We construct a variant of Karoubi’s relative Chern character for smooth separated schemes over the ring of integers in a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$-adic field, and prove a comparison with the rigid syntomic regulator. For smooth projective schemes, we further relate the relative Chern character to the étale $p$-adic regulator via the Bloch–Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers, and generalizes it to all smooth projective schemes as above.

We show how a theorem of Gabber on alterations can be used to apply the work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X) \otimes \mathbb{Z}[{1}/{p}]= 0$ for$n < {-}\! \dim X$ where $X$ is a quasi-excellent noetherian scheme, $p$ is a prime that is nilpotent on $X$, and $K_n$ is the $K$-theory of Bass–Thomason–Trobaugh. This gives a partial answer to a question of Weibel.

We calculate equivariant elliptic cohomology of the partial flag variety $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where $H\subseteq G$ are compact connected Lie groups of equal rank. We identify the ${\rm RO}(G)$-graded coefficients ${\mathcal{E}} ll_G^*$ as powers of Looijenga’s line bundle and prove that transfer along the map

$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$

$K$

We construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

Let $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

This paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to the classical Lüroth problem. By generalizing a method due to Peyre, we produce for any prime number $\ell $ and any integer $n\geq 2$, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree $n$ unramified étale cohomology class with $\ell $-torsion coefficients. When $\ell = 2$, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified étale cohomology class of lower degree.

We give a negative answer to the question raised by Mart Abel about whether his proposed definition of ${K}_{0} $ and ${K}_{1} $ groups in terms of quasi multiplication is indeed equivalent to the established ones in algebraic $K$-theory.

Consider the Mackey functor that assigns to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p > 0. Thévenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.

In this article we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

We prove the conjectural relations between Mahler measures and L-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for L-values of elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family (1+X) (1+Y )(X+Y )−αXY, α∈ℝ.

For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p])(which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p]and relate it with unramified cohomology.

In this article, we further the study of higher K-theory of differential graded (dg) categories via universal invariants, initiated in [G. Tabuada, Higher K-theory via universal invariants, Duke Math. J. 145 (2008), 121–206]. Our main result is the co-representability of non-connective K-theory by the base ring in the ‘universal localizing motivator’. As an application, we obtain for free higher Chern characters, respectively higher trace maps, from non-connective K-theory to cyclic homology, respectively to topological Hochschild homology.

Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this paper we show how to deform the Fredholm family in the sense of quantum groups. The family of Dirac-type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transforms covariantly under a central extension of the algebra.

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

For a central simple k-algebra A with indk (A) ∈ k×, Suslin defined a cohomological invariant for SK1 (A) [Sus2]. In this text, we generalise his invariant to any central simple k-algebra using a lift from positive characteristic to characteristic 0. To be able to define the invariant, we use Kato's cohomology of logarithmic differentials [Kat1].

We describe explicitly the continuous Hochschild and cyclic cohomology groups of certain tensor products of -algebras which are Fréchet spaces or nuclear DF-spaces. To this end we establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear DF-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear -algebras which are Fréchet spaces or DF-spaces for which all boundary maps of the standard homology complexes have closed ranges.