The Stacks project

Proof. A quadruple $(d, f' : T \to S', \mathcal{L}', \psi ' : (f')^*(\mathcal{A}')^{(d)} \to \bigoplus _{n \geq 0} (\mathcal{L}')^{\otimes n})$ is the same as a quadruple $(d, f, \mathcal{L}, \psi : f^*\mathcal{A}^{(d)} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n})$ together with a factorization of $f$ as $f = g \circ f'$. Namely, the correspondence is $f = g \circ f'$, $\mathcal{L} = \mathcal{L}'$ and $\psi = \psi '$ via the identifications $(f')^*(\mathcal{A}')^{(d)} = (f')^*g^*(\mathcal{A}^{(d)}) = f^*\mathcal{A}^{(d)}$. Hence the lemma. $\square$

Proof. Write $S = \mathop{\mathrm{Spec}}(R)$ and $A = \Gamma (S, \mathcal{A})$. Then $A$ is a graded $R$-algebra and $\mathcal{A} = \widetilde A$. To prove the lemma we have to identify the functor $F_ d$ with the functor $F_ d^{triples}$ of triples defined in Section 27.12.

Let $(d, f : T \to S, \mathcal{L}, \psi )$ be a quadruple. We may think of $\psi $ as a $\mathcal{O}_ S$-module map $\mathcal{A}^{(d)} \to \bigoplus _{n \geq 0} f_*\mathcal{L}^{\otimes n}$. Since $\mathcal{A}^{(d)}$ is quasi-coherent this is the same thing as an $R$-linear homomorphism of graded rings $A^{(d)} \to \Gamma (S, \bigoplus _{n \geq 0} f_*\mathcal{L}^{\otimes n})$. Clearly, $\Gamma (S, \bigoplus _{n \geq 0} f_*\mathcal{L}^{\otimes n}) = \Gamma _*(T, \mathcal{L})$. Thus we may associate to the quadruple the triple $(d, \mathcal{L}, \psi )$.

Conversely, let $(d, \mathcal{L}, \psi )$ be a triple. The composition $R \to A_0 \to \Gamma (T, \mathcal{O}_ T)$ determines a morphism $f : T \to S = \mathop{\mathrm{Spec}}(R)$, see Schemes, Lemma 26.6.4. With this choice of $f$ the map $A^{(d)} \to \Gamma (S, \bigoplus _{n \geq 0} f_*\mathcal{L}^{\otimes n})$ is $R$-linear, and hence corresponds to a $\psi $ which we can use for a quadruple $(d, f : T \to S, \mathcal{L}, \psi )$. We omit the verification that this establishes an isomorphism of functors $F_ d = F_ d^{triples}$. $\square$

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. $\square$

Proof. Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Assume (3) for some integer $m = ad = a'd'$. Let $m_0 = \text{lcm}(d, d')$ and write it as $m_0 = a_0d = a'_0d'$. We are given an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property described in (3). We want to find an isomorphism $\beta _0 : \mathcal{L}^{\otimes a_0} \to (\mathcal{L}')^{\otimes a'_0}$ having that property as well. Since by assumption the maps $\psi : f^*\mathcal{A}_ d \to \mathcal{L}$ and $\psi ' : (f')^*\mathcal{A}_{d'} \to \mathcal{L}'$ are surjective the same is true for the maps $\psi : f^*\mathcal{A}_{m_0} \to \mathcal{L}^{\otimes a_0}$ and $\psi ' : (f')^*\mathcal{A}_{m_0} \to (\mathcal{L}')^{\otimes a_0}$. Hence if $\beta _0$ exists it is uniquely determined by the condition that $\beta _0 \circ \psi = \psi '$. This means that we may work locally on $T$. Hence we may assume that $f = f' : T \to S$ maps into an affine open, in other words we may assume that $S$ is affine. In this case the result follows from the corresponding result for triples (see Lemma 27.12.4) and the fact that triples and quadruples correspond in the affine base case (see proof of Lemma 27.16.2). $\square$

Proof. Let $U_ d \to S$ be the scheme representing the functor $F_ d$ defined above. Let $\mathcal{L}_ d$, $\psi ^ d : \pi _ d^*\mathcal{A}^{(d)} \to \bigoplus _{n \geq 0} \mathcal{L}_ d^{\otimes n}$ be the universal object. If $d | d'$, then we may consider the quadruple $(d', \pi _ d, \mathcal{L}_ d^{\otimes d'/d}, \psi ^ d|_{\mathcal{A}^{(d')}})$ which determines a canonical morphism $U_ d \to U_{d'}$ over $S$. By construction this morphism corresponds to the transformation of functors $F_ d \to F_{d'}$ defined above.

For every affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ setting $A = \Gamma (V, \mathcal{A})$ we have a canonical identification of the base change $U_{d, V}$ with the corresponding open subscheme of $\text{Proj}(A)$, see Lemma 27.16.2. Moreover, the morphisms $U_{d, V} \to U_{d', V}$ constructed above correspond to the inclusions of opens in $\text{Proj}(A)$. Thus we conclude that $U_ d \to U_{d'}$ is an open immersion.

This allows us to construct $X$ by glueing the schemes $U_ d$ along the open immersions $U_ d \to U_{d'}$. Technically, it is convenient to choose a sequence $d_1 | d_2 | d_3 | \ldots $ such that every positive integer divides one of the $d_ i$ and to simply take $X = \bigcup U_{d_ i}$ using the open immersions above. It is then a simple matter to prove that $X$ represents the functor $F$. $\square$

Proof. Let $X$ be the scheme representing the functor $F$. Note that $X$ is a scheme over $S$ since the functor $F$ comes equipped with a natural transformation $F \to h_ S$. Write $Y = \underline{\text{Proj}}_ S(\mathcal{A})$. We have to show that $X \cong Y$ as $S$-schemes. We give two arguments.

The first argument uses the construction of $X$ as the union of the schemes $U_ d$ representing $F_ d$ in the proof of Lemma 27.16.5. Over each affine open of $S$ we can identify $X$ with the homogeneous spectrum of the sections of $\mathcal{A}$ over that open, since this was true for the opens $U_ d$. Moreover, these identifications are compatible with further restrictions to smaller affine opens. On the other hand, $Y$ was constructed by glueing these homogeneous spectra. Hence we can glue these isomorphisms to an isomorphism between $X$ and $\underline{\text{Proj}}_ S(\mathcal{A})$ as desired. Details omitted.

Here is the second argument. Lemma 27.15.5 shows that there exists a morphism of graded algebras

over $Y$ which on sections over affine opens of $S$ agrees with (27.10.1.3). Hence for every $y \in Y$ there exists an open neighbourhood $V \subset Y$ of $y$ and an integer $d \geq 1$ such that for $d | n$ the sheaf $\mathcal{O}_ Y(n)|_ V$ is invertible and the multiplication maps $\mathcal{O}_ Y(n)|_ V \otimes _{\mathcal{O}_ V} \mathcal{O}_ Y(m)|_ V \to \mathcal{O}_ Y(n + m)|_ V$ are isomorphisms. Thus $\psi $ restricted to the sheaf $\pi ^*\mathcal{A}^{(d)}|_ V$ gives an element of $F_ d(V)$. Since the opens $V$ cover $Y$ we see “$\psi $” gives rise to an element of $F(Y)$. Hence a canonical morphism $Y \to X$ over $S$. Because this construction is completely canonical to see that it is an isomorphism we may work locally on $S$. Hence we reduce to the case $S$ affine where the result is clear. $\square$

Proof. Note that the quadruple $(f : T \to S, d, \mathcal{L}, \psi )$ defines an element of $F_ d(T)$. Let $U_ d \subset \underline{\text{Proj}}_ S(\mathcal{A})$ be the locus where the sheaf $\mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(d)$ is invertible and generated by the image of $\psi _{univ} : \pi ^*\mathcal{A}_ d \to \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(d)$. Recall that $U_ d$ represents the functor $F_ d$, see the proof of Lemma 27.16.5. Hence the result will follow if we can show the quadruple $(U_ d \to S, d, \mathcal{O}_{U_ d}(d), \psi _{univ}|_{\mathcal{A}^{(d)}})$ is the universal family, i.e., the representing object in $F_ d(U_ d)$. We may do this after restricting to an affine open of $S$ because (a) the formation of the functors $F_ d$ commutes with base change (see Lemma 27.16.1), and (b) the pair $(\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ S(\mathcal{A})}(n), \psi _{univ})$ is constructed by glueing over affine opens in $S$ (see Lemma 27.15.5). Hence we may assume that $S$ is affine. In this case the functor of quadruples $F_ d$ and the functor of triples $F_ d$ agree (see proof of Lemma 27.16.2) and moreover Lemma 27.12.2 shows that $(d, \mathcal{O}_{U_ d}(d), \psi ^ d)$ is the universal triple over $U_ d$. Going backwards through the identifications in the proof of Lemma 27.16.2 shows that $(U_ d \to S, d, \mathcal{O}_{U_ d}(d), \psi _{univ}|_{\mathcal{A}^{(d)}})$ is the universal quadruple as desired. $\square$

Proof. To prove a morphism is separated we may work locally on the base, see Schemes, Section 26.21. By construction $\underline{\text{Proj}}_ S(\mathcal{A})$ is over any affine $U \subset S$ isomorphic to $\text{Proj}(A)$ with $A = \mathcal{A}(U)$. By Lemma 27.8.8 we see that $\text{Proj}(A)$ is separated. Hence $\text{Proj}(A) \to U$ is separated (see Schemes, Lemma 26.21.13) as desired. $\square$

Proof. This follows from Lemma 27.16.1 and the construction of $\underline{\text{Proj}}_ S(\mathcal{A})$ in Lemma 27.16.5 as the union of the schemes $U_ d$ representing the functors $F_ d$. In terms of the construction of relative Proj via glueing this isomorphism is given by the isomorphisms constructed in Lemma 27.11.6 which provides us with the isomorphism $\theta $. Some details omitted. $\square$

Proof. Under the assumptions of the lemma the sheaves $\mathcal{O}_{\underline{\text{Proj}}(\mathcal{A})}(n)$ are invertible and the multiplication maps isomorphisms by Lemma 27.16.5 and Lemma 27.12.3 over affine opens of $S$. Thus $X$ actually represents the functor $F_1$, see proof of Lemma 27.16.5. $\square$