hcju/stacksqa huggingface dataset:
Note that $Y$ is locally Noetherian by Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}. Because $f$ is surjective the fibre $Y_\xi$ is not empty. Pick $\xi' \in Y$ mapping to $\xi$. Let $Z' = \overline{\{\xi'\}}$. We may think of $Z' \subset Y$ as a reduced closed subscheme, see Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme}. Hence the sheaf $\mathcal{F} = (Z' \to Y)_*\mathcal{O}_{Z'}$ is a coherent sheaf on $Y$ (see Lemma \ref{lemma-finite-pushforward-coherent}). Look at the commutative diagram $$ \xymatrix{ Z' \ar[r]_{i'} \ar[d]_{f'} & Y \ar[d]^f \\ Z \ar[r]^i & X } $$ We see that $f_*\mathcal{F} = i_*f'_*\mathcal{O}_{Z'}$. Hence the stalk of $f_*\mathcal{F}$ at $\xi$ is the stalk of $f'_*\mathcal{O}_{Z'}$ at $\xi$. Note that since $Z'$ is integral with generic point $\xi'$ we have that $\xi'$ is the only point of $Z'$ lying over $\xi$, see Algebra, Lemmas \ref{algebra-lemma-finite-is-integral} and \ref{algebra-lemma-integral-no-inclusion}. Hence the stalk of $f'_*\mathcal{O}_{Z'}$ at $\xi$ equal $\mathcal{O}_{Z', \xi'} = \kappa(\xi')$. In particular the stalk of $f_*\mathcal{F}$ at $\xi$ is not zero. This combined with the fact that $f_*\mathcal{F}$ is of the form $i_*f'_*(\text{something})$ implies the lemma.
We will prove that under the assumptions of the lemma for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. This will in particular imply that $H^1(X, \mathcal{I}) = 0$ for every quasi-coherent sheaf of ideals of $\mathcal{O}_X$. Then it follows that $X$ is affine from either Lemma \ref{lemma-quasi-compact-h1-zero-covering} or Lemma \ref{lemma-quasi-separated-h1-zero-covering}. \medskip\noindent Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow H^1(X, \mathcal{F}) = 0. $$ We are going to apply Lemma \ref{lemma-property-higher-rank-cohomological}. Thus we have to verify (1), (2) and (3) of that lemma for $\mathcal{P}$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves. Property (2) follows since $H^1(X, -)$ is an additive functor. To see (3) let $Z \subset X$ be an integral closed subscheme with generic point $\xi$. Let $\mathcal{F}$ be a coherent sheaf on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi$ is annihilated by $\mathfrak m_\xi$, see Lemma \ref{lemma-finite-morphism-Noetherian}. We claim that taking $\mathcal{G} = f_*\mathcal{F}$ works. We only have to verify part (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological}. Hence assume that $\mathcal{J} \subset \mathcal{O}_X$ is a quasi-coherent sheaf of ideals such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$. A finite morphism is affine hence by Lemma \ref{lemma-affine-morphism-projection-ideal} we see that $\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})$. Also, as pointed out in the proof of Lemma \ref{lemma-affine-morphism-projection-ideal} the sheaf $f^{-1}\mathcal{J}\mathcal{F}$ is a quasi-coherent $\mathcal{O}_Y$-module. Since $Y$ is affine we see that $H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0$, see Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero}. Since $f$ is finite, hence affine, we see that $$ H^1(X, \mathcal{J}\mathcal{G}) = H^1(X, f_*(f^{-1}\mathcal{J}\mathcal{F})) = H^1(Y, f^{-1}\mathcal{J}\mathcal{F}) = 0 $$ by Lemma \ref{lemma-relative-affine-cohomology}. Hence the quasi-coherent subsheaf $\mathcal{G}' = \mathcal{J}\mathcal{G}$ satisfies $\mathcal{P}$. This verifies property (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological} as desired.
By assumption there exists a surjection of graded $A_0$-algebras $$ A_0[T_0, \ldots, T_n] \longrightarrow A $$ where $\deg(T_j) = 1$ for $j = 0, \ldots, n$. By Constructions, Lemma \ref{constructions-lemma-surjective-graded-rings-generated-degree-1-map-proj} this defines a closed immersion $i : X \to \mathbf{P}^n_{A_0}$ such that $i^*\mathcal{O}_{\mathbf{P}^n_{A_0}}(1) = \mathcal{O}_X(1)$. In particular, $X$ is Noetherian as a closed subscheme of the Noetherian scheme $\mathbf{P}^n_{A_0}$. We claim that the results of the lemma for $\mathcal{F}$ follow from the corresponding results of Lemma \ref{lemma-coherent-projective} for the coherent sheaf $i_*\mathcal{F}$ (Lemma \ref{lemma-i-star-equivalence}) on $\mathbf{P}^n_{A_0}$. For example, by this lemma there exists a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, r} \mathcal{O}_{\mathbf{P}^n_{A_0}}(d_j) \longrightarrow i_*\mathcal{F}. $$ By adjunction this corresponds to a map $\bigoplus_{j = 1, \ldots, r} \mathcal{O}_X(d_j) \longrightarrow \mathcal{F}$ which is surjective as well. The statements on cohomology follow from the fact that $H^p(X, \mathcal{F}(d)) = H^p(\mathbf{P}^n_{A_0}, i_*\mathcal{F}(d))$ by Lemma \ref{lemma-relative-affine-cohomology}.
Because $M$ is a finite $A$-module we see that $\widetilde{M}$ is a finite type $\mathcal{O}_X$-module, i.e., a coherent $\mathcal{O}_X$-module. Set $N = \bigoplus_{n \geq 0} \Gamma(X, \widetilde{M}(n))$. We have to show that the map $M \to N$ of graded $A$-modules is an isomorphism in all sufficiently large degrees. By Properties, Lemma \ref{properties-lemma-proj-quasi-coherent} we have a canonical isomorphism $\widetilde{N} \to \widetilde{M}$ such that $M_n \to N_n = \Gamma(X, \widetilde{M}(n))$ is the canonical map. Let $K = \Ker(M \to N)$ and $Q = \Coker(M \to N)$. Recall that the functor $M \mapsto \widetilde{M}$ is exact, see Constructions, Lemma \ref{constructions-lemma-proj-sheaves}. Hence we see that $\widetilde{K} = 0$ and $\widetilde{Q} = 0$. On the other hand, $A$ is a Noetherian ring and $M$ and $N$ are finitely generated $A$-modules (for $N$ this follows from the last part of Lemma \ref{lemma-coherent-on-proj}). Hence $K$ and $Q$ are finite $A$-modules. Thus it suffices to show that a finite $A$-module $K$ with $\widetilde{K} = 0$ has only finitely many nonzero homogeneous parts $K_d$. To do this, let $x_1, \ldots, x_r \in K$ be homogeneous generators say sitting in degrees $d_1, \ldots, d_r$. Let $f_1, \ldots, f_n \in A_1$ be elements generating $A$ over $A_0$. For each $i$ and $j$ there exists an $n_{ij} \geq 0$ such that $f_i^{n_{ij}} x_j = 0$ in $K_{d_j + n_{ij}}$: if not then $x_i/f_i^{d_i} \in K_{(f_i)}$ would not be zero, i.e., $\widetilde{K}$ would not be zero. Then we see that $K_d$ is zero for $d > \max_j(d_j + \sum_i n_{ij})$ as every element of $K_d$ is a sum of terms where each term is a monomials in the $f_i$ times one of the $x_j$ of total degree $d$.
We will prove this by reducing the statement to Lemma \ref{lemma-coherent-on-proj}. By Algebra, Lemmas \ref{algebra-lemma-graded-Noetherian} and \ref{algebra-lemma-S-plus-generated} the ring $A_0$ is Noetherian and $A$ is generated over $A_0$ by finitely many elements $f_1, \ldots, f_r$ homogeneous of positive degree. Let $d$ be a sufficiently divisible integer. Set $A' = A^{(d)}$ with notation as in Algebra, Section \ref{algebra-section-graded}. Then $A'$ is generated over $A'_0 = A_0$ by elements of degree $1$, see Algebra, Lemma \ref{algebra-lemma-uple-generated-degree-1}. Thus Lemma \ref{lemma-coherent-on-proj} applies to $X' = \text{Proj}(A')$. \medskip\noindent By Constructions, Lemma \ref{constructions-lemma-d-uple} there exist an isomorphism of schemes $i : X \to X'$ and isomorphisms $\mathcal{O}_X(nd) \to i^*\mathcal{O}_{X'}(n)$ compatible with the map $A' \to A$ and the maps $A_n \to H^0(X, \mathcal{O}_X(n))$ and $A'_n \to H^0(X', \mathcal{O}_{X'}(n))$. Thus Lemma \ref{lemma-coherent-on-proj} implies $X$ is Noetherian and that (1) and (2) hold. To see (3) and (4) we can use that for any fixed $k$, $p$, and $q$ we have $$ \bigoplus\nolimits_{dn + q \geq k} H^p(X, \mathcal{F}(dn + q)) = \bigoplus\nolimits_{dn + q \geq k} H^p(X', (i_*\mathcal{F}(q))(n) $$ by the compatibilities above. If $p > 0$, we have the vanishing of the right hand side for $k$ depending on $q$ large enough by Lemma \ref{lemma-coherent-on-proj}. Since there are only a finite number of congruence classes of integers modulo $d$, we see that (3) holds for $\mathcal{F}$ on $X$. If $p = 0$, then we have that the right hand side is a finite $A'$-module by Lemma \ref{lemma-coherent-on-proj}. Using the finiteness of congruence classes once more, we find that $\bigoplus_{n \geq k} H^0(X, \mathcal{F}(n))$ is a finite $A'$-module too. Since the $A'$-module structure comes from the $A$-module structure (by the compatibilities mentioned above), we conclude it is finite as an $A$-module as well.
By Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample} there exists a $d > 0$ and an immersion $i : X \to \mathbf{P}^n_R$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^n_R}(1)$. Since $X$ is proper over $R$ the morphism $i$ is a closed immersion (Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}). Thus we have $H^i(X, \mathcal{G}) = H^i(\mathbf{P}^n_R, i_*\mathcal{G})$ for any quasi-coherent sheaf $\mathcal{G}$ on $X$ (by Lemma \ref{lemma-relative-affine-cohomology} and the fact that closed immersions are affine, see Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-affine}). Moreover, if $\mathcal{G}$ is coherent, then $i_*\mathcal{G}$ is coherent as well (Lemma \ref{lemma-i-star-equivalence}). We will use these facts without further mention. \medskip\noindent Proof of (1). Set $S = R[T_0, \ldots, T_n]$ so that $\mathbf{P}^n_R = \text{Proj}(S)$. Observe that $A$ is an $S$-algebra (but the ring map $S \to A$ is not a homomorphism of graded rings because $S_n$ maps into $A_{dn}$). By the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we have $$ i_*(\mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{L}^{\otimes q}) \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(n) $$ for all $n \in \mathbf{Z}$. We conclude that $\bigoplus_{n \geq 0} A_{nd + q}$ is a finite graded $S$-module by Lemma \ref{lemma-coherent-projective}. Since $A = \bigoplus_{q \in \{0, \ldots, d - 1} \bigoplus_{n \geq 0} A_{nd + q}$ we see that $A$ is finite as an $S$-algebra, hence (1) is true. \medskip\noindent Proof of (2). This follows from Properties, Proposition \ref{properties-proposition-characterize-ample}. \medskip\noindent Proof of (3). Apply Lemma \ref{lemma-coherent-projective} and use $H^p(X, \mathcal{F}) = H^p(\mathbf{P}^n_R, i_*\mathcal{F})$. \medskip\noindent Proof of (4). Fix $p > 0$. By the projection formula we have $$ i_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes q}) \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(n) $$ for all $n \in \mathbf{Z}$. By Lemma \ref{lemma-coherent-projective} we conclude that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{nd + q}) = 0$ for $n \gg 0$. Since there are only finitely many congruence classes of integers modulo $d$ this proves (4). \medskip\noindent Proof of (5). Fix an integer $k$. Set $M = \bigoplus_{n \geq k} H^0(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. Arguing as above we conclude that $\bigoplus_{nd + q \geq k} A_{nd + q}$ is a finite graded $S$-module. Since $M = \bigoplus_{q \in \{0, \ldots, d - 1\}} \bigoplus_{nd + q \geq k} M_{nd + q}$ we see that $M$ is finite as an $S$-module. Since the $S$-module structure factors through the ring map $S \to A$, we conclude that $M$ is finite as an $A$-module.
Choose a finite affine open covering $S = \bigcup V_j$ and set $X_j = f^{-1}(V_j)$. Clearly, if we solve the question for each of the finitely many systems $(X_j \to V_j, \mathcal{L}|_{X_j}, \mathcal{F}|_{V_j})$ then the result follows. Thus we may assume $S$ is affine. In this case the vanishing of $R^pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes n})$ is equivalent to the vanishing of $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$, see Lemma \ref{lemma-quasi-coherence-higher-direct-images-application}. Thus the required vanishing follows from Lemma \ref{lemma-coherent-proper-ample} (which applies because $\mathcal{L}$ is ample on $X$ by Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample}).
We first remark that a locally projective morphism is proper (Morphisms, Lemma \ref{morphisms-lemma-locally-projective-proper}) and hence of finite type. In particular $X$ is locally Noetherian (Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}) and hence the statement makes sense. Moreover, by Lemma \ref{lemma-quasi-coherence-higher-direct-images} the sheaves $R^pf_*\mathcal{F}$ are quasi-coherent. \medskip\noindent Having said this the statement is local on $S$ (for example by Cohomology, Lemma \ref{cohomology-lemma-localize-higher-direct-images}). Hence we may assume $S = \Spec(R)$ is the spectrum of a Noetherian ring, and $X$ is a closed subscheme of $\mathbf{P}^n_R$ for some $n$, see Morphisms, Lemma \ref{morphisms-lemma-characterize-locally-projective}. In this case, the sheaves $R^pf_*\mathcal{F}$ are the quasi-coherent sheaves associated to the $R$-modules $H^p(X, \mathcal{F})$, see Lemma \ref{lemma-quasi-coherence-higher-direct-images-application}. Hence it suffices to show that $R$-modules $H^p(X, \mathcal{F})$ are finite $R$-modules (Lemma \ref{lemma-coherent-Noetherian}). This follows from Lemma \ref{lemma-coherent-proper-ample} (because the restriction of $\mathcal{O}_{\mathbf{P}^n_R}(1)$ to $X$ is ample on $X$).
The pullback of an ample invertible sheaf by a quasi-affine morphism is ample, see Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}. This proves one of the implications as a finite morphism is affine by definition. \medskip\noindent Assume that $f^*\mathcal{L}$ is ample. Let $P$ be the following property on coherent $\mathcal{O}_X$-modules $\mathcal{F}$: there exists an $n_0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. We will prove that $P$ holds for any coherent $\mathcal{O}_X$-module $\mathcal{F}$, which implies $\mathcal{L}$ is ample by Lemma \ref{lemma-vanshing-gives-ample}. We are going to apply Lemma \ref{lemma-property-higher-rank-cohomological}. Thus we have to verify (1), (2) and (3) of that lemma for $P$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since $H^p(X, -)$ is an additive functor. To see (3) let $Z \subset X$ be an integral closed subscheme with generic point $\xi$. Let $\mathcal{F}$ be a coherent sheaf on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi$ is annihilated by $\mathfrak m_\xi$, see Lemma \ref{lemma-finite-morphism-Noetherian}. We claim that taking $\mathcal{G} = f_*\mathcal{F}$ works. We only have to verify part (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological}. Hence assume that $\mathcal{J} \subset \mathcal{O}_X$ is a quasi-coherent sheaf of ideals such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi}$. A finite morphism is affine hence by Lemma \ref{lemma-affine-morphism-projection-ideal} we see that $\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})$. Also, as pointed out in the proof of Lemma \ref{lemma-affine-morphism-projection-ideal} the sheaf $f^{-1}\mathcal{J}\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module. As $\mathcal{L}$ is ample we see from Lemma \ref{lemma-vanshing-gives-ample} that there exists an $n_0$ such that $$ H^p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0, $$ for $n \geq n_0$ and $p > 0$. Since $f$ is finite, hence affine, we see that \begin{align*} H^p(X, \mathcal{J}\mathcal{G} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) & = H^p(X, f_*(f^{-1}\mathcal{J}\mathcal{F}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}) \\ & = H^p(X, f_*(f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n})) \\ & = H^p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes_{\mathcal{O}_Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*} Here we have used the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) and Lemma \ref{lemma-relative-affine-cohomology}. Hence the quasi-coherent subsheaf $\mathcal{G}' = \mathcal{J}\mathcal{G}$ satisfies $P$. This verifies property (3)(c) of Lemma \ref{lemma-property-higher-rank-cohomological} as desired.
Note that for $p = 0$ this is Properties, Lemma \ref{properties-lemma-invert-s-sections}. We will prove the statement using the induction principle (Lemma \ref{lemma-induction-principle}) where for $U \subset X$ quasi-compact open we let $P(U)$ be the property: for all $p \geq 0$ the map $$ H^p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^p(U_s, \mathcal{F}) $$ is an isomorphism. \medskip\noindent If $U$ is affine, then both sides of the arrow displayed above are zero for $p > 0$ by Lemma \ref{lemma-quasi-coherent-affine-cohomology-zero} and Properties, Lemma \ref{properties-lemma-affine-cap-s-open} and the statement is true. If $P$ is true for $U$, $V$, and $U \cap V$, then we can use the Mayer-Vietoris sequences (Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris}) to obtain a map of long exact sequences $$ \xymatrix{ H^{p - 1}_*(U \cap V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^p_*(U \cup V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \oplus H^p_*(V, \mathcal{L}, \mathcal{F})_{(s)} \ar[d] \\ H^{p - 1}(U_s \cap V_s, \mathcal{F}) \ar[r]& H^p(U_s \cup V_s, \mathcal{F}) \ar[r] & H^p(U_s, \mathcal{F}) \oplus H^p(V_s, \mathcal{F}) } $$ (only a snippet shown). Observe that $U_s \cap V_s = (U \cap V)_s$ and that $U_s \cup V_s = (U \cup V)_s$. Thus the left and right vertical maps are isomorphisms (as well as one more to the right and one more to the left which are not shown in the diagram). We conclude that $P(U \cup V)$ holds by the 5-lemma (Homology, Lemma \ref{homology-lemma-five-lemma}). This finishes the proof.
Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}. Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample, hence $\mathcal{O}_X$ is ample by Lemma \ref{lemma-ample-on-reduction}, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards.
We claim $H^p(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_X$-module and $p > 0$. Since $X$ is quasi-compact by Properties, Definition \ref{properties-definition-ample} the claim finishes the proof by Lemma \ref{lemma-quasi-compact-h1-zero-covering}. The scheme $X$ is separated by Properties, Lemma \ref{properties-lemma-ample-separated}. Say $X$ is covered by $e + 1$ affine opens. Then $H^p(X, \mathcal{F}) = 0$ for $p > e$, see Lemma \ref{lemma-vanishing-nr-affines}. Thus we may use descending induction on $p$ to prove the claim. Writing $\mathcal{F}$ as a filtered colimit of finite type quasi-coherent modules (Properties, Lemma \ref{properties-lemma-quasi-coherent-colimit-finite-type}) and using Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit} we may assume $\mathcal{F}$ is of finite type. Then we can choose $n > n_0$ such that $\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes n}$ is globally generated, see Properties, Proposition \ref{properties-proposition-characterize-ample}. This means there is a short exact sequence $$ 0 \to \mathcal{F}' \to \bigoplus\nolimits_{i \in I} \mathcal{L}^{\otimes -n} \to \mathcal{F} \to 0 $$ for some set $I$ (in fact we can choose $I$ finite). By induction hypothesis we have $H^{p + 1}(X, \mathcal{F}') = 0$ and by assumption (combined with the already used commutation of cohomology with colimits) we have $H^p(X, \bigoplus_{i \in I} \mathcal{L}^{\otimes -n}) = 0$. From the long exact cohomology sequence we conclude that $H^p(X, \mathcal{F}) = 0$ as desired.
This is just the affine case of Proposition \ref{proposition-proper-pushforward-coherent}. Namely, by Lemmas \ref{lemma-quasi-coherence-higher-direct-images} and \ref{lemma-quasi-coherence-higher-direct-images-application} we know that $R^if_*\mathcal{F}$ is the quasi-coherent sheaf associated to the $A$-module $H^i(X, \mathcal{F})$ and by Lemma \ref{lemma-coherent-Noetherian} this is a coherent sheaf if and only if $H^i(X, \mathcal{F})$ is an $A$-module of finite type.
To prove this we consider the fibre product diagram $$ \xymatrix{ X' = \Spec(B) \times_{\Spec(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^f \\ \Spec(B) \ar[r] & \Spec(A) } $$ Note that $f'$ is a proper morphism, see Morphisms, Lemma \ref{morphisms-lemma-base-change-proper}. Also, $B$ is a finitely generated $A$-algebra, and hence Noetherian (Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence}). This implies that $X'$ is a Noetherian scheme (Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}). Note that $X'$ is the relative spectrum of the quasi-coherent $\mathcal{O}_X$-algebra $\mathcal{B}$ by Constructions, Lemma \ref{constructions-lemma-spec-properties}. Since $\mathcal{F}$ is a quasi-coherent $\mathcal{B}$-module we see that there is a unique quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$ such that $\pi_*\mathcal{F}' = \mathcal{F}$, see Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules} Since $\mathcal{F}$ is finite type as a $\mathcal{B}$-module we conclude that $\mathcal{F}'$ is a finite type $\mathcal{O}_{X'}$-module (details omitted). In other words, $\mathcal{F}'$ is a coherent $\mathcal{O}_{X'}$-module (Lemma \ref{lemma-coherent-Noetherian}). Since the morphism $\pi : X' \to X$ is affine we have $$ H^p(X, \mathcal{F}) = H^p(X', \mathcal{F}') $$ by Lemma \ref{lemma-relative-affine-cohomology}. Thus (1) follows from Lemma \ref{lemma-proper-over-affine-cohomology-finite}. Given $\mathcal{L}$ as in (2) we set $\mathcal{L}' = \pi^*\mathcal{L}$. Note that $\mathcal{L}'$ is ample on $X'$ by Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}. By the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) we have $\pi_*(\mathcal{F}' \otimes \mathcal{L}') = \mathcal{F} \otimes \mathcal{L}$. Thus part (2) follows by the same reasoning as above from Lemma \ref{lemma-kill-by-twisting}.