The Devinatz-Hopkins Theorem via Algebraic Geometry

Abstract

In this note, we show how a continuous action of the Morava stabilizer group Gn on the Lubin-Tate spectrum En, satisfying the conclusion E hGn n ≃ L K(n) S of the Devinatz-Hopkins Theorem, may be obtained by monodromy on the stack of oriented deformations of formal groups in the context of formal spectral algebraic geometry.

Moduli stack of oriented formal groups and periodic complex bordism

“…Another topic we will return to in a follow-up to this paper, is the relationship between chromatic localizations of spectra and the algebraic geometry of the moduli stack of oriented formal groups M or FG . There we will also clarify the connection between [Gre21] and the present paper. Let us point out, however, that our work is no way alone in expanding upon the ideas from [Ell2] for chromatic applications; see for instance [Dav20], [Dav21], and [Dev18].…”

Moduli stack of oriented formal groups and the chromatic filtration

“…Finally, we relate this to our previous work in [Gre21a]. Let the map of ordinary stacks Spec(F p ) → M ♡,≤n FG classify a formal group of exact height n over F p .…”

“…The above equivalence of non-connective formal spectral stacks induces, in light of Theorem 8, on quasi-coherent sheaves the equivalence of ∞-categories 6) from Theorem 8 reveals them to be nothing but the arguments given to prove these results in [Gre21a, Proofs of Corollary 2.17 and Theorem 2.14]. It is in this sense that our prior work in the formal setting in [Gre21a] is compatible with the "global" results of this paper and its predecessor [Gre21b].…”

“…The issue is that the completion map A ′ ⊗ A B → A ′ ⊗A B, or more generally A → A ∧ I for any adic E ∞ -ring A with an ideal of definition I ⊆ π 0 (A), is in general not faithfully flat. By [Stacks, Tag 00MC], this issue can be overcome by restricting everywhere to adic E ∞ -rings A for which π 0 (A) is a local Noetherian ring with its usual adic topology -that is the approach we followed in [Gre21a]. But in the general case, it seems to be important to split up the story into an adic setting, which interacts well with descent, and a smaller formal setting, which does not.…”