAO YAMAZAKI For each i = 1,..., n, the section g i : L(C ) F i factors through a morphism ϕ i : A F i since F i is homotopy invariant. The homomorphism K(k; A,..., A) K(k; F 1,..., F n ) defined by (ϕ 1,..., ϕ n ) sends the image of (9.1) in K(k; A,..., A) to (9.2). Hence (9.2) vanishes by the assumption c). 10. Proper sheaves Definition. Let F be a Nisnevich sheaf with transfers. We call F proper if, for any smooth curve C over k and any closed point c C, the induced map F(O C,c ) F(k(C)) is surjective. We say that F is universally proper if the above condition holds when replacing k by any finitely generated extension K/k, and C by any regular K-curve Example. (1) A semi-abelian variety G over k is proper in the sense of Def if and only if G is an abelian variety. (2) Recall from [10] that F HI Nis is called birational if F(X) F(U) is bijective for any smooth k-variety X and any open dense subset U X. A birational sheaf F HI Nis is by definition proper. Examples of birational sheaves will be given in Lemma 11.2 b) below. In particular, if C is a smooth proper curve, then h Nis 0 (C) is proper. In fact: Lemma. Let F HI Nis. Then a) F is proper if and only if F(C) F(k(C)) for any smooth k-curve C. b) F is universally proper if and only if it is birational (see Example 10.2 (2)). Proof. Let us prove b), as the proof of a) is a subset of it. It is obvious from the definitions that birational sheaves are universally proper. Conversely, assume F to be universally proper. Let X be a smooth k-variety. By [31, Cor. 4.19], the map F(X) F(U) is injective for any dense open subset of X. Let x X (1) and let p : X A d1 be a dominant rational map defined at x, where d = dim X. (We may find such a p thanks to Noether s normalization theorem.) Applying the hypothesis to the generic fibre of p, we find that F(O X,x ) F(k(X)) is surjective. Since this is true for all points x X (1), we get the surjectivity of F(X) F(k(X)) from Voevodsky s Gersten resolution [31, Th. 4.37]. The following proposition is not necessary for the proof of the main theorem, but its proof is much simpler than the general case.

25 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY Proposition. Let F 1,..., F n HI Nis. Assume that F 1,..., F n1 are proper. Then the homomorphism (6.3) is bijective. Proof. Suppose that (C, f, (g i ) i=1,...,n ) is a relation datum of geometric type for (F 1,..., F n ). It suffices to show the vanishing of the image of M (10.1) v c (f) Tr k(c)/k (g 1 (c) g n (c)) (F 1... M F n )(k) c C in K(k; F 1,..., F n ). Let ḡ i be the image of g i in F(k(C)). By assumption we have ḡ i Im(F i (O C,c ) F i (k(c))) for all c C and i = 1,..., n 1. Hence (C, f, (ḡ i ) i=1,...,n ) is a relation datum of Somekawa type (with i(c) = n for all c C). By Corollary 4.11, the element (10.1) coincides with c C Tr k(c)/k (g 1 (c) g n1 (c) c (g n, f)), hence its image in K(k; F 1,..., F n ) vanishes by Definition Main theorem Definition. Let F HI Nis. We say that F is curve-like if there exists an exact sequence in HI Nis (11.1) 0 T F F 0 where F is proper (Definition 10.1) and T sits in an exact sequence in HI Nis (11.2) 0 R E1 /kg m R E2 /kg m T 0 where E 1 and E 2 are étale k-algebras. 5 This terminology is justified by the following lemma: Lemma. a) If C is a smooth curve over k, then h Nis 0 (C) is the Nisnevich sheaf associated to the presheaf of relative Picard groups U Pic( C U, D U) where C is the smooth projective completion of C, D = C \ C and U runs through smooth k-schemes. b) If X is a smooth projective variety over k, then, for any smooth variety U over k, we have (11.3) h Nis 0 (X)(U) = CH 0 (X k(u) ), 5 It then follows from Hilbert s theorem 90 applied to R E1/kG m that T = Tét, hence that T agrees with the cokernel of R E1/kG m R E2/kG m as tori; we shall not need this remark in the sequel.

26 26 BRUNO KAHN AND TAKAO YAMAZAKI where k(u) denotes the total ring of fractions of U. h Nis 0 (X) is birational. c) For any smooth curve C, h Nis 0 (C) is curve-like. In particular, Proof. a) and b) are proven in [29, Th. 3.1] and in [7, Th. 2.2] respectively. We prove c). This follows from b) if C is projective over k. We assume C is affine. With the notation of a), we have the Gysin exact triangle [32, Prop ] M(D)(1)[1] M(C) M( C) +1. By [32, Th ], we have h Nis i (C) = 0 for all i 0 and h Nis 1 ( C) = R E/k G m, where E = H 0 ( C, O C). Hence we get an exact sequence which proves c). 0 R E/k G m R D/k G m h Nis 0 (C) h Nis 0 ( C) 0, Remark. Let F HI Nis be curve-like. The sheaves T and F in (11.1) are uniquely determined by F up to unique isomorphism. Indeed, this amounts to showing that any morphism T F is trivial. This is reduced to the case T = R E/k G m as in (11.2), and further to T = G m by adjunction as in Lemma 8.8. Then we have Hom HINis (G m, F) F 1 (k) = 0 by definition (see (4.1) and Definition 10.1). We call T and F the toric and proper part of F respectively (see footnote 5) Lemma. a) Let F HI Nis be curve-like with toric part T, and let C be a smooth proper connected k-curve. Let Z be a proper closed subset of C, A = O C,Z and K = k(c). Then the sequence 0 T (A) f T (K) F(A) g F(K) 0 is exact, where f and g are given by f(a) = (a, a) and g(b, c) = b c, under the identification T (A) F(A) F(K) and T (A) T (K) F(K). b) Let F 1,..., F n HI Nis be curve-like with toric parts T 1,..., T n, and let C, Z, A, K be as in a). Then the group F 1 (K) F n (K) has the following presentation: Generators: for each subset I {1,..., n}, elements [I; f 1,..., f n ] with f i F i (A) if i I and f i T i (K) if i / I. Relations: Multilinearity: [I; f 1,..., f i + f i,..., f n ] = [I; f 1,..., f i,..., f n ] + [I; f 1,..., f i,..., f n ].

27 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY 27 Let I {1,..., n} and let i 0 / I. Let [I; f 1,..., f n ] be a generator. Suppose that f i0 T i0 (A). Then [I; f 1,..., f n ] = [I {i 0 }; f 1,..., f n ]. Proof. Consider the commutative diagram T (A) F(A) F(A) 0 a b 0 T (K) F(K) F(K) 0 c 0 T 1 (k i ) F 1 (k i ) F1 (k i ) Here the k i s run through the residue fields of points of Z and the (exact) vertical sequences are those from [31, Th. 4.37]. Since 0 T F F 0 is an exact sequence of Nisnevich sheaves and any field has Nisnevich cohomological dimension zero, all horizontal sequences are left exact and the middle one is also exact at F(K). (See 4.1 for the exactness of the bottom row.) Since F is proper, we have F 1 (k i ) = 0 (see the end of Remark 11.3); it follows that c is an isomorphism and that the upper horizontal sequence is also exact at F(A). Now a) follows from a diagram chase and b) follows from a) Remark. A shorter but more delicate proof is that the maps a, b, c have compatible retractions. Since C is a curve, this may be deduced from the proof of [32, Lemma 4.5] (see also [32, Cor. 4.18]) Proposition. Let C/k be a smooth proper connected curve, and let v C, K = k(c). Then there exists a unique law associating to a system (F 1,..., F n ) of n curve-like sheaves a homomorphism such that v : F 1 (K) F n (K) K K(k(v); F 1,..., F n )

28 28 BRUNO KAHN AND TAKAO YAMAZAKI (i) If σ is a permutation of {1,..., n}, the diagram F 1 (K) F n (K) K σ v K(k(v); F1,..., F n ) σ F σ(1) (K) F σ(n) (K) K (ii) v K(k(v); Fσ(1),..., F σ(n) ) commutes. If [I, f 1,..., f n ] is a generator of F 1 (K) F n (K) as in Lemma 11.4 b) for some Z containing v, with I = {1,..., i}, then v (f 1 f n f) = {f 1 (v),... f i (v), v ({f i+1,..., f n, f} K/K )} k(v)/k where v ({f i+1,..., f n, f} K/K ) is the residue (8.4). Proof. By Lemma 11.4 b), it suffice to check that v agrees on relations. Up to permutation, we may assume I = {1,..., i} and i 0 = i + 1. The claim then follows from Proposition Lemma. a) Keep the notation of Proposition Let L/K be a finite extension; write D for the smooth projective model of L and h : D C for the corresponding morphism. Let Z = h 1 (v). Write F n+1 = G m. Then, for any i {1,..., n + 1}, the diagram ( w) F 1 (L) F n+1 (L) K(k(w); F 1,..., F n ) u w Z F 1 (K) F i (L) F n+1 (K) (Tr k(w)/k(v) ) d F 1 (K) F n+1 (K) v K(k(v); F1,..., F n ) commutes, where u is given componentwise by functoriality for j i and by the identity for j = i, and d is given componentwise by the identity for j i and by Tr L/K for j = i. b) The homomorphisms v induce residue maps ) M M M v : (F 1... Fn Gm (K) K(k(v); F 1,..., F n ). which verify the compatibility of Lemma 4.2 b). Proof. a) For clarity, we distinguish two cases: i < n + 1 and i = n + 1. In the former case, up to permutation we may assume i = n. It is enough to check commutativity on generators in the style of Lemma

29 VOEVODSKY S MOTIVES AND WEIL RECIPROCITY b). Let T l denote the toric part of F l. In view of Lemma 11.4 a) and Proposition 11.6 (i), it suffices to check the commutativity for x = f 1 f n f when one of the following two conditions is satisfied: (i) for some j {0,..., n 1}, f l F l (O C,v ) (1 l j), f l T l (K) (j + 1 l n 1), f n T n (L) and f K. (ii) for some j {0,..., n 1}, f l F l (O C,v ) (1 l j), f l T l (K) (j + 1 l n 1), f n F n (O D,Z ) and f K. Let w Z. If (i) holds, we have w (u(x)) = {f 1 (w),..., f j (w), w ({f j+1,..., f n, f} L/L )} k(w)/k(w) and v (d(x)) = {f 1 (v),..., f j (v), v ({f j+1,..., Tr L/K (f n ), f} K/K )} k(v)/k(v). Observe that the restriction of f l (v) to k(w) is f l (w) for every w Z and l = 1,..., j. Since the residue maps ( w ) (8.4) verify the compatibility of Lemma 4.2, the commutativity for x follows. (Recall that Tr k(w)/k(v) ({a 1,..., a n } k(w)/k(w) ) = {a 1,..., a n } k(w)/k(v).) If (ii) holds, we have w (u(x)) = {f 1 (w),..., f j (w), w ({f j+1,..., f n1, f} L/L ), f n (w)} k(w)/k(w) and v (d(x)) = {f 1 (v),..., f j (v), v ({f j+1,... f n1, f} K/K ), Tr L/K (f n )(v)} k(v)/k(v). In addition to the observation mentioned in (i), we remark that the restriction of v ({f j+1,..., f n1, f} K/K ) to k(w) is w ({f j+1,..., f n1, f} L/L ) for every w Z. The commutativity for x follows from Lemma 4.2 b) applied to F n. If i = n + 1 the check is similar, the projection formula working on the last variable. Now b) follows from a) and the definition of M recalled in Lemma. The homomorphisms v of Lemma 11.7 induce residue maps v : K(K; F 1,..., F n, G m ) K(k(v); F 1,..., F n ). which verify the compatibility of Lemma 4.2 b). Proof. Set F n+1 = G m. Let i < j be two elements of {1,..., n + 1} and let χ i : G m F i, χ j : G m F j be two cocharacters. Let f K {1}. We must show that v vanishes on x = f 1 χ i (f) χ j (1 f) f n+1