Local parameter: Difference between revisions

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In the geometry of [[algebraic curves]], a '''local parameter''' for a curve ''C'' at a (smooth) point ''P'' is a rational function that has a simple zero at ''P''. They are used mainly to ''count properly'' (in a local way).

This has an algebraic ressemblanceresemblance 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 the local ring at a smooth point of an algebraic curve is always a discrete valuation ring.<ref>J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. p. 21</ref>. If this is the case of <math>P\in C</math>, then the maximal ideal of <math>\mathcal{O}_{C,P}</math> consists of all those regular functions defined around ''P'' which vanishes at ''P'', and finally a local parameter at ''P'' will be a uniformizing parameter for the DVR (<math>\mathcal{O}_{C,P}</math>, <math>m_P</math>) (where the valuation function is given by <math>\text{ord}_P(f)=\text{max}\{d=0,1,2,...: f\in m^d_P\}</math>).

==Definition==

Let ''C'' be an algebraic curve (defined over an algebraically closed field ''K''). The '''valuation''' on ''K''(''C'') (the field of rational functions of ''C'') corresponding to a smooth point <math>P\in C</math> is defined as

<math>\text{ord}_P(f/g)=\text{ord}_P(f)-\text{ord}_P(g)</math>. A '''local parameter''' for ''C'' at ''P'' is a function <math>t\in K(C)</math> such that <math>\text{ord}_P(t)=1</math>.

==See also==

* [[Discrete valuation ring]]

==References==

<references/>

[[Category:Commutative algebra]]