hcju/stacksqa huggingface dataset:
Write $M_n = H^p(X, I^n\mathcal{F})$ for $n \geq 1$ and $M_0 = H^p(X, \mathcal{F})$ so that we have maps $\ldots \to M_3 \to M_2 \to M_1 \to M_0$. Setting $B = \bigoplus_{n \geq 0} I^n$, then $M = \bigoplus_{n \geq 0} M_n$ is a finite graded $B$-module, see Lemma \ref{lemma-cohomology-powers-ideal-times-F}. Observe that the products $B_n \otimes M_m \to M_{m + n}$, $a \otimes m \mapsto a \cdot m$ are compatible with the maps in our inverse system in the sense that the diagrams $$ \xymatrix{ B_n \otimes_A M_m \ar[r] \ar[d] & M_{n + m} \ar[d] \\ B_n \otimes_A M_{m'} \ar[r] & M_{n + m'} } $$ commute for $n, m' \geq 0$ and $m \geq m'$. \medskip\noindent Proof of (1). Choose $d_1, \ldots, d_t \geq 0$ and $x_i \in M_{d_i}$ such that $M$ is generated by $x_1, \ldots, x_t$ over $B$. For any $c \geq \max\{d_i\}$ we conclude that $B_{n - c} \cdot M_c = M_n$ for $n \geq c$ and we conclude (1) is true. \medskip\noindent Proof of (2). Let $c$ be as in the proof of (1). Let $n + m \geq c$. We have $M_{n + m} = B_{n + m - c} \cdot M_c$. If $c > n$ then we use $M_c \to M_n$ and the compatibility of products with transition maps pointed out above to conclude that the image of $M_{n + m} \to M_n$ is contained in $I^{n + m - c}M_n$. If $c \leq n$, then we write $M_{n + m} = B_m \cdot B_{n - c} \cdot M_c = B_m \cdot M_n$ to see that the image is contained in $I^m M_n$. This proves (2). \medskip\noindent Let $K_n \subset M_n$ be the kernel of the map $M_n \to M_0$. The compatibility of products with transition maps pointed out above shows that $K = \bigoplus K_n \subset M$ is a graded $B$-submodule. As $B$ is Noetherian and $M$ is a finitely generated graded $B$-module, this shows that $K$ is a finitely generated graded $B$-module. Choose $d'_1, \ldots, d'_{t'} \geq 0$ and $y_i \in K_{d'_i}$ such that $K$ is generated by $y_1, \ldots, y_{t'}$ over $B$. Set $c = \max(d_i, d'_j)$. Since $y_i \in \Ker(M_{d'_i} \to M_0)$ we see that $B_n \cdot y_i \subset \Ker(M_{n + d_i} \to M_n)$. In this way we see that $K_n = \Ker(M_n \to M_{n - c})$ for $n \geq c$. This proves (3). \medskip\noindent Consider the following commutative solid diagram $$ \xymatrix{ I^n \otimes_A M_0 \ar[r] \ar[d] & I^nM_0 \ar[r] \ar@{..>}[d] & M_0 \ar[d] \\ M_n \ar[r] & M_{n - c} \ar[r] & M_0 } $$ Since the kernel of the surjective arrow $I^n \otimes_A M_0 \to I^nM_0$ maps into $K_n$ by the above we obtain the dotted arrow and the composition $I^nM_0 \to M_{n - c} \to M_0$ is the canonical map. Then clearly the composition $I^nM_0 \to M_{n - c} \to I^{n - 2c}M_0$ is the canonical map for $n \geq 2c$. Consider the composition $M_n \to I^{n - c}M_0 \to M_{n - 2c}$. The first map sends an element of the form $a \cdot m$ with $a \in I^{n - c}$ and $m \in M_c$ to $a m'$ where $m'$ is the image of $m$ in $M_0$. Then the second map sends this to $a \cdot m'$ in $M_{n - 2c}$ and we see (4) is true. \medskip\noindent Part (5) is an immediate consequence of (4) and the definition of morphisms of pro-objects.
Let $c = \max\{c_p, c_{p + 1}\}$, where $c_p, c_{p + 1}$ are the integers found in Lemma \ref{lemma-cohomology-powers-ideal-application} for $H^p$ and $H^{p + 1}$. \medskip\noindent Let us prove part (1). Consider the short exact sequence $$ 0 \to I^n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^n\mathcal{F} \to 0 $$ From the long exact cohomology sequence we see that $$ \Ker( H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F}) ) = \Im( H^p(X, I^n\mathcal{F}) \to H^p(X, \mathcal{F}) ) $$ Hence by Lemma \ref{lemma-cohomology-powers-ideal-application} part (2) we see that this is contained in $I^{n - c}H^p(X, \mathcal{F})$ for $n \geq c$. \medskip\noindent Note that part (3) implies part (2) by definition of the Mittag-Leffler systems. \medskip\noindent Let us prove part (3). Fix an $n$. Consider the commutative diagram $$ \xymatrix{ 0 \ar[r] & I^n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 } $$ This gives rise to the following commutative diagram $$ \xymatrix{ H^p(X, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}/I^n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^n\mathcal{F}) \ar[r] & H^{p + 1}(X, \mathcal{F}) \\ H^p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u]^\gamma & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^\alpha \ar[r]^-\beta & H^{p + 1}(X, \mathcal{F}) \ar[u]_1 } $$ with exact rows. By Lemma \ref{lemma-cohomology-powers-ideal-application} part (4) the kernel of $\beta$ is equal to the kernel of $\alpha$ for $m \geq c$. By a diagram chase this shows that the image of $\gamma$ is contained in the kernel of $\delta$ which shows that part (3) is true (set $k = n + m$ to get it).
This is a reformulation of the theorem on formal functions (Theorem \ref{theorem-formal-functions}) in the case of a complete Noetherian base ring. Namely, in this case the $A$-module $H^p(X, \mathcal{F})$ is finite (Lemma \ref{lemma-proper-over-affine-cohomology-finite}) hence $I$-adically complete (Algebra, Lemma \ref{algebra-lemma-completion-tensor}) and we see that completion on the left hand side is not necessary.
This is just a reformulation of a special case of the theorem on formal functions, Theorem \ref{theorem-formal-functions}. Let us spell it out. Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \Spec(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (\ref{schemes-equation-canonical-morphism}). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \Spec(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'$ by Lemma \ref{lemma-flat-base-change-cohomology}. Moreover, the infinitesimal neighbourhoods of the fibre $X_y$ and $X'_y$ are identified (verification omitted; hint: the morphisms $c_n$ factor through $c$). \medskip\noindent Hence we may assume that $Y = \Spec(A)$ is the spectrum of a Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $y \in Y$ corresponds to the closed point (i.e., to $\mathfrak m$). In particular it follows that $$ \left(R^pf_*\mathcal{F}\right)_y = \Gamma(Y, R^pf_*\mathcal{F}) = H^p(X, \mathcal{F}). $$ \medskip\noindent In this case also, the morphisms $c_n$ are each closed immersions. Hence their base changes $i_n$ are closed immersions as well. Note that $i_{n, *}\mathcal{F}_n = i_{n, *}i_n^*\mathcal{F} = \mathcal{F}/\mathfrak m^n\mathcal{F}$. By the Leray spectral sequence for $i_n$, and Lemma \ref{lemma-finite-pushforward-coherent} we see that $$ H^p(X_n, \mathcal{F}_n) = H^p(X, i_{n, *}\mathcal{F}) = H^p(X, \mathcal{F}/\mathfrak m^n\mathcal{F}) $$ Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win.
The fibre $X_y$ is finite, and by Morphisms, Lemma \ref{morphisms-lemma-finite-fibre} it is a finite discrete space. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_n$ is the same. Hence each of the schemes $X_n$ is affine according to Schemes, Lemma \ref{schemes-lemma-scheme-finite-discrete-affine}. Hence it follows that $H^p(X_n, \mathcal{F}_n) = 0$ for all $p > 0$. Hence we see that $(R^pf_*\mathcal{F})_y^\wedge = 0$ by Lemma \ref{lemma-formal-functions-stalk}. Note that $R^pf_*\mathcal{F}$ is coherent by Proposition \ref{proposition-proper-pushforward-coherent} and hence $R^pf_*\mathcal{F}_y$ is a finite $\mathcal{O}_{Y, y}$-module. By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) if the completion of a finite module over a local ring is zero, then the module is zero. Whence $(R^pf_*\mathcal{F})_y = 0$.
The fibre $X_y$ is of finite type over $\Spec(\kappa(y))$. Hence $X_y$ is a Noetherian scheme by Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}. Hence the underlying topological space of $X_y$ is Noetherian, see Properties, Lemma \ref{properties-lemma-Noetherian-topology}. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_n$ is the same as that of $X_y$. Hence $H^p(X_n, \mathcal{F}_n) = 0$ for all $p > d$ by Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}. Hence we see that $(R^pf_*\mathcal{F})_y^\wedge = 0$ by Lemma \ref{lemma-formal-functions-stalk} for $p > d$. Note that $R^pf_*\mathcal{F}$ is coherent by Proposition \ref{proposition-proper-pushforward-coherent} and hence $R^pf_*\mathcal{F}_y$ is a finite $\mathcal{O}_{Y, y}$-module. By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) if the completion of a finite module over a local ring is zero, then the module is zero. Whence $(R^pf_*\mathcal{F})_y = 0$.
A finite morphism is proper according to Morphisms, Lemma \ref{morphisms-lemma-finite-proper}. A finite morphism is quasi-finite according to Morphisms, Lemma \ref{morphisms-lemma-finite-quasi-finite}. A quasi-finite morphism has finite fibres, see Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}. Hence a finite morphism is proper and has finite fibres. \medskip\noindent Assume $f$ is proper with finite fibres. We want to show $f$ is finite. In fact it suffices to prove $f$ is affine. Namely, if $f$ is affine, then it follows that $f$ is integral by Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed} whereupon it follows from Morphisms, Lemma \ref{morphisms-lemma-finite-integral} that $f$ is finite. \medskip\noindent To show that $f$ is affine we may assume that $S$ is affine, and our goal is to show that $X$ is affine too. Since $f$ is proper we see that $X$ is separated and quasi-compact. Hence we may use the criterion of Lemma \ref{lemma-quasi-separated-h1-zero-covering} to prove that $X$ is affine. To see this let $\mathcal{I} \subset \mathcal{O}_X$ be a finite type ideal sheaf. In particular $\mathcal{I}$ is a coherent sheaf on $X$. By Lemma \ref{lemma-higher-direct-images-zero-finite-fibre} we conclude that $R^1f_*\mathcal{I}_s = 0$ for all $s \in S$. In other words, $R^1f_*\mathcal{I} = 0$. Hence we see from the Leray Spectral Sequence for $f$ that $H^1(X , \mathcal{I}) = H^1(S, f_*\mathcal{I})$. Since $S$ is affine, and $f_*\mathcal{I}$ is quasi-coherent (Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}) we conclude $H^1(S, f_*\mathcal{I}) = 0$ from Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero} as desired. Hence $H^1(X, \mathcal{I}) = 0$ as desired.
The morphism $f$ is quasi-finite at all the points of $f^{-1}(\{s\})$ by Morphisms, Lemma \ref{morphisms-lemma-finite-fibre}. By Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-points-open} the set of points at which $f$ is quasi-finite is an open $U \subset X$. Let $Z = X \setminus U$. Then $s \not \in f(Z)$. Since $f$ is proper the set $f(Z) \subset S$ is closed. Choose any open neighbourhood $V \subset S$ of $s$ with $Z \cap V = \emptyset$. Then $f^{-1}(V) \to V$ is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-locally-quasi-compact}), hence has finite fibres (Morphisms, Lemma \ref{morphisms-lemma-quasi-finite}), hence is finite by Lemma \ref{lemma-characterize-finite}.
Pick $d_0$ as in Lemma \ref{lemma-ample-on-fibre} for $\mathcal{F} = \mathcal{O}_X$. Pick $d \geq d_0$ so that we can find $r \geq 0$ and sections $s_{y, 0}, \ldots, s_{y, r} \in H^0(X_y, \mathcal{L}_y^{\otimes d})$ which define a closed immersion $$ \varphi_y = \varphi_{\mathcal{L}_y^{\otimes d}, (s_{y, 0}, \ldots, s_{y, r})} : X_y \to \mathbf{P}^r_{\kappa(y)}. $$ This is possible by Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample} but we also use Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed} to see that $\varphi_y$ is a closed immersion and Constructions, Section \ref{constructions-section-projective-space} for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of $d_0$, after replacing $Y$ by an open neighbourhood of $y$, we can choose $s_0, \ldots, s_r \in H^0(X, \mathcal{L}^{\otimes d})$ mapping to $s_{y, 0}, \ldots, s_{y, r}$. Let $X_{s_i} \subset X$ be the open subset where $s_i$ is a generator of $\mathcal{L}^{\otimes d}$. Since the $s_{y, i}$ generate $\mathcal{L}_y^{\otimes d}$ we see that $X_y \subset U = \bigcup X_{s_i}$. Since $X \to Y$ is closed, we see that there is an open neighbourhood $y \in V \subset Y$ such that $f^{-1}(V) \subset U$. After replacing $Y$ by $V$ we may assume that the $s_i$ generate $\mathcal{L}^{\otimes d}$. Thus we obtain a morphism $$ \varphi = \varphi_{\mathcal{L}^{\otimes d}, (s_0, \ldots, s_r)} : X \longrightarrow \mathbf{P}^r_Y $$ with $\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)$ whose base change to $y$ gives $\varphi_y$. \medskip\noindent We will finish the proof by a sleight of hand; the ``correct'' proof proceeds by directly showing that $\varphi$ is a closed immersion after base changing to an open neighbourhood of $y$. Namely, by Lemma \ref{lemma-proper-finite-fibre-finite-in-neighbourhood} we see that $\varphi$ is a finite over an open neighbourhood of the fibre $\mathbf{P}^r_{\kappa(y)}$ of $\mathbf{P}^r_Y \to Y$ above $y$. Using that $\mathbf{P}^r_Y \to Y$ is closed, after shrinking $Y$ we may assume that $\varphi$ is finite. Then $\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)$ is ample by the very general Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}.
Choose a finite affine open covering $X = \bigcup_{i = 1, \ldots, n} U_i$. By Lemmas \ref{lemma-separated-case-relative-cech} and \ref{lemma-base-change-complex} the {\v C}ech complex $K^\bullet = {\check C}^\bullet(\mathcal{U}, \mathcal{F})$ satisfies $$ K^\bullet \otimes_A A' = R\Gamma(X_{A'}, \mathcal{F}_{A'}) $$ for all ring maps $A \to A'$. Let $K_{alt}^\bullet = {\check C}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$ be the alternating {\v C}ech complex. By Cohomology, Lemma \ref{cohomology-lemma-alternating-usual} there is a homotopy equivalence $K_{alt}^\bullet \to K^\bullet$ of $A$-modules. In particular, we have $$ K_{alt}^\bullet \otimes_A A' = R\Gamma(X_{A'}, \mathcal{F}_{A'}) $$ as well. Since $\mathcal{F}$ is flat over $A$ we see that each $K_{alt}^n$ is flat over $A$ (see Morphisms, Lemma \ref{morphisms-lemma-flat-module-characterize}). Since moreover $K_{alt}^\bullet$ is bounded above (this is why we switched to the alternating {\v C}ech complex) $K_{alt}^\bullet \otimes_A A' = K_{alt}^\bullet \otimes_A^{\mathbf{L}} A'$ by the definition of derived tensor products (see More on Algebra, Section \ref{more-algebra-section-derived-tensor-product}). By Lemma \ref{lemma-proper-over-affine-cohomology-finite} the cohomology groups $H^i(K_{alt}^\bullet)$ are finite $A$-modules. As $K_{alt}^\bullet$ is bounded, we conclude that $K_{alt}^\bullet$ is pseudo-coherent, see More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-pseudo-coherent}. Given any $A$-module $M$ set $A' = A \oplus M$ where $M$ is a square zero ideal, i.e., $(a, m) \cdot (a', m') = (aa', am' + a'm)$. By the above we see that $K_{alt}^\bullet \otimes_A^\mathbf{L} A'$ has cohomology in degrees $0, \ldots, n$. Hence $K_{alt}^\bullet \otimes_A^\mathbf{L} M$ has cohomology in degrees $0, \ldots, n$. Hence $K_{alt}^\bullet$ has finite Tor dimension, see More on Algebra, Definition \ref{more-algebra-definition-tor-amplitude}. We win by More on Algebra, Lemma \ref{more-algebra-lemma-perfect}.
Let $\text{Mod}^{fg}_{A, I}$ be the category of inverse systems $(M_n)$ of finite $A$-modules satisfying: (1) $M_n$ is annihilated by $I^n$ and (2) $M_{n + 1}/I^nM_{n + 1} = M_n$. By the correspondence between coherent sheaves on $X$ and finite $A$-modules (Lemma \ref{lemma-coherent-Noetherian}) it suffices to show $\text{Mod}^{fg}_{A, I}$ is equivalent to the category of finite $A^\wedge$-modules. To see this it suffices to prove that given an object $(M_n)$ of $\text{Mod}^{fg}_{A, I}$ the module $$ M = \lim M_n $$ is a finite $A^\wedge$-module and that $M/I^nM = M_n$. As the transition maps are surjective, we see that $M \to M_1$ is surjective. Pick $x_1, \ldots, x_t \in M$ which map to generators of $M_1$. This induces a map of systems $(A/I^n)^{\oplus t} \to M_n$. By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) these maps are surjective. Let $K_n \subset (A/I^n)^{\oplus t}$ be the kernel. Property (2) implies that $K_{n + 1} \to K_n$ is surjective, in particular the system $(K_n)$ satisfies the Mittag-Leffler condition. By Homology, Lemma \ref{homology-lemma-Mittag-Leffler} we obtain an exact sequence $0 \to K \to (A^\wedge)^{\oplus t} \to M \to 0$ with $K = \lim K_n$. Hence $M$ is a finite $A^\wedge$-module. As $K \to K_n$ is surjective it follows that $$ M/I^nM = \Coker(K \to (A/I^n)^{\oplus t}) = (A/I^n)^{\oplus t}/K_n = M_n $$ as desired.
Let $\alpha =(\alpha_n) : (\mathcal{F}_n) \to (\mathcal{G}_n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. The cokernel of $\alpha$ is the inverse system $(\Coker(\alpha_n))$ (details omitted). To describe the kernel let $$ \mathcal{K}'_{l, m} = \Im(\Ker(\alpha_l) \to \mathcal{F}_m) $$ for $l \geq m$. We claim: \begin{enumerate} \item[(a)] the inverse system $(\mathcal{K}'_{l, m})_{l \geq m}$ is eventually constant, say with value $\mathcal{K}'_m$, \item[(b)] the system $(\mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m)_{m \geq n}$ is eventually constant, say with value $\mathcal{K}_n$, \item[(c)] the system $(\mathcal{K}_n)$ forms an object of $\textit{Coh}(X, \mathcal{I})$, and \item[(d)] this object is the kernel of $\alpha$. \end{enumerate} To see (a), (b), and (c) we may work affine locally, say $X = \Spec(A)$ and $\mathcal{I}$ corresponds to the ideal $I \subset A$. By Lemma \ref{lemma-inverse-systems-affine} $\alpha$ corresponds to a map $f : M \to N$ of finite $A^\wedge$-modules. Denote $K = \Ker(f)$. Note that $A^\wedge$ is a Noetherian ring (Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian}). Choose an integer $c \geq 0$ such that $K \cap I^n M \subset I^{n - c}K$ for $n \geq c$ (Algebra, Lemma \ref{algebra-lemma-Artin-Rees}) and which satisfies Algebra, Lemma \ref{algebra-lemma-map-AR} for the map $f$ and the ideal $I^\wedge = IA^\wedge$. Then $\mathcal{K}'_{l, m}$ corresponds to the $A$-module $$ K'_{l, m} = \frac{a^{-1}(I^lN) + I^mM}{I^mM} = \frac{K + I^{l - c}f^{-1}(I^cN) + I^mM}{I^mM} = \frac{K + I^mM}{I^mM} $$ where the last equality holds if $l \geq m + c$. So $\mathcal{K}'_m$ corresponds to the $A$-module $K/K \cap I^mM$ and $\mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m$ corresponds to $$ \frac{K}{K \cap I^mM + I^nK} = \frac{K}{I^nK} $$ for $m \geq n + c$ by our choice of $c$ above. Hence $\mathcal{K}_n$ corresponds to $K/I^nK$. \medskip\noindent We prove (d). It is clear from the description on affines above that the composition $(\mathcal{K}_n) \to (\mathcal{F}_n) \to (\mathcal{G}_n)$ is zero. Let $\beta : (\mathcal{H}_n) \to (\mathcal{F}_n)$ be a morphism such that $\alpha \circ \beta = 0$. Then $\mathcal{H}_l \to \mathcal{F}_l$ maps into $\Ker(\alpha_l)$. Since $\mathcal{H}_m = \mathcal{H}_l/\mathcal{I}^m\mathcal{H}_l$ for $l \geq m$ we obtain a system of maps $\mathcal{H}_m \to \mathcal{K}'_{l, m}$. Thus a map $\mathcal{H}_m \to \mathcal{K}_m'$. Since $\mathcal{H}_n = \mathcal{H}_m/\mathcal{I}^n\mathcal{H}_m$ we obtain a system of maps $\mathcal{H}_n \to \mathcal{K}'_m/\mathcal{I}^n\mathcal{K}'_m$ and hence a map $\mathcal{H}_n \to \mathcal{K}_n$ as desired. \medskip\noindent To finish the proof of (1) we still have to show that $\Coim = \Im$ in $\textit{Coh}(X, \mathcal{I})$. We have seen above that taking kernels and cokernels commutes, over affines, with the description of $\textit{Coh}(X, \mathcal{I})$ as a category of modules. Since $\Im = \Coim$ holds in the category of modules this gives $\Coim = \Im$ in $\textit{Coh}(X, \mathcal{I})$. Parts (2) and (3) of the lemma are immediate from our construction of kernels and cokernels.
It suffices to check this locally on $X$. Hence we may assume $X$ is affine, i.e., we have a situation as in Lemma \ref{lemma-inverse-systems-affine}. The functor is the functor $\text{Mod}^{fg}_A \to \text{Mod}^{fg}_{A^\wedge}$ which associates to a finite $A$-module $M$ the completion $M^\wedge$. Thus the result follows from Algebra, Lemma \ref{algebra-lemma-completion-flat}.
To prove this we may work affine locally on $X$. Hence we may assume $X = \Spec(A)$ and $\mathcal{F}$, $\mathcal{G}$ given by finite $A$-module $M$ and $N$. Then $\mathcal{H}$ corresponds to the finite $A$-module $H = \Hom_A(M, N)$. The statement of the lemma becomes the statement $$ H^\wedge = \Hom_{A^\wedge}(M^\wedge, N^\wedge) $$ via the equivalence of Lemma \ref{lemma-inverse-systems-affine}. By Algebra, Lemma \ref{algebra-lemma-completion-flat} (used 3 times) we have $$ H^\wedge = \Hom_A(M, N) \otimes_A A^\wedge = \Hom_{A^\wedge}(M \otimes_A A^\wedge, N \otimes_A A^\wedge) = \Hom_{A^\wedge}(M^\wedge, N^\wedge) $$ where the second equality uses that $A^\wedge$ is flat over $A$ (see More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherence-and-base-change-ext}). The lemma follows.
Proof of (1). The uniqueness implies it suffices to construct $(\mathcal{F}, a, \beta)$ Zariski locally on $X$. Thus we may assume $X = \Spec(A)$ and $\mathcal{I}$ corresponds to the ideal $I \subset A$. In this situation Lemma \ref{lemma-inverse-systems-affine} applies. Let $M'$ be the finite $A^\wedge$-module corresponding to $(\mathcal{F}_n)$. Let $N$ be the finite $A$-module corresponding to $\mathcal{G}$. Then $\alpha$ corresponds to a map $$ \varphi : M' \longrightarrow N^\wedge $$ whose kernel and cokernel are annihilated by $I^t$ for some $t$. Recall that $N^\wedge = N \otimes_A A^\wedge$ (Algebra, Lemma \ref{algebra-lemma-completion-tensor}). By More on Algebra, Lemma \ref{more-algebra-lemma-application-formal-glueing} there is an $A$-module map $\psi : M \to N$ whose kernel and cokernel are $I$-power torsion and an isomorphism $M \otimes_A A^\wedge = M'$ compatible with $\varphi$. As $N$ and $M'$ are finite modules, we conclude that $M$ is a finite $A$-module, see More on Algebra, Remark \ref{more-algebra-remark-formal-glueing-algebras}. Hence $M \otimes_A A^\wedge = M^\wedge$. We omit the verification that the triple $(M, N \to M, M^\wedge \to M')$ so obtained is unique up to unique isomorphism. \medskip\noindent The proof of (2) is exactly the same and we omit it.
Suppose $(\mathcal{F}_n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ which is annihilated by $\mathcal{I}^c$ for some $c \geq 1$. Then $\mathcal{F}_n \to \mathcal{F}_c$ is an isomorphism for $n \geq c$. Hence if we set $\mathcal{F} = \mathcal{F}_c$, then we see that $\mathcal{F}^\wedge \cong (\mathcal{F}_n)$. This proves the first assertion. \medskip\noindent Let $\mathcal{F}$, $\mathcal{G}$ be objects of $\textit{Coh}(\mathcal{O}_X)$ such that either $\mathcal{F}$ or $\mathcal{G}$ is annihilated by $\mathcal{I}^c$ for some $c \geq 1$. Then $\mathcal{H} = \SheafHom_{\mathcal{O}_X}(\mathcal{G}, \mathcal{F})$ is a coherent $\mathcal{O}_X$-module annihilated by $\mathcal{I}^c$. Hence we see that $$ \Hom_X(\mathcal{G}, \mathcal{F}) = H^0(X, \mathcal{H}) = \lim H^0(X, \mathcal{H}/\mathcal{I}^n\mathcal{H}) = \Mor_{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge, \mathcal{F}^\wedge). $$ see Lemma \ref{lemma-completion-internal-hom}. This proves the statement on homomorphisms. \medskip\noindent The notation $\Ext$ refers to extensions as defined in Homology, Section \ref{homology-section-extensions}. The injectivity of the map on $\Ext$'s follows immediately from the bijectivity of the map on $\Hom$'s. For surjectivity, assume $\mathcal{F}$ is annihilated by a power of $I$. Then part (1) of Lemma \ref{lemma-existence-easy} shows that given an extension $$ 0 \to \mathcal{G}^\wedge \to (\mathcal{E}_n) \to \mathcal{F}^\wedge \to 0 $$ in $\textit{Coh}(U, I\mathcal{O}_U)$ the morphism $\mathcal{G}^\wedge \to (\mathcal{E}_n)$ is isomorphic to $\mathcal{G} \to \mathcal{E}^\wedge$ for some $\mathcal{G} \to \mathcal{E}$ in $\textit{Coh}(\mathcal{O}_U)$. Similarly in the other case.