The Chromatic Nullstellensatz
Robert Burklund, Tomer M. Schlank, Allen Yuan

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 -local -rings as those that satisfy an analogue of Hilbert's Nullstellensatz. Furthermore, we show that for every -local -ring , the collection of -ring maps from to such Lubin-Tate theories jointly detect nilpotence. In particular, we deduce that every non-zero -local -ring admits an -ring map to such a Lubin-Tate theory. As consequences, we construct complex orientations of algebraically closed Lubin-Tate theories, compute the strict Picard spectra of such Lubin-Tate theories, and prove redshift for the algebraic -theory of arbitrary -rings.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
