David E Speyer. Source: Journal of Commutative Algebra, Volume 12, Number 4, 573--579.Abstract:
Let [math] and [math] be schemes of finite type over [math] and let [math] be a finite map. We show the following holds for all sufficiently large primes [math] : If [math] and [math] are any splittings on [math] and [math] , such that the restriction of [math] is compatible with [math] and [math] , and [math] is any compatibly split subvariety of [math] , then the reduction [math] is a compatibly split subvariety of [math] . This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.