Short five lemma: Difference between revisions
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== References == | == References == | ||
*{{cite book |first=Thomas W. |last=Hungerford |author1-link=Thomas W. Hungerford |title=Algebra |publisher=[[Springer-Verlag]] |location=Berlin |year=2003 | orig-date=1980 |page=176 |isbn=0-387-90518-9 | series=[[Graduate Texts in Mathematics]] | volume=73 | zbl=0442.00002 }} | *{{cite book |first=Thomas W. |last=Hungerford |author1-link=Thomas W. Hungerford |title=Algebra |publisher=[[Springer-Verlag]] |location=Berlin |year=2003 | orig-date=1980 |page=176 |isbn=0-387-90518-9 | series=[[Graduate Texts in Mathematics]] | volume=73 | zbl=0442.00002 }} | ||
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }} | * {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina|editor1-link=M. Cristina Pedicchio | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }} | ||
{{DEFAULTSORT:Short Five Lemma}} | {{DEFAULTSORT:Short Five Lemma}} | ||
[[Category:Homological algebra]] | [[Category:Homological algebra]] | ||
[[Category:Lemmas in category theory]] | [[Category:Lemmas in category theory]] | ||
Latest revision as of 19:55, 5 July 2025
Template:Short description In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.
It follows immediately from the five lemma.
The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object Template:Prime, and this homomorphism induces an isomorphism from a subobject A of B to a subobject Template:Prime of Template:Prime and also an isomorphism from the factor object B/A to Template:Prime/Template:Prime, then f itself is an isomorphism. Note however that the existence of f (such that the diagram commutes) has to be assumed from the start; two objects B and Template:Prime that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of abelian groups, B could be the cyclic group of order four and Template:Prime the Klein four-group).
References
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