Chow categories

This paper arose from an attempt to solve some questions which were posed at the seminar of A. N. Parchin when Deligne’s program ([D]) was reviewed. These problems are related to hypothetical functorial and metrical versions of the Riemann-Roch-Hirzebruch theorem. One of the problems posed by Deligne is, for instance, the following construction: Let a proper morphism of schemes X → S of relative dimension n and a polynomial P(c i (E j )) of absolute degree n + 1 (where deg(c i) = i) in the Chern classes of vector bundles E 1, ... , E k be given. Construct a functor which to the vector bundles E j . on X associates a line bundle on S 1 $${I_{x/s}}P\left( {{c_i}\left( {{E_j}} \right)} \right)$$ which is an ‘incarnation’ of ∫x/s P(c i (E j )) ∈ CH 1 (S). The functor (1) should be equipped with some natural transformations which correspond to well-known equalities between Chern classes (cf. [D, 2.1]). Further steps in Deligne’s program. are to equip the line bundles (1) with metrics, to prove a functorial version of the Riemann-Roch-Hirzebruch formula which provides an isomorphism between the determinant det(R p *(F)) of the cohomology of a vector bundle F and a certain line bundle of type (1); and (finally) to compare the metric on the right side of the Riemann-Roch isomorphism and the Quillen metric on the determinant of the cohomology.