The Stacks project

  • The choice above is such that if $M^\bullet $ has a left dual $N^\bullet $ as in Lemma 15.72.2, then we have a canonical isomorphism

    \[ \text{Tot}(K^\bullet \otimes _ R N^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , K^\bullet ) \]

    defined without the intervention of signs sending the summand $K^ p \otimes _ R N^ q$ to the summand $\mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, K^ p)$ via $N^ q = \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, R)$ and the canonical map $K^ p \otimes _ R \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, R) \to \mathop{\mathrm{Hom}}\nolimits _ R(M^{-q}, K^ p)$.

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