Lemma 13.19.9. Let \mathcal{A} be an abelian category. Assume \mathcal{A} has enough projectives. For any short exact sequence 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 of \text{Comp}^{+}(\mathcal{A}) there exists a commutative diagram in \text{Comp}^{+}(\mathcal{A})
\xymatrix{ 0 \ar[r] & P_1^\bullet \ar[r] \ar[d] & P_2^\bullet \ar[r] \ar[d] & P_3^\bullet \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A^\bullet \ar[r] & B^\bullet \ar[r] & C^\bullet \ar[r] & 0 }
where the vertical arrows are projective resolutions and the rows are short exact sequences of complexes. In fact, given any projective resolution P^\bullet \to C^\bullet we may assume P_3^\bullet = P^\bullet .
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