Mathematics

We prove some K -theoretic descent results for finite group actions on stable ∞ -categories, including the p -group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if R is an E ∞ -ring spectrum with L T(n) R=0 , then L T(n+1) K(R)=0 . Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.

The Lichtenbaum--Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni--Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the n -th Greek letter family is detected by a commutative ring spectrum R , then we conjecture that the n+1 -st Greek letter family will be detected by the algebraic K-theory of R . We prove this in the case n=1 for R=K( F q ) p modulo (p, v 1 ) where p≥5 and q is a prime power generator of the units in Z/ p 2 Z . In particular, we prove that the commutative ring spectrum K(K( F q ) p ) detects the part of p -primary β -family that survives mod (p, v 1 ) . The method of proof also implies that these β -elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to modular forms satisfying certain congruences.

We introduce a new perspective on the K -theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen Q -construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying dévissage and localization allows us to calculate a filtration on the K -theory of schemes of finite type. As an application of this theory we construct a comparison map showing that the two authors' definitions of the Grothendieck spectrum of varieties are equivalent.

The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is that producing HKR isomorphisms for other types of algebras is directly related to computing quasi-free resolutions in the category of left modules over an operad; we establish that an HKR-type result follows as soon as this resolution is diagonally pure. As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative and smooth brace algebras, respectively. We also prove an HKR theorem for operads obtained from a filtered distributive law, which recovers, in particular, all the aspects of the classical HKR theorem. Finally, we show that this property is Koszul dual to the operadic PBW property defined by V. Dotsenko and the second author (1804.06485).

In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT].

Dirac-Schrödinger operators on the real line coupled with a natural family of representations of the irrational rotation algebra are used to build a family of 2-dimensional spectral triples over the Schwartz algebra of the irrational torus. These spectral cycles (`Heisenberg cycles') fit easily into Connes' framework of Noncommutative Geometry: they are regular and give rise to meromorphic zeta functions, which we compute geometrically via Mehler's formula for the harmonic oscillator. We compute the Connes'-Moscovici Index formula for the Heisenberg cycles and determine their Chern characters, which are mixed-degree co-cochains with the standard trace in degree zero and Connes' curvature cocycle times the rotation parameter, in degree 2. We apply the resulting index computations to prove a result about the b-twist renormalization morphisms arising in a previous paper of the first author and A. Duwenig.

We construct distinguished free generators of the K 0 -group of the C*-algebra C( CP n T ) of the multipullback quantum complex projective space. To this end, first we prove a quantum-tubular-neighborhood lemma to overcome the difficulty of the lack of an embedding of CP n−1 T in CP n T . This allows us to compute K 0 (C( CP n T )) using the Mayer-Vietoris six-term exact sequence in K-theory. The same lemma also helps us to prove a comparison theorem identifying the K 0 -group of the C*-algebra C( CP n q ) of the Vaksman-Soibelman quantum complex projective space with K 0 (C( CP n T )) . Since this identification is induced by the restriction-corestriction of a U(1) -equivariant \mbox{*-homomorphism} from the C*-algebra C( S 2n+1 q ) of the (2n+1) -dimensional Vaksman-Soibelman quantum sphere to the C*-algebra C( S 2n+1 H ) of the (2n+1) -dimensional Heegaard quantum sphere, we conclude that there is a basis of K 0 (C( CP n T )) given by associated noncommutative vector bundles coming from the same representations that yield an associated-noncommutative-vector-bundle basis of the K 0 (C( CP n q )) . Finally, using identities in K-theory afforded by Toeplitz projections in C( CP n T ) , we prove noncommutative Atiyah-Todd identities.

A dévissage-type theorem in algebraic K-theory is a statement that identifies the K-theory of a Waldhausen category C in terms of the K-theories of a collection of Waldhausen subcategories of C when a dévissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between dévissage theorems of single type and of multiple type depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original dévissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of dévissage-type theorems and prove a general dévissage theorem of single type and a general dévissage theorem of multiple type.

We apply Goerss--Hopkins obstruction theory for motivic spectra to study the motivic Morava E -theories. We find that they always admit E ∞ structures, but that these may admit "exotic" E ∞ automorphisms not coming from the usual Morava stabilizer group.

We introduce in this work the notion of the category of pure E -Motives, where E is a motivic strict ring spectrum and construct twisted E -cohomology by using six functors formalism of J. Ayoub. In particular, we construct the category of pure Chow-Witt motives CHW(k ) Q over a field k and show that this category admits a fully faithful embedding into the geometric stable A 1 -derived category D A 1 ,gm (k ) Q .