Bernhard Köck

GALOIS STRUCTURE OF ZARISKI COHOMOLOGY FOR WEAKLY RAMIFIED COVERS OF CURVES

We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani’s computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case. Mathematics Subject Classification 2000. 14H30; 14F10; 11S20.We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kanis computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case.

Adams operations for projective modules over group rings

Let R be a commutative ring, Γ a group acting on R , and let k ∈ IN be invertible in R . Generalizing a definition of Kervaire we construct an Adams operation ψ on the Grothendieck group and on the higher K theory of projective modules over the twisted group ring R#Γ . For this we use generalizations of Atiyah’s cyclic power operations and shuffle products in higher K -theory. For the Grothendieck group we show that ψ is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψ for any other l which is invertible in R .

Shuffle products in higher K-theory

We construct shuffle products in higher K-theory. The fundamental observation for this is that the following assignments can be melted into one another in an entirely natural fashion. On the one hand to any locally free modules V1,…, Vk we assign the module ⊕σ(⊗r=1p)⊗ Vσ(r))⊗(⊗r=p+1k Vσ(r)) and on the other hand to any chain V1 ↪ · ↪ Vk = V of admissible monomorphisms we assign the submodule ∑σ(Λr=1p Vσ(r))⊗(Λr=p+1k Vσ(r)) of ΛpV ⊗ Λk−1V. In both cases the (direct) sum is taken over all (p, k − p)-shuffles σ. By means of these shuffle products we show that the exterior power operations in higher K-theory defined by D. Grayson are compatible (already on the simplicial level) with the direct sum and with the symmetric power operations in the expected way. Furthermore, we investigate the connection between the shuffle products and the usual products in higher K-theory.

Symmetric Powers of Galois Modules on Dedekind Schemes

We prove a certain Riemann–Roch-type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou–Noguès–Taylor operations.

Exterior power operations on higher K-groups via binary complexes

We use Graysons binary multicomplex presentation of algebraic

Modeling Volume Change and Mechanical Properties with Hydraulic Models

K

Chow motif and higher Chow theory ofG/P

-theory to give a new construction of exterior power operations on the higher

The Grothendieck-Riemann-Roch theorem for group scheme actions

K

Belyi's Theorem Revisited

-groups of a (quasi-compact) scheme. We show that these operations satisfy the axioms of a

Real Belyi theory

\lambda