hcju/stacksqa huggingface dataset:
(If $f$ is separated as well, please see Lemma \ref{lemma-base-change-complex}.) The assumptions imply in particular that $X$ is quasi-compact and quasi-separated as a scheme. Let $\mathcal{B}$ be the set of affine opens of $X$. By Hypercoverings, Lemma \ref{hypercovering-lemma-quasi-separated-quasi-compact-hypercovering} we can find a hypercovering $K = (I, \{U_i\})$ such that each $I_n$ is finite and each $U_i$ is an affine open of $X$. By Hypercoverings, Lemma \ref{hypercovering-lemma-cech-spectral-sequence} there is a spectral sequence with $E_2$-page $$ E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(X, \mathcal{F})$. Note that $\check{H}^p(K, \underline{H}^q(\mathcal{F}))$ is the $p$th cohomology group of the complex $$ \prod\nolimits_{i \in I_0} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_1} H^q(U_i, \mathcal{F}) \to \prod\nolimits_{i \in I_2} H^q(U_i, \mathcal{F}) \to \ldots $$ Since each $U_i$ is affine we see that this is zero unless $q = 0$ in which case we obtain $$ \prod\nolimits_{i \in I_0} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_1} \mathcal{F}(U_i) \to \prod\nolimits_{i \in I_2} \mathcal{F}(U_i) \to \ldots $$ Thus we conclude that $R\Gamma(X, \mathcal{F})$ is computed by this complex. \medskip\noindent For any $n$ and $i \in I_n$ denote $f_i : U_i \to S$ the restriction of $f$ to $U_i$. As $S$ is separated and $U_i$ is affine this morphism is affine. Consider the complex of quasi-coherent sheaves $$ \mathcal{K}^\bullet = ( \prod\nolimits_{i \in I_0} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_1} f_{i, *}\mathcal{F}|_{U_i} \to \prod\nolimits_{i \in I_2} f_{i, *}\mathcal{F}|_{U_i} \to \ldots ) $$ on $S$. As in Hypercoverings, Lemma \ref{hypercovering-lemma-cech-spectral-sequence} we obtain a map $\mathcal{K}^\bullet \to Rf_*\mathcal{F}$ in $D(\mathcal{O}_S)$ by choosing an injective resolution of $\mathcal{F}$ (details omitted). Consider any affine scheme $V$ and a morphism $g : V \to S$. Then the base change $X_V$ has a hypercovering $K_V = (I, \{U_{i, V}\})$ obtained by base change. Moreover, $g^*f_{i, *}\mathcal{F} = f_{i, V, *}(g')^*\mathcal{F}|_{U_{i, V}}$. Thus the arguments above prove that $\Gamma(V, g^*\mathcal{K}^\bullet)$ computes $R\Gamma(X_V, (g')^*\mathcal{F})$. This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on $S'$.
The implications (1) $\Rightarrow$ (2) and (1) $\Rightarrow$ (3) hold in general, see Modules, Lemma \ref{modules-lemma-coherent-finite-presentation}. If $\mathcal{F}$ is finitely presented then $\mathcal{F}$ is quasi-coherent, see Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-coherent}. Hence also (3) $\Rightarrow$ (2). \medskip\noindent Assume $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_X$-module. By Properties, Lemma \ref{properties-lemma-finite-type-module} we see that on any affine open $\Spec(A) = U \subset X$ we have $\mathcal{F}|_U = \widetilde M$ with $M$ a finite $A$-module. Since $A$ is Noetherian we see that $M$ has a finite resolution $$ A^{\oplus m} \to A^{\oplus n} \to M \to 0. $$ Hence $\mathcal{F}$ is of finite presentation by Properties, Lemma \ref{properties-lemma-finite-presentation-module}. In other words (2) $\Rightarrow$ (3). \medskip\noindent By Modules, Lemma \ref{modules-lemma-coherent-structure-sheaf} it suffices to show that $\mathcal{O}_X$ is coherent in order to show that (3) implies (1). Thus we have to show: given any open $U \subset X$ and any finite collection of sections $f_i \in \mathcal{O}_X(U)$, $i = 1, \ldots, n$ the kernel of the map $\bigoplus_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U$ is of finite type. Since being of finite type is a local property it suffices to check this in a neighbourhood of any $x \in U$. Thus we may assume $U = \Spec(A)$ is affine. In this case $f_1, \ldots, f_n \in A$ are elements of $A$. Since $A$ is Noetherian, see Properties, Lemma \ref{properties-lemma-locally-Noetherian} the kernel $K$ of the map $\bigoplus_{i = 1, \ldots, n} A \to A$ is a finite $A$-module. See for example Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}. As the functor\ $\widetilde{ }$\ is exact, see Schemes, Lemma \ref{schemes-lemma-spec-sheaves} we get an exact sequence $$ \widetilde K \to \bigoplus\nolimits_{i = 1, \ldots, n} \mathcal{O}_U \to \mathcal{O}_U $$ and by Properties, Lemma \ref{properties-lemma-finite-type-module} again we see that $\widetilde K$ is of finite type. We conclude that (1), (2) and (3) are all equivalent. \medskip\noindent It follows from Properties, Lemma \ref{properties-lemma-finite-type-module} that (2) implies (4). It is trivial that (4) implies (5). The discussion in Schemes, Section \ref{schemes-section-quasi-coherent} show that (5) implies that $\mathcal{F}$ is quasi-coherent and it is clear that (5) implies that $\mathcal{F}$ is of finite type. Hence (5) implies (2) and we win.
We may assume that $X$ is affine, say $X = \Spec(A)$. Properties, Lemma \ref{properties-lemma-locally-Noetherian} implies that $A$ is Noetherian. Lemma \ref{lemma-coherent-Noetherian} turns this into algebra. The algebraic counter part of the lemma is that a quotient, or a submodule of a finite $A$-module is a finite $A$-module, see for example Algebra, Lemma \ref{algebra-lemma-Noetherian-basic}.
Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{F}$ and let $\mathcal{G}$ be the finite type quasi-coherent sheaf on $Z$ such that $i_*\mathcal{G} \cong \mathcal{F}$. Since $Z = \text{Supp}(\mathcal{F})$ we see that the support is closed. The scheme $Z$ is locally Noetherian by Morphisms, Lemmas \ref{morphisms-lemma-immersion-locally-finite-type} and \ref{morphisms-lemma-finite-type-noetherian}. Finally, $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module by Lemma \ref{lemma-coherent-Noetherian}
The functor is fully faithful by Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence}. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module annihilated by $\mathcal{I}$. By Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. By Modules, Lemma \ref{modules-lemma-i-star-reflects-finite-type} we see that $\mathcal{G}$ is of finite type. Hence $\mathcal{G}$ is coherent by Lemma \ref{lemma-coherent-Noetherian}. Thus the functor is also essentially surjective as desired.
The higher direct images vanish by Lemma \ref{lemma-relative-affine-vanishing} and because a finite morphism is affine (by definition). Note that the assumptions imply that also $X$ is locally Noetherian (see Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}) and hence the statement makes sense. Let $\Spec(A) = V \subset Y$ be an affine open subset. By Morphisms, Definition \ref{morphisms-definition-integral} we see that $f^{-1}(V) = \Spec(B)$ with $A \to B$ finite. Lemma \ref{lemma-coherent-Noetherian} turns the statement of the lemma into the following algebra fact: If $M$ is a finite $B$-module, then $M$ is also finite viewed as a $A$-module, see Algebra, Lemma \ref{algebra-lemma-finite-module-over-finite-extension}.
By Lemma \ref{lemma-coherent-support-closed} we see that $\mathcal{F} = i_*\mathcal{G}$ where $i : Z \to X$ is the inclusion of the scheme theoretic support of $\mathcal{F}$ and where $\mathcal{G}$ is a coherent $\mathcal{O}_Z$-module. Since the dimension of $Z$ is $0$, we see $Z$ is a disjoint union of affines (Properties, Lemma \ref{properties-lemma-locally-Noetherian-dimension-0}). Hence $\mathcal{G}$ is globally generated and the higher cohomology groups of $\mathcal{G}$ are zero (Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}). Hence $\mathcal{F} = i_*\mathcal{G}$ is globally generated. Since the cohomologies of $\mathcal{F}$ and $\mathcal{G}$ agree (Lemma \ref{lemma-relative-affine-cohomology} applies as a closed immersion is affine) we conclude that the higher cohomology groups of $\mathcal{F}$ are zero.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the coherent sheaf of ideals cutting out the reduced induced scheme structure on $X \setminus U$. If $X$ is Noetherian, then Lemma \ref{lemma-homs-over-open} tells us that we can take $\mathcal{F}' = \mathcal{I}^n\mathcal{F}$ for some $n$. The general case will follow from this using Zorn's lemma. \medskip\noindent Consider the set of triples $(U', \mathcal{F}', \varphi')$ where $U \subset U' \subset X$ is open, $\mathcal{F}' \subset \mathcal{F}|_{U'}$ is a coherent subsheaf agreeing with $\mathcal{F}$ over $U$, and $\varphi' : \mathcal{F}' \to \mathcal{G}|_{U'}$ restricts to $\varphi$ over $U$. We say $(U'', \mathcal{F}'', \varphi'') \geq (U', \mathcal{F}', \varphi')$ if and only if $U'' \supset U'$, $\mathcal{F}''|_{U'} = \mathcal{F}'$, and $\varphi''|_{U'} = \varphi'$. It is clear that if we have a totally ordered collection of triples $(U_i, \mathcal{F}_i, \varphi_i)$, then we can glue the $\mathcal{F}_i$ to a subsheaf $\mathcal{F}'$ of $\mathcal{F}$ over $U' = \bigcup U_i$ and extend $\varphi$ to a map $\varphi' : \mathcal{F}' \to \mathcal{G}|_{U'}$. Hence any totally ordered subset of triples has an upper bound. Finally, suppose that $(U', \mathcal{F}', \varphi')$ is any triple but $U' \not = X$. Then we can choose an affine open $W \subset X$ which is not contained in $U'$. By the result of the first paragraph we can extend the subsheaf $\mathcal{F}'|_{W \cap U'}$ and the restriction $\varphi'|_{W \cap U'}$ to some subsheaf $\mathcal{F}'' \subset \mathcal{F}|_W$ and map $\varphi'' : \mathcal{F}'' \to \mathcal{G}|_W$. Of course the agreement between $(\mathcal{F}', \varphi')$ and $(\mathcal{F}'', \varphi'')$ over $W \cap U'$ exactly means that we can extend this to a triple $(U' \cup W, \mathcal{F}''', \varphi''')$. Hence any maximal triple $(U', \mathcal{F}', \varphi')$ (which exist by Zorn's lemma) must have $U' = X$ and the proof is complete.
Observe that $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is a coherent $\mathcal{O}_X$-module by Lemma \ref{lemma-tensor-hom-coherent}. Coherent modules are of finite presentation (Lemma \ref{lemma-coherent-Noetherian}) hence taking stalks commutes with taking $\SheafHom$ and $\Hom$, see Modules, Lemma \ref{modules-lemma-stalk-internal-hom}. Thus we reduce to the case of finite modules over local rings which is More on Algebra, Lemma \ref{more-algebra-lemma-hom-into-depth}.
Let $x \in X$. If (2) holds, then $\mathcal{F}_x$ is a free $\mathcal{O}_{X, x}$-module of rank $> 0$. Hence $\text{depth}(\mathcal{F}_x) = \dim(\mathcal{O}_{X, x})$ because a regular local ring is Cohen-Macaulay (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). Conversely, if (1) holds, then $\mathcal{F}_x$ is a maximal Cohen-Macaulay module over $\mathcal{O}_{X, x}$ (Algebra, Definition \ref{algebra-definition-maximal-CM}). Hence $\mathcal{F}_x$ is free by Algebra, Lemma \ref{algebra-lemma-regular-mcm-free}.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the sheaf of ideals defining the reduced induced closed subscheme structure on $Z$, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. Consider the subsheaves $\mathcal{G}'_n = \mathcal{I}^n\mathcal{F}$ and the quotients $\mathcal{G}_n = \mathcal{F}/\mathcal{I}^n\mathcal{F}$. For each $n$ we have a short exact sequence $$ 0 \to \mathcal{G}'_n \to \mathcal{F} \to \mathcal{G}_n \to 0 $$ For every point $x$ of $Z' \setminus Z$ we have $\mathcal{I}_x = \mathcal{O}_{X, x}$ and hence $\mathcal{G}_{n, x} = 0$. Thus we see that $\text{Supp}(\mathcal{G}_n) \subset Z$. Note that $X \setminus Z'$ is a Noetherian scheme. Hence by Lemma \ref{lemma-power-ideal-kills-sheaf} there exists an $n$ such that $\mathcal{G}'_n|_{X \setminus Z'} = \mathcal{I}^n\mathcal{F}|_{X \setminus Z'} = 0$. For such an $n$ we see that $\text{Supp}(\mathcal{G}'_n) \subset Z'$. Thus setting $\mathcal{G}' = \mathcal{G}'_n$ and $\mathcal{G} = \mathcal{G}_n$ works.
Let $\mathcal{J} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of local sections of $\mathcal{F}$ which are annihilated by $\mathcal{J}$. It is a quasi-coherent sheaf by Properties, Lemma \ref{properties-lemma-sections-annihilated-by-ideal}. Moreover, $\mathcal{F}'_\xi = \mathcal{F}_\xi$ because $\mathcal{J}_\xi = \mathfrak m_\xi$ and part (3) of Properties, Lemma \ref{properties-lemma-sections-annihilated-by-ideal}. By Lemma \ref{lemma-local-isomorphism} we see that $\mathcal{F}' \to \mathcal{F}$ induces an isomorphism in a neighbourhood of $\xi$. Hence we may replace $\mathcal{F}$ by $\mathcal{F}'$ and assume that $\mathcal{F}$ is annihilated by $\mathcal{J}$. \medskip\noindent Assume $\mathcal{J}\mathcal{F} = 0$. By Lemma \ref{lemma-i-star-equivalence} we can write $\mathcal{F} = i_*\mathcal{G}$ for some coherent sheaf $\mathcal{G}$ on $Z$. Suppose we can find a morphism $\mathcal{I}^{\oplus r} \to \mathcal{G}$ which is an isomorphism in a neighbourhood of the generic point $\xi$ of $Z$. Then applying $i_*$ (which is left exact) we get the result of the lemma. Hence we have reduced to the case $X = Z$. \medskip\noindent Suppose $Z = X$ is an integral Noetherian scheme with generic point $\xi$. Note that $\mathcal{O}_{X, \xi} = \kappa(\xi)$ is the function field of $X$ in this case. Since $\mathcal{F}_\xi$ is a finite $\mathcal{O}_\xi$-module we see that $r = \dim_{\kappa(\xi)} \mathcal{F}_\xi$ is finite. Hence the sheaves $\mathcal{O}_X^{\oplus r}$ and $\mathcal{F}$ have isomorphic stalks at $\xi$. By Lemma \ref{lemma-map-stalks-local-map} there exists a nonempty open $U \subset X$ and a morphism $\psi : \mathcal{O}_X^{\oplus r}|_U \to \mathcal{F}|_U$ which is an isomorphism at $\xi$, and hence an isomorphism in a neighbourhood of $\xi$ by Lemma \ref{lemma-local-isomorphism}. By Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} there exists a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ whose associated closed subscheme $Z \subset X$ is the complement of $U$. By Lemma \ref{lemma-homs-over-open} there exists an $n \geq 0$ and a morphism $\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) \to \mathcal{F}$ which recovers our $\psi$ over $U$. Since $\mathcal{I}^n(\mathcal{O}_X^{\oplus r}) = (\mathcal{I}^n)^{\oplus r}$ we get a map as in the lemma. It is injective because $X$ is integral and it is injective at the generic point of $X$ (easy proof omitted).
Consider the collection $$ \mathcal{T} = \left\{ \begin{matrix} Z \subset X \text{ closed such that there exists a coherent sheaf } \mathcal{F} \\ \text{ with } \text{Supp}(\mathcal{F}) = Z \text{ for which the lemma is wrong} \end{matrix} \right\} $$ We are trying to show that $\mathcal{T}$ is empty. If not, then because $X$ is Noetherian we can choose a minimal element $Z \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $Z$ and for which the lemma does not hold. Clearly $Z \not = \emptyset$ since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with $m = 0$). \medskip\noindent If $Z$ is not irreducible, then we can write $Z = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $Z$. Then we can apply Lemma \ref{lemma-prepare-filter-support} to get a short exact sequence of coherent sheaves $$ 0 \to \mathcal{G}_1 \to \mathcal{F} \to \mathcal{G}_2 \to 0 $$ with $\text{Supp}(\mathcal{G}_i) \subset Z_i$. By minimality of $Z$ each of $\mathcal{G}_i$ has a filtration as in the statement of the lemma. By considering the induced filtration on $\mathcal{F}$ we arrive at a contradiction. Hence we conclude that $Z$ is irreducible. \medskip\noindent Suppose $Z$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals cutting out the reduced induced closed subscheme structure of $Z$, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. By Lemma \ref{lemma-power-ideal-kills-sheaf} we see there exists an $n \geq 0$ such that $\mathcal{J}^n\mathcal{F} = 0$. Hence we obtain a filtration $$ 0 = \mathcal{J}^n\mathcal{F} \subset \mathcal{J}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{J}\mathcal{F} \subset \mathcal{F} $$ each of whose successive subquotients is annihilated by $\mathcal{J}$. Hence if each of these subquotients has a filtration as in the statement of the lemma then also $\mathcal{F}$ does. In other words we may assume that $\mathcal{J}$ does annihilate $\mathcal{F}$. \medskip\noindent In the case where $Z$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ we can apply Lemma \ref{lemma-prepare-filter-irreducible}. This gives a short exact sequence $$ 0 \to i_*(\mathcal{I}^{\oplus r}) \to \mathcal{F} \to \mathcal{Q} \to 0 $$ where $\mathcal{Q}$ is defined as the quotient. Since $\mathcal{Q}$ is zero in a neighbourhood of $\xi$ by the lemma just cited we see that the support of $\mathcal{Q}$ is strictly smaller than $Z$. Hence we see that $\mathcal{Q}$ has a filtration of the desired type by minimality of $Z$. But then clearly $\mathcal{F}$ does too, which is our final contradiction.
First note that if $\mathcal{F}$ is a coherent sheaf with a filtration $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F} $$ by coherent subsheaves such that each of $\mathcal{F}_i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$, then so does $\mathcal{F}$. This follows from the property (1) for $\mathcal{P}$. On the other hand, by Lemma \ref{lemma-coherent-filter} we can filter any $\mathcal{F}$ with successive subquotients as in (2). Hence the lemma follows.
First note that if $\mathcal{F}$ is a coherent sheaf with support contained in $Z_0$ with a filtration $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F} $$ by coherent subsheaves such that each of $\mathcal{F}_i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$, then so does $\mathcal{F}$. Or, if $\mathcal{F}$ has property $\mathcal{P}$ and all but one of the $\mathcal{F}_i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$ then so does the last one. This follows from assumption (1). \medskip\noindent As a first application we conclude that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. Namely, such a sheaf has a filtration (see Lemma \ref{lemma-coherent-filter}) whose subquotients have property $\mathcal{P}$ according to (2). \medskip\noindent Let $\mathcal{G}$ be as in (3). By Lemma \ref{lemma-prepare-filter-irreducible} there exist a sheaf of ideals $\mathcal{I}$ on $Z_0$, an integer $r \geq 1$, and a short exact sequence $$ 0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. By (3)(c) we see that $r = 1$. Since $\mathcal{Q}$ has property $\mathcal{P}$ too we conclude that $(Z_0 \to X)_*\mathcal{I}$ has property $\mathcal{P}$. \medskip\noindent Next, suppose that $\mathcal{I}' \not = 0$ is another quasi-coherent sheaf of ideals on $Z_0$. Then we can consider the intersection $\mathcal{I}'' = \mathcal{I}' \cap \mathcal{I}$ and we get two short exact sequences $$ 0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I} \to \mathcal{Q} \to 0 $$ and $$ 0 \to (Z_0 \to X)_*\mathcal{I}'' \to (Z_0 \to X)_*\mathcal{I}' \to \mathcal{Q}' \to 0. $$ Note that the support of the coherent sheaves $\mathcal{Q}$ and $\mathcal{Q}'$ are strictly contained in $Z_0$. Hence $\mathcal{Q}$ and $\mathcal{Q}'$ have property $\mathcal{P}$ (see above). Hence we conclude using (1) that $(Z_0 \to X)_*\mathcal{I}''$ and $(Z_0 \to X)_*\mathcal{I}'$ both have $\mathcal{P}$ as well. \medskip\noindent The final step of the proof is to note that any coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration (see Lemma \ref{lemma-coherent-filter} again) whose subquotients all have property $\mathcal{P}$ by what we just said.
According to Lemma \ref{lemma-property-initial} it suffices to show that for all integral closed subschemes $Z \subset X$ and all quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. If this fails, then since $X$ is Noetherian there is a minimal integral closed subscheme $Z_0 \subset X$ such that $\mathcal{P}$ fails for $(Z_0 \to X)_*\mathcal{I}_0$ for some quasi-coherent sheaf of ideals $\mathcal{I}_0 \subset \mathcal{O}_{Z_0}$, but $\mathcal{P}$ does hold for $(Z \to X)_*\mathcal{I}$ for all integral closed subschemes $Z \subset Z_0$, $Z \not = Z_0$ and quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_Z$. Since we have the existence of $\mathcal{G}$ for $Z_0$ by part (2), according to Lemma \ref{lemma-property-irreducible} this cannot happen.
Note that if $\mathcal{F}$ is a coherent sheaf with a filtration $$ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_m = \mathcal{F} $$ by coherent subsheaves such that each of $\mathcal{F}_i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$, then so does $\mathcal{F}$. This follows from assumption (1). \medskip\noindent As a first application we conclude that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. Namely, such a sheaf has a filtration (see Lemma \ref{lemma-coherent-filter}) whose subquotients have property $\mathcal{P}$ according to (3). \medskip\noindent Let us denote $i : Z_0 \to X$ the closed immersion. Consider a coherent sheaf $\mathcal{G}$ as in (4). By Lemma \ref{lemma-prepare-filter-irreducible} there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence $$ 0 \to i_*\mathcal{I}^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z_0$. Let $\mathcal{I}' \subset \mathcal{I}$ be any nonzero quasi-coherent sheaf of ideals on $Z_0$ contained in $\mathcal{I}$. Then we also get a short exact sequence $$ 0 \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{G} \to \mathcal{Q}' \to 0 $$ where $\mathcal{Q}'$ has support properly contained in $Z_0$. Let $\mathcal{J} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals cutting out the support of $\mathcal{Q}'$ (for example the ideal corresponding to the reduced induced closed subscheme structure on the support of $\mathcal{Q}'$). Then $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$. By Lemma \ref{lemma-power-ideal-kills-sheaf} we see that $\mathcal{J}^n\mathcal{Q}' = 0$ for some $n$. Hence $\mathcal{J}^n\mathcal{G} \subset i_*(\mathcal{I}')^{\oplus r}$. By assumption (4)(c) of the lemma we see there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}^n\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi$ for which property $\mathcal{P}$ holds. Hence we get a short exact sequence $$ 0 \to \mathcal{G}' \to i_*(\mathcal{I}')^{\oplus r} \to \mathcal{Q}'' \to 0 $$ where $\mathcal{Q}''$ has support properly contained in $Z_0$. Thus by our initial remarks and property (1) of the lemma we conclude that $i_*(\mathcal{I}')^{\oplus r}$ satisfies $\mathcal{P}$. Hence we see that $i_*\mathcal{I}'$ satisfies $\mathcal{P}$ by (2). Finally, for an arbitrary quasi-coherent sheaf of ideals $\mathcal{I}'' \subset \mathcal{O}_{Z_0}$ we can set $\mathcal{I}' = \mathcal{I}'' \cap \mathcal{I}$ and we get a short exact sequence $$ 0 \to i_*(\mathcal{I}') \to i_*(\mathcal{I}'') \to \mathcal{Q}''' \to 0 $$ where $\mathcal{Q}'''$ has support properly contained in $Z_0$. Hence we conclude that property $\mathcal{P}$ holds for $i_*\mathcal{I}''$ as well. \medskip\noindent The final step of the proof is to note that any coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z_0$ has a filtration (see Lemma \ref{lemma-coherent-filter} again) whose subquotients all have property $\mathcal{P}$ by what we just said.