# Homology of total complex of a double complex

The following question arose while studying chapter 12.5 of the book Categories and Sheaves by Kashiwara and Schapira, abbreviated in the following by [KS].

Let $$X$$ be a double complex (with cohomological convention, i.e. with differentials of degree $$1$$) in an abelian category such that for any $$n\in\mathbb Z$$, the set $$\{(k,l)\in\mathbb Z\times\mathbb Z~|~ k+l=n~\text{and}~X^{k,l}\neq 0\}$$ is finite. Denote by $$\operatorname{tot}X$$ the associated total complex (defined by $$(\operatorname{tot}X)^{n}=\oplus_{k+l=n} X^{k,l}$$ and with the usual differential).

Define $$H_I(X)$$ via $$H_I(X)^{k,l}=H^k(X^{\cdot,l})$$ with differential $$d_{II}$$ (in the second coordinate) induced by $$d_{II}$$ on $$X$$ and with the zero differential in the first coordinate. Define $$H_{II}$$ in a dual manner. Then we obtain a double complex $$H_{II}H_{I}(X)$$ with zero differentials in each coordinate. One then obtains ([KS], Theorem 12.5.4):

Let $$f\colon X\to Y$$ be a morphism of double complexes (both satisfying the finiteness constraint above) and suppose that $$H_{II}H_I(X)\to H_{II}H_I(Y)$$ is an isomorphism. Then $$\operatorname{tot} X\to \operatorname{tot} Y$$ is a quasi-isomorphism.

I understand this theorem as well as the proof. What I don't understand is the following corollary ([KS], Corollary 12.5.5 (i)):

Assume $$X$$ has exact rows (i.e. $$X^{k,\cdot}$$ is exact for each $$k$$). Then $$\operatorname{tot}X$$ is quasi-isomorphic to zero.

Replacing "rows" by "columns", the statement is clear. But I don't see how this can be true for exact rows, since in the above theorem we first take homology along the columns which does not necessarily vanish, and I don't think that the complex $$\cdots\to H^{k}(X^{\cdot,l-1})\to H^{k}(X^{\cdot,l})\to H^{k}(X^{\cdot,l+1})\to \cdots$$ is necessarily exact, given that $$\cdots\to X^{k,l-1}\to X^{k,l}\to X^{k,l+1}\to\cdots$$ is exact.

So is this statement even true or is it a typo?

Up to isomorphism, $$\operatorname{tot} X$$ doesn't change if you switch rows and columns. so whatever is true for columns is true for rows.
Alternatively, Theorem 12.5.4 is true, with the same proof, if you replace $$H_{II}H_I$$ with $$H_IH_{II}$$.