Complete intersection K-theory and Chern classes

Abstract

The purpose of this paper is to investigate the theory of complete intersection in noetherian commutative rings from the K-Theory point of view. (By complete intersection theory, we mean questions like when/whether an ideal is the image of a projective module of appropriate rank.) The paper has two parts. In part one (Section 1-5), we deal with the relationship between complete intersection and K-theory. The Part two (Section 6-8) is, essentially, devoted to construction projective modules with certain cycles as the total Chern class. Here Chern classes will take values in the Associated graded ring of the Grothedieck γ − filtration and as well in the Chow group in the smooth case. In this paper, all our rings are commutative and schemes are noetherian. To avoid unnecessary complications, we shall assume that all our schemes are connected. For a noetherian schemeX, K0(X) will denote the Grothendieck group of locally free sheaves of finite rank over X. Whenever it make sense, for a coherent sheaf M over X, [M ] will denote the class of M in K0(X). We shall mostly be concerned with X = SpecA, where A is a noetherian commutative ring and in this case we shall also use the notation K0(A) for K0(X).