Related papers: The $p$-completed cyclotomic trace in degree $2$

The uniform tracial completion of a C*-algebra A with compact non-empty trace space T(A) is obtained by completing the unit ball with respect to the uniform 2-seminorm $\|a\|_{2,T(A)}=\sup_{\tau \in T(A)} \tau(a^*a)^{1/2}$. The trace…

This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For…

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms phi:SL(2) --> G whose image is geometrically G-completely reducible…

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the…

We construct a trace map on the chiral homology of chiral Weyl algebra for any smooth Riemann surface. Our trace map can be viewed as a chiral version of the deformed HKR quasi-isomorphism. This also provides a mathematical rigorous…

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism B_n^1 -> (ZZ/N ZZ)^n rtimes S_n, where B_n^1 is a braid group of type B. The…

The Runge approximation theorem for holomorphic maps (U -> C) is a fundamental result in complex analysis. The aim of this article is to prove such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a compact…

There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over $\overline{\mathbb{Z}_p}$) whose spectra are inverse limits…

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as…

We study algebraic K-theory, syntomic cohomology, and prismatic cohomology of Cartier smooth rings. As an application, we provide an alternative proof of Kelly-Morrow's generalization of the Geisser-Levine theorem computing $p$-adic…

Let $p\in \mathbb Z$ be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum $\mathbb S$ admits an "eigensplitting" that generalizes known splittings on $K$-theory and $TC$. We identify the summands…

In this note we present a complete computation of the topological K-theory of the reduced C*-algebra of a semidirect product of the form $\Gamma=\mathbb{Z}^n\rtimes_\rho\mathbb{Z}/2$ with no further assumptions about of the conjugacy action…

We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

The classical Szeg\H{o}-Verblunsky theorem relates integrability of the logarithm of the absolutely continuous part of a probability measure on the circle to square summability of the sequence of recurrence coefficients for the orthogonal…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

Let $G$ be a reductive group over a field $k$ which is algebraically closed of characteristic $p \neq 0$. We prove a structure theorem for a class of subgroup schemes of $G$, for $p$ bounded below by the Coxeter number of $G$. As…

In this paper we continue the work of using the recent advances in algebraic $K$-theory to extend computations done in characteristic $p$ to the mixed characteristic setting using perfectoid rings. We extend the work of Hesselholt-Nikolaus…

We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness…

The classical trace map is a highly non-trivial map from algebraic K-theory to topological Hochschild homology (or topological cyclic homology) introduced by B\"okstedt, Hsiang and Madsen. It led to many computations of algebraic K-theory…

We construct a zig-zag from the once delooped space of pseudoisotopies of a closed $2n$-disc to the once looped algebraic $K$-theory space of the integers and show that the maps involved are $p$-locally $(2n-4)$-connected for $n>3$ and…