The Stacks project

Exercise 111.30.2. Let $\mathcal{A}$ be an abelian category. Let $I$ be a filtered object of $\mathcal{A}$. Assume that the filtration on $I$ is finite. Show the following are equivalent:

1. For any solid diagram

   \[ \xymatrix{ A \ar[r]_\alpha \ar[d] & B \ar@{-->}[ld] \\ I & } \]

   of filtered objects with (i) the filtrations on $A$ and $B$ are finite, and (ii) $\text{gr}(\alpha )$ injective the dotted arrow exists making the diagram commute.

2. Each $\text{gr}^ p I$ is injective.