The chromatic Nullstellensatz

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 Nellstellensatz. 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 K-theory of arbitrary $\mathbb{E}_\infty$-rings.

Authors

Robert Burklund

Department of Mathematical Sciences, University of Copenhagen, Denmark

Tomer M. Schlank

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

Allen Yuan

Department of Mathematics, Columbia University, New York, NY, USA