Duality and the topological filtration

Abstract

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

Detection by regular schemes in degree two

“…We prove the relation Sq 1 • Sq 1 = 0, which was apparently out of reach of the techniques used in [Hau13b]. The argument, as well as the 1 as pointed out by the referee, the minimal desingularisation input that we need is the following: if X is a quasi-excellent two-dimensional normal local scheme, then there is a regular scheme X ′ , and a proper birational morphism X ′ → X which induces an isomorphism over the complement of the closed point in X construction of Sq 1 , do not depend on the characteristic of the base field; in characteristic different from two, a different construction of the operation Sq 1 has been given in [Bro03,Voe03], and the relation Sq 1 • Sq 1 = 0 was known, being one of the Adem relations.…”

Integrality of the Chern character in small codimension

“…By construction, we have T i (K p (X)) = 0, hence a consequence of Proposition 8.3 is that T ′ i ∈ E, for all i. On the other hand when p = 2 and j > 1, then the cohomological Steenrod square S 2 j −1 does not belong to E (see [Hau10,Remark 8.4]).…”

“…The classical constructions of the operations modulo p ( [Boi08], [Bro03], [EKM08], [Lev07], [Voe03]) do not work over a field of characteristic p. In [Haub], we constructed a weak form of the operation for p = 2 and n = 1, over any field. We later obtained the full version of this operation in [Hau10]. In the present paper, we construct a weak form of the n-th Steenrod operation modulo p, for p an arbitrary prime and n = 1, · · · , p − 1, over any field.…”

Motivic Steenrod operations in characteristicp

“…It has also been an open problem to just define Steenrod operations on the mod Chow groups of smooth schemes over a field of characteristic . Haution made progress on this problem by constructing the first − 1 homological Steenrod operations on Chow groups mod and -primary torsion over any base field [12], defining the first Steenrod square on mod 2 Chow groups over any base field [13] and constructing weak forms of the second and third Steenrod squares over a field of characteristic 2 [15]. Note that in articles where Steenrod squares (or weak forms of Steenrod squares) on mod 2 Chow groups are used, the th Steenrod square on mod 2 Chow groups corresponds to the 2 th Steenrod square on mod 2 motivic cohomology, since the Bockstein homomorphism is 0 on mod 2 Chow groups.…”