en>AnomieBOT: Dating maintenance tags: {{Full}}

2012-11-28T18:11:24Z

Dating maintenance tags: {{Full}}

← Older revision		Revision as of 20:11, 28 November 2012
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	~~Hi there. Allow me begin by introducing~~ the ~~author~~, ~~her title~~ is ~~Myrtle Cleary~~. ~~Minnesota~~ is ~~exactly where he~~'s ~~been living for many years~~. ~~To collect coins~~ is ~~what his family~~ and ~~him appreciate. Managing people has been his day occupation for a while.~~<br><br>~~Feel free~~ to ~~visit my blog post :: home std test (~~[~~http~~://~~www~~.~~cam4teens~~.~~com~~/~~blog/84472 just click~~ the ~~next site~~])		In [[homological algebra]], the '''horseshoe lemma''', also called the '''simultaneous resolution theorem''', is a statement relating [[projective resolution\|resolution]]s of two objects <math>A'</math> and <math>A''</math> to resolutions of
			extensions of <math>A'</math> by <math>A''</math>. It says that if an object <math>A</math> is an extension of <math>A'</math> by <math>A''</math>, then a resolution of <math>A</math> can be built up [[mathematical induction\|inductively]] with the ''n''th item in the resolution equal to the [[coproduct]] of the ''n''th items in the resolutions of <math>A'</math> and <math>A''</math>. The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis.

			==Formal statement==
			Let <math>\mathcal{A}</math> be an [[abelian category]] with [[projective object\|enough projectives]]. If

			<center>[[Image:Horseshoe lemma.png]]</center>

			is a [[commutative diagram\|diagram]] in <math>\mathcal{A}</math> such that the column is [[exact sequence\|exact]] and the
			rows are projective resolutions of <math>A'</math> and <math>A''</math> respectively, then
			it can be completed to a commutative diagram

			<center>[[Image:Horseshoe lemma conclusion.png]]</center>

			where all columns are exact, the middle row is a projective resolution
			of <math>A</math>, and <math>P_n=P'_n\oplus P''_n</math> for all ''n''. If <math>\mathcal{A}</math> is an
			abelian category with [[injective object\|enough injectives]], the [[opposite category\|dual]] statement also holds.

			The lemma can be proved inductively. At each stage of the induction, the properties of projective objects are used to define maps in a projective resolution of <math>A</math>. Then the [[snake lemma]] is invoked to show that the simultaneous resolution constructed so far has exact rows.

			==See also==
			*[[Nine lemma]]

			==References==
			*[[Henri Cartan]] and [[Samuel Eilenberg]] ''Homological algebra'', Princeton University Press, 1956.
			*M. Scott Osborne, ''Basic homological algebra'', Springer-Verlag, 2000.

			{{PlanetMath attribution\|id=7799\|title=horseshoe lemma}}

			[[Category:Homological algebra]]
			[[Category:Lemmas]]

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2012-07-09T09:21:24Z

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Hi there. Allow me begin by introducing the author, her title is Myrtle Cleary. Minnesota is exactly where he's been living for many years. To collect coins is what his family and him appreciate. Managing people has been his day occupation for a while.<br><br>Feel free to visit my blog post :: home std test ([http://www.cam4teens.com/blog/84472 just click the next site])

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