## 27.10 Invertible sheaves on Proj

Recall from Algebra, Section 10.56 the construction of the twisted module $M(n)$ associated to a graded module over a graded ring.

Definition 27.10.1. 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 27.9.1 to construct some canonical maps. Since $S(n) \otimes _ S S(m) = S(n + m)$ we see that there are canonical maps

27.10.1.1
$$\label{constructions-equation-multiply} \mathcal{O}_ X(n) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(m) \longrightarrow \mathcal{O}_ X(n + m).$$

These maps are not isomorphisms in general, see the example in Remark 27.9.2. 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

27.10.1.2
$$\label{constructions-equation-multiply-on-sheaf} \mathcal{O}_ X(n) \otimes _{\mathcal{O}_ X} \mathcal{F}(m) \longrightarrow \mathcal{F}(n + m).$$

The maps (27.9.0.2) on global sections give a map of graded rings

27.10.1.3
$$\label{constructions-equation-global-sections} S \longrightarrow \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{O}_ X(n)).$$

And for an arbitrary $\mathcal{O}_ X$-module $\mathcal{F}$ the maps (27.10.1.2) give a graded module structure

27.10.1.4
$$\label{constructions-equation-global-sections-module} \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{O}_ X(n)) \times \bigoplus \nolimits _{m \in \mathbf{Z}} \Gamma (X, \mathcal{F}(m)) \longrightarrow \bigoplus \nolimits _{m \in \mathbf{Z}} \Gamma (X, \mathcal{F}(m))$$

and via (27.10.1.3) 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

27.10.1.5
$$\label{constructions-equation-multiply-more-generally} \widetilde M(n) = \widetilde M \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(n) \longrightarrow \widetilde{M(n)}.$$

On global sections (27.9.0.2) defines a map of graded $S$-modules

27.10.1.6
$$\label{constructions-equation-global-sections-more-generally} M \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \widetilde{M(n)}).$$

Here is an important fact which follows basically immediately from the definitions.

Lemma 27.10.2. 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 17.25.1). The maps (27.10.1.1) 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 (27.10.1.2) 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 (27.10.1.5) 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. The (not graded) $S$-module maps $S \to S(nd)$, and $M \to M(nd)$, given by $x \mapsto f^ n x$ become isomorphisms after inverting $f$. The first shows that $S_{(f)} \cong S(nd)_{(f)}$ which gives an isomorphism $\mathcal{O}_{D_{+}(f)} \cong \mathcal{O}_ X(nd)|_{D_{+}(f)}$. The second shows that the map $S(nd)_{(f)} \otimes _{S_{(f)}} M_{(f)} \to M(nd)_{(f)}$ is an isomorphism. The case of the map (27.10.1.2) is a consequence of the case of the map (27.10.1.1). $\square$

Lemma 27.10.3. 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 (27.10.1.1), (27.10.1.2), and (27.10.1.5) 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 (27.9.0.2) becomes a map

27.10.3.1
$$\label{constructions-equation-map-global-sections-degree-n-simplified} M_ n \longrightarrow \Gamma (X, \widetilde{M}(n))$$

and (27.10.1.6) becomes a map

27.10.3.2
$$\label{constructions-equation-global-sections-more-generally-simplified} M \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \widetilde{M}(n)).$$

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 27.10.2 above. $\square$

Lemma 27.10.4. 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 27.10.1.1) 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 27.10.1.5) 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 27.10.2.

Conversely, suppose that $\mathcal{O}_ X(d)$ is free of rank 1 in an open neighbourhood $V$ of $x \in X$ and all the multiplication maps $\mathcal{O}_ X(nd)|_ V \otimes _{\mathcal{O}_ V} \mathcal{O}_ X(md)|_ V \to \mathcal{O}_ X(nd + md)|_ V$ are isomorphisms. We may choose $h \in S_{+}$ homogeneous such that $x \in D_{+}(h) \subset V$. 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 27.10.1.5) are isomorphisms. Compare with the proof of Lemma 27.10.2. $\square$

Recall from Modules, Lemma 17.25.10 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 27.10.5. 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 27.10.4. 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 (27.10.1.3) 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 27.10.4 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. $\square$

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

Lemma 27.10.6. 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 27.10.4,

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 (27.10.1.1) 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 27.10.1.5) are isomorphisms,

5. given $f \in S_{nd}$ denote $s \in \Gamma (Y, \mathcal{O}_ Y(nd))$ the image of $f$ via (27.10.1.3) 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 27.10.4 or by (1) using Lemma 27.10.2. Note that (1) implies (2), (3) and (4) by Lemma 27.10.4. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma 27.10.5. Assertions (6) and (7) follow because the open subsets $D_{+}(f)$ form a basis for the topology of $X$ and are affine. $\square$

Lemma 27.10.7. 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 (27.10.1.4) and (27.10.1.3). 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 27.8.4. 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 27.10.2 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 26.7.1. 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. $\square$

Comment #8507 by Amina Šišić on

I think the reference to the map 27.10.1.1 that gives rise to the global map is wrong, and should instead be 27.9.0.2.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).