Cyclic homology, tight crossed products, and small stabilizations

The Cyclic Homology and K-theory of Certain Adelic Crossed Products

The multiplicative group of a global field acts on its adele ring by multiplication. We consider the crossed product algebra of the resulting action on the space of Schwartz functions on the adele ring and compute its Hochschild, cyclic and periodic cyclic homology. We also compute the topological K-theory of the C-algebra crossed product.

Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

We introduce the cylindrical module A♮H, where H is a Hopf algebra with S2 = idH and A is a Hopf module algebra over H. We show that there exists a cyclic map between the cyclic module of the crossed product algebra A⋊H and ∆(A♮H), the cyclic module related to the diagonal of A♮H. In the cocommutative case, ∆(A♮H) ∼= C•(A ⋊H). Finally we approximate ∆(A♮H) by a spectral sequence and we give an ...

The Homology of Cyclic Products

The Periodic Cyclic Homology of Crossed Products of Finite Type Algebras

We study the periodic cyclic homology groups of the cross-product of a finite type algebra A by a discrete group Γ. In case A is commutative and Γ is finite, our results are complete and given in terms of the singular cohomology of the sets of fixed points. These groups identify our cyclic homology groups with the “orbifold cohomology” of the underlying (algebraic) orbifold. The proof is based ...

Free Products, Cyclic Homology, and the Gauss-manin Connection

We present a new approach to cyclic homology that does not involve Connes’ differential and is based on (Ω q A)[u], d + u · ı∆, a noncommutative equivariant de Rham complex of an associative algebra A. Here d is the Karoubi-de Rham differential, which replaces the Connes differential, and ı∆ is an operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit ...