Relative $K_0$ and relative cycle class map

Abstract

A → B be an exact functor between small exact categories. We study the zeroth homotopy group K 0 (F ) of the homotopy fiber of the map K(A) → K(B) between K-theory spectra. Under the assumption that F is a cofinal and that B is split exact, we give an explicit description of K 0 (F ) in terms of the triangulated functor D b (A) → D b (B) between the derived categories.We apply it to the pair (X, D) of a scheme X and an affine closed subscheme D of X, and get a description of the relative K 0 -group K 0 (X, D) in terms of perfect complexes; it is generated by pairs of two perfect complexes of X together with quasi-isomorphisms along D. This description makes it possible to assign a cycle class in K 0 (X, D) to a cycle on X not meeting D in an intuitive way. When X is a separated regular scheme of finite type over a field and D is an affine effective Cartier divisor on X, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus CH * (X|D) defined by Binda-Saito to a suitable subquotient of K 0 (X, D).