Constructions of Schemes

Introduction

In this chapter we introduce ways of constructing schemes out of others. A basic reference is [EGA].

Relative glueing

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.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{B}\) be a basis for the topology of \(S\). Suppose given the following data:

1. For every \(U \in \mathcal{B}\) a scheme \(f_U : X_U \to U\) over \(U\).

2. For \(U, V \in \mathcal{B}\) with \(V \subset U\) a morphism \(\rho^U_V : X_V \to X_U\) over \(U\).

Assume that

1. each \(\rho^U_V\) induces an isomorphism \(X_V \to f_U^{-1}(V)\) of schemes over \(V\),

2. whenever \(W, V, U \in \mathcal{B}\), with \(W \subset V \subset U\) we have \(\rho^U_W = \rho^U_V \circ \rho ^V_W\).

Then there exists a morphism \(f : X \to S\) of schemes and isomorphisms \(i_U : f^{-1}(U) \to X_U\) over \(U \in \mathcal{B}\) such that for \(V, U \in \mathcal{B}\) with \(V \subset U\) 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 \(\rho^U_V\). Moreover \(X\) is unique up to unique isomorphism over \(S\).

Proof

To prove this we will use Schemes, Lemma 01JJ. 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 \(g(T) \subset W\).

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 \(h_U : g^{-1}(U) \to X_U\). It is easy to verify that the system \((g, \{h_U\})\) is an element of \(F(T)\). Hence \(F\) satisfies the sheaf property for the Zariski topology.

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.

Next, we verify each \(F_W \subset F\) is representable by open immersions. This is clear from the definitions.

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 \(S\).

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.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{B}\) be a basis for the topology of \(S\). Suppose given the following data:

1. For every \(U \in \mathcal{B}\) a scheme \(f_U : X_U \to U\) over \(U\).

2. For every \(U \in \mathcal{B}\) a quasi-coherent sheaf \(\mathcal{F}_U\) over \(X_U\).

3. For every pair \(U, V \in \mathcal{B}\) such that \(V \subset U\) a morphism \(\rho^U_V : X_V \to X_U\).

4. For every pair \(U, V \in \mathcal{B}\) such that \(V \subset U\) a morphism \(\theta^U_V : (\rho^U_V)^*\mathcal{F}_U \to \mathcal{F}_V\).

Assume that

1. each \(\rho^U_V\) induces an isomorphism \(X_V \to f_U^{-1}(V)\) of schemes over \(V\),

2. each \(\theta^U_V\) is an isomorphism,

3. whenever \(W, V, U \in \mathcal{B}\), with \(W \subset V \subset U\) we have \(\rho^U_W = \rho^U_V \circ \rho ^V_W\),

4. whenever \(W, V, U \in \mathcal{B}\), with \(W \subset V \subset U\) we have \(\theta^U_W = \theta^V_W \circ (\rho^V_W)^*\theta^U_V\).

Then there exists a morphism of schemes \(f : X \to S\) together with a 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 \(U \in \mathcal{B}\) such that for \(V, U \in \mathcal{B}\) with \(V \subset U\) 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 [01LJ]\[\begin{equation} (\rho^U_V)^*\mathcal{F}_U = (i_V^{-1})^*((i_U^*\mathcal{F}_U)|_{f^{-1}(V)}) \xrightarrow{\theta_U|_{f^{-1}(V)}} (i_V^{-1})^*(\mathcal{F}|_{f^{-1}(V)}) \xrightarrow{\theta_V^{-1}} \mathcal{F}_V \end{equation}\] is equal to \(\theta^U_V\). Moreover \((X, \mathcal{F})\) is unique up to unique isomorphism over \(S\).

Proof

By Lemma 01LH 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 00AK). By Sheaves, Lemma 00AL we get our \(\mathcal{F}\) on \(X\). We omit the verification of (01LJ).

Remark

There is a functoriality property for the constructions explained in Lemmas 01LH and 01LI. 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 01LH. 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 \((g_U : Y_U \to U, \sigma^U_V)\).

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 01LI. 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 \((g_U : Y_U \to U, \mathcal{G}_U, \sigma^U_V, \eta^U_V)\).

We omit the verifications and we omit a suitable formulation of “equivalence of categories” between relative glueing data and relative objects.

Relative spectrum via glueing

Situation

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.

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 01LH.

Lemma

In Situation 01LM. 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.

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 01I9 for example). The result follows from the description of the fibre product of affine schemes in Schemes, Lemma 01I4.

In particular the morphism \(\Spec(A) \to \Spec(A')\) of the lemma is an open immersion.

Lemma

In Situation 01LM. 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 01LN gives the morphism \(\Spec(A) \to \Spec(A'')\) of Lemma 01LN.

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).

Lemma

In Situation 01LM. There exists a morphism of schemes \[\pi : \underline{\Spec}_S(\mathcal{A}) \longrightarrow S\] with the following properties:

1. for every affine open \(U \subset S\) there exists an isomorphism \(i_U : \pi^{-1}(U) \to \Spec(\mathcal{A}(U))\) over \(U\), and

2. 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 01LN above.

Moreover, \(\underline{\Spec}_S(\mathcal{A})\) is unique up to unique isomorphism over \(S\).

Proof

Follows immediately from Lemmas 01LH, 01LN, and 01LO. Uniqueness is stated in the last sentence of Lemma 01LH.

Relative spectrum as a functor

We place ourselves in Situation 01LM, i.e., \(S\) is a scheme and \(\mathcal{A}\) is a quasi-coherent sheaf of \(\mathcal{O}_S\)-algebras.

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 008A. This is also the same as giving an \(\mathcal{O}_S\)-algebra map \(\varphi : \mathcal{A} \to f_*\mathcal{O}_T\), see Sheaves, Lemma 008Y. We will use all three ways of thinking about \(\varphi\), without further mention.

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 [01LR]\[\begin{eqnarray} F : \Sch^{opp} & \longrightarrow & \textit{Sets} \\ T & \longmapsto & F(T) = \{\text{pairs }(f, \varphi) \text{ as above}\} \nonumber \end{eqnarray}\]

Lemma

In Situation 01LM. 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.

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.

Lemma

In Situation 01LM. Let \(F\) be the functor associated to \((S, \mathcal{A})\) above. If \(S\) is affine, then \(F\) is representable by the affine scheme \(\Spec(\Gamma(S, \mathcal{A}))\).

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 01I9. Hence there is a canonical map \[\varphi_{univ} : f_{univ}^*\mathcal{A} = \widetilde{A \otimes_R A} \longrightarrow \widetilde A = \mathcal{O}_{\Spec(A)}\] coming from the \(A\)-module map \(A \otimes_R A \to A\), \(a \otimes a' \mapsto aa'\). We claim that the pair \((f_{univ}, \varphi_{univ})\) represents \(F\) in this case. In other words we claim that for any scheme \(T\) the map \[\Mor(T, \Spec(A)) \longrightarrow \{\text{pairs } (f, \varphi)\},\quad a \longmapsto (f_{univ} \circ a, a^*\varphi_{univ})\] is bijective.

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 01I1.

The verification that this map is inverse to the map displayed above is omitted.

Lemma

In Situation 01LM. The functor \(F\) is representable by a scheme.

Proof

We are going to use Schemes, Lemma 01JJ.

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 \(f|_{U_i} = f_i\), see Schemes, Section 01JA. 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 00AK 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.

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 \(f(T) \subset U_i\).

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 01LS. Thus the result follows from Lemma 01LT.

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 have \(a^*(f, \varphi) \in F_i(T')\) if and only if \(a(T') \subset V_i\). This is what we were required to show.

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.

Lemma

In Situation 01LM. The scheme \(\pi : \underline{\Spec}_S(\mathcal{A}) \to S\) constructed in Lemma 01LP and the scheme representing the functor \(F\) are canonically isomorphic as schemes over \(S\).

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 01LP part (1). For \(U \subset U' \subset S\) open these isomorphisms are compatible with the restriction mappings; this follows from Lemma 01LP part (2). Hence by Sheaves, Lemma 009U these isomorphisms result from an isomorphism of \(\mathcal{O}_S\)-algebras \(\varphi : \mathcal{A} \to \mathcal{R}\). Hence this gives an element \((\pi, \varphi) \in F(\underline{\Spec}_S(\mathcal{A}))\). Since \(X\) represents the functor \(F\) we get a corresponding morphism of schemes \(can : \underline{\Spec}_S(\mathcal{A}) \to X\) over \(S\).

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 01LT. 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 \(U\).

Definition

Let \(S\) be a scheme. Let \(\mathcal{A}\) be a quasi-coherent sheaf of \(\mathcal{O}_S\)-algebras. The relative spectrum of \(\mathcal{A}\) over \(S\), or simply the spectrum of \(\mathcal{A}\) over \(S\) is the scheme constructed in Lemma 01LP which represents the functor \(F\) (01LR), see Lemma 01LV. We denote it \(\pi : \underline{\Spec}_S(\mathcal{A}) \to S\). The “universal family” is a morphism of \(\mathcal{O}_S\)-algebras \[\mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})}\]

The following lemma says among other things that forming the relative spectrum commutes with base change.

Lemma

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 \(S\).

1. For every affine open \(U \subset S\) the inverse image \(\pi^{-1}(U)\) is affine.

2. For every morphism \(g : S' \to S\) we have \(S' \times_S \underline{\Spec}_S(\mathcal{A}) = \underline{\Spec}_{S'}(g^*\mathcal{A})\).

3. The universal map \[\mathcal{A} \longrightarrow \pi_*\mathcal{O}_{\underline{\Spec}_S(\mathcal{A})}\] is an isomorphism of \(\mathcal{O}_S\)-algebras.

Proof

Part (1) comes from the description of the relative spectrum by glueing, see Lemma 01LP. Part (2) follows immediately from Lemma 01LS. Part (3) follows because it is local on \(S\) and it is clear in case \(S\) is affine by Lemma 01LT for example.

Lemma

Let \(f : X \to S\) be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma 01LC 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 01I1.

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 01LT.

Affine n-space

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.

Definition

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 affine \(n\)-space over \(S\). If \(S = \Spec(R)\) is affine then we also call this affine \(n\)-space over \(R\) and we denote it \(\mathbf{A}^n_R\).

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 \(h_1, \ldots, h_n \in \Gamma(X, \mathcal{O}_X)\).

Vector bundles

Let \(S\) be a scheme. Let \(\mathcal{E}\) be a quasi-coherent sheaf of \(\mathcal{O}_S\)-modules. By Modules, Lemma 01CL 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 relative spectrum.

Definition

Let \(S\) be a scheme. Let \(\mathcal{E}\) be a quasi-coherent \(\mathcal{O}_S\)-module¹. The vector bundle associated to \(\mathcal{E}\) is \[\mathbf{V}(\mathcal{E}) = \underline{\Spec}_S(\text{Sym}(\mathcal{E})).\]

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.

Definition

Let \(S\) be a scheme. A 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 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.

An example of a vector bundle over \(S\) is affine \(n\)-space \(\mathbf{A}^n_S\) over \(S\), see Definition 01M0. This is true because \(\mathcal{O}_S[T_1, \ldots, T_n] = \text{Sym}(\mathcal{O}_S^{\oplus n})\).

Lemma

The category of vector bundles over a scheme \(S\) is anti-equivalent to the category of quasi-coherent \(\mathcal{O}_S\)-modules.

Proof

Omitted. Hint: In one direction one uses the functor \(\underline{\Spec}_S(\text{Sym}^*_{\mathcal{O}_S}(-))\) and in the other the functor \((\pi : V \to S) \leadsto (\pi_*\mathcal{O}_V)_1\) where the subscript indicates we take the degree \(1\) part.

Cones

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 00JL.

Definition

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 isomorphism². The cone associated to \(\mathcal{A}\) or the affine cone associated to \(\mathcal{A}\) is \[C(\mathcal{A}) = \underline{\Spec}_S(\mathcal{A}).\]

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.

Definition

Let \(S\) be a scheme. A 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 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.

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 \(S\).

Proj of a graded ring

In this section we construct Proj of a graded ring following [EGA, II, Section 2].

Let \(S\) be a graded ring. Consider the topological space \(\text{Proj}(S)\) associated to \(S\), see Algebra, Section 00JM. 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.

Recall that \(\text{Proj}(S)\) has a basis of open sets \(D_{+}(f)\), \(f \in S_d\), \(d \geq 1\) which we call standard opens, see Algebra, Section 00JM. 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.

Lemma

Let \(S\) be a graded ring. Let \(f \in S\) homogeneous of positive degree.

1. If \(g\in S\) homogeneous of positive degree and \(D_{+}(g) \subset D_{+}(f)\), then

   1. \(f\) is invertible in \(S_g\), and \(f^{\deg(g)}/g^{\deg(f)}\) is invertible in \(S_{(g)}\),

   2. \(g^e = af\) for some \(e \geq 1\) and \(a \in S\) homogeneous,

   3. there is a canonical \(S\)-algebra map \(S_f \to S_g\),

   4. there is a canonical \(S_0\)-algebra map \(S_{(f)} \to S_{(g)}\) compatible with the map \(S_f \to S_g\),

   5. the map \(S_{(f)} \to S_{(g)}\) induces an isomorphism \[(S_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong S_{(g)},\]

   6. 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,

   7. 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

   8. the map \(M_{(f)} \to M_{(g)}\) induces an isomorphism \[(M_{(f)})_{g^{\deg(f)}/f^{\deg(g)}} \cong M_{(g)}.\]

2. Any open covering of \(D_{+}(f)\) can be refined to a finite open covering of the form \(D_{+}(f) = \bigcup_{i = 1}^n D_{+}(g_i)\).

3. 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 \(S_{(f)}\).

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 00JP. 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 00E0. 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 00JP. The vertical arrows are open immersions since the left one is the inclusion of an open subset.

The open \(D_{+}(f)\) is quasi-compact because it is homeomorphic to \(\Spec(S_{(f)})\), see Algebra, Lemma 00E8. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology.

The third statement follows directly from Algebra, Lemma 00E0.

In Sheaves, Section 009H 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 009N and 009Q. Moreover, we showed in Sheaves, Lemma 009L 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.

Definition

Let \(S\) be a graded ring. Suppose that \(D_{+}(f) \subset \text{Proj}(S)\) is a standard open. A 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.

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 01M4 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 01M4 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.

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 00JM we defined \(M_{(\mathfrak p)}\) as the module 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 \[\widetilde M_x = M_{(\mathfrak p)}.\]

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 01M4 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 00EK 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.

Thus we conclude from the material in Sheaves, Section 009H 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 00JR the stalks of this sheaf of rings are all local rings.

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 009T.

Definition

Let \(S\) be a graded ring.

1. The 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.

2. The locally ringed space \((\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})\) is called the homogeneous spectrum of \(S\) and denoted \(\text{Proj}(S)\).

3. 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.

We summarize the results obtained so far.

Lemma

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 \(M\).

1. For every \(f \in S\) homogeneous of positive degree we have \[\Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}.\]

2. For every \(f\in S\) homogeneous of positive degree we have \(\Gamma(D_{+}(f), \widetilde M) = M_{(f)}\) as an \(S_{(f)}\)-module.

3. 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 01M4.

4. 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 \(\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}\).

5. 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 \((\widetilde M)_x = M_{(\mathfrak p)}\) as an \(S_{(\mathfrak p)}\)-module.

6. 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.

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.

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 00JR) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 01AG.

Lemma

Let \(S\) be a Noetherian graded ring generated by \(S_1\) over \(S_0\), set \(X=\operatorname{Proj}(S)\), let \(M\) be a finite graded \(S\)-module, and let \(N\) be a graded \(S\)-module. There is a canonical isomorphism \[\widetilde{\text{GrHom}(M,N)} \longrightarrow \SheafHom_{\mathcal O_X}(\widetilde M,\widetilde N).\] Moreover, if \(x\in X\) corresponds to the homogeneous prime \(\mathfrak p\), then for every \(q\geq0\) there is a canonical isomorphism \[\left(\widetilde{\text{GrExt}^q_S(M,N)}\right)_x \longrightarrow \Ext^q_{\mathcal O_{X,x}}(\widetilde M_x,\widetilde N_x).\] For \(q=0\) this is the map on stalks induced by the first isomorphism.

Proof

On a homogeneous standard open, a degree-zero fraction represented by \(\varphi/g\) in the localization of \(\text{GrHom}(M,N)\) acts by \[\frac{m}{h}\longmapsto\frac{\varphi(m)}{gh}.\] The fraction relations show that this is well defined, and these maps commute with restriction to smaller standard opens. They therefore define the first canonical morphism.

Because \(S\) is Noetherian and \(M\) is finite, there is a presentation of \(M\) by two finite graded free modules. Applying the exact associated-sheaf functor of Lemma 01M7 gives a finite presentation of \(\widetilde M\) on every degree-one standard open; twists are free there. Thus \(\widetilde M\) is finitely presented. At the point \(x\), Lemma 01M7 and Algebra, Lemma algebra-lemma-graded-ext-proj-prime for \(q=0\) identify the source stalk with \[\Hom_{S_{(\mathfrak p)}} \left(M_{(\mathfrak p)},N_{(\mathfrak p)}\right).\] The target stalk is canonically the same module by Modules, Lemma 01CP, and the two identifications commute with the fraction formula above. Hence the first morphism is an isomorphism on every stalk and therefore an isomorphism.

For arbitrary \(q\), Lemma 01M7 identifies the source stalk with \(\text{GrExt}^q_S(M,N)_{(\mathfrak p)}\), while Algebra, Lemma algebra-lemma-graded-ext-proj-prime identifies this with the displayed local Ext group.

Remark

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 an algebraically closed field \(k\) and \(S = k[x, y, z]\) 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 \(M_0 = k\). In particular, the presheaf which assigns to an arbitrary open \(U\) the degree-zero fractions with a single homogeneous denominator nonvanishing everywhere on \(U\) is not even separated in this example: its value on \(\text{Proj}(S)\) contains \(M_0 = k\), whereas its restrictions to the covering by \(D_{+}(x)\), \(D_{+}(y)\), and \(D_{+}(z)\) all vanish.

Lemma

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 an \(h \in S\) homogeneous of positive degree such that \(D(g)\) corresponds to \(D_{+}(h) \subset D_{+}(f)\) via the homeomorphism of Algebra, Lemma 00JP. In fact we can take \(h\) such that \(g = h/f^n\) for some \(n\).

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 \(D(g) \subset \Spec(S_{(f)})\).

Lemma

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.

Proof

Consider a standard open \(D_{+}(f) \subset \text{Proj}(S)\). By Lemmas 01M4 and 01M7 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 an \(h \in S_{+}\) as in Lemma 01MA. Then \(\varphi^{-1}(D(g)) = D_{+}(h)\), and by Lemmas 01M7 and 01M4 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 01HR. 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.

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.

Lemma

Let \(S\) be a graded ring. The scheme \(\text{Proj}(S)\) is separated.

Proof

We have to show that the canonical morphism \(\text{Proj}(S) \to \Spec(\mathbf{Z})\) is separated. We will use Schemes, Lemma 01KP. 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.

Lemma

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 \(\text{Proj}(S) = D_+(f_1) \cup \ldots \cup D_+(f_n)\).

Proof

Given such a collection of elements the standard affine opens \(D_{+}(f_i)\) cover \(\text{Proj}(S)\) by Algebra, Lemma 00JP. 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.

Lemma

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 00JN.

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 01I1. 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 01M4. 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 \(S_0\).

Lemma

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 01KD.

Proof

The uniqueness part follows from the fact that \(\text{Proj}(S)\) is separated (Lemma 01MC and Schemes, Lemma 01KZ). 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 01KD. Denote \(v : K^* \to \Gamma\) the valuation of \(A\), see Algebra, Definition 00IE. 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 \[\begin{align*} d \deg(f) v(\varphi'(x_1^{e_1} \ldots x_r^{e_r}/x_0^{e_0})) & = d v(\varphi'(x_1^{e_1 \deg(f)} \ldots x_r^{e_r \deg(f)}/x_0^{e_0 \deg(f)})) \\ & = d \sum e_i v(\varphi'(x_i^{\deg(f)}/f^{d_i})) - e_0 v(\varphi'(x_0^{\deg(f)}/f^{d_0})) \\ & = \sum e_i d_i \gamma_i - e_0 d_0 \gamma_0 \\ & \geq \sum e_i d_i \gamma_0 - e_0 d_0 \gamma_0 = 0 \end{align*}\] 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.

We saw in the proof of Lemma 01MF 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 01KF) we see that \(\text{Proj}(S) \to \Spec(S_0)\) is universally closed in the situation of the lemma. We give several examples showing these results do not hold without some assumption on the graded ring \(S\).

Example

Let \(\mathbf{C}[X_1, X_2, X_3, \ldots]\) be the graded \(\mathbf{C}\)-algebra with each \(X_i\) in degree \(1\). 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 \(\mathfrak m = (t^\alpha ; \alpha > 0)\). Let \(K\) be the fraction field of \(A\). The above gives a morphism \(\Spec(K) \to \text{Proj}(\mathbf{C}[X_1, X_2, X_3, \ldots])\) 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 \(t^{1/(i + 1) - 1/i}\).

Example

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 01KF ).

Example

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 01ME) is an isomorphism.

Example

Let \(X = \Spec(A)\) be an affine scheme, and let \(U \subset X\) be an open subscheme. Grade \(A[T]\) by setting \(\deg T = 1\). Define \(S\) to be the subring of \(A[T]\) generated by \(A\) and all \(fT^i\), where \(i \ge 0\) and where \(f \in A\) is such that \(D(f) \subset U\). We claim that \(S\) is a graded ring with \(S_0 = A\) such that \(\text{Proj}(S) \cong U\), and this isomorphism identifies the canonical morphism \(\text{Proj}(S) \to \Spec(A)\) of Lemma 01ME with the inclusion \(U \subset X\).

Suppose \(\mathfrak p \in \text{Proj}(S)\) is such that every \(fT \in S_1\) is in \(\mathfrak p\). Then every generator \(fT^i\) with \(i \ge 1\) is in \(\mathfrak p\) because \((fT^i)^2 = (fT)(fT^{2i-1}) \in \mathfrak p\) and \(\mathfrak p\) is radical. But then \(\mathfrak p \supset S_+\), which is impossible. Consequently \(\text{Proj}(S)\) is covered by the standard open affine subsets \(\{D_+(fT)\}_{fT \in S_1}\).

Observe that, if \(fT \in S_1\), then the inclusion \(S \subset A[T]\) induces a graded isomorphism of \(S[(fT)^{-1}]\) with \(A[T, T^{-1}, f^{-1}]\). Hence the standard open subset \(D_+(fT) \cong \Spec(S_{(fT)})\) is isomorphic to \(\Spec(A[T, T^{-1}, f^{-1}]_0) = \Spec(A[f^{-1}])\). It is clear that this isomorphism is a restriction of the canonical morphism \(\text{Proj}(S) \to \Spec(A)\). If in addition \(gT \in S_1\), then \(S[(fT)^{-1}, (gT)^{-1}] \cong A[T, T^{-1}, f^{-1}, g^{-1}]\) as graded rings, so \(D_+(fT) \cap D_+(gT) \cong \Spec(A[f^{-1}, g^{-1}])\). Therefore \(\text{Proj}(S)\) is the union of open subschemes \(D_+(fT)\) which are isomorphic to the open subschemes \(D(f) \subset X\) under the canonical morphism, and these open subschemes intersect in \(\text{Proj}(S)\) in the same way they do in \(X\). We conclude that the canonical morphism is an isomorphism of \(\text{Proj}(S)\) with the union of all \(D(f) \subset U\), which is \(U\).

Quasi-coherent sheaves on Proj

Let \(S\) be a graded ring. Let \(M\) be a graded \(S\)-module. We saw in Lemma 01M7 how to construct a quasi-coherent sheaf of modules \(\widetilde{M}\) on \(\text{Proj}(S)\) and a map [0AG1]\[\begin{equation} M_0 \longrightarrow \Gamma(\text{Proj}(S), \widetilde{M}) \end{equation}\] 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 00JL) is equal to \(M_n\). Hence we can get maps [0AG2]\[\begin{equation} M_n \longrightarrow \Gamma(\text{Proj}(S), \widetilde{M(n)}). \end{equation}\] 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 01MN. In order to relate the two notions we will use the following lemma.

Lemma

Let \(S\) be a graded ring. Let \((X, \mathcal{O}_X) = (\text{Proj}(S), \mathcal{O}_{\text{Proj}(S)})\) be the scheme of Lemma 01MB. 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 \[\xymatrix{ M_0 \otimes_{S_0} N_0 \ar[r] \ar[d] & (M \otimes_S N)_0 \ar[d] \\ \Gamma(X, \widetilde M \otimes_{\mathcal{O}_X} \widetilde N) \ar[r] & \Gamma(X, \widetilde{M \otimes_S N}) }\] is commutative where the vertical maps are given by (0AG1).

Proof

To construct a morphism as displayed is the same as constructing a \(\mathcal{O}_X\)-bilinear map \[\widetilde M \times \widetilde N \longrightarrow \widetilde{M \otimes_S N}\] see Modules, Section 01CA. 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.

Remark

In general the map constructed in Lemma 01MK 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 00JL 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.

Invertible sheaves on Proj

Recall from Algebra, Section 00JL the construction of the twisted module \(M(n)\) associated to a graded module over a graded ring.

Definition

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\).

1. We define \(\mathcal{O}_X(n) = \widetilde{S(n)}\). This is called the \(n\)th twist of the structure sheaf of \(\text{Proj}(S)\).

2. For any sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) we set \(\mathcal{F}(n) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X(n)\).

We are going to use Lemma 01MK to construct some canonical maps. Since \(S(n) \otimes_S S(m) = S(n + m)\) we see that there are canonical maps [01MO]\[\begin{equation} \mathcal{O}_X(n) \otimes_{\mathcal{O}_X} \mathcal{O}_X(m) \longrightarrow \mathcal{O}_X(n + m). \end{equation}\] These maps are not isomorphisms in general, see the example in Remark 01ML. The same example shows that \(\mathcal{O}_X(n)\) is not an invertible sheaf on \(X\) in general. Tensoring with an arbitrary \(\mathcal{O}_X\)-module \(\mathcal{F}\) we get maps [03GJ]\[\begin{equation} \mathcal{O}_X(n) \otimes_{\mathcal{O}_X} \mathcal{F}(m) \longrightarrow \mathcal{F}(n + m). \end{equation}\] The maps (0AG2) on global sections give a map of graded rings [01MP]\[\begin{equation} S \longrightarrow \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n)). \end{equation}\] And for an arbitrary \(\mathcal{O}_X\)-module \(\mathcal{F}\) the maps (03GJ) give a graded module structure [03GK]\[\begin{equation} \bigoplus\nolimits_{n \geq 0} \Gamma(X, \mathcal{O}_X(n)) \times \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(X, \mathcal{F}(m)) \longrightarrow \bigoplus\nolimits_{m \in \mathbf{Z}} \Gamma(X, \mathcal{F}(m)) \end{equation}\] and via (01MP) 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 [01MQ]\[\begin{equation} \widetilde M(n) = \widetilde M \otimes_{\mathcal{O}_X} \mathcal{O}_X(n) \longrightarrow \widetilde{M(n)}. \end{equation}\] On global sections (0AG2) defines a map of graded \(S\)-modules [01MR]\[\begin{equation} M \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M(n)}). \end{equation}\] Here is an important fact which follows basically immediately from the definitions.

Lemma

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 01CS). The maps (01MO) 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 (03GJ) 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 (01MQ) 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 \(n, m \in \mathbf{Z}\).

Proof

After inverting \(f\), multiplication by \(f^n\) defines isomorphisms of graded modules \(S_f \to S(nd)_f\) and \(M_f \to M(nd)_f\) for every \(n \in \mathbf{Z}\). 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 (03GJ) is a consequence of the case of the map (01MO).

Lemma

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 (01MO), (03GJ), and (01MQ) 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 (0AG2) becomes a map [0AG3]\[\begin{equation} M_n \longrightarrow \Gamma(X, \widetilde{M}(n)) \end{equation}\] and (01MR) becomes a map [0AG4]\[\begin{equation} M \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \Gamma(X, \widetilde{M}(n)). \end{equation}\]

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 01MS above.

Lemma

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:

1. 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 01MO) are isomorphisms.

2. The union of the open subsets \(D_{+}(fg)\) with \(f, g \in S\) homogeneous and \(\deg(f) = \deg(g) + d\).

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 01MQ) are isomorphisms.

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 01MS.

Conversely, suppose that \(\mathcal{O}_X(dn)\) is free of rank 1 for all \(n\) 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 \(x \in D_{+}(h) \subset V\). 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).

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 01MQ) are isomorphisms. Compare with the proof of Lemma 01MS.

Recall from Modules, Lemma 01CY 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.

Lemma

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 01MU. 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 (01MP) restricted to \(W\). Then \[W_s = D_{+}(f) \cap W.\]

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 01MU 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.

The following lemma states the properties that we will later use to characterize schemes with an ample invertible sheaf.

Lemma

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

1. the subscheme \(Y\) is contained in the open \(W_d\) defined in Lemma 01MU,

2. the sheaf \(\mathcal{O}_Y(dn)\) is invertible for all \(n \in \mathbf{Z}\),

3. all the maps \(\mathcal{O}_Y(nd) \otimes_{\mathcal{O}_Y} \mathcal{O}_Y(m) \longrightarrow \mathcal{O}_Y(nd + m)\) of Equation (01MO) are isomorphisms,

4. 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 01MQ) are isomorphisms,

5. given \(f \in S_{nd}\) denote \(s \in \Gamma(Y, \mathcal{O}_Y(nd))\) the image of \(f\) via (01MP) restricted to \(Y\), then \(D_{+}(f) \cap Y = Y_s\),

6. 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

7. 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.

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 01MU or by (1) using Lemma 01MS. Note that (1) implies (2), (3) and (4) by Lemma 01MU. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma 01MV. Assertions (6) and (7) follow because the open subsets \(D_{+}(f)\) form a basis for the topology of \(X\) and are affine.

Lemma

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 (03GK) and (01MP). 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.

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 01M7. 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 01MS 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 01I7. 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.

Lemma

Let \(S\) be a graded ring and set \(X=\operatorname{Proj}(S)\). Assume that \(X\) is covered by standard opens \(D_{+}(f)\) with \(f \in S_1\). For a graded \(S\)-module \(M\), let \[a_M:M\longrightarrow N_M=\bigoplus_{n\in\mathbf{Z}}\Gamma(X,\widetilde M(n))\] be the map of Equation (0AG4). For a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\), set \[N_{\mathcal{F}}= \bigoplus_{n\in\mathbf{Z}}\Gamma(X,\mathcal{F}(n))\] and let \(b_{\mathcal{F}}:\widetilde{N_{\mathcal{F}}}\to\mathcal{F}\) be the map of Lemma 0B5I. Then the composites \[\widetilde M\longrightarrow\widetilde{N_M} \xrightarrow{b_{\widetilde M}}\widetilde M \quad\text{and}\quad N_{\mathcal{F}}\longrightarrow N_{\widetilde{N_{\mathcal{F}}}}\longrightarrow N_{\mathcal{F}}\] are the identity maps. In the second composite the first arrow is \(a_{N_{\mathcal{F}}}\) and the second is induced by \(b_{\mathcal{F}}\) in every degree.

Proof

It is enough to check the first composite over the standard opens in the given covering. Let \(f\in S_1\) and represent a section of \(\widetilde M\) over \(D_{+}(f)\) by a degree-zero fraction \(m/f^r\), with \(r\geq 0\) and \(m\in M_r\). The first arrow sends it to \(a_M(m)/f^r\). By the definition of the comparison map, the second arrow multiplies the section represented by \(a_M(m)\) by \(f^{-r}\), and hence returns \(m/f^r\).

For the second composite take a homogeneous element \(s\in (N_{\mathcal{F}})_n=\Gamma(X,\mathcal{F}(n))\). On each \(D_{+}(f)\) in the covering, the first arrow represents \(s\) by \(s/1\); the degree-\(n\) twist of \(b_{\mathcal{F}}\) sends this section back to \(s|_{D_{+}(f)}\). These opens cover \(X\), so the composite returns \(s\). This proves the identity in every degree.

Lemma

Let \(S\) be a graded ring and set \(X=\operatorname{Proj}(S)\). Let \(f_1,\ldots,f_r\in S_1\) be homogeneous elements such that \(X=\bigcup_{i=1}^rD_+(f_i)\). Let \(M\) be a graded \(S\)-module. Set \[N_M=\bigoplus_{n\in\mathbf{Z}}\Gamma(X,\widetilde M(n))\] and let \[a_M:M\longrightarrow N_M\] be the map of Equation (0AG4). Assume that \((a_M)_n\) is an isomorphism for all sufficiently large \(n\). Then \(a_M\) is an isomorphism if and only if the following conditions hold:

1. If \(m\in M\) and \(f_i m=0\) for every \(i\), then \(m=0\).

2. For every \(n\in\mathbf{Z}\) and elements \(m_i\in M_{n+1}\) such that \(f_jm_i=f_im_j\) for all \(i,j\), there exists an \(m\in M_n\) such that \(m_i=f_im\) for every \(i\).

Moreover, \(a_M\) is injective if and only if condition (constructions-item-comparison-proj-injective) holds.

Proof

We first verify both conditions for \(N_M\). It is enough to work degreewise. If \(s\in (N_M)_n\) and \(f_is=0\) for all \(i\), then \(s\) restricts to zero on \(D_+(f_i)\) because \(f_i\) trivializes \(\mathcal{O}_X(1)\) there. These opens cover \(X\), hence \(s=0\).

Suppose that \(s_i\in (N_M)_{n+1}\) and \(f_js_i=f_is_j\) for all \(i,j\). On \(D_+(f_i)\) the quotient \(s_i/f_i\) is a section of \(\widetilde M(n)\). The displayed relations show that these sections agree on overlaps, so they glue to an \(s\in (N_M)_n\). To see that \(f_is=s_i\), restrict to \(D_+(f_j)\) and observe that \[f_j(f_is-s_i)=f_i(f_js)-f_js_i=f_is_j-f_js_i=0.\] Since \(f_j\) is invertible on \(D_+(f_j)\), the desired equality holds there, and hence globally. Thus \(N_M\) satisfies both conditions. Consequently, if \(a_M\) is an isomorphism, then \(M\) satisfies both of them. If \(a_M\) is merely injective, the first condition for \(N_M\) still shows that the first condition holds for \(M\).

Conversely, assume condition (constructions-item-comparison-proj-injective). We prove that \((a_M)_n\) is injective by descending induction on \(n\), starting in the range where it is an isomorphism. If \(a_M(m)=0\) for \(m\in M_n\), then \(a_M(f_im)=f_ia_M(m)=0\). The induction hypothesis gives \(f_im=0\) for every \(i\), and condition (constructions-item-comparison-proj-injective) gives \(m=0\). This proves both the injectivity assertion and one implication of the final statement.

Now assume condition (constructions-item-comparison-proj-glue) as well. We prove surjectivity by the same descending induction. Given \(s\in (N_M)_n\), choose \(m_i\in M_{n+1}\) with \(a_M(m_i)=f_is\), using the induction hypothesis. The injectivity just proved and the equalities \[a_M(f_jm_i-f_im_j)=f_jf_is-f_if_js=0\] show that \(f_jm_i=f_im_j\) for all \(i,j\). The second condition gives an \(m\in M_n\) with \(m_i=f_im\). Hence \(f_i(s-a_M(m))=0\) for every \(i\). The first condition for \(N_M\) gives \(s=a_M(m)\), as desired.

Functoriality of Proj

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.

Lemma

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:

1. 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.

2. 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 \(\psi\).

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.

Lemma

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 \[U(\psi \circ \varphi) = r_\psi^{-1}(U(\varphi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}.\] In addition we have \[\theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi}\] with obvious notation.

Proof

Omitted.

Lemma

With hypotheses and notation as in Lemma 01MY above. Assume \(A_d \to B_d\) is surjective for all \(d \gg 0\). Then

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is a closed immersion, and

3. 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).

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 \(r_\psi^*\mathcal{O}_X(1)\) is not zero. (There are less silly examples.)

Lemma

With hypotheses and notation as in Lemma 01MY above. Assume \(A_d \to B_d\) is an isomorphism for all \(d \gg 0\). Then

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is an isomorphism, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are isomorphisms.

Proof

We have (1) by Lemma 01N0. 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.

Lemma

With hypotheses and notation as in Lemma 01MY 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

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is a closed immersion, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are isomorphisms.

Proof

By Lemmas 07ZE and 01MZ 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 01N0 we get (1) and (2) and surjectivity in (3). By Lemma 01MT we see that both \(\mathcal{O}_X(n)\) and \(\mathcal{O}_Y(n)\) are invertible. Hence \(\theta\) is an isomorphism.

Lemma

With hypotheses and notation as in Lemma 01MY 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

1. \(U(\psi) = Y\),

2. 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

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are isomorphisms.

Proof

This follows immediately by looking at what happens over the standard opens \(D_{+}(f)\) for \(f \in A_{+}\).

Lemma

With hypotheses and notation as in Lemma 01MY 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.

Proof

This is a special case of Lemma 01N2 above.

Lemma

Let \(S\) be a graded ring. Let \(d \geq 1\). Set \(S' = S^{(d)}\) with notation as in Algebra, Section 00JL. Set \(X = \text{Proj}(S)\) and \(X' = \text{Proj}(S')\). There is a canonical isomorphism \(i : X \to X'\) of schemes such that

1. for any graded \(S\)-module \(M\) setting \(M' = M^{(d)}\), we have a canonical isomorphism \(\widetilde{M} \to i^*\widetilde{M'}\),

2. we have canonical isomorphisms \(\mathcal{O}_{X}(nd) \to i^*\mathcal{O}_{X'}(n)\)

and these isomorphisms are compatible with the multiplication maps of Lemma 01MK and hence with the maps (01MO), (03GJ), (01MP), (03GK), (01MQ), and (01MR) (see proof for precise statements).

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 we define a homogeneous ideal \(\mathfrak p\subset S\) by declaring, for homogeneous \(g\in S\), that \(g\in\mathfrak p\) if and only if \(g^d\in\mathfrak p'\). If homogeneous \(g,h\in\mathfrak p\) have the same degree, then \((g+h)^{2d}\in\mathfrak p'\) by the binomial formula. Hence \((g+h)^d\in\mathfrak p'\) as \(\mathfrak p'\) is prime, and therefore \(g+h\in\mathfrak p\). Closure under multiplication by homogeneous elements is immediate. If homogeneous \(gh\in\mathfrak p\), then \(g^dh^d\in\mathfrak p'\), so \(g\in\mathfrak p\) or \(h\in\mathfrak p\). Thus \(\mathfrak p\) is a graded prime ideal of \(S\), it does not contain \(f\), and its image under \(j\) is \(\mathfrak p'\). 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 01M4, 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.

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. There is a natural graded map \[\beta:M'\otimes_{S'}N'\longrightarrow(M\otimes_SN)'\] induced by the tensor product. This map need not be an isomorphism of graded modules. The compatibility with the multiplication maps of Lemma 01MK is the commutativity of the diagram \[\xymatrix{ \widetilde M \otimes_{\mathcal{O}_X} \widetilde N \ar[d]_{(1) \otimes (1)} \ar[r] & \widetilde{M \otimes_S N} \ar[d]^{(1)} \\ i^*(\widetilde{M'} \otimes_{\mathcal{O}_{X'}} \widetilde{N'}) \ar[r] & i^*(\widetilde{(M \otimes_S N)'}) }\] where the lower arrow is the multiplication map followed by \(i^*\widetilde\beta\). 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.

Morphisms into Proj

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 [01N5]\[\begin{equation} U_d = \bigcup\nolimits_{f \in S_d} D_{+}(f) \quad\subset\quad X = \text{Proj}(S) \end{equation}\] 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 00JP. Let us write \(\mathcal{O}_{U_d}(n) = \mathcal{O}_X(n)|_{U_d}\). By Lemma 01MS 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 \(\mathcal{O}_{U_d}(nd) \otimes_{\mathcal{O}_{U_d}} \mathcal{O}_{U_d}(m) \to \mathcal{O}_{U_d}(nd + m)\) of (01MO) are isomorphisms. In particular we have \(\mathcal{O}_{U_d}(nd) \cong \mathcal{O}_{U_d}(d)^{\otimes n}\). The graded ring map (01MP) on global sections combined with restriction to \(U_d\) give a homomorphism of graded rings [01N6]\[\begin{equation} \psi^d : S^{(d)} \longrightarrow \Gamma_*(U_d, \mathcal{O}_{U_d}(d)). \end{equation}\] For the notation \(S^{(d)}\), see Algebra, Section 00JL. For the notation \(\Gamma_*\) see Modules, Definition 01CV. 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 01AM.

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 01CV 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 [01N7]\[\begin{equation} \psi_\varphi^d : S^{(d)} \longrightarrow \Gamma_*(Y, \varphi^*\mathcal{O}_X(d)) \end{equation}\] which has the property that the invertible sheaf \(\varphi^*\mathcal{O}_X(d)\) is globally generated by the sections in the image of \((S^{(d)})_1 = S_d \to \Gamma(Y, \varphi^*\mathcal{O}_X(d))\).

Lemma

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 exist 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.

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 01CY. 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 \(\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O}_Y)\) by mapping the fraction \(a/f^n\) to \(\psi(a)/s^n\). By Schemes, Lemma 01I1 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 \(Y\) replaced by \(Y_s\), \(\varphi\) replaced by \(\varphi_f\), and \(\alpha\) replaced by \(\alpha_f\); verification omitted.

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 01MW 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 \(S_{(ff')} \to \Gamma(Y_{ss'}, \mathcal{O}_Y)\) making the following diagram commute \[\xymatrix{ \Gamma(Y_s, \mathcal{O}_Y) \ar[r] & \Gamma(Y_{ss'}, \mathcal{O}_Y) & \Gamma(Y_{s, '} \mathcal{O}_Y) \ar[l]\\ S_{(f)} \ar[r] \ar[u]^{\psi_{(f)}} & S_{(ff')} \ar[u] & S_{(f')} \ar[l] \ar[u]^{\psi_{(f')}} }\] 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 01MC 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 \(\psi_{(f')}(f/f')\) 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.

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 01IS) 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.

We continue the discussion from above the lemma. Let \(S\) be a graded ring. Let \(Y\) be a scheme. We will consider triples \((d, \mathcal{L}, \psi)\) where

1. \(d \geq 1\) is an integer,

2. \(\mathcal{L}\) is an invertible \(\mathcal{O}_Y\)-module, and

3. \(\psi : S^{(d)} \to \Gamma_*(Y, \mathcal{L})\) is a graded ring homomorphism such that \(\mathcal{L}\) is generated by the global sections \(\psi(f)\), with \(f \in S_d\).

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 strictly equivalent if there exists an isomorphism \(\beta : \mathcal{L} \to \mathcal{L}'\) such that \(\beta \circ \psi = \psi'\) as graded ring maps \(S^{(d)} \to \Gamma_*(Y, \mathcal{L}')\).

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.

Lemma

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\). The open subscheme \(U_d \subset X\) (01N5) 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 01JF).

Proof

This is a reformulation of Lemma 01N8

Lemma

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

1. \(\mathcal{L}\) is an invertible \(\mathcal{O}_Y\)-module, and

2. \(\psi : S \to \Gamma_*(Y, \mathcal{L})\) is a graded ring homomorphism such that \(\mathcal{L}\) is generated by the global sections \(\psi(f)\), with \(f \in S_1\)

up to strict equivalence as above.

Proof

Under the assumptions of the lemma we have \(X = U_1\) and the lemma is a reformulation of Lemma 01N9 above.

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.

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 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 \(S^{(dd')} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes dd'})\).

Lemma

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:

1. 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 \(S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})\).

2. The triples \((d, \mathcal{L}, \psi)\) and \((d', \mathcal{L}', \psi')\) are equivalent.

3. 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 \(S^{(n)} \to \Gamma_*(Y, (\mathcal{L}')^{\otimes n})\).

4. The morphisms \(\varphi : T \to X\) and \(\varphi' : T \to X\) associated to \((d, \mathcal{L}, \psi)\) and \((d', \mathcal{L}', \psi')\) are equal.

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 01N8. 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 01N8 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.

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\). Over the open subscheme scheme \(U_d \subset X = \text{Proj}(S)\) (01N5) 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 01N8 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.

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 01NB 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 \[\begin{eqnarray*} F : \Sch^{opp} & \longrightarrow & \textit{Sets} \end{eqnarray*}\]

Lemma

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\). The functor \(F\) defined above is representable by the scheme \(X\).

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.

Projective space

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.

Lemma

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

1. \(\mathcal{L}\) is an invertible \(\mathcal{O}_Y\)-module, and

2. \(s_0, \ldots, s_n\) are global sections of \(\mathcal{L}\) which generate \(\mathcal{L}\)

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 \(i = 0, \ldots, n\).

Proof

This is a special case of Lemma 01NA 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 \(\Gamma(Y, \mathcal{L})\).

Definition

The scheme \(\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n])\) is called projective \(n\)-space over \(\mathbf{Z}\). Its base change \(\mathbf{P}^n_S\) to a scheme \(S\) is called projective \(n\)-space over \(S\). If \(R\) is a ring the base change to \(\Spec(R)\) is denoted \(\mathbf{P}^n_R\) and called projective \(n\)-space over \(R\).

Given a scheme \(Y\) over \(S\) and a pair \((\mathcal{L}, (s_0, \ldots, s_n))\) as in Lemma 01NE 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 \(\mathbf{P}^n_S = \mathbf{P}^n_{\mathbf{Z}} \times S\). Via Lemma 01O3 we can identify \(\mathbf{P}^n_S\) with a relative Proj of the graded \(\mathcal{O}_S\)-algebra \(\mathcal{O}_S[T_0, \ldots, T_n]\) and we can characterize \(\varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}\) as the unique \(S\)-morphism such that \[\mathcal{L} = \varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}^*\mathcal{O}_{\mathbf{P}^n_S}(1) \quad \text{and} \quad s_i = \varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}^*T_i\] where we think of \(T_i\) as a global section of \(\mathcal{O}_{\mathbf{P}^n_S}(1)\) via the map \(\psi\) of Lemma 01NR; details omitted.

Lemma

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 \(\mathbf{Z}\).

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.

Lemma

Let \(S\) be a scheme. The structure morphism \(\mathbf{P}^n_S \to S\) is

1. separated,

2. quasi-compact,

3. satisfies the existence and uniqueness parts of the valuative criterion, and

4. universally closed.

Proof

All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas 01KU and 01K5). Hence it suffices to prove them for the morphism \(\mathbf{P}^n_{\mathbf{Z}} \to \Spec(\mathbf{Z})\). Separatedness is Lemma 01MC. Quasi-compactness follows from Lemma 01NG. Existence and uniqueness of the valuative criterion follow from Lemma 01MF. Universally closed follows from the above and Schemes, Proposition 01KF.

Remark

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.)

Lemma

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 Segre embedding.

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 01NE. 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 01HL and Example 01IG.)

The following two lemmas are special cases of more general results later, but perhaps it makes sense to prove these directly here now. The first one is a special case of Divisors, Lemma 0801.

Lemma

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 \(Z = \text{Proj}(R[T_0, \ldots, T_n]/I)\).

Proof

It is clear that \(I\) is a graded ideal. Set \(Z' = \text{Proj}(R[T_0, \ldots, T_n]/I)\). By Lemma 01N1 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 \(J' \subset J\).

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.

The following lemma is a special case of the more general Properties, Lemmas 01QJ or 0AG5.

Lemma

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 \(\mathcal{F} = \widetilde{M}\).

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 (03GK). 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 \(T_i/T_j\).

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.

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.

Lemma

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.

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 \(\mathcal{F} \otimes \mathcal{L}\).

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.

Invertible sheaves and morphisms into Proj

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 01CY. We can view the following lemma as a slight generalization of Lemma 01NA. It also is a generalization of Lemma 01MY.

Lemma

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:

1. For \(f \in A_{+}\) homogeneous we have \(r_{\mathcal{L}, \psi}^{-1}(D_{+}(f)) = T_{\psi(f)}\).

2. For every \(d \geq 0\) the diagram \[\xymatrix{ A_d \ar[d]_{(\href{constructions.html#constructions-equation-global-sections}{01MP})} \ar[r]_{\psi} & \Gamma(T, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(X, \mathcal{O}_X(d)) \ar[r]^{\theta} & \Gamma(U(\psi), \mathcal{L}^{\otimes d}) }\] is commutative.

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 01N8 associated to \(\psi|_{A^{(d)}} : A^{(d)} \to \Gamma_*(V, \mathcal{L}^{\otimes d})\).

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 00JM and Schemes, Lemma 01IS). 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.

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.

Let us show the last statement of the lemma. According to Lemma 01N8 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.

Remark

Assumptions as in Lemma 01NK 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 \(A \mapsto X_0\), \(B \mapsto X_1^2\), \(C \mapsto X_2^3\) and the second map is (01MP). 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 01ML we showed that the sheaf \(\mathcal{O}_X(1)\) is not invertible at all points of \(X\).

Relative Proj via glueing

Situation

Here \(S\) is a scheme, and \(\mathcal{A}\) is a quasi-coherent graded \(\mathcal{O}_S\)-algebra.

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 01LH.

Lemma

In Situation 01NN. 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.

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 01I9 for example). Hence the lemma follows from Lemma 01N2.

In particular the morphism \(\text{Proj}(A) \to \text{Proj}(A')\) of the lemma is an open immersion.

Lemma

In Situation 01NN. 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 01NO gives the morphism \(r'' : \text{Proj}(A) \to \text{Proj}(A'')\) of Lemma 01NO. A similar statement holds for the isomorphisms \(\theta\).

Proof

This follows from Lemma 01MZ since the map \(A'' \to A\) is the composition of \(A'' \to A'\) and \(A' \to A\).

Lemma

In Situation 01NN. There exists a morphism of schemes \[\pi : \underline{\text{Proj}}_S(\mathcal{A}) \longrightarrow S\] with the following properties:

1. for every affine open \(U \subset S\) there exists an isomorphism \(i_U : \pi^{-1}(U) \to \text{Proj}(A)\) with \(A = \mathcal{A}(U)\), and

2. for \(U \subset U' \subset S\) affine open the composition \[\xymatrix{ \text{Proj}(A) \ar[r]^{i_U^{-1}} & \pi^{-1}(U) \ar[rr]^{inclusion} & & \pi^{-1}(U') \ar[r]^{i_{U'}} & \text{Proj}(A') }\] with \(A = \mathcal{A}(U)\), \(A' = \mathcal{A}(U')\) is the open immersion of Lemma 01NO above.

Proof

Follows immediately from Lemmas 01LH, 01NO, and 01NP.

Lemma

In Situation 01NN. The morphism \(\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S\) of Lemma 01NQ comes with the following additional structure. There exists a quasi-coherent \(\mathbf{Z}\)-graded sheaf of \(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}\)-algebras \(\bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\), and a morphism of graded \(\mathcal{O}_S\)-algebras \[\psi : \mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \pi_*\left(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\right)\] uniquely determined by the following property: For every affine open \(U \subset S\) with \(A = \mathcal{A}(U)\) there is an isomorphism \[\theta_U : i_U^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n) \right) \longrightarrow \left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n) \right)|_{\pi^{-1}(U)}\] of \(\mathbf{Z}\)-graded \(\mathcal{O}_{\pi^{-1}(U)}\)-algebras such that \[\xymatrix{ A_n \ar[rr]_\psi \ar[dr]_-{(\href{constructions.html#constructions-equation-global-sections}{01MP})} & & \Gamma(\pi^{-1}(U), \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)) \\ & \Gamma(\text{Proj}(A), \mathcal{O}_{\text{Proj}(A)}(n)) \ar[ru]_-{\theta_U} & }\] is commutative.

Proof

We are going to use Lemma 01LI to glue the sheaves of \(\mathbf{Z}\)-graded algebras \(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\text{Proj}(A)}(n)\) for \(A = \mathcal{A}(U)\), \(U \subset S\) affine open over the scheme \(\underline{\text{Proj}}_S(\mathcal{A})\). We have constructed the data necessary for this in Lemma 01NO and we have checked condition (d) of Lemma 01LI in Lemma 01NP. Hence we get the sheaf of \(\mathbf{Z}\)-graded \(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}\)-algebras \(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\) together with the isomorphisms \(\theta_U\) for all \(U \subset S\) affine open and all \(n \in \mathbf{Z}\). For every affine open \(U \subset S\) with \(A = \mathcal{A}(U)\) we have a map \(A \to \Gamma(\text{Proj}(A), \bigoplus_{n \geq 0} \mathcal{O}_{\text{Proj}(A)}(n))\). Hence the map \(\psi\) exists by functoriality of relative glueing, see Remark 01LK. The diagram of the lemma commutes by construction. This characterizes the sheaf of \(\mathbf{Z}\)-graded \(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}\)-algebras \(\bigoplus \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\) because the proof of Lemma 01MY shows that having these diagrams commute uniquely determines the maps \(\theta_U\). Some details omitted.

Relative Proj as a functor

We place ourselves in Situation 01NN. So \(S\) is a scheme and \(\mathcal{A} = \bigoplus_{d \geq 0} \mathcal{A}_d\) is a quasi-coherent graded \(\mathcal{O}_S\)-algebra. In this section we relativize the construction of \(\text{Proj}\) by constructing a functor which the relative homogeneous spectrum will represent. As a result we will construct a morphism of schemes \[\underline{\text{Proj}}_S(\mathcal{A}) \longrightarrow S\] which above affine opens of \(S\) will look like the homogeneous spectrum of a graded ring. The discussion will be modeled after our discussion of the relative spectrum in Section 01LQ. The easier method using glueing schemes of the form \(\text{Proj}(A)\), \(A = \Gamma(U, \mathcal{A})\), \(U \subset S\) affine open, is explained in Section 01NM, and the result in this section will be shown to be isomorphic to that one.

Fix for the moment an integer \(d \geq 1\). We denote \(\mathcal{A}^{(d)} = \bigoplus_{n \geq 0} \mathcal{A}_{nd}\) similarly to the notation in Algebra, Section 00JL. Let \(T\) be a scheme. Let us consider quadruples \((d, f : T \to S, \mathcal{L}, \psi)\) over \(T\) where

1. \(d\) is the integer we fixed above,

2. \(f : T \to S\) is a morphism of schemes,

3. \(\mathcal{L}\) is an invertible \(\mathcal{O}_T\)-module, and

4. \(\psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0}\mathcal{L}^{\otimes n}\) is a homomorphism of graded \(\mathcal{O}_T\)-algebras such that \(f^*\mathcal{A}_d \to \mathcal{L}\) is surjective.

Given a morphism \(h : T' \to T\) and a quadruple \((d, f, \mathcal{L}, \psi)\) over \(T\) we can pull it back to the quadruple \((d, f \circ h, h^*\mathcal{L}, h^*\psi)\) over \(T'\). Given two quadruples \((d, f, \mathcal{L}, \psi)\) and \((d, f', \mathcal{L}', \psi')\) over \(T\) with the same integer \(d\) we say they are strictly equivalent if \(f = f'\) and there exists an isomorphism \(\beta : \mathcal{L} \to \mathcal{L}'\) such that \(\beta \circ \psi = \psi'\) as graded \(\mathcal{O}_T\)-algebra maps \(f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes n}\).

For each integer \(d \geq 1\) we define \[\begin{eqnarray*} F_d : \Sch^{opp} & \longrightarrow & \textit{Sets}, \\ T & \longmapsto & \{\text{strict equivalence classes of } (d, f : T \to S, \mathcal{L}, \psi) \text{ as above}\} \end{eqnarray*}\] with pullbacks as defined above.

Lemma

In Situation 01NN. Let \(d \geq 1\). Let \(F_d\) 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 \(F_d'\) be the functor associated to \((S', \mathcal{A}')\) above. Then there is a canonical isomorphism \[F'_d \cong h_{S'} \times_{h_S} F_d\] of functors.

Proof

A quadruple \((d, f' : T \to S', \mathcal{L}', \psi' : (f')^*(\mathcal{A}')^{(d)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes n})\) is the same as a quadruple \((d, f, \mathcal{L}, \psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n})\) together with a factorization of \(f\) as \(f = g \circ f'\). Namely, the correspondence is \(f = g \circ f'\), \(\mathcal{L} = \mathcal{L}'\) and \(\psi = \psi'\) via the identifications \((f')^*(\mathcal{A}')^{(d)} = (f')^*g^*(\mathcal{A}^{(d)}) = f^*\mathcal{A}^{(d)}\). Hence the lemma.

Lemma

In Situation 01NN. Let \(F_d\) be the functor associated to \((d, S, \mathcal{A})\) above. If \(S\) is affine, then \(F_d\) is representable by the open subscheme \(U_d\) (01N5) of the scheme \(\text{Proj}(\Gamma(S, \mathcal{A}))\).

Proof

Write \(S = \Spec(R)\) and \(A = \Gamma(S, \mathcal{A})\). Then \(A\) is a graded \(R\)-algebra and \(\mathcal{A} = \widetilde A\). To prove the lemma we have to identify the functor \(F_d\) with the functor \(F_d^{triples}\) of triples defined in Section 01N4.

Let \((d, f : T \to S, \mathcal{L}, \psi)\) be a quadruple. We may think of \(\psi\) as a \(\mathcal{O}_S\)-module map \(\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}\). Since \(\mathcal{A}^{(d)}\) is quasi-coherent this is the same thing as an \(R\)-linear homomorphism of graded rings \(A^{(d)} \to \Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n})\). Clearly, \(\Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}) = \Gamma_*(T, \mathcal{L})\). Thus we may associate to the quadruple the triple \((d, \mathcal{L}, \psi)\).

Conversely, let \((d, \mathcal{L}, \psi)\) be a triple. The composition \(R \to A_0 \to \Gamma(T, \mathcal{O}_T)\) determines a morphism \(f : T \to S = \Spec(R)\), see Schemes, Lemma 01I1. With this choice of \(f\) the map \(A^{(d)} \to \Gamma(S, \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n})\) is \(R\)-linear, and hence corresponds to a \(\psi\) which we can use for a quadruple \((d, f : T \to S, \mathcal{L}, \psi)\). We omit the verification that this establishes an isomorphism of functors \(F_d = F_d^{triples}\).

Lemma

In Situation 01NN. The functor \(F_d\) is representable by a scheme.

Proof

We are going to use Schemes, Lemma 01JJ.

First we check that \(F_d\) 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 \((d, f_i, \mathcal{L}_i, \psi_i) \in F_d(U_i)\) such that \((d, f_i, \mathcal{L}_i, \psi_i)|_{U_i \cap U_j}\) and \((d, f_j, \mathcal{L}_j, \psi_j)|_{U_i \cap U_j}\) are strictly equivalent. 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 01JA. Thus \(f_i^*\mathcal{A}^{(d)} = f^*\mathcal{A}^{(d)}|_{U_i}\). It also implies there exist isomorphisms \(\beta_{ij} : \mathcal{L}_i|_{U_i \cap U_j} \to \mathcal{L}_j|_{U_i \cap U_j}\) such that \(\beta_{ij} \circ \psi_i = \psi_j\) on \(U_i \cap U_j\). Note that the isomorphisms \(\beta_{ij}\) are uniquely determined by this requirement because the maps \(f_i^*\mathcal{A}_d \to \mathcal{L}_i\) are surjective. In particular we see that \(\beta_{jk} \circ \beta_{ij} = \beta_{ik}\) on \(U_i \cap U_j \cap U_k\). Hence by Sheaves, Section 00AK the invertible sheaves \(\mathcal{L}_i\) glue to an invertible \(\mathcal{O}_T\)-module \(\mathcal{L}\) and the morphisms \(\psi_i\) glue to morphism of \(\mathcal{O}_T\)-algebras \(\psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}\). This proves that \(F_d\) satisfies the sheaf condition with respect to the Zariski topology.

Let \(S = \bigcup_{i \in I} U_i\) be an affine open covering. Let \(F_{d, i} \subset F_d\) be the subfunctor consisting of those pairs \((f : T \to S, \varphi)\) such that \(f(T) \subset U_i\).

We have to show each \(F_{d, i}\) is representable. This is the case because \(F_{d, i}\) is identified with the functor associated to \(U_i\) equipped with the quasi-coherent graded \(\mathcal{O}_{U_i}\)-algebra \(\mathcal{A}|_{U_i}\) by Lemma 01NT. Thus the result follows from Lemma 01NU.

Next we show that \(F_{d, i} \subset F_d\) is representable by open immersions. Let \((f : T \to S, \varphi) \in F_d(T)\). Consider \(V_i = f^{-1}(U_i)\). It follows from the definition of \(F_{d, i}\) that given \(a : T' \to T\) we have \(a^*(f, \varphi) \in F_{d, i}(T')\) if and only if \(a(T') \subset V_i\). This is what we were required to show.

Finally, we have to show that the collection \((F_{d, i})_{i \in I}\) covers \(F_d\). Let \((f : T \to S, \varphi) \in F_d(T)\). 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 \((f, \varphi)|_{V_i} \in F_{d, i}(V_i)\). This finishes the proof of the lemma.

At this point we can redo the material at the end of Section 01N4 in the current relative setting and define a functor which is representable by \(\underline{\text{Proj}}_S(\mathcal{A})\). To do this we introduce the notion of equivalence between two quadruples \((d, f : T \to S, \mathcal{L}, \psi)\) and \((d', f' : T \to S, \mathcal{L}', \psi')\) with possibly different values of the integers \(d, d'\). Namely, we say these are equivalent if \(f = f'\), and there exists an isomorphism \(\beta : \mathcal{L}^{\otimes d'} \to (\mathcal{L}')^{\otimes d}\) such that \(\beta \circ \psi|_{f^*\mathcal{A}^{(dd')}} = \psi'|_{f^*\mathcal{A}^{(dd')}}\). The following lemma implies that this defines an equivalence relation. (This is not a complete triviality.)

Lemma

In Situation 01NN. Let \(T\) be a scheme. Let \((d, f, \mathcal{L}, \psi)\), \((d', f', \mathcal{L}', \psi')\) be two quadruples over \(T\). The following are equivalent:

1. Let \(m = \text{lcm}(d, d')\). Write \(m = ad = a'd'\). We have \(f = f'\) and there exists an isomorphism \(\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}\) with the property that \(\beta \circ \psi|_{f^*\mathcal{A}^{(m)}}\) and \(\psi'|_{f^*\mathcal{A}^{(m)}}\) agree as graded ring maps \(f^*\mathcal{A}^{(m)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes mn}\).

2. The quadruples \((d, f, \mathcal{L}, \psi)\) and \((d', f', \mathcal{L}', \psi')\) are equivalent.

3. We have \(f = f'\) and for some positive integer \(m = ad = a'd'\) there exists an isomorphism \(\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}\) with the property that \(\beta \circ \psi|_{f^*\mathcal{A}^{(m)}}\) and \(\psi'|_{f^*\mathcal{A}^{(m)}}\) agree as graded ring maps \(f^*\mathcal{A}^{(m)} \to \bigoplus_{n \geq 0} (\mathcal{L}')^{\otimes mn}\).

Proof

Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Assume (3) for some integer \(m = ad = a'd'\). Let \(m_0 = \text{lcm}(d, d')\) and write it as \(m_0 = a_0d = a'_0d'\). We are given an isomorphism \(\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}\) with the property described in (3). We want to find an isomorphism \(\beta_0 : \mathcal{L}^{\otimes a_0} \to (\mathcal{L}')^{\otimes a'_0}\) having that property as well. Since by assumption the maps \(\psi : f^*\mathcal{A}_d \to \mathcal{L}\) and \(\psi' : (f')^*\mathcal{A}_{d'} \to \mathcal{L}'\) are surjective the same is true for the maps \(\psi : f^*\mathcal{A}_{m_0} \to \mathcal{L}^{\otimes a_0}\) and \(\psi' : (f')^*\mathcal{A}_{m_0} \to (\mathcal{L}')^{\otimes a_0}\). Hence if \(\beta_0\) exists it is uniquely determined by the condition that \(\beta_0 \circ \psi = \psi'\). This means that we may work locally on \(T\). Hence we may assume that \(f = f' : T \to S\) maps into an affine open, in other words we may assume that \(S\) is affine. In this case the result follows from the corresponding result for triples (see Lemma 01NB) and the fact that triples and quadruples correspond in the affine base case (see proof of Lemma 01NU).

Suppose \(d' = ad\). Consider the transformation of functors \(F_d \to F_{d'}\) which assigns to the quadruple \((d, f, \mathcal{L}, \psi)\) over \(T\) the quadruple \((d', f, \mathcal{L}^{\otimes a}, \psi|_{f^*\mathcal{A}^{(d')}})\). One of the implications of Lemma 01NW 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 [01NX]\[\begin{equation} F : \Sch^{opp} \longrightarrow \textit{Sets} \end{equation}\]

Lemma

In Situation 01NN. The functor \(F\) above is representable by a scheme.

Proof

Let \(U_d \to S\) be the scheme representing the functor \(F_d\) defined above. Let \(\mathcal{L}_d\), \(\psi^d : \pi_d^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} \mathcal{L}_d^{\otimes n}\) be the universal object. If \(d | d'\), then we may consider the quadruple \((d', \pi_d, \mathcal{L}_d^{\otimes d'/d}, \psi^d|_{\mathcal{A}^{(d')}})\) which determines a canonical morphism \(U_d \to U_{d'}\) over \(S\). By construction this morphism corresponds to the transformation of functors \(F_d \to F_{d'}\) defined above.

For every affine open \(\Spec(R) = V \subset S\) setting \(A = \Gamma(V, \mathcal{A})\) we have a canonical identification of the base change \(U_{d, V}\) with the corresponding open subscheme of \(\text{Proj}(A)\), see Lemma 01NU. Moreover, the morphisms \(U_{d, V} \to U_{d', V}\) constructed above correspond to the inclusions of opens in \(\text{Proj}(A)\). Thus we conclude that \(U_d \to U_{d'}\) is an open immersion.

This allows us to construct \(X\) by glueing the schemes \(U_d\) along the open immersions \(U_d \to U_{d'}\). Technically, it is convenient to choose a sequence \(d_1 | d_2 | d_3 | \ldots\) such that every positive integer divides one of the \(d_i\) and to simply take \(X = \bigcup U_{d_i}\) using the open immersions above. It is then a simple matter to prove that \(X\) represents the functor \(F\).

Lemma

In Situation 01NN. The scheme \(\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S\) constructed in Lemma 01NQ and the scheme representing the functor \(F\) are canonically isomorphic as schemes over \(S\).

Proof

Let \(X\) be the scheme representing the functor \(F\). Note that \(X\) is a scheme over \(S\) since the functor \(F\) comes equipped with a natural transformation \(F \to h_S\). Write \(Y = \underline{\text{Proj}}_S(\mathcal{A})\). We have to show that \(X \cong Y\) as \(S\)-schemes. We give two arguments.

The first argument uses the construction of \(X\) as the union of the schemes \(U_d\) representing \(F_d\) in the proof of Lemma 01NY. Over each affine open of \(S\) we can identify \(X\) with the homogeneous spectrum of the sections of \(\mathcal{A}\) over that open, since this was true for the opens \(U_d\). Moreover, these identifications are compatible with further restrictions to smaller affine opens. On the other hand, \(Y\) was constructed by glueing these homogeneous spectra. Hence we can glue these isomorphisms to an isomorphism between \(X\) and \(\underline{\text{Proj}}_S(\mathcal{A})\) as desired. Details omitted.

Here is the second argument. Lemma 01NR shows that there exists a morphism of graded algebras \[\psi : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_Y(n)\] over \(Y\) which on sections over affine opens of \(S\) agrees with (01MP). Hence for every \(y \in Y\) there exists an open neighbourhood \(V \subset Y\) of \(y\) and an integer \(d \geq 1\) such that for \(d | n\) the sheaf \(\mathcal{O}_Y(n)|_V\) is invertible and the multiplication maps \(\mathcal{O}_Y(n)|_V \otimes_{\mathcal{O}_V} \mathcal{O}_Y(m)|_V \to \mathcal{O}_Y(n + m)|_V\) are isomorphisms. Thus \(\psi\) restricted to the sheaf \(\pi^*\mathcal{A}^{(d)}|_V\) gives an element of \(F_d(V)\). Since the opens \(V\) cover \(Y\) we see “\(\psi\)” gives rise to an element of \(F(Y)\). Hence a canonical morphism \(Y \to X\) over \(S\). Because this construction is completely canonical to see that it is an isomorphism we may work locally on \(S\). Hence we reduce to the case \(S\) affine where the result is clear.

Definition

Let \(S\) be a scheme. Let \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_S\)-algebras. The relative homogeneous spectrum of \(\mathcal{A}\) over \(S\), or the homogeneous spectrum of \(\mathcal{A}\) over \(S\), or the relative Proj of \(\mathcal{A}\) over \(S\) is the scheme constructed in Lemma 01NQ which represents the functor \(F\) (01NX), see Lemma 01NZ. We denote it \(\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S\).

The relative Proj comes equipped with a quasi-coherent sheaf of \(\mathbf{Z}\)-graded algebras \(\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\) (the twists of the structure sheaf) and a “universal” homomorphism of graded algebras \[\psi_{univ} : \mathcal{A} \longrightarrow \pi_*\left( \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n) \right)\] see Lemma 01NR. We may also think of this as a homomorphism \[\psi_{univ} : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n)\] if we like. The following lemma is a formulation of the universality of this object.

Lemma

In Situation 01NN. Let \((f : T \to S, d, \mathcal{L}, \psi)\) be a quadruple. Let \(r_{d, \mathcal{L}, \psi} : T \to \underline{\text{Proj}}_S(\mathcal{A})\) be the associated \(S\)-morphism. There exists an isomorphism of \(\mathbf{Z}\)-graded \(\mathcal{O}_T\)-algebras \[\theta : r_{d, \mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(nd) \right) \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\] such that the following diagram commutes \[\xymatrix{ \mathcal{A}^{(d)} \ar[rr]_-{\psi} \ar[rd]_-{\psi_{univ}} & & f_*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n} \right) \\ & \pi_*\left( \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(nd) \right) \ar[ru]_\theta }\] The commutativity of this diagram uniquely determines \(\theta\).

Proof

Note that the quadruple \((f : T \to S, d, \mathcal{L}, \psi)\) defines an element of \(F_d(T)\). Let \(U_d \subset \underline{\text{Proj}}_S(\mathcal{A})\) be the locus where the sheaf \(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)\) is invertible and generated by the image of \(\psi_{univ} : \pi^*\mathcal{A}_d \to \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)\). Recall that \(U_d\) represents the functor \(F_d\), see the proof of Lemma 01NY. Hence the result will follow if we can show the quadruple \((U_d \to S, d, \mathcal{O}_{U_d}(d), \psi_{univ}|_{\mathcal{A}^{(d)}})\) is the universal family, i.e., the representing object in \(F_d(U_d)\). We may do this after restricting to an affine open of \(S\) because (a) the formation of the functors \(F_d\) commutes with base change (see Lemma 01NT), and (b) the pair \((\bigoplus_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(n), \psi_{univ})\) is constructed by glueing over affine opens in \(S\) (see Lemma 01NR). Hence we may assume that \(S\) is affine. In this case the functor of quadruples \(F_d\) and the functor of triples \(F_d\) agree (see proof of Lemma 01NU) and moreover Lemma 01N9 shows that \((d, \mathcal{O}_{U_d}(d), \psi^d)\) is the universal triple over \(U_d\). Going backwards through the identifications in the proof of Lemma 01NU shows that \((U_d \to S, d, \mathcal{O}_{U_d}(d), \psi_{univ}|_{\mathcal{A}^{(d)}})\) is the universal quadruple as desired.

Lemma

Let \(S\) be a scheme and \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_S\)-algebras. The morphism \(\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S\) is separated.

Proof

To prove a morphism is separated we may work locally on the base, see Schemes, Section 01KH. By construction \(\underline{\text{Proj}}_S(\mathcal{A})\) is over any affine \(U \subset S\) isomorphic to \(\text{Proj}(A)\) with \(A = \mathcal{A}(U)\). By Lemma 01MC we see that \(\text{Proj}(A)\) is separated. Hence \(\text{Proj}(A) \to U\) is separated (see Schemes, Lemma 01KV) as desired.

Lemma

Let \(S\) be a scheme and \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_S\)-algebras. Let \(g : S' \to S\) be any morphism of schemes. Then there is a canonical isomorphism \[r : \underline{\text{Proj}}_{S'}(g^*\mathcal{A}) \longrightarrow S' \times_S \underline{\text{Proj}}_S(\mathcal{A})\] as well as a corresponding isomorphism \[\theta : r^*\text{pr}_2^*\left(\bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)\right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}(d)\] of \(\mathbf{Z}\)-graded \(\mathcal{O}_{\underline{\text{Proj}}_{S'}(g^*\mathcal{A})}\)-algebras.

Proof

This follows from Lemma 01NT and the construction of \(\underline{\text{Proj}}_S(\mathcal{A})\) in Lemma 01NY as the union of the schemes \(U_d\) representing the functors \(F_d\). In terms of the construction of relative Proj via glueing this isomorphism is given by the isomorphisms constructed in Lemma 01N2 which provides us with the isomorphism \(\theta\). Some details omitted.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{A}\) be a quasi-coherent sheaf of graded \(\mathcal{O}_S\)-modules generated as an \(\mathcal{A}_0\)-algebra by \(\mathcal{A}_1\). In this case the scheme \(X = \underline{\text{Proj}}_S(\mathcal{A})\) represents the functor \(F_1\) which associates to a scheme \(f : T \to S\) over \(S\) the set of pairs \((\mathcal{L}, \psi)\), where

1. \(\mathcal{L}\) is an invertible \(\mathcal{O}_T\)-module, and

2. \(\psi : f^*\mathcal{A} \to \bigoplus_{n \geq 0} \mathcal{L}^{\otimes n}\) is a graded \(\mathcal{O}_T\)-algebra homomorphism such that \(f^*\mathcal{A}_1 \to \mathcal{L}\) is surjective

up to strict equivalence as above. Moreover, in this case all the quasi-coherent sheaves \(\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)\) are invertible \(\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}\)-modules and the multiplication maps induce isomorphisms \(\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n) \otimes_{\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}} \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(m) = \mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n + m)\).

Proof

Under the assumptions of the lemma the sheaves \(\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)\) are invertible and the multiplication maps isomorphisms by Lemma 01NY and Lemma 01NA over affine opens of \(S\). Thus \(X\) actually represents the functor \(F_1\), see proof of Lemma 01NY.

Quasi-coherent sheaves on relative Proj

We briefly discuss how to deal with graded modules in the relative setting.

We place ourselves in Situation 01NN. So \(S\) is a scheme, and \(\mathcal{A}\) is a quasi-coherent graded \(\mathcal{O}_S\)-algebra. Let \(\mathcal{M} = \bigoplus_{n \in \mathbf{Z}} \mathcal{M}_n\) be a graded \(\mathcal{A}\)-module, quasi-coherent as an \(\mathcal{O}_S\)-module. We are going to describe the associated quasi-coherent sheaf of modules on \(\underline{\text{Proj}}_S(\mathcal{A})\). We first describe the value of this sheaf on schemes \(T\) mapping into the relative Proj.

Let \(T\) be a scheme. Let \((d, f : T \to S, \mathcal{L}, \psi)\) be a quadruple over \(T\), as in Section 01NS. We define a quasi-coherent sheaf \(\widetilde{\mathcal{M}}_T\) of \(\mathcal{O}_T\)-modules as follows [01O6]\[\begin{equation} \widetilde{\mathcal{M}}_T = \left( f^*\mathcal{M}^{(d)} \otimes_{f^*\mathcal{A}^{(d)}} \left(\bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\right) \right)_0 \end{equation}\] So \(\widetilde{\mathcal{M}}_T\) is the degree \(0\) part of the tensor product of the graded \(f^*\mathcal{A}^{(d)}\)-modules \(\mathcal{M}^{(d)}\) and \(\bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\). Note that the sheaf \(\widetilde{\mathcal{M}}_T\) depends on the quadruple even though we suppressed this in the notation. This construction has the pleasing property that given any morphism \(g : T' \to T\) we have \(\widetilde{\mathcal{M}}_{T'} = g^*\widetilde{\mathcal{M}}_T\) where \(\widetilde{\mathcal{M}}_{T'}\) denotes the quasi-coherent sheaf associated to the pullback quadruple \((d, f \circ g, g^*\mathcal{L}, g^*\psi)\).

Since all sheaves in (01O6) are quasi-coherent we can spell out the construction over an affine open \(\Spec(C) = V \subset T\) which maps into an affine open \(\Spec(R) = U \subset S\). Namely, suppose that \(\mathcal{A}|_U\) corresponds to the graded \(R\)-algebra \(A\), that \(\mathcal{M}|_U\) corresponds to the graded \(A\)-module \(M\), and that \(\mathcal{L}|_V\) corresponds to the invertible \(C\)-module \(L\). The map \(\psi\) gives rise to a graded \(R\)-algebra map \(\gamma : A^{(d)} \to \bigoplus_{n \geq 0} L^{\otimes n}\). (Tensor powers of \(L\) over \(C\).) Then \((\widetilde{\mathcal{M}}_T)|_V\) is the quasi-coherent sheaf associated to the \(C\)-module \[N_{R, C, A, M, \gamma} = \left( M^{(d)} \otimes_{A^{(d)}, \gamma} \left(\bigoplus\nolimits_{n \in \mathbf{Z}} L^{\otimes n}\right) \right)_0\] By assumption we may even cover \(T\) by affine opens \(V\) such that there exists some \(a \in A_d\) such that \(\gamma(a) \in L\) is a \(C\)-basis for the module \(L\). In that case any element of \(N_{R, C, A, M, \gamma}\) is a sum of pure tensors \(\sum m_i \otimes \gamma(a)^{-n_i}\) with \(m_i \in M_{n_id}\). In fact we may multiply each \(m_i\) with a suitable positive power of \(a\) and collect terms to see that each element of \(N_{R, C, A, M, \gamma}\) can be written as \(m \otimes \gamma(a)^{-n}\) with \(m \in M_{nd}\) and \(n \gg 0\). In other words we see that in this case \[N_{R, C, A, M, \gamma} = M_{(a)} \otimes_{A_{(a)}} C\] where the map \(A_{(a)} \to C\) is the map \(x/a^n \mapsto \gamma(x)/\gamma(a)^n\). In other words, this is the value of \(\widetilde{M}\) on \(D_{+}(a) \subset \text{Proj}(A)\) pulled back to \(\Spec(C)\) via the morphism \(\Spec(C) \to D_{+}(a)\) coming from \(\gamma\).

Lemma

In Situation 01NN. For any quasi-coherent sheaf of graded \(\mathcal{A}\)-modules \(\mathcal{M}\) on \(S\), there exists a canonical associated sheaf of \(\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}\)-modules \(\widetilde{\mathcal{M}}\) with the following properties:

1. Given a scheme \(T\) and a quadruple \((T \to S, d, \mathcal{L}, \psi)\) over \(T\) corresponding to a morphism \(h : T \to \underline{\text{Proj}}_S(\mathcal{A})\) there is a canonical isomorphism \(\widetilde{\mathcal{M}}_T = h^*\widetilde{\mathcal{M}}\) where \(\widetilde{\mathcal{M}}_T\) is defined by (01O6).

2. The isomorphisms of (1) are compatible with pullbacks.

3. There is a canonical map \[\pi^*\mathcal{M}_0 \longrightarrow \widetilde{\mathcal{M}}.\]

4. The construction \(\mathcal{M} \mapsto \widetilde{\mathcal{M}}\) is functorial in \(\mathcal{M}\).

5. The construction \(\mathcal{M} \mapsto \widetilde{\mathcal{M}}\) is exact.

6. There are canonical maps \[\widetilde{\mathcal{M}} \otimes_{\mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}} \widetilde{\mathcal{N}} \longrightarrow \widetilde{\mathcal{M} \otimes_\mathcal{A} \mathcal{N}}\] as in Lemma 01MK.

7. There exist canonical maps \[\pi^*\mathcal{M} \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \widetilde{\mathcal{M}(n)}\] generalizing (01MR).

8. The formation of \(\widetilde{\mathcal{M}}\) commutes with base change.

Proof

Omitted. We should split this lemma into parts and prove the parts separately.

Functoriality of relative Proj

This section is the analogue of Section 01MX for the relative Proj. Let \(S\) be a scheme. A graded \(\mathcal{O}_S\)-algebra map \(\psi : \mathcal{A} \to \mathcal{B}\) does not always give rise to a morphism of associated relative Proj. The correct result is stated as follows.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{A}\), \(\mathcal{B}\) be two graded quasi-coherent \(\mathcal{O}_S\)-algebras. Set \(p : X = \underline{\text{Proj}}_S(\mathcal{A}) \to S\) and \(q : Y = \underline{\text{Proj}}_S(\mathcal{B}) \to S\). Let \(\psi : \mathcal{A} \to \mathcal{B}\) be a homomorphism of graded \(\mathcal{O}_S\)-algebras. There is a canonical open \(U(\psi) \subset Y\) and a canonical morphism of schemes \[r_\psi : U(\psi) \longrightarrow X\] over \(S\) 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 property that for any affine open \(W \subset S\) the triple \[(U(\psi) \cap p^{-1}W,\quad r_\psi|_{U(\psi) \cap p^{-1}W} : U(\psi) \cap p^{-1}W \to q^{-1}W,\quad \theta|_{U(\psi) \cap p^{-1}W})\] is equal to the triple associated to \(\psi : \mathcal{A}(W) \to \mathcal{B}(W)\) in Lemma 01MY via the identifications \(p^{-1}W = \text{Proj}(\mathcal{A}(W))\) and \(q^{-1}W = \text{Proj}(\mathcal{B}(W))\) of Section 01NM.

Proof

This lemma proves itself by glueing the local triples.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{A}\), \(\mathcal{B}\), and \(\mathcal{C}\) be quasi-coherent graded \(\mathcal{O}_S\)-algebras. Set \(X = \underline{\text{Proj}}_S(\mathcal{A})\), \(Y = \underline{\text{Proj}}_S(\mathcal{B})\) and \(Z = \underline{\text{Proj}}_S(\mathcal{C})\). Let \(\varphi : \mathcal{A} \to \mathcal{B}\), \(\psi : \mathcal{B} \to \mathcal{C}\) be graded \(\mathcal{O}_S\)-algebra 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.

Proof

Omitted.

Lemma

With hypotheses and notation as in Lemma 07ZG above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is surjective for \(d \gg 0\). Then

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is a closed immersion, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are surjective but not isomorphisms in general (even if \(\mathcal{A} \to \mathcal{B}\) is surjective).

Proof

Follows on combining Lemma 07ZG with Lemma 01N0.

Lemma

With hypotheses and notation as in Lemma 07ZG above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is an isomorphism for all \(d \gg 0\). Then

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is an isomorphism, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are isomorphisms.

Proof

Follows on combining Lemma 07ZG with Lemma 07ZE.

Lemma

With hypotheses and notation as in Lemma 07ZG above. Assume \(\mathcal{A}_d \to \mathcal{B}_d\) is surjective for \(d \gg 0\) and that \(\mathcal{A}\) is generated by \(\mathcal{A}_1\) over \(\mathcal{A}_0\). Then

1. \(U(\psi) = Y\),

2. \(r_\psi : Y \to X\) is a closed immersion, and

3. the maps \(\theta : r_\psi^*\mathcal{O}_X(n) \to \mathcal{O}_Y(n)\) are isomorphisms.

Proof

Follows on combining Lemma 07ZG with Lemma 01N1.

Invertible sheaves and morphisms into relative Proj

It seems that we may need the following lemma somewhere. The situation is the following:

1. Let \(S\) be a scheme.

2. Let \(\mathcal{A}\) be a quasi-coherent graded \(\mathcal{O}_S\)-algebra.

3. Denote \(\pi : \underline{\text{Proj}}_S(\mathcal{A}) \to S\) the relative homogeneous spectrum over \(S\).

4. Let \(f : X \to S\) be a morphism of schemes.

5. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module.

6. Let \(\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}\) be a homomorphism of graded \(\mathcal{O}_X\)-algebras.

Given this data set \[U(\psi) = \bigcup\nolimits_{(U, V, a)} U_{\psi(a)}\] where \((U, V, a)\) satisfies:

1. \(V \subset S\) affine open,

2. \(U = f^{-1}(V)\), and

3. \(a \in \mathcal{A}(V)_{+}\) is homogeneous.

Namely, then \(\psi(a) \in \Gamma(U, \mathcal{L}^{\otimes \deg(a)})\) and \(U_{\psi(a)}\) is the corresponding open (see Modules, Lemma 01CY).

Lemma

With assumptions and notation as above. The morphism \(\psi\) induces a canonical morphism of schemes over \(S\) \[r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow \underline{\text{Proj}}_S(\mathcal{A})\] together with a map of graded \(\mathcal{O}_{U(\psi)}\)-algebras \[\theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d) \right) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi)}\] characterized by the following properties:

1. For every open \(V \subset S\) and every \(d \geq 0\) the diagram \[\xymatrix{ \mathcal{A}_d(V) \ar[d]_{\psi} \ar[r]_{\psi} & \Gamma(f^{-1}(V), \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(\pi^{-1}(V), \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(d)) \ar[r]^{\theta} & \Gamma(f^{-1}(V) \cap U(\psi), \mathcal{L}^{\otimes d}) }\] is commutative.

2. For any \(d \geq 1\) and any open subscheme \(W \subset X\) such that \(\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W\) is surjective the restriction of the morphism \(r_{\mathcal{L}, \psi}\) agrees with the morphism \(W \to \underline{\text{Proj}}_S(\mathcal{A})\) which exists by the construction of the relative homogeneous spectrum, see Definition 01O0.

3. For any affine open \(V \subset S\), the restriction \[(U(\psi) \cap f^{-1}(V), r_{\mathcal{L}, \psi}|_{U(\psi) \cap f^{-1}(V)}, \theta|_{U(\psi) \cap f^{-1}(V)})\] agrees via \(i_V\) (see Lemma 01NQ) with the triple \((U(\psi'), r_{\mathcal{L}, \psi'}, \theta')\) of Lemma 01NK associated to the map \(\psi' : A = \mathcal{A}(V) \to \Gamma_*(f^{-1}(V), \mathcal{L}|_{f^{-1}(V)})\) induced by \(\psi\).

Proof

Use characterization (3) to construct the morphism \(r_{\mathcal{L}, \psi}\) and \(\theta\) locally over \(S\). Use the uniqueness of Lemma 01NK to show that the construction glues. Details omitted.

Twisting by invertible sheaves and relative Proj

Let \(S\) be a scheme. Let \(\mathcal{A} = \bigoplus_{d \geq 0} \mathcal{A}_d\) be a quasi-coherent graded \(\mathcal{O}_S\)-algebra. Let \(\mathcal{L}\) be an invertible sheaf on \(S\). In this situation we obtain another quasi-coherent graded \(\mathcal{O}_S\)-algebra, namely \[\mathcal{B} = \bigoplus\nolimits_{d \geq 0} \mathcal{A}_d \otimes_{\mathcal{O}_S} \mathcal{L}^{\otimes d}\] It turns out that \(\mathcal{A}\) and \(\mathcal{B}\) have isomorphic relative homogeneous spectra.

Lemma

With notation \(S\), \(\mathcal{A}\), \(\mathcal{L}\) and \(\mathcal{B}\) as above. There is a canonical isomorphism \[\xymatrix{ P = \underline{\text{Proj}}_S(\mathcal{A}) \ar[rr]_g \ar[rd]_\pi & & \underline{\text{Proj}}_S(\mathcal{B}) = P' \ar[ld]^{\pi'} \\ & S & }\] with the following properties

1. There are isomorphisms \(\theta_n : g^*\mathcal{O}_{P'}(n) \to \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n}\) which fit together to give an isomorphism of \(\mathbf{Z}\)-graded algebras \[\theta : g^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{P'}(n) \right) \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n}\]

2. For every open \(V \subset S\) the diagrams \[\xymatrix{ \mathcal{A}_n(V) \otimes \mathcal{L}^{\otimes n}(V) \ar[r]_{multiply} \ar[d]^{\psi \otimes \pi^*} & \mathcal{B}_n(V) \ar[dd]^\psi \\ \Gamma(\pi^{-1}V, \mathcal{O}_P(n)) \otimes \Gamma(\pi^{-1}V, \pi^*\mathcal{L}^{\otimes n}) \ar[d]^{multiply} \\ \Gamma(\pi^{-1}V, \mathcal{O}_P(n) \otimes \pi^*\mathcal{L}^{\otimes n}) & \Gamma(\pi'^{-1}V, \mathcal{O}_{P'}(n)) \ar[l]_-{\theta_n} }\] are commutative.

3. Add more here as necessary.

Proof

This is the identity map when \(\mathcal{L} \cong \mathcal{O}_S\). In general choose an open covering of \(S\) such that \(\mathcal{L}\) is trivialized over the pieces and glue the corresponding maps. Details omitted.

Projective bundles

Let \(S\) be a scheme. Let \(\mathcal{E}\) be a quasi-coherent sheaf of \(\mathcal{O}_S\)-modules. By Modules, Lemma 01CL the symmetric algebra \(\text{Sym}(\mathcal{E})\) of \(\mathcal{E}\) over \(\mathcal{O}_S\) is a quasi-coherent sheaf of \(\mathcal{O}_S\)-algebras. Note that it is generated in degree \(1\) over \(\mathcal{O}_S\). Hence it makes sense to apply the construction of the previous section to it, specifically Lemmas 01NY and 01O4.

Definition

Let \(S\) be a scheme. Let \(\mathcal{E}\) be a quasi-coherent \(\mathcal{O}_S\)-module³. We denote \[\pi : \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_S(\text{Sym}(\mathcal{E})) \longrightarrow S\] and we call it the projective bundle associated to \(\mathcal{E}\). The symbol \(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)\) indicates the invertible \(\mathcal{O}_{\mathbf{P}(\mathcal{E})}\)-module of Lemma 01O4 and is called the \(n\)th twist of the structure sheaf.

According to Lemma 01NR there are canonical \(\mathcal{O}_S\)-module homomorphisms \[\text{Sym}^n(\mathcal{E}) \longrightarrow \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \quad\text{equivalently}\quad \pi^*\text{Sym}^n(\mathcal{E}) \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(n)\] for all \(n \geq 0\). In particular, for \(n = 1\) we have \[\mathcal{E} \longrightarrow \pi_*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1) \quad\text{equivalently}\quad \pi^*\mathcal{E} \longrightarrow \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\] and the map \(\pi^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\) is a surjection by Lemma 01O4. This is a good way to remember how we have normalized our construction of \(\mathbf{P}(\mathcal{E})\).

Warning: In some references the scheme \(\mathbf{P}(\mathcal{E})\) is only defined for \(\mathcal{E}\) finite locally free on \(S\). Moreover sometimes \(\mathbf{P}(\mathcal{E})\) is actually defined as our \(\mathbf{P}(\mathcal{E}^\vee)\) where \(\mathcal{E}^\vee\) is the dual of \(\mathcal{E}\) (and this is done only when \(\mathcal{E}\) is finite locally free).

Let \(S\), \(\mathcal{E}\), \(\mathbf{P}(\mathcal{E}) \to S\) be as in Definition 01OB. Let \(f : T \to S\) be a scheme over \(S\). Let \(\psi : f^*\mathcal{E} \to \mathcal{L}\) be a surjection where \(\mathcal{L}\) is an invertible \(\mathcal{O}_T\)-module. The induced graded \(\mathcal{O}_T\)-algebra map \[f^*\text{Sym}(\mathcal{E}) = \text{Sym}(f^*\mathcal{E}) \to \text{Sym}(\mathcal{L}) = \bigoplus\nolimits_{n \geq 0} \mathcal{L}^{\otimes n}\] corresponds to a morphism \[\varphi_{\mathcal{L}, \psi} : T \longrightarrow \mathbf{P}(\mathcal{E})\] over \(S\) by our construction of the relative Proj as the scheme representing the functor \(F\) in Section 01NS. On the other hand, given a morphism \(\varphi : T \to \mathbf{P}(\mathcal{E})\) over \(S\) we can set \(\mathcal{L} = \varphi^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\) and \(\psi : f^*\mathcal{E} \to \mathcal{L}\) equal to the pullback by \(\varphi\) of the canonical surjection \(\pi^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\). By Lemma 01O4 these constructions are inverse bijections between the set of isomorphism classes of pairs \((\mathcal{L}, \psi)\) and the set of morphisms \(\varphi : T \to \mathbf{P}(\mathcal{E})\) over \(S\). Thus we see that \(\mathbf{P}(\mathcal{E})\) represents the functor which associates to \(f : T \to S\) the set of \(\mathcal{O}_T\)-module quotients of \(f^*\mathcal{E}\) which are locally free of rank \(1\).

Example

Let \(k\) be a field. Let \(V\) be a \(k\)-vector space. The corresponding projective space is the \(k\)-scheme \[\mathbf{P}(V) = \text{Proj}(\text{Sym}(V))\] where \(\text{Sym}(V)\) is the symmetric algebra on \(V\) over \(k\). Of course we have \(\mathbf{P}(V) \cong \mathbf{P}^n_k\) if \(\dim(V) = n + 1\) because then the symmetric algebra on \(V\) is isomorphic to a polynomial ring in \(n + 1\) variables. If we think of \(V\) as a quasi-coherent module on \(\Spec(k)\), then \(\mathbf{P}(V)\) is the corresponding projective space bundle over \(\Spec(k)\). By the discussion above a \(k\)-valued point \(p\) of \(\mathbf{P}(V)\) corresponds to a surjection of \(k\)-vector spaces \(V \to L_p\) with \(\dim(L_p) = 1\). More generally, let \(X\) be a scheme over \(k\), let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module, and let \(\psi : V \to \Gamma(X, \mathcal{L})\) be a \(k\)-linear map such that \(\mathcal{L}\) is generated as an \(\mathcal{O}_X\)-module by the sections in the image of \(\psi\). Then the discussion above gives a canonical morphism \[\varphi_{\mathcal{L}, \psi} : X \longrightarrow \mathbf{P}(V)\] of schemes over \(k\) such that there is an isomorphism \(\theta : \varphi_{\mathcal{L}, \psi}^*\mathcal{O}_{\mathbf{P}(V)}(1) \to \mathcal{L}\) and such that \(\psi\) agrees with the composition \[V \to \Gamma(\mathbf{P}(V), \mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma(X, \varphi_{\mathcal{L}, \psi}^*\mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma(X, \mathcal{L})\] See Lemma 01NK. If \(V \subset \Gamma(X, \mathcal{L})\) is a subspace, then we will denote the morphism constructed above simply as \(\varphi_{\mathcal{L}, V}\). If \(\dim(V) = n + 1\) and we choose a basis \(v_0, \ldots, v_n\) of \(V\) then the diagram \[\xymatrix{ X \ar@{=}[d] \ar[rr]_{\varphi_{\mathcal{L}, \psi}} & & \mathbf{P}(V) \ar[d]^{\cong} \\ X \ar[rr]^{\varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}} & & \mathbf{P}^n_k }\] is commutative, where \(s_i = \psi(v_i) \in \Gamma(X, \mathcal{L})\), where \(\varphi_{(\mathcal{L}, (s_0, \ldots, s_n))}\) is as in Section 01ND, and where the right vertical arrow corresponds to the isomorphism \(k[T_0, \ldots, T_n] \to \text{Sym}(V)\) sending \(T_i\) to \(v_i\).

Example

The map \(\text{Sym}^n(\mathcal{E}) \to \pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n))\) is an isomorphism if \(\mathcal{E}\) is locally free, but in general need not be an isomorphism. In fact we will give an example where this map is not injective for \(n = 1\). Set \(S = \Spec(A)\) with \[A = k[u, v, s_1, s_2, t_1, t_2]/I\] where \(k\) is a field and \[I = (-us_1 + vt_1 + ut_2, vs_1 + us_2 - vt_2, vs_2, ut_1).\] Denote \(\overline{u}\) the class of \(u\) in \(A\) and similarly for the other variables. Let \(M = (Ax \oplus Ay)/A(\overline{u}x + \overline{v}y)\) so that \[\text{Sym}(M) = A[x, y]/(\overline{u}x + \overline{v}y) = k[x, y, u, v, s_1, s_2, t_1, t_2]/J\] where \[J = (-us_1 + vt_1 + ut_2, vs_1 + us_2 - vt_2, vs_2, ut_1, ux + vy).\] In this case the projective bundle associated to the quasi-coherent sheaf \(\mathcal{E} = \widetilde{M}\) on \(S = \Spec(A)\) is the scheme \[P = \text{Proj}(\text{Sym}(M)).\] Note that this scheme as an affine open covering \(P = D_{+}(x) \cup D_{+}(y)\). Consider the element \(m \in M\) which is the image of the element \(us_1x + vt_2y\). Note that \[x(us_1x + vt_2y) = (s_1x + s_2y)(ux + vy) \bmod I\] and \[y(us_1x + vt_2y) = (t_1x + t_2y)(ux + vy) \bmod I.\] The first equation implies that \(m\) maps to zero as a section of \(\mathcal{O}_P(1)\) on \(D_{+}(x)\) and the second that it maps to zero as a section of \(\mathcal{O}_P(1)\) on \(D_{+}(y)\). This shows that \(m\) maps to zero in \(\Gamma(P, \mathcal{O}_P(1))\). On the other hand we claim that \(m \not = 0\), so that \(m\) gives an example of a nonzero global section of \(\mathcal{E}\) mapping to zero in \(\Gamma(P, \mathcal{O}_P(1))\). Assume \(m = 0\) to get a contradiction. In this case there exists an element \(f \in k[u, v, s_1, s_2, t_1, t_2]\) such that \[us_1x + vt_2y = f(ux + vy) \bmod I\] Since \(I\) is generated by homogeneous polynomials of degree \(2\) we may decompose \(f\) into its homogeneous components and take the degree 1 component. In other words we may assume that \[f = au + bv + \alpha_1s_1 + \alpha_2s_2 + \beta_1t_1 + \beta_2t_2\] for some \(a, b, \alpha_1, \alpha_2, \beta_1, \beta_2 \in k\). The resulting conditions are that \[\begin{matrix} us_1 - u(au + bv + \alpha_1s_1 + \alpha_2s_2 + \beta_1t_1 + \beta_2t_2) \in I \\ vt_2 - v(au + bv + \alpha_1s_1 + \alpha_2s_2 + \beta_1t_1 + \beta_2t_2) \in I \end{matrix}\] There are no terms \(u^2, uv, v^2\) in the generators of \(I\) and hence we see \(a = b = 0\). Thus we get the relations \[\begin{matrix} us_1 - u(\alpha_1s_1 + \alpha_2s_2 + \beta_1t_1 + \beta_2t_2) \in I \\ vt_2 - v(\alpha_1s_1 + \alpha_2s_2 + \beta_1t_1 + \beta_2t_2) \in I \end{matrix}\] We may use the first generator of \(I\) to replace any occurrence of \(us_1\) by \(vt_1 + ut_2\), the second generator of \(I\) to replace any occurrence of \(vs_1\) by \(-us_2 + vt_2\), the third generator to remove occurrences of \(vs_2\) and the third to remove occurrences of \(ut_1\). Then we get the relations \[\begin{matrix} (1 - \alpha_1)vt_1 + (1 - \alpha_1)ut_2 - \alpha_2us_2 - \beta_2ut_2 = 0 \\ (1 - \alpha_1)vt_2 + \alpha_1us_2 - \beta_1vt_1 - \beta_2vt_2 = 0 \end{matrix}\] This implies that \(\alpha_1\) should be both \(0\) and \(1\) which is a contradiction as desired.

Lemma

Let \(S\) be a scheme. The structure morphism \(\mathbf{P}(\mathcal{E}) \to S\) of a projective bundle over \(S\) is separated.

Proof

Immediate from Lemma 01O2.

Lemma

Let \(S\) be a scheme. Let \(n \geq 0\). Then \(\mathbf{P}^n_S\) is a projective bundle over \(S\).

Proof

Note that \[\mathbf{P}^n_{\mathbf{Z}} = \text{Proj}(\mathbf{Z}[T_0, \ldots, T_n]) = \underline{\text{Proj}}_{\Spec(\mathbf{Z})} \left(\widetilde{\mathbf{Z}[T_0, \ldots, T_n]}\right)\] where the grading on the ring \(\mathbf{Z}[T_0, \ldots, T_n]\) is given by \(\deg(T_i) = 1\) and the elements of \(\mathbf{Z}\) are in degree \(0\). Recall that \(\mathbf{P}^n_S\) is defined as \(\mathbf{P}^n_{\mathbf{Z}} \times_{\Spec(\mathbf{Z})} S\). Moreover, forming the relative homogeneous spectrum commutes with base change, see Lemma 01O3. For any scheme \(g : S \to \Spec(\mathbf{Z})\) we have \(g^*\mathcal{O}_{\Spec(\mathbf{Z})}[T_0, \ldots, T_n] = \mathcal{O}_S[T_0, \ldots, T_n]\). Combining the above we see that \[\mathbf{P}^n_S = \underline{\text{Proj}}_S(\mathcal{O}_S[T_0, \ldots, T_n]).\] Finally, note that \(\mathcal{O}_S[T_0, \ldots, T_n] = \text{Sym}(\mathcal{O}_S^{\oplus n + 1})\). Hence we see that \(\mathbf{P}^n_S\) is a projective bundle over \(S\).

Grassmannians

In this section we introduce the standard Grassmannian functors and we show that they are represented by schemes. Pick integers \(k\), \(n\) with \(0 < k < n\). We will construct a functor [089S]\[\begin{equation} G(k, n) : \Sch \longrightarrow \textit{Sets} \end{equation}\] which will loosely speaking parametrize \(k\)-dimensional subspaces of \(n\)-space. However, for technical reasons it is more convenient to parametrize \((n - k)\)-dimensional quotients and this is what we will do.

More precisely, \(G(k, n)\) associates to a scheme \(S\) the set \(G(k, n)(S)\) of isomorphism classes of surjections \[q : \mathcal{O}_S^{\oplus n} \longrightarrow \mathcal{Q}\] where \(\mathcal{Q}\) is a finite locally free \(\mathcal{O}_S\)-module of rank \(n - k\). Note that this is indeed a set, for example by Modules, Lemma 01BC or by the observation that the isomorphism class of the surjection \(q\) is determined by the kernel of \(q\) (and given a sheaf there is a set of subsheaves). Given a morphism of schemes \(f : T \to S\) we let \(G(k, n)(f) : G(k, n)(S) \to G(k, n)(T)\) which sends the isomorphism class of \(q : \mathcal{O}_S^{\oplus n} \longrightarrow \mathcal{Q}\) to the isomorphism class of \(f^*q : \mathcal{O}_T^{\oplus n} \longrightarrow f^*\mathcal{Q}\). This makes sense since (1) \(f^*\mathcal{O}_S = \mathcal{O}_T\), (2) \(f^*\) is additive, (3) \(f^*\) preserves locally free modules (Modules, Lemma 01C8), and (4) \(f^*\) is right exact (Modules, Lemma 01AJ).

Lemma

Let \(0 < k < n\). The functor \(G(k, n)\) of (089S) is representable by a scheme.

Proof

Set \(F = G(k, n)\). To prove the lemma we will use the criterion of Schemes, Lemma 01JJ. The reason \(F\) satisfies the sheaf property for the Zariski topology is that we can glue sheaves, see Sheaves, Section 00AK (some details omitted).

The family of subfunctors \(F_i\). Let \(I\) be the set of subsets of \(\{1, \ldots, n\}\) of cardinality \(n - k\). Given a scheme \(S\) and \(j \in \{1, \ldots, n\}\) we denote \(e_j\) the global section \[e_j = (0, \ldots, 0, 1, 0, \ldots, 0)\quad(1\text{ in }j\text{th spot})\] of \(\mathcal{O}_S^{\oplus n}\). Of course these sections freely generate \(\mathcal{O}_S^{\oplus n}\). Similarly, for \(j \in \{1, \ldots, n - k\}\) we denote \(f_j\) the global section of \(\mathcal{O}_S^{\oplus n - k}\) which is zero in all summands except the \(j\)th where we put a \(1\). For \(i \in I\) we let \[s_i : \mathcal{O}_S^{\oplus n - k} \longrightarrow \mathcal{O}_S^{\oplus n}\] which is the direct sum of the coprojections \(\mathcal{O}_S \to \mathcal{O}_S^{\oplus n}\) corresponding to elements of \(I\). More precisely, if \(i = \{i_1, \ldots, i_{n - k}\}\) with \(i_1 < i_2 < \ldots < i_{n - k}\) then \(s_i\) maps \(f_j\) to \(e_{i_j}\) for \(j \in \{1, \ldots, n - k\}\). With this notation we can set \[F_i(S) = \{q : \mathcal{O}_S^{\oplus n} \to \mathcal{Q} \in F(S) \mid q \circ s_i \text{ is surjective}\} \subset F(S)\] Given a morphism \(f : T \to S\) of schemes the pullback \(f^*s_i\) is the corresponding map over \(T\). Since \(f^*\) is right exact (Modules, Lemma 01AJ) we conclude that \(F_i\) is a subfunctor of \(F\).

Representability of \(F_i\). To prove this we may assume (after renumbering) that \(i = \{1, \ldots, n - k\}\). This means \(s_i\) is the inclusion of the first \(n - k\) summands. Observe that if \(q \circ s_i\) is surjective, then \(q \circ s_i\) is an isomorphism as a surjective map between finite locally free modules of the same rank (Modules, Lemma 089Q). Thus if \(q : \mathcal{O}_S^{\oplus n} \to \mathcal{Q}\) is an element of \(F_i(S)\), then we can use \(q \circ s_i\) to identify \(\mathcal{Q}\) with \(\mathcal{O}_S^{\oplus n - k}\). After doing so we obtain \[q : \mathcal{O}_S^{\oplus n} \longrightarrow \mathcal{O}_S^{\oplus n - k}\] mapping \(e_j\) to \(f_j\) (notation as above) for \(j = 1, \ldots, n - k\). To determine \(q\) completely we have to fix the images \(q(e_{n - k + 1}), \ldots, q(e_n)\) in \(\Gamma(S, \mathcal{O}_S^{\oplus n - k})\). It follows that \(F_i\) is isomorphic to the functor \[S \longmapsto \prod\nolimits_{j = n - k + 1, \ldots, n} \Gamma(S, \mathcal{O}_S^{\oplus n - k})\] This functor is isomorphic to the \(k(n - k)\)-fold self product of the functor \(S \mapsto \Gamma(S, \mathcal{O}_S)\). By Schemes, Example 01JH the latter is representable by \(\mathbf{A}^1_\mathbf{Z}\). It follows \(F_i\) is representable by \(\mathbf{A}^{k(n - k)}_\mathbf{Z}\) since fibred product over \(\Spec(\mathbf{Z})\) is the product in the category of schemes.

The inclusion \(F_i \subset F\) is representable by open immersions. Let \(S\) be a scheme and let \(q : \mathcal{O}_S^{\oplus n} \to \mathcal{Q}\) be an element of \(F(S)\). By Modules, Lemma 01B8. the set \(U_i = \{s \in S \mid (q \circ s_i)_s\text{ surjective}\}\) is open in \(S\). Since \(\mathcal{O}_{S, s}\) is a local ring and \(\mathcal{Q}_s\) a finite \(\mathcal{O}_{S, s}\)-module by Nakayama’s lemma (Algebra, Lemma 00DV) we have \[s \in U_i \Leftrightarrow \left( \text{the map } \kappa(s)^{\oplus n - k} \to \mathcal{Q}_s/\mathfrak m_s\mathcal{Q}_s \text{ induced by } (q \circ s_i)_s \text{ is surjective} \right)\] Let \(f : T \to S\) be a morphism of schemes and let \(t \in T\) be a point mapping to \(s \in S\). We have \((f^*\mathcal{Q})_t = \mathcal{Q}_s \otimes_{\mathcal{O}_{S, s}} \mathcal{O}_{T, t}\) (Sheaves, Lemma 0098) and so on. Thus the map \[\kappa(t)^{\oplus n - k} \to (f^*\mathcal{Q})_t/\mathfrak m_t(f^*\mathcal{Q})_t\] induced by \((f^*q \circ f^*s_i)_t\) is the base change of the map \(\kappa(s)^{\oplus n - k} \to \mathcal{Q}_s/\mathfrak m_s\mathcal{Q}_s\) above by the field extension \(\kappa(t)/\kappa(s)\). It follows that \(s \in U_i\) if and only if \(t\) is in the corresponding open for \(f^*q\). In particular \(T \to S\) factors through \(U_i\) if and only if \(f^*q \in F_i(T)\) as desired.

The collection \(F_i\), \(i \in I\) covers \(F\). Let \(q : \mathcal{O}_S^{\oplus n} \to \mathcal{Q}\) be an element of \(F(S)\). We have to show that for every point \(s\) of \(S\) there exists an \(i \in I\) such that \(s_i\) is surjective in a neighbourhood of \(s\). Thus we have to show that one of the compositions \[\kappa(s)^{\oplus n - k} \xrightarrow{s_i} \kappa(s)^{\oplus n} \rightarrow \mathcal{Q}_s/\mathfrak m_s\mathcal{Q}_s\] is surjective (see previous paragraph). As \(\mathcal{Q}_s/\mathfrak m_s\mathcal{Q}_s\) is a vector space of dimension \(n - k\) this follows from the theory of vector spaces.

Definition

Let \(0 < k < n\). The scheme \(\mathbf{G}(k, n)\) representing the functor \(G(k, n)\) is called Grassmannian over \(\mathbf{Z}\). Its base change \(\mathbf{G}(k, n)_S\) to a scheme \(S\) is called Grassmannian over \(S\). If \(R\) is a ring the base change to \(\Spec(R)\) is denoted \(\mathbf{G}(k, n)_R\) and called Grassmannian over \(R\).

The definition makes sense as we’ve shown in Lemma 089T that these functors are indeed representable.

Lemma

Let \(n \geq 1\). There is a canonical isomorphism \(\mathbf{G}(n, n + 1) = \mathbf{P}^n_\mathbf{Z}\).

Proof

According to Lemma 01NE the scheme \(\mathbf{P}^n_\mathbf{Z}\) represents the functor which assigns to a scheme \(S\) the set of isomorphisms classes of pairs \((\mathcal{L}, (s_0, \ldots, s_n))\) consisting of an invertible module \(\mathcal{L}\) and an \((n + 1)\)-tuple of global sections generating \(\mathcal{L}\). Given such a pair we obtain a quotient \[\mathcal{O}_S^{\oplus n + 1} \longrightarrow \mathcal{L},\quad (h_0, \ldots, h_n) \longmapsto \sum h_i s_i.\] Conversely, given an element \(q : \mathcal{O}_S^{\oplus n + 1} \to \mathcal{Q}\) of \(G(n, n + 1)(S)\) we obtain such a pair, namely \((\mathcal{Q}, (q(e_1), \ldots, q(e_{n + 1})))\). Here \(e_i\), \(i = 1, \ldots, n + 1\) are the standard generating sections of the free module \(\mathcal{O}_S^{\oplus n + 1}\). We omit the verification that these constructions define mutually inverse transformations of functors.

Lemma

Let \(0 < k < n\) and set \(d = n - k\) and \(N = \binom{n}{d}\). There is a closed immersion \[\mathbf{G}(k, n) \longrightarrow \mathbf{P}^{N - 1}_{\mathbf{Z}}\] which sends a quotient \(q : \mathcal{O}_S^{\oplus n} \to \mathcal{Q}\) to the quotient \[\wedge^d q : \wedge^d\mathcal{O}_S^{\oplus n} \longrightarrow \det(\mathcal{Q}).\] The pullback of \(\mathcal{O}_{\mathbf{P}^{N - 1}_{\mathbf{Z}}}(1)\) is the determinant of the universal quotient on \(\mathbf{G}(k, n)\).

Proof

The displayed quotient defines the morphism by Lemma 01NE. The assertion about the invertible module follows from the same description of projective space.

Let \(I \subset \{1, \ldots, n\}\) have cardinality \(d\). The inverse image of the standard open where the Plücker coordinate indexed by \(I\) is invertible is the open \(F_I\) used in the proof of Lemma 089T. On this open we may identify the quotient with a matrix whose columns indexed by \(I\) form the identity matrix. The Plücker coordinates obtained by replacing one element of \(I\) by one element of its complement are, up to sign, the entries of the remaining columns. Consequently, the map from the coordinate ring of the standard open of projective space to the coordinate ring of \(F_I \cong \mathbf{A}^{kd}_{\mathbf{Z}}\) is surjective. Thus the morphism is a closed immersion over each of these standard opens. Since these opens cover projective space, the result follows.