arXiv subjects
Publications and source records attributed to Shay Ben-Moshe.
We study the algebraic K-theory of quotients of even ring spectra. Building on Efimov's work, we prove that the limit of the K-theories of these quotients is the continuous K-theory of nuclear modules. For quotients by elements in the height $n$ chromatic ideal, we show that this agrees with the K-theory of the ring spectrum, after chromatic localization at heights $\geq n{+}1$.
We give a new proof that algebraic K-theory increases chromatic height by at most one, originally established by Clausen-Mathew-Naumann-Noel. Our argument proceeds by descent from the Lubin-Tate spectrum, for which the required vanishing follows from Hahn-Wilson's Quillen-Lichtenbaum result.
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 using the Ravenel-Wilson computation of the Morava K-theory of Eilenberg-MacLane spaces.
We prove an ambidexterity result for $\infty$-categories of $\infty$-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable $\infty$-categories indexed by a space, and the $\infty$-semiadditivity of the $\infty$-category of $\infty$-categories with $\pi$-finite colimits proven by Harpaz. Our proof employs Stefanich's universal property for the higher category of iterated spans, which encodes ambidexterity phenomena in a coherent fashion.
We study the compatibility of higher semiadditivity across different chromatic heights. We prove that the categorified transchromatic character map assembles into a parameterized semiadditive functor, showing that it is higher semiadditive up to a free loop shift. By decategorification, this implies that the transchromatic character map is compatible with integration maps, generalizing the well-known formula for the character of an induced representation. Through a further decategorification process, we compute the higher semiadditive cardinalities of $\pi$-finite spaces at Lubin-Tate spectra.
We make Hinich's $\infty$-categorical enriched Yoneda embedding natural. To do so, we exhibit it as the unit of a partial adjunction between the functor taking enriched presheaves and Heine's functor taking a tensored category to an enriched category. Furthermore, we study a finiteness condition of objects in a tensored category called being atomic, and show that the partial adjunction restricts to a (non-partial) adjunction between taking enriched presheaves and taking atomic objects.
Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $π$-finite $p$-space $A$ at the Lubin-Tate spectrum $E_n$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^n A$. We explain how this allows one to bypass the Ravenel-Wilson computation in the proof of the $\infty$-semiadditivity of the $T(n)$-local categories.
We show that the Yoneda embedding extends to an $(\infty,2)$-natural transformation. Furthermore, as such, it is uniquely determined by its value at the trivial $\infty$-category. We also study the naturality of the Yoneda lemma in its arguments, showing that it is an isomorphism of $(\infty,2)$-natural transformations.
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. Using this, we show that it sends $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions. Furthermore, we show that it sends cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$, extending a result of Bhatt-Clausen-Mathew for $n=0$. As a consequence, we deduce that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring. Counterexamples to such cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory were constructed by Burklund, Hahn, Levy and the third author, thereby disproving the telescope conjecture.
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 the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$, which are satisfied for example by the Lubin-Tate spectrum $E_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $T(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson-Wilson spectrum $\widehat{E(n)}$.
Weak value is increasingly acknowledged as an important research tool for probing quantum pre- and post-selected ensembles, where some extraordinary phenomena occur. We generalize this concept to the broader notion of "weak potential" which enables predicting the interactions between particles when one of them is pre-/post-selected and the interaction potential is small. A harmonic oscillator is considered, undergoing weak position and momentum measurements between strong position measurement, and shown to possess peculiar physical properties, affecting the momentum rather than the position of another oscillator interacting with it.