The Galois Module Structure Of The Integers In Wildly Ramified Extensions

Author : Sugil Lee
Publisher :
ISBN 13 :
Total Pages : 59 pages
Book Rating : 4.6/5 (647 download)

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Download or read book Galois Module Structure of Weakly Ramified Covers of Curves written by Sugil Lee and published by . This book was released on 2020 with total page 59 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 ``simplest'' 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 |G| is odd, then the different ideal $\\frak{D}_{N/Q}$ is a square, and the square root of the inverse different is a locally free Z[G]-module if and only if N/Q is weakly ramified. Kock classified all fractional ideals of weakly ramified local rings that have normal integral bases. We generalize both of the results to curves over Spec Z to construct projective Euler characteristic for certain equivariant sheaves on weakly ramified covers of curves.