Lemma 21.17.8. Let (\mathcal{C}, \mathcal{O}) be a ringed site. A bounded above complex of flat \mathcal{O}-modules is K-flat.
Proof. Let \mathcal{K}^\bullet be a bounded above complex of flat \mathcal{O}-modules. Let \mathcal{L}^\bullet be an acyclic complex of \mathcal{O}-modules. Note that \mathcal{L}^\bullet = \mathop{\mathrm{colim}}\nolimits _ m \tau _{\leq m}\mathcal{L}^\bullet where we take termwise colimits. Hence also
\text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ) = \mathop{\mathrm{colim}}\nolimits _ m \text{Tot}( \mathcal{K}^\bullet \otimes _\mathcal {O} \tau _{\leq m}\mathcal{L}^\bullet )
termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since \tau _{\leq m}\mathcal{L}^\bullet is acyclic this reduces us to the case where \mathcal{L}^\bullet is bounded above. In this case the spectral sequence of Homology, Lemma 12.25.3 has
{}'E_1^{p, q} = H^ p(\mathcal{L}^\bullet \otimes _ R \mathcal{K}^ q)
which is zero as \mathcal{K}^ q is flat and \mathcal{L}^\bullet acyclic. Hence we win. \square
Comments (0)