Galois module structure of weakly ramified covers of curves
The main theme of our study is the obstruction to the existence of a normal integral basis for certain Galois modules of geometric origin. When G is a finite group acting on a projective scheme X over \\Spec Z and F is a Gequivariant coherent sheaf of O_Xmodules, the sheaf cohomology groups H. i(X, \\F) are Gmodules, and one asks if its equivariant Euler characteristic$$\\chi(X, F) := \\sum_i (1). i [H. i(X, F)]$$can be calculated using a bounded complex of finitely generated free modules over Z[G]. Then we say that the cohomology of F has a normal integral basis. The obstruction to the existence of a normal integral basis has been of great interest in the classical case of number fields: As conjectured by Frohlich and proven by Taylor, when N/Q is a finite tamely ramified Galois extension with Galois group G, the Galois module structure of the ring of integers O_N is determined (up to stable isomorphism) by the root numbers appearing in the functional equations of Artin Lfunctions associated to symplectic representations of G. Chinburg started a generalization of the theory to some schemes with tame group actions by introducing the reduced projective Euler characteristic classes $\\overline{\\chi. P(X, F)$.These Euler characteristics are elements of the class group $Cl(Z[G])$ and give the obstruction to the existence of normal integral basis.Our aim is to generalize the theory to the 03000300simplest'' kind of wild ramification, namely to weakly ramified covers of curves over Spec Z. If N/Q is wildly ramified, then O_N is not a free Z[G]module. Erez showed that when the order
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Electronic Theses & Dissertations
 Copyright Status
 Attribution 4.0 International
 Material Type

Theses
 Authors

Lee, Sugil
 Thesis Advisors

Pappas, Georgios
 Committee Members

Kulkarni, Rajesh
Shapiro, Michael
Levin, Aaron
 Date
 2020
 Program of Study

Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 viii, 50 pages
 ISBN

9798664744880
 Permalink
 https://doi.org/doi:10.25335/80k2h283