A locally Nagata PID that is not Nagata

Abstract

It has previously been shown by an example of dimension one that a locally excellent integral domain need not be excellent or even Nagata. A related example showed that a locally excellent normal domain of dimension at least two need not be excellent, though that example is Nagata. Here we present a locally excellent principal ideal domain which is neither excellent nor Nagata. A second example is a two dimensional locally excellent regular ring which is Japanese but not Nagata.