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The Chromatic Nullstellensatz

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arxiv 2207.09929 v1 pith:NB4KB5CE submitted 2022-07-20 math.AT math.KT

Robert Burklund , Tomer M. Schlank , Allen Yuan This is my paper · ORCID
classification math.ATmath.KT
keywords inftymathbblubin-tateringtheorieslocalalgebraicallyclosed
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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.

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Cited by 4 Pith papers

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