Standard homology result on bicomplexes

Question

Suppose you have got a bicomplex 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?

Answer 1

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.

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Answer 2

I see no reason to cite it when the proof is so easy! The homology of the associated total complex can be computed using the spectral sequence of a double complex in two different ways: either you start by taking homology of rows or of columns. In both cases the sequence obviously collapses, and the homology of $E^\infty $ is given by the homology of the bottom row and the leftmost column respectively.