Let $k$ be a field. Let $X$ be a smooth projective equidimensional variety over $k$. How to show the following proposition?



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> For any coherent sheaf $F\_1$, $F\_2$ on $X$, $\operatorname{Ext}^i(F\_1,F\_2)=0$ if $i\notin [0,n]$.
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This question is from Lemma 5.6 in this [paper](https://arxiv.org/pdf/1901.09461)



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> **Lemma 5.6.** Suppose that $X$ is a smooth projective equidimensional variety over $k$. Then
> $$
> \overline{\operatorname{Sdim}} D^{b}(\operatorname{coh} X)=\underline{\operatorname{Sdim}} D^{b}(\operatorname{coh} X)=\operatorname{dim} X.
> $$
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> **Proof.** Let $n=\operatorname{dim} X$. Recall that the Serre functor on $D^{b}(\operatorname{coh} X)$ is isomorphic to $(-) \otimes \omega\_{X}[n] .$ Recall also that **for any coherent sheaves $F\_{1}, F\_{2}$ on $X$ we have $\operatorname{Ext}^{i}\left(F\_{1}, F\_{2}\right)=0$ if $i \notin[0, n]$.**
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Thank you very much!



For any coherent sheaf $F_1$, $F_2$ on $X$, $\operatorname{Ext}^i(F_1,F_2)=0$ if $i\notin [0,n]$
First, $\mathrm{Ext}^i(F\_1,F\_2)$ for $i < 0$ vanish by definition. Second, for $i > n$ the vanishing follows from Serre duality
$$
\mathrm{Ext}^i(F\_1,F\_2) \cong \mathrm{Ext}^{n-i}(F\_2,F\_1 \otimes \omega\_X)^\vee
$$
and the above vanishing.


