hcju/stacksqa huggingface dataset:
You can deduce this from Lemma \ref{lemma-unramified-over-field} via Lemma \ref{lemma-etale-smooth-unramified} above. Here is a direct proof. \medskip\noindent We will use Algebra, Lemma \ref{algebra-lemma-etale-over-field}. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \Spec(k)$ is \'etale. Conversely, suppose that $X \to \Spec(k)$ is \'etale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma \ref{lemma-algebraic-residue-field-extension-closed-point-fibre} for example). Thus $X$ is a discrete space and we win.
You can deduce this from Algebra, Lemma \ref{algebra-lemma-characterize-etale}. Here is another proof. \medskip\noindent By Lemma \ref{lemma-etale-over-field} a fibre $X_s$ is \'etale and hence smooth over $s$. By Lemma \ref{lemma-smooth-flat-smooth-fibres} we see that $X \to S$ is smooth. By Lemma \ref{lemma-unramified-etale-fibres} we see that $f$ is unramified. We conclude by Lemma \ref{lemma-etale-smooth-unramified}.
Use Lemma \ref{lemma-etale-locally-standard-etale} and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) -- (10) combine Lemmas \ref{lemma-smooth-at-point} and \ref{lemma-unramified-at-point} to see that (1) holds by showing $f$ is both smooth and unramified at $x$ and applying Lemma \ref{lemma-etale-smooth-unramified}. Some details omitted.
Assume first that $f$ locally of finite type. Consider the set $$ T = \{x \in X \mid f\text{ is unramified at }x\} $$ and the corresponding set $T' \subset X'$ for $f'$. Then $T' = (g')^{-1}(T)$ by Lemma \ref{lemma-set-points-where-fibres-unramified}. \medskip\noindent Thus case (1) follows because in case (1) $T$ is the (open) set of points where $f$ is \'etale by Lemma \ref{lemma-flat-unramified-etale}. \medskip\noindent In case (2) let $x' \in W'$. Then $g'$ is flat at $x'$ (Lemma \ref{lemma-base-change-module-flat}) and $g \circ f'$ is flat at $x'$ (Lemma \ref{lemma-composition-module-flat}). It follows that $f$ is flat at $x = g'(x')$ by Lemma \ref{lemma-flat-permanence}. On the other hand, since $x' \in T'$ (Lemma \ref{lemma-base-change-smooth}) we see that $x \in T$. Hence $f$ is \'etale at $x$ by Lemma \ref{lemma-etale-at-point}.
By Lemma \ref{lemma-smooth-locally-standard-smooth} we can find an affine open $U$ as in the lemma such that $\varphi|_U : U \to V$ is standard smooth. Write $U = \Spec(A)$ and $V = \Spec(R)$ so that we can write $$ A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ with $$ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_c & \partial f_2/\partial x_c & \ldots & \partial f_c/\partial x_c \end{matrix} \right) $$ mapping to an invertible element of $A$. Then it is clear that $R[x_{c + 1}, \ldots, x_n] \to A$ is standard smooth of relative dimension $0$. Hence it is smooth of relative dimension $0$. In other words the ring map $R[x_{c + 1}, \ldots, x_n] \to A$ is \'etale. As $\mathbf{A}^{n - c}_V = \Spec(R[x_{c + 1}, \ldots, x_n])$ the lemma with $d = n - c$.
Assume $\mathcal{L}$ is $f$-ample and $\mathcal{M}$ ample. By assumption $Y$ and $f$ are quasi-compact (see Definition \ref{definition-relatively-ample} and Properties, Definition \ref{properties-definition-ample}). Hence $X$ is quasi-compact. Pick $x \in X$. We can choose $m \geq 1$ and $t \in \Gamma(Y, \mathcal{M}^{\otimes m})$ such that $Y_t$ is affine and $f(x) \in Y_t$. Since $\mathcal{L}$ restricts to an ample invertible sheaf on $f^{-1}(Y_t) = X_{f^*t}$ we can choose $n \geq 1$ and $s \in \Gamma(X_{f^*t}, \mathcal{L}^{\otimes n})$ with $x \in (X_{f^*t})_s$ with $(X_{f^*t})_s$ affine. By Properties, Lemma \ref{properties-lemma-invert-s-sections} there exists an integer $e \geq 1$ and a section $s' \in \Gamma(X, \mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes em})$ which restricts to $s(f^*t)^e$ on $X_{f^*t}$. For any $b > 0$ consider the section $s'' = s'(f^*t)^b$ of $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$. Then $X_{s''} = (X_{f^*t})_s$ is an affine open of $X$ containing $x$. Picking $b$ such that $n$ divides $e + b$ we see $\mathcal{L}^{\otimes n} \otimes f^*\mathcal{M}^{\otimes (e + b)m}$ is the $n$th power of $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ for some $a$ and we can get any $a$ divisible by $m$ and big enough. Since $X$ is quasi-compact a finite number of these affine opens cover $X$. We conclude that for some $a$ sufficiently divisible and large enough the invertible sheaf $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $X$. On the other hand, we know that $\mathcal{M}^{\otimes c}$ (and hence its pullback to $X$) is globally generated for all $c \gg 0$ by Properties, Proposition \ref{properties-proposition-characterize-ample}. Thus $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a + c}$ is ample (Properties, Lemma \ref{properties-lemma-ample-tensor-globally-generated}) for $c \gg 0$ and (1) is proved. \medskip\noindent Part (2) follows from Lemma \ref{lemma-quasi-affine-O-ample}, Properties, Lemma \ref{properties-lemma-ample-power-ample}, and part (1).
Let $S = \bigcup_{i = 1, \ldots, n} V_i$ be a finite affine open covering. By Lemma \ref{lemma-characterize-relatively-ample} it suffices to prove that  $\mathcal{L} \otimes f^*\mathcal{M}^{\otimes a}$ is ample on $(g \circ f)^{-1}(V_i)$ for $i = 1, \ldots, n$. Thus the lemma follows from Lemma \ref{lemma-pullback-ample-tensor-relatively-ample}.
By Lemma \ref{lemma-characterize-relatively-ample} it suffices to find an affine open covering $S' = \bigcup U'_i$ such that $\mathcal{L}'$ restricts to an ample invertible sheaf on $(f')^{-1}(U_i')$ for all $i$. We may choose $U'_i$ mapping into an affine open $U_i \subset S$. In this case the morphism $(f')^{-1}(U'_i) \to f^{-1}(U_i)$ is affine as a base change of the affine morphism $U'_i \to U_i$ (Lemma \ref{lemma-base-change-affine}). Thus $\mathcal{L}'|_{(f')^{-1}(U'_i)}$ is ample by Lemma \ref{lemma-pullback-ample-tensor-relatively-ample}.
Assume $f$ is quasi-compact and $\mathcal{L}$ is $g \circ f$-ample. Let $U \subset S$ be an affine open and let $V \subset Y$ be an affine open with $g(V) \subset U$. Then $\mathcal{L}|_{(g \circ f)^{-1}(U)}$ is ample on $(g \circ f)^{-1}(U)$ by assumption. Since $f^{-1}(V) \subset (g \circ f)^{-1}(U)$ we see that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)$ by Properties, Lemma \ref{properties-lemma-ample-on-locally-closed}. Namely, $f^{-1}(V) \to (g \circ f)^{-1}(U)$ is a quasi-compact open immersion by Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence} as $(g \circ f)^{-1}(U)$ is separated (Properties, Lemma \ref{properties-lemma-ample-separated}) and $f^{-1}(V)$ is quasi-compact (as $f$ is quasi-compact). Thus we conclude that $\mathcal{L}$ is $f$-ample by Lemma \ref{lemma-characterize-relatively-ample}.
By definition there exists quasi-coherent $\mathcal{O}_S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Set $\mathcal{A} = \text{Sym}(\mathcal{E})$, so $\mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_S(\mathcal{A})$ by definition. The graded $\mathcal{O}_S$-algebra $\mathcal{A}$ comes equipped with a map $$ \psi : \mathcal{A} \to \bigoplus\nolimits_{n \geq 0} \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \to \bigoplus\nolimits_{n \geq 0} f_*\mathcal{L}^{\otimes n} $$ where the second arrow uses the identification $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. By adjointness of $f_*$ and $f^*$ we get a morphism $\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0}\mathcal{L}^{\otimes n}$. We omit the verification that the morphism $r_{\mathcal{L}, \psi}$ associated to this map is exactly the immersion $i$. Hence the result follows from part (6) of Lemma \ref{lemma-characterize-relatively-ample}.
Being separated is local on the base (see Schemes, Section \ref{schemes-section-separation-axioms}). An immersion is separated (see Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}). Hence the lemma follows since locally $X$ has an immersion into the homogeneous spectrum of a graded ring which is separated, see Constructions, Lemma \ref{constructions-lemma-proj-separated}.
It is clear that (1) implies (2). It is also clear that (4) implies (1); the hypothesis of quasi-separation in (4) is used to guarantee that $f_*\mathcal{L}$ is quasi-coherent via Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. \medskip\noindent Assume (2). We will prove (4). Let $S = \bigcup V_j$ be an open covering as in (2). Set $X_j = f^{-1}(V_j)$ and $f_j : X_j \to V_j$ the restriction of $f$. We see that $f$ is separated by Lemma \ref{lemma-relatively-very-ample-separated} (as being separated is local on the base). By assumption there exists a quasi-coherent $\mathcal{O}_{V_j}$-module $\mathcal{E}_j$ and an immersion $i_j : X_j \to \mathbf{P}(\mathcal{E}_j)$ with $\mathcal{L}|_{X_j} \cong i_j^*\mathcal{O}_{\mathbf{P}(\mathcal{E}_j)}(1)$. The morphism $i_j$ corresponds to a surjection $f_j^*\mathcal{E}_j \to \mathcal{L}|_{X_j}$, see Constructions, Section \ref{constructions-section-projective-bundle}. This map is adjoint to a map $\mathcal{E}_j \to f_*\mathcal{L}|_{V_j}$ such that the composition $$ f_j^*\mathcal{E}_j \to (f^*f_*\mathcal{L})|_{X_j} \to \mathcal{L}|_{X_j} $$ is surjective. We conclude that $\psi : f^*f_*\mathcal{L} \to \mathcal{L}$ is surjective. Let $r_{\mathcal{L}, \psi} : X \to \mathbf{P}(f_*\mathcal{L})$ be the associated morphism. We still have to show that $r_{\mathcal{L}, \psi}$ is an immersion; we urge the reader to prove this for themselves. The $\mathcal{O}_{V_j}$-module map $\mathcal{E}_j \to f_*\mathcal{L}|_{V_j}$ determines a homomorphism on symmetric algebras, which in turn defines a morphism $$ \mathbf{P}(f_*\mathcal{L}|_{V_j}) \supset U_j \longrightarrow \mathbf{P}(\mathcal{E}_j) $$ where $U_j$ is the open subscheme of Constructions, Lemma \ref{constructions-lemma-morphism-relative-proj}. The compatibility of $\psi$ with $\mathcal{E}_j \to f_*\mathcal{L}|_{V_j}$ shows that $r_{\mathcal{L}, \psi}(X_j) \subset U_j$ and that there is a factorization $$ \xymatrix{ X_j \ar[r]^-{r_{\mathcal{L}, \psi}} & U_j \ar[r] & \mathbf{P}(\mathcal{E}_j) } $$ We omit the verification. This shows that $r_{\mathcal{L}, \psi}$ is an immersion. \medskip\noindent At this point we see that (1), (2) and (4) are equivalent. Clearly (4) implies (3). Assume (3). We will prove (1). Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_S$-algebras generated in degree $1$ over $\mathcal{O}_S$. Consider the map of graded $\mathcal{O}_S$-algebras $\text{Sym}(\mathcal{A}_1) \to \mathcal{A}$. This is surjective by hypothesis and hence induces a closed immersion $$ \underline{\text{Proj}}_S(\mathcal{A}) \longrightarrow \mathbf{P}(\mathcal{A}_1) $$ which pulls back $\mathcal{O}(1)$ to $\mathcal{O}(1)$, see Constructions, Lemma \ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}. Hence it is clear that (3) implies (1).
By Definition \ref{definition-very-ample} there exists there exist a quasi-coherent $\mathcal{O}_S$-module $\mathcal{E}$ and an immersion $i : X \to \mathbf{P}(\mathcal{E})$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. The base change of $\mathbf{P}(\mathcal{E})$ to $S'$ is the projective bundle associated to the pullback $\mathcal{E}'$ of $\mathcal{E}$ and the pullback of $\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$ is $\mathcal{O}_{\mathbf{P}(\mathcal{E}')}(1)$, see Constructions, Lemma \ref{constructions-lemma-relative-proj-base-change}. Finally, the base change of an immersion is an immersion (Schemes, Lemma \ref{schemes-lemma-base-change-immersion}).
Assume (1), (2) and (3). Condition (3) means $S = \Spec(R)$ for some ring $R$. Condition (1) means by definition there exists a quasi-coherent $\mathcal{O}_S$-module $\mathcal{E}$ and an immersion $\alpha : X \to \mathbf{P}(\mathcal{E})$ such that $\mathcal{L} = \alpha^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Write $\mathcal{E} = \widetilde{M}$ for some $R$-module $M$. Thus we have $$ \mathbf{P}(\mathcal{E}) = \text{Proj}(\text{Sym}_R(M)). $$ Since $\alpha$ is an immersion, and since the topology of $\text{Proj}(\text{Sym}_R(M))$ is generated by the standard opens $D_{+}(f)$, $f \in \text{Sym}_R^d(M)$, $d \geq 1$, we can find for each $x \in X$ an $f \in \text{Sym}_R^d(M)$, $d \geq 1$, with $\alpha(x) \in D_{+}(f)$ such that $$ \alpha|_{\alpha^{-1}(D_{+}(f))} : \alpha^{-1}(D_{+}(f)) \to D_{+}(f) $$ is a closed immersion. Condition (2) implies $X$ is quasi-compact. Hence we can find a finite collection of elements $f_j \in \text{Sym}_R^{d_j}(M)$, $d_j \geq 1$ such that for each $f = f_j$ the displayed map above is a closed immersion and such that $\alpha(X) \subset \bigcup D_{+}(f_j)$. Write $U_j = \alpha^{-1}(D_{+}(f_j))$. Note that $U_j$ is affine as a closed subscheme of the affine scheme $D_{+}(f_j)$. Write $U_j = \Spec(A_j)$. Condition (2) also implies that $A_j$ is of finite type over $R$, see Lemma \ref{lemma-locally-finite-type-characterize}. Choose finitely many $x_{j, k} \in A_j$ which generate $A_j$ as a $R$-algebra. Since $\alpha|_{U_j}$ is a closed immersion we see that $x_{j, k}$ is the image of an element $$ f_{j, k}/f_j^{e_{j, k}} \in \text{Sym}_R(M)_{(f_j)} = \Gamma(D_{+}(f_j), \mathcal{O}_{\text{Proj}(\text{Sym}_R(M))}). $$ Finally, choose $n \geq 1$ and elements $y_0, \ldots, y_n \in M$ such that each of the polynomials $f_j, f_{j, k} \in \text{Sym}_R(M)$ is a polynomial in the elements $y_t$ with coefficients in $R$. Consider the graded ring map $$ \psi : R[Y_0, \ldots, Y_n] \longrightarrow \text{Sym}_R(M), \quad Y_i \longmapsto y_i. $$ Denote $F_j$, $F_{j, k}$ the elements of $R[Y_0, \ldots, Y_n]$ such that $\psi(F_j) = f_j$ and $\psi(F_{j, k}) = f_{j, k}$. By Constructions, Lemma \ref{constructions-lemma-morphism-proj} we obtain an open subscheme $$ U(\psi) \subset \text{Proj}(\text{Sym}_R(M)) $$ and a morphism $r_\psi : U(\psi) \to \mathbf{P}^n_R$. This morphism satisfies $r_\psi^{-1}(D_{+}(F_j)) = D_{+}(f_j)$, and hence we see that $\alpha(X) \subset U(\psi)$. Moreover, it is clear that $$ i = r_\psi \circ \alpha : X \longrightarrow \mathbf{P}^n_R $$ is still an immersion since $i^\sharp(F_{j, k}/F_j^{e_{j, k}}) = x_{j, k} \in A_j = \Gamma(U_j, \mathcal{O}_X)$ by construction. Moreover, the morphism $r_\psi$ comes equipped with a map $\theta : r_\psi^*\mathcal{O}_{\mathbf{P}^n_R}(1) \to \mathcal{O}_{\text{Proj}(\text{Sym}_R(M))}(1)|_{U(\psi)}$ which is an isomorphism in this case (for construction $\theta$ see lemma cited above; some details omitted). Since the original map $\alpha$ was assumed to have the property that $\mathcal{L} = \alpha^*\mathcal{O}_{\text{Proj}(\text{Sym}_R(M))}(1)$ we win.
Let $A = \Gamma(X, \mathcal{O}_X)$. By assumption $X$ is quasi-compact and is identified with an open subscheme of $\Spec(A)$, see Properties, Lemma \ref{properties-lemma-quasi-affine}. Moreover, the set of opens $X_f$, for those $f \in A$ such that $X_f$ is affine, forms a basis for the topology of $X$, see the proof of Properties, Lemma \ref{properties-lemma-quasi-affine}. Hence we can find a finite number of $f_j \in A$, $j = 1, \ldots, m$ such that $X = \bigcup X_{f_j}$, and such that $\pi(X_{f_j}) \subset V_j$ for some affine open $V_j \subset S$. By Lemma \ref{lemma-locally-finite-type-characterize} the ring maps $\mathcal{O}(V_j) \to \mathcal{O}(X_{f_j}) = A_{f_j}$ are of finite type. Thus we may choose $a_1, \ldots, a_N \in A$ such that the elements $a_1, \ldots, a_N, 1/f_j$ generate $A_{f_j}$ over $\mathcal{O}(V_j)$ for each $j$. Take $n = m + N$ and let $$ i : X \longrightarrow \mathbf{A}^n_S $$ be the morphism given by the global sections $f_1, \ldots, f_m, a_1, \ldots, a_N$ of the structure sheaf of $X$. Let $D(x_j) \subset \mathbf{A}^n_S$ be the open subscheme where the $j$th coordinate function is nonzero. Then for $1 \leq j \leq m$ we have $i^{-1}(D(x_j)) = X_{f_j}$ and the induced morphism $X_{f_j} \to D(x_j)$ factors through the affine open $\Spec(\mathcal{O}(V_j)[x_1, \ldots, x_n, 1/x_j])$ of $D(x_j)$. Since the ring map $\mathcal{O}(V_j)[x_1, \ldots, x_n, 1/x_j] \to A_{f_j}$ is surjective by construction we conclude that $i^{-1}(D(x_j)) \to D(x_j)$ is an immersion as desired.
The equivalence of (1) and (2) is Lemma \ref{lemma-ample-over-affine}. The implication (2) $\Rightarrow$ (6) is Lemma \ref{lemma-quasi-projective-finite-type-over-S}. Trivially (6) implies (5). As $\mathbf{P}^n_S$ is a projective bundle over $S$ (see Constructions, Lemma \ref{constructions-lemma-projective-space-bundle}) we see that (5) implies (3) and (6) implies (4) from the definition of a relatively very ample sheaf. Trivially (4) implies (3). To finish we have to show that (3) implies (2) which follows from Lemma \ref{lemma-ample-very-ample} and Lemma \ref{lemma-ample-power-ample}.
Trivially (3) implies (2). Lemma \ref{lemma-ample-very-ample} guarantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that $\mathcal{L}$ is $f$-ample. Choose a finite affine open covering $S = V_1 \cup \ldots \cup V_m$. Write $X_i = f^{-1}(V_i)$. By Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} above we see there exists a $d_0$ such that $\mathcal{L}^{\otimes d}$ is relatively very ample on $X_i/V_i$ for all $d \geq d_0$. Hence we conclude (1) implies (3) by Lemma \ref{lemma-relatively-very-ample}.
By Lemma \ref{lemma-characterize-very-ample-on-finite-type} we reduce to the case $S$ is affine. Combining Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} and Properties, Proposition \ref{properties-proposition-characterize-ample} we can find an integer $d_0$ such that $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ is globally generated. Choose global sections $s_0, \ldots, s_n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ which generate it. This determines a morphism $j : X \to \mathbf{P}^n_S$ over $S$. By Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} we can also pick an integer $d_1$ such that for all $d \geq d_1$ there exist sections $t_{d, 0}, \ldots, t_{d, n(d)}$ of $\mathcal{L}^{\otimes d}$ which generate it and define an immersion $$ j_d = \varphi_{\mathcal{L}^{\otimes d}, t_{d, 0}, \ldots, t_{d, n(d)}} : X \longrightarrow \mathbf{P}^{n(d)}_S $$ over $S$. Then for $d \geq d_0 + d_1$ we can consider the morphism $$ \varphi_{\mathcal{N} \otimes \mathcal{L}^{\otimes d}, s_j \otimes t_{d - d_0, i}} : X \longrightarrow \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S $$ This morphism is an immersion as it is the composition $$ S \to \mathbf{P}^n_S \times_S \mathbf{P}^{n(d - d_0)}_S \to \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S $$ where the first morphism is $(j, j_{d - d_0})$ and the second is the Segre embedding (Constructions, Lemma \ref{constructions-lemma-segre-embedding}). Since $j$ is an immersion, so is $(j, j_{d - d_0})$ (apply Lemma \ref{lemma-immersion-permanence}). We have a composition of immersions and hence an immersion (Schemes, Lemma \ref{schemes-lemma-composition-immersion}).
By Lemma \ref{lemma-H-quasi-projective-quasi-projective} we see that (2) implies (1). Assume (1). The question is local on $S$ and hence we may assume $S$ is affine, $X$ of finite type over $S$ and $\mathcal{L}$ is a relatively ample invertible sheaf on $X/S$. By Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} we may assume $\mathcal{L}$ is ample on $X$. By Lemma \ref{lemma-quasi-projective-finite-type-over-S} we see that there exists an immersion of $X$ into a projective space over $S$, i.e., $X$ is H-quasi-projective over $S$ as desired.
A composition of closed morphisms is closed. If $X \to Y \to Z$ are universally closed morphisms and $Z' \to Z$ is any morphism, then we see that $Z' \times_Z X = (Z' \times_Z Y) \times_Y X  \to Z' \times_Z Y$ is closed and $Z' \times_Z Y \to Z'$ is closed. Hence the result for universally closed morphisms. We have seen that ``separated'' and ``finite type'' are preserved under compositions (Schemes, Lemma \ref{schemes-lemma-separated-permanence} and Lemma \ref{lemma-composition-finite-type}). Hence the result for proper morphisms.
The base change of a closed immersion is a closed immersion (Schemes, Lemma \ref{schemes-lemma-base-change-immersion}). Hence it is universally closed. A closed immersion is separated (Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}). A closed immersion is of finite type (Lemma \ref{lemma-immersion-locally-finite-type}). Hence a closed immersion is proper.
Assume that $X \to S$ is universally closed (resp.\ proper). We factor the morphism as $X \to X \times_S Y \to Y$. The first morphism is a closed immersion, see Schemes, Lemma \ref{schemes-lemma-semi-diagonal}. Hence the first morphism is proper (Lemma \ref{lemma-closed-immersion-proper}). The projection $X \times_S Y \to Y$ is the base change of a universally closed (resp.\ proper) morphism and hence universally closed (resp.\ proper), see Lemma \ref{lemma-base-change-proper}. Thus $X \to Y$ is universally closed (resp.\ proper) as the composition of universally closed (resp.\ proper) morphisms (Lemma \ref{lemma-composition-proper}).
Let $f : X \to S$ be a morphism. Assume that $f$ is not quasi-compact. Our goal is to show that $f$ is not universally closed. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} there exists an affine open $V \subset S$ such that $f^{-1}(V)$ is not quasi-compact. To achieve our goal it suffices to show that $f^{-1}(V) \to V$ is not universally closed, hence we may assume that $S = \Spec(A)$ for some ring $A$. \medskip\noindent Write $X = \bigcup_{i \in I} X_i$ where the $X_i$ are affine open subschemes of $X$.  Let $T = \Spec(A[y_i ; i \in I])$. Let $T_i = D(y_i) \subset T$. Let $Z$ be the closed set $(X \times_S T) - \bigcup_{i \in I} (X_i \times_S T_i)$.  It suffices to prove that the image $f_T(Z)$ of $Z$ under $f_T : X \times_S T \to T$ is not closed. \medskip\noindent There exists a point $s \in S$ such that there is no neighborhood $U$ of $s$ in $S$ such that $X_U$ is quasi-compact. Otherwise we could cover $S$ with finitely many such $U$ and Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} would imply $f$ quasi-compact. Fix such an $s \in S$. \medskip\noindent First we check that $f_T(Z_s) \ne T_s$.  Let $t \in T$ be the point lying over $s$ with $\kappa(t) = \kappa(s)$ such that $y_i = 1$ in $\kappa(t)$ for all $i$. Then $t \in T_i$ for all $i$, and the fiber of $Z_s \to T_s$ above $t$ is isomorphic to $(X - \bigcup_{i \in I} X_i)_s$, which is empty. Thus $t \in T_s - f_T(Z_s)$. \medskip\noindent Assume $f_T(Z)$ is closed in $T$. Then there exists an element $g \in A[y_i; i \in I]$ with $f_T(Z) \subset V(g)$ but $t \not \in V(g)$. Hence the image of $g$ in $\kappa(t)$ is nonzero. In particular some coefficient of $g$ has nonzero image in $\kappa(s)$. Hence this coefficient is invertible on some neighborhood $U$ of $s$. Let $J$ be the finite set of $j \in I$ such that $y_j$ appears in $g$. Since $X_U$ is not quasi-compact, we may choose a point $x \in X - \bigcup_{j \in J} X_j$ lying above some $u \in U$. Since $g$ has a coefficient that is invertible on $U$, we can find a point $t' \in T$ lying above $u$ such that $t' \not \in V(g)$ and $t' \in V(y_i)$ for all $i \notin J$. This is true because $V(y_i; i \in I, i \not\in J) = \Spec(A[t_j; j\in J])$ and the set of points of this scheme lying over $u$ is bijective with $\Spec(\kappa(u)[t_j; j \in J])$. In other words $t' \notin T_i$ for each $i \notin J$. By Schemes, Lemma \ref{schemes-lemma-points-fibre-product} we can find a point $z$ of $X \times_S T$ mapping to $x \in X$ and to $t' \in T$. Since $x \not \in X_j$ for $j \in J$ and $t' \not \in T_i$ for $i \in I \setminus J$ we see that $z \in Z$. On the other hand $f_T(z) = t' \not \in V(g)$ which contradicts $f_T(Z) \subset V(g)$. Thus the assumption ``$f_T(Z)$ closed'' is wrong and we conclude indeed that $f_T$ is not closed, as desired.