Over projective space, it is well-known that given a graded $S^\bullet$-module $M\_\bullet$, where $S^\bullet = k[x\_0, \dots, x\_N]$, there is a unique minimal free resolution
$$
 \cdots \to \bigoplus\_q S^\bullet(-q) \otimes B\_{p, q} \to \cdots \to \bigoplus\_q S^\bullet(-q) \otimes B\_{1, q} \to \bigoplus\_q S^\bullet(-q) \otimes B\_{0, q} \to M \to 0
$$


Over an arbitrary variety $X$ over $k$ (assumed to be projective and smooth if necessary), if I define a locally free resolution of a coherent $\mathcal O\_X$-module $\mathcal F$
$$
 \mathcal E^\bullet \to \mathcal F \to 0
$$
to be **minimal** if for each local ring $\mathcal O\_{X, x}$, the complex $\mathcal E^\bullet \otimes\_{\mathcal O\_{X, x}} \kappa(x)$ has zero differentiations (Matsumura, (18.E)).


I wonder whether a minimal locally free resolution of $\mathcal F$ exists, and whether it is unique if exists.



Minimal free resolution over arbitrary varieties
This is not a good definition, because the complex $\mathcal{E}^\bullet \otimes\_{\mathcal{O}\_{X,x}} \kappa(x)$ computes $\mathrm{Tor}\_i^{\mathcal{O}\_{X,x}}(\mathcal{F}, \kappa(x))$, so if this complex has zero differential and at least two terms, it follows that
$$
\mathrm{Tor}\_1^{\mathcal{O}\_{X,x}}(\mathcal{F}, \kappa(x)) \ne 0
$$
for any point $x \in X$. But any coherent sheaf is locally free at general point, which contradicts the above inequality.


