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.

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 $\bigoplus f^*\mathcal{O}_ Y(n) \to \bigoplus \mathcal{L}^{\otimes n}$.

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 17.24.7 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

$f^*\widetilde{M} \longrightarrow \mathcal{F}$

functorial in $\mathcal{F}$ such that $M_0 \to \Gamma (\text{Proj}(S), \widetilde{M}) \to \Gamma (X, \mathcal{F})$ is the identity map.

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

$\widetilde{M} \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(n) \longrightarrow \widetilde{M(n)}$

see Constructions, Equation (27.10.1.5). Pull this back to $X$ and consider

$\xymatrix{ f^*\widetilde{M} \otimes _{\mathcal{O}_ X} f^*\mathcal{O}_ Y(n) \ar[r] \ar[d]_{\theta _\mathcal {F} \otimes \theta _{\mathcal{L}^{\otimes n}}} & f^*\widetilde{M(n)} \ar[d]^{\theta _{\mathcal{F} \otimes \mathcal{L}^{\otimes n}}} \\ \mathcal{F} \otimes \mathcal{L}^{\otimes n} \ar[r]^{\text{id}} & \mathcal{F} \otimes \mathcal{L}^{\otimes n} }$

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.

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

$\widetilde{M} \longrightarrow \mathcal{F}$

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.

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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