Related papers: Excision in algebraic K-theory revisited

To any pullback square of ring spectra we associate a new ring spectrum and use it to describe the failure of excision in algebraic $K$-theory. The construction of this new ring spectrum is categorical and hence allows to determine the…

We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Corti\~nas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of Andr\'e--Quillen, Hochschild, and cyclic homology. Our key…

We prove an adelic descent result for localizing invariants: for each Noetherian scheme $X$ of finite Krull dimension and any localizing invariant $E$, e.g., algebraic K-theory of Bass-Thomason, there is an equivalence $E(X)\simeq \lim…

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite…

We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the…

We prove the excision theorem for the $K$-theory of perfect complexes on Deligne-Mumford stacks. This is then used to study the Nisnevich site of such stacks. We prove the Nisnevich descent for the $K$-theory of perfect complexes. We also…

Algebraic $K$-theory is a homology theory that behaves very well on sufficiently nice objects such as stable $C^*$-algebras or smooth algebraic varieties, and very badly in singular situations. This survey explains how to exploit this to…

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…

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the…

For a given ring (domain) in $\overline{\mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $n\ge 3\,,$ the…

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision…

In this short note we present an elementary matrix-constructive proof of Quillen-Suslin theorem for Ore extensions: If $K$ is a division ring and $A:=K[x;\sigma,\delta]$ is an Ore extension, with $\sigma$ bijective, then every finitely…

We generalize Dirichlet's $S$-unit theorem from the usual group of $S$-units of a number field $K$ to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over $S$. Specifically, we demonstrate…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…

A result of Andr\'e Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\mathrm{GL}_n(\mathbb{A})$ of regular matrices over the ring of ad\`eles (over…

In his 1973 paper Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of a paper by Nenashev, we are able to give an algebraic proof of Quillen's Resolution…

Suppose X is a projective toric scheme defined over a commutative ring R equipped with an ample line bundle L. We prove that its K-theory has k+1 direct summands K(R) where k is minimal among non-negative integers such that the twisted line…

For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is closely related to canonical central extensions of loop groups. Just like classical loop group…

We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is…

Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $\mathbb{E}_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence,…