$F$-pure and $F$-injective singularities in equal characteristic zero

Inspired by Schoutens’ results, we introduce a variant of sharp $F$-purity and sharp $F$-injectivity in equal characteristic zero via ultraproducts. As an application, we show that if $R\to S$ is pure and $S$ is of dense $F$-pure type, then $R$ is of dense $F$-pure type.