Related papers: Hyperdescent and \'etale K-theory

Building on the Waldhausen and Quillen models of higher algebraic $K$-theory for exact categories and Waldhausen categories attached to a non-commutative $n$-ary $\Ga$-semiring $(T,\Ga)$, we establish the fundamental formal properties of…

This paper continues the study of K-theoretic invariants for semigroup C*-algebras attached to ax+b-semigroups over rings of algebraic integers in number fields. We show that from the semigroup C*-algebra together with its canonical…

We prove a non-linear version of a theorem of Grayson which is an analogue of the Fundamental Theorem of Algebraic $K$-theory and identify the $K$-theory of the endomorphism category over a space $X$ in terms of reduced $K$-theory of a…

We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…

In this article, the theory of sheaves is studied from a categorical point of view. This perspective vastly generalizes the usual theory of sheaves of sets to a more abstract setting which allows us to investigate the theory of sheaves with…

We construct a notion of derived completion which applies to homomorphisms of commutative S-algebras. We study the relationship of the construction with other constructions of completions, and prove various invariance properties. The…

We show that equivariant K-theory satisfies descent with respect to the isovariant Nisnevich topology. The main step is to show that the isovariant Nisnevich topology is a regular, complete and bounded cd topology.

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic…

In this paper we establish a universal characterization of higher algebraic K-theory in the setting of small stable infinity categories. Specifically, we prove that connective algebraic K-theory is the universal additive invariant, i.e.,…

Let $\Omega \subseteq \mathbb C^m$ be a bounded connected open set and $\mathcal H \subseteq \mathcal O(\Omega)$ be an analytic Hilbert module, i.e., the Hilbert space $\mathcal H$ possesses a reproducing kernel $K$, the polynomial ring…

We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site C. We use this to prove homotopy invariance and generalized descent statements for categories of…

Let $SL_{2n}$, $Sp_{2n}$, $E_6 = G^{sc}(E_6)$, $F_4 = G(F_4)$ be simply connected split algebraic groups over an arbitrary field $F$. Algebraic K-theory of affine homogeneous varieties $SL_{2n}/Sp_{2n}$ and $E_6/F_4$ is computed. Moreover,…

We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant…

We give a direct description of the category of sheaves on Lichtenbaum's Weil-\'etale site of a number ring. Then we apply this result to define a spectral sequence relating Weil-\'etale cohomology to Artin-Verdier \'etale cohomology.…

We show that the classification of the symmetric spaces can be achieved by K-theoretical methods. We focus on Hermitian symmetric spaces of non-compact type, and define K-theory for JB*-triples along the lines of C*-theory. K-groups have to…

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute…

We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of…

This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras…

Let $C^*(E)$ be the graph $C^*$-algebra associated to a graph E and let J be a gauge invariant ideal in $C^*(E)$. We compute the cyclic six-term exact sequence in $K$-theory of the associated extension in terms of the adjacency matrix…

In this note, we prove a generalization of Efimov's computation for the universal localizing invariant of categories of sheaves with certain microsupport constraints. The proof is based on certain categorical equivalences given by the…