On the Galois module structure of semisimple holomorphic differentials
type: Item
instance of: scholarly article
title: On the Galois module structure of semisimple holomorphic differentials
zbMATH Open document ID: 0962.11035
DOI: 10.1007/BF02773225
author: Pedro Ricardo López Bautista
author: Gabriel D. Villa Salvador
published in: Israel Journal of Mathematics
publication date: 2000-09-19T00:00:00Z
review text: The topic of this paper is the \(k[G]\)-module structure of the module \(\Omega^s_L(0)\) of semisimple holomorphic differentials, where \(L/K\) is a \(G\)-Galois extension of function fields over the algebraically closed constant field \(k\) of characteristic \(p\), with \(G\) a \(p\)-group. The semisimple differentials are defined as the semisimple part of \(\Omega_L(0)\) under the Cartier operator. \textit{S. Nakajima} [Math. Z. 190, 559-566 (1985; Zbl 0559.14022)] obtained a fairly concrete description of the \(k[G]\)-module in question by exhibiting a short exact sequence with first term (not counting the zeros!) \(\Omega^s_L(0)\), middle term a free \(k[G]\)-module of known rank, and end term \(\Omega(M)\) where \(M\) is quite explicitly given in terms of the ramification of \(L/K\), and \(\Omega\) is Heller's ``loop space operator'', a rather canonical operation on \(k[G]\)-modules. The main content of the paper is now making the description even more explicit, using techniques of Al Weiss. The final outcome gives \(\Omega^s_L(0)\) as a direct sum of indecomposables. This takes over a page to state (Theorem 1 on p. 363f.), and we will not repeat it here.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 1445787
zbMATH Keywords: holomorphic differentials
zbMATH Keywords: Cartier operator
zbMATH Keywords: Heller operation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773225
OpenAlex ID: W2062110021
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description: scientific article; zbMATH DE number 1445787