arXiv subjects
Publications and source records attributed to Bruno Kahn.
We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".
We start developing a notion of reciprocity sheaves, generalizing Voevodsky's homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope reciprocity sheaves will eventually lead to the definition of a larger triangulated category of motivic nature, encompassing non homotopy invariant phenomena.
Let $A$ be an abelian variety over the function field of a smooth projective curve $C$ over an algebraically closed field $k$. We compute the $l$-adic cohomology groups of $C$ with coefficients in the locally constant sheaf associated to $H^1(\bar A,\mathbf{Q}_l)$ in terms of arithmetico-geometric invariants of $A$. We apply this, when $k$ is the algebraic closure of a finite field, to a motivic computation of the $L$-function of $A$.
This is a considerably expanded version of the "pure" part of our 2002 preprint. We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by "killing" the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch's decomposition of the diagonal. We study the induced Chow-Künneth decompositions in this category, and establish relationships with Rost's cycle modules and the Albanese functor for smooth projective varieties.
We introduce the notion of a separator for a morphism of schemes f:T\to S; in particular, it is universal among morphisms from T to separated S-schemes. A separator is a local isomorphism; this property conveys the intuition of gluing some affine covering more, in order to make the scheme separated. When f is quasi-separated, its separator exists if and only if the schematic closure of the diagonal projects on both factors by flat morphisms of finite type. In particular, f admits a separator if T is Noetherian Dedekind and S=Spec(Z), or if f is étale of finite presentation and S is normal. Any normal scheme of finite type over a Noetherian ring admits an open subset containing all the points of codimension 1, which has a separator. A contrario, we give several examples of morphisms f that do not admit a separator. As an application, we attach to every smooth scheme T over a normal base S a morphism to a separated étale S-scheme of finite presentation, which is universal (a kind of separated alternative for "scheme of connected components of the fibres"). This simultaneously generalizes the classical case where the base is a field, and the case of a smooth and proper morphism (Stein factorisation).
We show that the torsion in the group of indecomposable $(2,1)$-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as $b_2-ρ>0$. We derive a new insight into Roitman's theorem on torsion $0$-cycles over a surface.
Let G be a finite group of exponent m and let k be a field of characteristic prime to m, containing the m-th roots of unity. For any Rost cycle module M over k, we construct exact sequences detecting the unramified elements in Serre's group of invariants of G with values in M in terms of "residue" morphisms associated to pairs (D,g), where D runs through the subgroups of G and g runs through the homomorphisms μ_m \to G whose image centralises D. This allows us to recover results of Bogomolov and Peyre on the unramified cohomology of fields of invariants of G, and to generalise them.
This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.
New version, including a variant of Quillen's proof of the Solomon-Tits theorem.
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.
This text was written 20 years ago, inspired by M. Somekawa's paper on K-groups attached to semi-abelian varieties (K-Theory 4 (1990), 105--119) and before Voevodsky's theory of presheaves with transfers. The reason why it only had a limited circulation will be obvious towards the end. In view of recent developments, I thought it could be useful to make it generally available.
We present a triangulated version of the conjectures of Tate and Beilinson on algebraic cycles over a finite field. This sheds a new light on Lichtenbaum's Weil-etale cohomology.
Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ implies a local-global principle of Brauer--Manin type for the Chow group of zero-cycles on $V$. This sheds a new light on work started thirty years ago. ----- Soit $X$ une variété projective et lisse sur un corps fini $\F$. On s'intéresse au groupe de cohomologie non ramifiée $H^3_\nr(X,\Q/\Z(2))$. Un faisceau de conjectures implique que ce groupe est fini. Pour certaines classes de solides, on a $H^3_\nr(X,\Q/\Z(2))=0$. Savoir si c'est le cas pour tout solide est un problème ouvert. Lorsqu'un solide $X$ est fibré au-dessus d'une courbe $C$, de fibre générique une surface projective et lisse $V$ sur le corps global $\F(C)$, la combinaison de $H^3_\nr(X,\Q/\Z(2))=0$ et de la conjecture de Tate pour $X$ a pour conséquence un principe local-global de type Brauer--Manin pour le groupe de Chow des zéro-cycles de la fibre générique $V$. Ceci éclaire d'un jour nouveau des investigations commencées il y a trente ans.
We relate R-equivalence on tori with Voevodsky's theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.
We observe that the Galois action on local constants associated to Galois representations of a local field yields information on their arithmetic nature, for example provides an upper bound to their order when they are roots of unity. It also yields information on the effect of Adams operations on these constants.
Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (étale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Thélène and Voisin. If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. If k is the separable closure of a finite field and X comes from a variety over a finite field as described above, then D = 0 as soon as H^3(X,Q_l) is entirely of coniveau > 0, but an example of Schoen shows that this condition is not necessary.
We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.
We construct a surjective homomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic complexes.