On a decomposition theorem in equivariant generalized homology theories for finite group actions
Francesco Sala
TL;DR
This work extends the decomposition paradigm for equivariant theories to a broad algebraic context by leveraging the Burnside algebra action and a generalized inertia framework. It unifies Vistoli’s rational equivariant K-theory with analogous splits in broader cohomology theories, via a decomposition indexed by subgroup data and localized Burnside pieces. The paper applies these ideas to algebraic K-theory of DM stacks and to modular K-theory, introducing Auslander algebras and a corresponding coherent module category to define $K'_*( [X/G] )$, with a localization theorem proving isomorphisms between fixed-point data and the global theory under stratifications by gerbes. The results yield explicit descriptions of the decomposition, including inverses in favorable cases, and establish a robust localization mechanism for modular K-theory with concrete subgroup- and inertia-based structure, broadening both the theoretical framework and computability in equivariant algebraic geometry.
Abstract
A. Vistoli proved a decomposition theorem for the rational equivariant algebraic K-theory of a variety under the action of a finite group $G$. We generalize his result to more general algebraic (co)homology theories having the Mackey property and admitting localization long exact sequences. In general, the pieces are indexed by conjugacy classes of subgroups of $G$. Our construction is based on some result about a decomposition of the rational Burnside ring of a finite group, which stands behind the classical splitting theorems for equivariant spectra in stable equivariant homotopy theory. Applying this result to the case of Borne's modular K-theory we exhibit a case where the splitting is indexed by not necessarily abelian subgroups.