Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

The crossed coproduct theorem and Galois cohomology

We give a cohomological interpretation of the Brauer group of a coalgebra in terms of Galois coextensions and Galois cohomology. There is a crossed coproduct structure theorem, and the co-version of the classical splitting theorem holds for the Brauer group of an irreducible coreflexive coalgebra but it does not hold in general.     

Hereditary crossed products

We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.

     

The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.     

Thom elements and bivariant cyclic cohomology

We consider smooth dynamical systems with unital algebra. We show that the corresponding crossed product isH-unital, and use differentiable Thom elements for proving a Thom isomorphism in bivariant periodic cohomology.     

Nilpotent extensions and entire cyclic cohomology

We prove nilpotent ideals can be excised in entire cyclic cohomology. The proof is based on the construction of a certain Connes-Chern character map and uses the Cuntz-Quillen theory ofX-complexes.     

Preunits and weak crossed products

Using the notion of a preunit and the properties of idempotent morphisms, we give a general notion of a crossed product of an algebra A and an object V both living in a monoidal category C. We endow A⊗V with a multiplication and an idempotent morphism, whose image inherits the multiplication. Sufficient conditions for these multiplications to be associative are given. If the product on A⊗V has a preunit, the related idempotent is given in terms of the preunit, and its image has an algebra structure. A characterization of crossed products with preunit is given, and it is used to recover classical examples of crossed products and to study crossed products in weak contexts. Finally crossed products of an algebra by a weak bialgebra are recovered using this theory.     

Non-commutative shifts and crossed products

When A is a unital simple AF C∗-algebra and has a unique tracial state, it is shown that the crossed product of the two-sided infinite tensor product by the shift is a tracially AF C∗-algebra. A similar result is given to the crossed product of a certain non-unital two-sided infinite tensor product by the shift. Applying a far-reaching classification result of such C∗-algebras by H. Lin, we obtain an example of a one-parameter automorphism group on some AF C∗-algebra which is not approximately inner, a counter-example to the AF version of the so-called Powers-Sakai conjecture [23].     

Excision in entire cyclic cohomology

We prove that entire and periodic cyclic cohomology satisfy excision for extensions of bornological algebras with a bounded linear section. That is, for such an extension we obtain a six term exact sequence in cohomology. Received December 16, 1999 / final version received April 18, 2001?Published online May 29, 2001     

Group cohomology and control of p-fusion

We show that if an inclusion of finite groups H≤G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.     

2- 上循环交叉积

利用 von Neumann 代数 M 和带有正规的2- 上循环 μ 的离散群 G, 定义了2- 上循环交叉积, 推广了经典的离散交叉积, 并证明了2- 上循环交叉积具有结合律.     

Periodic cyclic cohomology of group rings

We generalize previous results ([2], [3], [4], etc.) relative to the cyclic homology and cohomology of the group algebra of G. In many cases, we express them in terms of the (co)homology of the discrete groups(u) = Z (u)/C (u), where 〈u〉 runs through the set of conjugacy classes of G and where Z(u) (resp. C(u)) denotes the centralizer of u (resp. the cyclic group generated by u).     

Continuous-trace groupoid crossed products

Let be a second countable, locally compact groupoid with Haar system, and let be a bundle of -algebras defined over the unit space of on which acts continuously. We determine conditions under which the associated crossed product is a continuous trace -algebra.