Lemma of the day

Let (C,O) be a ringed site. Let (K_n)_{n ∈ N} be a system of perfect objects of D(O). Let K= hocolim K_n be the derived colimit (Definition Tag 090Z). For E in D(O) we have

RHom(K, E) = Rlim E ⊗ L_n

where L_n = RHom(K_n, O) is the inverse system of duals. See Lemma Tag 0A0A.

Slogan: Trivial duality for systems of perfect objects.