Relative effective Cartier divisor

In algebraic geometry, 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.

An effective Cartier divisor as the zero-locus of a section of a line bundle

Let L be a line bundle on X and s a section of it such that $s:{\mathcal {O}}_{X}\hookrightarrow L$ (in other words, s is a ${\mathcal {O}}_{X}(U)$-regular element for any open subset U.)

Choose some open cover $\{U_{i}\}$ of X such that $L|_{U_{i}}\simeq {\mathcal {O}}_{X}|_{U_{i}}$. For each i, through the isomorphisms, the restriction $s|_{U_{i}}$ corresponds to a nonzerodivisor $f_{i}$ of ${\mathcal {O}}_{X}(U_{i})$. Now, define the closed subscheme $\{s=0\}$ of X (called the zero-locus of the section s) by

$\{s=0\}\cap U_{i}=\{f_{i}=0\},$

where the right-hand side means the closed subscheme of $U_{i}$ given by the ideal sheaf generated by $f_{i}$. This is well-defined (i.e., they agree on the overlaps) since $f_{i}/f_{j}|_{U_{i}\cap U_{j}}$ is a unit element. For the similar reason, the closed subscheme $\{s=0\}$ is independent of the choice of local trivializations.

This construction actually exhausts all effective Cartier divisors on X as follows. Let D be an effective Cartier divisor and $I(D)$ denote the ideal sheaf of D. Because of flatness, 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

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

In particular, 1 in $\Gamma (X,{\mathcal {O}}_{X})$ can be identified with a section in $\Gamma (X,I(D)^{-1})$, which we denote by $s_{D}$.

Now we can repeat the early argument with $L=I(D)^{-1}$. Since D is an effective Cartier divisor, D is locally of the form $\{f=0\}$ on $U=\operatorname {Spec} (A)$ for some nonzerodivisor f in A. The trivialization $L|_{U}=Af^{-1}{\overset {\sim }{\to }}A$ is given by multiplication by f; in particular, 1 corresponds to f. Hence, the zero-locus of $s_{D}$ is D.

Properties

• 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.

Effective Cartier divisors on a relative curve

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 $\operatorname {deg} (f^{*}D)=\operatorname {deg} (f)\operatorname {deg} (D)$. On the other hand, a base change does not change degree: $\operatorname {deg} (D\times _{R}R')=\operatorname {deg} (D)$.

A closed subscheme D of X is finite, flat and of finite presentation if and only if it is an effective Cartier divisor that is proper over R.

References

• Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5.