Mathematics

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.

We develop a generalization of quantitative K -theory, which we call controlled K -theory. It is powerful enough to study the K -theory of crossed product of C ∗ -algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum-Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.

The goal of this paper is to associate functorially to every symmetric monoidal additive category A with a strict G -action a lax symmetric monoidal functor V G A :GBornCoarse→ Add ∞ from the symmetric monoidal category of G -bornological coarse spaces GBornCoarse to the symmetric monoidal ∞ -category of additive categories Add ∞ . This allows to refine equivariant coarse algebraic K -homology to a lax symmetric monoidal functor.

We associate to every G -bornological coarse space X and every left-exact ∞ -category with G -action a left-exact infinity-category of equivariant X -controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact ∞ -categories.

We show that a complete hereditary cotorsion pair $(\C,\C^\bot)$ in an exact category $\E$, together with a subcategory $\Z\subseteq\E$ containing $\C^\bot$, determines a Waldhausen category structure on the exact category $\C$, in which $\Z$ is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the K -theory of exact categories $\A\subseteq\B$ to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require $\A$ to be a Serre subcategory, which produces new examples. Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.

In an intriguing paper arXiv:math/0509083 Khovanov proposed a generalization of homological algebra, called Hopfological algebra. Since then, several attempts have been made to import tools and techiniques from homological algebra to Hopfological algebra. For example, Qi arXiv:1205.1814 introduced the notion of cofibrant objects in the category C H A,H of H -equivariant modules over an H -module algebra A , which is a counterpart to the category of modules over a dg algebra, although he did not define a model structure on C H A,H . In this paper, we show that there exists an Abelian model structure on C H A,H in which cofibrant objects agree with Qi's cofibrant objects under a slight modification. This is done by constructing cotorsion pairs in C H A,H which form a Hovey triple in the sense of Gillespie arXiv:1512.06001. This can be regarded as a Hopfological analogues of the works of Enochs, Jenda, and Xu and of Avramov, Foxby, and Halperin. By restricting to compact cofibrant objects, we obtain a Waldhausen category P erf H A,H of perfect objects. By taking invariants of this Waldhausen category, such as algebraic K -theory, Hochschild homology, cyclic homology, and so on, we obtain Hopfological analogues of these invariants.

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of H S 3 iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding H S 2 and abelian extensions of groups.

In this article we prove that there exists an explicit bijection between nice d -pre-Calabi-Yau algebras and d -double Poisson differential graded algebras, where d∈Z , extending a result proved by N. Iyudu and M. Kontsevich. We also show that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of d -double Poisson dg algebras to the partial category of d -pre-Calabi-Yau algebras. Finally, we further generalize it to include double P ∞ -algebras, as introduced by T. Schedler.

We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to be zero in cohomology and hence the interesting structure is not given by this e_2-algebra structure but rather by the resulting e_3-algebra structure, which is expressed in terms of the cup product and B.

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any ring k⊃Q . In the second part, we extend the results to actions on locally convex algebras. We then deal with crossed-products associated with group actions on manifolds and smooth varieties. For the finite order components, the results are expressed in terms of what we call "mixed equivariant cohomology". This "mixed" theory mediates between group homology and de Rham cohomology. It is naturally related to equivariant cohomology, and so we obtain explicit constructions of cyclic cycles out of equivariant characteristic classes. For the infinite order components, we simplify and correct the misidentification of Crainic. An important new homological tool is the notion of "triangular S -module". This is a natural generalization of the cylindrical complexes of Getzler-Jones. It combines the mixed complexes of Burghelea-Kassel and parachain complexes of Getzler-Jones with the S -modules of Kassel-Jones. There are spectral sequences naturally associated with triangular S -modules. In particular, this allows us to recover spectral sequences Feigin-Tsygan and Getzler-Jones and leads us to a new spectral sequence.