Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.5.1.7 (Two-out-of-Three). Let $\operatorname{\mathcal{A}}$ be an abelian category and suppose we are given a commutative diagram of chain complexes

\[ \xymatrix { 0 \ar [r] & C'_{\ast } \ar [d]^{f'} \ar [r] & C_{\ast } \ar [d]^{f} \ar [r] & C''_{\ast } \ar [d]^{f''} \ar [r] & 0 \\ 0 \ar [r] & D'_{\ast } \ar [r] & D_{\ast } \ar [r] & D''_{\ast } \ar [r] & 0 } \]

in which the rows are exact. If any two of the chain maps $f$, $f'$, and $f''$ are quasi-isomorphisms, then so is the third. This follows by comparing the long exact homology sequences associated to the upper and lower rows (see Construction ).