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Bernhard Köck

Publications and source records attributed to Bernhard Köck.

At least 19 recordsLinked to original sources
arXiv2026

Exterior power operations on relative K-theory

We algebraically construct exterior power operations on higher relative algebraic K-groups and prove their desired properties such as the expected behaviour with respect to (tensor) products and composition. This builds on Grayson's description of relative K-groups in terms of explicit generators and relations and on work by Harris, the first author and Taelman for (absolute) K-groups. Among the new features in our approach is the observation that the product axiom in the classical notion of a lambda-ring is redundant.

math.KT↗
arXiv2026

The Galois Structure of the Spaces of polydifferentials on the Drinfeld Curve

Let $C$ be a smooth projective curve over an algebraically closed field ${\mathbb{F}}$ equipped with the action of a finite group $G$. When $p =\textrm{char}(\mathbb{F})$ divides the order of $G$, the long-standing problem of computing the induced representation of $G$ on the space $H^0(C,Ω^{\otimes m}_C)$ of globally holomorphic polydifferentials remains unsolved in general. In this paper, we study the case of the group $G = \mathrm{SL}_2(\mathbb{F}_q)$ (where $q$ is a power of~$p$) acting on the Drinfeld curve $C$ which is the projective plane curve given by the equation $XY^q-X^qY-Z^{q+1} = 0$. When $q = p$, we fully decompose $H^0(C,Ω^{\otimes m}_C)$ as a direct sum of indecomposable $\mathbb{F}[G]$-modules. For arbitrary $q$, we give a partial decomposition in terms of an explicit $\mathbb{F}$-basis of $H^0(C,Ω^{\otimes m}_C)$. Finally, in the appendix, we compute the $a$-number and $p$-rank of the Drinfeld curve.

math.AG↗
arXiv2026

On presentations of K-groups by generators and relations

In Grayson's combinatorial description of higher K-groups, the generators are bounded acyclic binary multi-complexes of arbitrary size. Generalising work by Kasprowski, Winges and the author, we show in this paper that multi-complexes of bounded size suffice and we provide the corresponding relations. Furthermore, we report on the progress in our attempt to algebraically prove the surjectivity of Quillen's dévissage isomorphism for K_1 and we give an elementary and fairly simple example in the codomain which appears to require a more sophisticated approach.

math.KT↗
arXiv2025

Comparison of Exterior Power Operations on Higher K-Theory of Schemes

Exterior power operations provide an additional structure on K-groups of schemes which lies at the heart of Grothendieck's Riemann-Roch theory. Over the past decades, various authors have constructed such operations on higher K-theory. In this paper, we prove that these constructions actually yield the same operations, ultimately matching up the explicit combinatorial description by Harris, the first author and Taelman on the one hand and the recent, conceptually clear-cut construction by Barwick, Glasman, Mathew and Nikolaus on the other hand. This also leads to the proof of a conjecture by the first author about composition of these operations in the equivariant context, completing the proof that higher equivariant K-groups satisfy all axioms of a lambda-ring.

math.KT↗
arXiv2024

The canonical representation of the Drinfeld curve

If $C$ is a smooth projective curve over an algebraically closed field $\mathbb{F}$ and $G$ is a subgroup of automorphisms of $C$, then $G$ acts linearly on the $\mathbb{F}$-vector space of holomorphic differentials $H^0\big(C,Ω_C\big)$ by pulling back differentials. In other words, $H^0\big(C,Ω_C\big)$ is a representation of $G$ over the field $\mathbb{F}$, called $\textit{the canonical representation}$ of $C$. Computing its decomposition as a direct sum of indecomposable representations is still an open problem when the ramification of the cover of curves $C \longrightarrow C/G$ is wild. In this paper, we compute this decomposition for $C$ the Drinfeld curve ${XY^q-X^qY-Z^{q+1}=0}$, $\mathbb{F}=\bar{\mathbb{F}}_q$, and ${G=SL_2\big(\mathbb{F}_q\big)}$ where $q$ is a prime power.

math.AG↗
arXiv2019

$K_1$-groups via binary complexes of fixed length

We modify Grayson's model of $K_1$ of an exact category to give a presentation whose generators are binary acyclic complexes of length at most $k$ for any given $k \ge 2$. As a corollary, we obtain another, very short proof of the identification of Nenashev's and Grayson's presentations.

math.KT↗
arXiv2019

Galois-module theory for wildly ramified covers of curves over finite fields

Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.

math.NT↗
arXiv2019

Two Formulae for Exterior power operations on higher $K$-groups

Exterior power operations on the higher $K$-groups of a quasi-compact scheme have recently been constructed by Taelman and the authors by purely algebraic means. In this paper, we prove two formulae that help to compute these operations. The first is a formula for exterior powers of external products. The second is a formula for exterior powers of $n$-cubes, i.e., of acyclic binary multi-complexes supported on $[0,1]^n$. These formulae provide evidence for the expectation that our exterior power operations agree with those defined by Hiller.

math.KT↗
arXiv2018

On the de-Rham cohomology of hyperelliptic curves

For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of Čech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.

math.AG↗
arXiv2016

Exterior power operations on higher $K$-groups via binary complexes

We use Grayson's binary multicomplex presentation of algebraic $K$-theory to give a new construction of exterior power operations on the higher $K$-groups of a (quasi-compact) scheme. We show that these operations satisfy the axioms of a $λ$-ring, including the product and composition laws. To prove the composition law we show that the Grothendieck group of the exact category of integral polynomial functors is the universal $λ$-ring on one generator.

math.KT↗
arXiv2014

Belyi's theorem revisited

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.

math.AG↗
arXiv2014

Faithful action on the space of global differentials of an algebraic curve

Given a faithful action of a finite group on an algebraic curve of genus at least 2, we prove that the induced action on the space of global holomorphic differentials is faithful as well, except in the following very special case: the given action is not tame, the genus of the quotient curve is 0 and the characteristic of the base field is 2.

math.AG↗
arXiv2014

Faithfulness of actions on Riemann-Roch spaces

Given a faithful action of a finite group G on an algebraic curve X of genus g_X > 1, we give explicit criteria for the induced action of G on the Riemann-Roch space H^0(X,O_X(D)) to be faithful, where D is a G-invariant divisor on X of degree at least 2g_X-2. This leads to a concise answer to the question when the action of G on the space H^0(X, Ω_X^m) of global holomorphic polydifferentials of order m is faithful. If X is hyperelliptic, we furthermore provide an explicit basis of H^0(X, Ω_X^m). Finally, we give applications in deformation theory and in coding theory and we discuss the analogous problem for the action of G on the first homology H_1(X, Z/mZ) if X is a Riemann surface.

math.AG↗
arXiv2011

Quadratic differentials and equivariant deformation theory of curves

Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when the action of G on X is weakly ramified. Moreover we determine certain subrepresentations of V, called p-rank representations.

math.AG↗
arXiv2009

An algorithmic approach to Dold-Puppe complexes

A Dold-Puppe complex is the image NFΓ(C.) of a chain complex C. under the composition of the functors Γ, F and N where Γand N are given by the Dold-Kan correspondence and F is a not-necessarily linear functor between two abelian categories. The first half of this paper gives an algorithm that streamlines the calculation of Γ(C.). The second half gives an algorithm that allows the explicit calculation of the Dold-Puppe complex NFΓ(C.) in terms of the cross-effect functors of F.

math.AC↗
arXiv2008

Equivariant Riemann-Roch theorems for curves over perfect fields

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. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank.

math.AG↗
arXiv2006

Real Belyi theory

We develop a Belyi type theory that applies to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular we extend Belyi's famous theorem from Riemann surfaces to Klein surfaces.

math.AG↗