Related papers: D\'{e}vissage for Waldhausen K-theory

Walker's cancellation theorem says that if B+Z is isomorphic to C+Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the…

It is shown that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau variety birationally equivalent to it. The key idea is to construct suitable homomorphisms…

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts $S$ and a fixed…

A central question in equivariant algebraic K-theory asks whether there exists an equivariant K-theory machine from genuine symmetric monoidal G-categories to orthogonal G-spectra that preserves equivariant algebraic structures. We answer…

Type theory can be described as a generalised algebraic theory. This automatically gives a notion of model and the existence of the syntax as the initial model, which is a quotient inductive-inductive type. Algebraic definitions of type…

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture…

We prove a twisting theorem for nodal classes in permutation-equivariant quantum $K$-theory, and combine it with existing theorems of Givental to obtain a twisting result for general characteristic classes of the virtual tangent bundle.…

We study the structure of singularities in the discrete Korteweg-deVries (d-KdV) equation. Four different types of singularities are identified. The first type corresponds to localised, `confined', singularities, the confinement constraints…

For any simply-laced type simple Lie algebra $\mathfrak{g}$ and any height function $\xi$ adapted to an orientation $Q$ of the Dynkin diagram of $\mathfrak{g}$, Hernandez-Leclerc introduced a certain category $\mathcal{C}^{\leq \xi}$ of…

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different "sorts." We prove a rigidification result for simplicial algebras over these theories, showing…

A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending the…

Let X --> B be a proper submersion with a Riemannian structure. Given a differential K-theory class on X, we define its analytic and topological indices as differential K-theory classes on B. We prove that the two indices are the same.

Let $W$ be a finite dimensional algebraic structure (e.g. an algebra) over a field $K$ of characteristic zero. We study forms of $W$ by using Deligne's Theory of symmetric monoidal categories. We construct a category $\mathcal{C}_W$, which…

In this paper, we apply quantitative operator K-theory to develop an algorithm for computing K-theory for the class of filtered C *-algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the K{\"u}nneth formula…

Let K be a subgroup of the inhomogeneous Hecke group G_5 of finite index. Suppose that the geometric level of K is r. Then K is congruence if and only if K contains the principal congruence subgroup of level 2r.

Given a maximal rigid object $T$ of the cluster tube, we determine the objects finitely presented by $T$. We then use the method of Keller and Reiten to show that the endomorphism algebra of $T$ is Gorenstein and of finite representation…

Let $k$ be an algebraically closed field of characteristic $p>0$, $W$ the ring of Witt vectors over $k$ and ${R}$ the integral closure of $W$ in the algebraic closure ${\bar{K}}$ of $K:=Frac(W)$; let moreover $X$ be a smooth, connected and…

We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction. This construction yields a quasicategory whose weak homotopy type is a delooping of the K-theory space. We show…

For an additive Waldhausen category linear over a ring $k$, the corresponding $K$-theory spectrum is a module spectrum over the $K$-theory spectrum of $k$. Thus if $k$ is a finite field of characteristic $p$, then after localization at $p$,…

In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded $\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert modules with…