Detection by regular schemes in degree two

Using Lipman's results on resolution of two-dimensional singularities, we provide a form of resolution of singularities in codimension two for reduced quasi-excellent schemes. We deduce that operations of degree less than two on algebraic cycles are characterised by their values on classes of regular schemes. We provide several applications of this "detection principle", when the base is an arbitrary regular excellent scheme: integrality of the Chern character in codimension less than three, existence of weak forms of the second and third Steenrod squares, Adem relation for the first Steenrod square, commutativity and Poincar\'e duality for bivariant Chow groups in small degrees. We also provide an application to the possible values of the Witt indices of non-degenerate quadratic forms in characteristic two.Comment: final versio

Duality and the topological filtration

We investigate some relations between the duality and the topological filtration in algebraic K-theory. As a result, we obtain a construction of the first Steenrod square for Chow groups modulo two of varieties over a field of arbitrary characteristic. This improves previously obtained results, in the sense that it is not anymore needed to mod out the image modulo two of torsion integral cycles. Along the way we construct a lifting of the first Steenrod square to algebraic connective K-theory with integral coefficients, and homological Adams operations in this theory. Finally we provide some applications to the Chow groups of quadrics.Comment: To appear in Math. Ann. The numbering of the statements has been modified, in order to be compatible with the published versio

Degree formula for the Euler characteristic

We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill Zainoulline. The arguments used here are considerably simpler, and allow us to remove all restrictions on the characteristic of the base field

Reduced Steenrod operations and resolution of singularities

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.Comment: Final version, to appear in J. K-theor

The stable Adams operations on Hermitian K-theory

We prove that exterior powers of (skew-)symmetric bundles induce a $\lambda$-ring structure on the ring $GW^0(X) \oplus GW^2(X)$, when $X$ is a scheme where $2$ is invertible. Using this structure, we define stable Adams operations on Hermitian $K$-theory. As a byproduct of our methods, we also compute the ternary laws associated to Hermitian $K$-theory

Odd rank vector bundles in eta-periodic motivic homotopy theory

We observe that, in the eta-periodic motivic stable homotopy category, odd rank vector bundles behave to some extent as if they had a nowhere vanishing section. We discuss some consequences concerning SLc-orientations of motivic ring spectra, and the etale classifying spaces of certain algebraic groups. In particular, we compute the classifying spaces of diagonalisable groups in the eta-periodic motivic stable homotopy category

Involutions and Chern numbers of varieties

Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism. We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but not all, are analogues of theorems in algebraic topology obtained by Conner-Floyd and Boardman in the sixties. We include versions of our results concerning the vanishing loci of idempotent global derivations in characteristic two. Our approach to cobordism, following Merkurjev's, is elementary, in the sense that it does not involve resolution of singularities or homotopical methods