Exotic principal ideal domains - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:26:05Z http://mathoverflow.net/feeds/question/56513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56513/exotic-principal-ideal-domains Exotic principal ideal domains Qiaochu Yuan 2011-02-24T10:54:01Z 2011-02-25T01:01:27Z <p>Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that happen to have trivial class group, localizations of these, and completions of localizations of these at a prime. Are there more exotic examples? Is there anything like a classification? </p> http://mathoverflow.net/questions/56513/exotic-principal-ideal-domains/56517#56517 Answer by Emil Jeřábek for Exotic principal ideal domains Emil Jeřábek 2011-02-24T12:38:18Z 2011-02-24T12:38:18Z <p><a href="http://dx.doi.org/10.1006/jabr.1993.1154" rel="nofollow">Smith</a> constructed a PID which is a nonstandard model of open induction. That should be exotic enough. (Note that nonstandard models of just slightly stronger theories of arithmetic, such as $IE_1$, are never even UFDs.)</p> http://mathoverflow.net/questions/56513/exotic-principal-ideal-domains/56526#56526 Answer by Pete L. Clark for Exotic principal ideal domains Pete L. Clark 2011-02-24T15:05:05Z 2011-02-24T15:10:23Z <p>No, to the best of my knowledge there is nothing like a general classification of PIDs. Despite their easy definition, they turn out to be rather a finicky class of rings, as for instance Gauss conjectured that there are infinitely many PIDs among rings of integers of real quadratic fields, but more than $200$ years later we have not been able to prove that there are infinitely many PIDs among rings of integers of <em>all</em> number fields. And, as came out in the comments to Emil's answer, the property of being a PID is not first order, so is not very robust in a model-theoretic sense. In that regard, the better class of rings are the <a href="http://en.wikipedia.org/wiki/B%C3%A9zout_domain" rel="nofollow">Bézout domains</a>, i.e., domains in which every finitely generated ideal is principal. A theorem of Kaplansky which can be used to show that various "big" domains (e.g. $\overline{\mathbb{Z}}$, the ring of all algebraic integers) are Bézout can be found at the end of the section on overrings in <a href="http://math.uga.edu/~pete/integral.pdf" rel="nofollow">these notes</a>. (I am now giving less precise citations to my often-changing commutative algebra notes in the hope that they will take longer to become obsolete.)</p> <p>There are some interesting papers on construction of PIDs with various properties. The one I want to read next is <a href="http://projecteuclid.org.proxy-remote.galib.uga.edu/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.dmj/1077310578" rel="nofollow">this 1974 paper of Raymond C. Heitmann</a>: given any countable collection $\mathcal{F}$ of countable fields containing only finitely many fields of any given positive characteristic, Heitmann constructs a countable PID of characteristic $0$ with residue fields precisely the elements of $\mathcal{F}$. </p> <p><b>Added</b>: note that $\overline{\mathbb{Z}}$ is also an <strong>antimatter domain</strong>, i.e., it has no irreducible elements (which specialists in the field tend to call "atoms"). Thus this gives an example of a Bézout domain which is not an ultraproduct of PIDs. </p> http://mathoverflow.net/questions/56513/exotic-principal-ideal-domains/56536#56536 Answer by Georges Elencwajg for Exotic principal ideal domains Georges Elencwajg 2011-02-24T17:13:50Z 2011-02-24T17:13:50Z <p>Dear Qiaochu, if $A$ is a discrete valuation ring and if $B$ is an étale algebra over $A$, then $B$ is a discrete valuation ring. In a related vein, the henselization of a discrete valuation ring $A$ is a discrete valuation ring $A^h$ (however it is not étale over $A$, for example because it is not finitely generated ).If $A$ is the local ring of a point on a curve in the Zariski topology, then $A^h$ is the local ring of that point in the étale topology.</p> <p>A very concrete example: the henselization of the local ring $A=\mathcal O_{\mathbb A^1,0}$ of the complex affine line at the origin is the subring of the ring of formal series $\mathbb C [[T]]$ consisting of those series that are algebraic over $A$. </p> <p>These seem to be examples not on your list, but I'll let you be the judge of their exotism....</p> http://mathoverflow.net/questions/56513/exotic-principal-ideal-domains/56554#56554 Answer by anonymous for Exotic principal ideal domains anonymous 2011-02-24T19:30:56Z 2011-02-25T01:01:27Z <p>Fontaine's ring $B_{cris}^{\varphi=1}$ is a PID, and no expert in the field would have bet on it in the first place (this led to some very nice recent developments by Fargues and Fontaine).</p> <p><a href="http://www.math.u-psud.fr/~fargues/Courbe.pdf" rel="nofollow">http://www.math.u-psud.fr/~fargues/Courbe.pdf</a></p>