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 ∞-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.

Chromatic homotopy theory is asymptotically algebraic

“…The next result summarizes the key features of local duality contexts that we will use throughout this paper; for the proofs, see [HPS97] and [BHV18b].…”

“…• Mod (A ⊗s n,p ) denotes the symmetric monoidal ∞-category of modules over the completion of A ⊗s n,p as an object in Fr n,p . • Torsion objects in this context refers to torsion objects in the sense of local duality contexts [BHV18b], and the corresponding categories of torsion objects are indicated by a superscript "tors". • The protoproduct of the form Pic F is a generalization of the Pic-protoproduct of [BSS17, Section 3.5] to the ∞-categories of interest in this paper.…”

“…follows from applying the protoproduct construction to the local duality equivalence of Proposition 3.11, following [BHV18b].…”

“…Mod (E ⊗s n,p ) * and the second step was to deduce the main theorem by descent along these equivalences. From the point of view of local duality contexts [BHV18b], the K(n)-local category can be realized as a torsion subcategory in the E(n)-local category. Thus, to get the monochromatic analogue of the equivalence (1.3), we restrict it to an equivalence between suitable torsion subcategories.…”

Derived completion for comodules

“…This determines a closed substack Z of X, along which we can consider both completion and torsion functors on QCoh(X) together with their corresponding derived functors. In light of the symmetric monoidal equivalence between Comod Ψ and QCoh(X) [BHV18,Prop. 5.37], and the definition of Z in terms of I, we see that the corresponding subcategories of complete objects, resp.…”