By Jean-Pierre Serre
The aim of this ebook is to provide neighborhood category box concept from the cohomo logical standpoint, following the tactic inaugurated via Hochschild and built by way of Artin-Tate. This thought is ready extensions-primarily abelian-of "local" (i.e., whole for a discrete valuation) fields with finite residue box. for instance, such fields are got via finishing an algebraic quantity box; that's one of many facets of "localisation". The chapters are grouped in "parts". There are 3 initial components: the 1st at the normal concept of neighborhood fields, the 3rd on staff coho mology. neighborhood category box concept, strictly conversing, doesn't look until eventually the fourth half. here's a extra distinct define of the contents of those 4 components: the 1st comprises simple definitions and effects on discrete valuation jewelry, Dedekind domain names (which are their "globalisation") and the of entirety approach. The prerequisite for this half is an information of basic notions of algebra and topology, that could be came across for example in Bourbaki. the second one half is worried with ramification phenomena (different, discriminant, ramification teams, Artin representation). simply as within the first half, no assumptions are made the following in regards to the residue fields. it's during this surroundings that the "norm" map is studied; i've got expressed the consequences by way of "additive polynomials" and of "multiplicative polynomials", considering the fact that utilizing the language of algebraic geometry could have led me too some distance off beam.
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