Local duality for structured ring spectra

Abstract

Abstract. We use the abstract framework constructed in our earlier paper [BHV15] to study local duality for Noetherian E∞-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.

Stratification and duality for homotopical groups

“…Local cohomology and support. In this section we summarize some of the constructions of [BIK08]; for an alternative approach, one can apply the techniques of [BHV17]. We refer to these sources for the basic terminology of localization and colocalization sequences.…”

“…4.8]. By [BHV17,Prop. 4.7], algebraically Gorenstein ring spectra are always absolute Gorenstein in the sense of Definition 6.1.…”

The algebraic chromatic splitting conjecture for Noetherian ring spectra

“…Conventions. Throughout this paper, we will work in the setting of ∞-categories as developed in [Lur09,Lur17], and will use the local duality framework described in [BHV15,BHV18]. In particular, all constructions will implicitly be assumed to be derived.…”

“…7.23] and [BS16, Prop. 3.7] combined with Lemma 2.2, the chromatic localization functors L n−1 and L K(n) restricted to Mod BP are equivalent to the algebraic localization functors L V(In) and Λ V(In) L V(In) , respectively (see also [BHV18,Prop. 3.12]).…”

“…The goal of this paper is to study this question for the category Mod R of module spectra over a structured ring spectrum R. To this end, we need to work in the context of the companion paper [BHV18], which in turn is based on [BIK08] and [BHV15]. If V ⊆ Spec h (π * R) is a specialization closed subset, there is a quadruple of functors (Γ V , L V , ∆ V , Λ V ) on Mod R (see [BHV18,Thm. 3.9]) which captures the part of Mod R supported on V.…”