In the proof of Theorem 4.27 in the paper (Characteristic p Techniques in Commutative Algebra and Algebraic Geometry Math 732 - Winter 2019 Karen E Smith1), it is written that Without loss of generality, we can assume that (R,m,K) is a complete local F-regular ring.
May I know Why we can assume that?
The only condition we have on R is that R is F-regular, so it is Frobenius-finite with prime characteristic.
Thanks