Objectivity (frame invariance): Difference between revisions

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In [[mathematics]], especially in the area of [[abstract algebra|algebra]] known as [[module theory]], '''Schanuel's lemma''', named after [[Stephen Schanuel]], allows one to compare how far modules depart from being [[projective module|projective]]. It is useful in defining the Heller operator in the stable category, and in giving elementary descriptions of [[projective dimension|dimension shifting]].
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==Statement==
 
'''Schanuel's lemma''' is the following statement:
 
If 0 &nbsp;&rarr;&nbsp; ''K'' &nbsp;&rarr;&nbsp;''P''&nbsp;&rarr;&nbsp; ''M''&nbsp;&rarr;&nbsp; 0 and 0 &nbsp;&rarr;&nbsp;''K''' &nbsp;&rarr;&nbsp; ''P'' ' &nbsp;&rarr;&nbsp; ''M'' &nbsp;&rarr;&nbsp;0 are [[short exact sequence]]s of ''R''-modules and ''P'' and ''P'' ' are projective, then ''K'' &oplus; ''P'' ' is [[Isomorphism|isomorphic]] to ''K'' ' &oplus; ''P.''
 
==Proof==
 
Define the following [[submodule]] of ''P'' &oplus; ''P'' ', where &phi; : ''P'' &rarr; ''M'' and &phi;' : ''P'' ' &rarr; ''M'':
 
:<math> X = \{ (p,q) \in P \oplus P^\prime : \phi(p) = \phi^\prime(q) \}. </math>
 
The map &pi; : ''X'' &rarr; ''P'', where &pi; is defined as the projection of the first coordinate of ''X'' into ''P'', is surjective. Since &phi;' is surjective,  for any ''p'' <math>\in</math> ''P'', one may find a ''q'' <math>\in</math> ''P'' ' such that &phi;(''p'') = &phi; '(''q''). This gives (''p'',''q'') <math>\in</math> ''X'' with &pi; (''p'',''q'') = ''p''.  Now examine the [[kernel (algebra)|kernel]] of the map &pi; :
 
<math>
\begin{align}
\text{ker} \; \pi &= \{ (0,q): (0,q) \in X \} \\
& = \{ (0,q): \phi^\prime(q) =0 \} \\
& \cong \; \text{ker} \; \phi^\prime \cong K^\prime.
\end{align} </math>
 
We may conclude that there is a short exact sequence
 
:<math> 0 \rightarrow K^\prime \rightarrow X \rightarrow P \rightarrow 0. </math>
 
Since ''P'' is projective this sequence splits, so ''X'' &cong; ''K'' ' &oplus; ''P'' .  Similarly, we can write another map &pi; : ''X'' &rarr; ''P'' ', and the same argument as above shows that there is another short exact sequence
 
:<math> 0 \rightarrow K \rightarrow X \rightarrow P^\prime \rightarrow 0, </math>
 
and so ''X'' &cong; ''P'' ' &oplus; ''K''. Combining the two equivalences for ''X'' gives the desired result.
 
==Long exact sequences==
 
The above argument may also be generalized to [[long exact sequence]]s.<ref>{{cite book | author = Lam, T.Y. | title = Lectures on Modules and Rings | publisher = Springer | year = 1999 | id =  ISBN 0-387-98428-3}} pgs. 165&ndash;167.</ref>
 
==Origins==
 
[[Stephen Schanuel]] discovered the argument in [[Irving Kaplansky]]'s [[homological algebra]] course at the [[University of Chicago]] in Autumn of 1958.  Kaplansky writes:
:''Early in the course I formed a one-step projective resolution of a module, and remarked that if the kernel was projective in one resolution it was projective in all.  I added that, although the statement was so simple and straightforward, it would be a while before we proved it.  Steve Shanuel spoke up and told me and the class that it was quite easy, and thereupon sketched what has come to be known as "Schanuel's lemma." '' <ref>{{cite book | author = Kaplansky, Irving. | title = Fields and Rings | publisher = University Of Chicago Press | year = 1972 | id = ISBN 0-226-42451-0}} pgs. 165&ndash;168.</ref>
 
==Notes==
{{reflist}}
 
[[Category:Homological algebra]]
[[Category:Module theory]]

Latest revision as of 12:31, 10 December 2014

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