hcju/stacksqa huggingface dataset:
We will use the criterion of Lemma \ref{lemma-unwinding-conditions}. Let $y \in U \cap Y$ with $\dim(\overline{\{y\}} = 1$ and let $\mathfrak p$ be a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$. Condition (4) shows that $\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) = \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p)$. Thus we have to prove $$ \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) > 2 $$ Let $\mathfrak p_0 \subset A$ be the image of $\mathfrak p$. Let $\mathfrak q \subset A$ be the prime corresponding to $y$. By Local Cohomology, Lemma \ref{local-cohomology-lemma-change-completion} we have \begin{align*} \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) & = \text{depth}(A_{\mathfrak p_0}) + \dim((A/\mathfrak p_0)_\mathfrak q) \\ & = \text{depth}(A_{\mathfrak p_0}) + \dim(A/\mathfrak p_0) - 1 \end{align*} If (5)(a) holds, then we get that this is $$ \geq \min(2, \dim(A_{\mathfrak p_0})) + \dim(A/\mathfrak p_0) - 1 $$ Note that in any case $\dim(A/\mathfrak p_0) \geq 2$. Hence if we get $2$ for the minimum, then we are done. If not we get $$ \dim(A_{\mathfrak p_0}) + \dim(A/\mathfrak p_0) - 1 \geq 4 - 1 $$ because every component of $\Spec(A)$ passing through $\mathfrak p_0$ has dimension $\geq 4$. If (5)(b) holds, then we win immediately.
For every $n$ we let $\mathcal{F}_n \to \mathcal{F}'_n$ be the surjection constructed in Local Cohomology, Lemma \ref{local-cohomology-lemma-get-depth-1-along-Z}. Since this is the quotient of $\mathcal{F}_n$ by the subsheaf of sections supported on $T$ we see that we get canonical maps $\mathcal{F}'_{n + 1} \to \mathcal{F}'_n$ such that we obtain a map $(\mathcal{F}_n) \to (\mathcal{F}_n')$ of inverse systems of coherent $\mathcal{O}_X$-modules. Property (1) holds by construction. \medskip\noindent To prove properties (2) and (3) we may assume that $X = \Spec(A_0)$ is affine and $A_0$ has a dualizing complex. Let $I_0 \subset A_0$ be the ideal corresponding to $Y$. Let $A, I$ be the $I$-adic completions of $A_0, I_0$. For later use we observe that $A$ has a dualizing complex (Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}). Let $M$ be the finite $A$-module corresponding to $(\mathcal{F}_n)$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-affine}. Then $\mathcal{F}_n$ corresponds to $M_n = M/I^nM$. Recall that $\mathcal{F}'_n$ corresponds to the quotient $M'_n = M_n / H^0_T(M_n)$, see Local Cohomology, Lemma \ref{local-cohomology-lemma-get-depth-1-along-Z} and its proof. \medskip\noindent Set $s = 0$ and $d = \text{cd}(A, I)$. We claim that $A, I, T, M, s, d$ satisfy assumptions (1), (3), (4), (6) of Situation \ref{situation-bootstrap}. Namely, (1) and (3) are immediate from the above, (4) is the empty condition as $s = 0$, and (6) is the assumption we made in the statement of the lemma. \medskip\noindent By Theorem \ref{theorem-final-bootstrap} we see that $\{H^0_T(M_n)\}$ is Mittag-Leffler, that $\lim H^0_T(M_n) = H^0_T(M)$, and that $H^0_T(M)$ is killed by a power of $I$. Thus the limit of the short exact sequences $0 \to H^0_T(M_n) \to M_n \to M'_n \to 0$ is the short exact sequence $$ 0 \to H^0_T(M) \to M \to \lim M'_n \to 0 $$ Setting $M' = \lim M'_n$ we find that $\mathcal{G}_n$ corresponds to the finite $A_0$-module $M'/I^nM'$. To finish the prove we have to show that the canonical map $\{M'/I^nM'\} \to \{M'_n\}$ is a pro-isomorphism. This is equivalent to saying that $\{H^0_T(M) + I^nM\} \to \{\ker(M \to M'_n)\}$ is a pro-isomorphism. Which in turn says that $\{H^0_T(M)/H^0_T(M) \cap I^nM\} \to \{H^0_T(M_n)\}$ is a pro-isomorphism. This is true because $\{H^0_T(M_n)\}$ is Mittag-Leffler, $\lim H^0_T(M_n) = H^0_T(M)$, and $H^0_T(M)$ is killed by a power of $I$ (so that Artin-Rees tells us that $H^0_T(M) \cap I^nM = 0$ for $n$ large enough).
As in Lemma \ref{lemma-divide-torsion-formal-coherent-module} and its proof for every $n$ we let $\mathcal{F}_n \to \mathcal{F}'_n$ be the surjection constructed in Local Cohomology, Lemma \ref{local-cohomology-lemma-get-depth-1-along-Z}. Next, we let $\mathcal{F}'_n \to \mathcal{F}''_n$ be the injection constructed in Local Cohomology, Lemma \ref{local-cohomology-lemma-make-S2-along-T} and its proof. The constructions show that we get canonical maps $\mathcal{F}''_{n + 1} \to \mathcal{F}''_n$ such that we obtain maps $$ (\mathcal{F}_n) \longrightarrow (\mathcal{F}_n') \longrightarrow (\mathcal{F}''_n) $$ of inverse systems of coherent $\mathcal{O}_X$-modules. Properties (1) and (2) hold by construction. \medskip\noindent To prove properties (3) and (4) we may assume that $X = \Spec(A_0)$ is affine and $A_0$ has a dualizing complex. Let $I_0 \subset A_0$ be the ideal corresponding to $Y$. Let $A, I$ be the $I$-adic completions of $A_0, I_0$. For later use we observe that $A$ has a dualizing complex (Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}). Let $M$ be the finite $A$-module corresponding to $(\mathcal{F}_n)$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-affine}. Then $\mathcal{F}_n$ corresponds to $M_n = M/I^nM$. Recall that $\mathcal{F}'_n$ corresponds to the quotient $M'_n = M_n / H^0_T(M_n)$. Also, recall that $M' = \lim M'_n$ is the quotient of $M$ by $H^0_T(M)$ and that $\{M'_n\}$ and $\{M'/I^nM'\}$ are isomorphic as pro-systems. Finally, we see that $\mathcal{F}''_n$ corresponds to an extension $$ 0 \to M'_n \to M''_n \to H^1_{T'}(M'_n) \to 0 $$ see proof of Local Cohomology, Lemma \ref{local-cohomology-lemma-make-S2-along-T}. \medskip\noindent Set $s = 1$. We claim that $A, I, T', M', s, d$ satisfy assumptions (1), (3), (4), (6) of Situation \ref{situation-bootstrap}. Namely, (1) and (3) are immediate, (4) is implied by (c), and (6) follows from (d). We omit the details of the verification (c) $\Rightarrow$ (4). \medskip\noindent By Theorem \ref{theorem-final-bootstrap} we see that $\{H^1_{T'}(M'/I^nM')\}$ is Mittag-Leffler, that $H^1_{T'}(M') = \lim H^1_{T'}(M'/I^nM')$, and that $H^1_{T'}(M')$ is killed by a power of $I$. We deduce $\{H^1_{T'}(M'_n)\}$ is Mittag-Leffler and $H^1_{T'}(M') = \lim H^1_{T'}(M'_n)$. Thus the limit of the short exact sequences displayed above is the short exact sequence $$ 0 \to M' \to \lim M''_n \to H^1_{T'}(M') \to 0 $$ Set $M'' = \lim M''_n$. It follows from Local Cohomology, Proposition \ref{local-cohomology-proposition-finiteness} that $H^1_{T'}(M')$ and hence $M''$ are finite $A$-modules. Thus we find that $\mathcal{H}_n$ corresponds to the finite $A_0$-module $M''/I^nM''$. To finish the prove we have to show that the canonical map $\{M''/I^nM''\} \to \{M''_n\}$ is a pro-isomorphism. Since we already know that $\{M'/I^nM'\}$ is pro-isomorphic to $\{M'_n\}$ the reader verifies (omitted) this is equivalent to asking $\{H^1_{T'}(M')/I^nH^1_{T'}(M')\} \to \{H^1_{T'}(M'_n)\}$ to be a pro-isomorphism. This is true because $\{H^1_{T'}(M'_n)\}$ is Mittag-Leffler, $H^1_{T'}(M') = \lim H^1_{T'}(M'_n)$, and $H^1_{T'}(M')$ is killed by a power of $I$.
The existence and properties (2), (3), (4) follow immediately from Lemma \ref{lemma-improvement-formal-coherent-module-better} applied to $U$, $U \cap Y$, $T = \{y \in U \cap Y : \delta^Y_Z(y) \leq 2\}$, $T' = \{y \in U \cap Y : \delta^Y_Z(y) \leq 1\}$, and $(\mathcal{F}_n)$. The finiteness of the modules $H^0(U, \mathcal{F}''_n)$ and $H^1(U, \mathcal{F}''_n)$ follows from Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-Rjstar} and the elementary properties of the function $\delta^Y_Z(-)$ proved in Lemma \ref{lemma-discussion}.
We will show that the hypotheses (1), (2), (3)(b) of Proposition \ref{proposition-d-generators} are satisfied. This is clear for (1) and (2). \medskip\noindent Let $y \in U \cap Y$ and let $\mathfrak p$ be a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not \in V(I\mathcal{O}_{X, y}^\wedge)$. The last condition shows that $\text{depth}((\mathcal{F}_y^\wedge)_\mathfrak p) = \text{depth}((\mathcal{O}_{X, y}^\wedge)_\mathfrak p)$. Since $X \setminus Y$ is Cohen-Macaulay we see that $(\mathcal{O}_{X, y}^\wedge)_\mathfrak p$ is Cohen-Macaulay. Thus we see that \begin{align*} & \text{depth}((\mathcal{F}^\wedge_y)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) \\ & = \dim((\mathcal{O}_{X, y}^\wedge)_\mathfrak p) + \dim(\mathcal{O}_{X, y}^\wedge/\mathfrak p) + \delta^Y_Z(y) \\ & = \dim(\mathcal{O}_{X, y}^\wedge) + \delta^Y_Z(y) \end{align*} The final equality because $\mathcal{O}_{X, y}$ is equidimensional by the second condition. Let $\delta(y) = \dim(\overline{\{y\}})$. This is a dimension function as $A$ is a catenary local ring. By Lemma \ref{lemma-discussion} we have $\delta^Y_Z(y) \geq \delta(y) - \dim(Z)$. Since $X$ is equidimensional we get $$ \dim(\mathcal{O}_{X, y}^\wedge) + \delta^Y_Z(y) \geq \dim(\mathcal{O}_{X, y}^\wedge) + \delta(y) - \dim(Z) = \dim(X) - \dim(Z) $$ Thus we get the desired inequality and we win.
It follows from Lemma \ref{lemma-fully-faithful-general} that the functor is fully faithful (details omitted). Let us prove essential surjectivity. Let $(\mathcal{F}_n)$ be a finite locally free object of $\textit{Coh}(U, f\mathcal{O}_U)$. By either Lemma \ref{lemma-algebraization-principal-bis} or Proposition \ref{proposition-cd-1} there exists a coherent $\mathcal{O}_U$-module $\mathcal{F}$ such that $(\mathcal{F}_n)$ is the completion of $\mathcal{F}$. Namely, for the application of either result the only thing to check is that $(\mathcal{F}_n)$ satisfies the $(2, 3)$-inequalities. This is done in Lemma \ref{lemma-unwinding-conditions-bis}. If $y \in U_0$, then the $f$-adic completion of the stalk $\mathcal{F}_y$ is isomorphic to a finite free module over the $f$-adic completion of $\mathcal{O}_{U, y}$. Hence $\mathcal{F}$ is finite locally free in an open neighbourhood $U'$ of $U_0$. This finishes the proof.
Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent $\mathcal{O}_X$-module associated to $M$, see Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}. Since $\mathcal{F}$ is $f$-torsion free, we may replace $M$ by its quotient by $f$-power torsion. On the other hand, let $M_0 = \Gamma(X_0, \mathcal{F}_0)$ so that $\mathcal{F}_0$ is the coherent $\mathcal{O}_{X_0}$-module associated to the finite $A/fA$-module $M_0$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} there exists an $n$ such that the isomorphism $\alpha_0$ corresponds to an $A/fA$-module homomorphism $\mathfrak m^n M/fM \to M_0$ (whose kernel and cokernel are annihilated by a power of $\mathfrak m$, but we don't need this). Thus if we take $M' = \mathfrak m^n M$ and we let $\mathcal{F}'$ be the coherent $\mathcal{O}_X$-module associated to $M'$, then the lemma is clear.
Let $\mathcal{F}', \alpha', \alpha'_0$ and $\mathcal{F}'', \alpha'', \alpha''_0$ be two such choices. For $n > 0$ set $\mathcal{F}'_n = \mathfrak m^n \mathcal{F}'$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} for some $n$ there exists an $\mathcal{O}_X$-module map $\mathcal{F}'_n \to \mathcal{F}''$ agreeing with the identification $\mathcal{F}''|_U = \mathcal{F}'|_U$ determined by $\alpha'$ and $\alpha''$. Then the diagram $$ \xymatrix{ \mathcal{F}'_n/f\mathcal{F}'_n \ar[r] \ar[d] & \mathcal{F}'/f\mathcal{F}' \ar[d]^{\alpha_0'} \\ \mathcal{F}''/f\mathcal{F}'' \ar[r]^{\alpha_0''} & \mathcal{F}_0 } $$ is commutative after restricting to $U_0$. Hence by Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} it is commutative after restricting to $\mathfrak m^l(\mathcal{F}'_n/f\mathcal{F}'_n)$ for some $l > 0$. Since $\mathcal{F}'_{n + l}/f\mathcal{F}'_{n + l} \to \mathcal{F}'_n/f\mathcal{F}'_n$ factors through $\mathfrak m^l(\mathcal{F}'_n/f\mathcal{F}'_n)$ we see that after replacing $n$ by $n + l$ the diagram is commutative. In other words, we have found a third choice $\mathcal{F}''', \alpha''', \alpha'''_0$ such that there are maps $\mathcal{F}''' \to \mathcal{F}''$ and $\mathcal{F}''' \to \mathcal{F}'$ over $X$ compatible with the maps over $U$ and $X_0$. This reduces us to the case discussed in the next paragraph. \medskip\noindent Assume we have a map $\mathcal{F}'' \to \mathcal{F}'$ over $X$ compatible with $\alpha', \alpha''$ over $U$ and with $\alpha'_0, \alpha''_0$ over $X_0$. Observe that $\mathcal{F}'' \to \mathcal{F}'$ is injective as it is an isomorphism over $U$ and since $f : \mathcal{F}'' \to \mathcal{F}''$ is injective. Clearly $\mathcal{F}'/\mathcal{F}''$ is supported on $\{\mathfrak m\}$ hence has finite length. We have the maps of coherent $\mathcal{O}_{X_0}$-modules $$ \mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}' \xrightarrow{\alpha'_0} \mathcal{F}_0 $$ whose composition is $\alpha''_0$ and which are isomorphisms over $U_0$. Elementary homological algebra gives a $6$-term exact sequence $$ \begin{matrix} 0 \to \Ker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}') \to \Ker(\alpha''_0) \to \Ker(\alpha'_0) \to \\ \Coker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}') \to \Coker(\alpha''_0) \to \Coker(\alpha'_0) \to 0 \end{matrix} $$ By additivity of lengths (Algebra, Lemma \ref{algebra-lemma-length-additive}) we find that it suffices to show that $$ \text{length}_A( \Coker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}')) - \text{length}_A( \Ker(\mathcal{F}''/f\mathcal{F}'' \to \mathcal{F}'/f\mathcal{F}')) = 0 $$ This follows from applying the snake lemma to the diagram $$ \xymatrix{ 0 \ar[r] & \mathcal{F}'' \ar[r]_f \ar[d] & \mathcal{F}'' \ar[r] \ar[d] & \mathcal{F}''/f\mathcal{F}'' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}' \ar[r]^f & \mathcal{F}' \ar[r] & \mathcal{F}'/f\mathcal{F}' \ar[r] & 0 } $$ and the fact that $\mathcal{F}'/\mathcal{F}''$ has finite length.
The equivalence of the depth conditions follows from Algebra, Lemma \ref{algebra-lemma-depth-drops-by-one}. By the depth condition we see that $\Gamma(U, \mathcal{O}_U) = A$ and $\Gamma(U_0, \mathcal{O}_{U_0}) = A/fA$, see Dualizing Complexes, Lemma \ref{dualizing-lemma-depth} and Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}. Using Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-for-finite-locally-free} we find that $M = \Gamma(U, \mathcal{L})$ is a finite $A$-module. This in turn implies $\text{depth}(M) \geq 2$ for example by part (4) of Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local} or by Divisors, Lemma \ref{divisors-lemma-depth-pushforward}. Also, we have $\mathcal{L}_0 \cong \mathcal{O}_{X_0}$ as $X_0$ is a local scheme. Hence we also see that $M_0 = \Gamma(X_0, \mathcal{L}_0) = \Gamma(U_0, \mathcal{L}_0|_{U_0})$ and that this module is isomorphic to $A/fA$. \medskip\noindent By the above $\mathcal{F}' = \widetilde{M}$ is a coherent $\mathcal{O}_X$-module whose restriction to $U$ is isomorphic to $\mathcal{L}$. The isomorphism $\lambda : \mathcal{L}/f\mathcal{L} \to \mathcal{L}_0|_{U_0}$ determines a map $M/fM \to M_0$ on global sections which is an isomorphism over $U_0$. Since $\text{depth}(M) \geq 2$ we see that $H^0_\mathfrak m(M/fM) = 0$ and it follows that $M/fM \to M_0$ is injective. Thus by definition $$ \chi(\mathcal{L}, \mathcal{L}_0, \lambda) = \text{length}_A \Coker(M/fM \to M_0) $$ which gives the first statement of the lemma. \medskip\noindent Finally, if this length is $0$, then $M \to M_0$ is surjective. Hence we can find $s \in M = \Gamma(U, \mathcal{L})$ mapping to a trivializing section of $\mathcal{L}_0$. Consider the finite $A$-modules $K$, $Q$ defined by the exact sequence $$ 0 \to K \to A \xrightarrow{s} M \to Q \to 0 $$ The supports of $K$ and $Q$ do not meet $U_0$ because $s$ is nonzero at points of $U_0$. Using Algebra, Lemma \ref{algebra-lemma-depth-in-ses} we see that $\text{depth}(K) \geq 2$ (observe that $As \subset M$ has $\text{depth} \geq 1$ as a submodule of $M$). Thus the support of $K$ if nonempty has dimension $\geq 2$ by Algebra, Lemma \ref{algebra-lemma-bound-depth}. This contradicts $\text{Supp}(M) \cap V(f) \subset \{\mathfrak m\}$ unless $K = 0$. When $K = 0$ we find that $\text{depth}(Q) \geq 2$ and we conclude $Q = 0$ as before. Hence $A \cong M$ and $\mathcal{L}$ is trivial.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_U$-module. Observe that $\mathcal{L}$ maps to $0$ in $\text{Pic}(U_0)$ if and only if we can extend $\mathcal{L}$ to an invertible coherent triple $(\mathcal{L}, \mathcal{L}_0, \lambda)$ as in Section \ref{section-coherent-triples}. By Proposition \ref{proposition-hilbert-triple} the function $$ n \longmapsto \chi((\mathcal{L}, \mathcal{L}_0, \lambda)^{\otimes n}) $$ is a polynomial. By Lemma \ref{lemma-nonnegative-chi-triple} the value of this polynomial is zero if and only if $\mathcal{L}^{\otimes n}$ is trivial. Thus if $\mathcal{L}$ is torsion, then this polynomial has infinitely many zeros, hence is identically zero, hence $\mathcal{L}$ is trivial.
Let $U_0 \subset U_n \subset U$ be the $n$th infinitesimal neighbourhood of $U_0$. Observe that the ideal sheaf of $U_n$ in $U_{n + 1}$ is isomorphic to $\mathcal{O}_{U_0}$ as $U_0 \subset U$ is the principal closed subscheme cut out by the nonzerodivisor $f$. Hence we have an exact sequence of abelian groups $$ \Pic(U_{n + 1}) \to \Pic(U_n) \to H^2(U_0, \mathcal{O}_{U_0}) = H^3_\mathfrak m(A/fA) = 0 $$ see More on Morphisms, Lemma \ref{more-morphisms-lemma-picard-group-first-order-thickening}. Thus every invertible $\mathcal{O}_{U_0}$-module is the restriction of an invertible coherent formal module, i.e., an invertible object of $\textit{Coh}(U, f\mathcal{O}_U)$. We conclude by applying Lemma \ref{lemma-equivalence}.
Let $\mathcal{L}_0 \in \Pic(U_0)$. By Lemma \ref{lemma-surjective-Pic-first} there exists an open $U_0 \subset U' \subset U$ and $\mathcal{L}' \in \Pic(U')$ whose restriction to $U_0$ is $\mathcal{L}_0$. Since $U' \supset U_0$ we see that $U \setminus U'$ consists of points corresponding to prime ideals $\mathfrak p_1, \ldots, \mathfrak p_n$ as in (2). By assumption we can find invertible modules $\mathcal{L}'_i$ on $\Spec(A_{\mathfrak p_i})$ agreeing with $\mathcal{L}'$ over the punctured spectrum $U' \times_U \Spec(A_{\mathfrak p_i})$ since trivial invertible modules always extend. By Limits, Lemma \ref{limits-lemma-glueing-near-closed-point-modules} applied $n$ times we see that $\mathcal{L}'$ extends to an invertible module on $U$.
Let $\mathcal{L}$ be an invertible $\mathcal{O}_U$-module with pullback $\mathcal{L}^\wedge$ on $U^\wedge$. We have $H^0(U, \mathcal{O}_U) = A$ by our assumption on depth and Dualizing Complexes, Lemma \ref{dualizing-lemma-depth} and Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}. Thus $\mathcal{L}$ is trivial if and only if $M = H^0(U, \mathcal{L})$ is isomorphic to $A$ as an $A$-module. (Details omitted.) Since $A \to A^\wedge$ is flat we have $M \otimes_A A^\wedge = \Gamma(U^\wedge, \mathcal{L}^\wedge)$ by flat base change, see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. Finally, it is easy to see that $M \cong A$ if and only if $M \otimes_A A^\wedge \cong A^\wedge$.
By Proposition \ref{proposition-lefschetz} this is true for $V = X$. Thus it suffices to show that the map $\Gamma(V, \mathcal{F}) \to \lim \Gamma(Y_n, \mathcal{F}|_{Y_n})$ is injective. If $\sigma \in \Gamma(V, \mathcal{F})$ maps to zero, then its support is disjoint from $Y$ (details omitted; hint: use Krull's intersection theorem). Then the closure $T \subset X$ of $\text{Supp}(\sigma)$ is disjoint from $Y$. Whence $T$ is proper over $k$ (being closed in $X$) and affine (being closed in the affine scheme $X \setminus Y$, see Morphisms, Lemma \ref{morphisms-lemma-proper-ample-delete-affine}) and hence finite over $k$ (Morphisms, Lemma \ref{morphisms-lemma-finite-proper}). Thus $T$ is a finite set of closed points of $X$. Thus $\text{depth}(\mathcal{F}_x) \geq 2$ is at least $1$ for $x \in T$ by our assumption. We conclude that $\Gamma(V, \mathcal{F}) \to \Gamma(V \setminus T, \mathcal{F})$ is injective and $\sigma = 0$ as desired.
Let $K \to \mathcal{O}$ in $D(\mathcal{O})$ be as constructed in Lemma \ref{lemma-global-extended-cech-complex}. Let $E \in D(\mathcal{O})$. Then $E^\wedge = R\SheafHom(K, E)$ together with the map $E \to E^\wedge$ will do the job. Namely, locally on the site $\mathcal{C}$ we recover the adjoint of Lemma \ref{lemma-derived-completion}. This shows that $E^\wedge$ is always derived complete and that $E \to E^\wedge$ is an isomorphism if $E$ is derived complete.
We may assume $s > 0$ as the case $s = 0$ was done in Proposition \ref{proposition-application-H0}. \medskip\noindent Choose a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction to $U$ of the coherent module associated to $M$, see Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}. Set $d = \text{cd}(A, I)$. Let $\mathfrak p$ be a prime of $A$ not contained in $V(I)$ and let $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$. Then either $\text{depth}(M_\mathfrak p) \geq s + 1 > s$ or we have $\dim((A/\mathfrak p)_\mathfrak q) > d + s + 1$ by (2). By Lemma \ref{lemma-bootstrap-bis-bis} we conclude that the assumptions of Situation \ref{situation-bootstrap} are satisfied for $A, I, V(\mathfrak a), M, s, d$. On the other hand, the hypotheses of Lemma \ref{lemma-algebraize-local-cohomology-general} are satisfied for $s + 1$ and $d$; this is where condition (3) is used. \medskip\noindent Applying Lemma \ref{lemma-algebraize-local-cohomology-general} we find there exists an ideal $J_0 \subset \mathfrak a$ with $V(J_0) \cap V(I) = V(\mathfrak a)$ such that for any $J \subset J_0$ with $V(J) \cap V(I) = V(\mathfrak a)$ the maps $$ H^i_J(M) \longrightarrow H^i(R\Gamma_\mathfrak a(M)^\wedge) $$ is an isomorphism for $i \leq s + 1$. \medskip\noindent For $i \leq s$ the map $H^i_\mathfrak a(M) \to H^i_J(M)$ is an isomorphism by Lemmas \ref{lemma-bootstrap-inherited} and \ref{lemma-kill-colimit-support-general}. Using the comparison of cohomology and local cohomology (Local Cohomology, Lemma \ref{local-cohomology-lemma-local-cohomology}) we deduce $H^i(U, \mathcal{F}) \to H^i(V,\mathcal{F})$ is an isomorphism for $V = \Spec(A) \setminus V(J)$ and $i < s$. \medskip\noindent By Theorem \ref{theorem-final-bootstrap} we have $H^i_\mathfrak a(M) = \lim H^i_\mathfrak a(M/I^nM)$ for $i \leq s$. By Lemma \ref{lemma-combine-two} we have $H^{s + 1}_\mathfrak a(M) = \lim H^{s + 1}_\mathfrak a(M/I^nM)$. \medskip\noindent The isomorphism $H^0(U, \mathcal{F}) = H^0(V, \mathcal{F}) = \lim H^0(U, \mathcal{F}/I^n\mathcal{F})$ follows from the above and Proposition \ref{proposition-application-H0}. For $0 < i < s$ we get the desired isomorphisms $H^i(U, \mathcal{F}) = H^i(V, \mathcal{F}) = \lim H^i(U, \mathcal{F}/I^n\mathcal{F})$ in the same manner using the relation between local cohomology and cohomology; it is easier than the case $i = 0$ because for $i > 0$ we have $$ H^i(U, \mathcal{F}) = H^{i + 1}_\mathfrak a(M), \quad H^i(V, \mathcal{F}) = H^{i + 1}_J(M), \quad H^i(R\Gamma(U, \mathcal{F})^\wedge) =  H^{i + 1}(R\Gamma_\mathfrak a(M)^\wedge) $$ Similarly for the final statement.
By Lemma \ref{lemma-map-kernel-cokernel-on-closed} we may replace $(\mathcal{F}_n)$ by the object $(\mathcal{H}_n)$ of $\textit{Coh}(U, I\mathcal{O}_U)$ found in Lemma \ref{lemma-improvement-application}. Thus we may assume that $(\mathcal{F}_n)$ is pro-isomorphic to a inverse system $(\mathcal{F}_n'')$ with the properties mentioned in Lemma \ref{lemma-improvement-application}. In Lemma \ref{lemma-cd-1-canonical} we proved that $(\mathcal{F}_n)$ canonically extends to $X$. The final statement follows from Lemma \ref{lemma-canonically-algebraizable}.
We may assume $I = (f_1, \ldots, f_d)$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-ideals-equivalence}. Then we see that all fibres of the blowup of $X$ in $I$ have dimension at most $d - 1$. Thus we get the extension from Lemma \ref{lemma-blowup}. The final statement follows from Lemma \ref{lemma-essential-image-completion}.