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If  for
 are objects in an abelian category (for example, modules over a ring  ) such that there is a commutative diagram
with the rows exact, and  is surjective,  is injective, and  and  are isomorphisms, then  is an isomorphism as well.
A special case of this is the short 5-lemma, in which
 are the trivial objects of the category (hence the rows form short exact sequences). In this case, we have that if  and  are injective (resp. surjective, isomorphisms), then  is also injective (resp. surjective, an isomorphism).
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"5-lemma" is owned by mathcam. [ full author list (2) | owner history (1) ]
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(view preamble)
See Also: 9-lemma
| Other names: |
five-lemma, 5-lemma, short five lemma, short 5-lemma, five lemma, 5 lemma |
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Cross-references: short exact sequences, category, isomorphisms, injective, surjective, rows, commutative diagram, ring, modules, abelian category, objects
There are 5 references to this entry.
This is version 5 of 5-lemma, born on 2003-08-15, modified 2006-05-19.
Object id is 4598, canonical name is 5Lemma.
Accessed 7656 times total.
Classification:
| AMS MSC: | 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes) |
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Pending Errata and Addenda
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![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{A_1\ar[r]\ar[d]^{\gamma_1}&A_2\ar... ...A_5\ar[d]^{\gamma_5}\ B_1\ar[r]&B_2\ar[r]&B_3\ar[r]&B_4\ar[r]&B_5} } \end{xy}$](http://images.planetmath.org:8080/cache/objects/4598/l2h/img4.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{A_1\ar[r]\ar[d]^{\gamma_1}&A_2\ar... ...A_5\ar[d]^{\gamma_5}\ B_1\ar[r]&B_2\ar[r]&B_3\ar[r]&B_4\ar[r]&B_5} } \end{xy}$")