Lemma 15.71.3. Let $R$ be a ring. Let $M^\bullet $ be a complex of $R$-modules. Let $N^\bullet , \eta , \epsilon $ be a left dual of $M^\bullet $ in the monoidal category of complexes of $R$-modules. Then

$M^\bullet $ and $N^\bullet $ are bounded,

$M^ n$ and $N^ n$ are finite projective $R$-modules,

writing $\epsilon = \sum \epsilon _ n$ with $\epsilon _ n : N^{-n} \otimes _ R M^ n \to R$ and $\eta = \sum \eta _ n$ with $\eta _ n : R \to M^ n \otimes _ R N^{-n}$ then $(N^{-n}, \eta _ n, \epsilon _ n)$ is the left dual of $M^ n$ as in Lemma 15.71.2,

the differential $d_ N^ n : N^ n \to N^{n + 1}$ is equal to $-(-1)^ n$ times the map

\[ N^ n = \mathop{\mathrm{Hom}}\nolimits _ R(M^{-n}, R) \xrightarrow {d_ M^{-n - 1}} \mathop{\mathrm{Hom}}\nolimits _ R(M^{-n - 1}, R) = N^{n + 1} \]where the equality signs are the identifications from Lemma 15.71.2 part (2).

Conversely, given a bounded complex $M^\bullet $ of finite projective $R$-modules, setting $N^ n = \mathop{\mathrm{Hom}}\nolimits _ R(M^{-n}, R)$ with differentials as above, setting $\epsilon = \sum \epsilon _ n$ with $\epsilon _ n : N^{-n} \otimes _ R M^ n \to R$ given by evaluation, and setting $\eta = \sum \eta _ n$ with $\eta _ n : R \to M^ n \otimes _ R N^{-n}$ mapping $1$ to $\text{id}_{M_ n}$ we obtain a left dual of $M^\bullet $ in the monoidal category of complexes of $R$-modules.

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