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Assume that $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ be two ringed spaces and $U\subset X, V\subset Y$ be two open sets with an isomorphism of sheaves $\varphi: (U,\mathcal{O}|_U)\rightarrow (V,\mathcal{O}|_V)$. Then we can glue these two topological spaces along these open subsets with $Z:=X\sqcup Y/\sim$ where $x\sim y$ iff $x=y$ or $x\in U, y\in V$ and $\varphi(x)=y$.

Now my textbook says that we can define a new sheaf by defining $\mathcal{O}_{Z,p}:=\mathcal{O}_{X,p}$ if $p\in X$ and $\mathcal{O}_{Z,p}:=\mathcal{O}_{Y,p}$ if $p\in Y$ where $\mathcal{O}_{X,p}$ denotes the stalk of $(X,\mathcal{O}_X)$ on $p\in X$.

Showing that this gives a sheaf is left as an exercise. I have a problem with the identity axiom. Namely, I take an open cover $Z=\bigcup_{i\in I} U_i$ and two functions $f,g\in\mathcal{O}(Z)$ with res$_{Z,U_i}(f)=$res$_{Z,U_i}(g)$ on $U_i$ for all $i\in I$. I need to show that $f=g$. Using the identity axiom of $X$ and $Y$, it was easy to show that $f=g$ on $X$ and $Y$. But I have no idea how I can show that $f=g$ on whole $Z$? Thanks.

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    How did you describe $\mathcal{O}_Z$, explicitly? – 2017-01-02
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    Well, the textbook does not. It just defines stalks. But it is enough to define the stalks, right? – 2017-01-02
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    No, because there are nonisomorphic sheaves with isomorphic stalks. – 2017-01-02
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    Can you give an example? – 2017-01-02
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    There are some examples in this link: http://math.stackexchange.com/questions/43314/failure-of-isomorphisms-on-stalks-to-arise-from-an-isomorphism-of-sheaves – 2017-01-02
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    Also, you didn't start out the proof of the identity axiom correctly. You're supposed to show that if $V$ is an open subset of $Z$, $\{U_i\}$ an open cover of $V$, and $f,g\in \mathcal{O}_Z(V)$ with $\operatorname{res}_{V,U_i}(f) = \operatorname{res}_{V,U_i}(g)$ for all $i$, then $f = g$. – 2017-01-02
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    You are right, thanks. Then what should be the correct description of $\mathcal{O}_Z?$ – 2017-01-02
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    Let $\iota_X : X \to Z$ and $\iota_Y : Y \to Z$ be inclusions. For an open subset $W$ of $Z$, define $\mathcal{O}_Z(W)$ to be the set of all pairs $(s,t)\in \mathcal{O}_X(\iota_X^{-1}(W)) \times \mathcal{O}_Y(\iota_Y^{-1}(W))$ such that $\varphi(s|\iota_X^{-1}(W)\cap U) = t|\iota_Y^{-1}(W)\cap V$. – 2017-01-02

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