The Stacks project

Proof. Proof of (1). We say $M$ has affine locally finite injective dimension if $X$ has an open covering by affines $U = \mathop{\mathrm{Spec}}(A)$ such that the object of $D(A)$ corresponding to $M|_ U$ (Derived Categories of Schemes, Lemma 36.3.5) has finite injective dimension¹. To prove the lemma we may replace $X$ by $U$, i.e., we may assume $X = \mathop{\mathrm{Spec}}(A)$ for some Noetherian ring $A$. Observe that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.5. Moreover, the formation of the left and right hand side of the arrow commutes with the functor $D(A) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 48.2.3 and Derived Categories of Schemes, Lemma 36.10.8 (to be sure this uses the assumptions on $K$, $L$, $M$ and what we just proved about $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$). Then finally the arrow is an isomorphism by More on Algebra, Lemmas 15.98.1 part (2).

Proof of (2). We argue as above. A small change is that here we get $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ in $D_{\textit{Coh}}(\mathcal{O}_ X)$ because affine locally (which is allowable by Lemma 48.2.3) we may appeal to Dualizing Complexes, Lemma 47.15.2. Then we finally conclude by More on Algebra, Lemma 15.98.2. $\square$