hcju/stacksqa huggingface dataset:
The final statement is clear from the equivalence of (1) and (2). It is also clear that (2) implies (1). Assume (1) holds. Let $S = \bigcup V_i$ and $f^{-1}(V_i) = \bigcup U_{ij}$ be affine open coverings as in Definition \ref{definition-relative-pseudo-coherence}. Let $U \subset X$ and $V \subset S$ be as in (2). By Lemma \ref{lemma-glue-relative-pseudo-coherent} it suffices to find a standard open covering $U = \bigcup U_k$ of $U$ such that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pairs $(U_k \to V, E|_{U_k})$. In other words, for every $u \in U$ it suffices to find a standard affine open $u \in U' \subset U$ such that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U' \to V, E|_{U'})$. Pick $i$ such that $f(u) \in V_i$ and then pick $j$ such that $u \in U_{ij}$. By Schemes, Lemma \ref{schemes-lemma-standard-open-two-affines} we can find $v \in V' \subset V \cap V_i$ which is standard affine open in $V'$ and $V_i$. Then $f^{-1}V'  \cap U$, resp.\ $f^{-1}V' \cap U_{ij}$ are standard affine opens of $U$, resp.\ $U_{ij}$. Applying the lemma again we can find $u \in U' \subset f^{-1}V' \cap U \cap U_{ij}$ which is standard affine open in both $f^{-1}V'  \cap U$ and $f^{-1}V' \cap U_{ij}$. Thus $U'$ is also a standard affine open of $U$ and $U_{ij}$. By Lemma \ref{lemma-localize-relative-pseudo-coherent} the assumption that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U_{ij} \to V_i, E|_{U_{ij}})$ implies that the equivalent conditions of Lemma \ref{lemma-relatively-pseudo-coherent} are satisfied for the pair $(U' \to V, E|_{U'})$.
Let $U$ and $V$ be as in (2) and choose a closed immersion $i : U \to \mathbf{A}^n_V$. A formal argument, using Lemma \ref{lemma-relative-pseudo-coherence-characterize}, shows it suffices to prove that $Ri_*(E|_U)$ is $m$-pseudo-coherent if and only if $R\Gamma(U, E)$ is $m$-pseudo-coherent relative to $\mathcal{O}_S(V)$. Say $U = \Spec(A)$, $V = \Spec(R)$, and $\mathbf{A}^n_V = \Spec(R[x_1, \ldots, x_n]$. By the remarks preceding the lemma, $E|_U$ is quasi-isomorphic to the complex of quasi-coherent sheaves on $U$ associated to the object $R\Gamma(U, E)$ of $D(A)$. Note that $R\Gamma(U, E) = R\Gamma(\mathbf{A}^n_V, Ri_*(E|_U))$ as $i$ is a closed immersion (and hence $i_*$ is exact). Thus $Ri_*E$ is associated to $R\Gamma(U, E)$ viewed as an object of $D(R[x_1, \ldots, x_n])$. We conclude as $m$-pseudo-coherence of $Ri_*(E|_U)$ is equivalent to $m$-pseudo-coherence of $R\Gamma(U, E)$ in $D(R[x_1, \ldots, x_n])$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent-affine} which is equivalent to $R\Gamma(U, E)$ is $m$-pseudo-coherent relative to $R = \mathcal{O}_S(V)$ by definition.
By Morphisms, Lemma \ref{morphisms-lemma-homeomorphism-affine} the morphism $i$ is affine. Thus we may assume $S$, $Y$, and $X$ are affine. Say $S = \Spec(R)$, $Y = \Spec(A)$, and $X = \Spec(B)$. The condition means that $A/\text{rad}(A) \to B/\text{rad}(B)$ is surjective; here $\text{rad}(A)$ and $\text{rad}(B)$ denote the Jacobson radical of $A$ and $B$. As $B$ is of finite type over $A$, we can find $b_1, \ldots, b_m \in \text{rad}(B)$ which generate $B$ as an $A$-algebra. Say $b_j^N = 0$ for all $j$. Consider the diagram of rings $$ \xymatrix{ B & R[x_i, y_j]/(y_j^N) \ar[l] & R[x_i, y_j] \ar[l] \\ A \ar[u] & R[x_i] \ar[l] \ar[u] \ar[ru] } $$ which translates into a diagram $$ \xymatrix{ X \ar[d] \ar[r] & T \ar[d] \ar[r] & \mathbf{A}^{n + m}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S } $$ of affine schemes. By Lemma \ref{lemma-relative-pseudo-coherence-characterize} we see that $E$ is $m$-pseudo-coherent relative to $S$ if and only if its pushforward to $\mathbf{A}^{n + m}_S$ is $m$-pseudo-coherent.  By Derived Categories of Schemes, Lemma \ref{perfect-lemma-closed-push-pseudo-coherent} we see that this is true if and only if its pushforward to $T$ is $m$-pseudo-coherent. The same lemma shows that this holds if and only if the pushforward to $\mathbf{A}^n_S$ is $m$-pseudo-coherent. Again by Lemma \ref{lemma-relative-pseudo-coherence-characterize} this holds if and only if $Ri_*E$ is $m$-pseudo-coherent relative to $S$.
Translation of the result of More on Algebra, Lemma \ref{more-algebra-lemma-finite-extension-pseudo-coherent} into the language of schemes. Observe that $R\pi_*$ indeed maps $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-image}. To do the translation use Lemma \ref{lemma-relative-pseudo-coherence-characterize}.
Part (1) is immediate from the definition. To see part (3) we may work locally on $X$ (both properties are local). Thus we may assume $X$ and $S$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^n_S$. Then we see that $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $i_*\mathcal{F}$ is $(-1)$-pseudo-coherent, which is true if and only if $i_*\mathcal{F}$ is an $\mathcal{O}_{\mathbf{A}^n_S}$-module of finite presentation, see Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}. A module of finite presentation is quasi-coherent, see Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-coherent}. By Morphisms, Lemma \ref{morphisms-lemma-i-star-equivalence} we see that $\mathcal{F}$ is quasi-coherent if and only if $i_*\mathcal{F}$ is quasi-coherent. Having said this part (3) follows. The proof of (2) is similar but less involved.
The problem is local on $X$ and $X'$ hence we may assume $X$, $S$, $S'$, and $X'$ are affine. Choose a closed immersion $i : X \to \mathbf{A}^n_S$ and denote $i' : X' \to \mathbf{A}^n_{S'}$ the base change to $S'$. Denote $g : X' \to X$ and $g' : \mathbf{A}^n_{S'} \to \mathbf{A}^n_S$ the projections, so $E' = Lg^*E$. Since $X$ and $S'$ are tor-independent over $S$, the base change map (Cohomology, Remark \ref{cohomology-remark-base-change}) induces an isomorphism $$ Ri'_*(Lg^*E) = L(g')^*Ri_*E $$ Namely, for a point $x' \in X'$ lying over $x \in X$ the base change map on stalks at $x'$ is the map $$ E_x \otimes_{\mathcal{O}_{\mathbf{A}^n_S, x}}^\mathbf{L} \mathcal{O}_{\mathbf{A}^n_{S'}, x'} \longrightarrow E_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \mathcal{O}_{X', x'} $$ coming from the closed immersions $i$ and $i'$. Note that the source is quasi-isomorphic to a localization of $E_x \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}$ which is isomorphic to the target as $\mathcal{O}_{X', x'}$ is isomorphic to (the same) localization of $\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{S, s}}^\mathbf{L} \mathcal{O}_{S', s'}$ by assumption. We conclude the lemma holds by an application of Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
The problem is local on $X$. Thus we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma \ref{more-algebra-lemma-pull-relative-pseudo-coherent} we can find a commutative diagram $$ \xymatrix{ X \ar[r]_i \ar[d]_f & \mathbf{A}^d_Y \ar[r]_j \ar[ld]^p & \mathbf{A}^{n + d}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S } $$ Observe that $$ Ri_* Lf^*E = Ri_* Li^* Lp^*E = Lp^*E \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}}^\mathbf{L} Ri_*\mathcal{O}_X $$ by Cohomology, Lemma \ref{cohomology-lemma-projection-formula-closed-immersion}. By assumption and the fact that $Y$ is affine, we can represent $Ri_*\mathcal{O}_X = i_*\mathcal{O}_X$ by a complexes of finite free $\mathcal{O}_{\mathbf{A}_Y^n}$-modules $\mathcal{F}^\bullet$, with $\mathcal{F}^q = 0$ for $q > 0$ (details omitted; use Derived Categories of Schemes, Lemma \ref{perfect-lemma-pseudo-coherent-affine} and More on Algebra, Lemma \ref{more-algebra-lemma-rel-n-pseudo-module}). By assumption $E$ is bounded above, say $H^q(E) = 0$ for $q > a$. Represent $E$ by a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_Y$-modules with $\mathcal{E}^q = 0$ for $q > a$. Then the derived tensor product above is represented by $\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)$. \medskip\noindent Since $j$ is a closed immersion, the functor $j_*$ is exact and $Rj_*$ is computed by applying $j_*$ to any representating complex of sheaves. Thus we have to show that $j_*\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_{\mathbf{A}^{n + m}_S}$-modules. Note that $\text{Tot}(p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^\bullet)$ has a filtration by subcomplexes with successive quotients the complexes $p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]$. Note that for $q \ll 0$ the complexes $p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]$ have zero cohomology in degrees $\leq m$ and hence are $m$-pseudo-coherent. Hence, applying Lemma \ref{lemma-cone-relatively-pseudo-coherent} and induction, it suffices to show that $p^*\mathcal{E}^\bullet \otimes_{\mathcal{O}_{\mathbf{A}_Y^n}} \mathcal{F}^q[-q]$ is pseudo-coherent relative to $S$ for all $q$. Note that $\mathcal{F}^q = 0$ for $q > 0$. Since also $\mathcal{F}^q$ is finite free this reduces to proving that $p^*\mathcal{E}^\bullet$ is $m$-pseudo-coherent relative to $S$ which follows from Lemma \ref{lemma-base-change-relative-pseudo-coherent} for instance.
The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma \ref{more-algebra-lemma-pull-relative-pseudo-coherent} we can find a commutative diagram $$ \xymatrix{ X \ar[r]_i \ar[d]_f & \mathbf{A}^m_Y \ar[r]_j \ar[ld]^p & \mathbf{A}^{n + m}_S \ar[ld] \\ Y \ar[r] & \mathbf{A}^n_S } $$ The assumption that $\mathcal{O}_Y$ is pseudo-coherent relative to $S$ implies that $\mathcal{O}_{\mathbf{A}^m_Y}$ is pseudo-coherent relative to $\mathbf{A}^m_S$ (by flat base change; this can be seen by using for example Lemma \ref{lemma-base-change-relative-pseudo-coherent}). This in turn implies that $j_*\mathcal{O}_{\mathbf{A}^n_Y}$ is pseudo-coherent as an $\mathcal{O}_{\mathbf{A}^{n + m}_S}$-module. Then the equivalence of the lemma follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-closed-push-pseudo-coherent}.
The equivalence of (1) and (2) is Lemma \ref{lemma-finite-morphism-relative-pseudo-coherence}. The equivalence of (2) and (3) follows from Lemma \ref{lemma-composition-relative-pseudo-coherent} applied to $\text{id} : P \to P$ provided we can show that $\mathcal{O}_P$ is pseudo-coherent relative to $S$. This follows from More on Algebra, Lemma \ref{more-algebra-lemma-flat-finite-presentation-perfect} and the definitions.
To see the equivalence of (1) and (2) it suffices to check conditions (1)(a), (b), (c) of Morphisms, Definition \ref{morphisms-definition-property-local} for the property of being a pseudo-coherent ring map. These properties follow (using localization is flat) from More on Algebra, Lemmas \ref{more-algebra-lemma-base-change-relative-pseudo-coherent}, \ref{more-algebra-lemma-localize-relative-pseudo-coherent}, and \ref{more-algebra-lemma-glue-relative-pseudo-coherent}. \medskip\noindent If (1) holds, then $f$ is locally of finite type as a pseudo-coherent ring map is of finite type by definition. Moreover, (1) implies via Lemma \ref{lemma-qcoh-relative-pseudo-coherence-characterize} and the definitions that $\mathcal{O}_X$ is pseudo-coherent relative to $S$. Conversely, if (3) holds, then we see that for every $U$ and $V$ as in (2) the ring $\mathcal{O}_X(U)$ is of finite type over $\mathcal{O}_S(V)$ and $\mathcal{O}_X(U)$ is as a module pseudo-coherent relative to $\mathcal{O}_S(V)$, see Lemmas \ref{lemma-relative-pseudo-coherence-characterize} and \ref{lemma-qcoh-relative-pseudo-coherence-characterize}. This is the definition of a pseudo-coherent ring map, hence (2) and (1) hold.
Translated into algebra this lemma says the following: If $R \to A$ is a finite ring map, then $R \to A$ is pseudo-coherent as a ring map (which means by definition that $A$ as an $A$-module is pseudo-coherent relative to $R$) if and only if $A$ is pseudo-coherent as an $R$-module. This follows from the more general More on Algebra, Lemma \ref{more-algebra-lemma-finite-extension-pseudo-coherent}.
This translates into the following algebra result: If $R \to A$ is a finite type ring map with $R$ Noetherian, then $R \to A$ is pseudo-coherent if and only if $R \to A$ is of finite type. To see this, note that a pseudo-coherent ring map is of finite type by definition. Conversely, if $R \to A$ is of finite type, then we can write $A = R[x_1, \ldots, x_n]/I$ and it follows from More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-pseudo-coherent} that $A$ is pseudo-coherent as an $R[x_1, \ldots, x_n]$-module, i.e., $R \to A$ is a pseudo-coherent ring map.
We will use the criterion of Descent, Lemma \ref{descent-lemma-descending-properties-morphisms} to prove this. By Definition \ref{definition-pseudo-coherent} being pseudo-coherent is Zariski local on the base. By Lemma \ref{lemma-flat-base-change-pseudo-coherent} being pseudo-coherent is preserved under flat base change. The final hypothesis (3) of Descent, Lemma \ref{descent-lemma-descending-properties-morphisms} translates into the following algebra statement: Let $A \to B$ be a faithfully flat ring map. Let $C = A[x_1, \ldots, x_n]/I$ be an $A$-algebra. If $C \otimes_A B$ is pseudo-coherent as an $B[x_1, \ldots, x_n]$-module, then $C$ is pseudo-coherent as a $A[x_1, \ldots, x_n]$-module. This is More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-pseudo-coherent}.
Choose a presentation $B = A[x_1, \ldots, x_n]/J$. Note that $B$ is pseudo-coherent as an $A[x_1, \ldots, x_n]$-module because $A \to B$ is a pseudo-coherent ring map by Lemma \ref{lemma-flat-finite-presentation-pseudo-coherent}. Note that $A \to B/I$ is pseudo-coherent if and only if $B/I$ is pseudo-coherent as an $A[x_1, \ldots, x_n]$-module. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-push-pseudo-coherent} we see this is equivalent to the condition that $B/I$ is pseudo-coherent as an $B$-module. This proves the lemma as the short exact sequence $0 \to I \to B \to B/I \to 0$ shows that $I$ is pseudo-coherent if and only if $B/I$ is (see More on Algebra, Lemma \ref{more-algebra-lemma-two-out-of-three-pseudo-coherent}).
We will use the criterion of Descent, Lemma \ref{descent-lemma-properties-morphisms-local-source} to prove this. It follows from Lemmas \ref{lemma-flat-finite-presentation-pseudo-coherent} and \ref{lemma-composition-pseudo-coherent} that being pseudo-coherent is preserved under precomposing with flat morphisms locally of finite presentation, in particular under precomposing with syntomic morphisms (see Morphisms, Lemmas \ref{morphisms-lemma-syntomic-flat} and \ref{morphisms-lemma-syntomic-locally-finite-presentation}). It is clear from Definition \ref{definition-pseudo-coherent} that being pseudo-coherent is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma \ref{descent-lemma-properties-morphisms-local-source} it suffices to prove the following: Suppose $X' \to X \to Y$ are morphisms of affine schemes with $X' \to X$ syntomic and $X' \to Y$ pseudo-coherent. Then $X \to Y$ is pseudo-coherent. To see this, note that in any case $X \to Y$ is of finite presentation by Descent, Lemma \ref{descent-lemma-flat-finitely-presented-permanence-algebra}. Choose a closed immersion $X \to \mathbf{A}^n_Y$. By Algebra, Lemma \ref{algebra-lemma-lift-syntomic} we can find an affine open covering $X' = \bigcup_{i = 1, \ldots, n} X'_i$ and syntomic morphisms $W_i \to \mathbf{A}^n_Y$ lifting the morphisms $X'_i \to X$, i.e., such that there are fibre product diagrams $$ \xymatrix{ X'_i \ar[d] \ar[r] & W_i \ar[d] \\ X \ar[r] & \mathbf{A}^n_Y } $$ After replacing $X'$ by $\coprod X'_i$ and setting $W = \coprod W_i$ we obtain a fibre product diagram $$ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^h \\ X \ar[r] & \mathbf{A}^n_Y } $$ with $W \to \mathbf{A}^n_Y$ flat and of finite presentation and $X' \to Y$ still pseudo-coherent. Since $W \to \mathbf{A}^n_Y$ is open (see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}) and $X' \to X$ is surjective we can find $f \in \Gamma(\mathbf{A}^n_Y, \mathcal{O})$ such that $X \subset D(f) \subset \Im(h)$. Write $Y = \Spec(R)$, $X = \Spec(A)$, $X' = \Spec(A')$ and $W = \Spec(B)$, $A = R[x_1, \ldots, x_n]/I$ and $A' = B/IB$. Then $R \to A'$ is pseudo-coherent. Picture $$ \xymatrix{ A' = B/IB & B \ar[l] \\ A = R[x_1, \ldots, x_n]/I \ar[u] & R[x_1, \ldots, x_n] \ar[l] \ar[u] } $$ By Lemma \ref{lemma-quotient-of-flat-finitely-presented} we see that $IB$ is pseudo-coherent as a $B$-module. The ring map $R[x_1, \ldots, x_n]_f \to B_f$ is faithfully flat by our choice of $f$ above. This implies that $I_f \subset R[x_1, \ldots, x_n]_f$ is pseudo-coherent, see More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-pseudo-coherent}. Applying Lemma \ref{lemma-quotient-of-flat-finitely-presented} one more time we see that $R \to A$ is pseudo-coherent.
Let $f : X \to S$ be a morphism of schemes. Let $\{g_i : X_i \to X\}$ be an fppf covering such that each composition $f \circ g_i$ is pseudo-coherent. According to Lemma \ref{lemma-dominate-fppf} there exist \begin{enumerate} \item a Zariski open covering $X = \bigcup U_j$, \item surjective finite locally free morphisms $W_j \to U_j$, \item Zariski open coverings $W_j = \bigcup_k W_{j, k}$, \item surjective finite locally free morphisms $T_{j, k} \to W_{j, k}$ \end{enumerate} such that the fppf covering $\{h_{j, k} : T_{j, k} \to X\}$ refines the given covering $\{X_i \to X\}$. Denote $\psi_{j, k} : T_{j, k} \to X_{\alpha(j, k)}$ the morphisms that witness the fact that $\{T_{j, k} \to X\}$ refines the given covering $\{X_i \to X\}$. Note that $T_{j, k} \to X$ is a flat, locally finitely presented morphism, so both $X_i$ and $T_{j, k}$ are pseudo-coherent over $X$ by Lemma \ref{lemma-flat-finite-presentation-pseudo-coherent}. Hence $\psi_{j, k} : T_{j, k} \to X_i$ is pseudo-coherent, see Lemma \ref{lemma-permanence-pseudo-coherent}. Hence $T_{j, k} \to S$ is pseudo coherent as the composition of $\psi_{j, k}$ and $f \circ g_{\alpha(j, k)}$, see Lemma \ref{lemma-composition-pseudo-coherent}. Thus we see we have reduced the lemma to the case of a Zariski open covering (which is OK) and the case of a covering given by a single surjective finite locally free morphism which we deal with in the following paragraph. \medskip\noindent Assume that $X' \to X \to S$ is a sequence of morphisms of schemes with $X' \to X$ surjective finite locally free and $X' \to Y$ pseudo-coherent. Our goal is to show that $X \to S$ is pseudo-coherent. Note that by Descent, Lemma \ref{descent-lemma-flat-finitely-presented-permanence} the morphism $X \to S$ is locally of finite presentation. It is clear that the problem reduces to the case that $X'$, $X$ and $S$ are affine and $X' \to X$ is free of some rank $r > 0$. The corresponding algebra problem is the following: Suppose $R \to A \to A'$ are ring maps such that $R \to A'$ is pseudo-coherent, $R \to A$ is of finite presentation, and $A' \cong A^{\oplus r}$ as an $A$-module. Goal: Show $R \to A$ is pseudo-coherent. The assumption that $R \to A'$ is pseudo-coherent means that $A'$ as an $A'$-module is pseudo-coherent relative to $R$. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-extension-pseudo-coherent} this implies that $A'$ as an $A$-module is pseudo-coherent relative to $R$. Since $A' \cong A^{\oplus r}$ as an $A$-module we see that $A$ as an $A$-module is pseudo-coherent relative to $R$, see More on Algebra, Lemma \ref{more-algebra-lemma-summands-relative-pseudo-coherent}. This by definition means that $R \to A$ is pseudo-coherent and we win.
Assume (1) and let $U, V$ be as in (2). It follows from Lemma \ref{lemma-pseudo-coherent} that $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is pseudo-coherent. Hence it suffices to prove that the property of a ring map being "of finite tor dimension" satisfies conditions (1)(a), (b), (c) of Morphisms, Definition \ref{morphisms-definition-property-local}. These properties follow from More on Algebra, Lemmas \ref{more-algebra-lemma-flat-push-tor-amplitude}, \ref{more-algebra-lemma-flat-base-change-finite-tor-dimension}, and \ref{more-algebra-lemma-glue-tor-amplitude}. Some details omitted.
This translates into the following algebra result: Let $A \to B$ be a perfect ring map. Let $A \to A'$ be flat. Then $A' \to B \otimes_A A'$ is perfect. This result for pseudo-coherent ring maps we have seen in Lemma \ref{lemma-flat-base-change-pseudo-coherent}. The corresponding fact for finite tor dimension follows from More on Algebra, Lemma \ref{more-algebra-lemma-flat-base-change-finite-tor-dimension}.
This translates into the following algebra result: If $A \to B \to C$ are composable perfect ring maps then $A \to C$ is perfect. We have seen this is the case for pseudo-coherent in Lemma \ref{lemma-composition-pseudo-coherent} and its proof. By assumption there exist integers $n$, $m$ such that $B$ has tor dimension $\leq n$ over $A$ and $C$ has tor dimension $\leq m$ over $B$. Then for any $A$-module $M$ we have $$ M \otimes_A^{\mathbf{L}} C = (M \otimes_A^{\mathbf{L}} B) \otimes_B^{\mathbf{L}} C $$ and the spectral sequence of More on Algebra, Example \ref{more-algebra-example-tor} shows that $\text{Tor}^A_p(M, C) = 0$ for $p > n + m$ as desired.
Since a regular immersion is a Koszul-regular immersion, see Divisors, Lemma \ref{divisors-lemma-regular-quasi-regular-immersion}, it suffices to prove the second statement. This translates into the following algebraic statement: Suppose that $I \subset A$ is an ideal generated by a Koszul-regular sequence $f_1, \ldots, f_r$ of $A$. Then $A \to A/I$ is a perfect ring map. Since $A \to A/I$ is surjective this is a presentation of $A/I$ by a polynomial algebra over $A$. Hence it suffices to see that $A/I$ is pseudo-coherent as an $A$-module and has finite tor dimension. By definition of a Koszul sequence the Koszul complex $K(A, f_1, \ldots, f_r)$ is a finite free resolution of $A/I$. Hence $A/I$ is a perfect complex of $A$-modules and we win.
We can factor $f$ as the composition $$ X \longrightarrow X \times_S Y \longrightarrow Y $$ where the first morphism is the map $i = (1, f)$ and the second morphism is the projection. Since $Y \to S$ is flat, see Morphisms, Lemma \ref{morphisms-lemma-smooth-flat}, we see that $X \times_S Y \to Y$ is perfect by Lemma \ref{lemma-flat-base-change-perfect}. As $Y \to S$ is smooth, also $X \times_S Y \to X$ is smooth, see Morphisms, Lemma \ref{morphisms-lemma-base-change-smooth}. Hence $i$ is a section of a smooth morphism, therefore $i$ is a regular immersion, see Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion}. This implies that $i$ is perfect, see Lemma \ref{lemma-regular-immersion-perfect}. We conclude that $f$ is perfect because the composition of perfect morphisms is perfect, see Lemma \ref{lemma-composition-perfect}.
We will use the criterion of Descent, Lemma \ref{descent-lemma-descending-properties-morphisms} to prove this. By Definition \ref{definition-perfect} being perfect is Zariski local on the base. By Lemma \ref{lemma-flat-base-change-perfect} being perfect is preserved under flat base change. The final hypothesis (3) of Descent, Lemma \ref{descent-lemma-descending-properties-morphisms} translates into the following algebra statement: Let $A \to B$ be a faithfully flat ring map. Let $C = A[x_1, \ldots, x_n]/I$ be an $A$-algebra. If $C \otimes_A B$ is perfect as an $B[x_1, \ldots, x_n]$-module, then $C$ is perfect as a $A[x_1, \ldots, x_n]$-module. This is More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-perfect}.