For both assertions, we may assume that \(X = \Spec A\) is affine; write \(Z = \Spec A/I\text{.}\) Write \(nE\) for the subscheme of \(Y\) cut out by \(I^n\text{.}\)

For (1), we may assume \(\calF = \calO\text{.}\) By our hypotheses, we have \(\calO(X) \cong \calO(Y)\) and \(\calO(Z) \cong \calO(E)\) by Stein factorization, and similarly after taking perfections. Since \(X\) and \(Z\) are both affine, it remains to check that \(H^i(Y_{\perf}, \calO) \to H^i(E_{\perf}, \calO)\) is an isomorphism for each \(i \gt 0\text{.}\)

At this point, we follow

[27], Lemma 3.9 (which is written using the Nisnevich topology, but the Zariski topology works equally well). By

[117], tag 02OB, point (1), there exists a constant

\(c\) such that for

\(n \geq c\text{,}\)
\begin{equation*}
\ker(H^i(Y, \calO) \to H^i(E_n, \calO)) \subseteq I^{n-c} H^i (Y,\calO).
\end{equation*}

Note that \(H^i(Y, \calO)\) is a finitely generated \(A\)-module which, since \(f\) is a blowup and \(i \gt 0\text{,}\) is supported entirely on \(Z\text{.}\) Hence for \(n \gg 0\text{,}\) \(I^{n-c}\) annihilates \(H^i(Y, \calO)\) and so

\begin{equation}
H^i(Y, \calO) \hookrightarrow H^i(E_n, \calO) \qquad (n \gg 0).\tag{21.1}
\end{equation}

On the other hand, by

[117], tag 020B, point (3), for

\(m \gg n \gg 0\) we have

\begin{equation}
\im(H^i(E_m, \calO) \to H^i(E_n, \calO)) = \im(H^i(Y, \calO) \to H^i(E_n, \calO)).\tag{21.2}
\end{equation}

Fix a value

\(n \gg 0\) that is large enough for both

(21.1) and

(21.2) to hold. Then for

\(e \gg 0\text{,}\) the image of

\(\phi^e\colon H^i(E_n, \calO) \to H^i(E_n, \calO)\) is contained in the image of

\(H^i(Y, \calO) \to H^i(E_n, \calO)\text{:}\) to see this, refactor the former map as

\begin{equation*}
H^i(E_n, \calO) \stackrel{\phi^e}{\to} H^i(E_{p^e n}, \calO) \to H^i(E_n, \calO)
\end{equation*}

\begin{equation*}
\colim_\phi H^i(Y, \calO) = \colim_\phi H^i(E_n, \calO)
\end{equation*}

and hence

\begin{align*}
H^i(Y_{\perf}, \calO) &= \colim_\phi H^i(E_n, \calO)\\
&= \colim_\phi H^i(E, \calO) = H^i(E_{\perf}, \calO)
\end{align*}

as claimed.

For (2), we follow

[24], Lemma 4.6. By the Beauville-Laszlo theorem (see

Remark 21.2.7), we may assume that

\(A\) is (classically)

\(I\)-complete. We may also assume that we start with an object in

\(\Vect(Y) \times_{\Vect(E)} \Vect(Z)\text{.}\) Let

\(\calI\) be the inverse image ideal sheaf of

\(I\text{;}\) by the construction of the blowup,

\(\calI\) is an

*ample* invertible sheaf on

\(Y\text{.}\) Consequently, by Serre vanishing, we may choose some

\(n\) such that

\begin{equation}
H^i(Y, \calI^k/\calI^{k+1}) = 0 \qquad (k \geq n).\tag{21.3}
\end{equation}

Since

\(X\) is affine and complete along

\(Z\text{,}\) \(\Vect(X) \to \Vect(Z)\) is an equivalence of categories (

Exercise 6.7.9). We thus have objects

\(\calE \in \Vect(X)\text{,}\) \(\calF \in \Vect(Y)\) and an isomorphism

\(\psi\colon f^* \calE|_E \cong \calF|_E\text{.}\) By pulling back by a suitable power of

\(\phi\text{,}\) we may construct another isomorphism

\(\psi_n\colon f^* \calE|_{nE} \cong \calF|_{nE}\text{.}\)
We now observe that for \(m \geq n\text{,}\) an isomorphism \(\psi_m\colon f^* \calE|_{mE} \cong \calF|_{mE}\) can be promoted to an isomorphism \(\psi_{m+1}\colon f^* \calE|_{(m+1)E} \cong \calF|_{(m+1)E}\text{:}\) namely, the obstruction to lifting belongs to

\begin{equation*}
H^1(Y, \calI^m/\calI^{m+1} \otimes \sheafHom(f^* \calE, \calF))
\end{equation*}

which vanishes by

(21.3). Since

\begin{equation*}
\Vect(Y) \to \lim_m \Vect(mE)
\end{equation*}

is an equivalence by the formal existence theorem (

[117], tag 0885), we deduce the desired result.