Frobenius functors and invariant factors

Abstract

Let (R,m) be a noetherian local ring of prime characteristic and FR (-) be the Frobenius functor. We will show that for all finite R-modules Mthere is an n∈ N (depending on M) such that the fundamental theorem for modules over principal domains holds for i.e. there are r, s∈ N and elements if and only if Ris geometrically unibranch in the sense of Grothendieck [2] and dim R≤ 1.For exampleRis geometrically unibranch, if it is complete and R/m is an algebraically closed field.