Lattices and cohomological Mackey functors for finite cyclic p-groups

TL;DR

This paper characterizes permutation modules over cyclic p-groups using cohomology, shows all lattices can be built from permutation modules, and analyzes the global dimension of related categories.

Contribution

It provides a cohomological characterization of permutation modules and establishes the global dimension of the category of cohomological Mackey functors for cyclic p-groups.

Findings

R[G]-permutation modules characterized by vanishing first cohomology

Any R[G]-lattice can be presented by permutation modules

Category of cohomological Mackey functors has global dimension 3

Abstract

For a finite cyclic p-group G and a discrete valuation domain R of characteristic 0 with maximal ideal pR the R[G]-permutation modules are characterized in terms of the vanishing of first degree cohomology on all sub- groups (cf. Thm. A). As a consequence any R[G]-lattice can be presented by R[G]-permutation modules (cf. Thm. C). The proof of these results is based on a detailed analysis of the category of cohomological G-Mackey functors with values in the category of R-modules. It is shown that this category has global dimension 3 (cf. Thm. E). A crucial step in the proof of Theorem E is the fact that a gentle R-order category (with parameter p) has global dimension less or equal to 2 (cf. Thm. D).