Principal cartier divisor

Let $X$ be a scheme. For a global meromorphic unit $f\in \mathrm{\Gamma }(X,{\mathcal{K}}_X^{\times })$, its principal Cartier divisor is ${\mathrm{div}}_C(f):=q_X(f)\in \mathrm{CaDiv}(X),$ where $q_X$ is the global-section map induced by the quotient sheaf ${\mathcal{K}}_X^{\times }\to {\mathcal{K}}_X^{\times }\mathrm{/}{\mathcal{O}}_X^{\times }$ (Cartier divisor). Equivalently, use the single local equation $f$ on the open set $X$.

We use additive notation for Cartier divisors, even though their local equations multiply. The quotient map is a group homomorphism, so ${\mathrm{div}}_C(fg)={\mathrm{div}}_C(f)+{\mathrm{div}}_C(g)$, ${\mathrm{div}}_C(1)=0$, and ${\mathrm{div}}_C(f^{-1})=-{\mathrm{div}}_C(f)$. In particular, principal Cartier divisors form a subgroup of $\mathrm{CaDiv}(X)$. Multiplying $f$ by a global regular unit leaves its divisor unchanged, since that unit has zero image in the quotient sheaf.

The sign convention is zeros positive, poles negative: a regular local equation cutting out a zero contributes positively; replacing it by its inverse reverses the sign. This is the convention of Cartier divisor, without asserting that a numerical order exists at every point of an arbitrary scheme.

On the empty scheme the groups of units and of Cartier divisors are trivial, so this definition gives only the zero divisor.