Galois structure of Zariski cohomology for weakly ramified covers of curves

Abstract

Abstract. 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.

Equivariant Riemann–Roch theorems for curves over perfect fields

“…For curves over algebraically closed fields, our result is already proven, see Theorem 3.1 in [Kö2]. We have seen (Lemma 1.9) that the injective homomorphism β :…”

“…This equivariant Riemann-Roch formula coincides with Theorem 3.1 in [Kö2] when k is algebraically closed and in fact can be derived from this theorem by tensoring our formula with the algebraic closurē k over k and using various folklore facts from Algebraic Geometry and Representation Theory. All this is explained in the first two sections of this paper.…”

“…If k is algebraically closed, the theorem coincides with Theorem 2.1 in [Kö2]. We can deduce the general case using the following facts:…”

“…The second author has recently found a formula that does not need this last assumption and also provides new proofs for the results mentioned before (cf. [Kö1], [Kö2]). In this paper, we concentrate on the case where the underlying field k is perfect, which includes in particular all algebraically closed fields, all fields of characteristic zero and all finite fields.…”

“…When π is weakly ramified we furthermore derive from the corresponding result in [Kö2] the existence of the so-called ramification module N G,X , a certain projective k[G] -module which embodies a global relation between the (local) representations m P /m 2 P of the inertia group I P for P ∈ X . If moreover D is an equivariant divisor as above, our main result, Theorem 4.6,…”