On the Galois module structure of semisimple holomorphic differentials
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.
scientific article; zbMATH DE number 1445787
scientific article