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 \input{preamble} % OK, start here. % \begin{document} \title{Constructions of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we introduce ways of constructing schemes out of others. A basic reference is \cite{EGA}. \section{Relative glueing} \label{section-relative-glueing} \noindent The following lemma is relevant in case we are trying to construct a scheme $X$ over $S$, and we already know how to construct the restriction of $X$ to the affine opens of $S$. The actual result is completely general and works in the setting of (locally) ringed spaces, although our proof is written in the language of schemes. \begin{lemma} \label{lemma-relative-glueing} Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \begin{enumerate} \item For every $U \in \mathcal{B}$ a scheme $f_U : X_U \to U$ over $U$. \item For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho^U_V : X_V \to X_U$. \end{enumerate} Assume that \begin{enumerate} \item[(a)] each $\rho^U_V$ induces an isomorphism $X_V \to f_U^{-1}(V)$ of schemes over $V$, \item[(b)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho^U_W = \rho^U_V \circ \rho ^V_W$. \end{enumerate} Then there exists a unique scheme $f : X \to S$ over $S$ and isomorphisms $i_U : f^{-1}(U) \to X_U$ over $U$ such that for $V \subset U \subset S$ affine open the composition $$\xymatrix{ X_V \ar[r]^{i_V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_U} & X_U }$$ is the morphism $\rho^U_V$. \end{lemma} \begin{proof} To prove this we will use Schemes, Lemma \ref{schemes-lemma-glue-functors}. First we define a contravariant functor $F$ from the category of schemes to the category of sets. Namely, for a scheme $T$ we set $$F(T) = \left\{ \begin{matrix} (g, \{h_U\}_{U \in \mathcal{B}}), \ g : T \to S, \ h_U : g^{-1}(U) \to X_U, \\ f_U \circ h_U = g|_{g^{-1}(U)}, \ h_U|_{g^{-1}(V)} = \rho^U_V \circ h_V \ \forall\ V, U \in \mathcal{B}, V \subset U \end{matrix} \right\}.$$ The restriction mapping $F(T) \to F(T')$ given a morphism $T' \to T$ is just gotten by composition. For any $W \in \mathcal{B}$ we consider the subfunctor $F_W \subset F$ consisting of those systems $(g, \{h_U\})$ such that $g(T) \subset W$. \medskip\noindent First we show $F$ satisfies the sheaf property for the Zariski topology. Suppose that $T$ is a scheme, $T = \bigcup V_i$ is an open covering, and $\xi_i \in F(V_i)$ is an element such that $\xi_i|_{V_i \cap V_j} = \xi_j|_{V_i \cap V_j}$. Say $\xi_i = (g_i, \{h_{i, U}\})$. Then we immediately see that the morphisms $g_i$ glue to a unique global morphism $g : T \to S$. Moreover, it is clear that $g^{-1}(U) = \bigcup g_i^{-1}(U)$. Hence the morphisms $h_{i, U} : g_i^{-1}(U) \to X_U$ glue to a unique morphism $h_U : U \to X_U$. It is easy to verify that the system $(g, \{f_U\})$ is an element of $F(T)$. Hence $F$ satisfies the sheaf property for the Zariski topology. \medskip\noindent Next we verify that each $F_W$, $W \in \mathcal{B}$ is representable. Namely, we claim that the transformation of functors $$F_W \longrightarrow \Mor(-, X_W), \ (g, \{h_U\}) \longmapsto h_W$$ is an isomorphism. To see this suppose that $T$ is a scheme and $\alpha : T \to X_W$ is a morphism. Set $g = f_W \circ \alpha$. For any $U \in \mathcal{B}$ such that $U \subset W$ we can define $h_U : g^{-1}(U) \to X_U$ be the composition $(\rho^W_U)^{-1} \circ \alpha|_{g^{-1}(U)}$. This works because the image $\alpha(g^{-1}(U))$ is contained in $f_W^{-1}(U)$ and condition (a) of the lemma. It is clear that $f_U \circ h_U = g|_{g^{-1}(U)}$ for such a $U$. Moreover, if also $V \in \mathcal{B}$ and $V \subset U \subset W$, then $\rho^U_V \circ h_V = h_U|_{g^{-1}(V)}$ by property (b) of the lemma. We still have to define $h_U$ for an arbitrary element $U \in \mathcal{B}$. Since $\mathcal{B}$ is a basis for the topology on $S$ we can find an open covering $U \cap W = \bigcup U_i$ with $U_i \in \mathcal{B}$. Since $g$ maps into $W$ we have $g^{-1}(U) = g^{-1}(U \cap W) = \bigcup g^{-1}(U_i)$. Consider the morphisms $h_i = \rho^U_{U_i} \circ h_{U_i} : g^{-1}(U_i) \to X_U$. It is a simple matter to use condition (b) of the lemma to prove that $h_i|_{g^{-1}(U_i) \cap g^{-1}(U_j)} = h_j|_{g^{-1}(U_i) \cap g^{-1}(U_j)}$. Hence these morphisms glue to give the desired morphism $h_U : g^{-1}(U) \to X_U$. We omit the (easy) verification that the system $(g, \{h_U\})$ is an element of $F_W(T)$ which maps to $\alpha$ under the displayed arrow above. \medskip\noindent Next, we verify each $F_W \subset F$ is representable by open immersions. This is clear from the definitions. \medskip\noindent Finally we have to verify the collection $(F_W)_{W \in \mathcal{B}}$ covers $F$. This is clear by construction and the fact that $\mathcal{B}$ is a basis for the topology of $S$. \medskip\noindent Let $X$ be a scheme representing the functor $F$. Let $(f, \{i_U\}) \in F(X)$ be a universal family''. Since each $F_W$ is representable by $X_W$ (via the morphism of functors displayed above) we see that $i_W : f^{-1}(W) \to X_W$ is an isomorphism as desired. The lemma is proved. \end{proof} \begin{lemma} \label{lemma-relative-glueing-sheaves} Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data: \begin{enumerate} \item For every $U \in \mathcal{B}$ a scheme $f_U : X_U \to U$ over $U$. \item For every $U \in \mathcal{B}$ a quasi-coherent sheaf $\mathcal{F}_U$ over $X_U$. \item For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho^U_V : X_V \to X_U$. \item For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\theta^U_V : (\rho^U_V)^*\mathcal{F}_U \to \mathcal{F}_V$. \end{enumerate} Assume that \begin{enumerate} \item[(a)] each $\rho^U_V$ induces an isomorphism $X_V \to f_U^{-1}(V)$ of schemes over $V$, \item[(b)] each $\theta^U_V$ is an isomorphism, \item[(c)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho^U_W = \rho^U_V \circ \rho ^V_W$, \item[(d)] whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\theta^U_W = \theta^V_W \circ (\rho^V_W)^*\theta^U_V$. \end{enumerate} Then there exists a unique scheme $f : X \to S$ over $S$ together with a unique quasi-coherent sheaf $\mathcal{F}$ on $X$ and isomorphisms $i_U : f^{-1}(U) \to X_U$ and $\theta_U : i_U^*\mathcal{F}_U \to \mathcal{F}|_{f^{-1}(U)}$ over $U$ such that for $V \subset U \subset S$ affine open the composition $$\xymatrix{ X_V \ar[r]^{i_V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_U} & X_U }$$ is the morphism $\rho^U_V$, and the composition \begin{equation} \label{equation-glue} (\rho^U_V)^*\mathcal{F}_U = (i_V^{-1})^*((i_U^*\mathcal{F}_U)|_{f^{-1}(V)}) \xrightarrow{\theta_U|_{f^{-1}(V)}} (i_V^{-1})^*(\mathcal{F}|_{f^{-1}(V)}) \xrightarrow{\theta_V^{-1}} \mathcal{F}_V \end{equation} is equal to $\theta^U_V$. \end{lemma} \begin{proof} 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}). \end{proof} \begin{remark} \label{remark-relative-glueing-functorial} There is a functoriality property for the constructions explained in Lemmas \ref{lemma-relative-glueing} and \ref{lemma-relative-glueing-sheaves}. Namely, suppose given two collections of data $(f_U : X_U \to U, \rho^U_V)$ and $(g_U : Y_U \to U, \sigma^U_V)$ as in Lemma \ref{lemma-relative-glueing}. Suppose for every $U \in \mathcal{B}$ given a morphism $h_U : X_U \to Y_U$ over $U$ compatible with the restrictions $\rho^U_V$ and $\sigma^U_V$. Functoriality means that this gives rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_U$, where $f : X \to S$ is obtained from the datum $(f_U : X_U \to U, \rho^U_V)$ and $g : Y \to S$ is obtained from the datum $(g_U : Y_U \to U, \sigma^U_V)$. \medskip\noindent Similarly, suppose given two collections of data $(f_U : X_U \to U, \mathcal{F}_U, \rho^U_V, \theta^U_V)$ and $(g_U : Y_U \to U, \mathcal{G}_U, \sigma^U_V, \eta^U_V)$ as in Lemma \ref{lemma-relative-glueing-sheaves}. Suppose for every $U \in \mathcal{B}$ given a morphism $h_U : X_U \to Y_U$ over $U$ compatible with the restrictions $\rho^U_V$ and $\sigma^U_V$, and a morphism $\tau_U : h_U^*\mathcal{G}_U \to \mathcal{F}_U$ compatible with the maps $\theta^U_V$ and $\eta^U_V$. Functoriality means that these give rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_U$, and a morphism $h^*\mathcal{G} \to \mathcal{F}$ restricting back to the maps $h_U$ where $(f : X \to S, \mathcal{F})$ is obtained from the datum $(f_U : X_U \to U, \mathcal{F}_U, \rho^U_V, \theta^U_V)$ and where $(g : Y \to S, \mathcal{G})$ is obtained from the datum $(g_U : Y_U \to U, \mathcal{G}_U, \sigma^U_V, \eta^U_V)$. \medskip\noindent We omit the verifications and we omit a suitable formulation of equivalence of categories'' between relative glueing data and relative objects. \end{remark} \section{Relative spectrum via glueing} \label{section-spec-via-glueing} \begin{situation} \label{situation-relative-spec} Here $S$ is a scheme, and $\mathcal{A}$ is a quasi-coherent $\mathcal{O}_S$-algebra. This means that $\mathcal{A}$ is a sheaf of $\mathcal{O}_S$-algebras which is quasi-coherent as an $\mathcal{O}_S$-module. \end{situation} \noindent In this section we outline how to construct a morphism of schemes $$\underline{\Spec}_S(\mathcal{A}) \longrightarrow S$$ by glueing the spectra $\Spec(\Gamma(U, \mathcal{A}))$ where $U$ ranges over the affine opens of $S$. We first show that the spectra of the values of $\mathcal{A}$ over affines form a suitable collection of schemes, as in Lemma \ref{lemma-relative-glueing}. \begin{lemma} \label{lemma-spec-inclusion} In Situation \ref{situation-relative-spec}. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of rings $A' \to A$ induces a morphism $\Spec(A) \to \Spec(A')$, and the diagram $$\xymatrix{ \Spec(A) \ar[r] \ar[d] & \Spec(A') \ar[d] \\ U \ar[r] & U' }$$ is cartesian. \end{lemma} \begin{proof} 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}. \end{proof} \noindent In particular the morphism $\Spec(A) \to \Spec(A')$ of the lemma is an open immersion. \begin{lemma} \label{lemma-transitive-spec} In Situation \ref{situation-relative-spec}. Suppose $U \subset U' \subset U'' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ and $A'' = \mathcal{A}(U'')$. The composition of the morphisms $\Spec(A) \to \Spec(A')$, and $\Spec(A') \to \Spec(A'')$ of Lemma \ref{lemma-spec-inclusion} gives the morphism $\Spec(A) \to \Spec(A'')$ of Lemma \ref{lemma-spec-inclusion}. \end{lemma} \begin{proof} This follows as the map $A'' \to A$ is the composition of $A'' \to A'$ and $A' \to A$ (because $\mathcal{A}$ is a sheaf). \end{proof} \begin{lemma} \label{lemma-glue-relative-spec} In Situation \ref{situation-relative-spec}. There exists a morphism of schemes $$\pi : \underline{\Spec}_S(\mathcal{A}) \longrightarrow S$$ with the following properties: \begin{enumerate} \item for every affine open $U \subset S$ there exists an isomorphism $i_U : \pi^{-1}(U) \to \Spec(\mathcal{A}(U))$, and \item for $U \subset U' \subset S$ affine open the composition $$\xymatrix{ \Spec(\mathcal{A}(U)) \ar[r]^{i_U^{-1}} & \pi^{-1}(U) \ar[rr]^{inclusion} & & \pi^{-1}(U') \ar[r]^{i_{U'}} & \Spec(\mathcal{A}(U')) }$$ is the open immersion of Lemma \ref{lemma-spec-inclusion} above. \end{enumerate} \end{lemma} \begin{proof} Follows immediately from Lemmas \ref{lemma-relative-glueing}, \ref{lemma-spec-inclusion}, and \ref{lemma-transitive-spec}. \end{proof} \section{Relative spectrum as a functor} \label{section-spec} \noindent We place ourselves in Situation \ref{situation-relative-spec}, i.e., $S$ is a scheme and $\mathcal{A}$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. \medskip\noindent For any $f : T \to S$ the pullback $f^*\mathcal{A}$ is a quasi-coherent sheaf of $\mathcal{O}_T$-algebras. We are going to consider pairs $(f : T \to S, \varphi)$ where $f$ is a morphism of schemes and $\varphi : f^*\mathcal{A} \to \mathcal{O}_T$ is a morphism of $\mathcal{O}_T$-algebras. Note that this is the same as giving a $f^{-1}\mathcal{O}_S$-algebra homomorphism $\varphi : f^{-1}\mathcal{A} \to \mathcal{O}_T$, see Sheaves, Lemma \ref{sheaves-lemma-adjointness-tensor-restrict}. This is also the same as giving a $\mathcal{O}_S$-algebra map $\varphi : \mathcal{A} \to f_*\mathcal{O}_T$, see Sheaves, Lemma \ref{sheaves-lemma-adjoint-push-pull-modules}. We will use all three ways of thinking about $\varphi$, without further mention. \medskip\noindent Given such a pair $(f : T \to S, \varphi)$ and a morphism $a : T' \to T$ we get a second pair $(f' = f \circ a, \varphi' = a^*\varphi)$ which we call the pullback of $(f, \varphi)$. One way to describe $\varphi' = a^*\varphi$ is as the composition $\mathcal{A} \to f_*\mathcal{O}_T \to f'_*\mathcal{O}_{T'}$ where the second map is $f_*a^\sharp$ with $a^\sharp : \mathcal{O}_T \to a_*\mathcal{O}_{T'}$. In this way we have defined a functor \begin{eqnarray} \label{equation-spec} F : \Sch^{opp} & \longrightarrow & \textit{Sets} \\ T & \longmapsto & F(T) = \{\text{pairs }(f, \varphi) \text{ as above}\} \nonumber \end{eqnarray} \begin{lemma} \label{lemma-spec-base-change} In Situation \ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. Let $g : S' \to S$ be a morphism of schemes. Set $\mathcal{A}' = g^*\mathcal{A}$. Let $F'$ be the functor associated to $(S', \mathcal{A}')$ above. Then there is a canonical isomorphism $$F' \cong h_{S'} \times_{h_S} F$$ of functors. \end{lemma} \begin{proof} A pair $(f' : T \to S', \varphi' : (f')^*\mathcal{A}' \to \mathcal{O}_T)$ is the same as a pair $(f, \varphi : f^*\mathcal{A} \to \mathcal{O}_T)$ together with a factorization of $f$ as $f = g \circ f'$. Namely with this notation we have $(f')^* \mathcal{A}' = (f')^*g^*\mathcal{A} = f^*\mathcal{A}$. Hence the lemma. \end{proof} \begin{lemma} \label{lemma-spec-affine} In Situation \ref{situation-relative-spec}. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. If $S$ is affine, then $F$ is representable by the affine scheme $\Spec(\Gamma(S, \mathcal{A}))$. \end{lemma} \begin{proof} 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 $$\Mor(T, \Spec(A)) \longrightarrow \{\text{pairs } (f, \varphi)\},\quad a \longmapsto (a^*f_{univ}, a^*\varphi)$$ 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. \end{proof} \begin{lemma} \label{lemma-spec} In Situation \ref{situation-relative-spec}. The functor $F$ is representable by a scheme. \end{lemma} \begin{proof} 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. \end{proof} \begin{lemma} \label{lemma-glueing-gives-functor-spec} In Situation \ref{situation-relative-spec}. The scheme $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$ constructed in Lemma \ref{lemma-glue-relative-spec} and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$. \end{lemma} \begin{proof} 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 $U$. \end{proof} \begin{definition} \label{definition-relative-spec} Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. The {\it relative spectrum of $\mathcal{A}$ over $S$}, or simply the {\it spectrum of $\mathcal{A}$ over $S$} is the scheme constructed in Lemma \ref{lemma-glue-relative-spec} which represents the functor $F$ (\ref{equation-spec}), see Lemma \ref{lemma-glueing-gives-functor-spec}. We denote it $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$. The universal family'' is a morphism of $\mathcal{O}_S$-algebras $$\mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})}$$ \end{definition} \noindent The following lemma says among other things that forming the relative spectrum commutes with base change. \begin{lemma} \label{lemma-spec-properties} Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. Let $\pi : \underline{\Spec}_S(\mathcal{A}) \to S$ be the relative spectrum of $\mathcal{A}$ over $S$. \begin{enumerate} \item For every affine open $U \subset S$ the inverse image $\pi^{-1}(U)$ is affine. \item For every morphism $g : S' \to S$ we have $S' \times_S \underline{\Spec}_S(\mathcal{A}) = \underline{\Spec}_{S'}(g^*\mathcal{A})$. \item The universal map $$\mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})}$$ is an isomorphism of $\mathcal{O}_S$-algebras. \end{enumerate} \end{lemma} \begin{proof} 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. \end{proof} \begin{lemma} \label{lemma-canonical-morphism} Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. There is a canonical morphism $$can : X \longrightarrow \underline{\Spec}_S(f_*\mathcal{O}_X)$$ of schemes over $S$. For any affine open $U \subset S$ the restriction $can|_{f^{-1}(U)}$ is identified with the canonical morphism $$f^{-1}(U) \longrightarrow \Spec(\Gamma(f^{-1}(U), \mathcal{O}_X))$$ coming from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}. \end{lemma} \begin{proof} 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}. \end{proof} \section{Affine n-space} \label{section-affine-n-space} \noindent As an application of the relative spectrum we define affine $n$-space over a base scheme $S$ as follows. For any integer $n \geq 0$ we can consider the quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{O}_S[T_1, \ldots, T_n]$. It is quasi-coherent because as a sheaf of $\mathcal{O}_S$-modules it is just the direct sum of copies of $\mathcal{O}_S$ indexed by multi-indices. \begin{definition} \label{definition-affine-n-space} Let $S$ be a scheme and $n \geq 0$. The scheme $$\mathbf{A}^n_S = \underline{\Spec}_S(\mathcal{O}_S[T_1, \ldots, T_n])$$ over $S$ is called {\it affine $n$-space over $S$}. If $S = \Spec(R)$ is affine then we also call this {\it affine $n$-space over $R$} and we denote it $\mathbf{A}^n_R$. \end{definition} \noindent Note that $\mathbf{A}^n_R = \Spec(R[T_1, \ldots, T_n])$. For any morphism $g : S' \to S$ of schemes we have $g^*\mathcal{O}_S[T_1, \ldots, T_n] = \mathcal{O}_{S'}[T_1, \ldots, T_n]$ and hence $\mathbf{A}^n_{S'} = S' \times_S \mathbf{A}^n_S$ is the base change. Therefore an alternative definition of affine $n$-space is the formula $$\mathbf{A}^n_S = S \times_{\Spec(\mathbf{Z})} \mathbf{A}^n_{\mathbf{Z}}.$$ Also, a morphism from an $S$-scheme $f : X \to S$ to $\mathbf{A}^n_S$ is given by a homomorphism of $\mathcal{O}_S$-algebras $\mathcal{O}_S[T_1, \ldots, T_n] \to f_*\mathcal{O}_X$. This is clearly the same thing as giving the images of the $T_i$. In other words, a morphism from $X$ to $\mathbf{A}^n_S$ over $S$ is the same as giving $n$ elements $h_1, \ldots, h_n \in \Gamma(X, \mathcal{O}_X)$. \section{Vector bundles} \label{section-vector-bundle} \noindent Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent sheaf of $\mathcal{O}_S$-modules. By Modules, Lemma \ref{modules-lemma-whole-tensor-algebra-permanence} the symmetric algebra $\text{Sym}(\mathcal{E})$ of $\mathcal{E}$ over $\mathcal{O}_S$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. Hence it makes sense to apply the construction of the previous section to it. \begin{definition} \label{definition-vector-bundle} Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module\footnote{The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with \cite[II, Definition 1.7.8]{EGA}.}. The {\it vector bundle associated to $\mathcal{E}$} is $$\mathbf{V}(\mathcal{E}) = \underline{\Spec}_S(\text{Sym}(\mathcal{E})).$$ \end{definition} \noindent The vector bundle associated to $\mathcal{E}$ comes with a bit of extra structure. Namely, we have a grading $$\pi_*\mathcal{O}_{\mathbf{V}(\mathcal{E})} = \bigoplus\nolimits_{n \geq 0} \text{Sym}^n(\mathcal{E}).$$ which turns $\pi_*\mathcal{O}_{\mathbf{V}(\mathcal{E})}$ into a graded $\mathcal{O}_S$-algebra. Conversely, we can recover $\mathcal{E}$ from the degree $1$ part of this. Thus we define an abstract vector bundle as follows. \begin{definition} \label{definition-abstract-vector-bundle} Let $S$ be a scheme. A {\it vector bundle $\pi : V \to S$ over $S$} is an affine morphism of schemes such that $\pi_*\mathcal{O}_V$ is endowed with the structure of a graded $\mathcal{O}_S$-algebra $\pi_*\mathcal{O}_V = \bigoplus\nolimits_{n \geq 0} \mathcal{E}_n$ such that $\mathcal{E}_0 = \mathcal{O}_S$ and such that the maps $$\text{Sym}^n(\mathcal{E}_1) \longrightarrow \mathcal{E}_n$$ are isomorphisms for all $n \geq 0$. A {\it morphism of vector bundles over $S$} is a morphism $f : V \to V'$ such that the induced map $$f^* : \pi'_*\mathcal{O}_{V'} \longrightarrow \pi_*\mathcal{O}_V$$ is compatible with the given gradings. \end{definition} \noindent An example of a vector bundle over $S$ is affine $n$-space $\mathbf{A}^n_S$ over $S$, see Definition \ref{definition-affine-n-space}. This is true because $\mathcal{O}_S[T_1, \ldots, T_n] = \text{Sym}(\mathcal{O}_S^{\oplus n})$. \begin{lemma} \label{lemma-category-vector-bundles} The category of vector bundles over a scheme $S$ is anti-equivalent to the category of quasi-coherent $\mathcal{O}_S$-modules. \end{lemma} \begin{proof} Omitted. Hint: In one direction one uses the functor $\underline{\Spec}_S(-)$ and in the other the functor $(\pi : V \to S) \leadsto (\pi_*\mathcal{O}_V)_1$ (degree $1$ part). \end{proof} \section{Cones} \label{section-cone} \noindent In algebraic geometry cones correspond to graded algebras. By our conventions a graded ring or algebra $A$ comes with a grading $A = \bigoplus_{d \geq 0} A_d$ by the nonnegative integers, see Algebra, Section \ref{algebra-section-graded}. \begin{definition} \label{definition-cone} Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_S$-algebra. Assume that $\mathcal{O}_S \to \mathcal{A}_0$ is an isomorphism\footnote{Often one imposes the assumption that $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{O}_S$. We do not assume this in order to be consistent with \cite[II, (8.3.1)]{EGA}.}. The {\it cone associated to $\mathcal{A}$} or the {\it affine cone associated to $\mathcal{A}$} is $$C(\mathcal{A}) = \underline{\Spec}_S(\mathcal{A}).$$ \end{definition} \noindent The cone associated to a graded sheaf of $\mathcal{O}_S$-algebras comes with a bit of extra structure. Namely, we obtain a grading $$\pi_*\mathcal{O}_{C(\mathcal{A})} = \bigoplus\nolimits_{n \geq 0} \mathcal{A}_n$$ Thus we can define an abstract cone as follows. \begin{definition} \label{definition-abstract-cone} Let $S$ be a scheme. A {\it cone $\pi : C \to S$ over $S$} is an affine morphism of schemes such that $\pi_*\mathcal{O}_C$ is endowed with the structure of a graded $\mathcal{O}_S$-algebra $\pi_*\mathcal{O}_C = \bigoplus\nolimits_{n \geq 0} \mathcal{A}_n$ such that $\mathcal{A}_0 = \mathcal{O}_S$. A {\it morphism of cones} from $\pi : C \to S$ to $\pi' : C' \to S$ is a morphism $f : C \to C'$ such that the induced map $$f^* : \pi'_*\mathcal{O}_{C'} \longrightarrow \pi_*\mathcal{O}_C$$ is compatible with the given gradings. \end{definition} \noindent Any vector bundle is an example of a cone. In fact the category of vector bundles over $S$ is a full subcategory of the category of cones over $S$. \section{Proj of a graded ring} \label{section-proj} \noindent Let $S$ be a graded ring. Consider the topological space $\text{Proj}(S)$ associated to $S$, see Algebra, Section \ref{algebra-section-proj}. We will endow this space with a sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ such that the resulting pair $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ will be a scheme. \medskip\noindent Recall that $\text{Proj}(S)$ has a basis of open sets $D_{+}(f)$, $f \in S_d$, $d \geq 1$ which we call {\it standard opens}, see Algebra, Section \ref{algebra-section-proj}. This terminology will always imply that $f$ is homogeneous of positive degree even if we forget to mention it. In addition, the intersection of two standard opens is another: $D_{+}(f) \cap D_{+}(g) = D_{+}(fg)$, for $f, g \in S$ homogeneous of positive degree. \begin{lemma} \label{lemma-standard-open} Let $S$ be a graded ring. Let $f \in S$ homogeneous of positive degree. \begin{enumerate} \item If $g\in S$ homogeneous of positive degree and $D_{+}(g) \subset D_{+}(f)$, then \begin{enumerate} \item $f$ is invertible in $S_g$, and $f^{\deg(g)}/g^{\deg(f)}$ is invertible in $S_{(g)}$, \item $g^e = af$ for some $e \geq 1$ and $a \in S$ homogeneous, \item there is a canonical $S$-algebra map $S_f \to S_g$, \item there is a canonical $S_0$-algebra map $S_{(f)} \to S_{(g)}$ compatible with the map $S_f \to S_g$, \item the map $S_{(f)} \to S_{(g)}$ induces an isomorphism $$(S_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong S_{(g)},$$ \item these maps induce a commutative diagram of topological spaces $$\xymatrix{ D_{+}(g) \ar[d] & \{\mathbf{Z}\text{-graded primes of }S_g\} \ar[l] \ar[r] \ar[d] & \Spec(S_{(g)}) \ar[d] \\ D_{+}(f) & \{\mathbf{Z}\text{-graded primes of }S_f\} \ar[l] \ar[r] & \Spec(S_{(f)}) }$$ where the horizontal maps are homeomorphisms and the vertical maps are open immersions, \item there are compatible canonical $S_f$ and $S_{(f)}$-module maps $M_f \to M_g$ and $M_{(f)} \to M_{(g)}$ for any graded $S$-module $M$, and \item the map $M_{(f)} \to M_{(g)}$ induces an isomorphism $$(M_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong M_{(g)}.$$ \end{enumerate} \item Any open covering of $D_{+}(f)$ can be refined to a finite open covering of the form $D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)$. \item Let $g_1, \ldots, g_n \in S$ be homogeneous of positive degree. Then $D_{+}(f) \subset \bigcup D_{+}(g_i)$ if and only if $g_1^{\deg(f)}/f^{\deg(g_1)}, \ldots, g_n^{\deg(f)}/f^{\deg(g_n)}$ generate the unit ideal in $S_{(f)}$. \end{enumerate} \end{lemma} \begin{proof} 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}. \end{proof} \noindent In Sheaves, Section \ref{sheaves-section-bases} we defined the notion of a sheaf on a basis, and we showed that it is essentially equivalent to the notion of a sheaf on the space, see Sheaves, Lemmas \ref{sheaves-lemma-extend-off-basis} and \ref{sheaves-lemma-extend-off-basis-structures}. Moreover, we showed in Sheaves, Lemma \ref{sheaves-lemma-cofinal-systems-coverings-standard-case} that it is sufficient to check the sheaf condition on a cofinal system of open coverings for each standard open. By the lemma above it suffices to check on the finite coverings by standard opens. \begin{definition} \label{definition-standard-covering} Let $S$ be a graded ring. Suppose that $D_{+}(f) \subset \text{Proj}(S)$ is a standard open. A {\it standard open covering} of $D_{+}(f)$ is a covering $D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)$, where $g_1, \ldots, g_n \in S$ are homogeneous of positive degree. \end{definition} \noindent Let $S$ be a graded ring. Let $M$ be a graded $S$-module. We will define a presheaf $\widetilde M$ on the basis of standard opens. Suppose that $U \subset \text{Proj}(S)$ is a standard open. If $f, g \in S$ are homogeneous of positive degree such that $D_{+}(f) = D_{+}(g)$, then by Lemma \ref{lemma-standard-open} above there are canonical maps $M_{(f)} \to M_{(g)}$ and $M_{(g)} \to M_{(f)}$ which are mutually inverse. Hence we may choose any $f$ such that $U = D_{+}(f)$ and define $$\widetilde M(U) = M_{(f)}.$$ Note that if $D_{+}(g) \subset D_{+}(f)$, then by Lemma \ref{lemma-standard-open} above we have a canonical map $$\widetilde M(D_{+}(f)) = M_{(f)} \longrightarrow M_{(g)} = \widetilde M(D_{+}(g)).$$ Clearly, this defines a presheaf of abelian groups on the basis of standard opens. If $M = S$, then $\widetilde S$ is a presheaf of rings on the basis of standard opens. And for general $M$ we see that $\widetilde M$ is a presheaf of $\widetilde S$-modules on the basis of standard opens. \medskip\noindent Let us compute the stalk of $\widetilde M$ at a point $x \in \text{Proj}(S)$. Suppose that $x$ corresponds to the homogeneous prime ideal $\mathfrak p \subset S$. By definition of the stalk we see that $$\widetilde M_x = \colim_{f\in S_d, d > 0, f\not\in \mathfrak p} M_{(f)}$$ Here the set $\{f \in S_d, d > 0, f \not \in \mathfrak p\}$ is preordered by the rule $f \geq f' \Leftrightarrow D_{+}(f) \subset D_{+}(f')$. If $f_1, f_2 \in S \setminus \mathfrak p$ are homogeneous of positive degree, then we have $f_1f_2 \geq f_1$ in this ordering. In Algebra, Section \ref{algebra-section-proj} we defined $M_{(\mathfrak p)}$ as the ring whose elements are fractions $x/f$ with $x, f$ homogeneous, $\deg(x) = \deg(f)$, $f \not \in \mathfrak p$. Since $\mathfrak p \in \text{Proj}(S)$ there exists at least one $f_0 \in S$ homogeneous of positive degree with $f_0 \not\in \mathfrak p$. Hence $x/f = f_0x/ff_0$ and we see that we may always assume the denominator of an element in $M_{(\mathfrak p)}$ has positive degree. From these remarks it follows easily that $$\widetilde M_x = M_{(\mathfrak p)}.$$ \medskip\noindent Next, we check the sheaf condition for the standard open coverings. If $D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)$, then the sheaf condition for this covering is equivalent with the exactness of the sequence $$0 \to M_{(f)} \to \bigoplus M_{(g_i)} \to \bigoplus M_{(g_ig_j)}.$$ Note that $D_{+}(g_i) = D_{+}(fg_i)$, and hence we can rewrite this sequence as the sequence $$0 \to M_{(f)} \to \bigoplus M_{(fg_i)} \to \bigoplus M_{(fg_ig_j)}.$$ By Lemma \ref{lemma-standard-open} we see that $g_1^{\deg(f)}/f^{\deg(g_1)}, \ldots, g_n^{\deg(f)}/f^{\deg(g_n)}$ generate the unit ideal in $S_{(f)}$, and that the modules $M_{(fg_i)}$, $M_{(fg_ig_j)}$ are the principal localizations of the $S_{(f)}$-module $M_{(f)}$ at these elements and their products. Thus we may apply Algebra, Lemma \ref{algebra-lemma-cover-module} to the module $M_{(f)}$ over $S_{(f)}$ and the elements $g_1^{\deg(f)}/f^{\deg(g_1)}, \ldots, g_n^{\deg(f)}/f^{\deg(g_n)}$. We conclude that the sequence is exact. By the remarks made above, we see that $\widetilde M$ is a sheaf on the basis of standard opens. \medskip\noindent Thus we conclude from the material in Sheaves, Section \ref{sheaves-section-bases} that there exists a unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the standard opens. Note that by our computation of stalks above and Algebra, Lemma \ref{algebra-lemma-proj-prime} the stalks of this sheaf of rings are all local rings. \medskip\noindent Similarly, for any graded $S$-module $M$ there exists a unique sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules $\mathcal{F}$ which agrees with $\widetilde M$ on the standard opens, see Sheaves, Lemma \ref{sheaves-lemma-extend-off-basis-module}. \begin{definition} \label{definition-structure-sheaf} Let $S$ be a graded ring. \begin{enumerate} \item The {\it structure sheaf $\mathcal{O}_{\text{Proj}(S)}$ of the homogeneous spectrum of $S$} is the unique sheaf of rings $\mathcal{O}_{\text{Proj}(S)}$ which agrees with $\widetilde S$ on the basis of standard opens. \item The locally ringed space $(\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ is called the {\it homogeneous spectrum} of $S$ and denoted $\text{Proj}(S)$. \item The sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules extending $\widetilde M$ to all opens of $\text{Proj}(S)$ is called the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. This sheaf is denoted $\widetilde M$ as well. \end{enumerate} \end{definition} \noindent We summarize the results obtained so far. \begin{lemma} \label{lemma-proj-sheaves} Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$. \begin{enumerate} \item For every $f \in S$ homogeneous of positive degree we have $$\Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}.$$ \item For every $f\in S$ homogeneous of positive degree we have $\Gamma(D_{+}(f), \widetilde M) = M_{(f)}$ as an $S_{(f)}$-module. \item Whenever $D_{+}(g) \subset D_{+}(f)$ the restriction mappings on $\mathcal{O}_{\text{Proj}(S)}$ and $\widetilde M$ are the maps $S_{(f)} \to S_{(g)}$ and $M_{(f)} \to M_{(g)}$ from Lemma \ref{lemma-standard-open}. \item Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$. \item Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{F}_x = M_{(\mathfrak p)}$ as an $S_{(\mathfrak p)}$-module. \item \label{item-map} There is a canonical ring map $S_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde S)$ and a canonical $S_0$-module map $M_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde M)$ compatible with the descriptions of sections over standard opens and stalks above. \end{enumerate} Moreover, all these identifications are functorial in the graded $S$-module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of graded $S$-modules to the category of $\mathcal{O}_{\text{Proj}(S)}$-modules. \end{lemma} \begin{proof} 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}. \end{proof} \begin{remark} \label{remark-global-sections-not-isomorphism} The map from $M_0$ to the global sections of $\widetilde M$ is generally far from being an isomorphism. A trivial example is to take $S = k[x, y, z]$ with $1 = \deg(x) = \deg(y) = \deg(z)$ (or any number of variables) and to take $M = S/(x^{100}, y^{100}, z^{100})$. It is easy to see that $\widetilde M = 0$, but $M_0 = k$. \end{remark} \begin{lemma} \label{lemma-standard-open-proj} Let $S$ be a graded ring. Let $f \in S$ be homogeneous of positive degree. Suppose that $D(g) \subset \Spec(S_{(f)})$ is a standard open. Then there exists a $h \in S$ homogeneous of positive degree such that $D(g)$ corresponds to $D_{+}(h) \subset D_{+}(f)$ via the homeomorphism of Algebra, Lemma \ref{algebra-lemma-topology-proj}. In fact we can take $h$ such that $g = h/f^n$ for some $n$. \end{lemma} \begin{proof} Write $g = h/f^n$ for some $h$ homogeneous of positive degree and some $n \geq 1$. If $D_{+}(h)$ is not contained in $D_{+}(f)$ then we replace $h$ by $hf$ and $n$ by $n + 1$. Then $h$ has the required shape and $D_{+}(h) \subset D_{+}(f)$ corresponds to $D(g) \subset \Spec(S_{(f)})$. \end{proof} \begin{lemma} \label{lemma-proj-scheme} Let $S$ be a graded ring. The locally ringed space $\text{Proj}(S)$ is a scheme. The standard opens $D_{+}(f)$ are affine opens. For any graded $S$-module $M$ the sheaf $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules. \end{lemma} \begin{proof} 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. \end{proof} \begin{lemma} \label{lemma-proj-separated} Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is separated. \end{lemma} \begin{proof} 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. \end{proof} \begin{lemma} \label{lemma-proj-quasi-compact} Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ is quasi-compact if and only if there exist finitely many homogeneous elements $f_1, \ldots, f_n \in S_{+}$ such that $S_{+} \subset \sqrt{(f_1, \ldots, f_n)}$. In this case $\text{Proj}(S) = D_+(f_1) \cup \ldots \cup D_+(f_n)$. \end{lemma} \begin{proof} 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. \end{proof} \begin{lemma} \label{lemma-structure-morphism-proj} Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ has a canonical morphism towards the affine scheme $\Spec(S_0)$, agreeing with the map on topological spaces coming from Algebra, Definition \ref{algebra-definition-proj}. \end{lemma} \begin{proof} 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 $S_0$. \end{proof} \begin{lemma} \label{lemma-proj-valuative-criterion} Let $S$ be a graded ring. If $S$ is finitely generated as an algebra over $S_0$, then the morphism $\text{Proj}(S) \to \Spec(S_0)$ satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{lemma} \begin{proof} 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. \end{proof} \noindent We saw in the proof of Lemma \ref{lemma-proj-valuative-criterion} that, under the hypotheses of that lemma, the morphism $\text{Proj}(S) \to \Spec(S_0)$ is quasi-compact as well. Hence (by Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}) we see that $\text{Proj}(S) \to \Spec(S_0)$ is universally closed in the situation of the lemma. We give two examples showing these results do not hold without some assumption on the graded ring $S$. \begin{example} \label{example-not-existence-valuative-big-proj} Let $\mathbf{C}[X_1, X_2, X_3, \ldots]$ be the graded $\mathbf{C}$-algebra with each $X_i$ in degree $0$. Consider the ring map $$\mathbf{C}[X_1, X_2, X_3, \ldots] \longrightarrow \mathbf{C}[t^\alpha ; \alpha \in \mathbf{Q}_{\geq 0}]$$ which maps $X_i$ to $t^{1/i}$. The right hand side becomes a valuation ring $A$ upon localization at the ideal $\mathfrak m = (t^\alpha ; \alpha > 0)$. This gives a morphism from $\Spec(f.f.(A))$ to $\text{Proj}(\mathbf{C}[X_1, X_2, X_3, \ldots])$ which does not extend to a morphism defined on all of $\Spec(A)$. The reason is that the image of $\Spec(A)$ would be contained in one of the $D_{+}(X_i)$ but then $X_{i + 1}/X_i$ would map to an element of $A$ which it doesn't since it maps to $t^{1/(i + 1) - 1/i}$. \end{example} \begin{example} \label{example-not-existence-valuative-small-proj} Let $R = \mathbf{C}[t]$ and $$S = R[X_1, X_2, X_3, \ldots]/(X_i^2 - tX_{i + 1}).$$ The grading is such that $R = S_0$ and $\deg(X_i) = 2^{i - 1}$. Note that if $\mathfrak p \in \text{Proj}(S)$ then $t \not \in \mathfrak p$ (otherwise $\mathfrak p$ has to contain all of the $X_i$ which is not allowed for an element of the homogeneous spectrum). Thus we see that $D_{+}(X_i) = D_{+}(X_{i + 1})$ for all $i$. Hence $\text{Proj}(S)$ is quasi-compact; in fact it is affine since it is equal to $D_{+}(X_1)$. It is easy to see that the image of $\text{Proj}(S) \to \Spec(R)$ is $D(t)$. Hence the morphism $\text{Proj}(S) \to \Spec(R)$ is not closed. Thus the valuative criterion cannot apply because it would imply that the morphism is closed (see Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed} ). \end{example} \begin{example} \label{example-trivial-proj} Let $A$ be a ring. Let $S = A[T]$ as a graded $A$ algebra with $T$ in degree $1$. Then the canonical morphism $\text{Proj}(S) \to \Spec(A)$ (see Lemma \ref{lemma-structure-morphism-proj}) is an isomorphism. \end{example} \section{Quasi-coherent sheaves on Proj} \label{section-quasi-coherent-proj} \noindent Let $S$ be a graded ring. Let $M$ be a graded $S$-module. We saw in Lemma \ref{lemma-proj-sheaves} how to construct a quasi-coherent sheaf of modules $\widetilde{M}$ on $\text{Proj}(S)$ and a map \begin{equation} \label{equation-map-global-sections} M_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde{M}) \end{equation} of the degree $0$ part of $M$ to the global sections of $\widetilde{M}$. The degree $0$ part of the $n$th twist $M(n)$ of the graded module $M$ (see Algebra, Section \ref{algebra-section-graded}) is equal to $M_n$. Hence we can get maps \begin{equation} \label{equation-map-global-sections-degree-n} M_n \longrightarrow \Gamma(\text{Proj}(S), \widetilde{M(n)}). \end{equation} We would like to be able to perform this operation for any quasi-coherent sheaf $\mathcal{F}$ on $\text{Proj}(S)$. We will do this by tensoring with the $n$th twist of the structure sheaf, see Definition \ref{definition-twist}. In order to relate the two notions we will use the following lemma. \begin{lemma} \label{lemma-widetilde-tensor} Let $S$ be a graded ring. Let $(X, \mathcal{O}_X) = (\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})$ be the scheme of Lemma \ref{lemma-proj-scheme}. Let $f \in S_{+}$ be homogeneous. Let $x \in X$ be a point corresponding to the homogeneous prime $\mathfrak p \subset S$. Let $M$, $N$ be graded $S$-modules. There is a canonical map of $\mathcal{O}_{\text{Proj}(S)}$-modules $$\widetilde M \otimes_{\mathcal{O}_X} \widetilde N \longrightarrow \widetilde{M \otimes_S N}$$ which induces the canonical map $M_{(f)} \otimes_{S_{(f)}} N_{(f)} \to (M \otimes_S N)_{(f)}$ on sections over $D_{+}(f)$ and the canonical map $M_{(\mathfrak p)} \otimes_{S_{(\mathfrak p)}} N_{(\mathfrak p)} \to (M \otimes_S N)_{(\mathfrak p)}$ on stalks at $x$. Moreover, the following diagram $$\xymatrix{ M_0 \otimes_{S_0} N_0 \ar[r] \ar[d] & (M \otimes_S N)_0 \ar[d] \\ \Gamma(X, \widetilde M \otimes_{\mathcal{O}_X} \widetilde N) \ar[r] & \Gamma(X, \widetilde{M \otimes_R N}) }$$ is commutative where the vertical maps are given by (\ref{equation-map-global-sections}). \end{lemma} \begin{proof} To construct a morphism as displayed is the same as constructing a $\mathcal{O}_X$-bilinear map $$\widetilde M \times \widetilde N \longrightarrow \widetilde{M \otimes_R N}$$ see Modules, Section \ref{modules-section-tensor-product}. It suffices to define this on sections over the opens $D_{+}(f)$ compatible with restriction mappings. On $D_{+}(f)$ we use the $S_{(f)}$-bilinear map $M_{(f)} \times N_{(f)} \to (M \otimes_S N)_{(f)}$, $(x/f^n, y/f^m) \mapsto (x \otimes y)/f^{n + m}$. Details omitted. \end{proof} \begin{remark} \label{remark-not-isomorphism} In general the map constructed in Lemma \ref{lemma-widetilde-tensor} above is not an isomorphism. Here is an example. Let $k$ be a field. Let $S = k[x, y, z]$ with $k$ in degree $0$ and $\deg(x) = 1$, $\deg(y) = 2$, $\deg(z) = 3$. Let $M = S(1)$ and $N = S(2)$, see Algebra, Section \ref{algebra-section-graded} for notation. Then $M \otimes_S N = S(3)$. Note that \begin{eqnarray*} S_z & = & k[x, y, z, 1/z] \\ S_{(z)} & = & k[x^3/z, xy/z, y^3/z^2] \cong k[u, v, w]/(uw - v^3) \\ M_{(z)} & = & S_{(z)} \cdot x + S_{(z)} \cdot y^2/z \subset S_z \\ N_{(z)} & = & S_{(z)} \cdot y + S_{(z)} \cdot x^2 \subset S_z \\ S(3)_{(z)} & = & S_{(z)} \cdot z \subset S_z \end{eqnarray*} Consider the maximal ideal $\mathfrak m = (u, v, w) \subset S_{(z)}$. It is not hard to see that both $M_{(z)}/\mathfrak mM_{(z)}$ and $N_{(z)}/\mathfrak mN_{(z)}$ have dimension $2$ over $\kappa(\mathfrak m)$. But $S(3)_{(z)}/\mathfrak mS(3)_{(z)}$ has dimension $1$. Thus the map $M_{(z)} \otimes N_{(z)} \to S(3)_{(z)}$ is not an isomorphism. \end{remark} \section{Invertible sheaves on Proj} \label{section-invertible-on-proj} \noindent Recall from Algebra, Section \ref{algebra-section-graded} the construction of the twisted module $M(n)$ associated to a graded module over a graded ring. \begin{definition} \label{definition-twist} Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. \begin{enumerate} \item We define $\mathcal{O}_X(n) = \widetilde{S(n)}$. This is called the $n$th {\it twist of the structure sheaf of $\text{Proj}(S)$}. \item For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we set $\mathcal{F}(n) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X(n)$. \end{enumerate} \end{definition} \noindent We are going to use Lemma \ref{lemma-widetilde-tensor} to construct some canonical maps. Since $S(n) \otimes_S S(m) = S(n + m)$ we see that there are canonical maps \begin{equation} \label{equation-multiply} \mathcal{O}_X(n) \otimes_{\mathcal{O}_X} \mathcal{O}_X(m) \longrightarrow \mathcal{O}_X(n + m). \end{equation} These maps are not isomorphisms in general, see the example in Remark \ref{remark-not-isomorphism}. The same example shows that $\mathcal{O}_X(n)$ is {\it not} an invertible sheaf on $X$ in general. Tensoring with an arbitrary $\mathcal{O}_X$-module $\mathcal{F}$ we get maps \begin{equation} \label{equation-multiply-on-sheaf} \mathcal{O}_X(n) \otimes_{\mathcal{O}_X} \mathcal{F}(m) \longrightarrow \mathcal{F}(n + m). \end{equation} The maps (\ref{equation-multiply}) on global sections give a map of graded rings \begin{equation} \label{equation-global-sections} S \longrightarrow \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n)). \end{equation} And for an arbitrary $\mathcal{O}_X$-module $\mathcal{F}$ the maps (\ref{equation-multiply-on-sheaf}) give a graded module structure \begin{equation} \label{equation-global-sections-module} \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n)) \times \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(X, \mathcal{F}(m)) \longrightarrow \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(X, \mathcal{F}(m)) \end{equation} and via (\ref{equation-global-sections}) also a $S$-module structure. More generally, given any graded $S$-module $M$ we have $M(n) = M \otimes_S S(n)$. Hence we get maps \begin{equation} \label{equation-multiply-more-generally} \widetilde M(n) = \widetilde M \otimes_{\mathcal{O}_X} \mathcal{O}_X(n) \longrightarrow \widetilde{M(n)}. \end{equation} On global sections (\ref{equation-map-global-sections-degree-n}) defines a map of graded $S$-modules \begin{equation} \label{equation-global-sections-more-generally} M \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M(n)}). \end{equation} Here is an important fact which follows basically immediately from the definitions. \begin{lemma} \label{lemma-when-invertible} Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $f \in S$ be homogeneous of degree $d > 0$. The sheaves $\mathcal{O}_X(nd)|_{D_{+}(f)}$ are invertible, and in fact trivial for all $n \in \mathbf{Z}$ (see Modules, Definition \ref{modules-definition-invertible}). The maps (\ref{equation-multiply}) restricted to $D_{+}(f)$ $$\mathcal{O}_X(nd)|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_X(m)|_{D_{+}(f)} \longrightarrow \mathcal{O}_X(nd + m)|_{D_{+}(f)},$$ the maps (\ref{equation-multiply-on-sheaf}) restricted to $D_+(f)$ $$\mathcal{O}_X(nd)|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{F}(m)|_{D_{+}(f)} \longrightarrow \mathcal{F}(nd + m)|_{D_{+}(f)},$$ and the maps (\ref{equation-multiply-more-generally}) restricted to $D_{+}(f)$ $$\widetilde M(nd)|_{D_{+}(f)} = \widetilde M|_{D_{+}(f)} \otimes_{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_X(nd)|_{D_{+}(f)} \longrightarrow \widetilde{M(nd)}|_{D_{+}(f)}$$ are isomorphisms for all $n, m \in \mathbf{Z}$. \end{lemma} \begin{proof} The (not graded) $S$-module maps $S \to S(nd)$, and $M \to M(nd)$, given by $x \mapsto f^n x$ become isomorphisms after inverting $f$. The first shows that $S_{(f)} \cong S(nd)_{(f)}$ which gives an isomorphism $\mathcal{O}_{D_{+}(f)} \cong \mathcal{O}_X(nd)|_{D_{+}(f)}$. The second shows that the map $S(nd)_{(f)} \otimes_{S_{(f)}} M_{(f)} \to M(nd)_{(f)}$ is an isomorphism. The case of the map (\ref{equation-multiply-on-sheaf}) is a consequence of the case of the map (\ref{equation-multiply}). \end{proof} \begin{lemma} \label{lemma-apply-modules} Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Set $X = \text{Proj}(S)$. Assume $X$ is covered by the standard opens $D_+(f)$ with $f \in S_1$, e.g., if $S$ is generated by $S_1$ over $S_0$. Then the sheaves $\mathcal{O}_X(n)$ are invertible and the maps (\ref{equation-multiply}), (\ref{equation-multiply-on-sheaf}), and (\ref{equation-multiply-more-generally}) are isomorphisms. In particular, these maps induce isomorphisms $$\mathcal{O}_X(1)^{\otimes n} \cong \mathcal{O}_X(n) \quad \text{and} \quad \widetilde{M} \otimes_{\mathcal{O}_X} \mathcal{O}_X(n) = \widetilde{M}(n) \cong \widetilde{M(n)}$$ Thus (\ref{equation-map-global-sections-degree-n}) becomes a map \begin{equation} \label{equation-map-global-sections-degree-n-simplified} M_n \longrightarrow \Gamma(X, \widetilde{M}(n)) \end{equation} and (\ref{equation-global-sections-more-generally}) becomes a map \begin{equation} \label{equation-global-sections-more-generally-simplified} M \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M}(n)). \end{equation} \end{lemma} \begin{proof} Under the assumptions of the lemma $X$ is covered by the open subsets $D_{+}(f)$ with $f \in S_1$ and the lemma is a consequence of Lemma \ref{lemma-when-invertible} above. \end{proof} \begin{lemma} \label{lemma-where-invertible} Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. The following open subsets of $X$ are equal: \begin{enumerate} \item The largest open subset $W = W_d \subset X$ such that each $\mathcal{O}_X(dn)|_W$ is invertible and all the multiplication maps $\mathcal{O}_X(nd)|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(md)|_W \to \mathcal{O}_X(nd + md)|_W$ (see \ref{equation-multiply}) are isomorphisms. \item The union of the open subsets $D_{+}(fg)$ with $f, g \in S$ homogeneous and $\deg(f) = \deg(g) + d$. \end{enumerate} Moreover, all the maps $\widetilde M(nd)|_W = \widetilde M|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(nd)|_W \to \widetilde{M(nd)}|_W$ (see \ref{equation-multiply-more-generally}) are isomorphisms. \end{lemma} \begin{proof} If $x \in D_{+}(fg)$ with $\deg(f) = \deg(g) + d$ then on $D_{+}(fg)$ the sheaves $\mathcal{O}_X(dn)$ are generated by the element $(f/g)^n = f^{2n}/(fg)^n$. This implies $x$ is in the open subset $W$ defined in (1) by arguing as in the proof of Lemma \ref{lemma-when-invertible}. \medskip\noindent Conversely, suppose that $\mathcal{O}_X(d)$ is free of rank 1 in an open neighbourhood $V$ of $x \in X$ and all the multiplication maps $\mathcal{O}_X(nd)|_V \otimes_{\mathcal{O}_V} \mathcal{O}_X(md)|_V \to \mathcal{O}_X(nd + md)|_V$ are isomorphisms. We may choose $h \in S_{+}$ homogeneous such that $D_{+}(h) \subset V$. By the definition of the twists of the structure sheaf we conclude there exists an element $s$ of $(S_h)_d$ such that $s^n$ is a basis of $(S_h)_{nd}$ as a module over $S_{(h)}$ for all $n \in \mathbf{Z}$. We may write $s = f/h^m$ for some $m \geq 1$ and $f \in S_{d + m \deg(h)}$. Set $g = h^m$ so $s = f/g$. Note that $x \in D(g)$ by construction. Note that $g^d \in (S_h)_{-d\deg(g)}$. By assumption we can write this as a multiple of $s^{\deg(g)} = f^{\deg(g)}/g^{\deg(g)}$, say $g^d = a/g^e \cdot f^{\deg(g)}/g^{\deg(g)}$. Then we conclude that $g^{d + e + \deg(g)} = a f^{\deg(g)}$ and hence also $x \in D_{+}(f)$. So $x$ is an element of the set defined in (2). \medskip\noindent The existence of the generating section $s = f/g$ over the affine open $D_{+}(fg)$ whose powers freely generate the sheaves of modules $\mathcal{O}_X(nd)$ easily implies that the multiplication maps $\widetilde M(nd)|_W = \widetilde M|_W \otimes_{\mathcal{O}_W} \mathcal{O}_X(nd)|_W \to \widetilde{M(nd)}|_W$ (see \ref{equation-multiply-more-generally}) are isomorphisms. Compare with the proof of Lemma \ref{lemma-when-invertible}. \end{proof} \noindent Recall from Modules, Lemma \ref{modules-lemma-s-open} that given an invertible sheaf $\mathcal{L}$ on a locally ringed space $X$, and given a global section $s$ of $\mathcal{L}$ the set $X_s = \{x \in X \mid s \not \in \mathfrak m_x\mathcal{L}_x\}$ is open. \begin{lemma} \label{lemma-principal-open} Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Fix $d \geq 1$ an integer. Let $W = W_d \subset X$ be the open subscheme defined in Lemma \ref{lemma-where-invertible}. Let $n \geq 1$ and $f \in S_{nd}$. Denote $s \in \Gamma(W, \mathcal{O}_W(nd))$ the section which is the image of $f$ via (\ref{equation-global-sections}) restricted to $W$. Then $$W_s = D_{+}(f) \cap W.$$ \end{lemma} \begin{proof} Let $D_{+}(ab) \subset W$ be a standard affine open with $a, b \in S$ homogeneous and $\deg(a) = \deg(b) + d$. Note that $D_{+}(ab) \cap D_{+}(f) = D_{+}(abf)$. On the other hand the restriction of $s$ to $D_{+}(ab)$ corresponds to the element $f/1 = b^nf/a^n (a/b)^n \in (S_{ab})_{nd}$. We have seen in the proof of Lemma \ref{lemma-where-invertible} that $(a/b)^n$ is a generator for $\mathcal{O}_W(nd)$ over $D_{+}(ab)$. We conclude that $W_s \cap D_{+}(ab)$ is the principal open associated to $b^nf/a^n \in \mathcal{O}_X(D_{+}(ab))$. Thus the result of the lemma is clear. \end{proof} \noindent The following lemma states the properties that we will later use to characterize schemes with an ample invertible sheaf. \begin{lemma} \label{lemma-ample-on-proj} Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Let $Y \subset X$ be a quasi-compact open subscheme. Denote $\mathcal{O}_Y(n)$ the restriction of $\mathcal{O}_X(n)$ to $Y$. There exists an integer $d \geq 1$ such that \begin{enumerate} \item the subscheme $Y$ is contained in the open $W_d$ defined in Lemma \ref{lemma-where-invertible}, \item the sheaf $\mathcal{O}_Y(dn)$ is invertible for all $n \in \mathbf{Z}$, \item all the maps $\mathcal{O}_Y(nd) \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(m) \longrightarrow \mathcal{O}_Y(nd + m)$ of Equation (\ref{equation-multiply}) are isomorphisms, \item all the maps $\widetilde M(nd)|_Y = \widetilde M|_Y \otimes_{\mathcal{O}_Y} \mathcal{O}_X(nd)|_Y \to \widetilde{M(nd)}|_Y$ (see \ref{equation-multiply-more-generally}) are isomorphisms, \item given $f \in S_{nd}$ denote $s \in \Gamma(Y, \mathcal{O}_Y(nd))$ the image of $f$ via (\ref{equation-global-sections}) restricted to $Y$, then $D_{+}(f) \cap Y = Y_s$, \item a basis for the topology on $Y$ is given by the collection of opens $Y_s$, where $s \in \Gamma(Y, \mathcal{O}_Y(nd))$, $n \geq 1$, and \item a basis for the topology of $Y$ is given by those opens $Y_s \subset Y$, for $s \in \Gamma(Y, \mathcal{O}_Y(nd))$, $n \geq 1$ which are affine. \end{enumerate} \end{lemma} \begin{proof} Since $Y$ is quasi-compact there exist finitely many homogeneous $f_i \in S_{+}$, $i = 1, \ldots, n$ such that the standard opens $D_{+}(f_i)$ give an open covering of $Y$. Let $d_i = \deg(f_i)$ and set $d = d_1 \ldots d_n$. Note that $D_{+}(f_i) = D_{+}(f_i^{d/d_i})$ and hence we see immediately that $Y \subset W_d$, by characterization (2) in Lemma \ref{lemma-where-invertible} or by (1) using Lemma \ref{lemma-when-invertible}. Note that (1) implies (2), (3) and (4) by Lemma \ref{lemma-where-invertible}. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma \ref{lemma-principal-open}. Assertions (6) and (7) follow because the open subsets $D_{+}(f)$ form a basis for the topology of $X$ and are affine. \end{proof} \begin{lemma} \label{lemma-comparison-proj-quasi-coherent} Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Set $M = \bigoplus_{n \in \mathbf{Z}} \Gamma(X, \mathcal{F}(n))$ as a graded $S$-module, using (\ref{equation-global-sections-module}) and (\ref{equation-global-sections}). Then there is a canonical $\mathcal{O}_X$-module map $$\widetilde{M} \longrightarrow \mathcal{F}$$ functorial in $\mathcal{F}$ such that the induced map $M_0 \to \Gamma(X, \mathcal{F})$ is the identity. \end{lemma} \begin{proof} Let $f \in S$ be homogeneous of degree $d > 0$. Recall that $\widetilde{M}|_{D_{+}(f)}$ corresponds to the $S_{(f)}$-module $M_{(f)}$ by Lemma \ref{lemma-proj-sheaves}. Thus we can define a canonical map $$M_{(f)} \longrightarrow \Gamma(D_+(f), \mathcal{F}),\quad m/f^n \longmapsto m|_{D_+(f)} \otimes f|_{D_+(f)}^{-n}$$ which makes sense because $f|_{D_+(f)}$ is a trivializing section of the invertible sheaf $\mathcal{O}_X(d)|_{D_+(f)}$, see Lemma \ref{lemma-when-invertible} and its proof. Since $\widetilde{M}$ is quasi-coherent, this leads to a canonical map $$\widetilde{M}|_{D_+(f)} \longrightarrow \mathcal{F}|_{D_+(f)}$$ via Schemes, Lemma \ref{schemes-lemma-compare-constructions}. We obtain a global map if we prove that the displayed maps glue on overlaps. Proof of this is omitted. We also omit the proof of the final statement. \end{proof} \section{Functoriality of Proj} \label{section-functoriality-proj} \noindent A graded ring map $\psi : A \to B$ does not always give rise to a morphism of associated projective homogeneous spectra. The reason is that the inverse image $\psi^{-1}(\mathfrak q)$ of a homogeneous prime $\mathfrak q \subset B$ may contain the irrelevant prime $A_{+}$ even if $\mathfrak q$ does not contain $B_{+}$. The correct result is stated as follows. \begin{lemma} \label{lemma-morphism-proj} Let $A$, $B$ be two graded rings. Set $X = \text{Proj}(A)$ and $Y = \text{Proj}(B)$. Let $\psi : A \to B$ be a graded ring map. Set $$U(\psi) = \bigcup\nolimits_{f \in A_{+}\ \text{homogeneous}} D_{+}(\psi(f)) \subset Y.$$ Then there is a canonical morphism of schemes $$r_\psi : U(\psi) \longrightarrow X$$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi)}$-algebras $$\theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_X(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d).$$ The triple $(U(\psi), r_\psi, \theta)$ is characterized by the following properties: \begin{enumerate} \item For every $d \geq 0$ the diagram $$\xymatrix{ A_d \ar[d] \ar[rr]_{\psi} & & B_d \ar[d] \\ \Gamma(X, \mathcal{O}_X(d)) \ar[r]^-\theta & \Gamma(U(\psi), \mathcal{O}_Y(d)) & \Gamma(Y, \mathcal{O}_Y(d)) \ar[l] }$$ is commutative. \item For any $f \in A_{+}$ homogeneous we have $r_\psi^{-1}(D_{+}(f)) = D_{+}(\psi(f))$ and the restriction of $r_\psi$ to $D_{+}(\psi(f))$ corresponds to the ring map $A_{(f)} \to B_{(\psi(f))}$ induced by $\psi$. \end{enumerate} \end{lemma} \begin{proof} Clearly condition (2) uniquely determines the morphism of schemes and the open subset $U(\psi)$. Pick $f \in A_d$ with $d \geq 1$. Note that $\mathcal{O}_X(n)|_{D_{+}(f)}$ corresponds to the $A_{(f)}$-module $(A_f)_n$ and that $\mathcal{O}_Y(n)|_{D_{+}(\psi(f))}$ corresponds to the $B_{(\psi(f))}$-module $(B_{\psi(f)})_n$. In other words $\theta$ when restricted to $D_{+}(\psi(f))$ corresponds to a map of $\mathbf{Z}$-graded $B_{(\psi(f))}$-algebras $$A_f \otimes_{A_{(f)}} B_{(\psi(f))} \longrightarrow B_{\psi(f)}$$ Condition (1) determines the images of all elements of $A$. Since $f$ is an invertible element which is mapped to $\psi(f)$ we see that $1/f^m$ is mapped to $1/\psi(f)^m$. It easily follows from this that $\theta$ is uniquely determined, namely it is given by the rule $$a/f^m \otimes b/\psi(f)^e \longmapsto \psi(a)b/\psi(f)^{m + e}.$$ To show existence we remark that the proof of uniqueness above gave a well defined prescription for the morphism $r$ and the map $\theta$ when restricted to every standard open of the form $D_{+}(\psi(f)) \subset U(\psi)$ into $D_{+}(f)$. Call these $r_f$ and $\theta_f$. Hence we only need to verify that if $D_{+}(f) \subset D_{+}(g)$ for some $f, g \in A_{+}$ homogeneous, then the restriction of $r_g$ to $D_{+}(\psi(f))$ matches $r_f$. This is clear from the formulas given for $r$ and $\theta$ above. \end{proof} \begin{lemma} \label{lemma-morphism-proj-transitive} Let $A$, $B$, and $C$ be graded rings. Set $X = \text{Proj}(A)$, $Y = \text{Proj}(B)$ and $Z = \text{Proj}(C)$. Let $\varphi : A \to B$, $\psi : B \to C$ be graded ring maps. Then we have $$U(\psi \circ \varphi) = r_\varphi^{-1}(U(\psi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}.$$ In addition we have $$\theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi}$$ with obvious notation. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-surjective-graded-rings-map-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is surjective for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are surjective but not isomorphisms in general (even if $A \to B$ is surjective). \end{enumerate} \end{lemma} \begin{proof} Part (1) follows from the definition of $U(\psi)$ and the fact that $D_{+}(f) = D_{+}(f^n)$ for any $n > 0$. For $f \in A_{+}$ homogeneous we see that $A_{(f)} \to B_{(\psi(f))}$ is surjective because any element of $B_{(\psi(f))}$ can be represented by a fraction $b/\psi(f)^n$ with $n$ arbitrarily large (which forces the degree of $b \in B$ to be large). This proves (2). The same argument shows the map $$A_f \to B_{\psi(f)}$$ is surjective which proves the surjectivity of $\theta$. For an example where this map is not an isomorphism consider the graded ring $A = k[x, y]$ where $k$ is a field and $\deg(x) = 1$, $\deg(y) = 2$. Set $I = (x)$, so that $B = k[y]$. Note that $\mathcal{O}_Y(1) = 0$ in this case. But it is easy to see that $r_\psi^*\mathcal{O}_Y(1)$ is not zero. (There are less silly examples.) \end{proof} \begin{lemma} \label{lemma-eventual-iso-graded-rings-map-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is an isomorphism for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is an isomorphism, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate} \end{lemma} \begin{proof} We have (1) by Lemma \ref{lemma-surjective-graded-rings-map-proj}. Let $f \in A_{+}$ be homogeneous. The assumption on $\psi$ implies that $A_f \to B_f$ is an isomorphism (details omitted). Thus it is clear that $r_\psi$ and $\theta$ restrict to isomorphisms over $D_{+}(f)$. The lemma follows. \end{proof} \begin{lemma} \label{lemma-surjective-graded-rings-generated-degree-1-map-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume $A_d \to B_d$ is surjective for $d \gg 0$ and that $A$ is generated by $A_1$ over $A_0$. Then \begin{enumerate} \item $U(\psi) = Y$, \item $r_\psi : Y \to X$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate} \end{lemma} \begin{proof} By Lemmas \ref{lemma-eventual-iso-graded-rings-map-proj} and \ref{lemma-morphism-proj-transitive} we may replace $B$ by the image of $A \to B$ without changing $X$ or the sheaves $\mathcal{O}_X(n)$. Thus we may assume that $A \to B$ is surjective. By Lemma \ref{lemma-surjective-graded-rings-map-proj} we get (1) and (2) and surjectivity in (3). By Lemma \ref{lemma-apply-modules} we see that both $\mathcal{O}_X(n)$ and $\mathcal{O}_Y(n)$ are invertible. Hence $\theta$ is an isomorphism. \end{proof} \begin{lemma} \label{lemma-base-change-map-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume there exists a ring map $R \to A_0$ and a ring map $R \to R'$ such that $B = R' \otimes_R A$. Then \begin{enumerate} \item $U(\psi) = Y$, \item the diagram $$\xymatrix{ Y = \text{Proj}(B) \ar[r]_{r_\psi} \ar[d] & \text{Proj}(A) = X \ar[d] \\ \Spec(R') \ar[r] & \Spec(R) }$$ is a fibre product square, and \item the maps $\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)$ are isomorphisms. \end{enumerate} \end{lemma} \begin{proof} This follows immediately by looking at what happens over the standard opens $D_{+}(f)$ for $f \in A_{+}$. \end{proof} \begin{lemma} \label{lemma-localization-map-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume there exists a $g \in A_0$ such that $\psi$ induces an isomorphism $A_g \to B$. Then $U(\psi) = Y$, $r_\psi : Y \to X$ is an open immersion which induces an isomorphism of $Y$ with the inverse image of $D(g) \subset \Spec(A_0)$. Moreover the map $\theta$ is an isomorphism. \end{lemma} \begin{proof} This is a special case of Lemma \ref{lemma-base-change-map-proj} above. \end{proof} \begin{lemma} \label{lemma-d-uple} Let $S$ be a graded ring. Let $d \geq 1$. Set $S' = S^{(d)}$ with notation as in Algebra, Section \ref{algebra-section-graded}. Set $X = \text{Proj}(S)$ and $X' = \text{Proj}(S')$. There is a canonical isomorphism $i : X \to X'$ of schemes such that \begin{enumerate} \item for any graded $S$-module $M$ setting $M' = M^{(d)}$, we have a canonical isomorphism $\widetilde{M} \to i^*\widetilde{M'}$, \item we have canonical isomorphisms $\mathcal{O}_{X}(nd) \to i^*\mathcal{O}_{X'}(n)$ \end{enumerate} and these isomorphisms are compatible with the multiplication maps of Lemma \ref{lemma-widetilde-tensor} and hence with the maps (\ref{equation-multiply}), (\ref{equation-multiply-on-sheaf}), (\ref{equation-global-sections}), (\ref{equation-global-sections-module}), (\ref{equation-multiply-more-generally}), and (\ref{equation-global-sections-more-generally}) (see proof for precise statements. \end{lemma} \begin{proof} The injective ring map $S' \to S$ (which is not a homomorphism of graded rings due to our conventions), induces a map $j : \Spec(S) \to \Spec(S')$. Given a graded prime ideal $\mathfrak p \subset S$ we see that $\mathfrak p' = j(\mathfrak p) = S' \cap \mathfrak p$ is a graded prime ideal of $S'$. Moreover, if $f \in S_+$ is homogeneous and $f \not \in \mathfrak p$, then $f^d \in S'_+$ and $f^d \not \in \mathfrak p'$. Conversely, if $\mathfrak p' \subset S'$ is a graded prime ideal not containing some homogeneous element $f \in S'_+$, then $\mathfrak p = \{g \in S \mid g^d \in \mathfrak p'\}$ is a graded prime ideal of $S$ not containing $f$ whose image under $j$ is $\mathfrak p'$. To see that $\mathfrak p$ is an ideal, note that if $g, h \in \mathfrak p$, then $(g + h)^{2d} \in \mathfrak p'$ by the binomial formula and hence $g + h \in \mathfrak p'$ as $\mathfrak p'$ is a prime. In this way we see that $j$ induces a homeomorphism $i : X \to X'$. Moreover, given $f \in S_+$ homogeneous, then we have $S_{(f)} \cong S'_{(f^d)}$. Since these isomorphisms are compatible with the restrictions mappings of Lemma \ref{lemma-standard-open}, we see that there exists an isomorphism $i^\sharp : i^{-1}\mathcal{O}_{X'} \to \mathcal{O}_X$ of structure sheaves on $X$ and $X'$, hence $i$ is an isomorphism of schemes. \medskip\noindent Let $M$ be a graded $S$-module. Given $f \in S_+$ homogeneous, we have $M_{(f)} \cong M'_{(f^d)}$, hence in exactly the same manner as above we obtain the isomorphism in (1). The isomorphisms in (2) are a special case of (1) for $M = S(nd)$ which gives $M' = S'(n)$. Let $M$ and $N$ be graded $S$-modules. Then we have $$M' \otimes_{S'} N' = (M \otimes_S N)^{(d)} = (M \otimes_S N)'$$ as can be verified directly from the definitions. Having said this the compatibility with the multiplication maps of Lemma \ref{lemma-widetilde-tensor} is the commutativity of the diagram $$\xymatrix{ \widetilde M \otimes_{\mathcal{O}_X} \widetilde N \ar[d]_{(1) \otimes (1)} \ar[r] & \widetilde{M \otimes_S N} \ar[d]^{(1)} \\ i^*(\widetilde{M'} \otimes_{\mathcal{O}_{X'}} \widetilde{N'}) \ar[r] & i^*(\widetilde{M' \otimes_{S'} N'}) }$$ This can be seen by looking at the construction of the maps over the open $D_+(f) = D_+(f^d)$ where the top horizontal arrow is given by the map $M_{(f)} \times N_{(f)} \to (M \otimes_S N)_{(f)}$ and the lower horizontal arrow by the map $M'_{(f^d)} \times N'_{(f^d)} \to (M' \otimes_{S'} N')_{(f^d)}$. Since these maps agree via the identifications $M_{(f)} = M'_{(f^d)}$, etc, we get the desired compatibility. We omit the proof of the other compatibilities. \end{proof} \section{Morphisms into Proj} \label{section-morphisms-proj} \noindent Let $S$ be a graded ring. Let $X = \text{Proj}(S)$ be the homogeneous spectrum of $S$. Let $d \geq 1$ be an integer. Consider the open subscheme \begin{equation} \label{equation-Ud} U_d = \bigcup\nolimits_{f \in S_d} D_{+}(f) \quad\subset\quad X = \text{Proj}(S) \end{equation} Note that $d | d' \Rightarrow U_d \subset U_{d'}$ and $X = \bigcup_d U_d$. Neither $X$ nor $U_d$ need be quasi-compact, see Algebra, Lemma \ref{algebra-lemma-topology-proj}. Let us write $\mathcal{O}_{U_d}(n) = \mathcal{O}_X(n)|_{U_d}$. By Lemma \ref{lemma-when-invertible} we know that $\mathcal{O}_{U_d}(nd)$, $n \in \mathbf{Z}$ is an invertible $\mathcal{O}_{U_d}$-module and that all the multiplication maps $\mathcal{O}_{U_d}(nd) \otimes_{\mathcal{O}_{U_d}} \mathcal{O}_X(m) \to \mathcal{O}_{U_d}(nd + m)$ of (\ref{equation-multiply}) are isomorphisms. In particular we have $\mathcal{O}_{U_d}(nd) \cong \mathcal{O}_{U_d}(d)^{\otimes n}$. The graded ring map (\ref{equation-global-sections}) on global sections combined with restriction to $U_d$ give a homomorphism of graded rings \begin{equation} \label{equation-psi-d} \psi^d : S^{(d)} \longrightarrow \Gamma_*(U_d, \mathcal{O}_{U_d}(d)). \end{equation} For the notation $S^{(d)}$, see Algebra, Section \ref{algebra-section-graded}. For the notation $\Gamma_*$ see Modules, Definition \ref{modules-definition-gamma-star}. Moreover, since $U_d$ is covered by the opens $D_{+}(f)$, $f \in S_d$ we see that $\mathcal{O}_{U_d}(d)$ is globally generated by the sections in the image of $\psi^d_1 : S^{(d)}_1 = S_d \to \Gamma(U_d, \mathcal{O}_{U_d}(d))$, see Modules, Definition \ref{modules-definition-globally-generated}. \medskip\noindent Let $Y$ be a scheme, and let $\varphi : Y \to X$ be a morphism of schemes. Assume the image $\varphi(Y)$ is contained in the open subscheme $U_d$ of $X$. By the discussion following Modules, Definition \ref{modules-definition-gamma-star} we obtain a homomorphism of graded rings $$\Gamma_*(U_d, \mathcal{O}_{U_d}(d)) \longrightarrow \Gamma_*(Y, \varphi^*\mathcal{O}_X(d)).$$ The composition of this and $\psi^d$ gives a graded ring homomorphism \begin{equation} \label{equation-psi-phi-d} \psi_\varphi^d : S^{(d)} \longrightarrow \Gamma_*(Y, \varphi^*\mathcal{O}_X(d)) \end{equation} which has the property that the invertible sheaf $\varphi^*\mathcal{O}_X(d)$ is globally generated by the sections in the image of $(S^{(d)})_1 = S_d \to \Gamma(Y, \varphi^*\mathcal{O}_X(d))$. \begin{lemma} \label{lemma-converse-construction} Let $S$ be a graded ring, and $X = \text{Proj}(S)$. Let $d \geq 1$ and $U_d \subset X$ as above. Let $Y$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $\psi : S^{(d)} \to \Gamma_*(Y, \mathcal{L})$ be a graded ring homomorphism such that $\mathcal{L}$ is generated by the sections in the image of $\psi|_{S_d} : S_d \to \Gamma(Y, \mathcal{L})$. Then there exists a morphism $\varphi : Y \to X$ such that $\varphi(Y) \subset U_d$ and an isomorphism $\alpha : \varphi^*\mathcal{O}_{U_d}(d) \to \mathcal{L}$ such that $\psi_\varphi^d$ agrees with $\psi$ via $\alpha$: $$\xymatrix{ \Gamma_*(Y, \mathcal{L}) & \Gamma_*(Y, \varphi^*\mathcal{O}_{U_d}(d)) \ar[l]^-\alpha & \Gamma_*(U_d, \mathcal{O}_{U_d}(d)) \ar[l]^-{\varphi^*} \\ S^{(d)} \ar[u]^\psi & & S^{(d)} \ar[u]^{\psi^d} \ar[ul]^{\psi^d_\varphi} \ar[ll]_{\text{id}} }$$ commutes. Moreover, the pair $(\varphi, \alpha)$ is unique. \end{lemma} \begin{proof} Pick $f \in S_d$. Denote $s = \psi(f) \in \Gamma(Y, \mathcal{L})$. On the open set $Y_s$ where $s$ does not vanish multiplication by $s$ induces an isomorphism $\mathcal{O}_{Y_s} \to \mathcal{L}|_{Y_s}$, see Modules, Lemma \ref{modules-lemma-s-open}. We will denote the inverse of this map $x \mapsto x/s$, and similarly for powers of $\mathcal{L}$. Using this we define a ring map $\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O})$ by mapping the fraction $a/f^n$ to $\psi(a)/s^n$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} this corresponds to a morphism $\varphi_f : Y_s \to \Spec(S_{(f)}) = D_{+}(f)$. We also introduce the isomorphism $\alpha_f : \varphi_f^*\mathcal{O}_{D_{+}(f)}(d) \to \mathcal{L}|_{Y_s}$ which maps the pullback of the trivializing section $f$ over $D_{+}(f)$ to the trivializing section $s$ over $Y_s$. With this choice the commutativity of the diagram in the lemma holds with $Y$ replace by $Y_s$, $\varphi$ replaced by $\varphi_f$, and $\alpha$ replaced by $\alpha_f$; verification omitted. \medskip\noindent Suppose that $f' \in S_d$ is a second element, and denote $s' = \psi(f') \in \Gamma(Y, \mathcal{L})$. Then $Y_s \cap Y_{s'} = Y_{ss'}$ and similarly $D_{+}(f) \cap D_{+}(f') = D_{+}(ff')$. In Lemma \ref{lemma-ample-on-proj} we saw that $D_{+}(f') \cap D_{+}(f)$ is the same as the set of points of $D_{+}(f)$ where the section of $\mathcal{O}_X(d)$ defined by $f'$ does not vanish. Hence $\varphi_f^{-1}(D_{+}(f') \cap D_{+}(f)) = Y_s \cap Y_{s'} = \varphi_{f'}^{-1}(D_{+}(f') \cap D_{+}(f))$. On $D_{+}(f) \cap D_{+}(f')$ the fraction $f/f'$ is an invertible section of the structure sheaf with inverse $f'/f$. Note that $\psi_{(f')}(f/f') = \psi(f)/s' = s/s'$ and $\psi_{(f)}(f'/f) = \psi(f')/s = s'/s$. We claim there is a unique ring map $S_{(ff')} \to \Gamma(Y_{ss'}, \mathcal{O})$ making the following diagram commute $$\xymatrix{ \Gamma(Y_s, \mathcal{O}) \ar[r] & \Gamma(Y_{ss'}, \mathcal{O}) & \Gamma(Y_{s, '} \mathcal{O}) \ar[l]\\ S_{(f)} \ar[r] \ar[u]^{\psi_{(f)}} & S_{(ff')} \ar[u] & S_{(f')} \ar[l] \ar[u]^{\psi_{(f')}} }$$ It exists because we may use the rule $x/(ff')^n \mapsto \psi(x)/(ss')^n$, which works'' by the formulas above. Uniqueness follows as $\text{Proj}(S)$ is separated, see Lemma \ref{lemma-proj-separated} and its proof. This shows that the morphisms $\varphi_f$ and $\varphi_{f'}$ agree over $Y_s \cap Y_{s'}$. The restrictions of $\alpha_f$ and $\alpha_{f'}$ agree over $Y_s \cap Y_{s'}$ because the regular functions $s/s'$ and $\psi_{(f')}(f)$ agree. This proves that the morphisms $\psi_f$ glue to a global morphism from $Y$ into $U_d \subset X$, and that the maps $\alpha_f$ glue to an isomorphism satisfying the conditions of the lemma. \medskip\noindent We still have to show the pair $(\varphi, \alpha)$ is unique. Suppose $(\varphi', \alpha')$ is a second such pair. Let $f \in S_d$. By the commutativity of the diagrams in the lemma we have that the inverse images of $D_{+}(f)$ under both $\varphi$ and $\varphi'$ are equal to $Y_{\psi(f)}$. Since the opens $D_{+}(f)$ are a basis for the topology on $X$, and since $X$ is a sober topological space (see Schemes, Lemma \ref{schemes-lemma-scheme-sober}) this means the maps $\varphi$ and $\varphi'$ are the same on underlying topological spaces. Let us use $s = \psi(f)$ to trivialize the invertible sheaf $\mathcal{L}$ over $Y_{\psi(f)}$. By the commutativity of the diagrams we have that $\alpha^{\otimes n}(\psi^d_{\varphi}(x)) = \psi(x) = (\alpha')^{\otimes n}(\psi^d_{\varphi'}(x))$ for all $x \in S_{nd}$. By construction of $\psi^d_{\varphi}$ and $\psi^d_{\varphi'}$ we have $\psi^d_{\varphi}(x) = \varphi^\sharp(x/f^n) \psi^d_{\varphi}(f^n)$ over $Y_{\psi(f)}$, and similarly for $\psi^d_{\varphi'}$. by the commutativity of the diagrams of the lemma we deduce that $\varphi^\sharp(x/f^n) = (\varphi')^\sharp(x/f^n)$. This proves that $\varphi$ and $\varphi'$ induce the same morphism from $Y_{\psi(f)}$ into the affine scheme $D_{+}(f) = \Spec(S_{(f)})$. Hence $\varphi$ and $\varphi'$ are the same as morphisms. Finally, it remains to show that the commutativity of the diagram of the lemma singles out, given $\varphi$, a unique $\alpha$. We omit the verification. \end{proof} \noindent We continue the discussion from above the lemma. Let $S$ be a graded ring. Let $Y$ be a scheme. We will consider {\it triples} $(d, \mathcal{L}, \psi)$ where \begin{enumerate} \item $d \geq 1$ is an integer, \item $\mathcal{L}$ is an invertible $\mathcal{O}_Y$-module, and \item $\psi : S^{(d)} \to \Gamma_*(Y, \mathcal{L})$ is a graded ring homomorphism such that $\mathcal{L}$ is generated by the global sections $\psi(f)$, with $f \in S_d$. \end{enumerate} Given a morphism $h : Y' \to Y$ and a triple $(d, \mathcal{L}, \psi)$ over $Y$ we can pull it back to the triple $(d, h^*\mathcal{L}, h^* \circ \psi)$. Given two triples $(d, \mathcal{L}, \psi)$ and $(d, \mathcal{L}', \psi')$ with the same integer $d$ we say they are {\it strictly equivalent} if there exists an isomorphism $\beta : \mathcal{L} \to \mathcal{L}'$ such that $\beta \circ \psi = \psi'$ as graded ring maps $S^{(d)} \to \Gamma_*(Y, \mathcal{L}')$. \medskip\noindent For each integer $d \geq 1$ we define \begin{eqnarray*} F_d : \Sch^{opp} & \longrightarrow & \textit{Sets}, \\ Y & \longmapsto & \{\text{strict equivalence classes of triples } (d, \mathcal{L}, \psi) \text{ as above}\} \end{eqnarray*} with pullbacks as defined above. \begin{lemma} \label{lemma-proj-functor-strict} Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. The open subscheme $U_d \subset X$ (\ref{equation-Ud}) represents the functor $F_d$ and the triple $(d, \mathcal{O}_{U_d}(d), \psi^d)$ defined above is the universal family (see Schemes, Section \ref{schemes-section-representable}). \end{lemma} \begin{proof} This is a reformulation of Lemma \ref{lemma-converse-construction} \end{proof} \begin{lemma} \label{lemma-apply} Let $S$ be a graded ring generated as an $S_0$-algebra by the elements of $S_1$. In this case the scheme $X = \text{Proj}(S)$ represents the functor which associates to a scheme $Y$ the set of pairs $(\mathcal{L}, \psi)$, where \begin{enumerate} \item $\mathcal{L}$ is an invertible $\mathcal{O}_Y$-module, and \item $\psi : S \to \Gamma_*(Y, \mathcal{L})$ is a graded ring homomorphism such that $\mathcal{L}$ is generated by the global sections $\psi(f)$, with $f \in S_1$ \end{enumerate} up to strict equivalence as above. \end{lemma} \begin{proof} Under the assumptions of the lemma we have $X = U_1$ and the lemma is a reformulation of Lemma \ref{lemma-proj-functor-strict} above. \end{proof} \noindent We end this section with a discussion of a functor corresponding to $\text{Proj}(S)$ for a general graded ring $S$. We advise the reader to skip the rest of this section. \medskip\noindent Fix an arbitrary graded ring $S$. Let $T$ be a scheme. We will say two triples $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ over $T$ with possibly different integers $d$, $d'$ are {\it equivalent} if there exists an isomorphism $\beta : \mathcal{L}^{\otimes d'} \to (\mathcal{L}')^{\otimes d}$ of invertible sheaves over $T$ such that $\beta \circ \psi|_{S^{(dd')}}$ and $\psi'|_{S^{(dd')}}$ agree as graded ring maps $S^{(dd')} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes dd'})$. \begin{lemma} \label{lemma-equivalent} Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $T$ be a scheme. Let $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ be two triples over $T$. The following are equivalent: \begin{enumerate} \item Let $n = \text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{S^{(n)}}$ and $\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})$. \item The triples $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ are equivalent. \item For some positive integer $n = ad = a'd'$ there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi|_{S^{(n)}}$ and $\psi'|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})$. \item The morphisms $\varphi : T \to X$ and $\varphi' : T \to X$ associated to $(d, \mathcal{L}, \psi)$ and $(d', \mathcal{L}', \psi')$ are equal. \end{enumerate} \end{lemma} \begin{proof} Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Also (3) implies (4) by the uniqueness statement in Lemma \ref{lemma-converse-construction}. Thus we have to prove that (4) implies (1). Assume (4), in other words $\varphi = \varphi'$. Note that this implies that we may write $\mathcal{L} = \varphi^*\mathcal{O}_X(d)$ and $\mathcal{L}' = \varphi^*\mathcal{O}_X(d')$. Moreover, via these identifications we have that the graded ring maps $\psi$ and $\psi'$ correspond to the restriction of the canonical graded ring map $$S \longrightarrow \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n))$$ to $S^{(d)}$ and $S^{(d')}$ composed with pullback by $\varphi$ (by Lemma \ref{lemma-converse-construction} again). Hence taking $\beta$ to be the isomorphism $$(\varphi^*\mathcal{O}_X(d))^{\otimes a} = \varphi^*\mathcal{O}_X(n) = (\varphi^*\mathcal{O}_X(d'))^{\otimes a'}$$ works. \end{proof} \noindent Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Over the open subscheme scheme $U_d \subset X = \text{Proj}(S)$ (\ref{equation-Ud}) we have the triple $(d, \mathcal{O}_{U_d}(d), \psi^d)$. Clearly, if $d | d'$ the triples $(d, \mathcal{O}_{U_d}(d), \psi^d)$ and $(d', \mathcal{O}_{U_{d'}}(d'), \psi^{d'})$ are equivalent when restricted to the open $U_d$ (which is a subset of $U_{d'}$). This, combined with Lemma \ref{lemma-converse-construction} shows that morphisms $Y \to X$ correspond roughly to equivalence classes of triples over $Y$. This is not quite true since if $Y$ is not quasi-compact, then there may not be a single triple which works. Thus we have to be slightly careful in defining the corresponding functor. \medskip\noindent Here is one possible way to do this. Suppose $d' = ad$. Consider the transformation of functors $F_d \to F_{d'}$ which assigns to the triple $(d, \mathcal{L}, \psi)$ over $T$ the triple $(d', \mathcal{L}^{\otimes a}, \psi|_{S^{(d')}})$. One of the implications of Lemma \ref{lemma-equivalent} is that the transformation $F_d \to F_{d'}$ is injective! For a quasi-compact scheme $T$ we define $$F(T) = \bigcup\nolimits_{d \in \mathbf{N}} F_d(T)$$ with transition maps as explained above. This clearly defines a contravariant functor on the category of quasi-compact schemes with values in sets. For a general scheme $T$ we define $$F(T) = \lim_{V \subset T\text{ quasi-compact open}} F(V).$$ In other words, an element $\xi$ of $F(T)$ corresponds to a compatible system of choices of elements $\xi_V \in F(V)$ where $V$ ranges over the quasi-compact opens of $T$. We omit the definition of the pullback map $F(T) \to F(T')$ for a morphism $T' \to T$ of schemes. Thus we have defined our functor \begin{eqnarray*} F : \Sch^{opp} & \longrightarrow & \textit{Sets} \end{eqnarray*} \begin{lemma} \label{lemma-proj-functor} Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. The functor $F$ defined above is representable by the scheme $X$. \end{lemma} \begin{proof} We have seen above that the functor $F_d$ corresponds to the open subscheme $U_d \subset X$. Moreover the transformation of functors $F_d \to F_{d'}$ (if $d | d'$) defined above corresponds to the inclusion morphism $U_d \to U_{d'}$ (see discussion above). Hence to show that $F$ is represented by $X$ it suffices to show that $T \to X$ for a quasi-compact scheme $T$ ends up in some $U_d$, and that for a general scheme $T$ we have $$\Mor(T, X) = \lim_{V \subset T\text{ quasi-compact open}} \Mor(V, X).$$ These verifications are omitted. \end{proof} \section{Projective space} \label{section-projective-space} \noindent Projective space is one of the fundamental objects studied in algebraic geometry. In this section we just give its construction as $\text{Proj}$ of a polynomial ring. Later we will discover many of its beautiful properties. \begin{lemma} \label{lemma-projective-space} Let $S = \mathbf{Z}[T_0, \ldots, T_n]$ with $\deg(T_i) = 1$. The scheme $$\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(S)$$ represents the functor which associates to a scheme $Y$ the pairs $(\mathcal{L}, (s_0, \ldots, s_n))$ where \begin{enumerate} \item $\mathcal{L}$ is an invertible $\mathcal{O}_Y$-module, and \item $s_0, \ldots, s_n$ are global sections of $\mathcal{L}$ which generate $\mathcal{L}$ \end{enumerate} up to the following equivalence: $(\mathcal{L}, (s_0, \ldots, s_n)) \sim (\mathcal{N}, (t_0, \ldots, t_n))$ $\Leftrightarrow$ there exists an isomorphism $\beta : \mathcal{L} \to \mathcal{N}$ with $\beta(s_i) = t_i$ for $i = 0, \ldots, n$. \end{lemma} \begin{proof} This is a special case of Lemma \ref{lemma-apply} above. Namely, for any graded ring $A$ we have \begin{eqnarray*} \Mor_{graded rings}(\mathbf{Z}[T_0, \ldots, T_n], A) & = & A_1 \times \ldots \times A_1 \\ \psi & \mapsto & (\psi(T_0), \ldots, \psi(T_n)) \end{eqnarray*} and the degree $1$ part of $\Gamma_*(Y, \mathcal{L})$ is just $\Gamma(Y, \mathcal{L})$. \end{proof} \begin{definition} \label{definition-projective-space} The scheme $\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])$ is called {\it projective $n$-space over $\mathbf{Z}$}. Its base change $\mathbf{P}^n_S$ to a scheme $S$ is called {\it projective $n$-space over $S$}. If $R$ is a ring the base change to $\Spec(R)$ is denoted $\mathbf{P}^n_R$ and called {\it projective $n$-space over $R$}. \end{definition} \noindent Given a scheme $Y$ over $S$ and a pair $(\mathcal{L}, (s_0, \ldots, s_n))$ as in Lemma \ref{lemma-projective-space} the induced morphism to $\mathbf{P}^n_S$ is denoted $$\varphi_{(\mathcal{L}, (s_0, \ldots, s_n))} : Y \longrightarrow \mathbf{P}^n_S$$ This makes sense since the pair defines a morphism into $\mathbf{P}^n_{\mathbf{Z}}$ and we already have the structure morphism into $S$ so combined we get a morphism into $\mathbf{P}^n_S = \mathbf{P}^n_{\mathbf{Z}} \times S$. Note that this is the $S$-morphism characterized by $$\mathcal{L} = \varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}^*\mathcal{O}_{\mathbf{P}^n_R}(1) \quad \text{and} \quad s_i = \varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}^*T_i$$ where we think of $T_i$ as a global section of $\mathcal{O}_{\mathbf{P}^n_S}(1)$ via (\ref{equation-global-sections}). \begin{lemma} \label{lemma-standard-covering-projective-space} Projective $n$-space over $\mathbf{Z}$ is covered by $n + 1$ standard opens $$\mathbf{P}^n_{\mathbf{Z}} = \bigcup\nolimits_{i = 0, \ldots, n} D_{+}(T_i)$$ where each $D_{+}(T_i)$ is isomorphic to $\mathbf{A}^n_{\mathbf{Z}}$ affine $n$-space over $\mathbf{Z}$. \end{lemma} \begin{proof} This is true because $\mathbf{Z}[T_0, \ldots, T_n]_{+} = (T_0, \ldots, T_n)$ and since $$\Spec \left( \mathbf{Z} \left[\frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i} \right] \right) \cong \mathbf{A}^n_{\mathbf{Z}}$$ in an obvious way. \end{proof} \begin{lemma} \label{lemma-projective-space-separated} Let $S$ be a scheme. The structure morphism $\mathbf{P}^n_S \to S$ is \begin{enumerate} \item separated, \item quasi-compact, \item satisfies the existence and uniqueness parts of the valuative criterion, and \item universally closed. \end{enumerate} \end{lemma} \begin{proof} All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas \ref{schemes-lemma-separated-permanence} and \ref{schemes-lemma-quasi-compact-preserved-base-change}). Hence it suffices to prove them for the morphism $\mathbf{P}^n_{\mathbf{Z}} \to \Spec(\mathbf{Z})$. Separatedness is Lemma \ref{lemma-proj-separated}. Quasi-compactness follows from Lemma \ref{lemma-standard-covering-projective-space}. Existence and uniqueness of the valuative criterion follow from Lemma \ref{lemma-proj-valuative-criterion}. Universally closed follows from the above and Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}. \end{proof} \begin{remark} \label{remark-missing-finite-type} What's missing in the list of properties above? Well to be sure the property of being of finite type. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. (Another property which is missing is smoothness''. And I'm sure there are many more you can think of.) \end{remark} \begin{lemma}[Segre embedding] \label{lemma-segre-embedding} Let $S$ be a scheme. There exists a closed immersion $$\mathbf{P}^n_S \times_S \mathbf{P}^m_S \longrightarrow \mathbf{P}^{nm + n + m}_S$$ called the {\it Segre embedding}. \end{lemma} \begin{proof} It suffices to prove this when $S = \Spec(\mathbf{Z})$. Hence we will drop the index $S$ and work in the absolute setting. Write $\mathbf{P}^n = \text{Proj}(\mathbf{Z}[X_0, \ldots, X_n])$, $\mathbf{P}^m = \text{Proj}(\mathbf{Z}[Y_0, \ldots, Y_m])$, and $\mathbf{P}^{nm + n + m} = \text{Proj}(\mathbf{Z}[Z_0, \ldots, Z_{nm + n + m}])$. In order to map into $\mathbf{P}^{nm + n + m}$ we have to write down an invertible sheaf $\mathcal{L}$ on the left hand side and $(n + 1)(m + 1)$ sections $s_i$ which generate it. See Lemma \ref{lemma-projective-space}. The invertible sheaf we take is $$\mathcal{L} = \text{pr}_1^*\mathcal{O}_{\mathbf{P}^n}(1) \otimes \text{pr}_2^*\mathcal{O}_{\mathbf{P}^m}(1)$$ The sections we take are $$s_0 = X_0Y_0, \ s_1 = X_1Y_0, \ldots, \ s_n = X_nY_0, \ s_{n + 1} = X_0Y_1, \ldots, \ s_{nm + n + m} = X_nY_m.$$ These generate $\mathcal{L}$ since the sections $X_i$ generate $\mathcal{O}_{\mathbf{P}^n}(1)$ and the sections $Y_j$ generate $\mathcal{O}_{\mathbf{P}^m}(1)$. The induced morphism $\varphi$ has the property that $$\varphi^{-1}(D_{+}(Z_{i + (n + 1)j})) = D_{+}(X_i) \times D_{+}(Y_j).$$ Hence it is an affine morphism. The corresponding ring map in case $(i, j) = (0, 0)$ is the map $$\mathbf{Z}[Z_1/Z_0, \ldots, Z_{nm + n + m}/Z_0] \longrightarrow \mathbf{Z}[X_1/X_0, \ldots, X_n/X_0, Y_1/Y_0, \ldots, Y_n/Y_0]$$ which maps $Z_i/Z_0$ to the element $X_i/X_0$ for $i \leq n$ and the element $Z_{(n + 1)j}/Z_0$ to the element $Y_j/Y_0$. Hence it is surjective. A similar argument works for the other affine open subsets. Hence the morphism $\varphi$ is a closed immersion (see Schemes, Lemma \ref{schemes-lemma-closed-local-target} and Example \ref{schemes-example-closed-immersion-affines}.) \end{proof} \noindent The following two lemmas are special cases of more general results later, but perhaps it makes sense to prove these directly here now. \begin{lemma} \label{lemma-closed-in-projective-space} Let $R$ be a ring. Let $Z \subset \mathbf{P}^n_R$ be a closed subscheme. Let $$I_d = \Ker\left( R[T_0, \ldots, T_n]_d \longrightarrow \Gamma(Z, \mathcal{O}_{\mathbf{P}^n_R}(d)|_Z)\right)$$ Then $I = \bigoplus I_d \subset R[T_0, \ldots, T_n]$ is a graded ideal and $Z = \text{Proj}(R[T_0, \ldots, T_n]/I)$. \end{lemma} \begin{proof} It is clear that $I$ is a graded ideal. Set $Z' = \text{Proj}(R[T_0, \ldots, T_n]/I)$. By Lemma \ref{lemma-surjective-graded-rings-generated-degree-1-map-proj} we see that $Z'$ is a closed subscheme of $\mathbf{P}^n_R$. To see the equality $Z = Z'$ it suffices to check on an standard affine open $D_{+}(T_i)$. By renumbering the homogeneous coordinates we may assume $i = 0$. Say $Z \cap D_{+}(T_0)$, resp.\ $Z' \cap D_{+}(T_0)$ is cut out by the ideal $J$, resp.\ $J'$ of $R[T_1/T_0, \ldots, T_n/T_0]$. Then $J'$ is the ideal generated by the elements $F/T_0^{\deg(F)}$ where $F \in I$ is homogeneous. Suppose the degree of $F \in I$ is $d$. Since $F$ vanishes as a section of $\mathcal{O}_{\mathbf{P}^n_R}(d)$ restricted to $Z$ we see that $F/T_0^d$ is an element of $J$. Thus $J' \subset J$. \medskip\noindent Conversely, suppose that $f \in J$. If $f$ has total degree $d$ in $T_1/T_0, \ldots, T_n/T_0$, then we can write $f = F/T_0^d$ for some $F \in R[T_0, \ldots, T_n]_d$. Pick $i \in \{1, \ldots, n\}$. Then $Z \cap D_{+}(T_i)$ is cut out by some ideal $J_i \subset R[T_0/T_i, \ldots, T_n/T_i]$. Moreover, $$J \cdot R\left[ \frac{T_1}{T_0}, \ldots, \frac{T_n}{T_0}, \frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i} \right] = J_i \cdot R\left[ \frac{T_1}{T_0}, \ldots, \frac{T_n}{T_0}, \frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i} \right]$$ The left hand side is the localization of $J$ with respect to the element $T_i/T_0$ and the right hand side is the localization of $J_i$ with respect to the element $T_0/T_i$. It follows that $T_0^{d_i}F/T_i^{d + d_i}$ is an element of $J_i$ for some $d_i$ sufficiently large. This proves that $T_0^{\max(d_i)}F$ is an element of $I$, because its restriction to each standard affine open $D_{+}(T_i)$ vanishes on the closed subscheme $Z \cap D_{+}(T_i)$. Hence $f \in J'$ and we conclude $J \subset J'$ as desired. \end{proof} \noindent The following lemma is a special case of the more general Properties, Lemmas \ref{properties-lemma-ample-quasi-coherent} or \ref{properties-lemma-proj-quasi-coherent}. \begin{lemma} \label{lemma-quasi-coherent-projective-space} Let $R$ be a ring. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathbf{P}^n_R$. For $d \geq 0$ set $$M_d = \Gamma(\mathbf{P}^n_R, \mathcal{F} \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{O}_{\mathbf{P}^n_R}(d)) = \Gamma(\mathbf{P}^n_R, \mathcal{F}(d))$$ Then $M = \bigoplus_{d \geq 0} M_d$ is a graded $R[T_0, \ldots, R_n]$-module and there is a canonical isomorphism $\mathcal{F} = \widetilde{M}$. \end{lemma} \begin{proof} The multiplication maps $$R[T_0, \ldots, R_n]_e \times M_d \longrightarrow M_{d + e}$$ come from the natural isomorphisms $$\mathcal{O}_{\mathbf{P}^n_R}(e) \otimes_{\mathcal{O}_{\mathbf{P}^n_R}} \mathcal{F}(d) \longrightarrow \mathcal{F}(e + d)$$ see Equation (\ref{equation-global-sections-module}). Let us construct the map $c : \widetilde{M} \to \mathcal{F}$. On each of the standard affines $U_i = D_{+}(T_i)$ we see that $\Gamma(U_i, \widetilde{M}) = (M[1/T_i])_0$ where the subscript ${}_0$ means degree $0$ part. An element of this can be written as $m/T_i^d$ with $m \in M_d$. Since $T_i$ is a generator of $\mathcal{O}(1)$ over $U_i$ we can always write $m|_{U_i} = m_i \otimes T_i^d$ where $m_i \in \Gamma(U_i, \mathcal{F})$ is a unique section. Thus a natural guess is $c(m/T_i^d) = m_i$. A small argument, which is omitted here, shows that this gives a well defined map $c : \widetilde{M} \to \mathcal{F}$ if we can show that $$(T_i/T_j)^d m_i|_{U_i \cap U_j} = m_j|_{U_i \cap U_j}$$ in $M[1/T_iT_j]$. But this is clear since on the overlap the generators $T_i$ and $T_j$ of $\mathcal{O}(1)$ differ by the invertible function $T_i/T_j$. \medskip\noindent Injectivity of $c$. We may check for injectivity over the affine opens $U_i$. Let $i \in \{0, \ldots, n\}$ and let $s$ be an element $s = m/T_i^d \in \Gamma(U_i, \widetilde{M})$ such that $c(m/T_i^d) = 0$. By the description of $c$ above this means that $m_i = 0$, hence $m|_{U_i} = 0$. Hence $T_i^em = 0$ in $M$ for some $e$. Hence $s = m/T_i^d = T_i^e/T_i^{e + d} = 0$ as desired. \medskip\noindent Surjectivity of $c$. We may check for surjectivity over the affine opens $U_i$. By renumbering it suffices to check it over $U_0$. Let $s \in \mathcal{F}(U_0)$. Let us write $\mathcal{F}|_{U_i} = \widetilde{N_i}$ for some $R[T_0/T_i, \ldots, T_0/T_i]$-module $N_i$, which is possible because $\mathcal{F}$ is quasi-coherent. So $s$ corresponds to an element $x \in N_0$. Then we have that $$(N_i)_{T_j/T_i} \cong (N_j)_{T_i/T_j}$$ (where the subscripts mean principal localization at'') as modules over the ring $$R\left[ \frac{T_0}{T_i}, \ldots, \frac{T_n}{T_i}, \frac{T_0}{T_j}, \ldots, \frac{T_n}{T_j} \right].$$ This means that for some large integer $d$ there exist elements $s_i \in N_i$, $i = 1, \ldots, n$ such that $$s = (T_i/T_0)^d s_i$$ on $U_0 \cap U_i$. Next, we look at the difference $$t_{ij} = s_i - (T_j/T_i)^d s_j$$ on $U_i \cap U_j$, $0 < i < j$. By our choice of $s_i$ we know that $t_{ij}|_{U_0 \cap U_i \cap U_j} = 0$. Hence there exists a large integer $e$ such that $(T_0/T_i)^et_{ij} = 0$. Set $s_i' = (T_0/T_i)^es_i$, and $s_0' = s$. Then we will have $$s_a' = (T_b/T_a)^{e + d} s_b'$$ on $U_a \cap U_b$ for all $a, b$. This is exactly the condition that the elements $s'_a$ glue to a global section $m \in \Gamma(\mathbf{P}^n_R, \mathcal{F}(e + d))$. And moreover $c(m/T_0^{e + d}) = s$ by construction. Hence $c$ is surjective and we win. \end{proof} \begin{lemma} \label{lemma-globally-generated-omega-twist-1} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf and let $s_0, \ldots, s_n$ be global sections of $\mathcal{L}$ which generate it. Let $\mathcal{F}$ be the kernel of the induced map $\mathcal{O}_X^{\oplus n + 1} \to \mathcal{L}$. Then $\mathcal{F} \otimes \mathcal{L}$ is globally generated. \end{lemma} \begin{proof} In fact the result is true if $X$ is any locally ringed space. The sheaf $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module of rank $n$. The elements $$s_{ij} = (0, \ldots, 0, s_j, 0, \ldots, 0, s_i, 0, \ldots, 0) \in \Gamma(X, \mathcal{L}^{\oplus n + 1})$$ with $s_j$ in the $i$th spot and $s_i$ in the $j$th spot map to zero in $\mathcal{L}^{\otimes 2}$. Hence $s_{ij} \in \Gamma(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L})$. A local computation shows that these sections generate $\mathcal{F} \otimes \mathcal{L}$. \medskip\noindent Alternative proof. Consider the morphism $\varphi : X \to \mathbf{P}^n_\mathbf{Z}$ associated to the pair $(\mathcal{L}, (s_0, \ldots, s_n))$. Since the pullback of $\mathcal{O}(1)$ is $\mathcal{L}$ and since the pullback of $T_i$ is $s_i$, it suffices to prove the lemma in the case of $\mathbf{P}^n_\mathbf{Z}$. In this case the sheaf $\mathcal{F}$ corresponds to the graded $S = \mathbf{Z}[T_0, \ldots, T_n]$ module $M$ which fits into the short exact sequence $$0 \to M \to S^{\oplus n + 1} \to S(1) \to 0$$ where the second map is given by $T_0, \ldots, T_n$. In this case the statement above translates into the statement that the elements $$T_{ij} = (0, \ldots, 0, T_j, 0, \ldots, 0, T_i, 0, \ldots, 0) \in M(1)_0$$ generate the graded module $M(1)$ over $S$. We omit the details. \end{proof} \section{Invertible sheaves and morphisms into Proj} \label{section-invertible-proj} \noindent Let $T$ be a scheme and let $\mathcal{L}$ be an invertible sheaf on $T$. For a section $s \in \Gamma(T, \mathcal{L})$ we denote $T_s$ the open subset of points where $s$ does not vanish. See Modules, Lemma \ref{modules-lemma-s-open}. We can view the following lemma as a slight generalization of Lemma \ref{lemma-apply}. It also is a generalization of Lemma \ref{lemma-morphism-proj}. \begin{lemma} \label{lemma-invertible-map-into-proj} Let $A$ be a graded ring. Set $X = \text{Proj}(A)$. Let $T$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_T$-module. Let $\psi : A \to \Gamma_*(T, \mathcal{L})$ be a homomorphism of graded rings. Set $$U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)}$$ The morphism $\psi$ induces a canonical morphism of schemes $$r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow X$$ together with a map of $\mathbf{Z}$-graded $\mathcal{O}_T$-algebras $$\theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_X(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{L}^{\otimes d}|_{U(\psi)}.$$ The triple $(U(\psi), r_{\mathcal{L}, \psi}, \theta)$ is characterized by the following properties: \begin{enumerate} \item For $f \in A_{+}$ homogeneous we have $r_{\mathcal{L}, \psi}^{-1}(D_{+}(f)) = T_{\psi(f)}$. \item For every $d \geq 0$ the diagram $$\xymatrix{ A_d \ar[d]_{(\ref{equation-global-sections})} \ar[r]_{\psi} & \Gamma(T, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(X, \mathcal{O}_X(d)) \ar[r]^{\theta} & \Gamma(U(\psi), \mathcal{L}^{\otimes d}) }$$ is commutative. \end{enumerate} Moreover, for any $d \geq 1$ and any open subscheme $V \subset T$ such that the sections in $\psi(A_d)$ generate $\mathcal{L}^{\otimes d}|_V$ the morphism $r_{\mathcal{L}, \psi}|_V$ agrees with the morphism $\varphi : V \to \text{Proj}(A)$ and the map $\theta|_V$ agrees with the map $\alpha : \varphi^*\mathcal{O}_X(d) \to \mathcal{L}^{\otimes d}|_V$ where $(\varphi, \alpha)$ is the pair of Lemma \ref{lemma-converse-construction} associated to $\psi|_{A^{(d)}} : A^{(d)} \to \Gamma_*(V, \mathcal{L}^{\otimes d})$. \end{lemma} \begin{proof} Suppose that we have two triples $(U, r : U \to X, \theta)$ and $(U', r' : U' \to X, \theta')$ satisfying (1) and (2). Property (1) implies that $U = U' = U(\psi)$ and that $r = r'$ as maps of underlying topological spaces, since the opens $D_{+}(f)$ form a basis for the topology on $X$, and since $X$ is a sober topological space (see Algebra, Section \ref{algebra-section-proj} and Schemes, Lemma \ref{schemes-lemma-scheme-sober}). Let $f \in A_{+}$ be homogeneous. Note that $\Gamma(D_{+}(f), \bigoplus_{n \in \mathbf{Z}} \mathcal{O}_X(n)) = A_f$ as a $\mathbf{Z}$-graded algebra. Consider the two $\mathbf{Z}$-graded ring maps $$\theta, \theta' : A_f \longrightarrow \Gamma(T_{\psi(f)}, \bigoplus \mathcal{L}^{\otimes n}).$$ We know that multiplication by $f$ (resp.\ $\psi(f)$) is an isomorphism on the left (resp.\ right) hand side. We also know that $\theta(x/1) = \theta'(x/1) = \psi(x)|_{T_{\psi(f)}}$ by (2) for all $x \in A$. Hence we deduce easily that $\theta = \theta'$ as desired. Considering the degree $0$ parts we deduce that $r^\sharp = (r')^\sharp$, i.e., that $r = r'$ as morphisms of schemes. This proves the uniqueness. \medskip\noindent Now we come to existence. By the uniqueness just proved, it is enough to construct the pair $(r, \theta)$ locally on $T$. Hence we may assume that $T = \Spec(R)$ is affine, that $\mathcal{L} = \mathcal{O}_T$ and that for some $f \in A_{+}$ homogeneous we have $\psi(f)$ generates $\mathcal{O}_T = \mathcal{O}_T^{\otimes \deg(f)}$. In other words, $\psi(f) = u \in R^*$ is a unit. In this case the map $\psi$ is a graded ring map $$A \longrightarrow R[x] = \Gamma_*(T, \mathcal{O}_T)$$ which maps $f$ to $ux^{\deg(f)}$. Clearly this extends (uniquely) to a $\mathbf{Z}$-graded ring map $\theta : A_f \to R[x, x^{-1}]$ by mapping $1/f$ to $u^{-1}x^{-\deg(f)}$. This map in degree zero gives the ring map $A_{(f)} \to R$ which gives the morphism $r : T = \Spec(R) \to \Spec(A_{(f)}) = D_{+}(f) \subset X$. Hence we have constructed $(r, \theta)$ in this special case. \medskip\noindent Let us show the last statement of the lemma. According to Lemma \ref{lemma-converse-construction} the morphism constructed there is the unique one such that the displayed diagram in its statement commutes. The commutativity of the diagram in the lemma implies the commutativity when restricted to $V$ and $A^{(d)}$. Whence the result. \end{proof} \begin{remark} \label{remark-not-in-invertible-locus} Assumptions as in Lemma \ref{lemma-invertible-map-into-proj} above. The image of the morphism $r_{\mathcal{L}, \psi}$ need not be contained in the locus where the sheaf $\mathcal{O}_X(1)$ is invertible. Here is an example. Let $k$ be a field. Let $S = k[A, B, C]$ graded by $\deg(A) = 1$, $\deg(B) = 2$, $\deg(C) = 3$. Set $X = \text{Proj}(S)$. Let $T = \mathbf{P}^2_k = \text{Proj}(k[X_0, X_1, X_2])$. Recall that $\mathcal{L} = \mathcal{O}_T(1)$ is invertible and that $\mathcal{O}_T(n) = \mathcal{L}^{\otimes n}$. Consider the composition $\psi$ of the maps $$S \to k[X_0, X_1, X_2] \to \Gamma_*(T, \mathcal{L}).$$ Here the first map is $A \mapsto X_0^6$, $B \mapsto X_1^3$, $C \mapsto X_2^3$ and the second map is (\ref{equation-global-sections}). By the lemma this corresponds to a morphism $r_{\mathcal{L}, \psi} : T \to X = \text{Proj}(S)$ which is easily seen to be surjective. On the other hand, in Remark \ref{remark-not-isomorphism} we showed that the sheaf $\mathcal{O}_X(1)$ is not invertible at all points of $X$. \end{remark} \section{Relative Proj via glueing} \label{section-relative-proj-via-glueing} \begin{situation} \label{situation-relative-proj} Here $S$ is a scheme, and $\mathcal{A}$ is a quasi-coherent graded $\mathcal{O}_S$-algebra. \end{situation} \noindent In this section we outline how to construct a morphism of schemes $$\underline{\text{Proj}}_S(\mathcal{A}) \longrightarrow S$$ by glueing the homogeneous spectra $\text{Proj}(\Gamma(U, \mathcal{A}))$ where $U$ ranges over the affine opens of $S$. We first show that the homogeneous spectra of the values of $\mathcal{A}$ over affines form a suitable collection of schemes, as in Lemma \ref{lemma-relative-glueing}. \begin{lemma} \label{lemma-proj-inclusion} In Situation \ref{situation-relative-proj}. Suppose $U \subset U' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$ and $A' = \mathcal{A}(U')$. The map of graded rings $A' \to A$ induces a morphism $r : \text{Proj}(A) \to \text{Proj}(A')$, and the diagram $$\xymatrix{ \text{Proj}(A) \ar[r] \ar[d] & \text{Proj}(A') \ar[d] \\ U \ar[r] & U' }$$ is cartesian. Moreover there are canonical isomorphisms $\theta : r^*\mathcal{O}_{\text{Proj}(A')}(n) \to \mathcal{O}_{\text{Proj}(A)}(n)$ compatible with multiplication maps. \end{lemma} \begin{proof} 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). Hence the lemma follows from Lemma \ref{lemma-base-change-map-proj}. \end{proof} \noindent In particular the morphism $\text{Proj}(A) \to \text{Proj}(A')$ of the lemma is an open immersion. \begin{lemma} \label{lemma-transitive-proj} In Situation \ref{situation-relative-proj}. Suppose $U \subset U' \subset U'' \subset S$ are affine opens. Let $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$ and $A'' = \mathcal{A}(U'')$. The composition of the morphisms $r : \text{Proj}(A) \to \text{Proj}(A')$, and $r' : \text{Proj}(A') \to \text{Proj}(A'')$ of Lemma \ref{lemma-proj-inclusion} gives the morphism $r'' : \text{Proj}(A) \to \text{Proj}(A'')$ of Lemma \ref{lemma-proj-inclusion}. A similar statement holds for the isomorphisms $\theta$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-morphism-proj-transitive} since the map $A'' \to A$ is the composition of $A'' \to A'$ and $A' \to A$. \end{proof} \begin{lemma} \label{lemma-glue-relative-proj} In Situation \ref{situation-relative-proj}. There exists a morphism of schemes