For an algebraic variety X we introduce generalized first Chern classes, which are defined for coherent sheaves on X with support in codimension p and take values in CH p (X). We use them to provide an explicit formula for the differentialsin the Quillen spectral sequence.

Let ϕ be a rational map P 2 P 2 which preserves the rational volume form dxx ∧ dy y . Sergey Galkin conjectured that in this case ϕ is necessarily birational. We show that such a map preserves the element {x, y} of the second K-group K 2 (k(x, y)) up to multiplication by a constant, and restate this condition explicitly in terms of mutual intersections of the divisors of coordinates of ϕ in a way suitable for computations.