Kahler differentials and excision for curves

Abstract

In this paper we continue the study of excision for K1 of algebraic curves begun in [4]. If A c B are commutative rings and I is a B-ideal contained in A, then excision holds if the natural map K1(A, I) + K1(B, I) is an isomorphism. By “curve” we mean an algebra A of finite type over a field k, such that if P is a minimal prime ideal of A, then A/P is of Krull dimension 1. This is an abuse of notation, but is shorter than “co-ordinate ring of an affine curve.” All the curves that we consider in the paper are reduced and irreducible, i.e. A is already a domain. We have not thought about non-reduced curves. For reduced curves the additional assumption of irreducibility involves no loss of generality, as can be seen from the proof of Theorem 9 in [4]. The methods we use in the curve case sometimes work for subrings of R[t], R a commutative ring. This paper is a revision and extension of the unpublished manuscript [S]. In order to prove that excision holds we use the exact sequence of Swan (see Section 1) or its improvement due to Vorst (see Section 2). Usually we find explicit generators of R *,,., &I/I2 and then show that these generators map to 1 in K1(A, I). However in Section 4 we use a trick of Swan [15, p. 2381. Using these methods we show that excision holds if enough integers are invertible (Theorem 3.1, Theorem 3.3) or if the ground field is perfect (Theorem 4.2). We also give counterexamples to excision in all characteristics. In order to prove that elements in the kernel of the excision homomorphism are non-zero we use in this paper only one method, the “12-trick” (see proof of Theorem 6.3). However we have