The Ring of Polynomial Functors of Prime Degree

TL;DR

This paper establishes an equivalence between a category of polynomial functors of prime degree from free abelian groups to p-adic modules and modules over a Green order, confirming a conjecture by Yuri Drozd.

Contribution

It proves the conjecture that such polynomial functors are categorically equivalent to modules over a Green order, providing a new understanding of their structure.

Findings

Category of polynomial functors is equivalent to modules over Green order.

Confirms Drozd's conjecture on polynomial functors of prime degree.

Provides a classification framework for these functors.

Abstract

Let $\hat\Z_p$ be the ring of $p$-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to $\hat \Z_p$-modules of degree at most $p$ is equivalent to the category of finitely generated modules over a particularly well understood ring, called Green order. That this is the case was conjectured by Yuri Drozd.