Mathematics

We prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and σ . We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.

We prove that the ideal used in recent works to categorify the cyclotomic integers is generated by a cyclotomic polynomial. Moreover, we publish a proof by T. Ekedahl that the q -binomial relations used in the tensor product of N -complexes makes it necessary for the category to be enriched over the cyclotomic integers.

We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch and Mesland on the self-adjointness and regularity of the sum of two weakly anticommuting operators, we show that we still have an equivalence between indefinite Kasparov modules and pairs of Kasparov modules. Importantly, the weakened version of indefinite Kasparov modules now includes the main motivating example of the Dirac operator on a pseudo-Riemannian manifold. The appendix contains a construction of an approximate identity for weakly commuting operators, which is due to Lesch and Mesland.

We introduce and study the notion of an orthogonal sum of a (possibly infinite) family of objects in a C ∗ -category. Furthermore, we construct reduced crossed products of C ∗ -categories with groups. We axiomatize the basic properties of the K -theory for C ∗ -categories in the notion of a homological functor. We then study various rigidity properties of homological functors in general, and special additional features of the K -theory of C ∗ -categories. As an application we construct and study interesting functors on the orbit category of a group from C ∗ -categorical data.

In this paper, we evaluate the algebraic K -groups of a planar cuspidal curve over a perfect F p -algebra relative to the cusp point. A conditional calculation of these groups was given earlier by Hesselholt, assuming a conjecture on the structure of certain polytopes. Our calculation here, however, is unconditional and illustrates the advantage of the new setup for topological cyclic homology by Nikolaus-Scholze, which is used throughout. The only input necessary for our calculation is the evaluation by the Buenos Aires Cyclic Homology group and by Larsen of the structure of Hochschild complex of the coordinate ring as a mixed complex, that is, as an object of the infinity category of chain complexes with circle action.

We give a concise introduction to the Farrell-Jones Conjecture in algebraic K -theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.

The aim of this work is to give an algebraic weak version of the Atiyah-Singer index theorem. We compute then a few small examples with the elliptic differential operator of order ?? coming from the Atiyah class in Ext 1 O X ( O X , Ω 1 X/k ) , where X?�Spec(k) is a smooth projective scheme over a perfect field k .

We study the algebraic K -theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic K -theory space of an integral monoid scheme X in terms of its Picard group Pic(X) and pointed monoid of regular functions Γ(X, O X ) and a description of the Grothendieck-Witt space of X in terms of an additional involution on Pic(X) . We also prove space-level projective bundle formulae in both settings.

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free produts in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups.

We construct a semi-orthogonal decomposition on the category of perfect complexes on the blow-up of a derived Artin stack in a quasi-smooth centre. This gives a generalization of Thomason's blow-up formula in algebraic K-theory to derived stacks. We also provide a new criterion for descent in Voevodsky's cdh topology, which we use to give a direct proof of Cisinski's theorem that Weibel's homotopy invariant K-theory satisfies cdh descent.