Situation 28.28.1. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Set $S = \Gamma _*(X, \mathcal{L})$ as a graded ring. Set $Y = \text{Proj}(S)$. Let $f : X \to Y$ be the canonical morphism of Lemma 28.26.9. It comes equipped with a $\mathbf{Z}$-graded $\mathcal{O}_ X$-algebra map $\bigoplus f^*\mathcal{O}_ Y(n) \to \bigoplus \mathcal{L}^{\otimes n}$.
28.28 Quasi-coherent sheaves and ample invertible sheaves
Theme of this section: in the presence of an ample invertible sheaf every quasi-coherent sheaf comes from a graded module.
The following lemma is really a special case of the next lemma but it seems like a good idea to point out its validity first.
Lemma 28.28.2. In Situation 28.28.1. The canonical morphism $f : X \to Y$ maps $X$ into the open subscheme $W = W_1 \subset Y$ where $\mathcal{O}_ Y(1)$ is invertible and where all multiplication maps $\mathcal{O}_ Y(n) \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(m) \to \mathcal{O}_ Y(n + m)$ are isomorphisms (see Constructions, Lemma 27.10.4). Moreover, the maps $f^*\mathcal{O}_ Y(n) \to \mathcal{L}^{\otimes n}$ are all isomorphisms.
Proof. By Proposition 28.26.13 there exists an integer $n_0$ such that $\mathcal{L}^{\otimes n}$ is globally generated for all $n \geq n_0$. Let $x \in X$ be a point. By the above we can find $a \in S_{n_0}$ and $b \in S_{n_0 + 1}$ such that $a$ and $b$ do not vanish at $x$. Hence $f(x) \in D_{+}(a) \cap D_{+}(b) = D_{+}(ab)$. By Constructions, Lemma 27.10.4 we see that $f(x) \in W_1$ as desired. By Constructions, Lemma 27.14.1 which was used in the construction of the map $f$ the maps $f^*\mathcal{O}_ Y(n_0) \to \mathcal{L}^{\otimes n_0}$ and $f^*\mathcal{O}_ Y(n_0 + 1) \to \mathcal{L}^{\otimes n_0 + 1}$ are isomorphisms in a neighbourhood of $x$. By compatibility with the algebra structure and the fact that $f$ maps into $W$ we conclude all the maps $f^*\mathcal{O}_ Y(n) \to \mathcal{L}^{\otimes n}$ are isomorphisms in a neighbourhood of $x$. Hence we win. $\square$
Recall from Modules, Definition 17.25.7 that given a locally ringed space $X$, an invertible sheaf $\mathcal{L}$, and a $\mathcal{O}_ X$-module $\mathcal{F}$ we have the graded $\Gamma _*(X, \mathcal{L})$-module
\[ \Gamma _*(X, \mathcal{L}, \mathcal{F}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}). \]
The following lemma says that, in Situation 28.28.1, we can recover a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ from this graded module. Take a look also at Constructions, Lemma 27.13.8 where we prove this lemma in the special case $X = \mathbf{P}^ n_ R$.
Lemma 28.28.3. In Situation 28.28.1. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Set $M = \Gamma _*(X, \mathcal{L}, \mathcal{F})$ as a graded $S$-module. There are isomorphisms functorial in $\mathcal{F}$ such that $M_0 \to \Gamma (\text{Proj}(S), \widetilde{M}) \to \Gamma (X, \mathcal{F})$ is the identity map.
\[ f^*\widetilde{M} \longrightarrow \mathcal{F} \]
Proof. Let $s \in S_{+}$ be homogeneous such that $X_ s$ is affine open in $X$. Recall that $\widetilde{M}|_{D_{+}(s)}$ corresponds to the $S_{(s)}$-module $M_{(s)}$, see Constructions, Lemma 27.8.4. Recall that $f^{-1}(D_{+}(s)) = X_ s$. As $X$ carries an ample invertible sheaf it is quasi-compact and quasi-separated, see Section 28.26. By Lemma 28.17.2 there is a canonical isomorphism $M_{(s)} = \Gamma _*(X, \mathcal{L}, \mathcal{F})_{(s)} \to \Gamma (X_ s, \mathcal{F})$. Since $\mathcal{F}$ is quasi-coherent this leads to a canonical isomorphism
\[ f^*\widetilde{M}|_{X_ s} \to \mathcal{F}|_{X_ s} \]
Since $\mathcal{L}$ is ample on $X$ we know that $X$ is covered by the affine opens of the form $X_ s$. Hence it suffices to prove that the displayed maps glue on overlaps. Proof of this is omitted. $\square$
Remark 28.28.4. With assumptions and notation of Lemma 28.28.3. Denote the displayed map of the lemma by $\theta _\mathcal {F}$. Note that the isomorphism $f^*\mathcal{O}_ Y(n) \to \mathcal{L}^{\otimes n}$ of Lemma 28.28.2 is just $\theta _{\mathcal{L}^{\otimes n}}$. Consider the multiplication maps see Constructions, Equation (27.10.1.5). Pull this back to $X$ and consider Here we have used the obvious identification $M(n) = \Gamma _*(X, \mathcal{L}, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. This diagram commutes. Proof omitted.
\[ \widetilde{M} \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(n) \longrightarrow \widetilde{M(n)} \]
\[ \xymatrix{ f^*\widetilde{M} \otimes _{\mathcal{O}_ X} f^*\mathcal{O}_ Y(n) \ar[r] \ar[d]_{\theta _\mathcal {F} \otimes \theta _{\mathcal{L}^{\otimes n}}} & f^*\widetilde{M(n)} \ar[d]^{\theta _{\mathcal{F} \otimes \mathcal{L}^{\otimes n}}} \\ \mathcal{F} \otimes \mathcal{L}^{\otimes n} \ar[r]^{\text{id}} & \mathcal{F} \otimes \mathcal{L}^{\otimes n} } \]
It should be possible to deduce the following lemma from Lemma 28.28.3 (or conversely) but it seems simpler to just repeat the proof.
Lemma 28.28.5. Let $S$ be a graded ring such that $X = \text{Proj}(S)$ is quasi-compact. 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, see Constructions, Section 27.10. The map of Constructions, Lemma 27.10.7 is an isomorphism. If $X$ is covered by standard opens $D_+(f)$ where $f$ has degree $1$, then the induced maps $M_ n \to \Gamma (X, \mathcal{F}(n))$ are the identity maps.
\[ \widetilde{M} \longrightarrow \mathcal{F} \]
Proof. Since $X$ is quasi-compact we can find homogeneous elements $f_1, \ldots , f_ n \in S$ of positive degrees such that $X = D_+(f_1) \cup \ldots \cup D_+(f_ n)$. Let $d$ be the least common multiple of the degrees of $f_1, \ldots , f_ n$. After replacing $f_ i$ by a power we may assume that each $f_ i$ has degree $d$. Then we see that $\mathcal{L} = \mathcal{O}_ X(d)$ is invertible, the multiplication maps $\mathcal{O}_ X(ad) \otimes \mathcal{O}_ X(bd) \to \mathcal{O}_ X((a + b)d)$ are isomorphisms, and each $f_ i$ determines a global section $s_ i$ of $\mathcal{L}$ such that $X_{s_ i} = D_+(f_ i)$, see Constructions, Lemmas 27.10.4 and 27.10.5. Thus $\Gamma (X, \mathcal{F}(ad)) = \Gamma (X, \mathcal{F} \otimes \mathcal{L}^{\otimes a})$. Recall that $\widetilde{M}|_{D_{+}(f_ i)}$ corresponds to the $S_{(f_ i)}$-module $M_{(f_ i)}$, see Constructions, Lemma 27.8.4. Since the degree of $f_ i$ is $d$, the isomorphism class of $M_{(f_ i)}$ depends only on the homogeneous summands of $M$ of degree divisible by $d$. More precisely, the isomorphism class of $M_{(f_ i)}$ depends only on the graded $\Gamma _*(X, \mathcal{L})$-module $\Gamma _*(X, \mathcal{L}, \mathcal{F})$ and the image $s_ i$ of $f_ i$ in $\Gamma _*(X, \mathcal{L})$. The scheme $X$ is quasi-compact by assumption and separated by Constructions, Lemma 27.8.8. By Lemma 28.17.2 there is a canonical isomorphism
\[ M_{(f_ i)} = \Gamma _*(X, \mathcal{L}, \mathcal{F})_{(s_ i)} \to \Gamma (X_{s_ i}, \mathcal{F}). \]
The construction of the map in Constructions, Lemma 27.10.7 then shows that it is an isomorphism over $D_+(f_ i)$ hence an isomorphism as $X$ is covered by these opens. We omit the proof of the final statement. $\square$
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