  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.
