Effective cartier divisor

A Cartier divisor $D$ on a scheme $X$ is effective if it has a local-equation representation $(U_i,f_i)$ as in Cartier divisor with $f_i\in {\mathcal{O}}_X(U_i)$ and with multiplication by the germ $(f_i{)}_x$ injective on ${\mathcal{O}}_{X,x}$ for every $x\in U_i$. Thus each $f_i$ is a regular section in the precise sense of Sheaf total quotient rings; the condition uses injectivity of multiplication, including exclusion of the zero germ on a nonzero stalk.

The condition is independent of the representation. On overlaps two Cartier equations differ by a regular unit. Multiplication or division by such a unit preserves regularity as a section of ${\mathcal{O}}_X$ and preserves injectivity of multiplication at every stalk. These local conditions remain true on refinements and descend by sheaf locality.

The local principal ideal sheaves $f_i{\mathcal{O}}_{U_i}$ agree on overlaps because $f_i\mathrm{/}{f}_j$ is a regular unit. We denote the resulting ideal sheaf by ${\mathcal{I}}_D$. The construction of its associated closed subscheme is proved in the subsequent closed-immersion theorem.

A unit equation, in particular $f_i=1$, gives the zero Cartier divisor and the ideal sheaf ${\mathcal{O}}_X$. Locally its quotient ring is the zero ring, so its vanishing subscheme is empty. The zero Cartier divisor is therefore the empty effective divisor. The empty scheme has only this effective divisor.