Lemma 21.46.2. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{E}^\bullet be a bounded above complex of flat \mathcal{O}-modules with tor-amplitude in [a, b]. Then \mathop{\mathrm{Coker}}(d_{\mathcal{E}^\bullet }^{a - 1}) is a flat \mathcal{O}-module.
Proof. As \mathcal{E}^\bullet is a bounded above complex of flat modules we see that \mathcal{E}^\bullet \otimes _\mathcal {O} \mathcal{F} = \mathcal{E}^\bullet \otimes _\mathcal {O}^{\mathbf{L}} \mathcal{F} for any \mathcal{O}-module \mathcal{F}. Hence for every \mathcal{O}-module \mathcal{F} the sequence
\mathcal{E}^{a - 2} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^ a \otimes _\mathcal {O} \mathcal{F}
is exact in the middle. Since \mathcal{E}^{a - 2} \to \mathcal{E}^{a - 1} \to \mathcal{E}^ a \to \mathop{\mathrm{Coker}}(d^{a - 1}) \to 0 is a flat resolution this implies that \text{Tor}_1^\mathcal {O}(\mathop{\mathrm{Coker}}(d^{a - 1}), \mathcal{F}) = 0 for all \mathcal{O}-modules \mathcal{F}. This means that \mathop{\mathrm{Coker}}(d^{a - 1}) is flat, see Lemma 21.17.15. \square
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