Let $(X^{n,m}, d'^{n,m}, d''^{n,m})$ be a double complex. In Kashiwara, Schapira Categories and Sheaves as a corollary of
Theorem. Let $f: X \rightarrow Y$ be a morphism of bounded double complexes such that $f$ induces an isomorphism $f: H_{II}(H_{I}(X)) \rightarrow H_{II}(H_{I}(Y))$, then $f$ induces an isomorphism $\text{tot}(X) \cong \text{tot}(Y)$
it is proved that:
- If the rows of $X^{\bullet,\bullet}$ are exact then $\text{tot}(X^{\bullet, \bullet})$ is quasi isomorphic to $0$
- If the rows of $X^{j,\bullet}$ are exact except for $j = p$ then $\text{tot}(X^{\bullet,\bullet})$ is quasi isomorphic to $X^{p, \bullet}[-p]$
To prove 2 they say to apply the theorem to the morphisms $\sigma_{I}^{\geq p}X \rightarrow X$ and $\sigma_{I}^{\leq p} \sigma_{I}^{\geq p} X \rightarrow \sigma_{I}^{\geq p}X$, where $\sigma_{I}^{\geq p}$ is the stupid truncation which leaves the rows greater or equal than p. I don't understand why the hypotheses of the theorem are satisfied. Any help?
($H_I$ is the vertical cohomology and $H_{II}$ is the horizontal cohomology)