Categorizations of limits of Grothendieck groups over a Frobenius P-category

TL;DR

This paper explores the limits of Grothendieck groups over Frobenius P-categories, establishing their equivalence to certain module categories and providing a unified framework for understanding their structure.

Contribution

It introduces a new perspective on the limits of Grothendieck groups via perfect F-locality and regular central extensions, linking them to module categories over specific groups.

Findings

Inverse limits of Grothendieck groups are equivalent to true Grothendieck groups of module categories.

The existence of perfect F-locality enables a unified description of these limits.

The framework applies to categories associated with blocks and self-centralizing subgroups.

Abstract

In "Frobenius Categories versus Brauer Blocks" and in "Ordinary Grothendieck groups of a Frobenius P-category" we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics p and zero, obtained from a so-called "folded Frobenius P-category", which covers the case of the Frobenius P-categories associated with blocks, moreover, in "Affirmative answer to a question of Linckelmann" we show that a "folded Frobenius P-category" is actually equivalent to the choice of a regular central k*-extension of the Frobenius P-category restricted to the set of F-selfcentralizing subgroups of P. Here, taking advantage of the existence of the perfect F-locality L, recently proved, we exhibit those inverse limits as the true Grothendieck groups of the categories of K*\hat G- and k*\hat G-modules for a suitable k*-group \hat G associated to the k*-category obtained from the perfect F-locality L and \hat F, both restricted to the set of F-selfcentralizing subgroups of P.