27.2 Relative glueing

The following lemma is relevant in case we are trying to construct a scheme $X$ over $S$, and we already know how to construct the restriction of $X$ to the affine opens of $S$. The actual result is completely general and works in the setting of (locally) ringed spaces, although our proof is written in the language of schemes.

Lemma 27.2.1. Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data:

1. For every $U \in \mathcal{B}$ a scheme $f_ U : X_ U \to U$ over $U$.

2. For $U, V \in \mathcal{B}$ with $V \subset U$ a morphism $\rho ^ U_ V : X_ V \to X_ U$ over $U$.

Assume that

1. each $\rho ^ U_ V$ induces an isomorphism $X_ V \to f_ U^{-1}(V)$ of schemes over $V$,

2. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ U_ V \circ \rho ^ V_ W$.

Then there exists a morphism $f : X \to S$ of schemes and isomorphisms $i_ U : f^{-1}(U) \to X_ U$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition

$\xymatrix{ X_ V \ar[r]^{i_ V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_ U} & X_ U }$

is the morphism $\rho ^ U_ V$. Moreover $X$ is unique up to unique isomorphism over $S$.

Proof. To prove this we will use Schemes, Lemma 26.15.4. First we define a contravariant functor $F$ from the category of schemes to the category of sets. Namely, for a scheme $T$ we set

$F(T) = \left\{ \begin{matrix} (g, \{ h_ U\} _{U \in \mathcal{B}}), \ g : T \to S, \ h_ U : g^{-1}(U) \to X_ U, \\ f_ U \circ h_ U = g|_{g^{-1}(U)}, \ h_ U|_{g^{-1}(V)} = \rho ^ U_ V \circ h_ V \ \forall \ V, U \in \mathcal{B}, V \subset U \end{matrix} \right\} .$

The restriction mapping $F(T) \to F(T')$ given a morphism $T' \to T$ is just gotten by composition. For any $W \in \mathcal{B}$ we consider the subfunctor $F_ W \subset F$ consisting of those systems $(g, \{ h_ U\} )$ such that $g(T) \subset W$.

First we show $F$ satisfies the sheaf property for the Zariski topology. Suppose that $T$ is a scheme, $T = \bigcup V_ i$ is an open covering, and $\xi _ i \in F(V_ i)$ is an element such that $\xi _ i|_{V_ i \cap V_ j} = \xi _ j|_{V_ i \cap V_ j}$. Say $\xi _ i = (g_ i, \{ h_{i, U}\} )$. Then we immediately see that the morphisms $g_ i$ glue to a unique global morphism $g : T \to S$. Moreover, it is clear that $g^{-1}(U) = \bigcup g_ i^{-1}(U)$. Hence the morphisms $h_{i, U} : g_ i^{-1}(U) \to X_ U$ glue to a unique morphism $h_ U : g^{-1}(U) \to X_ U$. It is easy to verify that the system $(g, \{ h_ U\} )$ is an element of $F(T)$. Hence $F$ satisfies the sheaf property for the Zariski topology.

Next we verify that each $F_ W$, $W \in \mathcal{B}$ is representable. Namely, we claim that the transformation of functors

$F_ W \longrightarrow \mathop{\mathrm{Mor}}\nolimits (-, X_ W), \ (g, \{ h_ U\} ) \longmapsto h_ W$

is an isomorphism. To see this suppose that $T$ is a scheme and $\alpha : T \to X_ W$ is a morphism. Set $g = f_ W \circ \alpha$. For any $U \in \mathcal{B}$ such that $U \subset W$ we can define $h_ U : g^{-1}(U) \to X_ U$ be the composition $(\rho ^ W_ U)^{-1} \circ \alpha |_{g^{-1}(U)}$. This works because the image $\alpha (g^{-1}(U))$ is contained in $f_ W^{-1}(U)$ and condition (a) of the lemma. It is clear that $f_ U \circ h_ U = g|_{g^{-1}(U)}$ for such a $U$. Moreover, if also $V \in \mathcal{B}$ and $V \subset U \subset W$, then $\rho ^ U_ V \circ h_ V = h_ U|_{g^{-1}(V)}$ by property (b) of the lemma. We still have to define $h_ U$ for an arbitrary element $U \in \mathcal{B}$. Since $\mathcal{B}$ is a basis for the topology on $S$ we can find an open covering $U \cap W = \bigcup U_ i$ with $U_ i \in \mathcal{B}$. Since $g$ maps into $W$ we have $g^{-1}(U) = g^{-1}(U \cap W) = \bigcup g^{-1}(U_ i)$. Consider the morphisms $h_ i = \rho ^ U_{U_ i} \circ h_{U_ i} : g^{-1}(U_ i) \to X_ U$. It is a simple matter to use condition (b) of the lemma to prove that $h_ i|_{g^{-1}(U_ i) \cap g^{-1}(U_ j)} = h_ j|_{g^{-1}(U_ i) \cap g^{-1}(U_ j)}$. Hence these morphisms glue to give the desired morphism $h_ U : g^{-1}(U) \to X_ U$. We omit the (easy) verification that the system $(g, \{ h_ U\} )$ is an element of $F_ W(T)$ which maps to $\alpha$ under the displayed arrow above.

Next, we verify each $F_ W \subset F$ is representable by open immersions. This is clear from the definitions.

Finally we have to verify the collection $(F_ W)_{W \in \mathcal{B}}$ covers $F$. This is clear by construction and the fact that $\mathcal{B}$ is a basis for the topology of $S$.

Let $X$ be a scheme representing the functor $F$. Let $(f, \{ i_ U\} ) \in F(X)$ be a “universal family”. Since each $F_ W$ is representable by $X_ W$ (via the morphism of functors displayed above) we see that $i_ W : f^{-1}(W) \to X_ W$ is an isomorphism as desired. The lemma is proved. $\square$

Lemma 27.2.2. Let $S$ be a scheme. Let $\mathcal{B}$ be a basis for the topology of $S$. Suppose given the following data:

1. For every $U \in \mathcal{B}$ a scheme $f_ U : X_ U \to U$ over $U$.

2. For every $U \in \mathcal{B}$ a quasi-coherent sheaf $\mathcal{F}_ U$ over $X_ U$.

3. For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho ^ U_ V : X_ V \to X_ U$.

4. For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\theta ^ U_ V : (\rho ^ U_ V)^*\mathcal{F}_ U \to \mathcal{F}_ V$.

Assume that

1. each $\rho ^ U_ V$ induces an isomorphism $X_ V \to f_ U^{-1}(V)$ of schemes over $V$,

2. each $\theta ^ U_ V$ is an isomorphism,

3. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ U_ V \circ \rho ^ V_ W$,

4. whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\theta ^ U_ W = \theta ^ V_ W \circ (\rho ^ V_ W)^*\theta ^ U_ V$.

Then there exists a morphism of schemes $f : X \to S$ together with a quasi-coherent sheaf $\mathcal{F}$ on $X$ and isomorphisms $i_ U : f^{-1}(U) \to X_ U$ and $\theta _ U : i_ U^*\mathcal{F}_ U \to \mathcal{F}|_{f^{-1}(U)}$ over $U \in \mathcal{B}$ such that for $V, U \in \mathcal{B}$ with $V \subset U$ the composition

$\xymatrix{ X_ V \ar[r]^{i_ V^{-1}} & f^{-1}(V) \ar[rr]^{inclusion} & & f^{-1}(U) \ar[r]^{i_ U} & X_ U }$

is the morphism $\rho ^ U_ V$, and the composition

27.2.2.1
\begin{equation} \label{constructions-equation-glue} (\rho ^ U_ V)^*\mathcal{F}_ U = (i_ V^{-1})^*((i_ U^*\mathcal{F}_ U)|_{f^{-1}(V)}) \xrightarrow {\theta _ U|_{f^{-1}(V)}} (i_ V^{-1})^*(\mathcal{F}|_{f^{-1}(V)}) \xrightarrow {\theta _ V^{-1}} \mathcal{F}_ V \end{equation}

is equal to $\theta ^ U_ V$. Moreover $(X, \mathcal{F})$ is unique up to unique isomorphism over $S$.

Proof. By Lemma 27.2.1 we get the scheme $X$ over $S$ and the isomorphisms $i_ U$. Set $\mathcal{F}'_ U = i_ U^*\mathcal{F}_ U$ for $U \in \mathcal{B}$. This is a quasi-coherent $\mathcal{O}_{f^{-1}(U)}$-module. The maps

$\mathcal{F}'_ U|_{f^{-1}(V)} = i_ U^*\mathcal{F}_ U|_{f^{-1}(V)} = i_ V^*(\rho ^ U_ V)^*\mathcal{F}_ U \xrightarrow {i_ V^*\theta ^ U_ V} i_ V^*\mathcal{F}_ V = \mathcal{F}'_ V$

define isomorphisms $(\theta ')^ U_ V : \mathcal{F}'_ U|_{f^{-1}(V)} \to \mathcal{F}'_ V$ whenever $V \subset U$ are elements of $\mathcal{B}$. Condition (d) says exactly that this is compatible in case we have a triple of elements $W \subset V \subset U$ of $\mathcal{B}$. This allows us to get well defined isomorphisms

$\varphi _{12} : \mathcal{F}'_{U_1}|_{f^{-1}(U_1 \cap U_2)} \longrightarrow \mathcal{F}'_{U_2}|_{f^{-1}(U_1 \cap U_2)}$

whenever $U_1, U_2 \in \mathcal{B}$ by covering the intersection $U_1 \cap U_2 = \bigcup V_ j$ by elements $V_ j$ of $\mathcal{B}$ and taking

$\varphi _{12}|_{V_ j} = \left((\theta ')^{U_2}_{V_ j}\right)^{-1} \circ (\theta ')^{U_1}_{V_ j}.$

We omit the verification that these maps do indeed glue to a $\varphi _{12}$ and we omit the verification of the cocycle condition of a glueing datum for sheaves (as in Sheaves, Section 6.33). By Sheaves, Lemma 6.33.2 we get our $\mathcal{F}$ on $X$. We omit the verification of (27.2.2.1). $\square$

Remark 27.2.3. There is a functoriality property for the constructions explained in Lemmas 27.2.1 and 27.2.2. Namely, suppose given two collections of data $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $(g_ U : Y_ U \to U, \sigma ^ U_ V)$ as in Lemma 27.2.1. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$. Functoriality means that this gives rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, where $f : X \to S$ is obtained from the datum $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $g : Y \to S$ is obtained from the datum $(g_ U : Y_ U \to U, \sigma ^ U_ V)$.

Similarly, suppose given two collections of data $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$ as in Lemma 27.2.2. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$, and a morphism $\tau _ U : h_ U^*\mathcal{G}_ U \to \mathcal{F}_ U$ compatible with the maps $\theta ^ U_ V$ and $\eta ^ U_ V$. Functoriality means that these give rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, and a morphism $h^*\mathcal{G} \to \mathcal{F}$ restricting back to the maps $h_ U$ where $(f : X \to S, \mathcal{F})$ is obtained from the datum $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and where $(g : Y \to S, \mathcal{G})$ is obtained from the datum $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$.

We omit the verifications and we omit a suitable formulation of “equivalence of categories” between relative glueing data and relative objects.

Comment #2540 by Noah Olander on

In the last sentence of the statement of lemma 26.2.1 you have written "for $V \subset U \subset S$ affine open," when you should have written "for $V, U \in \mathcal{B}$ such that $V \subset U$"

Comment #2548 by Dario Weißmann on

In 26.2.1 and 26.2.2: I don't think the $X$ constructed is unique, just unique up to unique isomorphism. Noah Olander's comment should also apply to 26.2.2.

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