The Stacks project

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

These maps are not isomorphisms in general, see the example in Remark 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
\begin{equation} \label{constructions-equation-multiply-on-sheaf} \mathcal{O}_ X(n) \otimes _{\mathcal{O}_ X} \mathcal{F}(m) \longrightarrow \mathcal{F}(n + m). \end{equation}

The maps ( on global sections give a map of graded rings
\begin{equation} \label{constructions-equation-global-sections} S \longrightarrow \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{O}_ X(n)). \end{equation}

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

and via ( also a $S$-module structure. More generally, given any graded $S$-module $M$ we have $M(n) = M \otimes _ S S(n)$. Hence we get maps
\begin{equation} \label{constructions-equation-multiply-more-generally} \widetilde M(n) = \widetilde M \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(n) \longrightarrow \widetilde{M(n)}. \end{equation}

On global sections ( defines a map of graded $S$-modules
\begin{equation} \label{constructions-equation-global-sections-more-generally} M \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \widetilde{M(n)}). \end{equation}

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

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 17.24.1). The maps ( 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 ( 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 ( 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 ( is a consequence of the case of the map ( $\square$

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 (, (, and ( 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 ( becomes a map
\begin{equation} \label{constructions-equation-map-global-sections-degree-n-simplified} M_ n \longrightarrow \Gamma (X, \widetilde{M}(n)) \end{equation}

and ( becomes a map
\begin{equation} \label{constructions-equation-global-sections-more-generally-simplified} 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 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 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 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 $D_{+}(h) \subset V$. 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 are isomorphisms. Compare with the proof of Lemma 27.10.2. $\square$

Recall from Modules, Lemma 17.24.10 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 ( 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 ( 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 are isomorphisms,

  5. given $f \in S_{nd}$ denote $s \in \Gamma (Y, \mathcal{O}_ Y(nd))$ the image of $f$ via ( 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 ( and ( 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. $\square$

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