K理论与同调

For an infinite field $F$, we study the kernel of the map $H_{n}(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]) \to H_{n}(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big])$ and the cokernel of…

Let $A \rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let $\rho: A\times G\to G$ be the multiplication map. We show that there is a natural map $\varphi: H_1(\Sigma_2^\epsilon, {\rm…

For a central perfect extension of groups $A \rightarrowtail G\twoheadrightarrow Q$, first we study the natural image of $H_3(A,\mathbb{Z})$ in $H_3(G, \mathbb{Z})$. As a particular case, we show that if the extension is universal this…

We describe chromatic localisations of genuine L-spectra of discrete rings and deduce that the purity property of $K(1)$-local $K$-theory of rings established by Bhatt-Clausen-Mathew also holds in Grothendieck-Witt theory. In addition, we…

Let H = H (R,q) be an affine Hecke algebra with complex, possibly unequal parameters q, which are not roots of unity. We compute the Hochschild and the cyclic homology of H. It turns out that these are independent of q and that they admit…

Let $B\subset A$ be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that $B$ satisfies Han's conjecture if and only if $A$ does, regardless if the extension…

The present paper contains new geometric theorems in mixed characteristic case. We derive a bunch of cohomological consequences using these geometric theorems. Among them an isotropy result for quadratic spaces, a purity result for…

This paper formulates a group condition which is enjoyed by absolute Galois groups, and which guarantees that profinite groups satisfying the condition can be approximated as an inverse limit of groups which are profinite analogues of…

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence…

Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $\lambda \in \ell$ such that $\lambda + \lambda^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action…

The Baum-Connes conjecture predicts that a certain assembly map is an isomorphism. We identify the homotopy theoretical construction of the assembly map by Davis and L\"uck with the category theoretical construction by Meyer and Nest. This…

We prove that the hermitian Gersten-Witt complex is exact for Azumaya algebras with involution of the first- or second kind over a regular local ring, which is essentially smooth over a field, or over a discrete valuation ring.

We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper…

We introduce a differential extension of algebraic K-theory of an algebra using Karoubi's Chern character. In doing so, we develop a necessary theory of secondary transgression forms as well as a differential refinement of the smooth…

Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory…

We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small' $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of…

We compute the Hochschild cohomology ring of the algebras $A= k\langle X, Y\rangle/ (X^a, XY-qYX, Y^a)$ over a field $k$ where $a\geq 2$ and where $q\in k$ is a primitive $a$-th root of unity. We find the the dimension of $\mathrm{HH}^n(A)$…

We compute the equivariant Bauer-Furuta degree, when a finite group acts freely on a spin 4-manifold. In the case when the group is cyclic of order power of two, Bryan gave a formula and its applications. We have treated the case when the…

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $\mathbb Z,$ and study connections…

It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong \Lambda(A/p)\otimes \Gamma({}_pA)$ and for finitely generated abelian groups there is the…