Localization Meaning

"James has been responsible for the government works programme. He is the last expatriate senior official left in the administration, apart from the Attorney General, Jeremy Mathews. It has been a remarkably swift and comprehensive process of localization, and the competence of government in Hong Kong has not suffered an iota. His departure is a significant day for Hong Kong." — 2022, Chris Patten, The Hong Kong Diaries, Penguin UK, →ISBN:

"3) Geometric interpretation of the localization. Let V be an irreducible algebraic variety. Then P = J(V) is a prime ideal of #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93; and so #92;mathbb#123;C#125;#91;V#93;#61;#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#47;J(V) is an integral domain. The localization #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#95;P is a subring of #92;mathbb#123;C#125;(X#95;1,...,X#95;n) consisting of rational functions #92;#123;f#47;g#58;f,g#92;in#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;,g#92;notinP#92;#125; which are defined on a nonempty subset of V. If V = {x} is a point, then P is maximal and #92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;#95;P#61;#92;#123;f#47;g#58;f,g#92;in#92;mathbb#123;C#125;#91;X#95;1,...,X#95;n#93;,g(x)#92;ne 0#92;#125; consists of rational functions which are defined at x." — 2007, Ivan Fesenko, “Rings and modules”, in G13ALS Algebra 2, 2007/2008 @ maths.nottingham.ac.uk, page 27: