Let $X$ be a sufficiently regular space of dimension (n), for example a PL manifold or a regular variety. Is it true that $$gl.dim(Sh(X)):= \max_{\mathcal{F} \text{sheaf} } \min_{I \text{inj. res.}} \text{length of } I$$ equals to $n$?
Remark 1: this is similar to Serre theorem in the sense that for a regular ring, $gl.dim(R-mod)=dim(R)$.
Remark 2: In Methods of homological Algebra, Gelfand Manin, is showed that if $F$ is flat, it has a flat soft resolution of length $n$.