Finite permutation resolutions

Three real Artin-Tate motives

“…We can consider the big derived category of permutation modules DPerm(G; k), that admits our K(G; k) as its compact objects. In the case of a finite group G, we proved in [BG23b,§ 9] that DPerm(G; k) satisfies BHSstratification, following Barthel-Heard-Sanders [BHS23]. This means that its spectrum is reasonable, namely is a weakly noetherian space (it is even noetherian for G finite), and that the ensuing support theory on the big objects of DPerm(G; k) classifies all localizing tensor-ideals.…”

“…The spectrum. We already computed in [BG23b] the spectrum of the permutation category K(G; k) when G is a finite group. So our first task is to 'pass these results to the limit'.…”

“…We outline in Remark 3.22, without elaborating, some alternative descriptions of the topology of Spc(K(G; k)) via pro-elementary abelian groups or via twisted cohomology. These themes have been explored in depth in [BG23b] in the finite case and would appear repetitive here. We find it more instructive to discuss an example in some detail.…”

“…The p-Sylow of any finite quotient G/N is cyclic of order p n , for some n ≥ 0. We proved in [BG23b] that the spectrum of K(G/N ; k) is then the space…”

Permutation modules and cohomological singularity

“…The colimit theorem. To discuss the tt-geometry of K(G), it is instructive to keep in mind the bounded derived category of finitely generated kG-modules, D b (kG), which is a localization of our K(G) by [BG23,Theorem 5.13]. A theorem of Serre [Ser65], famously expanded by Quillen [Qui71], implies that Spc(D b (kG)) is the colimit of the Spc(D b (kE)), for E running through the elementary abelian p-subgroups of G; see [Bal16,§ 4].…”