Picard groups of singular affine curves over a perfect field

Abstract

This was proved in [W1] for the case of an algebraically closed field k; in that case the conditions force R to be a Dedekind domain. For an infinite perfect ground field this is no longer the case. For example, the conditions are satisfied for the restricted polynomial ring k + X K [X l, where K/k is a finite algebraic extension and X is an indeterminate. A sharpened form of the main theorem, proved in the next section, says that a not Dedekind domain satisfying the conditions of (0.1) has exactly one singularity, and its singularity is analytically isomorphic to that of a restricted polynomial ring. We wilt show by example that (0.1.1) no longer implies (0.1.3) if "domain" is replaced by "reduced ring", or if "infinite" is deleted from the hypotheses on k. (I don't know whether perfection of the ground field k is essential.) We obtain a partial result for reduced rings: If D(R) is finitely generated, then R is seminormal and the intersection graph of the irreducible components of spec (R) is acyclic.