Tight Closure and Differential Simplicity

The behavior of the Hasse–Schmidt algebra under étale extension is used to show that the Hasse–Schmidt algebra of a smooth algebra of finite type over a field equals the ring of differential operators. These techniques show that the formation of Hasse–Schmidt derivations does not commute with localization, providing a counterexample to a question of Brown and Kuan; their conjecture is reformulated in terms of the Hasse–Schmidt algebra. These techniques also imply that a smooth domain R is differentially simple. Tight closure is used to show that the test ideal is Hasse–Schmidt stable. Indeed, differentially simple rings of prime characteristic are strongly F-regular.