Affineness and chromatic homotopy theory

By A. Mathew, Lennart Meier. 2013

47 citations recorded by the source corpus.

Given an algebraic stack X, one may compare the derived category of quasi‐coherent sheaves on X with the category of dg‐modules over the dg‐ring of functions on X. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber‐exact homology theories. We identify a condition (quasi‐affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and vanishing results for Tate spectra.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cdcf9b9f681f94855905684ab6a592cc4fba10e8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

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

In Mathematical Physics

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

Citations in the corpus

• The Galois group of a stable homotopy theory (2014, background): “At least away from the prime 2 (this restriction is removed in [57]), it is a result of Meier, proved in [55], that the adjunction is an equivalence: TMF-modules are equivalent to quasi-coherent Otop-modules.”
• Nilpotence and descent in equivariant stable homotopy theory (2015, methodology): “When combined with techniques from [53] for G = U(n), we will be able to prove that the equivariant complexification map KOG → KUG is a faithful C2-Galois extension of E∞-rings in SpU(n) (which we will then deduce for any compact Lie group G).”
• The Picard group of topological modular forms via descent theory (2014, methodology): “For instance, Meier [48] has studied TMF–modules which become free when certain level structures are introduced; these can be thought of as locally free sheaves with respect to a predetermined cover.”
• The C2–spectrum Tmf1(3) and its invertible modules (2015, background): “By the main result of [50], the derived stack .”