Related papers: A universal characterization of higher algebraic K…

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A-->B be the localisation with respect to a…

Waldhausen's $K$-theory of the sphere spectrum (closely related to the algebraic $K$-theory of the integers) is a naturally augmented $S^0$-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification…

Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of…

We continue the work initiated in arXiv:1206.3645, where we introduced a new stable symmetric monoidal $(\infty,1)$-category $SH_{nc}$ encoding a motivic stable homotopy theory for the noncommutative spaces of Kontsevich and obtained a…

Utilizing simplicial Waldhausen theory, we prove that the geometric realization of the topologized category of bounded chain complexes over complex numbers (resp. real numbers) is an infinite loop space that represents connective complex…

Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We…

The purpose of this paper is twofold. First, we use the motivic Landweber exact functor theorem to deduce that the Bott inverted infinite projective space is homotopy algebraic $K$-theory. The argument is considerably shorther than any…

Let G be a finite group. We systematically exploit general homological methods in order to reduce the computation of G-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor assigning to a…

In this paper we study the "holomorphic K-theory" of a projective variety, which is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson,…

We give an explicit point-set construction of the Dennis trace map from the $K$-theory of endomorphisms $K\mathrm{End}(\mathcal{C})$ to topological Hochschild homology $\mathrm{THH}(\mathcal{C})$ for any spectral Waldhausen category…

We show that for a coconnective ring spectrum satisfying regularity and flatness assumptions, its algebraic K-theory agrees with that of its $\pi_0$. We prove this as a consequence of a more general devissage result for stable infinity…

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous…

We develop the foundations of $G$-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce…

We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and A^1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and…

Higson proved that every homotopy invariant, stable and split exact functor from the category of $C^*$-algebras to an additive category factors through Kasparov's $KK$-theory. By adapting a group equivariant generalization of this result by…

The main objective of this paper is to propose a definition of non-connective K-theory for a wide class of relative exact categories which, in general, do not satisfy the factorization axiom and confirm that it agrees with the…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the…

We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for…

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the…