hcju/stacksqa huggingface dataset:
Since $\text{Gal}(k'/k) = \text{Aut}(k'/k)$ it acts (from the right) on $\Spec(k')$, hence it acts (from the right) on $X_{k'} = X \times_{\Spec(k)} \Spec(k')$, and since $\Pic(-)$ is a contravariant functor, it acts (from the left) on $\Pic(X_{k'})$. If $k'/k$ is an infinite Galois extension, then we write $k' = \colim k'_\lambda$ as a filtered colimit of finite Galois extensions, see Fields, Lemma \ref{fields-lemma-infinite-galois-limit}. Then $X_{k'} = \lim X_{k_\lambda}$ (as in Limits, Section \ref{limits-section-limits}) and we obtain $$ \Pic(X_{k'}) = \colim \Pic(X_{k_\lambda}) $$ by Limits, Lemma \ref{limits-lemma-descend-invertible-modules}. Moreover, the transition maps in this system of abelian groups are injective by Varieties, Lemma \ref{varieties-lemma-change-fields-pic}. It follows that every element of $\Pic(X_{k'})$ is fixed by one of the open subgroups $\text{Gal}(k'/k_\lambda)$, which exactly means that the action is continuous. Injectivity of the transition maps implies that it suffices to prove the statement on fixed points in the case that $k'/k$ is finite Galois. \medskip\noindent Assume $k'/k$ is finite Galois with Galois group $G = \text{Gal}(k'/k)$. Let $\mathcal{L}$ be an element of $\Pic(X_{k'})$ fixed by $G$. We will use Galois descent (Descent, Lemma \ref{descent-lemma-galois-descent}) to prove that $\mathcal{L}$ is the pullback of an invertible sheaf on $X$. Recall that $f_\sigma = \text{id}_X \times \Spec(\sigma) : X_{k'} \to X_{k'}$ and that $\sigma$ acts on $\Pic(X_{k'})$ by pulling back by $f_\sigma$. Hence for each $\sigma \in G$ we can choose an isomorphism $\varphi_\sigma : \mathcal{L} \to f_\sigma^*\mathcal{L}$ because $\mathcal{L}$ is a fixed by the $G$-action. The trouble is that we don't know if we can choose $\varphi_\sigma$ such that the cocycle condition $\varphi_{\sigma\tau} = f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ holds. To see that this is possible we use that $X$ has a $k$-rational point $x \in X(k)$. Of course, $x$ similarly determines a $k'$-rational point $x' \in X_{k'}$ which is fixed by $f_\sigma$ for all $\sigma$. Pick a nonzero element $s$ in the fibre of $\mathcal{L}$ at $x'$; the fibre is the $1$-dimensional $k' = \kappa(x')$-vector space $$ \mathcal{L}_{x'} \otimes_{\mathcal{O}_{X_{k'}, x'}} \kappa(x'). $$ Then $f_\sigma^*s$ is a nonzero element of the fibre of $f_\sigma^*\mathcal{L}$ at $x'$. Since we can multiply $\varphi_\sigma$ by an element of $(k')^*$ we may assume that $\varphi_\sigma$ sends $s$ to $f_\sigma^*s$. Then we see that both $\varphi_{\sigma\tau}$ and $f_\sigma^*\varphi_\tau \circ \varphi_\sigma$ send $s$ to $f_{\sigma\tau}^*s = f_\tau^*f_\sigma^*s$. Since $H^0(X_{k'}, \mathcal{O}_{X_{k'}}) = k'$ these two isomorphisms have to be the same (as one is a global unit times the other and they agree in $x'$) and the proof is complete.
Either using relative glueing (Constructions, Section \ref{constructions-section-relative-glueing}) or using the functorial point of view (Schemes, Lemma \ref{schemes-lemma-glue-functors}) we reduce to the case where $S$ is affine. Details omitted. \medskip\noindent Assume $S$ is affine. For $U \subset X$ affine open, denote $F_U \subset \Hilbfunctor^d_{X/S}$ the subfunctor such that for a scheme $T/S$ an element $Z \in \Hilbfunctor^d_{X/S}(T)$ is in $F_U(T)$ if and only if $Z \subset U_T$. We will use Schemes, Lemma \ref{schemes-lemma-glue-functors} and the subfunctors $F_U$ to conclude. \medskip\noindent Condition (1) is Lemma \ref{lemma-hilb-d-sheaf}. \medskip\noindent Condition (2)(a) follows from the fact that $F_U = \Hilbfunctor^d_{U/S}$ and that this is representable by Lemma \ref{lemma-hilb-d-An}. Namely, if $Z \in F_U(T)$, then $Z$ can be viewed as a closed subscheme of $U_T$ which is finite locally free of degree $d$ over $T$ and hence $Z \in \Hilbfunctor^d_{U/S}(T)$. Conversely, if $Z \in \Hilbfunctor^d_{U/S}(T)$ then $Z \to U_T \to X_T$ is a closed immersion\footnote{This is clear if $X \to S$ is separated as in this case Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed} tells us that the immersion $\varphi : Z \to X_T$ has closed image and hence is a closed immersion by Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}. We suggest the reader skip the rest of this footnote as we don't know of any instance where the assumptions on $X \to S$ hold but $X \to S$ is not separated. In the general case, let $x \in X_T$ be a point in the closure of $\varphi(Z)$. We have to show that $x \in \varphi(Z)$. Let $t \in T$ be the image of $x$. By assumption on $X \to S$ we can choose an affine open $W \subset X_T$ containing $x$ and $\varphi(Z_t)$. Then $\varphi^{-1}(W)$ is an open containing the whole fibre $Z_t$ and since $Z \to T$ is closed, we may after replacing $T$ by an open neighbourhood of $t$ assume that $Z = \varphi^{-1}(W)$. Then $\varphi(Z) \subset W$ is closed by the separated case (as $W \to T$ is separated) and we conclude $x \in \varphi(Z)$.} and we may view $Z$ as an element of $F_U(T)$. \medskip\noindent Let $Z \in \Hilbfunctor^d_{X/S}(T)$ for some scheme $T$ over $S$. Let $$ B = (Z \to T)\left((Z \to X_T \to X)^{-1}(X \setminus U)\right) $$ This is a closed subset of $T$ and it is clear that over the open $T_{Z, U} = T \setminus B$ the restriction $Z_{t'}$ maps into $U_{T'}$. On the other hand, for any $b \in B$ the fibre $Z_b$ does not map into $U$. Thus we see that given a morphism $T' \to T$ we have $Z_{T'} \in F_U(T')$ $\Leftrightarrow$ $T' \to T$ factors through the open $T_{Z, U}$. This proves condition (2)(b). \medskip\noindent Condition (2)(c) follows from our assumption on $X/S$. All we have to do is show the following: If $T$ is the spectrum of a field and $Z \subset X_T$ is a closed subscheme, finite flat of degree $d$ over $T$, then $Z \to X_T \to X$ factors through an affine open $U$ of $X$. This is clear because $Z$ will have at most $d$ points and these will all map into the fibre of $X$ over the image point of $T \to S$.
The functors $\Hilbfunctor^d_{X/k}$ are representable by Proposition \ref{proposition-hilb-d-representable} (see also Remark \ref{remark-when-proposition-applies}) and the fact that $X$ is projective (Varieties, Lemma \ref{varieties-lemma-dim-1-proper-projective}). The schemes $\underline{\Hilbfunctor}^d_{X/k}$ are separated over $k$ by Lemma \ref{lemma-hilb-d-separated}. The schemes $\underline{\Hilbfunctor}^d_{X/k}$ are smooth over $k$ by Lemma \ref{lemma-hilb-d-smooth}. Starting with $X = \underline{\Hilbfunctor}^1_{X/k}$, the morphisms of Lemma \ref{lemma-universal-object}, and induction we find a morphism $$ X^d = X \times_k X \times_k \ldots \times_k X \longrightarrow \underline{\Hilbfunctor}^d_{X/k},\quad (x_1, \ldots, x_d) \longrightarrow x_1 + \ldots + x_d $$ which is finite locally free of degree $d!$. Since $X$ is proper over $k$, so is $X^d$, hence $\underline{\Hilbfunctor}^d_{X/k}$ is proper over $k$ by Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}. Since $X$ is geometrically irreducible over $k$, the product $X^d$ is irreducible (Varieties, Lemma \ref{varieties-lemma-bijection-irreducible-components}) hence the image is irreducible (in fact geometrically irreducible). This proves (1). Part (2) follows from the definitions. Part (3) follows from the commutative diagram $$ \xymatrix{ X^{d_1} \times_k X^{d_2} \ar[d] \ar@{=}[r] & X^{d_1 + d_2} \ar[d] \\ \underline{\Hilbfunctor}^{d_1}_{X/k} \times_k \underline{\Hilbfunctor}^{d_2}_{X/k} \ar[r] & \underline{\Hilbfunctor}^{d_1 + d_2}_{X/k} } $$ and multiplicativity of degrees of finite locally free morphisms.
Since $k$ is separably closed there exists a $k$-rational point $\sigma$ of $X$, see Varieties, Lemma \ref{varieties-lemma-smooth-separable-closed-points-dense}. As discussed above, it suffices to show that the functor $\Picardfunctor_{X/k, \sigma}$ classifying invertible modules trivial along $\sigma$ is representable. To do this we will check conditions (1), (2)(a), (2)(b), and (2)(c) of Lemma \ref{lemma-criterion}. \medskip\noindent The functor $\Picardfunctor_{X/k, \sigma}$ satisfies the sheaf condition for the fppf topology because it is isomorphic to $\Picardfunctor_{X/k}$. It would be more correct to say that we've shown the sheaf condition for $\Picardfunctor_{X/k, \sigma}$ in the proof of Lemma \ref{lemma-flat-geometrically-connected-fibres-with-section} which applies by Lemma \ref{lemma-check-conditions}. This proves condition (1) \medskip\noindent As our subfunctor we use $F$ as defined in Lemma \ref{lemma-define-open}. Condition (2)(b) follows. Condition (2)(a) is Lemma \ref{lemma-open-representable}. Condition (2)(c) is Lemma \ref{lemma-twist-with-general-divisor}.
By Lemma \ref{lemma-relative-glueing} we get the scheme $X$ over $S$ and the isomorphisms $i_U$. Set $\mathcal{F}'_U = i_U^*\mathcal{F}_U$ for $U \in \mathcal{B}$. This is a quasi-coherent $\mathcal{O}_{f^{-1}(U)}$-module. The maps $$ \mathcal{F}'_U|_{f^{-1}(V)} = i_U^*\mathcal{F}_U|_{f^{-1}(V)} = i_V^*(\rho^U_V)^*\mathcal{F}_U \xrightarrow{i_V^*\theta^U_V} i_V^*\mathcal{F}_V = \mathcal{F}'_V $$ define isomorphisms $(\theta')^U_V : \mathcal{F}'_U|_{f^{-1}(V)} \to \mathcal{F}'_V$ whenever $V \subset U$ are elements of $\mathcal{B}$. Condition (d) says exactly that this is compatible in case we have a triple of elements $W \subset V \subset U$ of $\mathcal{B}$. This allows us to get well defined isomorphisms $$ \varphi_{12} : \mathcal{F}'_{U_1}|_{f^{-1}(U_1 \cap U_2)} \longrightarrow \mathcal{F}'_{U_2}|_{f^{-1}(U_1 \cap U_2)} $$ whenever $U_1, U_2 \in \mathcal{B}$ by covering the intersection $U_1 \cap U_2 = \bigcup V_j$ by elements $V_j$ of $\mathcal{B}$ and taking $$ \varphi_{12}|_{V_j} = \left((\theta')^{U_2}_{V_j}\right)^{-1} \circ (\theta')^{U_1}_{V_j}. $$ We omit the verification that these maps do indeed glue to a $\varphi_{12}$ and we omit the verification of the cocycle condition of a glueing datum for sheaves (as in Sheaves, Section \ref{sheaves-section-glueing-sheaves}). By Sheaves, Lemma \ref{sheaves-lemma-glue-sheaves} we get our $\mathcal{F}$ on $X$. We omit the verification of (\ref{equation-glue}).
Let $R = \mathcal{O}_S(U)$ and $R' = \mathcal{O}_S(U')$. Note that the map $R \otimes_{R'} A' \to A$ is an isomorphism as $\mathcal{A}$ is quasi-coherent (see Schemes, Lemma \ref{schemes-lemma-widetilde-pullback} for example). The result follows from the description of the fibre product of affine schemes in Schemes, Lemma \ref{schemes-lemma-fibre-product-affine-schemes}.
Write $S = \Spec(R)$ and $A = \Gamma(S, \mathcal{A})$. Then $A$ is an $R$-algebra and $\mathcal{A} = \widetilde A$. The ring map $R \to A$ gives rise to a canonical map $$ f_{univ} : \Spec(A) \longrightarrow S = \Spec(R). $$ We have $f_{univ}^*\mathcal{A} =  \widetilde{A \otimes_R A}$ by Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}. Hence there is a canonical map $$ \varphi_{univ} : f_{univ}^*\mathcal{A} = \widetilde{A \otimes_R A} \longrightarrow \widetilde A = \mathcal{O}_{\Spec(A)} $$ coming from the $A$-module map $A \otimes_R A \to A$, $a \otimes a' \mapsto aa'$. We claim that the pair $(f_{univ}, \varphi_{univ})$ represents $F$ in this case. In other words we claim that for any scheme $T$ the map $$ \Mor(T, \Spec(A)) \longrightarrow \{\text{pairs } (f, \varphi)\},\quad a \longmapsto (f_{univ} \circ a, a^*\varphi) $$ is bijective. \medskip\noindent Let us construct the inverse map. For any pair $(f : T \to S, \varphi)$ we get the induced ring map $$ \xymatrix{ A = \Gamma(S, \mathcal{A}) \ar[r]^{f^*} & \Gamma(T, f^*\mathcal{A}) \ar[r]^{\varphi} & \Gamma(T, \mathcal{O}_T) } $$ This induces a morphism of schemes $T \to \Spec(A)$ by Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. \medskip\noindent The verification that this map is inverse to the map displayed above is omitted.
We are going to use Schemes, Lemma \ref{schemes-lemma-glue-functors}. \medskip\noindent First we check that $F$ satisfies the sheaf property for the Zariski topology. Namely, suppose that $T$ is a scheme, that $T = \bigcup_{i \in I} U_i$ is an open covering, and that $(f_i, \varphi_i) \in F(U_i)$ such that $(f_i, \varphi_i)|_{U_i \cap U_j} = (f_j, \varphi_j)|_{U_i \cap U_j}$. This implies that the morphisms $f_i : U_i \to S$ glue to a morphism of schemes $f : T \to S$ such that $f|_{I_i} = f_i$, see Schemes, Section \ref{schemes-section-glueing-schemes}. Thus $f_i^*\mathcal{A} = f^*\mathcal{A}|_{U_i}$ and by assumption the morphisms $\varphi_i$ agree on $U_i \cap U_j$. Hence by Sheaves, Section \ref{sheaves-section-glueing-sheaves} these glue to a morphism of $\mathcal{O}_T$-algebras $f^*\mathcal{A} \to \mathcal{O}_T$. This proves that $F$ satisfies the sheaf condition with respect to the Zariski topology. \medskip\noindent Let $S = \bigcup_{i \in I} U_i$ be an affine open covering. Let $F_i \subset F$ be the subfunctor consisting of those pairs $(f : T \to S, \varphi)$ such that $f(T) \subset U_i$. \medskip\noindent We have to show each $F_i$ is representable. This is the case because $F_i$ is identified with the functor associated to $U_i$ equipped with the quasi-coherent $\mathcal{O}_{U_i}$-algebra $\mathcal{A}|_{U_i}$, by Lemma \ref{lemma-spec-base-change}. Thus the result follows from Lemma \ref{lemma-spec-affine}. \medskip\noindent Next we show that $F_i \subset F$ is representable by open immersions. Let $(f : T \to S, \varphi) \in F(T)$. Consider $V_i = f^{-1}(U_i)$. It follows from the definition of $F_i$ that given $a : T' \to T$ we gave $a^*(f, \varphi) \in F_i(T')$ if and only if $a(T') \subset V_i$. This is what we were required to show. \medskip\noindent Finally, we have to show that the collection $(F_i)_{i \in I}$ covers $F$. Let $(f : T \to S, \varphi) \in F(T)$. Consider $V_i = f^{-1}(U_i)$. Since $S = \bigcup_{i \in I} U_i$ is an open covering of $S$ we see that $T = \bigcup_{i \in I} V_i$ is an open covering of $T$. Moreover $(f, \varphi)|_{V_i} \in F_i(V_i)$. This finishes the proof of the lemma.
Let $X \to S$ be the scheme representing the functor $F$. Consider the sheaf of $\mathcal{O}_S$-algebras $\mathcal{R} = \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})}$. By construction of $\underline{\Spec}_S(\mathcal{A})$ we have isomorphisms $\mathcal{A}(U) \to \mathcal{R}(U)$ for every affine open $U \subset S$; this follows from Lemma \ref{lemma-glue-relative-spec} part (1). For $U \subset U' \subset S$ open these isomorphisms are compatible with the restriction mappings; this follows from Lemma \ref{lemma-glue-relative-spec} part (2). Hence by Sheaves, Lemma \ref{sheaves-lemma-restrict-basis-equivalence-modules} these isomorphisms result from an isomorphism of $\mathcal{O}_S$-algebras $\varphi : \mathcal{A} \to \mathcal{R}$. Hence this gives an element $(\underline{\Spec}_S(\mathcal{A}), \varphi) \in F(\underline{\Spec}_S(\mathcal{A}))$. Since $X$ represents the functor $F$ we get a corresponding morphism of schemes $can : \underline{\Spec}_S(\mathcal{A}) \to X$ over $S$. \medskip\noindent Let $U \subset S$ be any affine open. Let $F_U \subset F$ be the subfunctor of $F$ corresponding to pairs $(f, \varphi)$ over schemes $T$ with $f(T) \subset U$. Clearly the base change $X_U$ represents $F_U$. Moreover, $F_U$ is represented by $\Spec(\mathcal{A}(U)) = \pi^{-1}(U)$ according to Lemma \ref{lemma-spec-affine}. In other words $X_U \cong \pi^{-1}(U)$. We omit the verification that this identification is brought about by the base change of the morphism $can$ to $U$.
Part (1) comes from the description of the relative spectrum by glueing, see Lemma \ref{lemma-glue-relative-spec}. Part (2) follows immediately from Lemma \ref{lemma-spec-base-change}. Part (3) follows because it is local on $S$ and it is clear in case $S$ is affine by Lemma \ref{lemma-spec-affine} for example.
The morphism comes, via the definition of $\underline{\Spec}$ as the scheme representing the functor $F$, from the canonical map $\varphi : f^*f_*\mathcal{O}_X \to \mathcal{O}_X$ (which by adjointness of push and pull corresponds to $\text{id} : f_*\mathcal{O}_X \to f_*\mathcal{O}_X$). The statement on the restriction to $f^{-1}(U)$ follows from the description of the relative spectrum over affines, see Lemma \ref{lemma-spec-affine}.
Recall that $D_{+}(g) = \Spec(S_{(g)})$ with identification given by the ring maps $S \to S_g \leftarrow S_{(g)}$, see Algebra, Lemma \ref{algebra-lemma-topology-proj}. Thus $f^{\deg(g)}/g^{\deg(f)}$ is an element of $S_{(g)}$ which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma \ref{algebra-lemma-Zariski-topology}. We conclude that (a) holds. Write the inverse of $f$ in $S_g$ as $a/g^d$. We may replace $a$ by its homogeneous part of degree $d\deg(g) - \deg(f)$. This means $g^d - af$ is annihilated by a power of $g$, whence $g^e = af$ for some $a \in S$ homogeneous of degree $e\deg(g) - \deg(f)$. This proves (b). For (c), the map $S_f \to S_g$ exists by (a) from the universal property of localization, or we can define it by mapping $b/f^n$ to $a^nb/g^{ne}$. This clearly induces a map of the subrings $S_{(f)} \to S_{(g)}$ of degree zero elements as well. We can similarly define $M_f \to M_g$ and $M_{(f)} \to M_{(g)}$ by mapping $x/f^n$ to $a^nx/g^{ne}$. The statements writing $S_{(g)}$ resp.\ $M_{(g)}$ as principal localizations of $S_{(f)}$ resp.\ $M_{(f)}$ are clear from the formulas above. The maps in the commutative diagram of topological spaces correspond to the ring maps given above. The horizontal arrows are homeomorphisms by Algebra, Lemma \ref{algebra-lemma-topology-proj}. The vertical arrows are open immersions since the left one is the inclusion of an open subset. \medskip\noindent The open $D_{+}(f)$ is quasi-compact because it is homeomorphic to $\Spec(S_{(f)})$, see  Algebra, Lemma \ref{algebra-lemma-quasi-compact}. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology. \medskip\noindent The third statement follows directly from Algebra, Lemma \ref{algebra-lemma-Zariski-topology}.
Assertions (1) - (5) are clear from the discussion above. We see (6) since there are canonical maps $M_0 \to M_{(f)}$, $x \mapsto x/1$ compatible with the restriction maps described in (3). The exactness of the functor $M \mapsto \widetilde M$ follows from the fact that the functor $M \mapsto M_{(\mathfrak p)}$ is exact (see Algebra, Lemma \ref{algebra-lemma-proj-prime}) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma \ref{modules-lemma-abelian}.
Consider a standard open $D_{+}(f) \subset \text{Proj}(S)$. By Lemmas \ref{lemma-standard-open} and \ref{lemma-proj-sheaves} we have $\Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}$, and we have a homeomorphism $\varphi : D_{+}(f) \to \Spec(S_{(f)})$. For any standard open $D(g) \subset \Spec(S_{(f)})$ we may pick a $h \in S_{+}$ as in Lemma \ref{lemma-standard-open-proj}. Then $\varphi^{-1}(D(g)) = D_{+}(h)$, and by Lemmas \ref{lemma-proj-sheaves} and \ref{lemma-standard-open} we see $$ \Gamma(D_{+}(h), \mathcal{O}_{\text{Proj}(S)}) = S_{(h)} = (S_{(f)})_{h^{\deg(f)}/f^{\deg(h)}} = (S_{(f)})_g = \Gamma(D(g), \mathcal{O}_{\Spec(S_{(f)})}). $$ Thus the restriction of $\mathcal{O}_{\text{Proj}(S)}$ to $D_{+}(f)$ corresponds via the homeomorphism $\varphi$ exactly to the sheaf $\mathcal{O}_{\Spec(S_{(f)})}$ as defined in Schemes, Section \ref{schemes-section-affine-schemes}. We conclude that $D_{+}(f)$ is an affine scheme isomorphic to $\Spec(S_{(f)})$ via $\varphi$ and hence that $\text{Proj}(S)$ is a scheme. \medskip\noindent In exactly the same way we show that $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules. Namely, the argument above will show that $$ \widetilde M|_{D_{+}(f)} \cong \varphi^*\left(\widetilde{M_{(f)}}\right) $$ which shows that $\widetilde M$ is quasi-coherent.
We have to show that the canonical morphism $\text{Proj}(S) \to \Spec(\mathbf{Z})$ is separated. We will use Schemes, Lemma \ref{schemes-lemma-characterize-separated}. Thus it suffices to show given any pair of standard opens $D_{+}(f)$ and $D_{+}(g)$ that $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$ is affine (clear) and that the ring map $$ S_{(f)} \otimes_{\mathbf{Z}} S_{(g)} \longrightarrow S_{(fg)} $$ is surjective. Any element $s$ in $S_{(fg)}$ is of the form $s = h/(f^ng^m)$ with $h \in S$ homogeneous of degree $n\deg(f) + m\deg(g)$. We may multiply $h$ by a suitable monomial $f^ig^j$ and assume that $n = n' \deg(g)$, and $m = m' \deg(f)$. Then we can rewrite $s$ as $s = h/f^{(n' + m')\deg(g)} \cdot f^{m'\deg(g)}/g^{m'\deg(f)}$. So $s$ is indeed in the image of the displayed arrow.
Given such a collection of elements the standard affine opens $D_{+}(f_i)$ cover $\text{Proj}(S)$ by Algebra, Lemma \ref{algebra-lemma-topology-proj}. Conversely, if $\text{Proj}(S)$ is quasi-compact, then we may cover it by finitely many standard opens $D_{+}(f_i)$, $i = 1, \ldots, n$ and we see that $S_{+} \subset \sqrt{(f_1, \ldots, f_n)}$ by the lemma referenced above.
We saw above that our construction of $\widetilde S$, resp.\ $\widetilde M$ gives a sheaf of $S_0$-algebras, resp.\ $S_0$-modules. Hence we get a morphism by Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. This morphism, when restricted to $D_{+}(f)$ comes from the canonical ring map $S_0 \to S_{(f)}$. The maps $S \to S_f$, $S_{(f)} \to S_f$ are $S_0$-algebra maps, see Lemma \ref{lemma-standard-open}. Hence if the homogeneous prime $\mathfrak p \subset S$ corresponds to the $\mathbf{Z}$-graded prime $\mathfrak p' \subset S_f$ and the (usual) prime $\mathfrak p'' \subset S_{(f)}$, then each of these has the same inverse image in $S_0$.
The uniqueness part follows from the fact that $\text{Proj}(S)$ is separated (Lemma \ref{lemma-proj-separated} and Schemes, Lemma \ref{schemes-lemma-separated-implies-valuative}). Choose $x_i \in S_{+}$ homogeneous, $i = 1, \ldots, n$ which generate $S$ over $S_0$. Let $d_i = \deg(x_i)$ and set $d = \text{lcm}\{d_i\}$. Suppose we are given a diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & \text{Proj}(S) \ar[d] \\ \Spec(A) \ar[r] & \Spec(S_0) } $$ as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. Denote $v : K^* \to \Gamma$ the valuation of $A$, see Algebra, Definition \ref{algebra-definition-value-group}. We may choose an $f \in S_{+}$ homogeneous such that $\Spec(K)$ maps into $D_{+}(f)$. Then we get a commutative diagram of ring maps $$ \xymatrix{ K & S_{(f)} \ar[l]^{\varphi} \\ A \ar[u] & S_0 \ar[l] \ar[u] } $$ After renumbering we may assume that $\varphi(x_i^{\deg(f)}/f^{d_i})$ is nonzero for $i = 1, \ldots, r$ and zero for $i = r + 1, \ldots, n$. Since the open sets $D_{+}(x_i)$ cover $\text{Proj}(S)$ we see that $r \geq 1$. Let $i_0 \in \{1, \ldots, r\}$ be an index minimizing $\gamma_i = (d/d_i)v(\varphi(x_i^{\deg(f)}/f^{d_i}))$ in $\Gamma$. For convenience set $x_0 = x_{i_0}$ and $d_0 = d_{i_0}$. The ring map $\varphi$ factors though a map $\varphi' : S_{(fx_0)} \to K$ which gives a ring map $S_{(x_0)} \to S_{(fx_0)} \to K$. The algebra $S_{(x_0)}$ is generated over $S_0$ by the elements $x_1^{e_1} \ldots x_n^{e_n}/x_0^{e_0}$, where $\sum e_i d_i = e_0 d_0$. If $e_i > 0$ for some $i > r$, then $\varphi'(x_1^{e_1} \ldots x_n^{e_n}/x_0^{e_0}) = 0$. If $e_i = 0$ for $i > r$, then we have \begin{align*} \deg(f) v(\varphi'(x_1^{e_1} \ldots x_r^{e_r}/x_0^{e_0})) & = v(\varphi'(x_1^{e_1 \deg(f)} \ldots x_r^{e_r \deg(f)}/x_0^{e_0 \deg(f)})) \\ & = \sum e_i v(\varphi'(x_i^{\deg(f)}/f^{d_i})) - e_0 v(\varphi'(x_0^{\deg(f)}/f^{d_0})) \\ & = \sum e_i d_i \gamma_i - e_0 d_0 \gamma_0 \\ & \geq \sum e_i d_i \gamma_0 - e_0 d_0 \gamma_0 = 0 \end{align*} because $\gamma_0$ is minimal among the $\gamma_i$. This implies that $S_{(x_0)}$ maps into $A$ via $\varphi'$. The corresponding morphism of schemes $\Spec(A) \to \Spec(S_{(x_0)}) = D_{+}(x_0) \subset \text{Proj}(S)$ provides the morphism fitting into the first commutative diagram of this proof.