hcju/stacksqa huggingface dataset:
Consider a prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. Let $x \in X$ and $s = f(x) \in S$ be the corresponding points. Then $(f^{-1}\mathcal{O}_S)_x = \mathcal{O}_{S, s} = R_\mathfrak p$ and $E_x = M^\bullet_\mathfrak q$. Keeping this in mind we can see the equivalence as follows. \medskip\noindent If $M^\bullet$ has tor amplitude in $[a, b]$ as a complex of $R$-modules, then the same is true for the localization of $M^\bullet$ at any prime of $A$. Then we conclude by Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk} that $E$ has tor amplitude in $[a, b]$ as a complex of sheaves of $f^{-1}\mathcal{O}_S$-modules. Conversely, assume that $E$ has tor amplitude in $[a, b]$ as an object of $D(f^{-1}\mathcal{O}_S)$. We conclude (using the last cited lemma) that $M^\bullet_\mathfrak q$ has tor amplitude in $[a, b]$ as a complex of $R_\mathfrak p$-modules for every prime $\mathfrak q \subset A$ lying over $\mathfrak p \subset R$. By More on Algebra, Lemma \ref{more-algebra-lemma-tor-amplitude-localization} we find that $M^\bullet$ has tor amplitude in $[a, b]$ as a complex of $R$-modules. This finishes the proof of (1). \medskip\noindent Since $X$ is quasi-compact, if $E$ locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_S$-modules, then actually $E$ has tor amplitude in $[a, b]$ for some $a, b$ as a complex of $f^{-1}\mathcal{O}_S$-modules. Thus (2) follows from (1).
It is clear that (1) implies (2). Assume (2). Let $U \subset X$ be an affine open. As $X$ is quasi-separated the morphism $j : U \to X$ is quasi-compact and separated, hence $j_*$ transforms quasi-coherent modules into quasi-coherent modules (Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}). Thus the functor $\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)$ is essentially surjective. It follows that condition (2) implies the vanishing of $H^i(E|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \mathcal{G})$ for $i \not \in [a, b]$ for all quasi-coherent $\mathcal{O}_U$-modules $\mathcal{G}$. Write $U = \Spec(A)$ and let $M^\bullet$ be the complex of $A$-modules corresponding to $E|_U$ by Lemma \ref{lemma-affine-compare-bounded}. We have just shown that $M^\bullet \otimes_A^\mathbf{L} N$ has vanishing cohomology groups outside the range $[a, b]$, in other words $M^\bullet$ has tor amplitude in $[a, b]$. By Lemma \ref{lemma-tor-dimension-affine} we conclude that $E|_U$ has tor amplitude in $[a, b]$. This proves the lemma.
The construction of the internal hom in the derived category of $\mathcal{O}_X$ commutes with localization (see Cohomology, Section \ref{cohomology-section-internal-hom}). Hence to prove (1) and (2) we may replace $X$ by an affine open. By Lemmas \ref{lemma-affine-compare-bounded}, \ref{lemma-pseudo-coherent-affine}, and \ref{lemma-perfect-affine} in order to prove (1) and (2) it suffices to prove (3). \medskip\noindent Part (3) follows from the computation of the internal hom of Cohomology, Lemma \ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free} by representing $K$ by a bounded above (resp.\ finite) complex of finite projective $A$-modules and $L$ by a bounded below (resp.\ arbitrary) complex of $A$-modules. \medskip\noindent To prove (4) recall that on any ringed space the $n$th cohomology sheaf of $R\SheafHom(A, B)$ is the sheaf associated to the presheaf $$ U \mapsto \Hom_{D(U)}(A|_U, B|_U[n]) = \Ext^n_{D(\mathcal{O}_U)}(A|_U, B|_U) $$ See Cohomology, Section \ref{cohomology-section-internal-hom}. On the other hand, the restriction of $\widetilde{K}$ to a principal open $D(f)$ is the image of $K \otimes_A A_f$ and similarly for $L$. Hence (4) follows from the equivalence of categories of Lemma \ref{lemma-affine-compare-bounded}.
Lemma \ref{lemma-quasi-coherence-internal-hom} reduces cases (1) and (3) to the affine case which is treated in More on Algebra, Lemma \ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism}. (You also have to use Lemmas \ref{lemma-pseudo-coherent-affine}, \ref{lemma-perfect-affine}, and \ref{lemma-tor-dimension-affine} to do the translation into algebra.) If $K$ is perfect but no other assumptions are made, then we do not know that either side of the arrow is in $D_\QCoh(\mathcal{O}_X)$ but the result is still true because we can work locally and reduce to the case that $K$ is a finite complex of finite free modules in which case it is clear.
Observe that $\textit{Coh}(\mathcal{O}_X) \subset \QCoh(\mathcal{O}_X)$ is a Serre subcategory, see Homology, Definition \ref{homology-definition-serre-subcategory} and Lemma \ref{homology-lemma-characterize-serre-subcategory} and Cohomology of Schemes, Lemmas \ref{coherent-lemma-coherent-abelian-Noetherian} and \ref{coherent-lemma-coherent-Noetherian-quasi-coherent-sub-quotient}. On the other hand, if $\mathcal{G} \to \mathcal{F}$ is a surjection from a quasi-coherent $\mathcal{O}_X$-module to a coherent $\mathcal{O}_X$-module, then there exists a coherent submodule $\mathcal{G}' \subset \mathcal{G}$ which surjects onto $\mathcal{F}$. Namely, we can write $\mathcal{G}$ as the filtered union of its coherent submodules by Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type} and then one of these will do the job. Thus the lemma follows from Derived Categories, Lemma \ref{derived-lemma-fully-faithful-embedding}.
Consider the spectral sequence $$ R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E $$ see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. By assumption and Cohomology of Schemes, Lemma \ref{coherent-lemma-support-proper-over-base-pushforward} the sheaves $R^pf_*H^q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $Rf_*E \in D_{\textit{Coh}}(\mathcal{O}_S)$. Boundedness from below is trivial. Boundedness from above follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images} or from Lemma \ref{lemma-quasi-coherence-direct-image}.
It suffices to prove this when $X$ is the spectrum of a Noetherian ring $A$. By Lemma \ref{lemma-identify-pseudo-coherent-noetherian} we see that $K$ is pseudo-coherent. Then we can use Lemma \ref{lemma-quasi-coherence-internal-hom} to translate the problem into the following algebra problem: for $L \in D^+_{\textit{Coh}}(A)$ and $K$ in $D^-_{\textit{Coh}}(A)$, then $R\Hom_A(K, L)$ is in $D^+_{\textit{Coh}}(A)$. Since $L$ is bounded below and $K$ is bounded below there is a convergent spectral sequence $$ \Ext^p_A(K, H^q(L)) \Rightarrow \text{Ext}^{p + q}_A(K, L) $$ and there are convergent spectral sequences $$ \Ext^i_A(H^{-j}(K), H^q(L)) \Rightarrow \text{Ext}^{i + j}_A(K, H^q(L)) $$ See Injectives, Remarks \ref{injectives-remark-spectral-sequences-ext} and \ref{injectives-remark-spectral-sequences-ext-variant}. This finishes the proof as the modules $\Ext^p_A(M, N)$ are finite for finite $A$-modules $M$, $N$ by Algebra, Lemma \ref{algebra-lemma-ext-noetherian}.
Since $X$ is quasi-compact, each of these statements can be checked over the members of any open covering of $X$. Thus we may assume $E$ is represented by a bounded complex $\mathcal{E}^\bullet$ of finite free modules, see Cohomology, Lemma \ref{cohomology-lemma-perfect-on-locally-ringed}. In this case each of the statements is clear as both $R\SheafHom_{\mathcal{O}_X}(E, L)$ and $E \otimes_{\mathcal{O}_X}^\mathbf{L} L$ can be computed on the level of complexes using $\mathcal{E}^\bullet$, see Cohomology, Lemmas \ref{cohomology-lemma-Rhom-strictly-perfect} and \ref{cohomology-lemma-bounded-flat-K-flat}. Some details omitted.
Recall that $$ \Ext_{\mathcal{O}_X}^n(K, L) = H^n(X, R\SheafHom_{\mathcal{O}_X}(K, L)) = H^n(\Spec(A), Rf_*R\SheafHom_{\mathcal{O}_X}(K, L)) $$ see Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U} and Cohomology, Section \ref{cohomology-section-Leray}. Thus the result follows from Lemmas \ref{lemma-coherent-internal-hom} and \ref{lemma-direct-image-coherent-bdd-below}.
Pullback always preserves tor-amplitude, see Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-pullback}. We may check tor-amplitude in $[a, b]$ on stalks, see Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk}. A flat local ring homomorphism is faithfully flat by Algebra, Lemma \ref{algebra-lemma-local-flat-ff}. Thus the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-tor-amplitude}.
Pullback always preserves $m$-pseudo-coherence, see Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}. Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$. Let $U \subset X$ be an affine open. It suffices to prove that $E|_U$ is $m$-pseudo-coherent. Since $\{f_i : X_i \to X\}$ is an fpqc covering, we can find finitely many affine open $V_j \subset X_{a(j)}$ such that $f_{a(j)}(V_j) \subset U$ and $U = \bigcup f_{a(j)}(V_j)$. Set $V = \coprod V_i$. Thus we may replace $X$ by $U$ and $\{f_i : X_i \to X\}$ by $\{V \to U\}$ and assume that $X$ is affine and our covering is given by a single surjective flat morphism $\{f : Y \to X\}$ of affine schemes. In this case the result follows from More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-pseudo-coherent} via Lemmas \ref{lemma-affine-compare-bounded} and \ref{lemma-pseudo-coherent-affine}.
Pullback always preserves $m$-pseudo-coherence, see Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}. Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$. Let $U \subset X$ be an affine open. It suffices to prove that $E|_U$ is $m$-pseudo-coherent. Since $\{f_i : X_i \to X\}$ is an fppf covering, we can find finitely many affine open $V_j \subset X_{a(j)}$ such that $f_{a(j)}(V_j) \subset U$ and $U = \bigcup f_{a(j)}(V_j)$. Set $V = \coprod V_i$. Thus we may replace $X$ by $U$ and $\{f_i : X_i \to X\}$ by $\{V \to U\}$ and assume that $X$ is affine and our covering is given by a single surjective flat morphism $\{f : Y \to X\}$ of finite presentation. \medskip\noindent Since $f$ is flat the derived functor $Lf^*$ is just given by $f^*$ and $f^*$ is exact. Hence $H^i(Lf^*E) = f^*H^i(E)$. Since $Lf^*E$ is $m$-pseudo-coherent, we see that $Lf^*E \in D^-(\mathcal{O}_Y)$. Since $f$ is surjective and flat, we see that $E \in D^-(\mathcal{O}_X)$. Let $i \in \mathbf{Z}$ be the largest integer such that $H^i(E)$ is nonzero. If $i < m$, then we are done. Otherwise, $f^*H^i(E)$ is a finite type $\mathcal{O}_Y$-module by Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}. Then by Descent, Lemma \ref{descent-lemma-finite-type-descends-fppf} the $\mathcal{O}_X$-module $H^i(E)$ is of finite type. Thus, after replacing $X$ by the members of a finite affine open covering, we may assume there exists a map $$ \alpha : \mathcal{O}_X^{\oplus n}[-i] \longrightarrow E $$ such that $H^i(\alpha)$ is a surjection. Let $C$ be the cone of $\alpha$ in $D(\mathcal{O}_X)$. Pulling back to $Y$ and using Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent} we find that $Lf^*C$ is $m$-pseudo-coherent. Moreover $H^j(C) = 0$ for $j \geq i$. Thus by induction on $i$ we see that $C$ is $m$-pseudo-coherent. Using Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent} again we conclude.
Pullback always preserves perfect complexes, see Cohomology, Lemma \ref{cohomology-lemma-perfect-pullback}. Conversely, assume that $Lf_i^*E$ is perfect for all $i$. Then the cohomology sheaves of each $Lf_i^*E$ are quasi-coherent, see Lemma \ref{lemma-pseudo-coherent} and Cohomology, Lemma \ref{cohomology-lemma-perfect}. Since the morphisms $f_i$ is flat we see that $H^p(Lf_i^*E) = f_i^*H^p(E)$. Thus the cohomology sheaves of $E$ are quasi-coherent by Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}. Having said this the lemma follows formally from Cohomology, Lemma \ref{cohomology-lemma-perfect} and Lemmas \ref{lemma-tor-amplitude-descends} and \ref{lemma-pseudo-coherent-descends-fpqc}.
The argument preceding the lemma applies for the first case because $Rj_*$ maps $D_\QCoh(\mathcal{O}_U)$ into $D_\QCoh(\mathcal{O}_X)$ by Lemma \ref{lemma-quasi-coherence-direct-image}. It is clear that $Rj_*$ maps $D^+_\QCoh(\mathcal{O}_U)$ into $D^+_\QCoh(\mathcal{O}_X)$ which implies the statement on bounded below complexes. Finally, Lemma \ref{lemma-quasi-coherence-direct-image} guarantees that $Rj_*$ maps $D^-_\QCoh(\mathcal{O}_U)$ into $D^-_\QCoh(\mathcal{O}_X)$ if $X$ is quasi-compact. Combining these two we obtain the last statement.
Let $K$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_U)$. By Proposition \ref{proposition-DCoh} we can represent $K$ by a bounded complex $\mathcal{F}^\bullet$ of coherent $\mathcal{O}_U$-modules. Say $\mathcal{F}^i = 0$ for $i \not \in [a, b]$ for some $a \leq b$. Since $j$ is quasi-compact and separated, the terms of the bounded complex $j_*\mathcal{F}^\bullet$ are quasi-coherent modules on $X$, see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. We inductively pick a coherent submodule $\mathcal{G}^i \subset j_*\mathcal{F}^i$ as follows. For $i = a$ we pick any coherent submodule $\mathcal{G}^a \subset j_*\mathcal{F}^a$ whose restriction to $U$ is $\mathcal{F}^a$. This is possible by Properties, Lemma \ref{properties-lemma-extend}. For $i > a$ we first pick any coherent submodule $\mathcal{H}^i \subset j_*\mathcal{F}^i$ whose restriction to $U$ is $\mathcal{F}^i$ and then we set $\mathcal{G}^i = \Im(\mathcal{H}^i \oplus \mathcal{G}^{i - 1} \to j_*\mathcal{F}^i)$. It is clear that $\mathcal{G}^\bullet \subset j_*\mathcal{F}^\bullet$ is a bounded complex of coherent $\mathcal{O}_X$-modules whose restriction to $U$ is $\mathcal{F}^\bullet$ as desired.
Recall that $\Ext^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) = \Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n])$. We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris-hom}. Thus if $X = U \cup V$ and the result of the lemma holds for $E|_U$, $E|_V$, and $E|_{U \cap V}$ for some bound $n_0$, then the result holds for $E$ with bound $n_0 + 1$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent Assume $X = \Spec(A)$ is affine. Choose a complex of $A$-modules $M^\bullet$ whose associated complex of quasi-coherent modules represents $E$, see Lemma \ref{lemma-affine-compare-bounded}. Write $\mathcal{E} = \widetilde{P}$ for some $A$-module $P$. Since $\mathcal{E}$ is finite locally free, we see that $P$ is a finite projective $A$-module. We have \begin{align*} \Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n]) & =  \Hom_{D(A)}(P, M^\bullet[n]) \\ & = \Hom_{K(A)}(P, M^\bullet[n]) \\ & = \Hom_A(P, H^n(M^\bullet)) \end{align*} The first equality by Lemma \ref{lemma-affine-compare-bounded}, the second equality by Derived Categories, Lemma \ref{derived-lemma-morphisms-from-projective-complex}, and the final equality because $\Hom_A(P, -)$ is an exact functor. As $E$ and hence $M^\bullet$ is bounded we get zero for all sufficiently large $n$.
Say $X = \Spec(A)$. Recall that a perfect complex is pseudo-coherent, see Cohomology, Lemma \ref{cohomology-lemma-perfect}. By Lemma \ref{lemma-lift-pseudo-coherent} we can find a bounded above complex $\mathcal{F}^\bullet$ of finite free $A$-modules such that $E$ is isomorphic to $\mathcal{F}^\bullet|_U$ in $D(\mathcal{O}_U)$. By Cohomology, Lemma \ref{cohomology-lemma-perfect} and since $U$ is quasi-compact, we see that $E$ has finite tor dimension, say $E$ has tor amplitude in $[a, b]$. Pick $r < a$ and set $$ \mathcal{F} = \Ker(\mathcal{F}^{r} \to \mathcal{F}^{r + 1}) = \Im(\mathcal{F}^{r - 1} \to \mathcal{F}^r). $$ Since $E$ has tor amplitude in $[a, b]$ we see that $\mathcal{F}|_U$ is flat (Cohomology, Lemma \ref{cohomology-lemma-last-one-flat}). Hence $\mathcal{F}|_U$ is flat and of finite presentation, thus finite locally free (Properties, Lemma \ref{properties-lemma-finite-locally-free}). It follows that $$ (\mathcal{F} \to \mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )|_U $$ is a strictly perfect complex on $U$ representing $E$. We obtain a distinguished triangle $$ \mathcal{F}|_U[- r - 1] \to E \to (\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )|_U \to \mathcal{F}|_U[- r] $$ Note that $(\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )$ is a perfect complex on $X$. To finish the proof it suffices to pick $r$ such that the map $\mathcal{F}|_U[- r - 1] \to E$ is zero in $D(\mathcal{O}_U)$, see Derived Categories, Lemma \ref{derived-lemma-split}. By Lemma \ref{lemma-vanishing-ext} this holds if $r \ll 0$.
Write $X = \Spec(A)$. Write $U = D(f_1) \cup \ldots \cup D(f_r)$. Choose finite complex of finite projective $A$-modules $M^\bullet$ representing $E$ (Lemma \ref{lemma-perfect-affine}). Choose a complex of $A$-modules $(M')^\bullet$ representing $E'$ (Lemma \ref{lemma-affine-compare-bounded}). In this case the complex $H^\bullet = \Hom_A(M^\bullet, (M')^\bullet)$ is a complex of $A$-modules whose associated complex of quasi-coherent $\mathcal{O}_X$-modules represents $R\SheafHom(E, E')$, see Cohomology, Lemma \ref{cohomology-lemma-Rhom-strictly-perfect}. Then $\alpha$ determines an element $s$ of $H^0(U, R\SheafHom(E, E'))$, see Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U}. There exists an $e$ and a map $$ \xi : I^\bullet(f_1^e, \ldots, f_r^e) \to \Hom_A(M^\bullet, (M')^\bullet) $$ corresponding to $s$, see Proposition \ref{proposition-represent-cohomology-class-on-open}. Letting $E_1$ be the object corresponding to complex of quasi-coherent $\mathcal{O}_X$-modules associated to $$ \text{Tot}(I^\bullet(f_1^e, \ldots, f_r^e) \otimes_A M^\bullet) $$ we obtain $E_1 \to E$ using the canonical map $I^\bullet(f_1^e, \ldots, f_r^e) \to A$ and $E_1 \to E'$ using $\xi$ and Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U}.
By Lemma \ref{lemma-lift-perfect-complex-plus-locally-free} we can find a perfect object $E$ of $D(\mathcal{O}_X)$ such that $E|_U = \mathcal{F}[r] \oplus F$ for some finite locally free $\mathcal{O}_U$-module $\mathcal{F}$. By Lemma \ref{lemma-lift-map} we can find a morphism of perfect complexes $\alpha : E_1 \to E$ such that $(E_1)|_U \cong E|_U$ and such that $\alpha|_U$ is the map $$ \left( \begin{matrix} \text{id}_{\mathcal{F}[r]} & 0 \\ 0 & 0 \end{matrix} \right) : \mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F $$ Then the cone on $\alpha$ is a solution.
We have Mayer-Vietoris for morphisms in the derived category, see Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris-hom}. Thus if $X = U \cup V$ and the result of the lemma holds for $f|_U$, $f|_V$, and $f|_{U \cap V}$, then the result holds for $f$. Thus it suffices to prove the lemma when $X$ is affine, see Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}. \medskip\noindent Let $X = \Spec(A)$. Then $f \in A$. We will use the equivalence $D(A) = D_\QCoh(X)$ of Lemma \ref{lemma-affine-compare-bounded} without further mention. Represent $E$ by a finite complex of finite projective $A$-modules $P^\bullet$. This is possible by Lemma \ref{lemma-perfect-affine}. Let $t$ be the largest integer such that $P^t$ is nonzero. The distinguished triangle $$ P^t[-t] \to P^\bullet \to \sigma_{\leq t - 1}P^\bullet \to P^t[-t + 1] $$ shows that by induction on the length of the complex $P^\bullet$ we can reduce to the case where $P^\bullet$ has a single nonzero term. This and the shift functor reduces us to the case where $P^\bullet$ consists of a single finite projective $A$-module $P$ in degree $0$. Represent $E'$ by a complex $M^\bullet$ of $A$-modules. Then $\alpha$ corresponds to a map $P \to H^0(M^\bullet)$. Since the module $H^0(M^\bullet)$ is supported on $V(f)$ by assumption (2) we see that every element of $H^0(M^\bullet)$ is annihilated by a power of $f$. Since $P$ is a finite $A$-module the map $f^n\alpha : P \to H^0(M^\bullet)$ is zero for some $n$ as desired.
Say $T = V(g_1, \ldots, g_s)$. After replacing $g_j$ by a power we may assume multiplication by $g_j$ is zero on $F$, see Lemma \ref{lemma-perfect-into-support-on-T}. Choose $E$ as in Lemma \ref{lemma-lift-perfect-complex-plus-shift}. Note that $g_j : E \to E$ restricts to zero on $U$. Choose a distinguished triangle $$ E \xrightarrow{g_1} E \to C_1 \to E[1] $$ By Derived Categories, Lemma \ref{derived-lemma-split} the object $C_1$ restricts to $F \oplus F[1] \oplus F[1] \oplus F[2]$ on $U$. Moreover, $g_1 : C_1 \to C_1$ has square zero by Derived Categories, Lemma \ref{derived-lemma-third-map-square-zero}. Namely, the diagram $$ \xymatrix{ E \ar[r] \ar[d]_0 & C_1 \ar[d]_{g_1} \ar[r] & E[1] \ar[d]_0 \\ E \ar[r] & C_1 \ar[r] & E[1] } $$ is commutative since the compositions $E \xrightarrow{g_1} E \to C_1$ and $C_1 \to E[1] \xrightarrow{g_1} E[1]$ are zero. Continuing, setting $C_{i + 1}$ equal to the cone of the map $g_i : C_i \to C_i$ we obtain a perfect complex $C_s$ on $X$ supported on $T$ whose restriction to $U$ gives $$ F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] $$ Choose morphisms of perfect complexes $\beta : C' \to C_s$ and $\gamma : C' \to C_s$ as in Lemma \ref{lemma-lift-map} such that $\beta|_U$ is an isomorphism and such that $\gamma|_U \circ \beta|_U^{-1}$ is the morphism $$ F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] \to F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}} \oplus \ldots \oplus F[s] $$ which is the identity on all summands except for $F$ where it is zero. By Lemma \ref{lemma-lift-map} we also have $C' = C_s \otimes^\mathbf{L} I$ for some perfect complex $I$ on $X$. Hence the nullity of $g_j^2\text{id}_{C_s}$ implies the same thing for $C'$. Thus $C'$ is supported on $T$ as well. Then $\text{Cone}(\gamma)$ is a solution.
Since $X$ is quasi-compact there exists an integer $m$ such that $X = U \cup V_1 \cup \ldots \cup V_m$ for some affine opens $V_j$ of $X$. Arguing by induction on $m$ we see that we may assume $m = 1$. In other words, we may assume that $X = U \cup V$ with $V$ affine. By Lemma \ref{lemma-lift-perfect-complex-plus-shift-support} we can choose a perfect object $Q$ in $D(\mathcal{O}_V)$ supported on $T \cap V$ and an isomorphism $Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$. By Lemma \ref{lemma-lift-map} we can replace $Q$ by $Q \otimes^\mathbf{L} I$ (still supported on $T \cap V$) and assume that the map $$ Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V} \longrightarrow P|_{U \cap V} \longrightarrow E|_{U \cap V} $$ lifts to $Q \to E|_V$. By Cohomology, Lemma \ref{cohomology-lemma-glue} we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$ such that $a|_U$ is isomorphic to $P \oplus P[1] \to E|_U$ and $a|_V$ isomorphic to $Q \to E|_V$. Thus $R$ is perfect and supported on $T$ as desired.
Let $j : U \to X$ be the inclusion morphism. If $P \to E|_U$ is an approximation of the triple $(T, E|_U, m)$ over $U$, then $j_!P = Rj_*P \to j_!(E|_U) \to E$ is an approximation of $(T, E, m)$ over $X$. See Cohomology, Lemmas \ref{cohomology-lemma-pushforward-restriction} and \ref{cohomology-lemma-pushforward-perfect}.