Lemma 27.2.2 (01LI)—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:

For every $U \in \mathcal{B}$ a scheme $f_ U : X_ U \to U$ over $U$.
For every $U \in \mathcal{B}$ a quasi-coherent sheaf $\mathcal{F}_ U$ over $X_ U$.
For every pair $U, V \in \mathcal{B}$ such that $V \subset U$ a morphism $\rho ^ U_ V : X_ V \to X_ U$.
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

each $\rho ^ U_ V$ induces an isomorphism $X_ V \to f_ U^{-1}(V)$ of schemes over $V$,
each $\theta ^ U_ V$ is an isomorphism,
whenever $W, V, U \in \mathcal{B}$, with $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ U_ V \circ \rho ^ V_ W$,
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 (2)

Comment #8442 by Elías Guisado on May 31, 2023 at 12:51

In 27.2.2.1, I think the middle and last arrows should be labelled as $(i_V^{-1})^*(\theta_U|_{f^{-1}(V)})$ and $(i_V^{-1})^*\theta_V^{-1}$ , respectively.

Instead of saying "by covering the intersection $U_1\cap U_2=\bigcup V_j$ by elements $V_j$ of $\mathcal{B}$ and taking" I think it would be better to write "by considering any $V\in\mathcal{B}$ such that $V\subset U_1\cap U_2$ and defining", since actually using some covering is not enough to later construct $\varphi_{12}$ .

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

After "condition (d) says exactly that this is compatible in case we have a triple of elements $W\subset V\subset U$ of $\mathcal{B}$ ," add "i.e., $(\theta')^U_W=(\theta')^V_W\circ(\theta')^U_V|_{f^{-1}(W)}$ ."

If we used the notation $\varphi^{V_j}_{12}$ for what the proof currently denotes $\varphi_{12}|_{V_j}$ , then instead of "we omit the verification that these maps do indeed glue to a $\varphi_{12}$ " one could write "since $\varphi^V_{12}|_W=\varphi^W_{12}$ for $V,W\in\mathcal{B}$ with $W\subset V\subset U_1\cap U_2$ , by Sheaves, 6.33.1, we get a $\varphi_{12}$ ."

On the other hand, I think that for 27.2.2.1 to follow, in the lemma hypotheses we would need to add "suppose $\rho_U^U=\text{id}_{X_U}$ and $\theta^U_U=\text{id}_{\mathcal{F}_U}$ for $U\in\mathcal{B}$ ." The proof of 27.2.2.1 may be added after "we get our $\mathcal{F}$ on $X$ " by adding "along with isomorphisms $\theta_U:\mathcal{F}'_U\to\mathcal{F}|_{f^{-1}(U)}$ , $U\in\mathcal{B}$ , such that $\theta_U|_{f^{-1}(U\cap V)}=\theta_V|_{f^{-1}(U\cap V)}\circ \varphi_{U,V}$ for $U,V\in\mathcal{B}$ . In particular, for $U,V\in\mathcal{B}$ with $V\subset U$ , we have $\theta_V^{-1}\circ\theta_U|_{f^{-1}(V)}=\varphi_{U,V}=(\theta')^U_V=i^*_V\theta^U_V$ , which is 27.2.2.1 (in the middle equality we used the definition of $\varphi_{U,V}$ , plus $(\theta')^V_V=\text{id}_{\mathcal{F}'_V}$ )."

Comment #8443 by Elías Guisado on May 31, 2023 at 12:58

Okay, sorry, the conditions $\rho_U^U=\text{id}_{X_U}$ and $\theta^U_U=\text{id}_{\mathcal{F}_U}$ 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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