hcju/stacksqa huggingface dataset:
We will show that conditions (1) and (2) of Lemma \ref{lemma-property-initial} hold. This is clear for condition (1). To show that (2) holds, let $$ \mathcal{T} = \left\{ \begin{matrix} i : Z \to X \text{ reduced closed subspace with }|Z|\text{ irreducible such}\\ \text{ that }i_*\mathcal{I}\text{ does not have }\mathcal{P} \text{ for some quasi-coherent }\mathcal{I} \subset \mathcal{O}_Z \end{matrix} \right\} $$ If $\mathcal{T}$ is nonempty, then since $X$ is Noetherian, we can find an $i : Z \to X$ which is minimal in $\mathcal{T}$. We will show that this leads to a contradiction. \medskip\noindent Let $\mathcal{G}$ be the sheaf whose scheme theoretic support is $Z$ whose existence is assumed in assumption (3). Let $\varphi : i_*\mathcal{I}^{\oplus r} \to \mathcal{G}$ be as in Lemma \ref{lemma-prepare-filter-irreducible}. Let $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \Coker(\varphi) $$ be a filtration as in Lemma \ref{lemma-coherent-filter}. By minimality of $Z$ and assumption (1) we see that $\Coker(\varphi)$ has property $\mathcal{P}$. As $\varphi$ is injective we conclude using assumption (1) once more that $i_*\mathcal{I}^{\oplus r}$ has property $\mathcal{P}$. Using assumption (2) we conclude that $i_*\mathcal{I}$ has property $\mathcal{P}$. \medskip\noindent Finally, if $\mathcal{J} \subset \mathcal{O}_Z$ is a second quasi-coherent sheaf of ideals, set $\mathcal{K} = \mathcal{I} \cap \mathcal{J}$ and consider the short exact sequences $$ 0 \to \mathcal{K} \to \mathcal{I} \to \mathcal{I}/\mathcal{K} \to 0 \quad \text{and} \quad 0 \to \mathcal{K} \to \mathcal{J} \to \mathcal{J}/\mathcal{K} \to 0 $$ Arguing as above, using the minimality of $Z$, we see that $i_*\mathcal{I}/\mathcal{K}$ and $i_*\mathcal{J}/\mathcal{K}$ satisfy $\mathcal{P}$. Hence by assumption (1) we conclude that $i_*\mathcal{K}$ and then $i_*\mathcal{J}$ satisfy $\mathcal{P}$. In other words, $Z$ is not an element of $\mathcal{T}$ which is the desired contradiction.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are coherent $\mathcal{O}_X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another coherent $\mathcal{O}_X$-submodule which contains both of them (see Lemmas \ref{lemma-coherent-abelian-Noetherian} and \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}). In this way we see that the system is directed. Hence it now suffices to show that $\mathcal{F}$ can be written as a filtered colimit of coherent modules, as then we can take the images of these modules in $\mathcal{F}$ to conclude there are enough of them. \medskip\noindent Let $U$ be an affine scheme and $U \to X$ a surjective \'etale morphism. Set $R = U \times_X U$ so that $X = U/R$ as usual. By Properties of Spaces, Proposition \ref{spaces-properties-proposition-quasi-coherent} we see that $\QCoh(\mathcal{O}_X) = \QCoh(U, R, s, t, c)$. Hence we reduce to showing the corresponding thing for $\QCoh(U, R, s, t, c)$. Thus the result follows from the more general Groupoids, Lemma \ref{groupoids-lemma-colimit-coherent}.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Write $f_*\mathcal{F} = \colim \mathcal{H}_i$ with $\mathcal{H}_i$ a coherent $\mathcal{O}_Y$-module, see Lemma \ref{lemma-directed-colimit-coherent}. By Lemma \ref{lemma-coherent-Noetherian} the modules $\mathcal{H}_i$ are $\mathcal{O}_Y$-modules of finite presentation. Hence $f^*\mathcal{H}_i$ is an $\mathcal{O}_X$-module of finite presentation, see Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}. We claim the map $$ \colim f^*\mathcal{H}_i = f^*f_*\mathcal{F} \to \mathcal{F} $$ is surjective as $f$ is assumed affine, Namely, choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Set $U = X \times_Y V$. Then $U$ is a scheme, $f' : U \to V$ is affine, and $U \to X$ is surjective \'etale. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules} we see that $f'_*(\mathcal{F}|_U) = f_*\mathcal{F}|_V$ and similarly for pullbacks. Thus the restriction of $f^*f_*\mathcal{F} \to \mathcal{F}$ to $U$ is the map $$ f^*f_*\mathcal{F}|_U = (f')^*(f_*\mathcal{F})|_V) = (f')^*f'_*(\mathcal{F}|_U) \to \mathcal{F}|_U $$ which is surjective as $f'$ is an affine morphism of schemes. Hence the claim holds. \medskip\noindent We conclude that every quasi-coherent module on $X$ is a quotient of a filtered colimit of finitely presented modules. In particular, we see that $\mathcal{F}$ is a cokernel of a map $$ \colim_{j \in J} \mathcal{G}_j \longrightarrow \colim_{i \in I} \mathcal{H}_i $$ with $\mathcal{G}_j$ and $\mathcal{H}_i$ finitely presented. Note that for every $j \in I$ there exist $i \in I$ and a morphism $\alpha : \mathcal{G}_j \to \mathcal{H}_i$ such that $$ \xymatrix{ \mathcal{G}_j \ar[r]_\alpha \ar[d] & \mathcal{H}_i \ar[d] \\ \colim_{j \in J} \mathcal{G}_j \ar[r] & \colim_{i \in I} \mathcal{H}_i } $$ commutes, see Lemma \ref{lemma-finite-presentation-quasi-compact-colimit}. In this situation $\Coker(\alpha)$ is a finitely presented $\mathcal{O}_X$-module which comes endowed with a map $\Coker(\alpha) \to \mathcal{F}$. Consider the set $K$ of triples $(i, j, \alpha)$ as above. We say that $(i, j, \alpha) \leq (i', j', \alpha')$ if and only if $i \leq i'$, $j \leq j'$, and the diagram $$ \xymatrix{ \mathcal{G}_j \ar[r]_\alpha \ar[d] & \mathcal{H}_i \ar[d] \\ \mathcal{G}_{j'} \ar[r]^{\alpha'} & \mathcal{H}_{i'} } $$ commutes. It follows from the above that $K$ is a directed partially ordered set, $$ \mathcal{F} = \colim_{(i, j, \alpha) \in K} \Coker(\alpha), $$ and we win.
The equality $p_*(M \otimes_A \mathcal{O}_X) = \widetilde{M}$ follows from the equality $\Gamma(X, M \otimes_A \mathcal{O}_X) = M$ as $p_*(M \otimes_A \mathcal{O}_X)$ is a quasi-coherent module on $\Spec(A)$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}. Observe that $\Gamma(X, \bigoplus_{i \in I} \mathcal{O}_X) = \bigoplus_{i \in I} A$ by Lemma \ref{lemma-colimits}. Hence the lemma holds for free modules. Choose a short exact sequence $F_1 \to F_0 \to M$ where $F_0, F_1$ are free $A$-modules. Since $H^1(X, -)$ is zero the global sections functor is right exact. Moreover the pullback $p^*$ is right exact as well. Hence we see that $$ \Gamma(X, F_1 \otimes_A \mathcal{O}_X) \to \Gamma(X, F_0 \otimes_A \mathcal{O}_X) \to \Gamma(X, M \otimes_A \mathcal{O}_X) \to 0 $$ is exact. The result follows.
Part (1) follows from Lemma \ref{lemma-affine-vanishing-higher-direct-images} and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-Leray}). Let $A \to A'$ be as in (2). Then $X' \to X$ is affine because affine morphisms are preserved under base change (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-affine}) and the fact that a morphism of affine schemes is affine. The equality $\Gamma(X', \mathcal{O}_{X'}) = A'$ follows as $(X' \to X)_*\mathcal{O}_{X'} = A' \otimes_A \mathcal{O}_X$ by Lemma \ref{lemma-affine-base-change} and thus $$ \Gamma(X', \mathcal{O}_{X'}) = \Gamma(X, (X' \to X)_*\mathcal{O}_{X'}) = \Gamma(X, A' \otimes_A \mathcal{O}_X) = A' $$ by Lemma \ref{lemma-vanishing-compute}.
We may and do endow $Z_0$, $Z_1$ with the reduced induced subspace structure (Properties of Spaces, Definition \ref{spaces-properties-definition-reduced-induced-space}) and we denote $i_0 : Z_0 \to X$ and $i_1 : Z_1 \to X$ the corresponding closed immersions. Since $Z_0 \cap Z_1 = \emptyset$ we see that the canonical map of quasi-coherent $\mathcal{O}_X$-modules $$ \mathcal{O}_X \longrightarrow i_{0, *}\mathcal{O}_{Z_0} \oplus i_{1, *}\mathcal{O}_{Z_1} $$ is surjective (look at stalks at geometric points). Since $H^1(X, -)$ is zero on the kernel of this map the induced map of global sections is surjective. Thus we can find $a \in A$ which maps to the global section $(0, 1)$ of the right hand side.
Let $A \to k$ be a ring homomorphism where $k$ is a field. It suffices to show that $\Spec(k) \times_{\Spec(A)} X$ has at most one point (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-injective-local}). Using Lemma \ref{lemma-vanishing-base-change} we may assume that $A$ is a field and we have to show that $|X|$ has at most one point. \medskip\noindent Let's think of $X$ as an algebraic space over $\Spec(k)$ and let's use the notation $X(K)$ to denote $K$-valued points of $X$ for any extension $k \subset K$, see Morphisms of Spaces, Section \ref{spaces-morphisms-section-points-fields}. If $k \subset K$ is an algebraically closed field extension of large transcendence degree, then we see that $X(K) \to |X|$ is surjective, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-large-enough}. Hence, after replacing $k$ by $K$, we see that it suffices to prove that $X(k)$ is a singleton (in the case $A = k)$. \medskip\noindent Let $x, x' \in X(k)$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-algebraic-residue-field-extension-closed-point} we see that $x$ and $x'$ are closed points of $|X|$. Hence $x$ and $x'$ map to distinct points of $\Spec(k)$ if $x \not = x'$ by Lemma \ref{lemma-vanishing-separate-closed}. We conclude that $x = x'$ as desired.
By Decent Spaces, Lemma \ref{decent-spaces-lemma-there-is-a-scheme-integral-over} we can find a scheme $Y$ and a surjective integral morphism $Y \to X$. Since an integral morphism is affine, we can apply Lemma \ref{lemma-vanishing-base-change} to see that $H^1(Y, \mathcal{G}) = 0$ for every quasi-coherent $\mathcal{O}_Y$-module $\mathcal{G}$. Since $Y \to X$ is quasi-compact and $X$ is quasi-compact, we see that $Y$ is quasi-compact. Since $Y$ is a scheme, we may apply Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-compact-h1-zero-covering} to see that $Y$ is affine. Hence $Y$ is separated. Note that an integral morphism is affine and universally closed, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-integral-universally-closed}. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-universally-closed-separated} we see that $X$ is a separated algebraic space.
Observe that $f'$ is the base change of $f$ and hence is finite and surjective by Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-surjective} and \ref{spaces-morphisms-lemma-base-change-integral}. Note that $Y$, $Z$, and $Z'$ are locally Noetherian by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian} (and the fact that closed immersions and finite morphisms are of finite type). By Lemma \ref{lemma-finite-pushforward-coherent} we see that $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module. The support of $\mathcal{G}$ is closed in $|Z|$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}. Hence if the support of $\mathcal{G}$ is not equal to $|Z|$, then after replacing $X$ by an open subspace we may assume $\mathcal{G} = 0$ but $Z \not = \emptyset$. This would mean that $f'_*\mathcal{O}_{Z'} = 0$. In particular the section $1 \in \Gamma(Z', \mathcal{O}_{Z'}) = \Gamma(Z, f'_*\mathcal{O}_{Z'})$ would be zero which would imply $Z' = \emptyset$ is the empty algebraic space. This is impossible as $Z' \to Z$ is surjective.
The notation means the following. Since $f^{-1}$ is an exact functor we see that $f^{-1}\mathcal{I}$ is a sheaf of ideals of $f^{-1}\mathcal{O}_X$. Via the map $f^\sharp : f^{-1}\mathcal{O}_X \to \mathcal{O}_Y$ on $Y_\etale$ this acts on $\mathcal{F}$. Then $f^{-1}\mathcal{I}\mathcal{F}$ is the subsheaf generated by sums of local sections of the form $as$ where $a$ is a local section of $f^{-1}\mathcal{I}$ and $s$ is a local section of $\mathcal{F}$. It is a quasi-coherent $\mathcal{O}_Y$-submodule of $\mathcal{F}$ because it is also the image of a natural map $f^*\mathcal{I} \otimes_{\mathcal{O}_Y} \mathcal{F} \to \mathcal{F}$. \medskip\noindent Having said this the proof is straightforward. Namely, the question is \'etale local on $X$ and hence we may assume $X$ is an affine scheme. In this case the result is a consequence of the corresponding result for schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-morphism-projection-ideal}.
We will prove that under the assumptions of the lemma for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. This implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ by Lemmas \ref{lemma-directed-colimit-coherent} and \ref{lemma-colimits}. Then it follows that $X$ is affine from Proposition \ref{proposition-vanishing-affine}. \medskip\noindent Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow H^1(X, \mathcal{F}) = 0. $$ We are going to apply Lemma \ref{lemma-property-higher-rank-cohomological}. Thus we have to verify (1), (2) and (3) of that lemma for $\mathcal{P}$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves. Property (2) follows since $H^1(X, -)$ is an additive functor. To see (3) let $i : Z \to X$ be a reduced closed subspace with $|Z|$ irreducible. Let $i' : Z' \to Y$ and $f' : Z' \to Z$ be as in Lemma \ref{lemma-finite-morphism-Noetherian} and set $\mathcal{G} = f'_*\mathcal{O}_{Z'}$. We claim that $\mathcal{G}$ satisfies properties (3)(a) and (3)(b) of Lemma \ref{lemma-property-higher-rank-cohomological} which will finish the proof. Property (3)(a) we have seen in Lemma \ref{lemma-finite-morphism-Noetherian}. To see (3)(b) let $\mathcal{I}$ be a nonzero quasi-coherent sheaf of ideals on $Z$. Denote $\mathcal{I}' \subset \mathcal{O}_{Z'}$ the quasi-coherent ideal $(f')^{-1}\mathcal{I} \mathcal{O}_{Z'}$, i.e., the image of $(f')^*\mathcal{I} \to \mathcal{O}_{Z'}$. By Lemma \ref{lemma-affine-morphism-projection-ideal} we have $f_*\mathcal{I}' = \mathcal{I} \mathcal{G}$. We claim the common value $\mathcal{G}' = \mathcal{I} \mathcal{G} = f'_*\mathcal{I}'$ satisfies the condition expressed in (3)(b). First, it is clear that the support of $\mathcal{G}/\mathcal{G}'$ is contained in the support of $\mathcal{O}_Z/\mathcal{I}$ which is a proper subspace of $|Z|$ as $\mathcal{I}$ is a nonzero ideal sheaf on the reduced and irreducible algebraic space $Z$. The morphism $f'$ is affine, hence $R^1f'_*\mathcal{I}' = 0$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images}. As $Z'$ is affine (as a closed subscheme of an affine scheme) we have $H^1(Z', \mathcal{I}') = 0$. Hence the Leray spectral sequence (in the form Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}) implies that $H^1(Z, f'_*\mathcal{I}') = 0$. Since $i : Z \to X$ is affine we conclude that $R^1i_*f'_*\mathcal{I}' = 0$ hence $H^1(X, i_*f'_*\mathcal{I}') = 0$ by Leray again. In other words, we have $H^1(X, i_*\mathcal{G}') = 0$ as desired.
To prove $f$ is separated, we may work \'etale locally on $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-separated-local}). Choose an affine scheme $U$ and an \'etale morphism $U \to X \times_Y X$. Set $V = X \times_{\Delta, X \times_Y X} U$ which is quasi-compact because $f$ is quasi-separated. Consider a commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & U } $$ We can interpret the composition $\Spec(A) \to U \to X \times_Y X$ as a pair of morphisms $a, b : \Spec(A) \to X$ agreeing as morphisms into $Y$ and equal when restricted to $\Spec(K)$. Hence our assumption (3) guarantees $a = b$ and we find the dotted arrow in the diagram. By Limits, Lemma \ref{limits-lemma-Noetherian-dvr-valuative-proper} we conclude that $V \to U$ is proper. In other words, $\Delta$ is proper. Since $\Delta$ is a monomorphism, we find that $\Delta$ is a closed immersion (\'Etale Morphisms, Lemma \ref{etale-lemma-characterize-closed-immersion}) as desired.
It suffices to prove $f$ is universally closed because $f$ is separated by Lemma \ref{lemma-check-separated-dvr}. To do this we may work \'etale locally on $Y$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local}). Hence we may assume $Y = \Spec(A)$ is a Noetherian affine scheme. Choose $X' \to X$ as in the weak form of Chow's lemma (Lemma \ref{lemma-weak-chow}). We claim that $X' \to \Spec(A)$ is universally closed. The claim implies the lemma by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}. To prove this, according to Limits, Lemma \ref{limits-lemma-check-universally-closed-Noetherian} it suffices to prove that in every solid commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X' \ar[r] & X \ar[d] \\ \Spec(A) \ar[rr] \ar@{-->}[ru]^a \ar@{-->}[rru]_b & & Y } $$ where $A$ is a dvr with fraction field $K$ we can find the dotted arrow $a$. By assumption we can find the dotted arrow $b$. Then the morphism $X' \times_{X, b} \Spec(A) \to \Spec(A)$ is a proper morphism of schemes and by the valuative criterion for morphisms of schemes we can lift $b$ to the desired morphism $a$.
Checking whether $R^pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes d})$ is zero can be done \'etale locally on $Y$, see Equation (\ref{equation-representable-higher-direct-image}). Hence we may assume $Y$ is the spectrum of a Noetherian ring. In this case $X$ is a scheme and the result follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-kill-by-twisting}.
This is just the affine case of Lemma \ref{lemma-proper-pushforward-coherent}. Namely, by Lemma \ref{lemma-higher-direct-image} we know that $R^if_*\mathcal{F}$ is a quasi-coherent sheaf. Hence it is the quasi-coherent sheaf associated to the $A$-module $\Gamma(\Spec(A), R^if_*\mathcal{F}) = H^i(X, \mathcal{F})$. The equality holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray} and vanishing of higher cohomology groups of quasi-coherent modules on affine schemes (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}). By Lemma \ref{lemma-coherent-Noetherian} we see $R^if_*\mathcal{F}$ is a coherent sheaf if and only if $H^i(X, \mathcal{F})$ is an $A$-module of finite type. Hence Lemma \ref{lemma-proper-pushforward-coherent} gives us the conclusion.
To prove this we consider the fibre product diagram $$ \xymatrix{ X' = \Spec(B) \times_{\Spec(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) } $$ Note that $f'$ is a proper morphism, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}. Also, $B$ is a finitely generated $A$-algebra, and hence Noetherian (Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence}). This implies that $X'$ is a Noetherian algebraic space (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-noetherian}). Note that $X'$ is the relative spectrum of the quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{B}$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-equivalence-algebras}. Since $\mathcal{F}$ is a quasi-coherent $\mathcal{B}$-module we see that there is a unique quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$ such that $\pi_*\mathcal{F}' = \mathcal{F}$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-equivalence-modules}. Since $\mathcal{F}$ is finite type as a $\mathcal{B}$-module we conclude that $\mathcal{F}'$ is a finite type $\mathcal{O}_{X'}$-module (details omitted). In other words, $\mathcal{F}'$ is a coherent $\mathcal{O}_{X'}$-module (Lemma \ref{lemma-coherent-Noetherian}). Since the morphism $\pi : X' \to X$ is affine we have $$ H^p(X, \mathcal{F}) = H^p(X', \mathcal{F}') $$ by Lemma \ref{lemma-affine-vanishing-higher-direct-images} and Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}. Thus the lemma follows from Lemma \ref{lemma-proper-over-affine-cohomology-finite}.
The implication (1) $\Rightarrow$ (2) follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-vanshing-gives-ample}. The implication (2) $\Rightarrow$ (3) is trivial. The implication (3) $\Rightarrow$ (1) is Lemma \ref{lemma-Noetherian-h1-zero-invertible}.
By Lemma \ref{lemma-cohomology-powers-ideal-times-F} we can find $d_1, \ldots, d_t \geq 0$, and $x_i \in H^p(X, I^{d_i}\mathcal{F})$ such that $\bigoplus_{n \geq 0} H^p(X, I^n\mathcal{F})$ is generated by $x_1, \ldots, x_t$ over $B = \bigoplus_{n \geq 0} I^n$. Take $c = \max\{d_i\}$. It is clear that (1) holds. For (2) let $b = \max(0, n - c)$. Consider the commutative diagram of $A$-modules $$ \xymatrix{ I^{n + m - c - b} \otimes I^b \otimes H^p(X, I^c\mathcal{F}) \ar[r] \ar[d] & I^{n + m - c} \otimes H^p(X, I^c\mathcal{F}) \ar[r] & H^p(X, I^{n + m}\mathcal{F}) \ar[d] \\ I^{n + m - c - b} \otimes H^p(X, I^n\mathcal{F}) \ar[rr] & & H^p(X, I^n\mathcal{F}) } $$ By part (1) of the lemma the composition of the horizontal arrows is surjective if $n + m \geq c$. On the other hand, it is clear that $n + m - c - b \geq m - c$. Hence part (2).
Let $c_1 = \max\{c_p, c_{p + 1}\}$, where $c_p, c_{p +1}$ are the integers found in Lemma \ref{lemma-cohomology-powers-ideal-application} for $H^p$ and $H^{p + 1}$. We will use this constant in the proofs of (1), (2) and (3). \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 our choice of $c_1$ we see that this is contained in $I^{n - c_1}H^p(X, \mathcal{F})$ for $n \geq c_1$. \medskip\noindent Note that part (3) implies part (2) by definition of the Mittag-Leffler condition. \medskip\noindent Let us prove part (3). Fix an $n$ throughout the rest of the proof. 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 $$ \xymatrix{ H^p(X, I^n\mathcal{F}) \ar[r] & H^p(X, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}/I^n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^n\mathcal{F}) \\ H^p(X, I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^a } $$ If $m \geq c_1$ we see that the image of $a$ is contained in $I^{m - c_1} H^{p + 1}(X, I^n\mathcal{F})$. By the Artin-Rees lemma (see Algebra, Lemma \ref{algebra-lemma-map-AR}) there exists an integer $c_3(n)$ such that $$ I^N H^{p + 1}(X, I^n\mathcal{F}) \cap \Im(\delta) \subset \delta\left(I^{N - c_3(n)}H^p(X, \mathcal{F}/I^n\mathcal{F})\right) $$ for all $N \geq c_3(n)$. As $H^p(X, \mathcal{F}/I^n\mathcal{F})$ is annihilated by $I^n$, we see that if $m \geq c_3(n) + c_1 + n$, then $$ \Im(H^p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) = \Im(H^p(X, \mathcal{F}) \to H^p(X, \mathcal{F}/I^n\mathcal{F})) $$ In other words, part (3) holds with $c_2(n) = c_3(n) + c_1 + n$.