Mathematics

We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle A × f ρ H , of a weak module algebra A by a weak Hopf algebra H . This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle A × f ρ H , of a weak module algebra A by a weak Hopf algebra H whose unit cocommutes. This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When f takes its values in a separable subalgebra of A that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

Let k be a field of characteristic 0 and A a curved k -algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect A -module taking values in H N II 0 (A) , the negative cyclic homology of the second kind associated to A , when A satisfies a certain smoothness condition.

Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the Fundamental Theorem of K -theory from the category of simplicial rings to the category of S -algebras. As an intermediate step, we prove the Fundamental Theorem for simplicial rings appealing to recent results from the first author's thesis. This recovers as a special case the Fundamental Theorem for the K -theory of spaces appearing in Hüttemann-Klein-Vogell-Waldhausen-Williams.

We define a functorial "Artin map" attached to any small Z -linear stable ??-category, which in the case of perfect complexes over a global field recovers the usual Artin map from the idele class group to the abelianized absolute Galois group. In particular, this gives a new proof of the Artin reciprocity law.

We reframe Paradan-Vergne's approach to quantization commutes with reduction in KK-theory through a recent formalism introduced by Kasparov, focusing more especially the index theoretic parts that lead to their "Witten non-abelian localization formula". While our method uses the same ingredients as their's in spirit, interesting conceptual simplifications occur, and the relationship to the Ma-Tian-Zhang analytic approach becomes quite transparent.

We study the index theory of a class of perturbed Dirac operators on non-compact manifolds of the form D+ic(X) , where c(X) is a Clifford multiplication operator by an orbital vector field with respect to the action of a compact Lie group. Our main result is that the index class of such an operator factors as a KK-product of certain KK-theory classes defined by D and X . As a corollary we obtain the excision and cobordism-invariance properties first established by Braverman. An index theorem of Braverman relates the index of D+ic(X) to the index of a transversally elliptic operator. We explain how to deduce this theorem using a recent index theorem for transversally elliptic operators due to Kasparov.

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.

Let A be a C*-algebra that is the norm closure A= ∑ β∈α I β ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ of an arbitrary sum of C*-ideals I β ⊆A . We construct a homological spectral sequence that takes as input the K-theory of ⋂ j∈J I j for all finite nonempty index sets J⊆α and converges strongly to the K-theory of A . For a coarse space X , the Roe algebra C ∗ X encodes large-scale properties. Given a coarsely excisive cover { X β } β∈α of X , we reshape C ∗ X β as input for the spectral sequence. From the K-theory of C ∗ X( ⋂ j∈J X j ) for finite nonempty index sets J⊆α , we compute the K-theory of C ∗ X if α is finite, or of a direct limit C*-ideal of C ∗ X if α is infinite. Analogous spectral sequences exist for the algebra D ∗ X of pseudocompact finite-propagation operators that contains the Roe algebra as a C*-ideal, and for Q ∗ X= D ∗ X/ C ∗ X .

In this paper, we study weakly unital dg categories as they were defined by Kontsevich and Soibelman [KS, Sect.4]. We construct a cofibrantly generated Quillen model structure on the category Cat dgwu (k) of small weakly unital dg categories over a field k . Our model structure can be thought of as an extension of the model structure on the category Cat dg (k) of (strictly unital) small dg categories over k , due to Tabuada [Tab]. More precisely, we show that the imbedding of Cat dg (k) to Cat dgwu (k) is a right adjoint of a Quillen pair of functors. We prove that this Quillen pair is, in turn, a Quillen equivalence. In course of the proof, we study a non-symmetric dg operad O , governing the weakly unital dg categories, which is encoded in the Kontsevich-Soibelman definition. We prove that this dg operad is quasi-isomorphic to the operad Assoc + of unital associative algebras.