Localization for flat modules in algebraic K-theory

Abstract

There are two localization theorems in algebraic K-theory which yield long esact sequences for the K-groups of certain triples of exact categories: One handles the quotient of an Ahelian category [9], and the other relates a ring and its localization by nonzero divisors. We present here a localization theorem for certain categories of R-modules flat over a base ring d (Theorem 4.8). The problem encountered in [6] was the lack of suitable non-Abelian quotient categories for the filtration by relative codimension of support: We find a suitable quotient category and use techniques of Quillen to obtain the long exact sequence. The proof is longer than the proof in [9] of the localization theorem for Abelian catcgorics, and dots not retain the symmetry between monies and epis. This asymmetry leads to the introduction of relatively torsion-free modules, for they tend to map into or onto enough other modules. Useful tools include a generalization of Waldhauscn’s cofinality theorem and a modification of Quillen’s resolution theorem. The quotient category obtained is not very computable; a slightly larger exact category is more so. The K-groups are the same if cofinality can be established, but we can only do this for codimension <l (Theorem 5.2). This is good enough to show that the first two sheaves in the resolution of .X,(X) constructed in [6j are flasque (Proposition 6.4). If X is smooth over S of relative dimension 2, and S is finite, then we can see that A”(X) :: H’(X, Z,) is g enerated by classes of coherent C’,-modules finite and free over CS , and the relations can be described (Section 9) with no reference to higher K-theory. Recall that when 5’ is a field, -if’(X) is the group of zero cycles on X modulo rational equivalence. When S is not a field, A’(S) may yield information about the infinitesimal str-ucture of the cycle class group, as in [2]; moreover A”(X) is part of a graded ring A’(X) which might harbor Chern classes for vector bundles on the singular variety X. The relations for A2(X) can be found by computing boundary maps from higher K-theory and using the fact that K, of a local ring is generated by symbols. They are those given by exact sequences and those of the form c(y, f, g) q -= 0, 463