Suppose you have got a double complex in an abelian category with objects $(A\_{rs},d\_{rs})$ such that $A\_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A\_{r\bullet},d\_{r\bullet})$ for $r > 0$ and the columns $(A\_{\bullet s},d\_{\bullet s})$ for $s > 0$ are exact.


Then a standard result says: The complexes the top-most row $(A\_{0 \bullet},d\_{0\bullet})$ and the left-most column $(A\_{\bullet 0},d\_{\bullet 0})$ are isomorphic on homology.


I am not aware of a citable reference for this standard result. Can help me out with that?



Standard homology result on double complexes
This is Proposition 3.9 in Osborne's Basic Homological Algebra. I expect you could also find it somewhere in pretty much any homological algebra text, though it might be mentioned in passing rather than stated explicitly as a result.


${}{}{}{}{}{}$


