The Stacks project

27.14 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 17.24.10. We can view the following lemma as a slight generalization of Lemma 27.12.3. It also is a generalization of Lemma 27.11.1.

Lemma 27.14.1. 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

    \[ \xymatrix{ A_ d \ar[d]_{(01MP)} \ar[r]_{\psi } & \Gamma (T, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma (X, \mathcal{O}_ X(d)) \ar[r]^{\theta } & \Gamma (U(\psi ), \mathcal{L}^{\otimes d}) } \]

    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 27.12.1 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 10.57 and Schemes, Lemma 26.11.1). 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 $T = \mathop{\mathrm{Spec}}(R)$ 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 $r : T = \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(A_{(f)}) = D_{+}(f) \subset X$. Hence we have constructed $(r, \theta )$ in this special case.

Let us show the last statement of the lemma. According to Lemma 27.12.1 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. $\square$

Remark 27.14.2. Assumptions as in Lemma 27.14.1 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 ( 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 27.9.2 we showed that the sheaf $\mathcal{O}_ X(1)$ is not invertible at all points of $X$.

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