# Minimal free resolution over arbitrary varieties

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.

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.