Processing math: 100%

The Stacks project

27.12 Morphisms into Proj

Let S be a graded ring. Let X = \text{Proj}(S) be the homogeneous spectrum of S. Let d \geq 1 be an integer. Consider the open subscheme

27.12.0.1
\begin{equation} \label{constructions-equation-Ud} U_ d = \bigcup \nolimits _{f \in S_ d} D_{+}(f) \quad \subset \quad X = \text{Proj}(S) \end{equation}

Note that d | d' \Rightarrow U_ d \subset U_{d'} and X = \bigcup _ d U_ d. Neither X nor U_ d need be quasi-compact, see Algebra, Lemma 10.57.3. Let us write \mathcal{O}_{U_ d}(n) = \mathcal{O}_ X(n)|_{U_ d}. By Lemma 27.10.2 we know that \mathcal{O}_{U_ d}(nd), n \in \mathbf{Z} is an invertible \mathcal{O}_{U_ d}-module and that all the multiplication maps \mathcal{O}_{U_ d}(nd) \otimes _{\mathcal{O}_{U_ d}} \mathcal{O}_{U_ d}(m) \to \mathcal{O}_{U_ d}(nd + m) of (27.10.1.1) are isomorphisms. In particular we have \mathcal{O}_{U_ d}(nd) \cong \mathcal{O}_{U_ d}(d)^{\otimes n}. The graded ring map (27.10.1.3) on global sections combined with restriction to U_ d give a homomorphism of graded rings

27.12.0.2
\begin{equation} \label{constructions-equation-psi-d} \psi ^ d : S^{(d)} \longrightarrow \Gamma _*(U_ d, \mathcal{O}_{U_ d}(d)). \end{equation}

For the notation S^{(d)}, see Algebra, Section 10.56. For the notation \Gamma _* see Modules, Definition 17.25.7. Moreover, since U_ d is covered by the opens D_{+}(f), f \in S_ d we see that \mathcal{O}_{U_ d}(d) is globally generated by the sections in the image of \psi ^ d_1 : S^{(d)}_1 = S_ d \to \Gamma (U_ d, \mathcal{O}_{U_ d}(d)), see Modules, Definition 17.4.1.

Let Y be a scheme, and let \varphi : Y \to X be a morphism of schemes. Assume the image \varphi (Y) is contained in the open subscheme U_ d of X. By the discussion following Modules, Definition 17.25.7 we obtain a homomorphism of graded rings

\Gamma _*(U_ d, \mathcal{O}_{U_ d}(d)) \longrightarrow \Gamma _*(Y, \varphi ^*\mathcal{O}_ X(d)).

The composition of this and \psi ^ d gives a graded ring homomorphism

27.12.0.3
\begin{equation} \label{constructions-equation-psi-phi-d} \psi _\varphi ^ d : S^{(d)} \longrightarrow \Gamma _*(Y, \varphi ^*\mathcal{O}_ X(d)) \end{equation}

which has the property that the invertible sheaf \varphi ^*\mathcal{O}_ X(d) is globally generated by the sections in the image of (S^{(d)})_1 = S_ d \to \Gamma (Y, \varphi ^*\mathcal{O}_ X(d)).

Lemma 27.12.1. Let S be a graded ring, and X = \text{Proj}(S). Let d \geq 1 and U_ d \subset X as above. Let Y be a scheme. Let \mathcal{L} be an invertible sheaf on Y. Let \psi : S^{(d)} \to \Gamma _*(Y, \mathcal{L}) be a graded ring homomorphism such that \mathcal{L} is generated by the sections in the image of \psi |_{S_ d} : S_ d \to \Gamma (Y, \mathcal{L}). Then there exist a morphism \varphi : Y \to X such that \varphi (Y) \subset U_ d and an isomorphism \alpha : \varphi ^*\mathcal{O}_{U_ d}(d) \to \mathcal{L} such that \psi _\varphi ^ d agrees with \psi via \alpha :

\xymatrix{ \Gamma _*(Y, \mathcal{L}) & \Gamma _*(Y, \varphi ^*\mathcal{O}_{U_ d}(d)) \ar[l]^-\alpha & \Gamma _*(U_ d, \mathcal{O}_{U_ d}(d)) \ar[l]^-{\varphi ^*} \\ S^{(d)} \ar[u]^\psi & & S^{(d)} \ar[u]^{\psi ^ d} \ar[ul]^{\psi ^ d_\varphi } \ar[ll]_{\text{id}} }

commutes. Moreover, the pair (\varphi , \alpha ) is unique.

Proof. Pick f \in S_ d. Denote s = \psi (f) \in \Gamma (Y, \mathcal{L}). On the open set Y_ s where s does not vanish multiplication by s induces an isomorphism \mathcal{O}_{Y_ s} \to \mathcal{L}|_{Y_ s}, see Modules, Lemma 17.25.10. We will denote the inverse of this map x \mapsto x/s, and similarly for powers of \mathcal{L}. Using this we define a ring map \psi _{(f)} : S_{(f)} \to \Gamma (Y_ s, \mathcal{O}) by mapping the fraction a/f^ n to \psi (a)/s^ n. By Schemes, Lemma 26.6.4 this corresponds to a morphism \varphi _ f : Y_ s \to \mathop{\mathrm{Spec}}(S_{(f)}) = D_{+}(f). We also introduce the isomorphism \alpha _ f : \varphi _ f^*\mathcal{O}_{D_{+}(f)}(d) \to \mathcal{L}|_{Y_ s} which maps the pullback of the trivializing section f over D_{+}(f) to the trivializing section s over Y_ s. With this choice the commutativity of the diagram in the lemma holds with Y replaced by Y_ s, \varphi replaced by \varphi _ f, and \alpha replaced by \alpha _ f; verification omitted.

Suppose that f' \in S_ d is a second element, and denote s' = \psi (f') \in \Gamma (Y, \mathcal{L}). Then Y_ s \cap Y_{s'} = Y_{ss'} and similarly D_{+}(f) \cap D_{+}(f') = D_{+}(ff'). In Lemma 27.10.6 we saw that D_{+}(f') \cap D_{+}(f) is the same as the set of points of D_{+}(f) where the section of \mathcal{O}_ X(d) defined by f' does not vanish. Hence \varphi _ f^{-1}(D_{+}(f') \cap D_{+}(f)) = Y_ s \cap Y_{s'} = \varphi _{f'}^{-1}(D_{+}(f') \cap D_{+}(f)). On D_{+}(f) \cap D_{+}(f') the fraction f/f' is an invertible section of the structure sheaf with inverse f'/f. Note that \psi _{(f')}(f/f') = \psi (f)/s' = s/s' and \psi _{(f)}(f'/f) = \psi (f')/s = s'/s. We claim there is a unique ring map S_{(ff')} \to \Gamma (Y_{ss'}, \mathcal{O}) making the following diagram commute

\xymatrix{ \Gamma (Y_ s, \mathcal{O}) \ar[r] & \Gamma (Y_{ss'}, \mathcal{O}) & \Gamma (Y_{s, '} \mathcal{O}) \ar[l]\\ S_{(f)} \ar[r] \ar[u]^{\psi _{(f)}} & S_{(ff')} \ar[u] & S_{(f')} \ar[l] \ar[u]^{\psi _{(f')}} }

It exists because we may use the rule x/(ff')^ n \mapsto \psi (x)/(ss')^ n, which “works” by the formulas above. Uniqueness follows as \text{Proj}(S) is separated, see Lemma 27.8.8 and its proof. This shows that the morphisms \varphi _ f and \varphi _{f'} agree over Y_ s \cap Y_{s'}. The restrictions of \alpha _ f and \alpha _{f'} agree over Y_ s \cap Y_{s'} because the regular functions s/s' and \psi _{(f')}(f) agree. This proves that the morphisms \psi _ f glue to a global morphism from Y into U_ d \subset X, and that the maps \alpha _ f glue to an isomorphism satisfying the conditions of the lemma.

We still have to show the pair (\varphi , \alpha ) is unique. Suppose (\varphi ', \alpha ') is a second such pair. Let f \in S_ d. By the commutativity of the diagrams in the lemma we have that the inverse images of D_{+}(f) under both \varphi and \varphi ' are equal to Y_{\psi (f)}. Since the opens D_{+}(f) are a basis for the topology on X, and since X is a sober topological space (see Schemes, Lemma 26.11.1) this means the maps \varphi and \varphi ' are the same on underlying topological spaces. Let us use s = \psi (f) to trivialize the invertible sheaf \mathcal{L} over Y_{\psi (f)}. By the commutativity of the diagrams we have that \alpha ^{\otimes n}(\psi ^ d_{\varphi }(x)) = \psi (x) = (\alpha ')^{\otimes n}(\psi ^ d_{\varphi '}(x)) for all x \in S_{nd}. By construction of \psi ^ d_{\varphi } and \psi ^ d_{\varphi '} we have \psi ^ d_{\varphi }(x) = \varphi ^\sharp (x/f^ n) \psi ^ d_{\varphi }(f^ n) over Y_{\psi (f)}, and similarly for \psi ^ d_{\varphi '}. By the commutativity of the diagrams of the lemma we deduce that \varphi ^\sharp (x/f^ n) = (\varphi ')^\sharp (x/f^ n). This proves that \varphi and \varphi ' induce the same morphism from Y_{\psi (f)} into the affine scheme D_{+}(f) = \mathop{\mathrm{Spec}}(S_{(f)}). Hence \varphi and \varphi ' are the same as morphisms. Finally, it remains to show that the commutativity of the diagram of the lemma singles out, given \varphi , a unique \alpha . We omit the verification. \square

We continue the discussion from above the lemma. Let S be a graded ring. Let Y be a scheme. We will consider triples (d, \mathcal{L}, \psi ) where

  1. d \geq 1 is an integer,

  2. \mathcal{L} is an invertible \mathcal{O}_ Y-module, and

  3. \psi : S^{(d)} \to \Gamma _*(Y, \mathcal{L}) is a graded ring homomorphism such that \mathcal{L} is generated by the global sections \psi (f), with f \in S_ d.

Given a morphism h : Y' \to Y and a triple (d, \mathcal{L}, \psi ) over Y we can pull it back to the triple (d, h^*\mathcal{L}, h^* \circ \psi ). Given two triples (d, \mathcal{L}, \psi ) and (d, \mathcal{L}', \psi ') with the same integer d we say they are strictly equivalent if there exists an isomorphism \beta : \mathcal{L} \to \mathcal{L}' such that \beta \circ \psi = \psi ' as graded ring maps S^{(d)} \to \Gamma _*(Y, \mathcal{L}').

For each integer d \geq 1 we define

\begin{eqnarray*} F_ d : \mathit{Sch}^{opp} & \longrightarrow & \textit{Sets}, \\ Y & \longmapsto & \{ \text{strict equivalence classes of triples } (d, \mathcal{L}, \psi ) \text{ as above}\} \end{eqnarray*}

with pullbacks as defined above.

Lemma 27.12.2. Let S be a graded ring. Let X = \text{Proj}(S). The open subscheme U_ d \subset X (27.12.0.1) represents the functor F_ d and the triple (d, \mathcal{O}_{U_ d}(d), \psi ^ d) defined above is the universal family (see Schemes, Section 26.15).

Proof. This is a reformulation of Lemma 27.12.1 \square

Lemma 27.12.3. Let S be a graded ring generated as an S_0-algebra by the elements of S_1. In this case the scheme X = \text{Proj}(S) represents the functor which associates to a scheme Y the set of pairs (\mathcal{L}, \psi ), where

  1. \mathcal{L} is an invertible \mathcal{O}_ Y-module, and

  2. \psi : S \to \Gamma _*(Y, \mathcal{L}) is a graded ring homomorphism such that \mathcal{L} is generated by the global sections \psi (f), with f \in S_1

up to strict equivalence as above.

Proof. Under the assumptions of the lemma we have X = U_1 and the lemma is a reformulation of Lemma 27.12.2 above. \square

We end this section with a discussion of a functor corresponding to \text{Proj}(S) for a general graded ring S. We advise the reader to skip the rest of this section.

Fix an arbitrary graded ring S. Let T be a scheme. We will say two triples (d, \mathcal{L}, \psi ) and (d', \mathcal{L}', \psi ') over T with possibly different integers d, d' are equivalent if there exists an isomorphism \beta : \mathcal{L}^{\otimes d'} \to (\mathcal{L}')^{\otimes d} of invertible sheaves over T such that \beta \circ \psi |_{S^{(dd')}} and \psi '|_{S^{(dd')}} agree as graded ring maps S^{(dd')} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes dd'}).

Lemma 27.12.4. Let S be a graded ring. Set X = \text{Proj}(S). Let T be a scheme. Let (d, \mathcal{L}, \psi ) and (d', \mathcal{L}', \psi ') be two triples over T. The following are equivalent:

  1. Let n = \text{lcm}(d, d'). Write n = ad = a'd'. There exists an isomorphism \beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'} with the property that \beta \circ \psi |_{S^{(n)}} and \psi '|_{S^{(n)}} agree as graded ring maps S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n}).

  2. The triples (d, \mathcal{L}, \psi ) and (d', \mathcal{L}', \psi ') are equivalent.

  3. For some positive integer n = ad = a'd' there exists an isomorphism \beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'} with the property that \beta \circ \psi |_{S^{(n)}} and \psi '|_{S^{(n)}} agree as graded ring maps S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n}).

  4. The morphisms \varphi : T \to X and \varphi ' : T \to X associated to (d, \mathcal{L}, \psi ) and (d', \mathcal{L}', \psi ') are equal.

Proof. Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Also (3) implies (4) by the uniqueness statement in Lemma 27.12.1. Thus we have to prove that (4) implies (1). Assume (4), in other words \varphi = \varphi '. Note that this implies that we may write \mathcal{L} = \varphi ^*\mathcal{O}_ X(d) and \mathcal{L}' = \varphi ^*\mathcal{O}_ X(d'). Moreover, via these identifications we have that the graded ring maps \psi and \psi ' correspond to the restriction of the canonical graded ring map

S \longrightarrow \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{O}_ X(n))

to S^{(d)} and S^{(d')} composed with pullback by \varphi (by Lemma 27.12.1 again). Hence taking \beta to be the isomorphism

(\varphi ^*\mathcal{O}_ X(d))^{\otimes a} = \varphi ^*\mathcal{O}_ X(n) = (\varphi ^*\mathcal{O}_ X(d'))^{\otimes a'}

works. \square

Let S be a graded ring. Let X = \text{Proj}(S). Over the open subscheme scheme U_ d \subset X = \text{Proj}(S) (27.12.0.1) we have the triple (d, \mathcal{O}_{U_ d}(d), \psi ^ d). Clearly, if d | d' the triples (d, \mathcal{O}_{U_ d}(d), \psi ^ d) and (d', \mathcal{O}_{U_{d'}}(d'), \psi ^{d'}) are equivalent when restricted to the open U_ d (which is a subset of U_{d'}). This, combined with Lemma 27.12.1 shows that morphisms Y \to X correspond roughly to equivalence classes of triples over Y. This is not quite true since if Y is not quasi-compact, then there may not be a single triple which works. Thus we have to be slightly careful in defining the corresponding functor.

Here is one possible way to do this. Suppose d' = ad. Consider the transformation of functors F_ d \to F_{d'} which assigns to the triple (d, \mathcal{L}, \psi ) over T the triple (d', \mathcal{L}^{\otimes a}, \psi |_{S^{(d')}}). One of the implications of Lemma 27.12.4 is that the transformation F_ d \to F_{d'} is injective! For a quasi-compact scheme T we define

F(T) = \bigcup \nolimits _{d \in \mathbf{N}} F_ d(T)

with transition maps as explained above. This clearly defines a contravariant functor on the category of quasi-compact schemes with values in sets. For a general scheme T we define

F(T) = \mathop{\mathrm{lim}}\nolimits _{V \subset T\text{ quasi-compact open}} F(V).

In other words, an element \xi of F(T) corresponds to a compatible system of choices of elements \xi _ V \in F(V) where V ranges over the quasi-compact opens of T. We omit the definition of the pullback map F(T) \to F(T') for a morphism T' \to T of schemes. Thus we have defined our functor

\begin{eqnarray*} F : \mathit{Sch}^{opp} & \longrightarrow & \textit{Sets} \end{eqnarray*}

Lemma 27.12.5. Let S be a graded ring. Let X = \text{Proj}(S). The functor F defined above is representable by the scheme X.

Proof. We have seen above that the functor F_ d corresponds to the open subscheme U_ d \subset X. Moreover the transformation of functors F_ d \to F_{d'} (if d | d') defined above corresponds to the inclusion morphism U_ d \to U_{d'} (see discussion above). Hence to show that F is represented by X it suffices to show that T \to X for a quasi-compact scheme T ends up in some U_ d, and that for a general scheme T we have

\mathop{\mathrm{Mor}}\nolimits (T, X) = \mathop{\mathrm{lim}}\nolimits _{V \subset T\text{ quasi-compact open}} \mathop{\mathrm{Mor}}\nolimits (V, X).

These verifications are omitted. \square


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.