Related papers: Multifunctorial Inverse $K$-Theory

We construct a nontrivial inverse system of Cuntz algebras $\{{\cal O}_{n}:2\leq n<\infty\}$, whose inverse limit is *-isomorphic onto ${\cal O}_{\infty}$. By using this result, it is shown that the $K_{0}$-functor is discontinuous with…

We define an analytic index and prove a topological index theorem for a non-compact manifold $M\_0$ with poly-cylindrical ends. We prove that an elliptic operator $P$ on $M\_0$ has an invertible perturbation $P+R$ by a lower order operator…

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, matrix-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, kk_*(A,B),…

A family of partial functions of a class of algebras $\mathsf{K}$ is said to be an implicit operation of $\mathsf{K}$ when it is defined by a first order formula and it is preserved by homomorphisms. In this work, we develop the theory of…

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an…

We explore a wider theoretical framework that has quantum field theory built-in, taking the fact that quantum mechanics is reconstructed from quantum field theory as a hint. We formulate a quantum theory with an embedded structure by…

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…

The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture…

We use filtrations of the tangent bundle of a manifold starting with an integrable subbundle to define transverse symbols to the corresponding foliation, define a condition of transversally Rockland and prove that transversally Rockland…

Freed-Hopkins-Teleman expressed the Verlinde algebra as twisted equivariant K-theory. We study how to recover the full system (fusion algebra of defect lines), nimrep (cylindrical partition function), etc of modular invariant partition…

In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's…

We introduce bivariant K-theory for nonarchimedean bornological algebras over a complete discrete valuation ring $V$. This is the universal target for dagger homotopy invariant, matricially stable and excisive functors, similar to bivariant…

The aim of this paper is to describe the topological $K$-ring, in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring and hence the Grothendieck ring of algebraic…

In previous works, we introduced and studied certain categories called quasi-BPS categories associated to symmetric quivers with potential, preprojective algebras, and local surfaces. They have properties reminiscent of BPS invariants/…

We construct an equivariant coarse homology theory arising from the algebraic $K$-theory of spherical group rings and use this theory to derive split injectivity results for associated assembly maps. On the way, we prove that the…

Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$. We show that the multiplicative group $F^\times/k^\times$ endowed with the equivalence relation induced by algebraic…

This is a thorough investigation on the real $K$-theory of the sphere $S^d$ associated with the antipodal involution. We calculate the algebraic structure of real $K$-theory and united $K$-theory for all $d$, we write down explicit…

Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and…

We adapt Grayson's model of higher algebraic $K$-theory using binary acyclic complexes to the setting of stable $\infty$-categories. As an application, we prove that the $K$-theory of stable $\infty$-categories preserves infinite products.

Weibel proved that $p$-inverted K-theory is $\mathbb{A}^1$-invariant on $\mathbb{F}_p$-schemes and K-theory with $\mathbb{Z}/p$-coefficients is $\mathbb{A}^1$-invariant on $\mathbb{Z}[\frac{1}{p}]$-schemes. We extend this result to all…