Weil-etale Cohomology over Local Fields

Details Weil-etale Cohomology over Local Fields Weil-etale Cohomology over Local Fields

2011-11-29

arxiv.org/abs/1111.6710v2

Mathematics

Number Theory

Scientific paper

Recent work by Lichtenbaum et al has established the arithmetic utility of the Weil group of a finite field, and demonstrated a connection between certain Euler characteristics in Weil-etale cohomology and special values of zeta functions. With this paradigm in mind, we establish results for the cohomology Weil group of a local field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for modules for the Weil group, which implies the main theorem of Local Class Field Theory. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is the Weil analogue of a classical duality theorem, also due to Lichtenbaum, for such curves.

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