Categorizations of Limits of Grothendieck Groups over a Frobenius P-Category

Abstract

In [Frobenius Categories Versus Brauer Blocks, Progress in Math., 274] and in [Ordinary Grothendieck groups of a Frobenius [Formula: see text]-category, Algebra Colloq. 18 (2011) 1–76], we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics [Formula: see text] and zero, obtained from a folded Frobenius [Formula: see text]-category[Formula: see text], which covers the case of the Frobenius[Formula: see text]-categories associated with blocks; moreover, in [Beyond a question of Markus Linckelmann, arxiv.org/abs/1507.04278] we show that a folded Frobenius [Formula: see text]-categoryis actually equivalent to the choice of a regular central [Formula: see text]-extension [Formula: see text] of [Formula: see text]. Here, taking advantage of the existence of a perfect [Formula: see text] -locality [Formula: see text], we exhibit those inverse limits as the true Grothendieck groups of the categories of [Formula: see text]-and [Formula: see text]-modules for a suitable [Formula: see text]-group [Formula: see text] associated to the [Formula: see text]-category [Formula: see text] obtained from [Formula: see text] and [Formula: see text]. It depends on a vanishing cohomology result, given with more generality in the Appendix.