This question is motivated by a construction, unclear for me, related to Cartier divisors. But, in the end it can be reduced to a question involving only sheaves on topological spaces.
Let $X$ be a locally Noetherian scheme. Consider a Cartier divisor $D=\{(f_i,U_i)_i\}$ where as usual:
- $X=\bigcup_i U_i$ is an open cover
- $f_i\in\mathcal K_X^\times(U_i)$ and $\frac{f_i|_{U_i\cap U_j}}{f_j|_{U_i\cap U_j}}\in\mathcal O_X^\times$. Remember that $\mathcal K_X$ is the sheaf of stalks of meromorphic functions.
We want to define a subsheaf $\mathcal O_X(D)\subset K_X$ associated to $D$. Liu's and Hartshorne's book give the following definition:
$$\mathcal O_X(D)(U_i):=f_i^{-1}\mathcal O_X(U_i)$$
Therefore they define the sheaf only on the open stes $U_i$. Where is the definition of $\mathcal O_X(D)$ on the remaining open sets of $X$?
I'm sure that my question will be solved some argument of the type: "there exists a unique minimal subsheaf $\mathcal G$ of $\mathcal K_X$ such that $\mathcal G(U_i)=\mathcal O_X(D)(U_i)$".
Unfortunately I can't find any reference for this result.
Best Answer
The value of the sheaf $\mathcal O_X(D)$ at an arbitrary open subset $U\subset X$ is the sub- $\mathcal O_U$ -module $\mathcal O_X(D)(U)\subset K_X(U)$ of $K_X(U)$ characterized by $$\mathcal O_X(D)(U)=\{s\in K_X(U) : \forall i,s\vert U_i\cap U\in (f_i\vert U_i\cap U) ^{-1}\mathcal O_X(U_i\cap U) \}$$ nothing more and nothing less.
There are no isomorphisms, gluing nor cocycle conditions in this definition.