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If $A_i,B_i$ for $i=1,\ldots,5$ are objects in an abelian category (for example, modules over a ring $R$ ) such that there is a commutative diagram
with the rows exact, and $\gamma_1$ is surjective, $\gamma_5$ is injective, and $\gamma_2$ and $\gamma_4$ are isomorphisms, then $\gamma_3$ is an isomorphism as well.
A special case of this is the short 5-lemma, in which $A_1=A_5=B_1=B_5$ are the trivial objects of the category (hence the rows form short exact sequences). In this case, we have that if $\gamma_2$ and $\gamma_4$ are injective (resp. surjective, isomorphisms), then $\gamma_3$ 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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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 10079 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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![$\displaystyle \xymatrix{A_1\ar[r]\ar[d]^{\gamma_1}&A_2\ar[r]\ar[d]^{\gamma_2}& ... ...{\gamma_4}&A_5\ar[d]^{\gamma_5}\ B_1\ar[r]&B_2\ar[r]&B_3\ar[r]&B_4\ar[r]&B_5}$](http://images.planetmath.org:8080/cache/objects/4598/js/img1.png "$\displaystyle \xymatrix{A_1\ar[r]\ar[d]^{\gamma_1}&A_2\ar[r]\ar[d]^{\gamma_2}& ... ...{\gamma_4}&A_5\ar[d]^{\gamma_5}\ B_1\ar[r]&B_2\ar[r]&B_3\ar[r]&B_4\ar[r]&B_5}$")