Global dimension of the category of rational incomplete Mackey functors for a finite abelian group G

TL;DR

This paper investigates the global dimension of rational incomplete Mackey functors for finite abelian groups, providing bounds and exact calculations in specific cases using algebraic and combinatorial methods.

Contribution

It offers new bounds and exact computations for the global dimension of rational incomplete Mackey functors, connecting homological algebra with incidence algebras.

Findings

Upper bound on global dimension for disk-like transfer systems

Exact calculation of global dimension for abelian groups in the disk-like case

Connection established between Mackey functors and incidence algebras

Abstract

In this paper, we analyse the global dimension of the category of rational incomplete Mackey functors over a finite abelian group. Incomplete Mackey functors have recently risen to prominence in algebraic topology and hence it is valuable to understand their homological algebra invariants. When working over the rational numbers, results of Greenlees--May and Th\'evanez--Webb show that the homological algebra of complete Mackey functors is quite simple, but the incomplete case is more complicated. In this paper we use splitting results by the first, third and fourth authors to give an upper bound on the global dimension of rational incomplete Mackey functors where the incompleteness is governed by what is known as a disk-like transfer system. We then avail ourselves of a new connection to incidence algebras over posets to calculate the global dimension of rational incomplete Mackey functors in the disk-like case when the group is abelian.