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 (3)
Comment #8442 by Elías Guisado on
Comment #8443 by Elías Guisado on
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