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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.

Two Cartier divisors are linearly equivalent when their difference is a principal Cartier divisor. Their associated line bundles are then isomorphic, so their global section spaces have the same dimension. This definition works in every dimension.

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.

Suppose $D{\sim }_{\mathbb{R}}tA+E$ on an integral projective variety, where $t>0$, $A$ is ample Cartier and $E$ is effective real Cartier. Then $D$ is an actual positive combination of big Cartier divisors.

Here is a finite-dimensional proof. First assume $X$ is normal. Express $D$ and $E$ in finite Cartier bases and write $D-tA-E$ as a finite combination of principal Cartier divisors. The union of the supports of these finitely many divisors has finitely many prime components. Their integer multiplicities turn the equality into finitely many rational linear equations and effectivity into finitely many rational linear inequalities. The given coefficient tuple lies in a rational polyhedron. Take its smallest face; within that face it lies in the relative interior, and $t>0$ is an open condition. A small simplex with rational vertices in this relative interior contains the tuple. Each vertex gives $D_{\nu }{\sim }_{\mathbb{Q}}{t}_{\nu }A+E_{\nu }$ with $t_{\nu }>0$ and $E_{\nu }\ge 0$. Clearing denominators and applying Kodaira's lemma shows that $D_{\nu }$ is a positive rational multiple of a big Cartier divisor. Taking the original convex weights proves the required actual equality.

If $X$ is nonnormal, pull the finite Cartier bases and the relation to its finite normalization and impose the same rational equations and effectivity inequalities there. The vertex divisors remain rational Cartier divisors on $X$ because they were constructed in bases from $X$. They are big on the normalization, hence big on $X$ by bigness under finite normalization. This proves the same conclusion. Effectivity is used in the usual effective Cartier sense so that pullback is effective; arbitrary cycles on a nonnormal variety cannot be substituted without defining a compatible divisor theory.

Real linear equivalence means that $D-E$ is a finite real combination of principal Cartier divisors. Rational linear equivalence uses rational coefficients instead. Both imply numerical equivalence of divisors.