# For any coherent sheaf $F_1$, $F_2$ on $X$, $\operatorname{Ext}^i(F_1,F_2)=0$ if $i\notin [0,n]$

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

For any coherent sheaf $$F_1$$, $$F_2$$ on $$X$$, $$\operatorname{Ext}^i(F_1,F_2)=0$$ if $$i\notin [0,n]$$.

This question is from Lemma 5.6 in this paper

Lemma 5.6. Suppose that $$X$$ is a smooth projective equidimensional variety over $$k$$. Then $$\overline{\operatorname{Sdim}} D^{b}(\operatorname{coh} X)=\underline{\operatorname{Sdim}} D^{b}(\operatorname{coh} X)=\operatorname{dim} X.$$

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 $$i \notin[0, n]$$. ...

Thank you very much!

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.