Lions–Lax–Milgram theorem: Difference between revisions

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In [[mathematics]], particularly [[homological algebra]], the '''zig-zag lemma''' asserts the existence of a particular [[long exact sequence]] in the [[homology group]]s of certain [[chain complex]]es. The result is valid in every [[abelian category]].
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In an abelian category (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]), let <math>(\mathcal{A},\partial_{\bullet}), (\mathcal{B},\partial_{\bullet}')</math> and <math>(\mathcal{C},\partial_{\bullet}'')</math> be chain complexes that fit into the following [[short exact sequence]]:
 
: <math>0 \longrightarrow \mathcal{A} \stackrel{\alpha}{\longrightarrow} \mathcal{B} \stackrel{\beta}{\longrightarrow} \mathcal{C}\longrightarrow 0</math>
 
Such a sequence is shorthand for the following [[commutative diagram]]:
 
[[image:complex_ses_diagram.png|commutative diagram representation of a short exact sequence of chain complexes]]
 
where the rows are [[exact sequence]]s and each column is a [[chain complex|complex]].
 
The zig-zag lemma asserts that there is a collection of boundary maps
 
: <math> \delta_n : H_n(\mathcal{C}) \longrightarrow H_{n-1}(\mathcal{A}), </math>
 
that makes the following sequence exact:
 
[[image:complex_les.png|long exact sequence in homology, given by the Zig-Zag Lemma]]
 
The maps <math>\alpha_*^{ }</math> and <math>\beta_*^{ }</math> are the usual maps induced by homology.  The boundary maps <math>\delta_n^{ }</math> are explained below.  The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence.  In an unfortunate overlap in terminology, this theorem is also commonly known as the "[[snake lemma]]," although there is another result in homological algebra with that name.  Interestingly, the "other" snake lemma can be used to prove the zig-zag lemma, in a manner different from what is described below.
 
== Construction of the boundary maps ==
The maps <math>\delta_n^{ }</math> are defined using a standard diagram chasing argument.  Let <math>c \in C_n</math> represent a class in <math>H_n(\mathcal{C})</math>, so <math>\partial_n''(c) = 0</math>.  Exactness of the row implies that <math>\beta_n^{ }</math> is surjective, so there must be some <math>b \in B_n</math> with <math>\beta_n^{ }(b) = c</math>.  By commutativity of the diagram,
 
:<math> \beta_{n-1} \partial_n' (b) = \partial_n'' \beta_n(b) = \partial_n''(c) = 0. </math>
 
By exactness,
 
:<math>\partial_n'(b) \in \ker \beta_{n-1} = \mathrm{im} \alpha_{n-1}.</math>
 
Thus, since <math>\alpha_{n-1}^{}</math> is injective, there is a unique element <math>a \in A_{n-1}</math> such that <math>\alpha_{n-1}(a) = \partial_n'(b)</math>.  This is a cycle, since <math>\alpha_{n-2}^{ }</math> is injective and
 
:<math>\alpha_{n-2} \partial_{n-1}(a) = \partial_{n-1}' \alpha_{n-1}(a) = \partial_{n-1}' \partial_n'(b) = 0,</math>
 
since <math>\partial^2 = 0</math>. That is, <math>\partial_{n-1}(a) \in \ker \alpha_{n-2} = \{0\}</math>. This means <math>a</math> is a cycle, so it represents a class in <math>H_{n-1}(\mathcal{A})</math>. We can now define
 
:<math> \delta_{ }^{ }[c] = [a].\, </math>
 
With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of ''c'' and ''b'').  The proof uses diagram chasing arguments similar to that above.  Such arguments are also used to show that the sequence in homology is exact at each group.
 
== See also ==
* [[Mayer–Vietoris sequence]]
 
==References==
 
*{{cite book | first = Allen | last = Hatcher | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher  = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
*{{Lang Algebra}}
*{{cite book | first = James R. | last = Munkres | authorlink = James Munkres | year = 1993 | title = Elements of Algebraic Topology | publisher = Westview Press | location = New York | isbn = 0-201-62728-0}}
 
 
[[Category:Homological algebra]]
[[Category:Lemmas]]

Latest revision as of 05:31, 9 December 2014

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