Affineness and chromatic homotopy theory

Abstract

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 about Tate spectra.

46 pages. Some mistakes corrected in the treatment of Tmf with level structures

Cited by in corpus (11)

• Nilpotence and descent in equivariant stable homotopy theory
• The Picard group of topological modular forms via descent theory
• The $C_2$-spectrum $Tm{f}_1(3)$ and its invertible modules
• Twisted differential cohomology
• Picard groups of higher real $K$-theory spectra at height $p-1$
• Purity in chromatically localized algebraic $K$-theory
• The Balmer spectrum of the equivariant homotopy category of a finite abelian group
• Separable commutative algebras and Galois theory in stable homotopy theories
• $C_2$-equivariant topological modular forms
• Connective Models for Topological Modular Forms of Level $n$
• Vanishing lines in chromatic homotopy theory