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- Bernhard Köck
- 2004

We give an elementary, self-contained and quick proof of Belyi’s theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface. MSC 2000: 14H30, 14H25, 12F10

Using Quillen’s universal transformation we verify some (standard) properties of Adams operations on the higher K -theory of projective modules over group rings. Furthermore, we rather explicitly describe Adams operations on the Whitehead group K1(CΓ ) associated with the group ring CΓ of a finite group Γ over an algebraically closed field of characteristic… (More)

- Bernhard Köck
- 2003

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… (More)

Let Γ be a finite group and K a number field. We show that the operation ψk defined by Cassou-Noguès and Taylor on the locally free classgroup Cl(OKΓ ) is a symmetric power operation if gcd(k, ord(Γ )) = 1. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for… (More)

- Bernhard Köck
- 1998

Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each f. g. projective R/I-module V has an Rprojective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P1 → P0 for all n ≥ 1, if d = 1. This computation yields a new… (More)

- Bernhard Köck
- 2001

We prove an equivariant Grothendieck-Ogg-Shafarevich formula. This formula may be viewed as an étale analogue of well-known formulas for Zariski sheaves which were proved by Ellingsrud/Lønsted and Nakajima and for which we give a new approach in this paper. Mathematics Subject Classification 2000. 14F20; 14L30; 14H30.

- Bernhard Köck
- 1999

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. Introduction Let G be a finite group and E a number field. Let OE denote the ring of integers in E, Y := Spec(OE), and… (More)

- Helena Fischbacher-Weitz, Bernhard Köck
- 2007

We prove an equivariant Riemann-Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in Q . We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients.… (More)

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… (More)