We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$. This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field $k$ which contains all $\ell$-power roots of unity.

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

Let X be a smooth projective variety over a finite field . We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds 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 (C), the vanishing of H3nr(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 new light on work started thirty years ago.

In a recent paper, Merkurjev showed that for a smooth proper variety X over a field k, the functor M* ↦ A0(X, M0) from cycle modules to abelian groups is corepresented by a cycle module constructed on the Chow group of 0-cycles of X. We show that if “proper” is relaxed, the result still holds by replacing the Chow group of 0-cycles by the 0-th Suslin homology group of X.

We exhibit a smooth complex affine 5-fold whose Witt group of quadratic forms does not inject into the Witt group of its function field. The dimension 5 is the smallest possible. The example depends on relating Witt groups, mod 2 Chow groups, and Steenrod operations.

AMS 2000 Mathematics subject classification: Primary 19G12. Secondary 19E15

Let k be a local field with residue characteristic p. Let n be a prime-to-p integer. Consider a smooth, irreducible k-curve X. In this article, we study the subgroup of H3(k(X), μn[otimes]2) consisting of classes which are unramified on X. It is known to be isomorphic to H0(XZar, R3π*μn[otimes]2), where π: Xét → XZar is the canonical map. A purely topological description of this group is given, using the Berkovich analytification Xan of X. More precisely, denote by Y a smooth and irreducible Berkovich-analytic k-curve, by Ytop the underlying topological space, by Yét the étale-analytic site and by πan the canonical map Yét → Ytop. Let Δ be the skeleton of Y (it is a closed subset defined by Berkovich which is locally a finite graph). Then we show (th. 42) that H0(Ytop, R3πan*μn[otimes]2) is naturally isomorphic (through sort of a pointwise evaluation of cohomology classes) to the group of harmonic cochains defined on Δ with values in $Bbb Z$/n. Now, if X is a smooth algebraic k-curve the natural map H0(XZar, R3π*μn[otimes]2) → H0(Xantop, R3πan*μn[otimes]2) is shown to be an isomorphism (th. 5.2). It is a new formulation (for the case where X is proper) and a generalization (to open curves) of a result which was due to Kato. Moreover we use here some steps of Kato's proof, but not his whole result. So our method gives, for projective curves, a (partially) new proof of Kato's theorem.