The Stacks project

Proof. We are going to use Schemes, Lemma 26.15.4.

First we check that $F_ d$ satisfies the sheaf property for the Zariski topology. Namely, suppose that $T$ is a scheme, that $T = \bigcup _{i \in I} U_ i$ is an open covering, and that $(d, f_ i, \mathcal{L}_ i, \psi _ i) \in F_ d(U_ i)$ such that $(d, f_ i, \mathcal{L}_ i, \psi _ i)|_{U_ i \cap U_ j}$ and $(d, f_ j, \mathcal{L}_ j, \psi _ j)|_{U_ i \cap U_ j}$ are strictly equivalent. This implies that the morphisms $f_ i : U_ i \to S$ glue to a morphism of schemes $f : T \to S$ such that $f|_{I_ i} = f_ i$, see Schemes, Section 26.14. Thus $f_ i^*\mathcal{A}^{(d)} = f^*\mathcal{A}^{(d)}|_{U_ i}$. It also implies there exist isomorphisms $\beta _{ij} : \mathcal{L}_ i|_{U_ i \cap U_ j} \to \mathcal{L}_ j|_{U_ i \cap U_ j}$ such that $\beta _{ij} \circ \psi _ i = \psi _ j$ on $U_ i \cap U_ j$. Note that the isomorphisms $\beta _{ij}$ are uniquely determined by this requirement because the maps $f_ i^*\mathcal{A}_ d \to \mathcal{L}_ i$ are surjective. In particular we see that $\beta _{jk} \circ \beta _{ij} = \beta _{ik}$ on $U_ i \cap U_ j \cap U_ k$. Hence by Sheaves, Section 6.33 the invertible sheaves $\mathcal{L}_ i$ glue to an invertible $\mathcal{O}_ T$-module $\mathcal{L}$ and the morphisms $\psi _ i$ glue to morphism of $\mathcal{O}_ T$-algebras $\psi : f^*\mathcal{A}^{(d)} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}$. This proves that $F_ d$ satisfies the sheaf condition with respect to the Zariski topology.

Let $S = \bigcup _{i \in I} U_ i$ be an affine open covering. Let $F_{d, i} \subset F_ d$ be the subfunctor consisting of those pairs $(f : T \to S, \varphi )$ such that $f(T) \subset U_ i$.

We have to show each $F_{d, i}$ is representable. This is the case because $F_{d, i}$ is identified with the functor associated to $U_ i$ equipped with the quasi-coherent graded $\mathcal{O}_{U_ i}$-algebra $\mathcal{A}|_{U_ i}$ by Lemma 27.16.1. Thus the result follows from Lemma 27.16.2.

Next we show that $F_{d, i} \subset F_ d$ is representable by open immersions. Let $(f : T \to S, \varphi ) \in F_ d(T)$. Consider $V_ i = f^{-1}(U_ i)$. It follows from the definition of $F_{d, i}$ that given $a : T' \to T$ we gave $a^*(f, \varphi ) \in F_{d, i}(T')$ if and only if $a(T') \subset V_ i$. This is what we were required to show.

Finally, we have to show that the collection $(F_{d, i})_{i \in I}$ covers $F_ d$. Let $(f : T \to S, \varphi ) \in F_ d(T)$. Consider $V_ i = f^{-1}(U_ i)$. Since $S = \bigcup _{i \in I} U_ i$ is an open covering of $S$ we see that $T = \bigcup _{i \in I} V_ i$ is an open covering of $T$. Moreover $(f, \varphi )|_{V_ i} \in F_{d, i}(V_ i)$. This finishes the proof of the lemma. $\square$

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