en>Gatewaycat: /* Limit sets */ replace tex with inline

2013-10-29T23:24:21Z

Limit sets: replace tex with inline

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	~~Dalton~~ is [~~http~~:/~~/Data~~.~~Gov~~.~~uk/data/search?q~~=~~what%27s+written what~~'s ~~written~~] ~~from his birth certificate regrettably he never really favored~~ that ~~name~~. ~~The [http~~://~~Www~~.~~google~~.de/~~search?q~~=~~preference+hobby preference hobby]~~ for ~~him and his kids~~ is ~~landscaping but he~~'~~s been taking on new things~~ a ~~short time ago~~. ~~Auditing~~ is where ~~his primary income derives from~~. ~~Massachusetts~~ is ~~where that he~~ and ~~his wife survive~~. He's not ~~godd at design but you might want~~ to ~~check its website: http~~://~~circuspartypanama~~.~~com~~<br><br>~~Take~~ a ~~look at my blog~~ - [~~http:~~//~~circuspartypanama~~.~~com clash~~ of ~~clans hack tool android~~]		In [[algebraic geometry]], the '''localization of a module''' is a construction to introduce [[denominator]]s in a [[module (mathematics)\|module]] for a [[ring (mathematics)\|ring]]. More precisely, it is a systematic way to construct a new module ''S''<sup>−1</sup>''M'' out of a given module ''M'' containing [[algebraic fraction]]s
			:<math>\frac{m}{s}</math>.
			where the [[denominator]]s ''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 (mathematics)\|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 (mathematics)\|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''<sup>−1</sup>''M'', is defined to be the following module: as a set, it consists of [[equivalence relation\|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
			:<math>\frac{m}{s}</math>.

			To make this set a ''R''-module, define
			:<math>\frac{m}{s} + \frac{n}{t} := \frac{tm+sn}{st}</math>
			and
			:<math>a \cdot \frac{m}{s} := \frac{a m}{s}</math>
			(''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''<sub>''p''</sub> instead of (''R''\''p'')<sup>−1</sup>''M''. The '''[[support of a module\|support of the module]]''' ''M'' is the set of prime ideals ''p'' such that ''M''<sub>''p''</sub> ≠ 0. Viewing ''M'' as a function from the [[spectrum of a ring\|spectrum]] of ''R'' to ''R''-modules, mapping
			:<math>p \mapsto M_p</math>
			this corresponds to the [[support (mathematics)\|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.

			==Remarks==
			* The definition applies in particular to ''M''=''R'', and we get back the [[localization of a ring\|localized ring]] ''S''<sup>−1</sup>''R''.

			* There is a module homomorphism
			::φ: ''M'' → ''S''<sup>−1</sup>''M''
			:mapping
			::φ(''m'') = ''m'' / 1.
			:Here φ need not be [[injective]], in general, because there may be significant [[torsion (algebra)\|torsion]]. The additional ''u'' showing up in the definition of the above equivalence relation can not be dropped (otherwise the relation would not be transitive), unless the module is torsion-free.

			*Some authors allow not necessarily multiplicatively closed sets ''S'' and define localizations in this situation, too. However, saturating such a set, i.e. adding ''1'' and finite products of all elements, this comes down to the above definition.

			==Tensor product interpretation==
			By the very definitions, the localization of the module is tightly linked to the one of the ring via the [[tensor product]]
			:''S''<sup>−1</sup>''M'' = ''M'' ⊗<sub>''R''</sub>''S''<sup>−1</sup>''R'',
			This way of thinking about localising is often referred to as [[extension of scalars]].

			As a tensor product, the localization satisfies the usual [[universal property]].

			==Flatness==
			From the definition, one can see that localization of modules is an [[exact functor]], or in other words (reading this in the tensor product) that ''S''<sup>−1</sup>''R'' is a [[flat module]] over ''R''. This fact is foundational for the use of flatness in algebraic geometry, saying in particular that the inclusion of the [[open set]] Spec(''S''<sup>−1</sup>''R'') into Spec(''R'') (see [[spectrum of a ring]]) is a [[flat morphism]].

			==(Quasi-)coherent sheaves==
			In terms of localization of modules, one can define [[quasi-coherent sheaf\|quasi-coherent sheaves]] and [[coherent sheaf\|coherent sheaves]] on [[locally ringed space]]s. In algebraic geometry, the '''quasi-coherent''' ''O''<sub>''X''</sub>-'''modules''' for [[scheme (mathematics)\|scheme]]s ''X'' are those that are locally modelled on sheaves on Spec(''R'') of localizations of any ''R''-module ''M''. A '''coherent''' ''O''<sub>''X''</sub>-'''module''' is such a sheaf, locally modelled on a [[finitely-presented module]] over ''R''.

			== See also ==
			=== Localization ===
			[[:Category:Localization (mathematics)]]
			* [[Local analysis]]
			* [[Localization (algebra)]]
			* [[Localization of a category]]
			* [[Localization of a ring]]
			* [[Localization of a topological space]]

			==References==
			Any textbook on commutative algebra covers this topic, such as:
			* {{Citation \| last1=Eisenbud \| first1=David \| author1-link=David Eisenbud \| title=Commutative algebra \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| series=Graduate Texts in Mathematics \| isbn=978-0-387-94268-1; 978-0-387-94269-8 \| id={{MathSciNet \| id = 1322960}} \| year=1995 \| volume=150}}

			[[Category:Module theory]]
			[[Category:Localization (mathematics)]]

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2012-06-26T17:07:09Z

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New page

Dalton is [http://Data.Gov.uk/data/search?q=what%27s+written what's written] from his birth certificate regrettably he never really favored that name. The [http://Www.google.de/search?q=preference+hobby preference hobby] for him and his kids is landscaping but he's been taking on new things a short time ago. Auditing is where his primary income derives from. Massachusetts is where that he and his wife survive. He's not godd at design but you might want to check its website: http://circuspartypanama.com<br><br>Take a look at my blog - [http://circuspartypanama.com clash of clans hack tool android]

Fuchsian group - Revision history

en>Gatewaycat: /* Limit sets */ replace tex with inline

en>Giftlite: −sp