Mathematics

We prove several results showing that the algebraic K -theory of valuation rings behave as though such rings were regular Noetherian, in particular an analogue of the Geisser--Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic K -theory.

We modify Grayson's model of K 1 of an exact category to give a presentation whose generators are binary acyclic complexes of length at most k for any given k≥2 . As a corollary, we obtain another, very short proof of the identification of Nenashev's and Grayson's presentations.

We compute the group homology, the algebraic K - and L -groups, and the topological K -groups of right-angled Artin groups, right-angled Coxeter groups, and more generally, graph products.

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class groups.

We compute explicitly the K-groups of some boundary groupoid C*-algebras with exponential isotropy subgroups. Then we derive index formulas that computes the K-theoretic and Fredholm indexes of elliptic (respectively totally elliptic) pseudo-differential operators on these groupoids.

We show that the algebraic K-theory space of stable infinity-categories is canonically functorial in polynomial functors.

Given a henselian pair (R,I) of commutative rings, we show that the relative K -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace K→TC . This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod n coefficients, with n invertible in R ) and McCarthy's theorem on relative K -theory (when I is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between p -adic K -theory and topological cyclic homology for a large class of p -adic rings. In addition, we show that K -theory with finite coefficients satisfies continuity for complete noetherian rings which are F -finite modulo p . Our main new ingredient is a basic finiteness property of TC with finite coefficients.

We compute K-theory for the reduced group C*-algebras of generalized Lamplighter groups.

We study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes.

We give a K -theoretic criterion for a quasi-projective variety to be smooth. If L is a line bundle corresponding to an ample invertible sheaf on X , it suffices that K q (X)= K q (L) for all q≤dim(X)+1 .