Question on condition for a sheaf to be locally free in Orlov 2004

Question

In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free:

If for all closed points $t:x \hookrightarrow X$ we have $Ext^i(P_1, t_* \mathscr{O}_x)=0$ for all $i>0$, then $P_1$ is a locally free sheaf.

I cannot prove this nor find a reference. My only thought is to use adjunction to make this problem local; that is first work with $Ext^i(t^{-1}(P_1), \mathscr{O}_x)$, and then take a left resolution $P^{\cdot} \xrightarrow{\sim} P_1$. Next I'd try to use a spectral sequence such as $$ E_1^{i, j} = Ext_\mathcal{O}^j(P^{i}, \mathcal{O}_x) \Rightarrow Ext_\mathcal{O}^{i + j}(P^{\cdot}, \mathcal{O}_x)=Ext^{i+j}(P_1, \mathcal{O}_x). $$ but I'm not sure this spectral sequence is valid (I'm trying to use the 2nd spectral sequence on https://stacks.math.columbia.edu/tag/07A9 but derived in the first factor instead of the second; hopefully this introduces a change in sign).

Does anyone have a proof or a reference?

Answer 1

The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ for $i > 0$ then $M$ is free. Let $n = \dim(M/\mathfrak{m}M)$ and let $f:A^{\oplus n} \to M$ be a homomorphism that induces an isomorphism modulo $\mathfrak{m}$. Then $f$ is surjective by Nakayama lemma and setting $N = Ker(f)$ we have an exact sequence $$ 0 \to N \to A^{\oplus n} \to M \to 0. $$ Since $$ Hom(M,A/\mathfrak{m}) \cong Hom(M/\mathfrak{m}M,A/\mathfrak{m}) \cong (A/\mathfrak{m})^{\oplus n} \cong Hom(A^{\oplus n},A/\mathfrak{m}) $$ and $Ext^1(M,A/\mathfrak{m}) = 0$, it follows that $Hom(N,A/\mathfrak{m}) = 0$, hence $N/\mathfrak{m}N = 0$, hence by Nakayama lemma $N = 0$, hence $M$ is free.