en>Brirush: /* =Differential space */

2014-11-23T13:47:01Z

=Differential space

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	~~In algebra,~~ given a ~~[[ring homomorphism]]~~ <~~math~~>~~f: R \to S~~<~~/math~~>, ~~there are three ways to change~~ the ~~coefficient ring~~ of ~~a [[module~~ (~~mathematics~~)~~\|module]]; namely~~, ~~for~~ a ~~right ''R''-module ''M''~~ and ~~a right ''S''~~-~~module ''N'',~~		Property development might be attributed as the enhancement of a given area by an individual or a consultant or by a corporation as a whole. Any particular person or group concerned in the above job will possible be known as a Property Developer. Singapore is absolutely well known in Asia for its booming property.<br><br>In August 2008, the Property division acquired 17.7% stake in Allco Commercial REIT ("Allco REIT") and one hundred% of the REIT's manager. The REIT was renamed as Frasers Commercial Belief ("FCOT"), and has a market capitalisation of close to $500 million, with $2 billion of nine commercial properties positioned in Singapore, Australia and Japan. 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	*<~~math~~>~~f_! M = M \otimes_R S~~<~~/math~~>, the ~~[[induced module]]~~.
	<~~math~~>~~f_ M = \operatorname{Hom}_R~~(S, M)<~~/math~~>, the [~~[coinduced module]~~].
	<~~math~~>~~f^ N = N_R~~<~~/math~~>, the ~~[[restriction~~ of ~~scalars]]~~.
	~~They~~ are ~~related as [[adjoint functor]]s:~~
	:<~~math~~>f_! : \text{Mod}_R \leftrightarrows \text{Mod}_S : f^*<~~/math~~>
	and
	:<~~math~~>~~f^* : \text{Mod}_S \leftrightarrows \text{Mod}_R : f_*.~~<~~/math~~>
	~~This~~ is ~~related~~ to ~~[[Shapiro's lemma]]~~.

	~~== See~~ also ==
	*[[Six operations]]

	~~== References ==~~
	*J.P. May, ~~[http://www~~.~~math~~.~~uchicago~~.~~edu/~may/MISC/TorExt~~.~~pdf Notes on Tor and Ext]~~


	~~[[Category:Ring theory]]~~


	~~{{algebra-stub}}~~

en>Malcolma: added Category:Differential geometry; removed {{uncategorized}} using HotCat

2014-01-22T09:57:26Z

added Category:Differential geometry; removed {{uncategorized}} using HotCat

New page

In algebra, given a [[ring homomorphism]] <math>f: R \to S</math>, there are three ways to change the coefficient ring of a [[module (mathematics)|module]]; namely, for a right ''R''-module ''M'' and a right ''S''-module ''N'',
*<math>f_! M = M \otimes_R S</math>, the [[induced module]].
*<math>f_* M = \operatorname{Hom}_R(S, M)</math>, the [[coinduced module]].
*<math>f^* N = N_R</math>, the [[restriction of scalars]].
They are related as [[adjoint functor]]s:
:<math>f_! : \text{Mod}_R \leftrightarrows \text{Mod}_S : f^*</math>
and
:<math>f^* : \text{Mod}_S \leftrightarrows \text{Mod}_R : f_*.</math>
This is related to [[Shapiro's lemma]].

== See also ==
*[[Six operations]]

== References ==
*J.P. May, [http://www.math.uchicago.edu/~may/MISC/TorExt.pdf Notes on Tor and Ext]

[[Category:Ring theory]]

{{algebra-stub}}

Differentiable stack - Revision history

en>Brirush: /* =Differential space */

en>Malcolma: added Category:Differential geometry; removed {{uncategorized}} using HotCat