Related papers: A Goodwillie-type Theorem for Milnor K-Theory

Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor…

We introduce a notion of Milnor square of stable $\infty$-categories and prove a criterion under which algebraic K-theory sends such a square to a cartesian square of spectra. We apply this to prove Milnor excision and proper excision…

Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category ${\sf Perf}(X)$ consisting of perfect…

We show that Klemenc's stable envelope of exact $\infty$-categories induces an equivalence between stable $\infty$-categories with a bounded heart structure and weakly idempotent complete exact $\infty$-categories. Moreover, we generalise…

We study homological invariants of \'etale groupoids arising from Smale spaces, continuing on our previous work, but going beyond the stably disconnected case by incorporating resolutions in the space direction. We show that the homology…

We compute the subgroup of the monodromy group of a generalized Kummer variety associated to equivalences of derived categories of abelian surfaces. The result was previously announced in arXiv:1201.0031. Mongardi showed that the subgroup…

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves…

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of…

First an `irregular Riemann-Hilbert correspondence' is established for meromorphic connections on principal G-bundles over a disc, where G is any connected complex reductive group. Secondly, in the case of poles of order two, isomonodromic…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

We use recent results proved by Berrick and the author (math.KT/0509404) to improve the periodicity theorem in hermitian K-theory. We define also a new filtration of the classical Witt ring W(A), built from non degenerate quadratic forms…

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular…

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term…

We define Grothendieck-Witt spectra in the setting of Poincar\'e $\infty$-categories and show that they fit into an extension with a K- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and…

The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group $G$. For a $G$-space $X$, this theorem gives an isomorphism between a completion of the…

We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better…

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove…

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are…

The goal of the paper is to introduce a version of Schubert calculus for each dihedral reflection group W. That is, to each "sufficiently rich'' spherical building Y of type W we associate a certain cohomology theory and verify that, first,…

The authors establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. They consider fields K that are complete discrete…