I am trying to understand section 5.5 of the standard reference Differential geometry of complex vector bundles (S. Kobayashi). Let me set up some notation: let $X$ be a complex manifold, $x_0\in X$ and let $\mathcal{S}$ be a coherent analytic sheaf.
[Solved by KReiser] In Lemma 5.5.7 claims that a free resolution of $\mathcal{S}_{x_0}$ extends to a free resolution locally (of the same length). It seems to me that this cannot be true in general, e.g. consider the skyscraper sheaf of $\mathcal{O}_X$-modules $\mathcal{S}=\mathcal{O}_{x_0}$, which can be checked easily to be coherent. In this case we have an obvious free resolution of $\mathcal{O}_{x_0}$-modules: $$0\rightarrow \mathcal{O}_{x_0} \rightarrow \mathcal{S}_{x_0}\rightarrow 0$$ but a local free resolution of length $0$ would mean $\mathcal{S}$ is locally free, which is not the case.
In the next Theorem 5.5.8, the singularity set $S_{m}=\{x\in X | dh(\mathcal{S_x})\geq \mathrm{dim}_{\mathbb{C}} X-m\}$ ($dh$ stands for homological dimension) is locally identified with the set of points of $X$ where for the map $h$ in the free resolution $$...\rightarrow \mathcal{E}_{n-m}\overset{h}{\rightarrow}\mathcal{E}_{n-m-1}\rightarrow ... \rightarrow \mathcal{E}_0\rightarrow \mathcal{S}_U \rightarrow 0$$ regarded as a matrix in holomorphic functions, its rank is strictly smaller than the maximum value it reaches (which may not necessarily be $\mathrm{min}\{\mathrm{rk}(\mathcal{E}_{n-m}),\mathrm{rk}(\mathcal{E}_{n-m-1})\}$). I understand that maximal rank is an open condition and on such locus, say $W\subset X$ one can consider the free resolution of length $n-m-1$ $$0\rightarrow \mathcal{coker}(h)|_W\rightarrow ...\rightarrow \mathcal{E}_0|_{W}\rightarrow \mathcal{S}|_{W}\rightarrow 0$$ and so $W\cap S_m=\emptyset$. However, I don't see why $S_m=W^c$ (locally). Could not be the case that another free resolution $0\rightarrow \mathcal{E'}_{\bullet}\rightarrow \mathcal{S}\rightarrow 0$ provides another map $h'$ and another holomorphic constraint for $S_m$?