The Chromatic Nullstellensatz

TL;DR

This paper characterizes Lubin--Tate theories among $T(n)$-local $ ext{E}_ abla$-rings using an analogue of Hilbert's Nullstellensatz and shows that maps to these theories detect nilpotence, with applications to orientations, Picard spectra, and algebraic K-theory.

Contribution

It introduces a Nullstellensatz analogue for Lubin--Tate theories and demonstrates their role in detecting nilpotence among $T(n)$-local $ ext{E}_ abla$-rings, advancing understanding of their structure and applications.

Findings

Lubin--Tate theories characterized by Nullstellensatz analogue.

Maps to Lubin--Tate theories detect nilpotence.

Constructs $ ext{E}_ abla$ complex orientations and proves K-theory redshift.

Abstract

We show that Lubin--Tate theories attached to algebraically closed fields are characterized among $T(n)$-local $\mathbb{E}_{\infty}$-rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$, the collection of $\mathbb{E}_\infty$-ring maps from $R$ to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero $T(n)$-local $\mathbb{E}_{\infty}$-ring $R$ admits an $\mathbb{E}_\infty$-ring map to such a Lubin-Tate theory. As consequences, we construct $\mathbb{E}_{\infty}$ complex orientations of algebraically closed Lubin-Tate theories, compute the strict Picard spectra of such Lubin-Tate theories, and prove redshift for the algebraic $\mathrm{K}$-theory of arbitrary $\mathbb{E}_{\infty}$-rings.