Local duality in algebra and topology

TL;DR

This paper develops an abstract framework for local duality in algebraic and topological contexts, unifying classical results and extending to stacks and stable homotopy theory with computable spectral sequences.

Contribution

It introduces a general abstract setting for local duality in stable $mbda$-categories, unifies various classical and modern duality theorems, and applies to algebraic stacks and stable homotopy theory.

Findings

Constructed local cohomology and homology functors satisfying duality in abstract categories.

Reproduced classical local duality results for rings and schemes within the new framework.

Established local duality for algebraic stacks and stable homotopy theories, with spectral sequences for computations.

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 Tarr\'io, Jerem\'ias L\'opez, 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.