K-Theory and Homology

In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for To\"en and Vezzosi's secondary Chern character. As a main application, we establish…

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,…

We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the…

We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason's blow-up formula in algebraic K-theory to derived…

Letting $G=F/R$ be a finitely-presented group, Hopf's formula expresses the second integral homology of $G$ in terms of $F$ and $R$. Expanding on previous work, we explain how to find generators of $H_2(G;\mathbb{F}_p)$. The context of the…

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study…

Let $R$ be a left-Gorenstein ring. We show that there is a Quillen equivalence between singular contraderived model category and singular coderived model category. Consequently, an equivalence between the homotopy category of exact…

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift…

We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove…

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and prove that…

Let $n$ be an integer greater than or equal to $3$ and $(R,\Delta)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,\Delta)$ normalised by a relative…

For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results…

In the founding paper on unbounded $KK$-theory it was established by Baaj and Julg that the bounded transform, which associates a class in $KK$-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability…

This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a…

In this short note, we given a new proof of Mitchell's theorem that $L_{T\left(n\right)} K(Z) \cong 0$ for $n \geq 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology…

In this short note, for a morphism of Waldhausen categories $f\colon \mathbb{A} = (\mathcal{A} ,w_{\mathbb{A}}) \to \mathbb{B} = (\mathcal{B},w_{\mathbb{B}})$, we will define $\operatorname{Cone} f$ to be a Waldhausen category. There exists…

In this paper, we show local Gersten's conjecture for regular system of parameters. As its consequence we obtain Gersten's conjecture for a commutative regular local ring and smooth over a commutative discrete valuation ring.

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the…

This paper has been merged with arXiv:0908.3765

We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel…