Relative effective Cartier divisor

An effective Cartier divisor in a scheme X over a ring R is a closed subscheme D of X that (1) is flat over R and (2) the ideal sheaf $I(D)$ of D is locally free of rank one (i.e., invertible sheaf). Equivalently, a closed subscheme D of X is an effective Cartier divisor if there is an open affine cover $U_{i}=\operatorname {Spec} A_{i}$ of X and nonzerodivisors $f_{i}\in A_{i}$ such that the intersection $D\cap U_{i}$ is given by the equation $f_{i}=0$ (called local equations) and its ideal sheaf $I(D)|_{U_{i}}=A/f_{i}A$ is flat over R and such that they are compatible.

If D and D' are effective Cartier divisors, then the sum $D+D'$ is the effective Cartier divisor defined locally as $fg=0$ if f, g give local equations for D and D' .
If D is an effective Cartier divisor and $R\to R'$ is a ring homomorphism, then $D\times _{R}R'$ is an effective Cartier divisor in $X\times _{R}R'$ .
If D is an effective Cartier divisor and $f:X'\to X$ a flat morphism over R, then $D'=D\times _{X}X'$ is an effective Cartier divisor in X' with the ideal sheaf $I(D')=f^{*}(I(D))$ .

Taking $I(D)^{-1}\otimes _{{\mathcal {O}}_{X}}-$ of $0\to I(D)\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{D}\to 0$ gives the exact sequence

0\to {\mathcal {O}}_{X}\to I(D)^{-1}\to I(D)^{-1}\otimes {\mathcal {O}}_{D}\to 0

.

This allows one to see global sections of ${\mathcal {O}}_{X}$ as global sections of $I(D)^{-1}$ . In particular, the constant 1 on X can be thought of as a section of $I(D)^{-1}$ and D is then the zero locus of this section. Conversely, if $L$ is a line bundle on X and s a global section of it that is a nonzerodivisor on ${\mathcal {O}}_{X}$ and if $L/{\mathcal {O}}_{X}$ is flat over R, then $s=0$ defines an effective Cartier divisor whose ideal sheaf is isomorphic to the inverse of L.

From now on suppose X is a smooth curve (still over R). Let D be an effective Cartier divisor in X and assume it is proper over R (which is immediate if X is proper.) Then $\Gamma (D,{\mathcal {O}}_{D})$ is a locally free R-module of finite rank. This rank is called the degree of D and is denoted by $\operatorname {deg} D$ . It is a locally constant function on $\operatorname {Spec} R$ . If D and D' are proper effective Cartier divisors, then $D+D'$ is proper over R and $\operatorname {deg} (D+D')=\operatorname {deg} (D)+\operatorname {deg} (D')$ . Let $f:X'\to X$ be a finite flat morphism. Then $\operatorname {deg} (f^{*}D)=\operatorname {deg} (f)\operatorname {deg} (D)$ .^[1] On the other hand, a base change does not change degree: $\operatorname {deg} (D\times _{R}R')=\operatorname {deg} (D)$ .^[2]