Localization (commutative algebra): Difference between revisions

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== Terminology ==

The term ''localization'' originates in [[algebraic geometry]]: if ''R'' is a ring of [[function (mathematics)|function]]s defined on some geometric object ([[algebraic variety]]) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions which are non-zero at ''p'' and localizes ''R'' with respect to ''S''. The resulting ring ''R*'' contains only information about the behavior of ''V'' near ''p''. Cf. the example given at [[local ring]].

In [[number theory]] and [[algebraic topology]], one refers to the behavior of a ring or space ''at'' a number ''n'' or ''away'' from ''n''. "Away from ''n''" means "in a ring where ''n'' is invertible" (so a <math>\mathbf{Z}[\textstyle{\frac{1}{n}}]</math>-algebra). For instance, for a field, "away from ''p''" means "characteristic not equal to ''p''". <math>\mathbf{Z}[\textstyle{\frac{1}{2}}]</math> is "away from 2", but <math>\mathbf{F}_2</math> or <math>\mathbf{Z}</math> are not.