hcju/stacksqa huggingface dataset:
Let $X$ be the scheme representing the functor $F$. Note that $X$ is a scheme over $S$ since the functor $F$ comes equipped with a natural transformation $F \to h_S$. Write $Y = \underline{\text{Proj}}_S(\mathcal{A})$. We have to show that $X \cong Y$ as $S$-schemes. We give two arguments. \medskip\noindent The first argument uses the construction of $X$ as the union of the schemes $U_d$ representing $F_d$ in the proof of Lemma \ref{lemma-relative-proj}. Over each affine open of $S$ we can identify $X$ with the homogeneous spectrum of the sections of $\mathcal{A}$ over that open, since this was true for the opens $U_d$. Moreover, these identifications are compatible with further restrictions to smaller affine opens. On the other hand, $Y$ was constructed by glueing these homogeneous spectra. Hence we can glue these isomorphisms to an isomorphism between $X$ and $\underline{\text{Proj}}_S(\mathcal{A})$ as desired. Details omitted. \medskip\noindent Here is the second argument. Lemma \ref{lemma-glue-relative-proj-twists} shows that there exists a morphism of graded algebras $$ \psi : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_Y(n) $$ over $Y$ which on sections over affine opens of $S$ agrees with (\ref{equation-global-sections}). Hence for every $y \in Y$ there exists an open neighbourhood $V \subset Y$ of $y$ and an integer $d \geq 1$ such that for $d | n$ the sheaf $\mathcal{O}_Y(n)|_V$ is invertible and the multiplication maps $\mathcal{O}_Y(n)|_V \otimes_{\mathcal{O}_V} \mathcal{O}_Y(m)|_V \to \mathcal{O}_Y(n + m)|_V$ are isomorphisms. Thus $\psi$ restricted to the sheaf $\pi^*\mathcal{A}^{(d)}|_V$ gives an element of $F_d(V)$. Since the opens $V$ cover $Y$ we see ``$\psi$'' gives rise to an element of $F(Y)$. Hence a canonical morphism $Y \to X$ over $S$. Because this construction is completely canonical to see that it is an isomorphism we may work locally on $S$. Hence we reduce to the case $S$ affine where the result is clear.
Note that the quadruple $(f : T \to S, d, \mathcal{L}, \psi)$ defines an element of $F_d(T)$. Let $U_d \subset \underline{\text{Proj}}_S(\mathcal{A})$ be the locus where the sheaf $\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)$ is invertible and generated by the image of $\psi_{univ} : \pi^*\mathcal{A}_d \to \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)$. Recall that $U_d$ represents the functor $F_d$, see the proof of Lemma \ref{lemma-relative-proj}. Hence the result will follow if we can show the quadruple $(U_d \to S, d, \mathcal{O}_{U_d}(d), \psi_{univ}|_{\mathcal{A}^{(d)}})$ is the universal family, i.e., the representing object in $F_d(U_d)$. We may do this after restricting to an affine open of $S$ because (a) the formation of the functors $F_d$ commutes with base change (see Lemma \ref{lemma-proj-base-change}), and (b) the pair $(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n), \psi_{univ})$ is constructed by glueing over affine opens in $S$ (see Lemma \ref{lemma-glue-relative-proj-twists}). Hence we may assume that $S$ is affine. In this case the functor of quadruples $F_d$ and the functor of triples $F_d$ agree (see proof of Lemma \ref{lemma-relative-proj-affine}) and moreover Lemma \ref{lemma-proj-functor-strict} shows that $(d, \mathcal{O}_{U_d}(d), \psi^d)$ is the universal triple over $U_d$. Going backwards through the identifications in the proof of Lemma \ref{lemma-relative-proj-affine} shows that $(U_d \to S, d, \mathcal{O}_{U_d}(d), \psi_{univ}|_{\mathcal{A}^{(d)}})$ is the universal quadruple as desired.
To prove a morphism is separated we may work locally on the base, see Schemes, Section \ref{schemes-section-separation-axioms}. By construction $\underline{\text{Proj}}_S(\mathcal{A})$ is over any affine $U \subset S$ isomorphic to $\text{Proj}(A)$ with $A = \mathcal{A}(U)$. By Lemma \ref{lemma-proj-separated} we see that $\text{Proj}(A)$ is separated. Hence $\text{Proj}(A) \to U$ is separated (see Schemes, Lemma \ref{schemes-lemma-compose-after-separated}) as desired.
This follows from Lemma \ref{lemma-proj-base-change} and the construction of $\underline{\text{Proj}}_S(\mathcal{A})$ in Lemma \ref{lemma-relative-proj} as the union of the schemes $U_d$ representing the functors $F_d$. In terms of the construction of relative Proj via glueing this isomorphism is given by the isomorphisms constructed in Lemma \ref{lemma-base-change-map-proj} which provides us with the isomorphism $\theta$. Some details omitted.
Under the assumptions of the lemma the sheaves $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)$ are invertible and the multiplication maps isomorphisms by Lemma \ref{lemma-relative-proj} and Lemma \ref{lemma-apply} over affine opens of $S$. Thus $X$ actually represents the functor $F_1$, see proof of Lemma \ref{lemma-relative-proj}.
Note that $$ \mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n]) = \underline{\text{Proj}}_{\Spec(\mathbf{Z})} \left(\widetilde{\mathbf{Z}[T_0, \ldots, T_n]}\right) $$ where the grading on the ring $\mathbf{Z}[T_0, \ldots, T_n]$ is given by $\deg(T_i) = 1$ and the elements of $\mathbf{Z}$ are in degree $0$. Recall that $\mathbf{P}^n_S$ is defined as $\mathbf{P}^n_{\mathbf{Z}} \times_{\Spec(\mathbf{Z})} S$. Moreover, forming the relative homogeneous spectrum commutes with base change, see Lemma \ref{lemma-relative-proj-base-change}. For any scheme $g : S \to \Spec(\mathbf{Z})$ we have $g^*\mathcal{O}_{\Spec(\mathbf{Z})}[T_0, \ldots, T_n] = \mathcal{O}_S[T_0, \ldots, T_n]$. Combining the above we see that $$ \mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n]). $$ Finally, note that $\mathcal{O}_S[T_0, \ldots, T_n] = \text{Sym}(\mathcal{O}_S^{\oplus n + 1})$. Hence we see that $\mathbf{P}^n_S$ is a projective bundle over $S$.
According to Lemma \ref{lemma-projective-space} the scheme $\mathbf{P}^n_\mathbf{Z}$ represents the functor which assigns to a scheme $S$ the set of isomorphisms classes of pairs $(\mathcal{L}, (s_0, \ldots, s_n))$ consisting of an invertible module $\mathcal{L}$ and an $(n + 1)$-tuple of global sections generating $\mathcal{L}$. Given such a pair we obtain a quotient $$ \mathcal{O}_S^{\oplus n + 1} \longrightarrow \mathcal{L},\quad (h_0, \ldots, h_n) \longmapsto \sum h_i s_i. $$ Conversely, given an element $q : \mathcal{O}_S^{\oplus n + 1} \to \mathcal{Q}$ of $G(n, n + 1)(S)$ we obtain such a pair, namely $(\mathcal{Q}, (q(e_1), \ldots, q(e_{n + 1})))$. Here $e_i$, $i = 1, \ldots, n + 1$ are the standard generating sections of the free module $\mathcal{O}_S^{\oplus n + 1}$. We omit the verification that these constructions define mutually inverse transformations of functors.
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}. Then the assumptions of Lemma \ref{lemma-algebraize-local-cohomology} are satisfied. Pick $J_0$ as in that lemma and set $V_0 = X \setminus V(J_0)$. Then opens $V \subset V_0$ containing $U \cap Y$ correspond $1$-to-$1$ with ideals $J \subset J_0$ with $V(J) \cap V(I) = \{\mathfrak m\}$. Moreover, for such a choice we have a distinguished triangle $$ R\Gamma_J(M) \to M \to R\Gamma(V, \mathcal{F}) \to R\Gamma_J(M)[1] $$ We similarly have a distinguished triangle $$ R\Gamma_\mathfrak m(M)^\wedge \to M \to R\Gamma(U, \mathcal{F})^\wedge \to R\Gamma_\mathfrak m(M)^\wedge[1] $$ involving derived $I$-adic completions. The cohomology groups of $R\Gamma(U, \mathcal{F})^\wedge$ are equal to the limits in the statement of the theorem by Lemma \ref{lemma-compare-with-derived-completion}. The canonical map between these triangles and some easy arguments show that our theorem follows from the main Lemma \ref{lemma-algebraize-local-cohomology} (note that we have $i < s$ here whereas we have $i \leq s$ in the lemma; this is because of the shift). The finiteness of the cohomology groups (under the additional assumption) follows from Lemma \ref{lemma-kill-colimit}.
The proof is exactly the same as the proof of Lemma \ref{lemma-ML-general}. In fact, the result will follow from the arguments given there as soon as we show that $\bigoplus H^1_n$ is a graded $\bigoplus_{n \geq 0} I^n/I^{n + 1}$-submodule of $\bigoplus H^1(X, I^n\mathcal{F}_{n + 1})$ and that the boundary maps $\delta_n$ have image contained in $H^1_n$. \medskip\noindent Suppose that $\xi \in H^1_n$ and $f \in I^k$. Choose $m \gg n + k$. Choose $\xi' \in H^1(X, I^n\mathcal{F}_{m + 1})$ lifting $\xi$. We consider the diagram $$ \xymatrix{ 0 \ar[r] & I^n\mathcal{F}_{m + 1} \ar[d]_f \ar[r] & \mathcal{F}_{m + 1} \ar[d]_f \ar[r] & \mathcal{F}_n \ar[d]_f \ar[r] & 0 \\ 0 \ar[r] & I^{n + k}\mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{m + 1} \ar[r] & \mathcal{F}_{n + k} \ar[r] & 0 } $$ constructed as in the proof of Lemma \ref{lemma-ML-general}. We get an induced map on cohomology and we see that $f \xi' \in H^1(X, I^{n + k}\mathcal{F}_{m + 1})$ maps to $f \xi$. Since this is true for all $m \gg n + k$ we see that $f\xi$ is in $H^1_{n + k}$ as desired. \medskip\noindent To see the boundary maps $\delta_n$ have image contained in $H^1_n$ we consider the diagrams $$ \xymatrix{ 0 \ar[r] & I^n\mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_{m + 1} \ar[d] \ar[r] & \mathcal{F}_n \ar[d] \ar[r] & 0 \\ 0 \ar[r] & I^n\mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_{n + 1} \ar[r] & \mathcal{F}_n \ar[r] & 0 } $$ for $m \geq n$. Looking at the induced maps on cohomology we conclude.
Set $F^n = \Ker(M \to H^0(X, \mathcal{F}_n))$ for $n \geq 1$ and $F^0 = M$. Observe that $I F^n \subset F^{n + 1}$. In particular $I^n M \subset F^n$ and we are trying to show that given $n$ there exists an $m \geq n$ such that $F^m \subset I^nM$. We have an injective map of graded modules $$ \bigoplus\nolimits_{n \geq 0} F^n/F^{n + 1} \longrightarrow \bigoplus\nolimits_{n \geq 0} H^0(X, I^n\mathcal{F}_{n + 1}) $$ By assumption the left hand side is generated by finitely many homogeneous elements. Hence we can find $r$ and $c_1, \ldots, c_r \geq 0$ and $a_i \in F^{c_i}$ whose images in $\bigoplus F^n/F^{n + 1}$ generate. Set $c = \max(c_i)$. \medskip\noindent For $n \geq c$ we claim that $I F^n = F^{n + 1}$. Namely, suppose $a \in F^{n + 1}$. The image of $a$ in $F^{n + 1}/F^{n + 2}$ is a linear combination of our $a_i$. Therefore $a  - \sum f_i a_i \in F^{n + 2}$ for some $f_i \in I^{n + 1 - c_i}$. Since $I^{n + 1 - c_i} = I \cdot I^{n - c_i}$ as $n \geq c_i$ we can write $f_i = \sum g_{i, j} h_{i, j}$ with $g_{i, j} \in I$ and $h_{i, j}a_i \in F^n$. Thus we see that $F^{n + 1} = F^{n + 2} + IF^n$. A simple induction argument gives $F^{n + 1} = F^{n + e} + IF^n$ for all $e > 0$. It follows that $IF^n$ is dense in $F^{n + 1}$. Choose generators $k_1, \ldots, k_r$ of $I$ and consider the continuous map $$ u : (F^n)^{\oplus r} \longrightarrow F^{n + 1},\quad (x_1, \ldots, x_r) \mapsto \sum k_i x_i $$ (in the limit topology). By the above the image of $(F^m)^{\oplus r}$ under $u$ is dense in $F^{m + 1}$ for all $m \geq n$. By the open mapping lemma (More on Algebra, Lemma \ref{more-algebra-lemma-open-mapping}) we find that $u$ is open. Hence $u$ is surjective. Hence $IF^n = F^{n + 1}$ for $n \geq c$. This concludes the proof.
Parts (1), (2), and (3) are general facts about inverse systems of quasi-coherent modules with surjective transition maps, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-Rlim-quasi-coherent} and Cohomology, Lemma \ref{cohomology-lemma-RGamma-commutes-with-Rlim}. Next, assume $\mathcal{I}$ is of finite type. Let $U \subset X$ be affine open. Say $U = \Spec(A)$ and $\mathcal{I}|_U$ corresponds to $I \subset A$. Observe that $I$ is a finitely generated ideal. By the equivalence of categories between quasi-coherent $\mathcal{O}_U$-modules and $A$-modules (Schemes, Lemma \ref{schemes-lemma-equivalence-quasi-coherent}) we find that $M_n = \mathcal{F}_n(U)$ is an inverse system of $A$-modules with $M_n = M_{n + 1}/I^nM_{n + 1}$. Thus $$ M = \mathcal{F}(U) = \lim \mathcal{F}_n(U) = \lim M_n $$ is an $I$-adically complete module with $M/I^nM = M_n$ by Algebra, Lemma \ref{algebra-lemma-limit-complete}. This proves (4). Part (5) translates into the statement that $\lim_{m \geq n} I^nM/I^mM = I^nM$. Since $I^mM = I^{m - n} \cdot I^nM$ this is just the statement that $I^mM$ is $I$-adically complete. This follows from Algebra, Lemma \ref{algebra-lemma-hathat-finitely-generated} and the fact that $M$ is complete.
We omit the proof of the equivalence of (1) and (2). The condition that $\mathcal{G}$ is $f$-divisible means that $f : \mathcal{G} \to \mathcal{G}$ is surjective. Thus given $\mathcal{F}_n$ as in (1) we set $\mathcal{G} = \colim \mathcal{F}_n$ where the maps $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to \ldots$ are as in (1). This produces an $f$-divisible $\mathcal{O}_X$-module with $\mathcal{F}_n = \mathcal{G}[f^n]$ as can be seen by checking on stalks. The condition that $\mathcal{F}$ is $f$-torsion free means that $f : \mathcal{F} \to \mathcal{F}$ is injective. If $X$ is a scheme and $\mathcal{F}_n$ is quasi-coherent, then we set $\mathcal{F} = \lim \mathcal{F}_n$. Namely, for an affine open $U \subset X$ the transition maps $\mathcal{F}_{n + 1}(U) \to \mathcal{F}_n(U)$ are surjective by vanishing of higher cohomology. This produces an $f$-torsion free $\mathcal{O}_X$-module with $\mathcal{F}_n = \mathcal{F}/f^n\mathcal{F}$ (Lemma \ref{lemma-properties-system}).
By Lemma \ref{lemma-topology-I-adic-f} and its proof we have $M/fM \subset H^0(X, \mathcal{F}_1)$. From (1) and the Noetherian property of $A$ we get that $M/fM$ is a finite $A$-module. Observe that $\bigcap f^nM = 0$ as $f^nM$ maps to zero in $H^0(X, \mathcal{F}_n)$. By Algebra, Lemma \ref{algebra-lemma-finite-over-complete-ring} we conclude that $M$ is finite over $A$.
Set $I = (f)$. We will use the criterion of Lemma \ref{lemma-ML-general}. Observe that $f^n : \mathcal{F}_0 \to I^n\mathcal{F}_{n + 1}$ is an isomorphism for all $n \geq 0$. Thus it suffices to show that $$ \bigoplus\nolimits_{n \geq 1} H^1(X, \mathcal{F}_1) \cdot f^{n + 1} $$ is a graded $S = \bigoplus_{n \geq 0} A/(f) \cdot f^n$-module satisfying the ascending chain condition. If $A$ is not Noetherian, then $H^1(X, \mathcal{F}_1)$ has finite length and the result holds. If $A$ is Noetherian, then $S$ is a Noetherian ring and the result holds as the module is finite over $S$ by the assumed finiteness of $H^1(X, \mathcal{F}_1)$. Some details omitted.
Set $I = (f)$. We will use the criterion of Lemma \ref{lemma-ML-general-better} involving the modules $H^1_n$. For $m \geq n$ we have $I^n\mathcal{F}_{m + 1} = \mathcal{F}_{m + 1 - n}$. Thus we see that $$ H^1_n = \bigcap\nolimits_{m \geq 1} \Im\left( H^1(X, \mathcal{F}_m) \to H^1(X, \mathcal{F}_1) \right) $$ is independent of $n$ and $\bigoplus H^1_n = \bigoplus H^1_1 \cdot f^{n + 1}$. Thus we conclude exactly as in the proof of Lemma \ref{lemma-ML}.
We start with the canonical identifications \begin{align*} R\Gamma(X, \mathcal{F})^\wedge & = R\lim R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} (A \xrightarrow{f^n} A) \\ & = R\lim R\Gamma(X, \mathcal{F} \xrightarrow{f^n} \mathcal{F}) \\ & = R\Gamma(X, R\lim (\mathcal{F} \xrightarrow{f^n} \mathcal{F})) \end{align*} The first equality holds by More on Algebra, Lemma \ref{more-algebra-lemma-derived-completion-koszul}. The second by the projection formula, see  Cohomology, Lemma \ref{cohomology-lemma-projection-formula-perfect}. The third by Cohomology, Lemma \ref{cohomology-lemma-Rf-commutes-with-Rlim}. Note that by Derived Categories of Schemes, Lemma \ref{perfect-lemma-Rlim-quasi-coherent} we have $\lim \mathcal{F}/f^n\mathcal{F} = R\lim \mathcal{F}/f^n \mathcal{F}$. Thus to finish the proof of the first statement of the lemma it suffices to show that the pro-objects $(f^n : \mathcal{F} \to \mathcal{F})$ and $(\mathcal{F}/f^n \mathcal{F})$ are isomorphic. There is clearly a map from the first inverse system to the second. Suppose that $\mathcal{F}[f^c] = \mathcal{F}[f^{c + 1}] = \mathcal{F}[f^{c + 2}] = \ldots$. Then we can define an arrow of inverse systems in $D(\mathcal{O}_X)$ in the other direction by the diagrams $$ \xymatrix{ \mathcal{F}/\mathcal{F}[f^c] \ar[r]_-{f^{n + c}} \ar[d]_{f^c} & \mathcal{F} \ar[d]^1 \\ \mathcal{F} \ar[r]^{f^n} & \mathcal{F} } $$ Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $\mathcal{F}/f^{n + c}\mathcal{F}$. Some details omitted. \medskip\noindent Since $R\Gamma(X, -)$ commutes with derived limits (Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim}) we see that $$ R\Gamma(X, \lim \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, R\lim \mathcal{F}/f^n\mathcal{F}) = R\lim R\Gamma(X, \mathcal{F}/f^n\mathcal{F}) $$ (for first equality see first paragraph of proof). By More on Algebra, Remark \ref{more-algebra-remark-compare-derived-limit} we obtain exact sequences $$ 0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p(X, \lim \mathcal{F}/I^n\mathcal{F}) \to \lim H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0 $$ of $A$-modules. The second set of short exact sequences follow immediately from the discussion in More on Algebra, Example \ref{more-algebra-example-spectral-sequence-principal}.
Since $\text{cd}(A, I) = 1$ the complement $U = \Spec(A) \setminus V(I)$ is affine (Local Cohomology, Lemma \ref{local-cohomology-lemma-cd-is-one}). Say $U = \Spec(B)$. Then $IB = B$ and we can write $1 = \sum_{j = 1, \ldots, r} f_j b_j$ for some $b_j \in B$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} we can represent $b_j$ by maps $\varphi_j : I^c \to A$ for some $c \geq 0$. Then $\sum f_j \varphi_j : I^c \to I \subset A$ is the canonical embedding, after possibly replacing $c$ by a larger integer, by the same lemma.
The uniqueness is clear from the formula and the uniqueness of $\Phi$ in Lemma \ref{lemma-cd-one-extend}. Consider the Noetherian $A$-algebra $B' = B \oplus N$ where $N$ is an ideal of square zero. To show the existence of $\Phi_N$ it is enough (via Lemma \ref{lemma-cd-one}) to show that $\varphi_j$ extends to a map $\varphi'_j : I^cB' \to B'$ after possibly increasing $c$ to some $c'$ (and replacing $\varphi_j$ by the composition of the inclusion $I^{c'} \to I^c$ with $\varphi_j$). Recall that $\varphi_j$ corresponds to a section $$ h_j \in \Gamma(\Spec(A) \setminus V(I), \mathcal{O}_{\Spec(A)}) $$ see Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}. (This is in fact how we chose our $\varphi_j$ in the proof of Lemma \ref{lemma-cd-one}.) Let us use the same lemma to represent the pullback $$ h'_j \in \Gamma(\Spec(B') \setminus V(IB'), \mathcal{O}_{\Spec(B')}) $$ of $h_j$ by a $B'$-linear map $\varphi'_j : I^{c'}B' \to B'$ for some $c' \geq c$. The agreement with $\varphi_j$ will hold for $c'$ sufficiently large by a further application of the lemma: namely we can test agreement on a finite list of generators of $I^{c'}$. Small detail omitted.
The uniqueness is immediate from the $\mathcal{O}_X$-linearity and the requirement that $\Phi_\mathcal{F}(g s) = \Phi(g) s$ for $g \in I^{nc}$ and $s$ a local section of $\mathcal{F}$. Thus we may assume $X = \Spec(B)$ is affine. Observe that $(\mathcal{F}_n)$ is an object of the category $\textit{Coh}(X, I\mathcal{O}_X)$ introduced in Cohomology of Schemes, Section \ref{coherent-section-coherent-formal}. Let $B' = B^\wedge$ be the $I$-adic completion of $B$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-inverse-systems-affine} the object $(\mathcal{F}_n)$ corresponds to a finite $B'$-module $N$ in the sense that $\mathcal{F}_n$ is the coherent module associated to the finite $B$-module $N/I^n N$. Applying Lemma \ref{lemma-cd-one-extend-to-module} to $I \subset A \to B'$ and $N$ we see that, after possibly increasing $c$ and adjusting $\varphi_j$ accordingly, we get unique maps $$ \Phi_N : \bigoplus\nolimits_{n \geq 0} I^{nc}N \to N[T_1, \ldots, T_r] $$ with the corresponding properties. Note that in degree $n$ we obtain an inverse system of maps $N/I^mN \to \bigoplus_{e_1 + \ldots + e_r = n} N/I^{m - nc}N \cdot T_1^{e_1} \ldots T_r^{e_r}$ for $m \geq nc$. Translating back into coherent sheaves we see that $\Phi_N$ corresponds to a system of maps $$ \Phi^n_m : I^{nc}\mathcal{F}_m \longrightarrow \bigoplus\nolimits_{e_1 + \ldots + e_r = n} \mathcal{F}_{m - nc} \cdot T_1^{e_1} \ldots T_r^{e_r} $$ for varying $m \geq nc$ and $n \geq 1$. Taking the inverse limit of these maps over $m$ we obtain $\Phi_\mathcal{F} = \bigoplus_n \lim_m \Phi^n_m$. Note that $\lim_m I^t\mathcal{F}_m = I^t \mathcal{F}$ as can be seen by evaluating on affines for example, but in fact we don't need this because it is clear there is a map $I^t\mathcal{F} \to \lim_m I^t\mathcal{F}_m$.
First it is clear that $I^t \lim H^p(X, \mathcal{F}_n)$ maps to zero in $H^p(X, \mathcal{F}_t)$. Thus the $I$-adic topology is finer than the limit topology. For the converse we set $\mathcal{F} = \lim \mathcal{F}_n$, we pick generators $f_1, \ldots, f_r$ of $I$, we pick $c \geq 1$, and we choose $\Phi_\mathcal{F}$ as in Lemma \ref{lemma-cd-is-one-for-system}. We will use the results of Lemma \ref{lemma-properties-system} without further mention. In particular we have a short exact sequence $$ 0 \to R^1\lim H^{p - 1}(X, \mathcal{F}_n) \to H^p(X, \mathcal{F}) \to \lim H^p(X, \mathcal{F}_n) \to 0 $$ Thus we can lift any element $\xi$ of $\lim H^p(X, \mathcal{F}_n)$ to an element $\xi' \in H^p(X, \mathcal{F})$. Suppose $\xi$ maps to zero in $H^p(X, \mathcal{F}_{nc})$ for some $n$, in other words, suppose $\xi$ is ``small'' in the limit topology. We have a short exact sequence $$ 0 \to I^{nc}\mathcal{F} \to \mathcal{F} \to \mathcal{F}_{nc} \to 0 $$ and hence the assumption means we can lift $\xi'$ to an element $\xi'' \in H^p(X, I^{nc}\mathcal{F})$. Applying $\Phi_\mathcal{F}$ we get $$ \Phi_\mathcal{F}(\xi'') = \sum\nolimits_{e_1 + \ldots + e_r = n} \xi'_{e_1, \ldots, e_r} \cdot T_1^{e_1} \ldots T_r^{e_r} $$ for some $\xi'_{e_1, \ldots, e_r} \in H^p(X, \mathcal{F})$. Letting $\xi_{e_1, \ldots, e_r} \in \lim H^p(X, \mathcal{F}_n)$ be the images and using the final assertion of Lemma \ref{lemma-cd-is-one-for-system} we conclude that $$ \xi = \sum f_1^{e_1} \ldots f_r^{e_r} \xi_{e_1, \ldots, e_r} $$ is in $I^n \lim H^p(X, \mathcal{F}_n)$ as desired.
Proof of (1). Let $A^\wedge$ be the completion of $A$. Since $H^i_\mathfrak m(M)$ is $\mathfrak m$-power torsion, we see that $H^i_\mathfrak m(M) = H^i_\mathfrak m(M) \otimes_A A^\wedge$. Moreover, we have $H^i_\mathfrak m(M) \otimes_A A^\wedge = H^i_{\mathfrak mA^\wedge}(M \otimes_A A^\wedge)$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-torsion-change-rings}. Thus $$ H^i_\mathfrak m(M) = H^i_{\mathfrak mA^\wedge}(M \otimes_A A^\wedge) $$ and $A$-submodules of the left hand side are the same thing as $A^\wedge$-submodules of the right hand side. Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $A$ is complete. Then $A$ has a normalized dualizing complex $\omega_A^\bullet$ (Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}). By the local duality theorem (Dualizing Complexes, Lemma \ref{dualizing-lemma-special-case-local-duality}) we find an isomorphism $$ \Hom_A(H^i_\mathfrak m(M), E) = \text{Ext}^{-i}_A(M, \omega_A^\bullet)^\wedge $$ where $E$ is an injective hull of the residue field of $A$. The module $\text{Ext}^{-i}_A(M, \omega_A^\bullet)$ on the right hand side is a finite $A$-module by Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing}. Since $A$ is complete, the completion isn't necessary. Thus $H^i_\mathfrak m(M)$ has the descending chain condition by Matlis duality, see  Dualizing Complexes, Proposition \ref{dualizing-proposition-matlis} and its addendum Remark \ref{dualizing-remark-matlis}. \medskip\noindent Part (2) follows from (1) via Local Cohomology, Lemma \ref{local-cohomology-lemma-finiteness-pushforwards-and-H1-local}.