On the Geometry of $p$-Typical Covers in Characteristic $p$

http://dx.doi.org/10.4153/CJM-2008-006-8
Canad. J. Math. 60(2008), 140-163

  Published:2008-02-01
 Printed: Feb 2008

For $p$ a prime, a $p$-typical cover of a connected scheme on which $p=0$ is a finite \'etale cover whose monodromy group (\emph{i.e.,} the Galois group of its normal closure) is a $p$-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the $p$\nobreakdash-typi\-cal quotients of the \'etale fundamental groups, and a decomposition theorem for $p$-typical covers of polynomial rings over an algebraically closed field.