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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

$\displaystyle \xymatrix{ & 0\ar[d]&0\ar[d]&0\ar[d]&\ 0\ar[r] & A_1\ar[r]\ar[d... ... 0\ar[r] & A_3\ar[r]\ar[d]&B_3\ar[r]\ar[d]&C_3\ar[r]\ar[d]&0\ & 0 & 0 & 0 &}$
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


Attachments:
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 4011 times total.

Classification:
AMS MSC18G35 (Category theory; homological algebra :: Homological algebra :: Chain complexes)

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