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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., Comment: 28 pages; v2: referees comments incorporated, 31 pages, to appear in Math. Z

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 $H^0\big(C,\Omega_C\big)$ by pulling back differentials. In other words, $H^0\big(C,\Omega_C\big)$ 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., Comment: 9 pages, to appear in Mathematische Nachrichten

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., Comment: 13 pages; v2: paragraph about operations on Milnor K-theory added

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., Comment: 10 pages, minor changes following a referee report, to appear in HHA

For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of \v{C}ech 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., Comment: 23 pages; v2: references added; v3: comments of referees addressed, to appear in Res. Number Theory

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 $\lambda$-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 $\lambda$-ring on one generator., Comment: 35 pages; v2: reference to a correspondence between Deligne and Grothendieck added; v3: referee's comments incorporated, to appear in Annals of K-Theory

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., Comment: v2: Appendix by Bernhard K\"ock and Adriano Marmmora added, mistake in earlier generalised version of Lemma 2.6 removed, further mainly editorial changes; v3: this final, somewhat shortened version is to appear in Documenta Mathematica, 33p

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, \Omega_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, \Omega_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., Comment: 27 pages, 1 figure, to appear in Canadian Journal of Mathematics

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., Comment: 30 pages, to appear in Ann. Inst. Fourier (Grenoble)

A Dold-Puppe complex is the image NF\Gamma(C.) of a chain complex C. under the composition of the functors \Gamma, F and N where \Gamma and 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 \Gamma(C.). The second half gives an algorithm that allows the explicit calculation of the Dold-Puppe complex NF\Gamma(C.) in terms of the cross-effect functors of F., Comment: 21 pages

We compute the derived functors of the third symmetric-power functor and their cross-effects for certain values. These calculations match predictions by the first named author and largely prove them in general., Comment: 22 pages; to appear in Tbilisi Mathematical Journal

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., Comment: 7 pages. The content of this short preprint is now part of the larger preprint arXiv:1404.3135

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., Comment: 20 pages

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., Comment: 10 pages, 5 figures

We lift the Euler characteristic of a nearly perfect complex to a relative algebraic K-group by passing to its l-adic Euler characteristics., Comment: 24 pages

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., Comment: 18 pages; v2: 22 pages, further result added, paper restructured and revised

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., Comment: 12 pages; v2: minor changes

We prove an equivariant Grothendieck-Ogg-Shafarevich formula. This formula may be viewed as an \'etale analogue of well-known formulas for Zariski sheaves generalizing the classical Chevalley-Weil formula. We give a new approach to those formulas (first proved by Ellingsrud/L{\o}nsted, Nakajima, Kani and Ksir) which can also be applied in the \'etale case., Comment: 16 pages; v2: generalizations of a theorem of Ksir added and substantially revised

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-Nogu\`es-Taylor operations., Comment: 22 pages; Latex

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 projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the functors occurring in the Dold-Kan correspondence., Comment: 35 pages

Mapping a locally free module to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behaviour of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups., Comment: 42 pages, LaTex2e

Let G be a finite group and K a number field. We show that the "k-th Adams operation" defined by Cassou-Nogues and Taylor on the locally free class group Cl(Z_K G) is a symmetric power operation, if k is coprime to the order of G. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.