Detection by regular schemes in degree two

TL;DR

This paper extends resolution of singularities to codimension two for certain schemes and uses this to characterize algebraic cycle operations of degree less than two, leading to multiple applications in algebraic geometry.

Contribution

It introduces a new resolution of singularities in codimension two for reduced quasi-excellent schemes and applies this to characterize algebraic cycle operations.

Findings

Resolution of singularities in codimension two for quasi-excellent schemes

Characterization of degree less than two algebraic cycle operations

Applications to Chern characters, Steenrod squares, and quadratic forms

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 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.