Noetherian rings without finite normalization (Q2895444)

This paper is a nice survey of examples of local Noetherian domains without finite normalization, and the theory behind them. (It does not consider in any detail other ``bad'' classes of Noetherian rings.) The examples, which include rings of characteristic zero and \(p >0\), are arranged in three main groupings: how they arise between, above, and below naturally occurring Noetherian rings.NEWLINENEWLINEThe first set consists of one dimensional analytically ramified local Noetherian domains discovered by \textit{Y.~Akizuki} [Proc. phys.-math. Soc. Japan (3) 17, 327--336 (1935; JFM 61.1029.02)], \textit{F.~K.~Schmidt} [Math.~Z. 41, 443--450 (1936; JFM 62.0096.01)], \textit{M.~Nagata} [Interscience tracts in pure and applied mathematics 13, John-Wiley \& Sons, New York-London, (1962; Zbl 0123.03402)], and \textit{D.~Ferrand} and \textit{M.~Raynaud} [Ann.~Sci.~Éc.~Norm.~Supér (4) 3, 295--311 (1970; Zbl 0204.36601)]. These occur in an immediate extension of rank one discrete valuations rings; i.e., between naturally occurring Noetherian rings.NEWLINENEWLINEThe second set discusses examples birationally dominating a local ring; i.e., above naturally occurring Noetherian rings. The focus here is still on one dimensional rings, but the emphasis is on when examples can be found birationally dominating a given local Noetherian domain of dimension possibly larger than one. This includes a geometric example, due to \textit{A.~Reguera} [Am.~J.~Math. 131, 313--350 (2009; Zbl 1188.14010)], in the sense that it is a local ring of a point on the space of arcs associated to an irreducible curve.NEWLINENEWLINEThe third grouping of examples considers strongly twisted subrings of local Noetherian domains. The main technique is to locate Noetherian domains without finite normalization as subintegral extensions of known Noetherian domains; i.e., occurring below naturally occurring Noetherian rings.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].