Operations on categories of modules are given by Schur functors

TL;DR

This paper characterizes operations on categories of modules as Schur functors linked to symmetric group representations, extending Macdonald's classification of polynomial functors to a broader context.

Contribution

It demonstrates that natural operations on module categories compatible with base change are precisely Schur functors derived from symmetric group representations.

Findings

Operations correspond to Schur functors associated with symmetric group representations

Results extend Macdonald's classification of polynomial functors

Provides a framework for functors compatible with base changes in module categories

Abstract

Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to $k$-linear representations of symmetric groups. This result is closely related to Macdonald's classification of polynomial functors.