A Property of the Frobenius Map of a Polynomial Ring

TL;DR

This paper characterizes the Frobenius map's properties in polynomial rings over perfect fields, providing explicit isomorphisms between certain Hom modules, with potential applications in algebra.

Contribution

It explicitly describes an R-module isomorphism involving the Frobenius map for polynomial rings over perfect fields, advancing understanding of Frobenius actions.

Findings

Explicit R-module isomorphism for Hom_R(F_*(M),N) and Hom_R(M,F^*(N))

Application potential in algebraic structures and module theory

Enhanced understanding of Frobenius map properties in polynomial rings

Abstract

Let R be a ring of polynomials in a finite number of variables over a perfect field k of characteristic p>0 and let F:R\to R be the Frobenius map of R, i.e. F(r)=r^p. We explicitly describe an R-module isomorphism Hom_R(F_*(M),N)\cong Hom_R(M,F^*(N)) for all R-modules M and N. Some recent and potential applications are discussed.