The Stacks project

Lemma 13.18.7. Let $\mathcal{A}$ be an abelian category. Consider a solid diagram

\[ \xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_\gamma & L^\bullet \ar@{-->}[dl]^{\beta _ i} \\ I^\bullet } \]

where $I^\bullet $ is bounded below and consists of injective objects, and $\alpha $ is a quasi-isomorphism. Any two morphisms $\beta _1, \beta _2$ making the diagram commute up to homotopy are homotopic.

Proof. This follows from Remark 13.18.5. We also give a direct argument here.

Let $\tilde\alpha : K \to \tilde L^\bullet $, $\pi $, $s$ be as in Lemma 13.9.6. If we can show that $\beta _1 \circ \pi $ is homotopic to $\beta _2 \circ \pi $, then we deduce that $\beta _1 \sim \beta _2$ because $\pi \circ s$ is the identity. Hence we may assume $\alpha ^ n : K^ n \to L^ n$ is the inclusion of a direct summand for all $n$. Thus we get a short exact sequence of complexes

\[ 0 \to K^\bullet \to L^\bullet \to M^\bullet \to 0 \]

which is termwise split and such that $M^\bullet $ is acyclic. We choose splittings $L^ n = K^ n \oplus M^ n$, so we have $\beta _ i^ n : K^ n \oplus M^ n \to I^ n$ and $\gamma ^ n : K^ n \to I^ n$. In this case the condition on $\beta _ i$ is that there are morphisms $h_ i^ n : K^ n \to I^{n - 1}$ such that

\[ \gamma ^ n - \beta _ i^ n|_{K^ n} = d \circ h_ i^ n + h_ i^{n + 1} \circ d \]

Thus we see that

\[ \beta _1^ n|_{K^ n} - \beta _2^ n|_{K^ n} = d \circ (h_1^ n - h_2^ n) + (h_1^{n + 1} - h_2^{n + 1}) \circ d \]

Consider the map $h^ n : K^ n \oplus M^ n \to I^{n - 1}$ which equals $h_1^ n - h_2^ n$ on the first summand and zero on the second. Then we see that

\[ \beta _1^ n - \beta _2^ n - (d \circ h^ n + h^{n + 1}) \circ d) \]

is a morphism of complexes $L^\bullet \to I^\bullet $ which is identically zero on the subcomplex $K^\bullet $. Hence it factors as $L^\bullet \to M^\bullet \to I^\bullet $. Thus the result of the lemma follows from Lemma 13.18.4. $\square$


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