Related papers: Effective Redshift

We give a simple argument to detect chromatic redshift in the algebraic $K$-theory of $\mathbb{E}_{\infty}$-ring spectra and give two applications: we show for $n\geq 1$ that $K(E_n)$, the algebraic $K$-theory of any height $n$ Lubin-Tate…

It is known that the truncated Brown--Peterson spectra can be equipped with a certain nice algebra structure, by the work of J. Hahn and D. Wilson, and these ring spectra can be viewed as rings of integers of local fields in chromatic…

At each prime $p$ and height $n+1 \ge 2$, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for $\mathbb{Z}$ acting by Adams operations on $\mathrm{BP}\langle n \rangle$, we prove that the…

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the $K(n)$- and $T(n)$-local categories. We prove that it satisfies a form of…

We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $\pi$-finite $p$-group actions on stable $\infty$-categories of chromatic height up to $n$, extending a result of Clausen-Mathew-Naumann-Noel for finite $p$-groups.…

We prove a higher chromatic analogue of Snaith's theorem which identifies the K-theory spectrum as the localisation of the suspension spectrum of CP^\infty away from the Bott class; in this result, higher Eilenberg-MacLane spaces play the…

We prove a conjecture of Rognes by establishing a localization cofiber sequence of spectra, K(Z) to K(ku) to K(KU) to Sigma K(Z), for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence…

Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action…

We identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a negative answer to a question of Rognes…

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map…

Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $\pi$-finite $p$-space $A$ at the Lubin-Tate spectrum $E_n$ is equal to the higher semiadditive cardinality of the…

Ravenel proved the remarkable fact that the $K$-theoretic localization $L_K S^0$ of the sphere spectrum has $\mathbb{Q}/\mathbb{Z}$ as homotopy group in dimension -2. Mike Hopkins' chromatic splitting conjecture implies more generally that…

The chromatic redshift philosophy, introduced by Ausoni and Rognes, suggests that algebraic $K$-theory raises chromatic height by $1$. We show that the analogue of this philosophy fails in the case of rigid symmetric monoidal stable…

We introduce a family of twisted $K(n)$-local theories that behave analogous to twisted K-theory. Let $R_n= E_n^{hS\mathbb G_n}$, the homotopy fixed point spectrum under the action of the subgroup $S\mathbb G_n$ of the Morava stabilizer…

There are at least two ways to approach the homotopy theory of spaces `at chromatic height $n$': one may localize with respect to $T(n)$-homology or with respect to $v_n$-periodic homotopy groups. It was already observed by Bousfield that…

We give a new proof of the $\infty$-semiadditivity of $K(n)$-local spectra. The proof proceeds by induction on the height via algebraic K-theory, utilizing recent advances in chromatic homotopy theory and the redshift conjecture, instead of…

We discuss some general properties of $\mathrm{TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm{TR}$ of $H\mathbb{Z}$-algebras is a truncating invariant in the sense of Land--Tamme, and deduce $h$-descent…

We prove a conjecture of Hesselholt and Ausoni-Rognes, establishing localization cofiber sequences of spectra for THH(ku) and TC(ku). These sequences support Hesselholt's view of the map l to ku as a "tamely ramified" extension of ring…

We show that the map from $K({\mathbb S})$ to its chromatic completion is a connective cover and identify the fiber in $K$-theoretic terms. We combine this with recent work of Land-Mathew-Meier-Tamme to prove a form of "Waldhausen's…

We describe chromatic localisations of genuine L-spectra of discrete rings and deduce that the purity property of $K(1)$-local $K$-theory of rings established by Bhatt-Clausen-Mathew also holds in Grothendieck-Witt theory. In addition, we…