The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K ⊂ C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.
We give a simple universal property of the multiplicative structure on the Thom spectrum of an n‐fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax scriptO‐monoidal functor. This allows us to relate Thom spectra to En‐algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins–Mahowald theorem realizing Hdouble-struckFp and HZ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.
Abstract. We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use "algebraic" characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored, and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G-spaces, and diagram spectra among others.
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.
For a finite abelian group A, we determine the Balmer spectrum of Sp ω A , the compact objects in genuine A-spectra. This generalizes the case A = Z/pZ due to Balmer and Sanders [BS17], by establishing (a corrected version of) their logp-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions [Kuh04].
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