𝐹-purity and rational singularity

We investigate singularities which are F F -pure (respectively F F -pure type). A ring R R of characteristic p p is F F -pure if for every R R -module M , 0 → M ⊗ R → M ⊗ 1 R M,0 → M ⊗ R → M ⊗ \, ^1R is exact where 1 R ^1R denotes the R R -algebra structure induced on R R via the Frobenius map (if r ∈ R r ∈ R and s ∈ 1 R s ∈ \, ¹R , then r ⋅ s = r p s r · s = {r^p}s in 1 R