# Why do we have two definitions of Cartier divisor?

Why do we have several definitions of Cartier divisor? For example, I found in two books the following definitions:

Let $X$ be a scheme. We denote the group $H^0(X,\mathcal K_X^*/\mathcal O_X^*)$ by $\operatorname{Div}(X)$. The elements of $\operatorname{Div}(X)$ are called as Cartier divisors on $X$.

and

Let $X$ be a scheme.

1. An invertible sheaf on $X$ is an $\mathcal{O}_X$-module that is isomorphic to $\mathcal O_X$ locally on $X$.

2. A closed subscheme $D$ of $X$ is called a Cartier divisor if its defining ideal sheaf $\mathcal I_D$ is an invertible sheaf on $X$.

How can one prove that the definitions are equivalent?

• I blame the French. They have turned mathematics into definiology.
– Pp..
Jan 1, 2015 at 16:54

Closed subschemes $$D$$ with invertible defining ideal $$\mathcal I_D$$ do not define general Cartier divisors, but only effective ones. Here is how:
There exists a covering of $$X$$ by open subsets $$U_i$$ on which the ideal sheaf is of the form $$\mathcal I_D|U_i=f_i\mathcal O_{U_i}$$ with $$f_i\in \Gamma(U_i,\mathcal O _X)$$.
The collection of pairs $$(U_i,f_i)$$ then represents the required global section of the sheaf $$\mathcal K_X^*/\mathcal O_X^*$$.
The effectivity mentioned above reflects the fact that this construction only yields sections of $$\mathcal K_X^*/\mathcal O_X^*$$ represented by regular functions but misses those represented by non regular rational functions, i.e. those having poles.