Invariants of a quadratic form attached to a tame covering of schemes

Philippe Cassou-Noguès; Boas Erez; Martin J. Taylor

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 2, page 597-660
  • ISSN: 1246-7405

Abstract

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We build on preceeding work of Serre, Esnault-Kahn-Viehweg and Kahn to establish a relation between invariants, in modulo 2 étale cohomology, attached to a tamely ramified covering of schemes with odd ramification indices. The first type of invariant is constructed using a natural quadratic form obtained from the covering. In the case of an extension of Dedekind domains, mains, this form is the square root of the inverse different equipped with the trace form. In the case of a covering of Riemann surfaces, it arises from a theta characteristic. The second type of invariant is constructed using the representation of the tame fundamental group, which corresponds to the covering. Our formula is valid in arbitrary dimension. For unramified coverings the result was proved by the above authors. The two main contributions of our work consist in (1) showing how to eliminate ramification to reduce to the unramified case, in such a way that the reduction is possible in arbitrary dimension, and; (2) getting around the difficulties, caused by the presence of crossings in the ramification divisor, by introducing what we call “normalisation along a divisor”. Our approach relies on a detailed analysis of the local structure of tame coverings. We include a review of the relevant material from the theory of quadratic forms on schemes and of the basic simplicial techniques needed for our purposes.

How to cite

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Cassou-Noguès, Philippe, Erez, Boas, and Taylor, Martin J.. "Invariants of a quadratic form attached to a tame covering of schemes." Journal de théorie des nombres de Bordeaux 12.2 (2000): 597-660. <http://eudml.org/doc/248516>.

@article{Cassou2000,
abstract = {We build on preceeding work of Serre, Esnault-Kahn-Viehweg and Kahn to establish a relation between invariants, in modulo 2 étale cohomology, attached to a tamely ramified covering of schemes with odd ramification indices. The first type of invariant is constructed using a natural quadratic form obtained from the covering. In the case of an extension of Dedekind domains, mains, this form is the square root of the inverse different equipped with the trace form. In the case of a covering of Riemann surfaces, it arises from a theta characteristic. The second type of invariant is constructed using the representation of the tame fundamental group, which corresponds to the covering. Our formula is valid in arbitrary dimension. For unramified coverings the result was proved by the above authors. The two main contributions of our work consist in (1) showing how to eliminate ramification to reduce to the unramified case, in such a way that the reduction is possible in arbitrary dimension, and; (2) getting around the difficulties, caused by the presence of crossings in the ramification divisor, by introducing what we call “normalisation along a divisor”. Our approach relies on a detailed analysis of the local structure of tame coverings. We include a review of the relevant material from the theory of quadratic forms on schemes and of the basic simplicial techniques needed for our purposes.},
author = {Cassou-Noguès, Philippe, Erez, Boas, Taylor, Martin J.},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {2},
pages = {597-660},
publisher = {Université Bordeaux I},
title = {Invariants of a quadratic form attached to a tame covering of schemes},
url = {http://eudml.org/doc/248516},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Cassou-Noguès, Philippe
AU - Erez, Boas
AU - Taylor, Martin J.
TI - Invariants of a quadratic form attached to a tame covering of schemes
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 597
EP - 660
AB - We build on preceeding work of Serre, Esnault-Kahn-Viehweg and Kahn to establish a relation between invariants, in modulo 2 étale cohomology, attached to a tamely ramified covering of schemes with odd ramification indices. The first type of invariant is constructed using a natural quadratic form obtained from the covering. In the case of an extension of Dedekind domains, mains, this form is the square root of the inverse different equipped with the trace form. In the case of a covering of Riemann surfaces, it arises from a theta characteristic. The second type of invariant is constructed using the representation of the tame fundamental group, which corresponds to the covering. Our formula is valid in arbitrary dimension. For unramified coverings the result was proved by the above authors. The two main contributions of our work consist in (1) showing how to eliminate ramification to reduce to the unramified case, in such a way that the reduction is possible in arbitrary dimension, and; (2) getting around the difficulties, caused by the presence of crossings in the ramification divisor, by introducing what we call “normalisation along a divisor”. Our approach relies on a detailed analysis of the local structure of tame coverings. We include a review of the relevant material from the theory of quadratic forms on schemes and of the basic simplicial techniques needed for our purposes.
LA - eng
UR - http://eudml.org/doc/248516
ER -

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