The Stacks project

Lemma 15.86.4. Let $R$ be a ring. Let $P^\bullet $ be a bounded above complex of projective $R$-modules. Let $K^\bullet $ be a K-flat complex of $R$-modules. If $P^\bullet $ is a perfect object of $D(R)$, then $\mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , K^\bullet )$ is K-flat and represents $R\mathop{\mathrm{Hom}}\nolimits _ R(P^\bullet , K^\bullet )$.

Proof. The last statement is Lemma 15.68.2. Since $P^\bullet $ represents a perfect object, there exists a finite complex of finite projective $R$-modules $F^\bullet $ such that $P^\bullet $ and $F^\bullet $ are isomorphic in $D(R)$, see Definition 15.69.1. Then $P^\bullet $ and $F^\bullet $ are homotopy equivalent, see Derived Categories, Lemma 13.19.8. Then $\mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , K^\bullet )$ and $\mathop{\mathrm{Hom}}\nolimits ^\bullet (F^\bullet , K^\bullet )$ are homotopy equivalent. Hence the first is K-flat if and only if the second is (follows from Definition 15.57.3 and Lemma 15.57.1). It is clear that

\[ \mathop{\mathrm{Hom}}\nolimits ^\bullet (F^\bullet , K^\bullet ) = \text{Tot}(E^\bullet \otimes _ R K^\bullet ) \]

where $E^\bullet $ is the dual complex to $F^\bullet $ with terms $E^ n = \mathop{\mathrm{Hom}}\nolimits _ R(F^{-n}, R)$, see Lemma 15.69.14 and its proof. Since $E^\bullet $ is a bounded complex of projectives we find that it is K-flat by Lemma 15.57.9. Then we conclude by Lemma 15.57.6. $\square$


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