Trivial duality for systems of perfect objects.

Lemma 15.74.16. Let $A$ be a ring. Let $(K_ n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(A)$. Let $K = \text{hocolim} K_ n$ be the derived colimit (Derived Categories, Definition 13.33.1). Then for any object $E$ of $D(A)$ we have

$R\mathop{\mathrm{Hom}}\nolimits _ A(K, E) = R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_ A K_ n^\vee$

where $(K_ n^\vee )$ is the inverse system of dual perfect complexes.

Proof. By Lemma 15.74.15 we have $R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_ A K_ n^\vee = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E)$ which fits into the distinguished triangle

$R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) \to \prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) \to \prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E)$

Because $K$ similarly fits into the distinguished triangle $\bigoplus K_ n \to \bigoplus K_ n \to K$ it suffices to show that $\prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) = R\mathop{\mathrm{Hom}}\nolimits _ A(\bigoplus K_ n, E)$. This is a formal consequence of (15.73.0.1) and the fact that derived tensor product commutes with direct sums. $\square$

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