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9-lemma (Theorem)

If $ A_i, B_i, C_i$, for $ i=1,2,3$ are objects of an abelian category such that there is a commutative diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & 0\ar[d]&0\ar[d]&0\ar[d]&\ 0\... ... A_3\ar[r]\ar[d]&B_3\ar[r]\ar[d]&C_3\ar[r]\ar[d]&0\ & 0 & 0 & 0 &} } \end{xy}$
with the columns and bottom two rows are exact, then the top row is exact as well.



"9-lemma" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: 5-lemma


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proof of 9-lemma (Proof) by rm50
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Cross-references: rows, columns, commutative diagram, abelian category, objects
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This is version 3 of 9-lemma, born on 2003-08-15, modified 2007-01-08.
Object id is 4597, canonical name is 9Lemma.
Accessed 3045 times total.

Classification:
AMS MSC: 18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)
Pending Errata and Addenda
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