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On an integral scheme, for a Cartier divisor locally represented by $a_i\in K^{\ast }$, its divisor line bundle is the subsheaf of rational functions locally equal to $a_i^{-1}{\mathcal{O}}_X$. Unit ratios glue these free rank-one modules. Given a nonzero rational section of a line bundle $s=a_i{e}_i$, the maps $e_i\mapsto a_i^{-1}$ identify that invertible sheaf with ${\mathcal{O}}_X(D)$. Replacing $s$ by a nonzero rational multiple changes $D$ by a principal Cartier divisor.

Let $K$ be the function field of the integral scheme $X$ and choose a nonzero rational section of a line bundle $s\in {\mathcal{L}}_{\eta }$. Choose an open trivializing cover $U_i$ with generators $e_i$ for the invertible sheaf $\mathcal{L}$. On each nonempty $U_i$, write $s=a_i{e}_i$ with $a_i\in K^{\ast }$. If $e_i=u_{ij}{e}_j$ on an overlap, then $a_j=a_i{u}_{ij}$, so $a_i/a_j=u_{ij}^{-1}$ is a regular unit. Consequently the local rational functions $a_i$ define a Cartier divisor $D$.

The divisor line bundle ${\mathcal{O}}_X(D)$ restricts to $a_i^{-1}{\mathcal{O}}_{U_i}$. Define its local isomorphism from $\mathcal{L}$ by $e_i\mapsto a_i^{-1}$. On an overlap,so the maps glue. HenceThe rational section maps to the rational function $1$, which also fixes the sign convention for $D$. Choosing a different nonzero rational section changes $D$ by a principal Cartier divisor, so its linear equivalence of Cartier divisors class is unchanged. No projectivity or Noetherian hypothesis is required.