On the Picard Group of a Quasi-Affine Scheme

Abstract

In [6] the restriction map ϕf : Pic(R) → Pic(Rf) was studied for any noetherian integrally closed domain R and any prime element f ∈ R. In particular, it was proved in loc. cit that ϕf is always injective and several sufficient conditions for the surjectivity of ϕf were given. If should be noted that Pic(R) is just the Picard group of the affine scheme Spec(R), hereas Pic (R_p) is the Picard group of the open affine subscheme of Spec(R) determined by f. The obvious question in this context is of course : given X = Spec(R) and U c X, a not necessarily affine open subscheme of X, what can one say about the restriction map ϕU : Pic(X) → Pic(U). In particular, one may ask when ϕU is surjective or what Ker ϕU is. It is the purpose of this note to provide answers to these questions.