hcju/stacksqa huggingface dataset:
If $M$ is perfect on $X$, then $L\text{pr}_1^*M$ is perfect on $X \times_S Y$, see Cohomology, Lemma \ref{cohomology-lemma-perfect-pullback}. We will use this without further mention below. We will also use that if $X \to S$ is of finite type, or proper, or flat, or of finite presentation, then the same thing is true for the base change $\text{pr}_2 : X \times_S Y \to Y$, see Morphisms, Lemmas \ref{morphisms-lemma-base-change-finite-type}, \ref{morphisms-lemma-base-change-proper}, \ref{morphisms-lemma-base-change-flat}, and \ref{morphisms-lemma-base-change-finite-presentation}. \medskip\noindent Part (1) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-direct-image} combined with Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian}. \medskip\noindent Part (2) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-base-change-tensor-perfect}. \medskip\noindent Part (3) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. \medskip\noindent Part (4) follows from part (3) and the fact that a finite type morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}. \medskip\noindent Part (5) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-derived-pushforward-rel-perfect} combined with Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-relatively-perfect}. \medskip\noindent Part (6) follows from part (5) in the same way that part (4) follows from part (3).
By Lemma \ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\Phi$ as in the statement. Observe that $a(\mathcal{O}_Y)$ is in $D^+_\QCoh(\mathcal{O}_{X \times_S Y})$ by Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image-bounded-below}. Hence for $K$ pseudo-coherent we have $K' \in D_\QCoh(\mathcal{O}_{Y \times_S X})$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-internal-hom} we we obtain $\Phi'$ as indicated. \medskip\noindent We abbreviate $\otimes^\mathbf{L} = \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L}$ and $\SheafHom = R\SheafHom_{\mathcal{O}_{X \times_S Y}}$. Let $M$ be in $D_\QCoh(\mathcal{O}_X)$ and let $N$ be in $D_\QCoh(\mathcal{O}_Y)$. We have \begin{align*} \Hom_Y(\Phi(M), N) & = \Hom_Y(R\text{pr}_{2, *}(L\text{pr}_1^*M \otimes^\mathbf{L} K), N) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M \otimes^\mathbf{L} K, a(N)) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M, R\SheafHom(K, a(N))) \\ & = \Hom_X(M, R\text{pr}_{1, *}R\SheafHom(K, a(N))) \end{align*} where we have used Cohomology, Lemmas \ref{cohomology-lemma-internal-hom} and \ref{cohomology-lemma-adjoint}. There are canonical maps $$ L\text{pr}_2^*N \otimes^\mathbf{L} R\SheafHom(K, a(\mathcal{O}_Y)) \xrightarrow{\alpha} R\SheafHom(K, L\text{pr}_2^*N \otimes^\mathbf{L} a(\mathcal{O}_Y)) \xrightarrow{\beta} R\SheafHom(K, a(N)) $$ Here $\alpha$ is Cohomology, Lemma \ref{cohomology-lemma-internal-hom-diagonal-better} and $\beta$ is Duality for Schemes, Equation (\ref{duality-equation-compare-with-pullback}). Combining all of these arrows we obtain the functorial displayed arrow in the statement of the lemma. \medskip\noindent The arrow $\alpha$ is an isomorphism by Derived Categories of Schemes, Lemma \ref{perfect-lemma-internal-hom-evaluate-tensor-isomorphism} as soon as either $K$ or $N$ is perfect. The arrow $\beta$ is an isomorphism if $N$ is perfect by Duality for Schemes, Lemma \ref{duality-lemma-compare-with-pullback-perfect} or in general if $X \to S$ is flat proper of finite presentation by Duality for Schemes, Lemma \ref{duality-lemma-compare-with-pullback-flat-proper}.
By Lemma \ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\Phi$ as in the statement. \medskip\noindent Observe that formation of the relative dualizing complex commutes with base change in our setting, see Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}. Thus $L\text{pr}_2^*\omega^\bullet_{Y/S} = \omega^\bullet_{X \times_S Y/X}$. Moreover, we observe that $\omega^\bullet_{Y/S}$ is an invertible object of the derived category, see Duality for Schemes, Lemma \ref{duality-lemma-affine-flat-Noetherian-gorenstein}, and a fortiori perfect. \medskip\noindent To actually prove the lemma we're going to cheat. Namely, we will show that if we replace the roles of $X$ and $Y$ and $K$ and $K'$ then these are as in Lemma \ref{lemma-fourier-mukai-right-adjoint} and we get the result. It is clear that $K'$ is perfect as a tensor product of perfect objects so that the discussion in Lemma \ref{lemma-fourier-mukai-right-adjoint} applies to it. To show that the procedure of Lemma \ref{lemma-fourier-mukai-right-adjoint} applied to $K'$ on $Y \times_S X$ produces a complex isomorphic to $K$ it suffices (details omitted) to show that $$ R\SheafHom(R\SheafHom(K, \omega^\bullet_{X \times_S Y/X}), \omega^\bullet_{X \times_S Y/X}) = K $$ This is clear because $K$ is perfect and $\omega^\bullet_{X \times_S Y/X}$ is invertible; details omitted. Thus Lemma \ref{lemma-fourier-mukai-right-adjoint} produces a map $$ \Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$ which is an isomorphism because $K'$ is perfect. This finishes the proof.
Part (1) is immediate from Lemma \ref{lemma-fourier-mukai} part (4). \medskip\noindent Part (2) follows from Lemma \ref{lemma-compose-fourier-mukai} and the fact that $K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K')$ is perfect for example by Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image}. \medskip\noindent The adjointness in part (3) on all complexes with quasi-coherent cohomology sheaves follows from Lemma \ref{lemma-fourier-mukai-right-adjoint} with $K'$ equal to the pullback of $R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y))$ by $Y \times_S X \to X \times_S Y$ where $a$ is the right adjoint to $R\text{pr}_{2, *} : D_\QCoh(\mathcal{O}_{X \times_S Y}) \to D_\QCoh(\mathcal{O}_Y)$. Denote $f : X \to S$ the structure morphism of $X$. Since $f$ is proper the functor $f^! : D_\QCoh^+(\mathcal{O}_S) \to D_\QCoh^+(\mathcal{O}_X)$ is the restriction to $D_\QCoh^+(\mathcal{O}_S)$ of the right adjoint to $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$, see Duality for Schemes, Section \ref{duality-section-upper-shriek}. Hence the relative dualizing complex $\omega_{X/S}^\bullet$ as defined in Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex} is equal to $\omega_{X/S}^\bullet = f^!\mathcal{O}_S$. Since formation of the relative dualizing complex commutes with base change (see Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}) we see that $a(\mathcal{O}_Y) = L\text{pr}_1^*\omega_{X/S}^\bullet$. Thus $$ R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \cong L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee $$ by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. Finally, since $X \to S$ is assumed Gorenstein the relative dualizing complex is invertible: this follows from Duality for Schemes, Lemma \ref{duality-lemma-affine-flat-Noetherian-gorenstein}. We conclude that $\omega_{X/S}^\bullet$ is perfect (Cohomology, Lemma \ref{cohomology-lemma-invertible-derived}) and hence $K'$ is perfect. Therefore $\Phi_{K'}$ does indeed map $D_{perf}(\mathcal{O}_Y)$ into $D_{perf}(\mathcal{O}_X)$ which finishes the proof of (3). \medskip\noindent The proof of (4) is the same as the proof of (3) except one uses Lemma \ref{lemma-fourier-mukai-left-adjoint} instead of Lemma \ref{lemma-fourier-mukai-right-adjoint}.
Let $U \subset X$ and $V \subset Y$ be affine open subschemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of the reduced induced closed subscheme structure on $X \setminus U$. Similarly, let $\mathcal{I}' \subset \mathcal{O}_Y$ be the ideal sheaf of the reduced induced closed subscheme structure on $Y \setminus V$. Then the ideal sheaf $$ \mathcal{J} = \Im(\text{pr}_1^*\mathcal{I} \otimes_{\mathcal{O}_{X \times_R Y}} \text{pr}_2^*\mathcal{I}' \to \mathcal{O}_{X \times_R Y}) $$ satisfies $V(\mathcal{J}) = X \times_R Y \setminus U \times_R V$. For any section $s \in \mathcal{F}(U \times_R V)$ we can find an integer $n > 0$ and a map $\mathcal{J}^n \to \mathcal{F}$ whose restriction to $U \times_R V$ gives $s$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}. By assumption we can choose surjections $\mathcal{E} \to \mathcal{I}$ and $\mathcal{G} \to \mathcal{I}'$. These produce corresponding surjections $$ \mathcal{E} \boxtimes \mathcal{G} \to \mathcal{J} \quad\text{and}\quad \mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{J}^n $$ and hence a map $\mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{F}$ whose image contains the section $s$ over $U \times_R V$. Since we can cover $X \times_R Y$ by a finite number of affine opens of the form $U \times_R V$ and since $\mathcal{F}|_{U \times_R V}$ is generated by finitely many sections (Properties, Lemma \ref{properties-lemma-finite-type-module}) we conclude that there exists a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, N} \mathcal{E}_j^{\otimes n_j} \boxtimes \mathcal{G}_j^{\otimes n_j} \to \mathcal{F} $$ where $\mathcal{E}_j$ is finite locally free on $X$ and $\mathcal{G}_j$ is finite locally free on $Y$. Setting $\mathcal{E} = \bigoplus \mathcal{E}_j^{\otimes n_j}$ and $\mathcal{G} = \bigoplus \mathcal{G}_j^{\otimes n_j}$ we conclude that the lemma is true.
Follows from Lemma \ref{lemma-on-product} by a limit argument. We urge the reader to skip the proof. Since $X \times_R Y$ is a closed subscheme of $X \times_\mathbf{Z} Y$ it is harmless if we replace $R$ by $\mathbf{Z}$. We can write $\mathcal{F}$ as the quotient of a finitely presented $\mathcal{O}_{X \times_R Y}$-module by Properties, Lemma \ref{properties-lemma-finite-directed-colimit-surjective-maps}. Hence we may assume $\mathcal{F}$ is of finite presentation. Next we can write $X = \lim X_i$ with $X_i$ of finite presentation over $\mathbf{Z}$ and similarly $Y = \lim Y_j$, see Limits, Proposition \ref{limits-proposition-approximate}. Then $\mathcal{F}$ will descend to $\mathcal{F}_{ij}$ on some $X_i \times_R Y_j$ (Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}) and so does the property of having the resolution property (Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-descends}). Then we apply Lemma \ref{lemma-on-product} to $\mathcal{F}_{ij}$ and we pullback.
Since $X$ is separated, the diagonal morphism $\Delta$ is a closed immersion and hence $\mathcal{O}_\Delta$ is a coherent $\mathcal{O}_{X \times_R X}$-module (Cohomology of Schemes, Lemma \ref{coherent-lemma-i-star-equivalence}). Thus the lemma follows immediately from Lemma \ref{lemma-on-product}.
To prove (1) we use the spectral sequence $$ H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) \Rightarrow \Ext^{p + q}_X(\mathcal{F}, \mathcal{G}) $$ of Cohomology, Section \ref{cohomology-section-ext}. Recall that taking $\SheafExt$ of coherent modules agrees with the algebra construction affine locally, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-internal-hom}. Let $x \in X$ be a point. A regular local ring is Gorenstein, see Dualizing Complexes, Lemma \ref{dualizing-lemma-regular-gorenstein}. Hence $\mathcal{O}_{X, x}[d_x]$ where $d_x = \dim(\mathcal{O}_{X, x})$ is a normalized dualizing complex for $\mathcal{O}_{X, x}$. It follows that the modules $\SheafExt^q(\mathcal{F}, \mathcal{G})$ have support of dimension at most $d - q$, see Dualizing Complexes, Lemma \ref{dualizing-lemma-sitting-in-degrees}. Hence we have $H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) = 0$ for $p > d - q$ by Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}. This proves (1). \medskip\noindent Proof of (2). We may use induction on the number of nonzero cohomology sheaves of $K$ and $L$. The case where these numbers are $0, 1$ follows from (1). If the number of nonzero cohomology sheaves of $K$ is $> 1$, then we let $i \in \mathbf{Z}$ be minimal such that $H^i(K)$ is nonzero. We obtain a distinguished triangle $$ H^i(K)[-i] \to K \to \tau_{\geq i + 1}K $$ (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) and we get the vanishing of $\Hom(K, L)$ from the vanishing of $\Hom(H^i(K)[-i], L)$ and $\Hom(\tau_{\geq i + 1}K, L)$ by Derived Categories, Lemma \ref{derived-lemma-representable-homological}. Simlarly if $L$ has more than one nonzero cohomology sheaf.
We have $\tau_{\leq a}K = \tau_{\leq a + d - 1}K$ by the assumed vanishing of cohomology sheaves. By Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle} we have a distinguished triangle $$ \tau_{\leq a}K \to K \to \tau_{\geq a + d}K \xrightarrow{\delta} (\tau_{\leq a}K)[1] $$ By Derived Categories, Lemma \ref{derived-lemma-split} it suffices to show that the morphism $\delta$ is zero. This follows from Lemma \ref{lemma-Ext-0-regular}.
Recall that $X$ is regular by Varieties, Lemma \ref{varieties-lemma-smooth-regular}. Hence $X$ has the resolution property by Derived Categories of Schemes, Lemma \ref{perfect-lemma-regular-resolution-property}. Hence we may choose a resolution as in Lemma \ref{lemma-diagonal-resolution}. Say $\dim(X) = d$. Since $X \times X$ is smooth over $k$ it is regular. Hence $X \times X$ is a regular Noetherian scheme with $\dim(X \times X) = 2d$. The object $$ K = (\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0) $$ of $D_{perf}(\mathcal{O}_{X \times X})$ has cohomology sheaves $\mathcal{O}_\Delta$ in degree $0$ and $\Ker(\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \mathcal{E}_{2d-1} \boxtimes \mathcal{G}_{2d-1})$ in degree $-2d$ and zero in all other degrees. Hence by Lemma \ref{lemma-split-complex-regular} we see that $\mathcal{O}_\Delta$ is a summand of $K$ in $D_{perf}(\mathcal{O}_{X \times X})$. Clearly, the object $K$ is in $$ \left\langle \bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i \boxtimes \mathcal{G}_i \right\rangle \subset \left\langle \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i\right) \boxtimes \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{G}_i\right) \right\rangle $$ which finishes the proof. (The reader may consult Derived Categories, Lemmas \ref{derived-lemma-generated-by-E-explicit} and \ref{derived-lemma-in-cone-n} to see that our object is contained in this category.)
Using Lemma \ref{lemma-diagonal-trick} choose finite locally free $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle$ in $D(\mathcal{O}_{X \times X})$. We claim that $\mathcal{G}$ is a strong generator for $D_{perf}(\mathcal{O}_X)$. With notation as in Derived Categories, Section \ref{derived-section-operate-on-full} choose $m, n \geq 1$ such that $$ \mathcal{O}_\Delta \in smd(add(\mathcal{E} \boxtimes \mathcal{G}[-m, m])^{\star n}) $$ This is possible by Derived Categories, Lemma \ref{derived-lemma-find-smallest-containing-E}. Let $K$ be an object of $D_{perf}(\mathcal{O}_X)$. Since $L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} -$ is an exact functor and since $$ L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G}) = (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G} $$ we conclude from Derived Categories, Remark \ref{derived-remark-operations-functor} that $$ L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta \in smd(add( (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}[-m, m])^{\star n}) $$ Applying the exact functor $R\text{pr}_{2, *}$ and observing that $$ R\text{pr}_{2, *} \left((K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}\right) = R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G} $$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} we conclude that $$ K = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta) \in smd(add(R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G}[-m, m])^{\star n}) $$ The equality follows from the discussion in Example \ref{example-diagonal-fourier-mukai}. Since $K$ is perfect, there exist $a \leq b$ such that $H^i(X, K)$ is nonzero only for $i \in [a, b]$. Since $X$ is proper, each $H^i(X, K)$ is finite dimensional. We conclude that the right hand side is contained in $smd(add(\mathcal{G}[-m + a, m + b])^{\star n})$ which is itself contained in $\langle \mathcal{G} \rangle_n$ by one of the references given above. This finishes the proof.
In the proof of Lemma \ref{lemma-smooth-proper-strong-generator} we have shown that there exist $m', n \geq 1$ such that for any coherent $\mathcal{O}_X$-module $\mathcal{F}$, $$ \mathcal{F} \in smd(add(\mathcal{G}[-m' + a, m' + b])^{\star n}) $$ for any $a \leq b$ such that $H^i(X, \mathcal{F})$ is nonzero only for $i \in [a, b]$. Thus we can take $a = 0$ and $b = \dim(X)$. Taking $m = \max(m', m' + b)$ finishes the proof.
Let $Z$ be the scheme theoretic support of an $\mathcal{F}$ as in (2). Then $Z \to \Spec(k)$ is finite, hence $Z \times Y \to Y$ is finite. It follows that for an object $M$ of $D_\QCoh(\mathcal{O}_{X \times Y})$ with cohomology sheaves supported on $Z \times Y$ we have $H^i(R\text{pr}_{2, *}(M)) = \text{pr}_{2, *}H^i(M)$ and the functor $\text{pr}_{2, *}$ is faithful on quasi-coherent modules supported on $Z \times Y$; details omitted. Hence we see that the objects $$ \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} K_n $$ in $D_{perf}(\mathcal{O}_{X \times Y})$ have vanshing cohomology sheaves outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in $[-m, m]$. Let $z \in X \times Y$ be a closed point mapping to the closed point $x \in X$. Then we know that $$ K_{n, z} \otimes_{\mathcal{O}_{X \times Y, z}}^\mathbf{L} \mathcal{O}_{X \times Y, z}/\mathfrak m_x^t\mathcal{O}_{X \times Y, z} $$ has nonzero cohomology only in the intervals $[-m, m] \cup [-m - n, m - n]$. We conclude by More on Algebra, Lemma \ref{more-algebra-lemma-kollar-kovacs-pseudo-coherent} that $K_{n, z}$ only has nonzero cohomology in degrees $[-m, m] \cup [-m - n, m - n]$. Since this holds for all closed points of $X \times Y$, we conclude $K_n$ only has nonzero cohomology sheaves in degrees $[-m, m] \cup [-m - n, m - n]$. In exactly the same way we see that the maps $K_n \to K_{n + 1}$ are isomorphisms on cohomology sheaves in degrees $[-m, m]$ for $n > 2m$. \medskip\noindent If $X$ and $Y$ are smooth over $k$, then $X \times Y$ is smooth over $k$ and hence regular by Varieties, Lemma \ref{varieties-lemma-smooth-regular}. Thus we will obtain the direct sum decomposition of $K_n$ as soon as $n > 2m + \dim(X \times Y)$ from Lemma \ref{lemma-split-complex-regular}. The final statement is clear from this.
First we observe that since $\text{pr}_2 : X \times_R Y Y$ is affine (Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}) the functor $\text{pr}_{2, *}$ is exact (see for example Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}). Hence the functor (\ref{equation-FM-QCoh}) is right exact in this case. \medskip\noindent Let us construct the quasi-inverse to the construction. Let $F$ be as in (1). Say $X = \Spec(A)$. Consider the quasi-coherent $\mathcal{O}_Y$-module $\mathcal{G} = F(\mathcal{O}_X)$. Every element $a \in A$ induces an endomorphism of $\mathcal{G}$ and this defines an $R$-linear map $A \to \text{End}_{\mathcal{O}_Y}(\mathcal{G})$. Hence we see that $\mathcal{G}$ is a sheaf of modules over $$ A \otimes_R \mathcal{O}_Y = \text{pr}_{2, *}\mathcal{O}_{X \times_R Y} $$ By Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules} we find that there is a unique quasi-coherent module $\mathcal{K}$ on $X \times_R Y$ such that $\mathcal{G} = \text{pr}_{2, *}\mathcal{K}$ compatible with action of $A$ and $\mathcal{O}_Y$. Commutation with direct sums shows that $F(\bigoplus_{i \in I} \mathcal{O}_X) = \bigoplus_{i \in I} \mathcal{G}$. Finally, since $X = \Spec(A)$ for every quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ we can choose an exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{O}_X \to \bigoplus\nolimits_{i \in I} \mathcal{O}_X \to \mathcal{F} \to 0 $$ This leads to an exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{K} \to \bigoplus\nolimits_{i \in I} \mathcal{K} \to \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K} \to 0 $$ which using the exact functor $\text{pr}_{2, *}$ gives the exact sequence $$ \bigoplus\nolimits_{j \in J} \mathcal{G} \to \bigoplus\nolimits_{i \in I} \mathcal{G} \to \text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) \to 0 $$ which as $F$ commutes with direct sums we may rewrite as $$ F(\bigoplus\nolimits_{j \in J} \mathcal{O}_X) \to F(\bigoplus\nolimits_{i \in I} \mathcal{O}_X) \to \text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) \to 0 $$ By right exactness of $F$ we conclude $F$ is isomorphic to the functor (\ref{equation-FM-QCoh}).
For part (1) let $\mathcal{F}'$ be a quasi-coherent module on $X'$. With obvious notation we have \begin{align*} \text{pr}_{2, *}(\text{pr}_1^*f_*\mathcal{F}' \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) & = \text{pr}_{2, *}((f \times \text{id}_Y)_* (\text{pr}'_1)^*\mathcal{F}' \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) \\ & = \text{pr}_{2, *}(f \times \text{id}_Y)_* \left((\text{pr}'_1)^*\mathcal{F}' \otimes_{\mathcal{O}_{X' \times_R Y}} (f \times \text{id}_Y)^*\mathcal{K})\right)  \\ & = \text{pr}'_{2, *}((\text{pr}'_1)^*\mathcal{F}' \otimes_{\mathcal{O}_{X' \times_R Y}} (f \times \text{id}_Y)^*\mathcal{K}) \end{align*} Here the first equality is affine base change, see Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change}. The second equality hold by Remark \ref{remark-affine-morphism}. The third equality is functoriality of pushforwards for modules. For part (2) we have $$ g^*\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K}) = \text{pr}'_{2, *}((\text{pr}'_1)^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_R Y'}} (\text{id}_X \times g)^*\mathcal{K}) $$ by flat base change, see Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}. For part (3) we only have to remark that formation of $\text{pr}_2$ commutes with localization on the target.
By Lemma \ref{lemma-functor-quasi-coherent-from-affine-compose} we may assume $Y$ is affine. In this case we can translate the statement into algebra as follows: Given a ring $R$ and $R$-algebras $A$, $B$ for an $A \otimes_R B$-module $K$ the functor $\text{Mod}_A \to \text{Mod}_B, M \mapsto M \otimes_A K$ is exact if and only if $K$ is flat as an $A$-module. This is obvious.
The question is local on $T$ hence we may assume $T$ is affine. In this case we can consider the diagram $$ \xymatrix{ T \times_R X \ar[d] & T \times_R X \times_R Y \ar[d] \ar[l] \ar[r] & T \times_R Y \ar[d] \\ X & X \times_R Y \ar[l] \ar[r] & Y } $$ whose vertical arrows are affine. In particular the pushforward along $T \times_R Y \to Y$ is faithful and exact. Chasing around in the diagram using that higher direct images along affine morphisms vanish we see that it suffices to prove $$ R^q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F} \otimes_{\mathcal{O}_{T \times_R X \times_R Y}} \text{pr}_{23}^*\mathcal{K})) = R^q\text{pr}_{2, *}( \text{pr}_{23, *}(\text{pr}_{12}^*\mathcal{F}) \otimes_{\mathcal{O}_{X \times_R Y}} \mathcal{K})) $$ is zero which is true by assumption on $\mathcal{K}$. The equality holds by Remark \ref{remark-affine-morphism}.