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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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