The Stacks project

Proof. Proof of (2). If $\mathcal{C}$ has enough points and $\mathcal{K}_ p^\bullet$ is K-flat for all points $p$ of $\mathcal{C}$ 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 on Sites, Lemma 18.14.4.

Proof of (1). Assume $\mathcal{K}^\bullet$ is K-flat. Choose a quasi-isomorphism $a : \mathcal{L}^\bullet \to \mathcal{K}^\bullet$ such that $\mathcal{L}^\bullet$ is K-flat with flat terms, see Lemma 21.17.11. Any pullback of $\mathcal{L}^\bullet$ is K-flat, see Lemma 21.18.1. In particular the stalk $\mathcal{L}_ p^\bullet$ is a K-flat complex of $\mathcal{O}_ p$-modules. Thus the cone $C(a)$ on $a$ is a K-flat (Lemma 21.17.6) acyclic complex of $\mathcal{O}$-modules and it suffuces to show the stalk of $C(a)$ is K-flat (by More on Algebra, Lemma 15.59.5). Thus we may assume that $\mathcal{K}^\bullet$ is K-flat and acyclic.

Assume $\mathcal{K}^\bullet$ is acyclic and K-flat. Before continuing we replace the site $\mathcal{C}$ by another one as in Sites, Lemma 7.29.5 to insure that $\mathcal{C}$ has all finite limits. This implies the category of neighbourhoods of $p$ is filtered (Sites, Lemma 7.33.2) and the colimit defining the stalk of a sheaf is filtered. Let $M$ be a finitely presented $\mathcal{O}_ p$-module. It suffices to show that $\mathcal{K}^\bullet \otimes _{\mathcal{O}_ p} M$ is acyclic, see More on Algebra, Lemma 15.59.9. Since $\mathcal{O}_ p$ is the filtered colimit of $\mathcal{O}(U)$ where $U$ runs over the neighbourhoods of $p$, we can find a neighbourhood $(U, x)$ of $p$ and a finitely presented $\mathcal{O}(U)$-module $M'$ whose base change to $\mathcal{O}_ p$ is $M$, see Algebra, Lemma 10.127.6. By Lemma 21.17.4 we may replace $\mathcal{C}, \mathcal{O}, \mathcal{K}^\bullet$ by $\mathcal{C}/U, \mathcal{O}_ U, \mathcal{K}^\bullet |_ U$. We conclude that we may assume there exists an $\mathcal{O}$-module $\mathcal{F}$ such that $M \cong \mathcal{F}_ p$. Since $\mathcal{K}^\bullet$ is K-flat and acyclic, we see that $\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{F}$ is acyclic (as it computes the derived tensor product by definition). Taking stalks is an exact functor, hence we get that $\mathcal{K}^\bullet \otimes _{\mathcal{O}_ p} M$ is acyclic as desired. $\square$