Projective Lines Over One-Dimensional Semilocal Domains and Spectra of Birational Extensions

In [7], Nashier asked if the condition on a one-dimensional local domain R that each maximal ideal of the Laurent polynomial ring R[y, y-1] contracts to a maximal ideal in R[y] or in R[y-1] implies that R is Henselian. Motivated by this question, we consider the structure of the projective line Proj(R[s, t]) over a one-dimensional semilocal domain R (the projective line regarded as a topological space, or equivalently as a partially ordered set). In particular, we give an affirmative answer to Nashier’s question. (Nashier has also independently answered his question [9].) Nashier has also studied implications on the prime spectrum of the Henselian property in [8] as well as in the papers cited above.