Detection by regular schemes in degree two

Abstract

Abstract. 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 Poincaré 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.

A Bound for the Index of a Quadratic Form After Scalar Extension to the Function Field of a Quadric

“…However, a generalization of this result similar to the one just given for quasilinear forms is not yet known in that setting. 5…”

“…Here the notation v 2 (n) stands for the 2-adic order of the integer n. It is worth remarking that while Hoffmann's conjecture is essentially wide open for non-quasilinear forms in characteristic 2, non-trivial partial results have been obtained by Hoffmann-Laghribi [9] and also by Haution as a by-product of his efforts to develop the geometric machinery which is currently absent from the characteristic-2 setting (see [4,5]). Theorems 1.1 and 1.2 represent important landmarks for the theory of quadratic forms.…”

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.…”

Motivic Steenrod operations in characteristic $p$

“…Hence, over the base field k, H * , * (BS p , F p ) ∼ = H * , * (k, F p ) and so one cannot carry out Voevodsky's construction. It has also been an open problem to just define Steenrod operations on the mod p Chow groups of smooth schemes over a field of characteristic p. Haution made progress on this problem by constructing the first p − 1 homological Steenrod operations on Chow groups mod p and p-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 papers where Steenrod squares (or weak forms of Steenrod squares) on mod 2 Chow groups are used, the nth Steenrod square on mod 2 Chow groups corresponds to the 2nth Steenrod square on mod 2 motivic cohomology since the Bockstein homomorphism is 0 on mod 2 Chow groups.…”