The Stacks project

Exercise 111.30.11. Let $\mathcal{A}$ be an abelian category. Consider a solid diagram

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

of complexes of $\text{Fil}^ f(\mathcal{A})$. Assume $K^\bullet $, $L^\bullet $ and $I^\bullet $ are bounded below and assume each $I^ n$ is a filtered injective object. Also assume that $\alpha $ is a filtered quasi-isomorphism.

  1. There exists a map of complexes $\beta $ making the diagram commute up to homotopy.

  2. If $\alpha $ is a strict monomorphism in every degree then we can find a $\beta $ which makes the diagram commute.

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View Exercise 111.30.11 as pdf