Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$.
Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the category of sheaves of $k$-vector spaces on $X$. If $\underline{k} \in \mathrm{Shv}_k(X)$ is the constant sheaf, then the previous assertion states that $\mathrm{Ext}^i(\underline{k}, \mathcal{F}) = 0$ for $i > n$ for any $\mathcal{F} \in \mathrm{Shv}_k(X)$. More generally, if $U \subset X$ is an open subset, then the lower-shriek image $\underline{k}_U$ (i.e., $j_!( \underline{k})$ if $j: U \to X$ is the inclusion) has the property that $\mathrm{Ext}^i(\underline{k}_U, \mathcal{F}) = H^i(U; \mathcal{F}) = 0$ for $i > n$.
I think this means that the cohomological dimension of the category $\mathrm{Shv}_k(X)$ is at most $n +1$: a generating set of injections in $\mathrm{Shv}_k(X)$ consists of the maps $\underline{k}_U \to \underline{k}_V$, for $U \subset V$, whose cokernels have (by the above claim) projective dimension $\leq n+1$.
Can this be improved? Is the cohomological dimension of $\mathrm{Shv}_k(X)$ actually $n$?
