Question: Does every short exact sequence of injective bounded cochain complexes, $0\rightarrow I^\bullet\rightarrow J^\bullet\rightarrow K^\bullet\rightarrow 0$, split?
I am interested in a discrete setting. Let $\Pi$ be a finite poset, and fix a field $k$. The objects I am interested in are equivalently:
- a finitely dimensional $k$-representation of the poset $\Pi$,
- a finitely dimensional sheaf on $\Pi$ with Alexandrov topology with $k$ coefficients,
- a finitely dimensional module over the bound quiver algebra over $k$ corresponding to $\Pi$.
For any of those we can define injective objects, and bounded cochain complexes. An injective cochain complexes is a cochain complexes $I^{\bullet}$ such that each $I^{d}$ is injective.
Given a SES in the question, $0\rightarrow I^d \rightarrow J^d \rightarrow K^d \rightarrow 0$ splits for every $d$. But can this split be done consistently over all degrees so taht it is compatible with the cochain maps?
