On Quasi-$F$-Purity of Excellent Rings

Abstract

We introduce an analogue to Quasi-$F$-splittings, Quasi-$F$-purity, which is definable over rings that are not necessarily $F$-finite. We show that this property is equivalent to being Quasi-$F$-split in the complete local and $F$-finite case. We then exhibit that it is stable under completion, direct limit, and local/finite \'etale extension.