Right and Left Modules over the Frobenius Skew Polynomial Ring in the F-Finite Case

TL;DR

This paper establishes an equivalence between categories of certain left and right modules over the Frobenius skew polynomial ring in the F-finite case, using Matlis duality in a complete local ring setting.

Contribution

It proves that, under specific conditions, Matlis duality induces an equivalence between Artinian left modules and Noetherian right modules over the Frobenius skew polynomial ring.

Findings

Categories of Artinian left modules and Noetherian right modules are equivalent via Matlis duality.

The result applies to complete Noetherian local rings with finite Frobenius homomorphism.

Provides a new perspective on module theory over Frobenius skew polynomial rings in prime characteristic.

Abstract

The main purposes of this paper are to establish and exploit the result that, over a complete (Noetherian) local ring $R$ of prime characteristic for which the Frobenius homomorphism $f$ is finite, the appropriate restrictions of the Matlis-duality functor provide an equivalence between the category of left modules over the Frobenius skew polynomial ring $R[x,f]$ that are Artinian as $R$-modules and the category of right $R[x,f]$-modules that are Noetherian as $R$-modules.