Segal conjecture fails for truncated polynomial algebras over higher chromatic local fields, causing Lichtenbaum-Quillen property to fail while weak redshift conjecture holds.
Schlank, and Alle Yuan, The Chromatic Nullstellensatz , Annals of Mathematics (to appear)
4 Pith papers cite this work. Polarity classification is still indexing.
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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.
representative citing papers
Proves a nilpotence theorem for rational rigid 2-rings of moderate growth and shows such categories have enough tt-fields of the form Perf(L) for even 2-periodic fields L.
Compiles a list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions and proves persistence of the strong Nullstellensatz in large polynomial rings.
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Algebraic K-theory of finite algebras over higher local fields
Segal conjecture fails for truncated polynomial algebras over higher chromatic local fields, causing Lichtenbaum-Quillen property to fail while weak redshift conjecture holds.
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A Nilpotence Theorem for Rational Rigid 2-Rings of Moderate Growth
Proves a nilpotence theorem for rational rigid 2-rings of moderate growth and shows such categories have enough tt-fields of the form Perf(L) for even 2-periodic fields L.
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Infinite Versions of Hilbert's Nullstellensatz
Compiles a list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions and proves persistence of the strong Nullstellensatz in large polynomial rings.
- Higher Semiadditive Character Theory