Galois module structure of weakly ramified covers of curves
Lee, Sugil
author
Pappas, Georgios
thesis advisor
Kulkarni, Rajesh
degree committee member
Shapiro, Michael
degree committee member
Levin, Aaron
degree committee member
text
Text
Theses
No place, unknown, or undetermined
2020
2020
eng
English
application/pdf
viii, 50 pages
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 G-equivariant coherent sheaf of O_X-modules, the sheaf cohomology groups H. i(X, \\F) are G-modules, 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 L-functions 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
Sugil Lee
Thesis (Ph. D.)--Michigan State University. Mathematics, 2020
Includes bibliographical references (pages 48-50)
Geometry, Algebraic
Galois modules (Algebra)
Euler characteristic
Projective curves
Mathematics
Geometry, Algebraic
Galois modules (Algebra)
Euler characteristic
Projective curves
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https://doi.org/doi:10.25335/80k2-h283
aacr
MiEM
Michigan State University. Libraries
2020-11-19
2021-03-26
1201190170
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