Cartan maps, clean rings, and unique factorization

Abstract

Let R be a Noetherian ring; denote by A’(R) the (exact) category of finitely generated R-modules, and by 9(R) the (exact) category of finitely generated projective R-modules. The inclusion functor 9(R) + A(R) induces the Cartan maps c, : K,,(R) -+ K:(R) (where we denote K,(Y(R)) = K,,(R) and K,(A(R)) = K;(R)). It is a consequence of Quillen’s resolution theorem that the maps c, are isomorphisms if R is a regular ring. One of the main results of this paper is a partial converse to this theorem. In Section 1 we show that if R is a reduced local ring whose residue class field is not the 2-element field IF,, then a necessary condition for c, to be an isomorphism is that R be integrally closed in its total quotient ring. If R is one-dimensional and has finite integral closure, then this result is valid without the restriction on the residue class field. In particular, for such a ring, ci is an isomorphism if and only if R is a discrete valuation ring. In Section 3 we give an example showing that these results are invalid for nonlocal rings. In Section 2, we consider c,,. For local integral domains, questions about c, are necessarily connected with the “higher rank class groups” Wi introduced by Claborn and Fossum in [2], and with the notion of clean ring introduced by Gersten in [4]. To explain these notions, let R be a commutative Noetherian ring, and consider the filtration of d’(R) defined by “codimension of support.” That is, consider the Serre subcategory Xi(R) consisting of those finitely generated R-modules M such that MP = 0 for all primes p with ht p < i. W,(R) is defined to be the image of the homomorphism K,(&/&+ ‘) + K,(Mi‘/&’ ‘). Gersten defines R to be cZean if each of the maps K,(&) + K&M-i) is zero. An understanding of the properties of clean rings may prove helpful in understanding the more general notion of very clean ring. (R is very clean if all the maps K,(J’) --) K,,(Mip’) are zero.) Gersten’s Conjecture is that regular local rings are very clean.