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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?

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

1 Answer 1

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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.

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