Generic Rank of R^{1/p} - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T07:00:09Z http://mathoverflow.net/feeds/question/38179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38179/generic-rank-of-r1-p Generic Rank of R^{1/p} Kevin 2010-09-09T14:15:23Z 2010-09-09T14:45:54Z <p>Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the generic or torsion free rank of $R^{1/p}$ (i.e. the rank of this module after tensoring up to the fraction field) always equal to $[k:k^p] \cdot p^d$ (which is true at least when $R$ is complete)? What if, in addition, the completion of $R$ along its maximal ideal is also known to be a domain?</p>