Lemma 20.26.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{K}^\bullet$ be a complex of $\mathcal{O}_ X$-modules. Then $\mathcal{K}^\bullet$ is K-flat if and only if for all $x \in X$ the complex $\mathcal{K}_ x^\bullet$ of $\mathcal{O}_{X, x}$-modules is K-flat (More on Algebra, Definition 15.59.1).

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 $x \in X$ and let $M^\bullet$ 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. $\square$

Comment #2030 by FĂ©lix BB on

Just to signal a small tipo in the statement of the lemma: $\mathcal{O}_{X, x}$-modules. (The word modules was missing.)

Comment #8616 by nkym on

Between "Pick $x\in X$" and "$M$ be an acyclic complex", period and "Let" are needed.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).