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The '''snake lemma''' is a tool used in [[mathematics]], particularly [[homological algebra]], to  construct [[long exact sequence]]s. The snake lemma is valid in every [[abelian category]] and is a crucial tool in homological algebra and its applications, for instance in [[algebraic topology]]. Homomorphisms constructed with its help are generally called '''''connecting homomorphisms'''''.
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== Statement ==
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]]), consider a [[commutative diagram]]:
 
[[File:SnakeLemma01.png]]
 
where the rows are [[exact sequence]]s and 0 is the [[zero object]].
Then there is an exact sequence relating the [[kernel (category theory)|kernels]] and [[cokernel]]s of ''a'', ''b'', and ''c'':
 
[[File:SnakeLemma02.png]]
 
Furthermore, if the morphism ''f'' is a [[monomorphism]], then so is the morphism ker&nbsp;''a'' → ker&nbsp;''b'', and if ''g''' is an [[epimorphism]], then so is coker&nbsp;''b'' → coker&nbsp;''c''.
 
== Explanation of the name ==
To see where the snake lemma gets its name, expand the diagram above as follows:
 
[[File:SnakeLemma03.png]]
 
and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering [[snake]].
 
== Construction of the maps ==
The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a ''connecting homomorphism'' ''d'' exists which completes the exact sequence.
 
In the case of abelian groups or [[module (mathematics)|modules]] over some [[ring (mathematics)|ring]], the map ''d'' can be constructed as follows.
Pick an element ''x'' in ker&nbsp;''c'' and view it as an element of ''C''; since ''g'' is [[surjective]], there exists ''y'' in ''B'' with ''g''(''y'') = ''x''. Because of the commutativity of the diagram, we have <math> g'(b(y)) = c(g(y))=c(x)=0 \! </math> (since ''x'' is in the kernel of ''c''), and therefore ''b''(''y'') is in the kernel of ''g' ''. Since the bottom row is exact, we find an element ''z'' in ''A' '' with ''f''&nbsp;'(''z'') = ''b''(''y''). ''z'' is unique by injectivity of ''f''&nbsp;'. We then define ''d''(''x'') = ''z'' + ''im''(''a''). Now one has to check that ''d'' is well-defined (i.e. ''d''(''x'') only depends on ''x'' and not on the choice of ''y''), that it is a homomorphism, and that the resulting long sequence is indeed exact.
 
Once that is done, the theorem is proven for abelian groups or modules over a ring. For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements.  Alternatively, one may invoke [[Mitchell's embedding theorem]].
 
== Naturality ==
In the applications, one often needs to show that long exact sequences are "natural" (in the sense of [[natural transformation]]s). This follows from the naturality of the sequence produced by the snake lemma.
 
If
:[[File:snake lemma nat.png|commutative diagram with exact rows]]
is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form
:[[File:snake lemma nat2.png|commutative diagram with exact rows]]
 
==In popular culture==
* The proof of the snake lemma is being taught by [[Jill Clayburgh]] at the very beginning of the 1980 film ''[[It's My Turn (film)|It's My Turn]]''.
 
==See also==
* [[List of lemmas]]
 
==References==
* [[Serge Lang]]: ''Algebra''. 3rd edition, Springer 2002, ISBN 978-0-387-95385-4, pp.&nbsp;157–159 ({{Google books|Fge-BwqhqIYC|online copy|page=157}})
* [[Michael Francis Atiyah|M. F. Atiyah]]; [[Ian G. Macdonald|I. G. Macdonald]]: ''Introduction to Commutative Algebra''. Oxford 1969, Addison–Wesley Publishing Company, Inc. ISBN 0-201-00361-9.
*P. Hilton; U. Stammbach: ''A course in homological algebra.'' 2. Auflage, Springer Verlag, Graduate Texts in Mathematics, 1997, ISBN 0-387-94823-6, p.&nbsp;99 ({{Google books|2VNrHuweuokC|online copy|page=99}})
 
==External links==
*{{MathWorld|title=Snake Lemma|urlname=SnakeLemma}}
*[http://planetmath.org/encyclopedia/SnakeLemma.html Snake Lemma] at [[PlanetMath]]
*[http://www.youtube.com/watch?v=etbcKWEKnvg Proof of the Snake Lemma] in the film [http://www.imdb.com/title/tt0080936/ It's My Turn]
 
[[Category:Homological algebra]]
[[Category:Lemmas]]

Latest revision as of 11:52, 9 January 2015

The body mass chart is a valuable instrument for checking if you've a healthy fat for your height. Just remember it's just a guide. You'll be capable to furthermore choose any of the fat charts on this page to check when you're underweight, overweight or merely right. The BMI chart has absolutely completed the reckonings for you. It demonstrates the healthy, underweight, overweight plus fat grades for individual weights plus heights. Only find the point where a fat meets your height.

There are certain elements to seriously consider in a workout regimen plus you need to seriously consider the following about overall body fat. The ideal body fat for a wellness guy adult is about 10 - 12%. The perfect body fat for a healthy woman adult 14 16%. BMI is commonly an indicator of health plus is computed using charts which are based on age, height plus present weight. Even if the body fat falls in your BMI or Body Mass Index it is very nonetheless possible which we might not fall in the protocols of body fat mentioned above. A really extreme athlete may not even apply to a bmi chart and it will moreover rely on the sport. It actually is unique to each individual. However when the close to a BMI then closer to desired body fat you may be close to achieving the objective of flat hard washboard abs.

An exclusive milk diet for rapid gain of fat has been advocated by certain bmi chart men nature cure practitioners. At the beginning of this mode of treatment, the patient must quickly for 3 days on warm water plus lime juice thus because to cleanse the program.

Stop smoking and drinking is a superior idea. This is additionally true for men as smoking and drinking may affects the standard of their sperm. Get expecting quicker by struggling to keep a BMI of less than 29 considering being obese could lower fertility. There is a link between improved scrotal temperature plus reduced sperm quality, so it might be a wise idea for guys to wear loose fitting underwear.

Besides the obvious weight plus BMI measurements, we might like to consider acquiring a model that has separate methods for guy plus female consumers. Weight distribution for people are different plus scales that have a gender way usually offer more exact readings.

21. Running Clothes/Bra: I like my running dresses sporty--not funky, but this is clearly bmi chart women personal preference. A superior running bra will go a long means...avoid cotton at all cost. I have learned that running skirts are the most polarizing of all apparel goods. But, if you love wearing a skirt, head over to the Skirtchaser Race Series...looks like fun!

In addition, men generally have more muscle than women plus women have more fat inside their body than guys. For this reason, women could have their fat plus height ratio calculated through a BMI calculator for ladies.

If your BMI is elevated you will wish To talk to the doctor about the risk factors associated with weight and whether or not you need to lose weight. Your doctor should be capable to advise we plus get we on track to beginning a weight loss routine. There are numerous dangers associated with individuals who are obese or obese like excellent blood pressure, significant blood cholesterol or other lipid disorders, type 2 diabetes, heart condition, stroke, and certain cancers.

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