hcju/stacksqa huggingface dataset:
Let $x \in \text{Ass}(\mathcal{F})$. By Proposition \ref{proposition-kollar} it suffices to check that $A = \mathcal{O}_{\overline{\{x\}}, z}$ satisfies the condition of the proposition on associated primes of its completion if and only if $\dim(A) \geq 2$. Observe that $A$ is universally catenary (this is clear) and that its formal fibres are $(S_1)$ as follows from More on Algebra, Lemma \ref{more-algebra-lemma-formal-fibres-normal} and Proposition \ref{more-algebra-proposition-finite-type-over-P-ring}. Let $\mathfrak p' \subset A^\wedge$ be an associated prime. As $A \to A^\wedge$ is flat, by Algebra, Lemma \ref{algebra-lemma-bourbaki}, we find that $\mathfrak p'$ lies over $(0) \subset A$. The formal fibre $A^\wedge \otimes_A F$ is $(S_1)$ where $F$ is the fraction field of $A$. We conclude that $\mathfrak p'$ is a minimal prime, see Algebra, Lemma \ref{algebra-lemma-criterion-no-embedded-primes}. Since $A$ is universally catenary it is formally catenary by More on Algebra, Proposition \ref{more-algebra-proposition-ratliff}. Hence $\dim(A^\wedge/\mathfrak p') = \dim(A)$ which proves the equivalence.
By definition of depth we have $IM_\mathfrak p \not = M_\mathfrak p$ and there exists an $M_\mathfrak p$-regular sequence $f_1, \ldots, f_e \in IA_\mathfrak p$. After replacing $A$ by a principal localization we may assume $f_1, \ldots, f_e \in I$ form an $M$-regular sequence, see Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}. Consider the module $M' = M/IM$. Since $\mathfrak p \in \text{Supp}(M')$ and since the support of a finite module is closed, we find $V(\mathfrak p) \subset \text{Supp}(M')$. Thus for $\mathfrak q \in V(\mathfrak p)$ we get $IM_\mathfrak q \not = M_\mathfrak q$. Hence, using that localization is exact, we see that $\text{depth}_{IA_\mathfrak q}(M_\mathfrak q) \geq e$ for any $\mathfrak q \in V(I)$ by definition of depth.
By definition of depth we have $\mathfrak p M_\mathfrak p \not = M_\mathfrak p$ and there exists an $M_\mathfrak p$-regular sequence $f_1, \ldots, f_e \in \mathfrak p A_\mathfrak p$. After replacing $A$ by a principal localization we may assume $f_1, \ldots, f_e \in \mathfrak p$ form an $M$-regular sequence, see Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}. Consider the module $M' = M/(f_1, \ldots, f_e)M$. Since $\mathfrak p \in \text{Supp}(M')$ and since the support of a finite module is closed, we find $V(\mathfrak p) \subset \text{Supp}(M')$. Thus for $\mathfrak q \in V(\mathfrak p)$ we get $\mathfrak q M_\mathfrak q \not = M_\mathfrak q$. Hence, using that localization is exact, we see that $\text{depth}_{A_\mathfrak q}(M_\mathfrak q) \geq e$ for any $\mathfrak q \in V(I)$ by definition of depth. Moreover, as soon as $\mathfrak q$ is not an associated prime of the module $M'$, then the depth goes up. Thus we see that the final statement holds by Algebra, Lemma \ref{algebra-lemma-finite-ass}.
We will prove this lemma by induction on $k$. The base case $k = 0$ says that $\mathcal{F}$ has a finite number of embedded associated points, which follows from Divisors, Lemma \ref{divisors-lemma-finite-ass}. \medskip\noindent Assume $k > 0$ and the result holds for all smaller $k$. We can cover $X$ by finitely many affine opens, hence we may assume $X = \Spec(A)$ is affine. Then $\mathcal{F}$ is the coherent $\mathcal{O}_X$-module associated to a finite $A$-module $M$ which satisfies $(S_k)$. We will use Algebra, Lemmas \ref{algebra-lemma-one-equation-module} and \ref{algebra-lemma-depth-drops-by-one} without further mention. \medskip\noindent Let $f \in A$ be a nonzerodivisor on $M$. Then $M/fM$ has $(S_{k - 1})$. By induction we see that there are finitely many primes $\mathfrak p \in V(f)$ with $\text{depth}((M/fM)_\mathfrak p) = k - 1$ and $\dim(\text{Supp}((M/fM)_\mathfrak p)) > k - 1$. These are exactly the primes $\mathfrak p \in V(f)$ with $\text{depth}(M_\mathfrak p) = k$ and $\dim(\text{Supp}(M_\mathfrak p)) > k$. Thus we may replace $A$ by $A_f$ and $M$ by $M_f$ in trying to prove the finiteness statement. \medskip\noindent Since $M$ satisfies $(S_k)$ and $k > 0$ we see that $M$ has no embedded associated primes (Algebra, Lemma \ref{algebra-lemma-criterion-no-embedded-primes}). Thus $\text{Ass}(M)$ is the set of generic points of the support of $M$. Thus Dualizing Complexes, Lemma \ref{dualizing-lemma-CM-open} shows the set $U = \{\mathfrak q \mid M_\mathfrak q\text{ is Cohen-Macaulay}\}$ is an open containing $\text{Ass}(M)$. By prime avoidance (Algebra, Lemma \ref{algebra-lemma-silly}) we can pick $f \in A$ with $f \not \in \mathfrak p$ for $\mathfrak p \in \text{Ass}(M)$ such that $D(f) \subset U$. Then $f$ is a nonzerodivisor on $M$ (Algebra, Lemma \ref{algebra-lemma-ass-zero-divisors}). After replacing $A$ by $A_f$ and $M$ by $M_f$ (see above) we find that $M$ is Cohen-Macaulay. Thus for all $\mathfrak q \subset A$ we have $\dim(M_\mathfrak q) = \text{depth}(M_\mathfrak q)$ and hence the set described in the lemma is empty and a fortiori finite.
Parts (1), (2), and (3) are copies of the statements in Dualizing Complexes, Lemma \ref{dualizing-lemma-sitting-in-degrees}. For a prime $\mathfrak p$ of $A$ we have that $(\omega_A^\bullet)_\mathfrak p[-\dim(A/\mathfrak p)]$ is a normalized dualzing complex for $A_\mathfrak p$. See Dualizing Complexes, Lemma \ref{dualizing-lemma-dimension-function}. Thus $$ E^i_\mathfrak p = \text{Ext}^{-i}_A(M, \omega_A^\bullet)_\mathfrak p = \text{Ext}^{-i + \dim(A/\mathfrak p)}_{A_\mathfrak p} (M_\mathfrak p, (\omega_A^\bullet)_\mathfrak p[-\dim(A/\mathfrak p)]) $$ is zero for $i - \dim(A/\mathfrak p) < \text{depth}_{A_\mathfrak p}(M_\mathfrak p)$ and nonzero for $i = \dim(A/\mathfrak p) + \text{depth}_{A_\mathfrak p}(M_\mathfrak p)$ by part (3) over $A_\mathfrak p$. This proves part (4). If $E$ is an injective hull of the residue field of $A$, then we have $$ \Hom_A(H^i_\mathfrak m(M), E) = \text{Ext}^{-i}_A(M, \omega_A^\bullet)^\wedge = (E^i)^\wedge = E^i \otimes_A A^\wedge $$ by the local duality theorem (in the form of Dualizing Complexes, Lemma \ref{dualizing-lemma-special-case-local-duality}). Since $A \to A^\wedge$ is faithfully flat, we find (5) is true by Matlis duality (Dualizing Complexes, Proposition \ref{dualizing-proposition-matlis}).
Consider the sets $$ T = V(I) \quad\text{and}\quad T' = \bigcup\nolimits_{f \in A, f \not \in \mathfrak p} V(f) $$ These are subsets of $\Spec(A)$ stable under specialization. Observe that $T \subset T'$ and $\mathfrak p \not \in T'$. Assumption (3) says that hypothesis (2) of Proposition \ref{proposition-annihilator} holds. Hence we can find $J \subset A$ with $V(J) \subset T'$ such that $J H^i_{V(I)}(M) = 0$ for $i \leq s$. Choose $f \in A$, $f \not \in \mathfrak p$ with $V(J) \subset V(f)$. A power of $f$ annihilates $H^i_{V(I)}(M)$ for $i \leq s$.
Let $\mathfrak p \subset \mathfrak q \subset A$ be primes with $I \subset \mathfrak q$ and $I \not \subset \mathfrak p$. Since $A \to B$ is finite there are finitely many primes $\mathfrak p_i$ lying over $\mathfrak p$. By Algebra, Lemma \ref{algebra-lemma-depth-goes-down-finite} we have $$ \text{depth}(M_\mathfrak p) = \min \text{depth}(M_{\mathfrak p_i}) $$ Let $\mathfrak p_i \subset \mathfrak q_{ij}$ be primes lying over $\mathfrak q$. By going up for $A \to B$ (Algebra, Lemma \ref{algebra-lemma-integral-going-up}) there is at least one $\mathfrak q_{ij}$ for each $i$. Then we see that $$ \dim((B/\mathfrak p_i)_{\mathfrak q_{ij}}) = \dim((A/\mathfrak p)_\mathfrak q) $$ by the dimension formula, see Algebra, Lemma \ref{algebra-lemma-dimension-formula}. This implies that the minimum of the quantities used to define $s_{B, J}(M)$ for the pairs $(\mathfrak p_i, \mathfrak q_{ij})$ is equal to the quantity for the pair $(\mathfrak p, \mathfrak q)$. This proves the lemma.
We have $$ \text{depth}(M'_{\mathfrak p'}) = \text{depth}(M_\mathfrak p) + \text{depth}(A'_{\mathfrak p'}/\mathfrak p A'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p) $$ by flatness of $A \to A'$, see Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module}. Since the fibres of $A \to A'$ are Cohen-Macaulay (Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing-gorenstein-formal-fibres} and More on Algebra, Section \ref{more-algebra-section-properties-formal-fibres}) we see that $\text{depth}(A'_{\mathfrak p'}/\mathfrak p A'_{\mathfrak p'}) = \dim(A'_{\mathfrak p'}/\mathfrak p A'_{\mathfrak p'})$. Thus we obtain \begin{align*} \text{depth}(M'_{\mathfrak p'}) + \dim((A'/\mathfrak p')_{\mathfrak q'}) & = \text{depth}(M_\mathfrak p) + \dim(A'_{\mathfrak p'}/\mathfrak p A'_{\mathfrak p'}) + \dim((A'/\mathfrak p')_{\mathfrak q'}) \\ & = \text{depth}(M_\mathfrak p) + \dim((A'/\mathfrak p A')_{\mathfrak q'}) \\ & = \text{depth}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) \end{align*} Second equality because $A'$ is catenary and third equality by More on Algebra, Lemma \ref{more-algebra-lemma-completion-dimension} as $(A/\mathfrak p)_\mathfrak q$ and $(A'/\mathfrak p A')_{\mathfrak q'}$ have the same $I$-adic completions.
The statement is local on $X$, hence we may assume $X$ is affine. Say $X = \Spec(A)$ and $Z = V(I)$. Let $M$ be a finite $A$-module whose associated coherent $\mathcal{O}_X$-module restricts to $\mathcal{F}$ over $U$, see Lemma \ref{lemma-finiteness-pushforwards-and-H1-local}. This lemma also tells us that $R^pj_*\mathcal{F}$ is coherent if and only if $H^{p + 1}_Z(M)$ is a finite $A$-module. Observe that the minimum of the expressions $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_x) + \dim(\mathcal{O}_{\overline{\{x\}}, z})$ is the number $s_{A, I}(M)$ of (\ref{equation-cutoff}). Having said this the lemma follows from Theorem \ref{theorem-finiteness} as elucidated by Remark \ref{remark-astute-reader}.
This lemma is the natural generalization of Lemma \ref{lemma-check-finiteness-local-cohomology-by-annihilator} whose proof the reader should read first. Assume (1) is true. Recall that $H^i_J(K) = H^i_{V(J)}(K)$, see Dualizing Complexes, Lemma \ref{dualizing-lemma-local-cohomology-noetherian}. Thus $H^i_T(K) = \colim H^i_J(K)$ where the colimit is over ideals $J \subset A$ with $V(J) \subset T$, see Lemma \ref{lemma-adjoint-ext}. Since $H^i_T(K)$ is finitely generated for $i \leq n$ we can find a $J \subset A$ as in (2) such that $H^i_J(K) \to H^i_T(K)$ is surjective for $i \leq n$. Thus the finite list of generators are $J$-power torsion elements and we see that (2) holds with $J$ replaced by some power. \medskip\noindent Let $a \in \mathbf{Z}$ be an integer such that $H^i(K) = 0$ for $i < a$. We prove (2) $\Rightarrow$ (1) by descending induction on $a$. If $a > n$, then we have $H^i_T(K) = 0$ for $i \leq n$ hence both (1) and (2) are true and there is nothing to prove. \medskip\noindent Assume we have $J$ as in (2). Observe that $N = H^a_T(K) = H^0_T(H^a(K))$ is finite as a submodule of the finite $A$-module $H^a(K)$. If $n = a$ we are done; so assume $a < n$ from now on. By construction of $R\Gamma_T$ we find that $H^i_T(N) = 0$ for $i > 0$ and $H^0_T(N) = N$, see Remark \ref{remark-upshot}. Choose a distinguished triangle $$ N[-a] \to K \to K' \to N[-a + 1] $$ Then we see that $H^a_T(K') = 0$ and $H^i_T(K) = H^i_T(K')$ for $i > a$. We conclude that we may replace $K$ by $K'$. Thus we may assume that $H^a_T(K) = 0$. This means that the finite set of associated primes of $H^a(K)$ are not in $T$. By prime avoidance (Algebra, Lemma \ref{algebra-lemma-silly}) we can find $f \in J$ not contained in any of the associated primes of $H^a(K)$. Choose a distinguished triangle $$ L \to K \xrightarrow{f} K \to L[1] $$ By construction we see that $H^i(L) = 0$ for $i \leq a$. On the other hand we have a long exact cohomology sequence $$ 0 \to H^{a + 1}_T(L) \to H^{a + 1}_T(K) \xrightarrow{f} H^{a + 1}_T(K) \to H^{a + 2}_T(L) \to H^{a + 2}_T(K) \xrightarrow{f} \ldots $$ which breaks into the identification $H^{a + 1}_T(L) = H^{a + 1}_T(K)$ and short exact sequences $$ 0 \to H^{i - 1}_T(K) \to H^i_T(L) \to H^i_T(K) \to 0 $$ for $i \leq n$ since $f \in J$. We conclude that $J^2$ annihilates $H^i_T(L)$ for $i \leq n$. By induction hypothesis applied to $L$ we see that $H^i_T(L)$ is finite for $i \leq n$. Using the short exact sequence once more we see that $H^i_T(K)$ is finite for $i \leq n$ as desired. \medskip\noindent We omit the proof of the equivalence of (2) and (3) in case $T = V(I)$.
Condition (3) just means that $\text{Ass}(\mathcal{F}') \cap T = \emptyset$. Thus $\mathcal{F} \to \mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections whose support is contained in $T$. This proves uniqueness. The statement on pullbacks follows from Divisors, Lemma \ref{divisors-lemma-bourbaki} and the uniqueness. \medskip\noindent Existence of $\mathcal{F} \to \mathcal{F}'$. By the uniqueness it suffices to prove the existence and uniqueness locally on $X$; small detail omitted. Thus we may assume $X = \Spec(A)$ is affine and $\mathcal{F}$ is the coherent module associated to the finite $A$-module $M$. Set $M' = M / H^0_T(M)$ with $H^0_T(M)$ as in Section \ref{section-supports}. Then $M_\mathfrak p = M'_\mathfrak p$ for $\mathfrak p \not \in T$ which proves (1). On the other hand, we have $H^0_T(M) = \colim H^0_Z(M)$ where $Z$ runs over the closed subsets of $X$ contained in $T$. Thus by Dualizing Complexes, Lemmas \ref{dualizing-lemma-divide-by-torsion} we have $H^0_T(M') = 0$, i.e., no associated prime of $M'$ is in $T$. Therefore $\text{depth}(M'_\mathfrak p) \geq 1$ for $\mathfrak p \in T$.
We have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}'_x) \geq 2$ by Divisors, Lemma \ref{divisors-lemma-depth-pushforward} part (3). The uniqueness of $\mathcal{F} \to \mathcal{F}'$ follows from Divisors, Lemma \ref{divisors-lemma-depth-2-hartog}. The compatibility with flat pullbacks follows from flat base change, see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
Let $\mathcal{F} \to \mathcal{F}'$ be the map constructed in Lemma \ref{lemma-get-depth-1-along-Z} for the subset $Z$ of $X$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $Z$. \medskip\noindent We first prove uniqueness. Let $\mathcal{F} \to \mathcal{F}''$ be as in the lemma. We get a factorization $\mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ since $\text{Ass}(\mathcal{F}'') \cap Z = \emptyset$ by conditions (2) and (3). Let $U \subset X$ be a maximal open subscheme such that $\mathcal{F}'|_U \to \mathcal{F}''|_U$ is an isomorphism. We see that $U$ contains all the points as in (2). Then by Divisors, Lemma \ref{divisors-lemma-depth-2-hartog} we conclude that $\mathcal{F}'' = j_*(\mathcal{F}'|_U)$. In this way we get uniqueness (small detail: if we have two of these $\mathcal{F}''$ then we take the intersection of the opens $U$ we get from either). \medskip\noindent Proof of existence. Recall that $\text{Ass}(\mathcal{F}') = \{x_1, \ldots, x_n\}$ is finite and $x_i \not \in Z$. Let $Y_i$ be the closure of $\{x_i\}$. Let $Z_{i, j}$ be the irreducible components of $Z \cap Y_i$. Observe that $\text{Supp}(\mathcal{F}') \cap Z = \bigcup Z_{i, j}$. Let $z_{i, j} \in Z_{i, j}$ be the generic point. Let $$ d_{i, j} = \dim(\mathcal{O}_{\overline{\{x_i\}}, z_{i, j}}) $$ If $d_{i, j} = 1$, then $z_{i, j}$ is one of the points as in (2). Thus we do not need to modify $\mathcal{F}'$ at these points. Furthermore, still assuming $d_{i, j} = 1$, using Lemma \ref{lemma-depth-function} we can find an open neighbourhood $z_{i, j} \in V_{i, j} \subset X$ such that $\text{depth}_{\mathcal{O}_{X, z}}(\mathcal{F}'_z) \geq 2$ for $z \in Z_{i, j} \cap V_{i, j}$, $z \not = z_{i, j}$. Set $$ Z' = X \setminus \left( X \setminus Z \cup \bigcup\nolimits_{d_{i, j} = 1} V_{i, j}) \right) $$ Denote $j' : X \setminus Z' \to X$. By our choice of $Z'$ the assumptions of Lemma \ref{lemma-finiteness-pushforward-general} are satisfied. We conclude by setting $\mathcal{F}'' = j'_*(\mathcal{F}'|_{X \setminus Z'})$ and applying Lemma \ref{lemma-get-depth-2-along-Z}. \medskip\noindent The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module} and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted.
Let $\mathcal{F} \to \mathcal{F}'$ be the quotient of $\mathcal{F}$ constructed in Lemma \ref{lemma-get-depth-1-along-Z} using $T'$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $T'$. \medskip\noindent Proof of uniqueness. Let $\mathcal{F} \to \mathcal{F}''$ be as in the lemma. We get a factorization $\mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ since $\text{Ass}(\mathcal{F}'') \cap T' = \emptyset$ by condition (2). Let $U \subset X$ be a maximal open subscheme such that $\mathcal{F}'|_U \to \mathcal{F}''|_U$ is an isomorphism. We see that $U$ contains all the points of $T'$. Then by Divisors, Lemma \ref{divisors-lemma-depth-2-hartog} we conclude that $\mathcal{F}'' = j_*(\mathcal{F}'|_U)$. In this way we get uniqueness (small detail: if we have two of these $\mathcal{F}''$ then we take the intersection of the opens $U$ we get from either). \medskip\noindent Proof of existence. We will define $$ \mathcal{F}'' = \colim j_*(\mathcal{F}'|_V) $$ where $j : V \to X$ runs over the open subschemes such that $X \setminus V \subset T'$. Observe that the colimit is filtered as $T'$ is stable under specialization. Each of the maps $\mathcal{F}' \to j_*(\mathcal{F}'|_V)$ is injective as $\text{Ass}(\mathcal{F}')$ is disjoint from $T'$. Thus $\mathcal{F}' \to \mathcal{F}''$ is injective. \medskip\noindent Suppose $X = \Spec(A)$ is affine and $\mathcal{F}$ corresponds to the finite $A$-module $M$. Then $\mathcal{F}'$ corresponds to $M' = M / H^0_{T'}(M)$, see proof of Lemma \ref{lemma-get-depth-1-along-Z}. Applying Lemmas \ref{lemma-local-cohomology} and \ref{lemma-adjoint-ext} we see that $\mathcal{F}''$ corresponds to an $A$-module $M''$ which fits into the short exact sequence $$ 0 \to M' \to M'' \to H^1_{T'}(M') \to 0 $$ By Proposition \ref{proposition-finiteness} and our condition on immediate specializations in the statement of the lemma we see that $M''$ is a finite $A$-module. In this way we see that $\mathcal{F}''$ is coherent. \medskip\noindent The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module} and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted.
First, let $\mathcal{F} \to \mathcal{F}'$ be the quotient of $\mathcal{F}$ constructed in Lemma \ref{lemma-get-depth-1-along-Z} using $T$. Second, let $\mathcal{F}' \to \mathcal{F}''$ be the unique map of coherent modules construction in Lemma \ref{lemma-make-S2-along-T-simple} using $T'$. Then $\mathcal{F} \to \mathcal{F}''$ is as desired.
By Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$. We will use Lemma \ref{lemma-isomorphism} to show $$ H^d_{V(I')}(A) \to H^d_{V(I)}(A) $$ is surjective which will finish the proof. Pick $\mathfrak p \in V(I) \setminus V(I')$. By our assumption on $I'$ we see that $\mathfrak p$ is not a maximal ideal of $A$. Hence $\dim(A_\mathfrak p) < d$. Then $H^d_{\mathfrak pA_\mathfrak p}(A_\mathfrak p) = 0$ by Lemma \ref{lemma-cd-dimension}.
By Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$. Thus for any finite $A$-module $N$ we have $H^i_{V(I)}(N) = 0$ for $i > d$. Let us say property $\mathcal{P}$ holds for the finite $A$-module $N$ if $H^d_{V(I)}(N) = 0$. One of our assumptions is that $\mathcal{P}(M)$ holds. Observe that $\mathcal{P}(N_1 \oplus N_2) \Leftrightarrow (\mathcal{P}(N_1) \wedge \mathcal{P}(N_2))$. Observe that if $N \to N'$ is surjective, then $\mathcal{P}(N) \Rightarrow \mathcal{P}(N')$ as we have the vanishing of $H^{d + 1}_{V(I)}$ (see above). Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be the minimal primes of $A$ with $\dim(A/\mathfrak p_i) = d$. Observe that $\mathcal{P}(N)$ holds if the support of $N$ is disjoint from $\{\mathfrak p_1, \ldots, \mathfrak p_n\}$ for dimension reasons, see Lemma \ref{lemma-cd-dimension}. For each $i$ set $M_i = M/\mathfrak p_i M$. This is a finite $A$-module annihilated by $\mathfrak p_i$ whose support is equal to $V(\mathfrak p_i)$ (here we use the assumption on the support of $M$). Finally, if $J \subset A$ is an ideal, then we have $\mathcal{P}(JM_i)$ as $JM_i$ is a quotient of a direct sum of copies of $M$. Thus it follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-property-higher-rank-cohomological} that $\mathcal{P}$ holds for every finite $A$-module.
The support of $M/fM$ has dimension $< d$ by our assumption on $f$. Thus $H^d_{V(I)}(M/fM) = 0$ by Lemma \ref{lemma-cd-dimension}. Thus $H^d_{V(I)}(fM) \to H^d_{V(I)}(M)$ is surjective. Since by Lemma \ref{lemma-cd-dimension} we know $\text{cd}(A, I) \leq d$ we also see that the surjection $M \to fM$, $x \mapsto fx$ induces a surjection $H^d_{V(I)}(M) \to H^d_{V(I)}(fM)$.
Set $d = \dim(A)$. Let $\mathfrak p_1, \ldots, \mathfrak p_n \subset A$ be the minimal primes of dimension $d$. Recall that the finite $A$-module $H^{-i}(\omega_A^\bullet)$ is nonzero only for $i \in \{0, \ldots, d\}$ and that the support of $H^{-i}(\omega_A^\bullet)$ has dimension $\leq i$, see Lemma \ref{lemma-sitting-in-degrees}. Set $\omega_A = H^{-d}(\omega_A^\bullet)$. By prime avoidence (Algebra, Lemma \ref{algebra-lemma-silly}) we can find $f \in A$, $f \not \in \mathfrak p_i$ which annihilates $H^{-i}(\omega_A^\bullet)$ for $i < d$. Consider the distinguished triangle $$ \omega_A[d] \to \omega_A^\bullet \to \tau_{\geq -d + 1}\omega_A^\bullet \to \omega_A[d + 1] $$ See Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}. By Derived Categories, Lemma \ref{derived-lemma-trick-vanishing-composition} we see that $f^d$ induces the zero endomorphism of $\tau_{\geq -d + 1}\omega_A^\bullet$. Using the axioms of a triangulated category, we find a map $$ \omega_A^\bullet \to \omega_A[d] $$ whose composition with $\omega_A[d] \to \omega_A^\bullet$ is multiplication by $f^d$ on $\omega_A[d]$. Thus we conclude that $f^d$ annihilates $H^d_{V(I)}(\omega_A)$. By Lemma \ref{lemma-top-coh-divisible} we conlude $H^d_{V(I)}(\omega_A) = 0$. Then we conclude by Lemma \ref{lemma-cd-top-vanishing-some-module} and the fact that $(\omega_A)_{\mathfrak p_i}$ is nonzero (see for example Dualizing Complexes, Lemma \ref{dualizing-lemma-nonvanishing-generically-local}).
Choose a presentation $A^{\oplus m} \to A^{\oplus n} \to M$ which induces an isomorphism $\kappa^{\oplus n} \to M/\mathfrak m M$. Let $T = (a_{ij})$ be the matrix of the map $A^{\oplus m} \to A^{\oplus n}$. Observe that $a_{ij} \in \mathfrak m$. Applying base change by $F$, using right exactness of base change, we get a presentation $A^{\oplus m} \to A^{\oplus n} \to M$ where the matrix is $T = (a_{ij}^p)$. Thus we have a presentation with $a_{ij} \in \mathfrak m^p$. Repeating this construction we find that for each $e \geq 1$ there exists a presentation with $a_{ij} \in \mathfrak m^e$. This implies the fitting ideals (More on Algebra, Definition \ref{more-algebra-definition-fitting-ideal}) $\text{Fit}_k(M)$ for $k < n$ are contained in $\bigcap_{e \geq 1} \mathfrak m^e$. Since this is zero by Krull's intersection theorem (Algebra, Lemma \ref{algebra-lemma-intersect-powers-ideal-module-zero}) we conclude that $M$ is free of rank $n$ by More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-finite-locally-free}.
We claim there is an exact sequence $$ 0 \to A/(f_1, \ldots, f_{r - 1}, g_r) \xrightarrow{f_r} A/(f_1, \ldots, f_{r - 1}, f_rg_r) \to A/(f_1, \ldots, f_{r - 1}, f_r) \to 0 $$ Namely, if $a f_r \in (f_1, \ldots, f_{r - 1}, f_rg_r)$, then $\sum_{i < r} a_i f_i + (a + bg_r)f_r = 0$ for some $b, a_i \in A$. Hence $\sum_{i < r} a_i g_r f_i + (a + bg_r)g_rf_r = 0$ which implies $a + bg_r \in (f_1, \ldots, f_{r - 1}, f_rg_r)$ which means that $a$ maps to zero in $A/(f_1, \ldots, f_{r - 1}, g_r)$. This proves the claim. To finish use additivity of lengths (Algebra, Lemma \ref{algebra-lemma-length-additive}).