Related papers: Regularity of structured ring spectra and localiza…

We study a modified version of Rognes' logarithmic structures on structured ring spectra. In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra. The…

Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring $A$ essentially finitely presented over a valuation ring $V$. We discuss several properties of this notion of regularity for…

We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{\pi_1(X)}])$…

We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is…

Using a construction closely related to Waldhausen's $S_\bullet$-construction, we produce a spectrum $K(\mathbf{Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (\mathbf{Var}_{/k})$. We then…

This paper is partly a report on current knowledge concerning the structure of (generic) quantized coordinate rings and their prime spectra, and partly propaganda in support of the conjecture that since these algebras share many common…

Let M_d(k) denote the space of dxd-matrices with coefficients in an algebraically closed field k. Let X be an orbit closure in the product [M_d(k)]^t equipped with the action of the general linear group GL_d(k) by simultaneous conjugation.…

Algebraic K-theory is the stable homotopy theory of homotopy theories, and it interacts with algebraic structures accordingly. In particular, we prove the Deligne Conjecture for algebraic K-theory.

We prove an explicit combinatorial formula for the structure constants of the Grothendieck ring of a Grassmann variety with respect to its basis of Schubert structure sheaves. We furthermore relate K-theory of Grassmannians to a bialgebra…

We prove two basic structural properties of the algebraic $K$-theory of rings after $K(1)$-localization at an implicit prime $p$. Our first result (also recently obtained by Land--Meier--Tamme by different methods) states that $L_{K(1)}…

We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better…

For a cyclic group $C_n$, we identify Greenlees' equivariant connective K theory spectrum $kU_{C_n}$ as an $RO(C_n)$-graded localization of the actual connective cover of $KU_{C_n}$.

We introduce the notion of a bounded weight structure on a stable $\infty$-category and use this to prove the natural generalization of Waldhausen's sphere theorem: We show that the algebraic $K$-theory of a stable $\infty$-category with a…

The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established and we prove that the Quillen K- groups are isomorphic to smooth K-groups for monoid algebras…

We prove a mixed-characteristic analogue of Kunz's theorem in terms of perfectoid towers: a Noetherian local ring of residue characteristic $p$ is regular if and only if it admits a flat map to a Noetherian ring that extends to a perfectoid…

The aim of this short paper is to establish a spectral algebra analog of the Bousfield-Kan "fibration lemma" under appropriate conditions. We work in the context of algebraic structures that can be described as algebras over an operad…

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…

The goal of this note is to present Kaplansky's proof of the Regular Element Property and to explain how this argument can be adapted to the case of a coherent, strongly discrete and Noetherian (with an inductive definition of Noetherian)…

This paper establishes an analogue of the special chain theorem for the embedding dimension of polynomial rings, with direct application on the (embedding) codimension. In particular, we recover a classic result on the transfer of…

The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can…