hcju/stacksqa huggingface dataset:
Because $U \to X$ is separated the diagonal $U \to U \times_X U$ is a closed immersion. Since $U \to X$ is \'etale the diagonal $U \to U \times_X U$ is an open immersion, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-unramified} and \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}. Hence $W_p$ is an open and closed subspace of $U^{p + 1} = U \times_X \ldots \times_X U$. The action of $S_{p + 1}$ on $W_p$ is free as we've thrown out the fixed points of the action. By Lemma \ref{lemma-product-is-tensor-product} we see that $$ (f_!\underline{\mathbf{Z}})^{\otimes p + 1} = f^{p + 1}_!\underline{\mathbf{Z}} = (W_p \to X)_!\underline{\mathbf{Z}} \oplus Rest $$ where $f^{p + 1} : U^{p + 1} \to X$ is the structure morphism. Looking at stalks over a geometric point $\overline{x}$ of $X$ we see that $$ \left( \bigoplus\nolimits_{\overline{u} \mapsto \overline{x}} \mathbf{Z} \right)^{\otimes p + 1} \longrightarrow (W_p \to X)_!\underline{\mathbf{Z}}_{\overline{x}} $$ is the quotient whose kernel is generated by all tensors $1_{\overline{u}_0} \otimes \ldots \otimes 1_{\overline{u}_p}$ where $\overline{u}_i = \overline{u}_j$ for some $i \not = j$. Thus the quotient map $$ (f_!\underline{\mathbf{Z}})^{\otimes p + 1} \longrightarrow \wedge^{p + 1}f_!\underline{\mathbf{Z}} $$ factors through $(W_p \to X)_!\underline{\mathbf{Z}}$, i.e., we get $$ (f_!\underline{\mathbf{Z}})^{\otimes p + 1} \longrightarrow (W_p \to X)_!\underline{\mathbf{Z}} \longrightarrow \wedge^{p + 1}f_!\underline{\mathbf{Z}} $$ This already proves that $\Hom(K^p, \mathcal{F})$ is (functorially) a subgroup of $$ \Hom((W_p \to X)_!\underline{\mathbf{Z}}, \mathcal{F}) = \mathcal{F}(W_p) $$ To identify it with the $S_{p + 1}$-anti-invariants we have to prove that the surjection $(W_p \to X)_!\underline{\mathbf{Z}} \to \wedge^{p + 1}f_!\underline{\mathbf{Z}}$ is the maximal $S_{p + 1}$-anti-invariant quotient. In other words, we have to show that $\wedge^{p + 1}f_!\underline{\mathbf{Z}}$ is the quotient of $(W_p \to X)_!\underline{\mathbf{Z}}$ by the subsheaf generated by the local sections $s - \text{sign}(\sigma)\sigma(s)$ where $s$ is a local section of $(W_p \to X)_!\underline{\mathbf{Z}}$. This can be checked on the stalks, where it is clear.
Note that since the action of $S_{p + 1}$ on $W_p$ is over $X$ we do obtain a morphism $U_p \to X$. Since $W_p \to X$ is \'etale and since $W_p \to U_p$ is surjective \'etale, it follows that also $U_p \to X$ is \'etale, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-etale-local}. Therefore an injective object of $\textit{Ab}(X_\etale)$ restricts to an injective object of $\textit{Ab}(U_{p, \etale})$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}. Moreover, the functor $\mathcal{G} \mapsto \mathcal{G} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$ is an auto-equivalence of $\textit{Ab}(U_p)$, whence transforms injective objects into injective objects and is exact (because $\underline{\mathbf{Z}}(\chi_p)$ is an invertible $\underline{\mathbf{Z}}$-module). Thus given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ in $\textit{Ab}(X_\etale)$ the complex $$ \Gamma(U_p, \mathcal{I}^0|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \Gamma(U_p, \mathcal{I}^1|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \Gamma(U_p, \mathcal{I}^2|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \to \ldots $$ computes $H^*(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$. On the other hand, by Lemma \ref{lemma-twist} it is equal to the complex of $S_{p + 1}$-anti-invariants in $$ \Gamma(W_p, \mathcal{I}^0) \to \Gamma(W_p, \mathcal{I}^1) \to \Gamma(W_p, \mathcal{I}^2) \to \ldots $$ which by Lemma \ref{lemma-compute} is equal to the complex $$ \Hom(K^p, \mathcal{I}^0) \to \Hom(K^p, \mathcal{I}^1) \to \Hom(K^p, \mathcal{I}^2) \to \ldots $$ which computes $\Ext^*_{\textit{Ab}(X_\etale)}(K^p, \mathcal{F})$. Putting everything together we win.
Since $X$ is quasi-compact we can find a surjective \'etale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. By Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres} the fibres of $f$ are universally bounded, hence we can find $d$. We have $U_p = W_p/S_{p + 1}$ and $W_p \subset U \times_X \ldots \times_X U$ is open and closed. Since $X$ is quasi-separated the schemes $W_p$ are quasi-compact, hence $U_p$ is quasi-compact. Since $U$ is separated, the schemes $W_p$ are separated, hence $U_p$ is separated by (the absolute version of) Spaces, Lemma \ref{spaces-lemma-quotient-finite-separated}. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover} we can find the morphisms $V_p \to W_p$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres} we can find the integers $d_p$. \medskip\noindent At this point the proof uses the spectral sequence $$ E_1^{p, q} = H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p)) \Rightarrow H^{p + q}(X, \mathcal{F}) $$ see Lemma \ref{lemma-alternating-spectral-sequence}. By definition of the integer $d$ we see that $U_p = 0$ for $p \geq d$. By Proposition \ref{proposition-vanishing} and Lemma \ref{lemma-quasi-coherent-twist} we see that $H^q(U_p, \mathcal{F}|_{U_p} \otimes_\mathbf{Z} \underline{\mathbf{Z}}(\chi_p))$ is zero for $q \geq d_p$ for $p = 0, \ldots, d$. Whence the lemma.
Let $V \to Y$ be a surjective \'etale morphism where $V$ is an affine scheme, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. Suppose we prove the result for the base change $f_V : V \times_Y X \to V$. Then the result holds for $f$ with $n(X \to Y) = n(X_V \to V)$. Namely, if $Y' \to Y$ and $\mathcal{F}'$ are as in the lemma, then $R^if'_*\mathcal{F}'|_{V \times_Y Y'}$ is equal to $R^if'_{V, *}\mathcal{F}'|_{X'_V}$ where $f'_V : X'_V = V \times_Y Y' \times_Y X \to V \times_Y Y' = Y'_V$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}. Thus we may assume that $Y$ is an affine scheme. \medskip\noindent Moreover, to prove the vanishing for all $Y' \to Y$ and $\mathcal{F}'$ it suffices to do so when $Y'$ is an affine scheme. In this case, $R^if'_*\mathcal{F}'$ is quasi-coherent by Lemma \ref{lemma-higher-direct-image}. Hence it suffices to prove that $H^i(X', \mathcal{F}') = 0$, because $H^i(X', \mathcal{F}') = H^0(Y', R^if'_*\mathcal{F}')$ by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray} and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition \ref{proposition-vanishing}). \medskip\noindent Choose $U \to X$, $d$, $V_p \to U_p$ and $d_p$ as in Lemma \ref{lemma-vanishing-quasi-separated}. For any affine scheme $Y'$ and morphism $Y' \to Y$ denote $X' = Y' \times_Y X$, $U' = Y' \times_Y U$, $V'_p = Y' \times_Y V_p$. Then $U' \to X'$, $d' = d$, $V'_p \to U'_p$ and $d'_p = d$ is a collection of choices as in Lemma \ref{lemma-vanishing-quasi-separated} for the algebraic space $X'$ (details omitted). Hence we see that $H^i(X', \mathcal{F}') = 0$ for $i \geq \max(p + d_p)$ and we win.
Observe that for any abelian sheaf $\mathcal{G}$ on $Z_\etale$ we have $$ \Hom_Z(\mathcal{G}, \mathcal{H}_Z(\mathcal{F})) = \Hom_X(i_*\mathcal{G}, \mathcal{F}) $$ because after all any section of $i_*\mathcal{G}$ has support in $Z$. Since $i_*$ is exact (Lemma \ref{lemma-finite-higher-direct-image-zero}) and as $\mathcal{I}$ is injective on $X_\etale$ we conclude that $\mathcal{H}_Z(\mathcal{I})$ is injective on $Z_\etale$.
Recall that $i^{-1}$ and $i_*$ is an adjoint pair of exact functors such that $i^{-1}i_*$ is isomorphic to the identify functor on abelian sheaves. See Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-push-pull}. Thus $i_* : D(Z_\etale) \to D_Z(X_\etale)$ is fully faithful and $i^{-1}$ determines a left inverse. On the other hand, suppose that $K$ is an object of $D_Z(X_\etale)$ and consider the adjunction map $K \to i_*i^{-1}K$. Using exactness of $i_*$ and $i^{-1}$ this induces the adjunction maps $H^n(K) \to i_*i^{-1}H^n(K)$ on cohomology sheaves. Since these cohomology sheaves are supported on $Z$ we see these adjunction maps are isomorphisms and we conclude that $D(Z_\etale) \to D_Z(X_\etale)$ is an equivalence. \medskip\noindent To finish the proof we have to show that $R\mathcal{H}_Z(K) = i^{-1}K$ if $K$ is an object of $D_Z(X_\etale)$. To do this we can use that $K = i_*i^{-1}K$ as we've just proved this is the case. Then we can choose a K-injective representative $\mathcal{I}^\bullet$ for $i^{-1}K$. Since $i_*$ is the right adjoint to the exact functor $i^{-1}$, the complex $i_*\mathcal{I}^\bullet$ is K-injective (Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}). We see that $R\mathcal{H}_Z(K)$ is computed by $\mathcal{H}_Z(i_*\mathcal{I}^\bullet) = \mathcal{I}^\bullet$ as desired.
The morphism $g'$ is flat by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-flat}. Note that flatness of $g$ and $g'$ is equivalent to flatness of the morphisms of small \'etale ringed sites, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-morphism-sites}. Hence we can apply Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-base-change-map-flat-case} to obtain a base change map $$ g^*R^pf_*\mathcal{F} \longrightarrow R^pf'_*\mathcal{F}' $$ To prove this map is an isomorphism we can work locally in the \'etale topology on $Y'$. Thus we may assume that $Y$ and $Y'$ are affine schemes. Say $Y = \Spec(A)$ and $Y' = \Spec(B)$. In this case we are really trying to show that the map $$ H^p(X, \mathcal{F}) \otimes_A B \longrightarrow H^p(X_B, \mathcal{F}_B) $$ is an isomorphism where $X_B = \Spec(B) \times_{\Spec(A)} X$ and $\mathcal{F}_B$ is the pullback of $\mathcal{F}$ to $X_B$. In other words, it suffices to prove (2). \medskip\noindent Fix $A \to B$ a flat ring map and let $X$ be a quasi-compact and quasi-separated algebraic space over $A$. Note that $g' : X_B \to X$ is affine as a base change of $\Spec(B) \to \Spec(A)$. Hence the higher direct images $R^i(g')_*\mathcal{F}_B$ are zero by Lemma \ref{lemma-affine-vanishing-higher-direct-images}. Thus $H^p(X_B, \mathcal{F}_B) = H^p(X, g'_*\mathcal{F}_B)$, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-apply-Leray}. Moreover, we have $$ g'_*\mathcal{F}_B = \mathcal{F} \otimes_{\underline{A}} \underline{B} $$ where $\underline{A}$, $\underline{B}$ denotes the constant sheaf of rings with value $A$, $B$. Namely, it is clear that there is a map from right to left. For any affine scheme $U$ \'etale over $X$ we have \begin{align*} g'_*\mathcal{F}_B(U) & = \mathcal{F}_B(\Spec(B) \times_{\Spec(A)} U) \\ & = \Gamma(\Spec(B) \times_{\Spec(A)} U, (\Spec(B) \times_{\Spec(A)} U \to U)^*\mathcal{F}|_U) \\ & = B \otimes_A \mathcal{F}(U) \end{align*} hence the map is an isomorphism. Write $B = \colim M_i$ as a filtered colimit of finite free $A$-modules $M_i$ using Lazard's theorem, see Algebra, Theorem \ref{algebra-theorem-lazard}. We deduce that \begin{align*} H^p(X, g'_*\mathcal{F}_B) & = H^p(X, \mathcal{F} \otimes_{\underline{A}} \underline{B}) \\ & = H^p(X, \colim_i \mathcal{F} \otimes_{\underline{A}} \underline{M_i}) \\ & = \colim_i H^p(X, \mathcal{F} \otimes_{\underline{A}} \underline{M_i}) \\ & = \colim_i H^p(X, \mathcal{F}) \otimes_A M_i \\ & = H^p(X, \mathcal{F}) \otimes_A \colim_i M_i \\ & = H^p(X, \mathcal{F}) \otimes_A B \end{align*} The first equality because $g'_*\mathcal{F}_B = \mathcal{F} \otimes_{\underline{A}} \underline{B}$ as seen above. The second because $\otimes$ commutes with colimits. The third equality because cohomology on $X$ commutes with colimits (see Lemma \ref{lemma-colimits}). The fourth equality because $M_i$ is finite free (i.e., because cohomology commutes with finite direct sums). The fifth because $\otimes$ commutes with colimits. The sixth by choice of our system.
To be sure, if $X$ is a locally Noetherian algebraic space and $U \to X$ is an \'etale morphism, then $U$ is locally Noetherian, see Properties of Spaces, Section \ref{spaces-properties-section-types-properties}. The lemma then follows from the points (1) -- (5) made in Properties of Spaces, Section \ref{spaces-properties-section-properties-modules} and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}.
Choose a scheme $U$ and a surjective \'etale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (\ref{spaces-properties-equation-restrict-modules}). By Lemma \ref{lemma-coherent-Noetherian} we can check whether an $\mathcal{O}_X$-module $\mathcal{F}$ is coherent by checking whether $f^*\mathcal{F}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-abelian-Noetherian}.
Choose a scheme $U$ and a surjective \'etale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (\ref{spaces-properties-equation-restrict-modules}). By Lemma \ref{lemma-coherent-Noetherian} we can check whether an $\mathcal{O}_X$-module $\mathcal{G}$ is coherent by checking whether $f^*\mathcal{H}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.
Let $\varphi : U \to X$ be an \'etale morphism where $U$ is a scheme and let $u \in U$ be a point mapping to $x$. By Properties of Spaces, Lemmas \ref{spaces-properties-lemma-stalk-quasi-coherent} and \ref{spaces-properties-lemma-describe-etale-local-ring} as well as More on Algebra, Lemma \ref{more-algebra-lemma-dumb-properties-henselization} we see that $\varphi_{\overline{x}}$ is injective, surjective, or bijective if and only if $\varphi_u : \varphi^*\mathcal{F}_u \to \varphi^*\mathcal{G}_u$ has the corresponding property. Thus we can apply the schemes version of this lemma to see that (after possibly shrinking $U$) the map $\varphi^*\mathcal{F} \to \varphi^*\mathcal{G}$ is injective, surjective, or an isomorphism. Let $X' \subset X$ be the open subspace corresponding to $|\varphi|(|U|) \subset |X|$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. Since $\{U \to X'\}$ is a covering for the \'etale topology, we conclude that $\varphi|_{X'}$ is injective, surjective, or an isomorphism as desired. Finally, observe that (1) follows from (2) by looking at the map $\mathcal{F} \to 0$.
The statement of the lemma makes sense as a coherent module is in particular of finite type. Moreover, as $Z \to X$ is a closed immersion it is locally of finite type and hence $Z$ is locally Noetherian, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-immersion-locally-finite-type} and \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. Finally, as $\mathcal{G}$ is of finite type it is a coherent $\mathcal{O}_Z$-module by Lemma \ref{lemma-coherent-Noetherian}
The functor is fully faithful by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence}. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module annihilated by $\mathcal{I}$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence} we can write $\mathcal{F} = i_*\mathcal{G}$ for some quasi-coherent sheaf $\mathcal{G}$ on $Z$. To check that $\mathcal{G}$ is coherent we can work \'etale locally (Lemma \ref{lemma-coherent-Noetherian}). Choosing an \'etale covering by a scheme we conclude that $\mathcal{G}$ is coherent by the case of schemes (Cohomology of Schemes, Lemma \ref{coherent-lemma-i-star-equivalence}). Hence the functor is fully faithful and the proof is done.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Then $V \times_Y X \to V$ is a finite morphism of locally Noetherian schemes. By (\ref{equation-representable-higher-direct-image}) we reduce to the case of schemes which is Cohomology of Schemes, Lemma \ref{coherent-lemma-finite-pushforward-coherent}.
Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). If $\mathcal{F}_n|_U = \mathcal{F}_{n + 1}|_U = \ldots$ then $\mathcal{F}_n = \mathcal{F}_{n + 1} = \ldots$. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-acc-coherent}.
Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). Note that $\mathcal{I}^n\mathcal{F}|_U = 0$ if and only if $\mathcal{I}^n\mathcal{F} = 0$ and similarly for the condition on the support. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-power-ideal-kills-sheaf}.
Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$ (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). The equality of the lemma holds if and only if it holds after restricting to $U$. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma \ref{coherent-lemma-Artin-Rees}.
Let $W$ be an affine scheme and let $W \to X$ be a surjective \'etale morphism (see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}). Set $R = W \times_X W$. Then $W$ and $R$ are Noetherian schemes, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}. Hence the result hold for the restrictions of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{I}$, $U$, $Z$ to $W$ and $R$ by Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}. It follows formally that the result holds over $X$.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the sheaf of ideals defining the reduced induced closed subspace structure on $Z$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-closed-subspace}. 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 geometric point $\overline{x}$ of $Z' \setminus Z$ we have $\mathcal{I}_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$ and hence $\mathcal{G}_{n, \overline{x}} = 0$. Thus we see that $\text{Supp}(\mathcal{G}_n) \subset Z$. Note that $X \setminus Z'$ is a Noetherian algebraic space. 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.
By assumption there exists a coherent $\mathcal{O}_Z$-module $\mathcal{G}$ with support $Z$ and $\mathcal{F} \cong i_*\mathcal{G}$, see Lemma \ref{lemma-coherent-support-closed}. Hence it suffices to prove the lemma for the case $Z = X$ and $i = \text{id}$. \medskip\noindent By Properties of Spaces, Proposition \ref{spaces-properties-proposition-locally-quasi-separated-open-dense-scheme} there exists a dense open subspace $U \subset X$ which is a scheme. Note that $U$ is a Noetherian integral scheme. After shrinking $U$ we may assume that $\mathcal{F}|_U \cong \mathcal{O}_U^{\oplus r}$ (for example by Cohomology of Schemes, Lemma \ref{coherent-lemma-prepare-filter-irreducible} or by a direct algebra argument). Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals whose associated closed subspace is the complement of $U$ in $X$ (see for example Properties of Spaces, Section \ref{spaces-properties-section-reduced}).  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 isomorphism 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: namely, if $\sigma$ is a nonzero section of $\mathcal{I}^{\oplus r}$ over a scheme $W$ \'etale over $X$, then because $X$ hence $W$ is reduced the support of $\sigma$ contains a nonempty open of $W$. But the kernel of $(\mathcal{I}^n)^{\oplus r} \to \mathcal{F}$ is zero over a dense open, hence $\sigma$ cannot be a section of the kernel.
Consider the collection $$ \mathcal{T} = \left\{ \begin{matrix} T \subset |X| \text{ closed such that there exists a coherent sheaf } \mathcal{F} \\ \text{ with } \text{Supp}(\mathcal{F}) = T \text{ for which the lemma is wrong} \end{matrix} \right\} $$ We are trying to show that $\mathcal{T}$ is empty. If not, then because $|X|$ is Noetherian (Properties of Spaces, Lemma \ref{spaces-properties-lemma-Noetherian-topology}) we can choose a minimal element $T \in \mathcal{T}$. This means that there exists a coherent sheaf $\mathcal{F}$ on $X$ whose support is $T$ and for which the lemma does not hold. Clearly $T \not = \emptyset$ since the only sheaf whose support is empty is the zero sheaf for which the lemma does hold (with $m = 0$). \medskip\noindent If $T$ is not irreducible, then we can write $T = Z_1 \cup Z_2$ with $Z_1, Z_2$ closed and strictly smaller than $T$. 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 $T$ 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 $T$ is irreducible. \medskip\noindent Suppose $T$ is irreducible. Let $\mathcal{J}$ be the sheaf of ideals defining the reduced induced closed subspace structure on $T$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-reduced-closed-subspace}. 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 $$ 0 = \mathcal{I}^n\mathcal{F} \subset \mathcal{I}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{I}\mathcal{F} \subset \mathcal{F} $$ 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 Assume $T$ is irreducible and $\mathcal{J}\mathcal{F} = 0$ where $\mathcal{J}$ is as above. Then the scheme theoretic support of $\mathcal{F}$ is $T$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-i-star-equivalence}. Hence 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 the support of $\mathcal{Q}$ is a proper closed subset of $T$. Hence we see that $\mathcal{Q}$ has a filtration of the desired type by minimality of $T$. But then clearly $\mathcal{F}$ does too, which is our final contradiction.