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.

Let k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.

We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.

We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

Let p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.

Let B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B = ℤp[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ≅ H ⋊ ℤp, this set S is Venjakob's canonical Ore set. In particular, our result implies that

is split exact for each n ≥ 0. We also prove the corresponding result for the localisation of ℤp[[G]][] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.

We establish a presentation for K1 of any small exact category P, based on the notion of “mirror image sequence,” originally introduced by Grayson in 1979; as part of the proof, we show that every element of K1(P) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”

For a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.