K理论与同调

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of…

In the paper "Aquino, C., Jim\'enez, R., Mijangos, M., Morales Mel\'endez, Q.: On Invariant (co)homology of a group, preprint" are introduced two groups generated by the orbits of an action of a group on another group by automorphisms. One…

In this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities, and in two variables for the case of commutativity. It is considered a large amount of…

Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\Or(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair $(G,H)$ defined by Adamson and Takasu to have…

We prove two basic structural properties of the algebraic $K$-theory of rings after $K(1)$-localization at an implicit prime $p$. Our first result (also recently obtained by Land--Meier--Tamme by different methods) states that $L_{K(1)}…

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of…

We compare the invariants of flat vector bundles defined by Atiyah et al. and Jones et al. and prove that, up to weak homotopy, they induce the same map, denoted by $e$, from the $0$-connective algebraic $K$-theory space of the complex…

In this paper, we prove that both topological Hochschild homology and topological cyclic homology are sheaves for the fpqc topology on connective commutative ring spectra, by exploiting the May filtration on topological Hochschild homology.

Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the Fundamental Theorem of $K$-theory from the category of simplicial rings to the category of $\mathbb{S}$-algebras. As an intermediate step, we prove the Fundamental…

We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$, which defines a natural splitting map from the algebraic $K$-theory of spaces to stable homotopy, is natural up to \emph{weak} homotopy with respect to transfer…

If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra \[…

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac…

This is the first in a series of papers constructing geometric models of twisted differential K-theory. In this paper we construct a model of even twisted differential K-theory when the underlying topological twist represents a torsion…

This paper explores further the computation of the twisted K-theory and K-homology of compact simple Lie groups, previously studied by Hopkins, Moore, Maldacena-Moore-Seiberg, Braun, and Douglas, with a focus on groups of rank 2. We give a…

Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$…

We show that faithfully flat smooth extensions are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We…

We compute the Hochschild cohomology of the reduced incidence algebras such as the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We…

We develop an appropriate dihedral extension of the Connes-Moscovici characteristic map for Hopf *-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes…

In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld-Jimbo quantum enveloping algebra $U_q(\mathfrak{g})$ for an arbitrary semi-simple Lie algebra $\mathfrak{g}$ with coefficients in a modular…

We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in Andr\'e-Quillen…