Related papers: An Algebraic Proof of Quillen's Resolution Theorem…
Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful computational tool for calculating $K$-groups of exact categories. At the level of $K_0$, this result goes back to Grothendieck. In this article, we first establish an…
The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented…
We present news proofs of the additivity, resolution and cofinality theorems for the algebraic $K$-theory of exact categories. These proofs are entirely algebraic, based on Grayson's presentation of higher algebraic $K$-groups via binary…
We show that Quillen's resolution theorem for K-theory also applies to exact $\infty$-categories. We introduce heart structures on a stable $\infty$-category, generalizing weight structures, and using resolution ideas, we show that the…
We give a complete and careful proof of Quillen's theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.
We prove a localisation theorem for the K-theory of filtering subcategories of exact $\infty$-categories which subsumes the localisation theorem for stable $\infty$-categories, Quillen's localisation theorem for abelian categories, and…
We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative…
The additivity theorem for derivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity,…
Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action…
By a theorem of Suslin, a Tor-unital (not necessarily unital) ring satisfies excision in algebraic K-theory. We give a new and direct proof of Suslin's result based on an exact sequence of categories of perfect modules. In fact, we prove a…
We discuss the proof of a certain integral theorem obtained by C. G. Cullen, originally stated on the class of the analytic intrinsic functions on the quaternions. It is shown that this integral theorem is true for a larger class of…
We introduce a new notion of regularity for rings and exact categories and we show important results in algebraic K-theory. In particular we prove a strong vanishing theorem for Nil groups and give an explicit class of groups, much bigger…
In a previous paper I gave a presentation for the Quillen higher algebraic K-groups of an exact category in terms of "acyclic binary multicomplexes". In this paper I take that presentation as a definition of the higher K-groups, generalize…
An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in…
We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.
We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite…
We establish a long exact sequence for the homotopy K-theory groups of the algebraic Cuntz-Pimsner rings introduced by Carlsen and Ortega [CO11] by adapting Pimsner's original proof [Pim97] to Cuntz's formalism.
It is proven that in the universal splitexact equivariant algebraic $KK$-theory for algebras, the $K$-theory groups coincide with classical $K$-theory in the sense of Phillips. This partially answers a question raised by Kasparov.
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the…
We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable…