Coherent sheaves, Serre’s theorem and ext groups

Question

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$.

Then, the category $\operatorname{Coh}(X)$ of coherent sheaves on $X$ is equivalent to the category $\operatorname{qgr}(A):= \operatorname{gr}(A)/{\operatorname{tor}(A)}$, where $A:= \bigoplus_{i \geq 0} H^0(X,O_X(1))$ and $\operatorname{qgr}(A)$ is the Serre quotient category of the category $\operatorname{gr}(A)$ of finitely generated graded $A$-modules by the category $\operatorname{tor}(A)$ of finitely generated torsion $A$-modules.

Then, we have an isomorphism \begin{align} \operatorname{Hom}_{O_X}(\mathcal{F},\mathcal{G}) &= \lim_{n \to \infty} \operatorname{Hom}_{\operatorname{gr}(A)}(\Gamma_{*}(\mathcal{F})_{\geq n}, \Gamma_{*}(\mathcal{G})_{\geq n})\\ &(= \operatorname{Hom}_{\operatorname{qgr}(A)}(\Gamma_{*}(\mathcal{F}),\Gamma_{*}(\mathcal{G}))), \end{align} where $\mathcal{F},\mathcal{G} \in \operatorname{Coh}(X)$ and $\Gamma_{*}(\mathcal{F}):=\bigoplus_{i} H^0(X,\mathcal{F}(i))$, $\Gamma_{*}(\mathcal{G}):=\bigoplus_{i} H^0(X,\mathcal{G}(i))$. $\Gamma_{*}(\mathcal{F})_{\geq n}$ also means the truncation of $M$ at the degree $n$. Moreover, we have $$ \operatorname{Ext}^i_{O_X}(\mathcal{F},\mathcal{G}) = \lim_{n \to \infty} \operatorname{Ext}^i_{\operatorname{gr}(A)}(\Gamma_{*}(\mathcal{F})_{\geq n}, \Gamma_{*}(\mathcal{G})_{\geq n}) $$ according to Proposition 4.3.3 (a) in Ciocan-Fontanine and Kapranov - Derived Quot schemes.

On the other hand, the functor $$ \Gamma_{*} : \operatorname{Coh}(X) \rightarrow \operatorname{gr}(A) $$ has a right adjoint $$ \tilde{-}:\operatorname{gr}(A) \rightarrow \operatorname{Coh}(X),\ M \mapsto \tilde{M}. $$ Then, for any $M \in \operatorname{gr}(A)$, there exists $m \in \mathbb{N}$, such that $\Gamma_{*}(\tilde{M})_{\geq m} \simeq M_{\geq m}$ in $\operatorname{gr}(A)$.

(Perhaps, the following relative theory is not necessarily in my questions.) We also have relative statement (Section 3.1 of The derived moduli space of stable sheaves) : Let $S$ be a noetherian scheme. Then, we have a functor \begin{align*} \Gamma_{*} : \operatorname{Coh}(X \times_k S)& \rightarrow \{\text{the category of graded coherent sheaves of $A \otimes_k \mathcal{O}_S$-modules} \} \\ &\hat{\mathcal{F}} \mapsto \oplus_i {\pi_{S}}_*(\hat{\mathcal{F}} \otimes \pi_{X}^*\mathcal{O}_X(i)). \end{align*} There exists the right adjoint functor $$ \tilde{-} : \{\text{the category of graded coherent sheaves of $A \otimes_k \mathcal{O}_S$-modules} \} \rightarrow \operatorname{Coh}(X \times_k S) . $$

Questions : Are the following true ?

1. For each $\mathcal{F},\mathcal{G} \in \operatorname{Coh}(X)$, and each $i \geq 0$, there exists $m$ such that $$ \operatorname{Ext}^i_{O_X}(\mathcal{F},\mathcal{G}) = \operatorname{Ext}^i(\Gamma_{*}(\mathcal{F})_{\geq m}, \Gamma_{*}(\mathcal{G})_{\geq m}). $$

2. Let $S$ be a noetherian scheme over $k$. Let $\hat{\mathcal{F}},\hat{\mathcal{G}} \in \operatorname{Coh}(X \times_k S)$ be flat families over $S$. Then, there exists $m \geq 0$ such that for any $i \geq 0$ and any $s \in S$, $$ \operatorname{Ext}^i_{O_X}(\hat{\mathcal{F}}_s,\hat{\mathcal{G}}_s) = \operatorname{Ext}^i_{\operatorname{gr}(A)}(\Gamma_{*}(\hat{\mathcal{F}}_s)_{\geq m}, \Gamma_{*}(\hat{\mathcal{G}}_s)_{\geq m}), $$ where $\hat{\mathcal{F}}_s$ is the restriction of $\hat{\mathcal{F}}$ to the fiber $\{s\} \times_S (X \times_k S) \simeq X$. (In Prop 4.3.3 (b) of the above reference, a similar result appears, but I am not sure that it is true. At least, I think $m$ is dependent on $i$ in the question.)

3. Let $S$ be a noetherian scheme. Let $\mathcal{M}$ be a flat family of finitely generated graded $A$-modules over $S$ (so that $\mathcal{M}$ is vector bundle on $S$). Then, there exists $m \geq 0$ s.t. for any $s \in S$, $$ \Gamma_{*}(\tilde{\mathcal{M}_s})_{\geq m} \simeq (\mathcal{M}_s)_{\geq m} \text{ as graded modules}, $$ where $\mathcal{M}_s$ is the restriction of $\mathcal{M}$ to the fiber $\{s \} \times_S S \simeq \operatorname{Spec}(k)$ (so that $\mathcal{M}_s$ is a graded $A$-module).

Any comment is welcome.

(Please let me know if any conditions are missing.)

The same question is in MO. 　

Edit(2023,12/15) : Some notations are changed. Question 3 is added. I also added more information about the adjoint functor of $\Gamma_*$, a relative theory of the Serre's theorem and question 2.

Edit(2024,1/29) In Ciocan-Fontanine and Kapranov’s paper, they tried to construct derived quot schemes. However, their construction is not correct. Actually, they point out a problem(cf. derived quot stacks I,derived quot stacks II). In addition, the authors in “derived quot stacks” gave the answer to the problem. On the other hand, they did not mention our questions.