Local duality in algebra and topology

Abstract

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 $\infty$-category $\mathcal{C}$ together with a collection of compact objects $\mathcal{K}\subset \mathcal{C}$ 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.

Significantly revised version to appear in Advances in Mathematics. Section 6 has been removed, and an expanded version will appear in a separate paper

References in corpus (4)

• Higher Topos Theory
• Quasi-coherent sheaves on the moduli stack of formal groups
• A characterization of categories of coherent sheaves of certain algebraic stacks
• Local duality for structured ring spectra

Cited by in corpus (24)

• C_2-equivariant stable homotopy from real motivic stable homotopy
• Injective DG-modules over non-positive DG-rings
• Completion and torsion over commutative DG rings
• Local duality for structured ring spectra
• The Left Localization Principle, completions, and cofree $G$-spectra
• Derived completion for comodules
• The algebraic chromatic splitting conjecture for Noetherian ring spectra
• Duality pairs, phantom maps, and definability in triangulated categories
• The homotopy theory of complete modules
• Torsion models for tensor-triangulated categories: the one-step case
• The homotopy limit problem and the cellular Picard group of Hermitian $K$-theory
• Localizing the $E_2$ page of the Adams spectral sequence
• Moduli stack of oriented formal groups and periodic complex bordism
• The ${\mathrm{Sp}}_{k,n}$-local stable homotopy category
• Morava K-theory and Filtrations by Powers
• The spectrum of excisive functors
• The $v_1$-Periodic Region of the Complex-Motivic Ext
• Local Gorenstein duality for cochains on spaces
• Moduli stack of oriented formal groups and the chromatic filtration
• Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra
• Rational local systems and connected finite loop spaces
• Algebraicity in monochromatic homotopy theory
• Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf{C}$
• Stratifying integral representations of finite groups