Lemma 93.11.4. In Lemma 93.11.3 if $X$ is proper over $k$, then $\text{Inf}(\mathcal{D}\! \mathit{ef}_ X)$ and $T\mathcal{D}\! \mathit{ef}_ X$ are finite dimensional.
Proof. By the lemma we have to show $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ and $\mathop{\mathrm{Ext}}\nolimits ^0_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/k}, \mathcal{O}_ X)$ are finite dimensional. By More on Morphisms of Spaces, Lemma 76.21.5 and the fact that $X$ is Noetherian, we see that $\mathop{N\! L}\nolimits _{X/k}$ has coherent cohomology sheaves zero except in degrees $0$ and $-1$. By Derived Categories of Spaces, Lemma 75.8.4 the displayed $\mathop{\mathrm{Ext}}\nolimits $-groups are finite $k$-vector spaces and the proof is complete. $\square$
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