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In algebraic geometry, the localization of a module is a construction to introduce denominators in a module for a ring. More precisely, it is a systematic way to construct a new module ''S''−1''M'' out of a given module ''M'' containing algebraic fractions :. where the denominators ''s'' range in a given subset ''S'' of ''R''. The technique has become fundamental, particularly in algebraic geometry, as the link between modules and sheaf theory. Localization of a module generalizes localization of a ring. ==Definition== In this article, let ''R'' be a commutative ring and ''M'' an ''R''-module. Let ''S'' a multiplicatively closed subset of ''R'', i.e. 1 ∈ ''S'' and for any ''s'' and ''t'' ∈ ''S'', the product ''st'' is also in ''S''. Then the localization of ''M'' with respect to ''S'', denoted ''S''−1''M'', is defined to be the following module: as a set, it consists of equivalence classes of pairs (''m'', ''s''), where ''m'' ∈ ''M'' and ''s'' ∈ ''S''. Two such pairs (''m'', ''s'') and (''n'', ''t'') are considered equivalent if there is a third element ''u'' of ''S'' such that :''u''(''sn''-''tm'') = 0 It is common to denote these equivalence classes :. To make this set a ''R''-module, define : and : (''a'' ∈ ''R''). It is straightforward to check that the definition is well-defined, i.e. yields the same result for different choices of representatives of fractions. One interesting characterization of the equivalence relation is that it is the smallest relation (considered as a set) such that cancellation laws hold for elements in ''S''. That is, it is the smallest relation such that ''rs/us = r/u'' for all ''s'' in ''S''. One case is particularly important: if ''S'' equals the complement of a prime ideal ''p'' ⊂ ''R'' (which is multiplicatively closed by definition of prime ideals) then the localization is denoted ''M''''p'' instead of (''R''\''p'')−1''M''. The support of the module ''M'' is the set of prime ideals ''p'' such that ''M''''p'' ≠ 0. Viewing ''M'' as a function from the spectrum of ''R'' to ''R''-modules, mapping : this corresponds to the support of a function. Localization of a module at primes also reflects the "local properties" of the module. In particular, there are many cases where the more general situation can be reduced to a statement about localized modules. The reduction is because a ''R''-module ''M'' is trivial if and only if all its localizations at primes or maximal ideals are trivial. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「localization of a module」の詳細全文を読む スポンサード リンク
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