hcju/stacksqa huggingface dataset:
Using Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex} we see that it suffices to prove the result for the groups $H^i(U, K \otimes^\mathbf{L} E)$. We will use the induction principle to prove the lemma, see Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent First we show that it holds when $U = \Spec(A)$ is affine. Namely, we can represent $K$ by a complex of $A$-modules $K^\bullet$ and $E$ by a finite complex of finite projective $A$-modules $P^\bullet$. See Lemmas \ref{lemma-affine-compare-bounded} and \ref{lemma-perfect-affine} and our definition of perfect complexes of $A$-modules (More on Algebra, Definition \ref{more-algebra-definition-perfect}). Then $(E \otimes^\mathbf{L} K)|_U$ is represented by the total complex associated to the double complex $P^\bullet \otimes_A K^\bullet$ (Lemma \ref{lemma-quasi-coherence-tensor-product}). Using induction on the length of the complex $P^\bullet$ (or using a suitable spectral sequence) we see that it suffices to show that $H^i(P^a \otimes_A K^\bullet)$ is countable for each $a$. Since $P^a$ is a direct summand of $A^{\oplus n}$ for some $n$ this follows from the assumption that the cohomology group $H^i(K^\bullet)$ is countable. \medskip\noindent To finish the proof it suffices to show: if $U = V \cup W$ and the result holds for $V$, $W$, and $V \cap W$, then the result holds for $U$. This is an immediate consquence of the Mayer-Vietoris sequence, see Cohomology, Lemma \ref{cohomology-lemma-unbounded-mayer-vietoris}.
By Lemma \ref{lemma-cohomology-base-change} we have $$ R\Gamma(U, Lf^*K) = R\Gamma(X, Rf_*\mathcal{O}_U \otimes^\mathbf{L} K) $$ functorially in $K$. Observe that $R\Gamma(X, -)$ commutes with homotopy colimits because it commutes with direct sums by Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}. Similarly, $- \otimes^\mathbf{L} K$ commutes with derived colimits because $- \otimes^\mathbf{L} K$ commutes with direct sums (because direct sums in $D(\mathcal{O}_X)$ are given by direct sums of representing complexes). Hence to prove (2) it suffices to write $Rf_*\mathcal{O}_U = \text{hocolim} E_n$ for a system of perfect objects $E_n$ of $D(\mathcal{O}_X)$. Once this is done we obtain (1) by setting $L_n = E_n^\vee$, see Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. \medskip\noindent Write $A = \colim A_i$ with $A_i$ of finite type over $\mathbf{Z}$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an $i$ and morphisms $U_i \to X_i \to \Spec(A_i)$ of finite presentation whose base change to $\Spec(A)$ recovers $U \to X \to \Spec(A)$. After increasing $i$ we may assume that $f_i : U_i \to X_i$ is flat, see Limits, Lemma \ref{limits-lemma-descend-flat-finite-presentation}. By Lemma \ref{lemma-compare-base-change} the derived pullback of $Rf_{i, *}\mathcal{O}_{U_i}$ by $g : X \to X_i$ is equal to $Rf_*\mathcal{O}_U$. Since $Lg^*$ commutes with derived colimits, it suffices to prove what we want for $f_i$. Hence we may assume that $U$ and $X$ are of finite type over $\mathbf{Z}$. \medskip\noindent Assume $f : U \to X$ is a morphism of schemes of finite type over $\mathbf{Z}$. To finish the proof we will show that $Rf_*\mathcal{O}_U$ is a homotopy colimit of perfect complexes. To see this we apply Lemma \ref{lemma-countable}. Thus it suffices to show that $R^if_*\mathcal{O}_U$ has countable sets of sections over affine opens. This follows from Lemma \ref{lemma-countable-cohomology} applied to the structure sheaf.
Choose a complex $M^\bullet$ of right $R$-modules representing $M$. Since $M$ is pseudo-coherent we have $H^i(M) = 0$ for large enough $i$. Let $m$ be the smallest index such that $H^m(M)$ is nonzero. Then $H^m(M)$ is a finite $A$-module by More on Algebra, Lemma \ref{more-algebra-lemma-finite-cohomology}. Thus we can choose a finite free $R$-module $F^m$ and a map $F^m \to M^m$ such that $F^m \to M^m \to M^{m + 1}$ is zero and such that $F^m \to H^m(M)$ is surjective. Picture: $$ \xymatrix{ & F^m \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{m - 1} \ar[r] & M^m \ar[r] & M^{m + 1} \ar[r] & \ldots } $$ By descending induction on $n \leq m$ we are going to construct finite free $R$-modules $F^i$ for $i \geq n$, differentials $d^i : F^i \to F^{i + 1}$ for $i \geq n$, maps $\alpha : F^i \to K^i$ compatible with differentials, such that (1) $H^i(\alpha)$ is an isomorphism for $i > n$ and surjective for $i = n$, and (2) $F^i = 0$ for $i > m$. Picture $$ \xymatrix{ & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \ar[r] & F^i \ar[d]^\alpha \ar[r] & 0 \ar[d] \ar[r] & \ldots \\ M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots \ar[r] & M^i \ar[r] & M^{i + 1} \ar[r] & \ldots } $$ The base case is $n = m$ which we've done above. Induction step. Let $C^\bullet$ be the cone on $\alpha$ (Derived Categories, Definition \ref{derived-definition-cone}). The long exact sequence of cohomology shows that $H^i(C^\bullet) = 0$ for $i \geq n$. Observe that $F^\bullet$ is pseudo-coherent as a complex of $A$-modules because $R$ is finite free as an $A$-module. Hence by More on Algebra, Lemma \ref{more-algebra-lemma-cone-pseudo-coherent} we see that $C^\bullet$ is $(n - 1)$-pseudo-coherent as a complex of $A$-modules. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-cohomology} we see that $H^{n - 1}(C^\bullet)$ is a finite $A$-module. Choose a finite free $R$-module $F^{n - 1}$ and a map $\beta : F^{n - 1} \to C^{n - 1}$ such that the composition $F^{n - 1} \to C^{n - 1} \to C^n$ is zero and such that $F^{n - 1}$ surjects onto $H^{n - 1}(C^\bullet)$. Since $C^{n - 1} = M^{n - 1} \oplus F^n$ we can write $\beta = (\alpha^{n - 1}, -d^{n - 1})$. The vanishing of the composition $F^{n - 1} \to C^{n - 1} \to C^n$ implies these maps fit into a morphism of complexes $$ \xymatrix{ & F^{n - 1} \ar[d]^{\alpha^{n - 1}} \ar[r]_{d^{n - 1}} & F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\ \ldots \ar[r] & M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots } $$ Moreover, these maps define a morphism of distinguished triangles $$ \xymatrix{ (F^n \to \ldots) \ar[r] \ar[d] & (F^{n - 1} \to \ldots) \ar[r] \ar[d] & F^{n - 1} \ar[r] \ar[d]_\beta & (F^n \to \ldots)[1] \ar[d] \\ (F^n \to \ldots) \ar[r] & M^\bullet \ar[r] & C^\bullet \ar[r] & (F^n \to \ldots)[1] } $$ Hence our choice of $\beta$ implies that the map of complexes $(F^{n - 1} \to \ldots) \to M^\bullet$ induces an isomorphism on cohomology in degrees $\geq n$ and a surjection in degree $n - 1$. This finishes the proof of the lemma.
Recall that $$ R\Hom_{\mathbf{P}^n_A}(E, K) = R\Gamma(\mathbf{P}^n_A, R\SheafHom_{\mathcal{O}_{\mathbf{P}^n_A}}(E, K)) $$ by definition, see Cohomology, Section \ref{cohomology-section-global-RHom}. Thus parts (4) and (3) are equivalent by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. \medskip\noindent Since every perfect complex is pseudo-coherent, it is clear that (2) implies (3). \medskip\noindent Assume (1) holds. Then $E \otimes^\mathbf{L} K$ is pseudo-coherent for every pseudo-coherent $E$, see Cohomology, Lemma \ref{cohomology-lemma-tensor-pseudo-coherent}. By Lemma \ref{lemma-flat-proper-pseudo-coherent-direct-image-general} the direct image of such a pseudo-coherent complex is pseudo-coherent and we see that (2) is true. \medskip\noindent Part (3) implies (5) because we can take $E = \mathcal{O}_{\mathbf{P}^n_A}(d)$ for $d = 0, 1, \ldots, n$. \medskip\noindent To finish the proof we have to show that (5) implies (1). Let $P$ be as in (\ref{equation-generator-Pn}) and $R$ as in (\ref{equation-algebra-for-Pn}). By Lemma \ref{lemma-Pn-module-category} we have an equivalence $$ - \otimes^\mathbf{L}_R P : D(R) \longrightarrow D_\QCoh(\mathcal{O}_{\mathbf{P}^n_A}) $$ Let $M \in D(R)$ be an object such that $M \otimes^\mathbf{L} P = K$. By Differential Graded Algebra, Lemma \ref{dga-lemma-upgrade-tensor-with-complex-derived} there is an isomorphism $$ R\Hom(R, M) = R\Hom_{\mathbf{P}^n_A}(P, K) $$ in $D(A)$. Arguing as above we obtain $$ R\Hom_{\mathbf{P}^n_A}(P, K) = R\Gamma(\mathbf{P}^n_A, R\SheafHom_{\mathcal{O}_{\mathbf{P}^n_A}}(E, K)) = R\Gamma(\mathbf{P}^n_A, P^\vee \otimes^\mathbf{L}_{\mathcal{O}_{\mathbf{P}^n_A}} K). $$ Using that $P^\vee$ is the direct sum of $\mathcal{O}_{\mathbf{P}^n_A}(d)$ for $d = 0, 1, \ldots, n$ and (5) we conclude $R\Hom(R, M)$ is pseudo-coherent as a complex of $A$-modules. Of course $M = R\Hom(R, M)$ in $D(A)$. Thus $M$ is pseudo-coherent as a complex of $A$-modules. By Lemma \ref{lemma-pseudo-coherent-over-algebra} we may represent $M$ by a bounded above complex $F^\bullet$ of finite free $R$-modules. Then $F^\bullet = \bigcup_{p \geq 0} \sigma_{\geq p}F^\bullet$ is a filtration which shows that $F^\bullet$ is a differential graded $R$-module with property (P), see Differential Graded Algebra, Section \ref{dga-section-P-resolutions}. Hence $K = M \otimes^\mathbf{L}_R P$ is represented by $F^\bullet \otimes_R P$ (follows from the construction of the derived tensor functor, see for example the proof of Differential Graded Algebra, Lemma \ref{dga-lemma-tensor-with-complex-derived}). Since $F^\bullet \otimes_R P$ is a bounded above complex whose terms are direct sums of copies of $P$ we conclude that the lemma is true.
Being pseudo-coherent is a local property and ``locally having finite tor dimension'' is a local property. Hence this lemma immediately reduces to the statement: if $X$ and $S$ are affine, then $E$ is $S$-perfect if and only if $K = R\Gamma(X, E)$ is $\mathcal{O}_S(S)$-perfect. Say $X = \Spec(A)$, $S = \Spec(R)$ and $E$ corresponds to $K \in D(A)$, i.e., $K = R\Gamma(X, E)$, see Lemma \ref{lemma-affine-compare-bounded}. \medskip\noindent Observe that $K$ is $R$-perfect if and only if $K$ is pseudo-coherent and has finite tor dimension as a complex of $R$-modules (More on Algebra, Definition \ref{more-algebra-definition-relatively-perfect}). By Lemma \ref{lemma-pseudo-coherent-affine} we see that $E$ is pseudo-coherent if and only if $K$ is pseudo-coherent. By Lemma \ref{lemma-tor-dimension-rel-affine} we see that $E$ has finite tor dimension over $f^{-1}\mathcal{O}_S$ if and only if $K$ has finite tor dimension as a complex of $R$-modules.
First proof: reduce to the affine case using Lemma \ref{lemma-affine-locally-rel-perfect} and then apply More on Algebra, Lemma \ref{more-algebra-lemma-base-change-relatively-perfect}. \medskip\noindent Second proof: $L(g')^*K$ is pseudo-coherent by Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback} and the bounded tor dimension property follows from Lemma \ref{lemma-tor-independence-and-tor-amplitude}.
For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that $$ \colim_{i \geq 0} \Hom_{D(\mathcal{O}_{U_i})}(K_i|_{U_i}, L_i|_{U_i}) \longrightarrow \Hom_{D(\mathcal{O}_U)}(K|_U, L|_U) $$ is an isomorphism where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$. If $P$ holds for $U_0$, $V_0$ and $U_0 \cap V_0$, then it holds for $U_0 \cup V_0$ by Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris-hom}. \medskip\noindent Denote $\pi_0 : X_0 \to S_0$ the given morphism. Then we can first consider $U_0 = \pi_0^{-1}(W_0)$ with $W_0 \subset S_0$ quasi-compact open. By the induction principle of Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle} applied to quasi-compact opens of $S_0$ and the remark above, we find that it is enough to prove $P$ for $U_0 = \pi_0^{-1}(W_0)$ with $W_0$ affine. In other words, we have reduced to the case where $S_0$ is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of $X_0$, to reduce to the case where $X_0$ is affine as well. \medskip\noindent If $X_0$ and $S_0$ are affine, the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-colimit-relatively-perfect}. Namely, by Lemmas \ref{lemma-pseudo-coherent} and \ref{lemma-affine-compare-bounded} the statement is translated into computations of homs in the derived categories of modules. Then Lemma \ref{lemma-pseudo-coherent-affine} shows that the complex of modules corresponding to $K_0$ is pseudo-coherent. And Lemma \ref{lemma-tor-dimension-rel-affine} shows that the complex of modules corresponding to $L_0$ has finite tor dimension over $\mathcal{O}_{S_0}(S_0)$. Thus the assumptions of More on Algebra, Lemma \ref{more-algebra-lemma-colimit-relatively-perfect} are satisfied and we win.
For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that the functor $$ \colim_{i \geq 0} D_{S_i\text{-perfect}}(\mathcal{O}_{U_i}) \longrightarrow D_{S\text{-perfect}}(\mathcal{O}_U) $$ is an equivalence where $U = f_0^{-1}(U_0)$ and $U_i = f_{i0}^{-1}(U_0)$. We observe that we already know this functor is fully faithful by Lemma \ref{lemma-relative-descend-homomorphisms}. Thus it suffices to prove essential surjectivity. \medskip\noindent Suppose that $P$ holds for quasi-compact opens $U_0$, $V_0$ of $X_0$. We claim that $P$ holds for $U_0 \cup V_0$. We will use the notation $U_i = f_{i0}^{-1}U_0$, $U = f_0^{-1}U_0$, $V_i = f_{i0}^{-1}V_0$, and $V = f_0^{-1}V_0$ and we will abusively use the symbol $f_i$ for all the morphisms $U \to U_i$, $V \to V_i$, $U \cap V \to U_i \cap V_i$, and $U \cup V \to U_i \cup V_i$. Suppose $E$ is an $S$-perfect object of $D(\mathcal{O}_{U \cup V})$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $S_i$-perfect object $E_{U, i}$ on $U_i$, an $S_i$-perfect object $E_{V, i}$ on $V_i$, and isomorphisms $Lf_i^*E_{U, i} \to E|_U$ and $Lf_i^*E_{V, i} \to E|_V$. Let $$ a : E_{U, i} \to (Rf_{i, *}E)|_{U_i} \quad\text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_i} $$ the maps adjoint to the isomorphisms $Lf_i^*E_{U, i} \to E|_U$ and $Lf_i^*E_{V, i} \to E|_V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_i \cap V_i} \to E_{V, i}|_{U_i \cap V_i}$ which pulls back to the identifications  $$ Lf_i^*E_{U, i}|_{U \cap V} \to E|_{U \cap V} \to Lf_i^*E_{V, i}|_{U \cap V}. $$ Apply Cohomology, Lemma \ref{cohomology-lemma-glue} to get an object $E_i$ on $U_i \cup V_i$ and a map $d : E_i \to Rf_{i, *}E$ which restricts to the maps $a$ and $b$ over $U_i$ and $V_i$. Then it is clear that $E_i$ is $S_i$-perfect (because being relatively perfect is a local property) and that $d$ is adjoint to an isomorphism $Lf_i^*E_i \to E$. \medskip\noindent By exactly the same argument as used in the proof of Lemma \ref{lemma-relative-descend-homomorphisms} using the induction principle (Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}) we reduce to the case where both $X_0$ and $S_0$ are affine. (First work with opens in $S_0$ to reduce to $S_0$ affine, then work with opens in $X_0$ to reduce to $X_0$ affine.) In the affine case the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-colimit-relatively-perfect}. The translation into algebra is done by Lemma \ref{lemma-affine-locally-rel-perfect}.
The statement on base change is Lemma \ref{lemma-base-change-tensor} (with $\mathcal{G}^\bullet$ equal to $\mathcal{O}_X$ in degree $0$). Thus it suffices to show that $Rf_*E$ is a perfect object. We will reduce to the case where $S$ is Noetherian affine by a limit argument. \medskip\noindent The question is local on $S$, hence we may assume $S$ is affine. Say $S = \Spec(R)$. We write $R = \colim R_i$ as a filtered colimit of Noetherian rings $R_i$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} there exists an $i$ and a scheme $X_i$ of finite presentation over $R_i$ whose base change to $R$ is $X$. By Limits, Lemmas \ref{limits-lemma-eventually-proper} and \ref{limits-lemma-descend-flat-finite-presentation} we may assume $X_i$ is proper and flat over $R_i$. By Lemma \ref{lemma-descend-relatively-perfect} we may assume there exists a $R_i$-perfect object $E_i$ of $D(\mathcal{O}_{X_i})$ whose pullback to $X$ is $E$. Applying Lemma \ref{lemma-perfect-direct-image} to $X_i \to \Spec(R_i)$ and $E_i$ and using the base change property already shown we obtain the result.
Since everything is local we immediately reduce to the case that $X$ and $S$ are affine, see Lemma \ref{lemma-affine-locally-rel-perfect}. Say $X \to S$ corresponds to $\Spec(A) \to \Spec(R)$ and $E$ corresponds to $K$ in $D(A)$. If $s$ corresponds to the prime $\mathfrak p \subset R$, then $L(X_s \to X)^*E$ corresponds to $K \otimes_R^\mathbf{L} \kappa(\mathfrak p)$ as $R \to A$ is flat, see for example Lemma \ref{lemma-compare-base-change}. Thus we see that our lemma follows from the corresponding algebra result, see More on Algebra, Lemma \ref{more-algebra-lemma-bounded-on-fibres-relatively-perfect}.
Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{L}$ be an $f$-ample invertible module. Choose an affine open covering $Y = V_1 \cup \ldots \cup V_m$. Set $U_j = f^{-1}(V_j)$. By Properties, Proposition \ref{properties-proposition-characterize-ample} for each $j$ we know there exists finitely many maps $s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_j} \to \mathcal{F}|_{U_j}$ which are jointly surjective. Consider the quasi-coherent $\mathcal{O}_Y$-modules $$ \mathcal{H}_n = f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) $$ We may think of $s_{j, i}$ as a section over $V_j$ of the sheaf $\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally free $\mathcal{O}_Y$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}}$ such that $s_{j, i}$ is in the image. Then the corresponding maps $$ f^*\mathcal{E}_{i, j} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F} $$ are going to be jointly surjective and the lemma is proved. By Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} for each $i, j$ we can find a finite type quasi-coherent submodule $\mathcal{H}'_{i, j} \subset  \mathcal{H}_{-n_{j, i}}$ which contains the section $s_{i, j}$ over $V_j$. Thus using the resolution property of $Y$ to get surjections $\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$ and we conclude.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. The pushforward $f_*\mathcal{F}$ is quasi-coherent, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. The adjunction map $f^*f_*\mathcal{F} \to \mathcal{F}$ is surjective; this follows from Schemes, Lemma \ref{schemes-lemma-widetilde-pullback} after restricting to $f^{-1}(V)$ for $V \subset Y$ affine open. Write $f_*\mathcal{F} = \colim \mathcal{G}_i$ as a filtered colimit with $\mathcal{G}_i$ quasi-coherent $\mathcal{O}_Y$-modules of finite type, see Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type}. Then we see that $\colim f^*\mathcal{G}_i \to \mathcal{F}$ is surjective. Since $\mathcal{F}$ is of finite type and $X$ is quasi-compact, we conclude that for some $i$ the map $f^*\mathcal{G}_i \to \mathcal{F}$ is surjective (details omitted; look at generators on affines). Hence if $\mathcal{G}_i$ is a quotient of a finite locally free $\mathcal{O}_Y$-module, then $\mathcal{F}$ is a quotient of the pullback which is a finite locally free $\mathcal{O}_X$-module.
One direction of the lemma is trivial. For the other,  say $\mathcal{E}_j \to \mathcal{I}_j$ is a surjection with $\mathcal{E}_j$ finite locally free. In the next paragraph, we reduce to the Noetherian case; we suggest the reader skip it. \medskip\noindent The first observation is that $U_j \cap U_{j'}$ is quasi-compact as the complement of the zero scheme of the quasi-coherent finite type ideal $\mathcal{I}_{j'}|{U_j}$ on the affine scheme $U_j$, see Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. Hence $X$ is quasi-compact and quasi-separated, see Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated}. By Limits, Proposition \ref{limits-proposition-approximate} we can write $X = \lim X_i$ as the limit of a direct system of Noetherian schemes with affine transition morphisms. For each $j$ we can find an $i$ and a finite type quasi-coherent ideal sheaf $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_j}$ pulling back to $\mathcal{I}_j$, see Limits, Lemma \ref{limits-lemma-descend-invertible-modules}. Denoting $U_{i, j} \subset X_i$ the complementary opens, we may assume these are affine for all $i, j$, see Limits, Lemma \ref{limits-lemma-limit-affine}. Similarly, we may assume the maps $\mathcal{E}_j \to \mathcal{I}_j$ are the pullbacks of surjections $\mathcal{E}_{i, j} \to \mathcal{I}_{i, j}$ with $\mathcal{E}_{i, j}$ finite locally free on $X_i$, see Limits, Lemmas \ref{limits-lemma-descend-invertible-modules} and \ref{limits-lemma-descend-modules-finite-presentation}. Using this and Lemma \ref{lemma-resolution-property-goes-up-affine} we reduce to the case of a Noetherian scheme. \medskip\noindent Assume $X$ is Noetherian. For every coherent module $\mathcal{F}$ we can choose a finite list of sections $s_{jk} \in \mathcal{F}(U_j)$, $k = 1, \ldots, e_j$ which generate the restriction of $\mathcal{F}$ to $U_j$. By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open} we can extend $s_{jk}$ to a map $s'_{jk} : \mathcal{I}_i^{n_{jk}} \to \mathcal{F}$ for some $n_{jk} \geq 1$. Then we can consider the compositions $$ \mathcal{E}_j^{\otimes n_{jk}} \to \mathcal{I}_j^{n_{jk}} \to \mathcal{F} $$ to conclude.
Observe that $X$ is a finite disjoint union of integral schemes (Properties, Lemmas \ref{properties-lemma-regular-normal} and \ref{properties-lemma-normal-Noetherian}). Thus we may assume that $X$ is integral as well as Noetherian, regular, and having affine diagonal. Choose an affine open covering $X = U_1 \cup \ldots \cup U_m$. We may and do assume $U_j$ nonempty for all $j$. By More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD} the local rings of $X$ are UFDs and hence by Divisors, Lemma \ref{divisors-lemma-complement-open-affine-effective-cartier-divisor-bis} we can find an effective Cartier divisors $D_j \subset X$ whose complement is $U_j$. Then the ideal sheaf of $D_j$ is invertible, hence a finite locally free module and we conclude that $X$ has the resolution property by Lemma \ref{lemma-resolution-property-finite-number}.
If $X_i$ has the resolution property, then $X$ does by Lemma \ref{lemma-resolution-property-goes-up-affine}. Assume $X$ has the resolution property. Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_m$. For each $j$ choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}_j \subset \mathcal{O}_X$ such that $X \setminus V(\mathcal{I}_j) = U_j$, see Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}. Choose finite locally free $\mathcal{O}_X$-modules and surjections $\mathcal{E}_j \to \mathcal{I}_j$. By Limits, Lemmas \ref{limits-lemma-descend-invertible-modules} and \ref{limits-lemma-descend-modules-finite-presentation} we can find an $i$ and finite locally free $\mathcal{O}_{X_i}$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{O}_{X_i}$ whose base changes to $X$ recover the maps $\mathcal{E}_j \to \mathcal{I}_j$, $j = 1, \ldots, m$. Denote $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_i}$ the image of these maps. Set $U_{i, j} = X_i \setminus V(\mathcal{I}_{i, j})$. After increasing $i$ we may assume $U_{i, j}$ is affine, see Limits, Lemma \ref{limits-lemma-limit-affine}. Then we conclude that $X_i$ has the resolution property by Lemma \ref{lemma-resolution-property-finite-number}.