Lemma 33.38.8. Let X be a scheme. Let Z_1, \ldots , Z_ n \subset X be closed subschemes. Let \mathcal{L}_ i be an invertible sheaf on Z_ i. Assume that
X is reduced,
X = \bigcup Z_ i set theoretically, and
Z_ i \cap Z_ j is a discrete topological space for i \not= j.
Then there exists an invertible sheaf \mathcal{L} on X whose restriction to Z_ i is \mathcal{L}_ i. Moreover, if we are given sections s_ i \in \Gamma (Z_ i, \mathcal{L}_ i) which are nonvanishing at the points of Z_ i \cap Z_ j, then we can choose \mathcal{L} such that there exists a s \in \Gamma (X, \mathcal{L}) with s|_{Z_ i} = s_ i for all i.
Proof.
The existence of \mathcal{L} can be deduced from Lemma 33.38.7 but we will also give a direct proof and we will use the direct proof to see the statement about sections is true. Set T = \bigcup _{i \not= j} Z_ i \cap Z_ j. As X is reduced we have
X \setminus T = \bigcup (Z_ i \setminus T)
as schemes. Assumption (3) implies T is a discrete subset of X. Thus for each t \in T we can find an open U_ t \subset X with t \in U_ t but t' \not\in U_ t for t' \in T, t' \not= t. By shrinking U_ t if necessary, we may assume that there exist isomorphisms \varphi _{t, i} : \mathcal{L}_ i|_{U_ t \cap Z_ i} \to \mathcal{O}_{U_ t \cap Z_ i}. Furthermore, for each i choose an open covering
Z_ i \setminus T = \bigcup \nolimits _ j U_{ij}
such that there exist isomorphisms \varphi _{i, j} : \mathcal{L}_ i|_{U_{ij}} \cong \mathcal{O}_{U_{ij}}. Observe that
\mathcal{U} : X = \bigcup U_ t \cup \bigcup U_{ij}
is an open covering of X. We claim that we can use the isomorphisms \varphi _{t, i} and \varphi _{i, j} to define a 2-cocycle with values in \mathcal{O}_ X^* for this covering that defines \mathcal{L} as in the statement of the lemma.
Namely, if i \not= i', then U_{i, j} \cap U_{i', j'} = \emptyset and there is nothing to do. For U_{i, j} \cap U_{i, j'} we have \mathcal{O}_ X(U_{i, j} \cap U_{i, j'}) = \mathcal{O}_{Z_ i}(U_{i, j} \cap U_{i, j'}) by the first remark of the proof. Thus the transition function for \mathcal{L}_ i (more precisely \varphi _{i, j} \circ \varphi _{i, j'}^{-1}) defines the value of our cocycle on this intersection. For U_ t \cap U_{i, j} we can do the same thing. Finally, for t \not= t' we have
U_ t \cap U_{t'} = \coprod (U_ t \cap U_{t'}) \cap Z_ i
and moreover the intersection U_ t \cap U_{t'} \cap Z_ i is contained in Z_ i \setminus T. Hence by the same reasoning as before we see that
\mathcal{O}_ X(U_ t \cap U_{t'}) = \prod \mathcal{O}_{Z_ i}(U_ t \cap U_{t'} \cap Z_ i)
and we can use the transition functions for \mathcal{L}_ i (more precisely \varphi _{t, i} \circ \varphi _{t', i}^{-1}) to define the value of our cocycle on U_ t \cap U_{t'}. This finishes the proof of existence of \mathcal{L}.
Given sections s_ i as in the last assertion of the lemma, in the argument above, we choose U_ t such that s_ i|_{U_ t \cap Z_ i} is nonvanishing and we choose \varphi _{t, i} such that \varphi _{t, i}(s_ i|_{U_ t \cap Z_ i}) = 1. Then using 1 over U_ t and \varphi _{i, j}(s_ i|_{U_{i, j}}) over U_{i, j} will define a section of \mathcal{L} which restricts to s_ i over Z_ i.
\square
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