The Stacks project

Proof. If $\mathcal{K}_ x^\bullet $ is K-flat for all $x \in X$ then we see that $\mathcal{K}^\bullet $ is K-flat because $\otimes $ and direct sums commute with taking stalks and because we can check exactness at stalks, see Modules, Lemma 17.3.1. Conversely, assume $\mathcal{K}^\bullet $ is K-flat. Pick $x \in X$ $M^\bullet $ be an acyclic complex of $\mathcal{O}_{X, x}$-modules. Then $i_{x, *}M^\bullet $ is an acyclic complex of $\mathcal{O}_ X$-modules. Thus $\text{Tot}(i_{x, *}M^\bullet \otimes _{\mathcal{O}_ X} \mathcal{K}^\bullet )$ is acyclic. Taking stalks at $x$ shows that $\text{Tot}(M^\bullet \otimes _{\mathcal{O}_{X, x}} \mathcal{K}_ x^\bullet )$ is acyclic. $\square$