Minimal free resolution over arbitrary varieties

Question

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.

Answer 1

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.