# Different definitions of Cartier divisor and when they agree

On a scheme $$X$$, the most general definition of Cartier divisor is a global section in $$\Gamma(X, \mathcal{K}^{*}/\mathcal{O}^{*})$$, where $$\mathcal{K}^{*}$$ is the sheaf of invertible elements of the sheaf of total quotient rings.

Alternatively, some texts refer to a Cartier divisor as an equivalence class of pairs $$(\mathcal{L}, s)$$, where $$\mathcal{L}$$ is an invertible sheaf on $$X$$ and $$s$$ is a non-zero rational section of $$\mathcal{L}$$.

I have been able to show that these definitions are equivalent in the case that $$X$$ is a noetherian integral scheme. But how general does this equivalence go? Are they always the same? I expect the problem may occur when you drop reducedness.

• how do you define a rational section of a line bundle on a general scheme?
– user690882
Commented Jul 28, 2019 at 6:29
• Actually good question. I guess for an irreducible scheme it is fine, but I don't know how it would be defined more generally than that.
– Luke
Commented Jul 28, 2019 at 6:36

Cartier divisors can be defined on any scheme. And to any Cartier divisor $$D$$ on any scheme $$X$$ you can associate a line bundle (invertible $$\mathcal{O}_X$$-module) which is usually denoted by $$\mathcal{O}_X(D)$$. So there is always a map \begin{align} \mathrm{Div}(X) &\to \mathrm{Pic}(X) \\ D &\to \mathcal{O}_X(D), \end{align} from the group of Cartier divisors to the group of line bundles. This further induces an injective map $$\mathrm{DivCl}(X) \to \mathrm{Pic}(X)$$, where $$\mathrm{DivCl}(X)$$ denotes the group $$\mathrm{Div}(X)$$ modulo linear equivalence. This map is surjective if $$X$$ is an integral scheme, but for general schemes not all line bundles come from Cartier divisors.
Now to answer your question, for any scheme $$X$$ and a line bundle $$\mathcal{L}$$ on it, you have the following natural one to one correspondence $$\{ \text{effective cartier divisors } D \text{ such that } \mathcal{O}_{X}(D) \cong \mathcal{L} \} \leftrightarrow \{ \text{non-zero divisors of }\Gamma(X, \mathcal{L}) \}\big/\sim,$$ where $$s \sim s'$$ if and only if $$s' = us$$ for some $$u \in \Gamma(X, \mathcal{O}_X^{\times}).$$ Note that we are only considering effective Cartier divisors, because the associated line bundle of any divisor has a nonzero global section if and only if it is linearly equivalent to an effective one.
• If you want explicit details, see pages $302-305$ of Algebraic Geometry $1$ by Görtz, Wedhorn. Commented Jul 28, 2019 at 8:00