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Let $$\mathbb{C}{\text{, }}\mathbb{D} {\text{and }}\mathbb{E}$$ be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor $$\mathbb{C} \to \mathbb{D}$$ , i.e., a functor 2q $$ \otimes {\text{ }}\mathbb{C} \to \mathbb{D}$$ , induce a right q-transfor $$\mathbb{C} \to \mathbb{D}$$ , i.e., a functor...
We prove that the Assembly map in algebraic K-theory is split injective for groups of finite asymptotic dimension admitting a finite classifying space.
We establish the Hasse principle (local-global principle) in the context of the Baum–Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field.
We compute the K-theory groups of Melrose's algebra of 1-suspended pseudo-differential operators. The boundary map in the six-term long exact sequence turns out to be related to both the eta invariant of Melrose and to the index of elliptic operators. The proof is based on a new identity between the formal trace and the Wodzicki residue trace on the suspended algebra.
To deal with technical issues in noncommutative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. These algebras and modules allow us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral triples are closed under the C∞ functional calculus of self-adjoint elements. In the final...
We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L2-space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SUq(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant...
We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple (A, H, M) consisting of a Hopf algebra H, an H-comodule algebra A, an H-module M, and a compatible grouplike element σ in H, we define the cyclic module of invariant chains on A with coefficients in M and call its cyclic homology the invariant cyclic homology of A with coefficients in M. We also develop...
We show that the Atiyah–Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold is such a stratified vector bundle.
We observe that any regular Lie groupoid G over a manifold M fits into an extension K → G → E of a foliation groupoid E by a bundle of connected Lie groups K. If F is the foliation on M given by the orbits of E and T is a complete transversal to F , this extension restricts to T, as an extension KT → GT → ET of an étale groupoid ET by a bundle of connected groups KT. We break up the...
A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover,...
Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n ≥ d + 3. Let I ⊂ A[T] be an ideal of height n. Assume that I = (f1,...,fn) + (I2T). Under these assumptions, it is proved in this paper that I = (g1,...,gn) with fi − gi ⊂ (I2T), thus settling a question of Nori affirmatively.
One of the main properties of Hochschild homology of the algebra of smooth functions on a smooth manifold is its local character. In this paper, we consider subalgebras of smooth functions which are significant for singular spaces such that simplicial complexes or cones over smooth manifolds. We compute their Hochschild homology and investigate the local character. Our computations show that, in opposition...
To any bimodule that is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown that separable bimodules give rise to coseparable comatrix...
In this paper we develop a K-theory of log schemes by using vector bundles on the Ket site. Then, for a wide class of log varieties, we describe the structure of their K-groups in terms of the usual algebraic K-groups.
We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L2(R),...
In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.
The classical Abel–Jacobi map is used to geometrically motivate the construction of regulator maps from Milnor K-groups KnM(C(X)) to Deligne cohomology. These maps are given in terms of some new, explicit (n − 1)-currents, higher residues of which are defined and related to polylogarithms. We study their behavior in families Xs and prove a rigidity result for the regulator image of the Tame...
This paper classifies under a local stable rank condition for rings with form parameter, subgroups of the general linear group GL2l which contain the elementary unitary subgroup EU2l.
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