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The twisted Hochschild homology groups of generic quantum hyperplanes are calculated using the Koszul resolution. For the example of the two-dimensional quantum plane also the twisted cyclic homology groups are determined.
Let $$F = \mathbb{Q} (\sqrt{-d_1})$$ and $$E = \mathbb{Q} (\sqrt {-d_1}, \sqrt{d_2})$$ , d1 and d2 squarefree integers, be an imaginary field and a biquadratic field, respectively. Let S be the set consisting of all infinite primes, all dyadic primes and all finite primes which ramify in E. Suppose the 4-rank of the class group of F is zero and the S-ideal class group of F has odd order,...
We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C*-algebras. Then we employ this result to show that the K-groups of our family of...
We study K2 of one-dimensional local domains which are essentially of finite type over a field of characteristic 0. In particular, we show that Berger’s conjecture implies Geller’s conjecture for such rings. This verifies Geller’s conjecture in many new cases of interest.
We introduce a new version kkalg of bivariant K-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra W, i.e. the algebra generated by two elements satisfying the Heisenberg commutation relation, with the fine locally convex topology. We determine its kkalg-invariants using a natural extension for W. Using analogous methods the kkalg-invariants...
We construct a spectral sequence which starts with RO $$(\mathbb{Z}/2)$$ -graded equivariant cohomology and converges to Atiyah’s KR theory. This is the analog of a well-known spectral sequence connecting motivic cohomology to algebraic K-theory.
. We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.
Traces taking values in suitable ‘Hochschild complexes’ are defined in the context of symmetric monoidal categories and applied to various categories of chain complexes, simplicial abelian groups, and symmetric spectra. Topological applications to parametrized fixed point theory are given.
Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra $$\mathcal{N}$$ G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet...
. In this paper we compute the group H2(SL2(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H2(SL2, (F)) is the fiber product of λ*:K2(F)→ I2(F)/I3(F) and σ: I2(F) → I2(F)/I3(F) where λ* and σ map onto I...
. The notion of n-fold Čech derived functors is introduced and studied. This is illustrated using the n-fold Čech derived functors of the nilization functors Zk. This gives a new purely algebraic method for the investigation of the Brown–Ellis generalised Hopf formula for the higher integral group homology and for its further generalisation. The paper ends with an application to algebraic K-theory.
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