Local duality in algebra and topology

By T. Barthel, Drew Heard, Gabriel Valenzuela. Advances in Mathematics, 2015

50 citations recorded by the source corpus.

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 Tarrio, Jeremias Lopez, 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/617ad7fc50cc04805799242ae1a59d9d751c35be

How researchers see this paper

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In Algebra and Number Theory

1. A. Mathew
2. J. Noel
3. N. Axel Naumann

In Geometry and Topology

1. N. Axel Naumann
2. J. Noel
3. A. Mathew
4. Piotr Pstrągowski
5. B. Bhatt
6. Peter Scholze
7. Lennart Meier
8. M. Hill

In Mathematical Physics

1. N. Axel Naumann
2. J. Noel
3. A. Mathew
4. Piotr Pstrągowski
5. B. Bhatt
6. Peter Scholze
7. Lennart Meier
8. M. Hill

Citations in the corpus

• Nilpotence and descent in equivariant stable homotopy theory (2015, background): “Similar ideas have also been explored in recent work of Barthel–Heard–Valenzuela [10].”