K-Theory and Homology
In this article we classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\hat{{\mathbb{Z}}}$ one. Moreover, although…
Owing to the difference in $K$-theory, an example by Dugger and Shipley implies that the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for…
We introduce an operad which acts on the Gerstenhaber-Schack complex of a prestack as defined by Dinh Van and Lowen, and which in particular allows us to endow this complex with an underlying $L_{\infty}$-structure. We make use of the…
We prove cancellation theorems for reciprocity sheaves and cube-invariant modulus sheaves with transfers of Kahn--Saito--Yamazaki, generalizing Voevodsky's cancellation theorem for $\mathbf{A}^1$-invariant sheaves with transfers. As an…
We construct a monoidal structure on the category of assemblers. As an application of this, we construct a derived local zeta-function which takes a variety over a finite field to the set of points over the separable closure, and use the…
We develop the theory of Mackey profunctors, a version of Mackey functors for profinite groups.
In this paper, we generalize the Cantor function to $2$-dimensional cubes and construct a cyclic $2$-cocycle on the Cantor dust. This cocycle is non-trivial on the pullback of the smooth functions on the $2$-dimensional torus with the…
We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.
We show that mapping spaces in the p-local motivic stable category over an Fp-scheme are strictly commutative monoids (whence HZ-modules) in a canonical way.
The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second…
For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model…
In this note, we consider the Gersten complex for Milnor $K$-theory over a regular local Henselian domain $S$ and prove that in degrees $\geq \dim S\geq 1$, the Gersten complex of an essentially smooth Henselian local $S$-scheme is exact.
This paper is an introduction to Hopf cyclic cohomology with an emphasis on its most recent developments. We cover three major areas: the original definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of their study…
Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two main inroads: motivic and cyclic homology. I've been…
We formulate and prove an index theorem for loop spaces of compact manifolds in the framework of $KK$-theory. It is a strong candidate for the noncommutative geometrical definition (or the analytic counterpart) of the Witten genus. In order…
In this note, we provide a proof of the generalised Green-Julg theorem by using the language of twisted localization algebras introduced by G. Yu. This proof is for those who have interests in coarse geometry but not so familiar with…
We establish a formula for the L-theory spectrum of real $C^*$-algebras from which we deduce a presentation of the L-groups in terms of the topological K-groups, extending all previously known results of this kind. Along the way, we extend…
This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure w. We…
We prove a version of J.P. May's theorem on the additivity of traces, in symmetric monoidal stable $\infty$-categories. Our proof proceeds via a categorification, namely we use the additivity of topological Hochschild homology as an…
We extend the theory of Mackey 2-functors introduced in arXiv:1808.04902 by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical…