Equivariant recollements and singular equivalences
Miltiadis Karakikes, Aristeides Kontogeorgis, Chrysostomos Psaroudakis
TL;DR
The work develops a comprehensive framework for equivariant recollements across abelian and triangulated categories under finite group actions, providing necessary and sufficient conditions for lifting to equivariant contexts and connecting these liftings to derived and singularity categories. It proves that, under suitable invertibility and invariance hypotheses, quotient-induced singular equivalences lift to equivariant quotients, and constructs canonical compatibilities between derived and equivariant recollements. The results yield powerful applications: equivariant recollements for derived categories of schemes with group actions, Morita-type equivalences in skew group algebras, and new isomorphisms in equivariant singular Hochschild cohomology. Collectively, these contributions extend the Grothendieck six-functor framework to equivariant settings, with implications for geometry, representation theory, and noncommutative algebraic structures.
Abstract
In this paper we investigate equivariant recollements of abelian (resp. triangulated) categories. We first characterize when a recollement of abelian (resp. triangulated) categories induces an equivariant recollement, i.e. a recollement between the corresponding equivariant abelian (resp. triangulated) categories. We further investigate singular equivalences in the context of equivariant abelian recollements. In particular, we characterize when a singular equivalence induced by the quotient functor in an abelian recollement lift to a singular equivalence induced by the equivariant quotient functor. As applications of our results: (i) we construct equivariant recollements for the derived category of a quasi-compact, quasi-separated scheme where the action is coming from a subgroup of the automorphism group of the scheme and (ii) we derive new singular equivalences between certain skew group algebras.