Stratified Noncommutative Geometry

Author:

Ayala David,Mazel-Gee Aaron,Rozenblyum Nick

Abstract

We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable \infty -categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as E n \mathbb {E}_n -monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of Möbius inversion.

Our main application is to equivariant stable homotopy theory: for any compact Lie group G G , we give a symmetric monoidal stratification of genuine G G -spectra. In the case that G G is finite, this expresses genuine G G -spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions).

We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory.

Publisher

American Mathematical Society (AMS)

Reference92 articles.

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4. The Segal conjecture for elementary abelian 𝑝-groups;Adams, J. F.;Topology,1985

5. David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum, Cyclotomic spectra via stratifications, available at arXiv:1710.06416, (2017).

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