Editing Valuation ring (section)

=== Specialization of places ===
We say a '''place ''p'' ''specializes to'' a place ''{{prime|p}}'',''' denoted by <math>p \rightsquigarrow p'</math>, if the valuation ring of ''p'' contains the valuation ring of ''p{{'}}''. In algebraic geometry, we say a prime ideal <math>\mathfrak{p}</math> specializes to <math>\mathfrak{p}'</math> if <math>\mathfrak{p} \subseteq \mathfrak{p}'</math>. The two notions coincide: <math>p \rightsquigarrow p'</math> if and only if a prime ideal corresponding to ''p'' specializes to a prime ideal corresponding to ''{{prime|p}}'' in some valuation ring (recall that if <math>D \supseteq D'</math> are valuation rings of the same field, then ''D'' corresponds to a prime ideal of <math>D'</math>.)

==== Example ====
For example, in the function field <math>\mathbb{F}(X)</math> of some algebraic variety <math>X</math> every prime ideal <math>\mathfrak{p} \in \text{Spec}(R)</math> contained in a maximal ideal <math>\mathfrak{m}</math> gives a specialization <math>\mathfrak{p} \rightsquigarrow \mathfrak{m}</math>.

==== Remarks ====
It can be shown: if <math>p \rightsquigarrow p'</math>, then <math>p' = q \circ p|_{D'}</math> for some place ''q'' of the residue field <math>k(p)</math> of ''p''. (Observe <math>p(D')</math> is a valuation ring of <math>k(p)</math> and let ''q'' be the corresponding place; the rest is mechanical.) If ''D'' is a valuation ring of ''p'', then its Krull dimension is the cardinarity of the specializations other than ''p'' to ''p''. Thus, for any place ''p'' with valuation ring ''D'' of a field ''K'' over a field ''k'', we have:

:<math> \operatorname{tr.deg}_k k(p) + \dim D \le \operatorname{tr.deg}_k K</math>.

If ''p'' is a place and ''A'' is a subring of the valuation ring of ''p'', then <math>\operatorname{ker}(p) \cap A</math> is called the ''center'' of ''p'' in ''A''.