Abstract
As already recalled in the Introduction, localization and completion are the basic algebraic techniques for computing the algebraic K- and L-groups, by reducing the computation for a complicated ring to simpler rings (e.g. fields). The classic example of localization and completion is the Hasse-Minkowski principle by which quadratic forms over ℤ are related to quadratic forms over ℚ and the finite fields F p and the p-adic completions \({\widehat {\Bbb Z}_p}\), \({\widehat {\Bbb Q}_p}\) of ℤ, ℚ (p prime). The localization of polynomial rings is particularly relevant to knot theory, starting with the way in which the Blanchfield form takes its values in the localization of ℤ[z, z −1] inverting the Alexander polynomials.
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