Galois module structure of weakly ramified covers of curves
Lee, Sugil
Geometry, Algebraic
Galois modules (Algebra)
Euler characteristic
Projective curves
Mathematics
Thesis Ph. D. Michigan State University. Mathematics 2020.
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
Description based on online resource; title from PDF title page (viewed on Nov. 19, 2020)
Includes bibliographical references (pages 48-50).
Pappas, Georgios
Kulkarni, Rajesh
Shapiro, Michael
Levin, Aaron
2020
text
Electronic dissertations
application/pdf
1 online resource (viii, 50 pages) : illustrations
etd:48842
isbn:9798664744880
umi:28090700
local:Lee_grad.msu_0128D_17874
https://doi.org/doi:10.25335/80k2-h283
eng
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