The Stacks project

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). $\square$


Comments (3)

Comment #8442 by on

In 27.2.2.1, I think the middle and last arrows should be labelled as and , respectively.

Instead of saying "by covering the intersection by elements of and taking" I think it would be better to write "by considering any such that and defining", since actually using some covering is not enough to later construct .

To give just the minimum amount of hints to avoid the first use of "we omit," one could:

  1. After "condition (d) says exactly that this is compatible in case we have a triple of elements of ," add "i.e., ."
  2. If we used the notation for what the proof currently denotes , then instead of "we omit the verification that these maps do indeed glue to a " one could write "since for with , by Sheaves, 6.33.1, we get a ."

On the other hand, I think that for 27.2.2.1 to follow, in the lemma hypotheses we would need to add "suppose and for ." The proof of 27.2.2.1 may be added after "we get our on " by adding "along with isomorphisms , , such that for . In particular, for with , we have , which is 27.2.2.1 (in the middle equality we used the definition of , plus )."

Comment #8443 by on

Okay, sorry, the conditions and already follow from the current hypotheses (maybe one could mention this?) .

There are also:

  • 3 comment(s) on Section 27.2: Relative glueing

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