I'm trying to read a part of the proof of Lemma 2.2 in Fulton's Intersection theory. He wants to prove that every line bundle $L$ on a variety (integral, of finite type over $k$) $X$ is of the form $\mathcal{O}(D)$ form some Cartier divisor $D$.
Let $g_{\alpha \beta}$ be transition functions for $L$ for some affine open covering $\{U_\alpha\}$ of $X$. Fix one index $\alpha_0$ and define $f_\alpha = g_{\alpha \alpha_0}$. Then $f_\alpha / f_\beta = g_{\alpha \beta}$, so the data $(U_\alpha, f_\alpha)$ define a Cartier divisor $D$ with $\mathcal{O}(D) \cong L$.
By definition, the $f_\alpha$ should belong to $\mathcal{K}(U_\alpha)$, however $g_{\alpha \alpha_0}$ is an isomorphism $\mathcal{O}(U_\alpha \cap U_{\alpha_0}) \to \mathcal{O}(U_\alpha \cap U_{\alpha_0})$, so it is not defined on the whole of $U_\alpha$. So in a sense, $(U_\alpha \cap U_{\alpha_0}, f_\alpha)_\alpha$ defines a Cartier divisor on $U_{\alpha_0}$, not on $X$?
What is happening here? Also, does he use here that $X$ is a variety?
You are right that $g_{\alpha\alpha_0}\in \mathcal{O}(U_{\alpha}\cap U_{\alpha_0})^*$. But note that an invertible regular function $g_{\alpha\alpha_0}$ defined on nonempty open subset $U_{\alpha}\cap U_{\alpha_0}$ determines uniquely a nonzero rational function $f_{\alpha}$ on integral scheme $X$. Since $\mathcal{K}$ is a constant sheaf with rational functions on $X$ as global sections (again due to the fact that $X$ is a variety), we deduce that $f_{\alpha}$ in $\mathcal{K}(U_{\alpha})$.