Local parameter: Difference between revisions

Revision as of 01:30, 31 January 2009

Revision as of 01:34, 31 January 2009

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<math>\operatorname{ord}_P(f)=\max\{d=0,1,2,\ldots: f\in m^d_P\};</math>

this valuation can naturally be extended to ''K''(''C'') (which is the field of [[rational functions]] of C) because it is the [[field of fractions]] of <math>\mathcal{O}_{C,P}</math>. Hence the conceptidea of ''having a simple zero at a point P'' is now complete: it will be a rational function <math>f\in K(C)</math> such that its germ falls into <math>m_P^d</math>, with ''d'' at most 1.

This has an algebraic resemblance with the concept of a [[Discrete_valuation_ring#Uniformizing_parameter|uniformizing parameter]] (or just '''uniformizer''') found in the context of [[discrete valuation ring]]s in [[commutative algebra]]; a uniformizing parameter for the DVR (''R, m'') is just a generator of the maximal ideal ''m''. The link comes from the fact that a local parameter at ''P'' will be a uniformizing parameter for the DVR (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>), whence the name.