# Why is (line bundle, appropriate rational section) not a standard kind of divisor?

In algebraic geometry, there are two standard "kinds" of divisors: Weil divisors and Cartier divisors. Weil divisors provide better geometric intuition, while Cartier divisors are more general (if not precisely a generalization). In both of these kinds of divisors, the "key" results seem to be their relation to the Picard group (of isomorphism classes of line bundles).

However, one could also define a "divisor" to be an equivalence class of pairs $(s, \mathcal{L})$, where $s$ is an invertible rational section of $\mathcal{L}$. (By "invertible" I mean that there exists a rational section $s'$ of $\mathcal{L}^{\vee}$ such that $s' s = 1$.) We say $(s, \mathcal{L}) \sim (s', \mathcal{L}')$ if there is an isomorphism $\mathcal{L} \to \mathcal{L}'$ taking $s \mapsto s'$.

This definition works well at least for all Noetherian schemes (on which associated points behave nicely), and possibly more generally. It also seems less confusing than the definition of a Cartier divisor, and the relationship between divisors and line bundles is already embodied in the definition.

So why have I never seen this definition given as a kind of divisor?

[Note: I am aware of what "data" defines a Cartier divisor (the $(U_i, f_i)$, etc.) and how this provides a reasonably natural way to think of Cartier divisors geometrically (the "subscheme" defined locally by the $f_i$). So while I appreciate the thought, please don't waste your time writing a note for the sole purpose of explaining this.]

Edit: Since this issue has come up in an answer, I thought I would explain that by "rational section," I mean a (maximally extended) section over an open subset containing all the associated points; or equivalently, a section of $\mathcal{K} \otimes \mathcal{L}$, where $\mathcal{K}$ is the sheaf of total quotient rings of $\mathcal{O}_X$. (Actually, $\mathcal{K} \otimes \mathcal{L}$ is isomorphic to $\mathcal{K}$, but not naturally so.) A rational section can fail to be invertible if it vanishes on one or more associated points.

• You should look in Fulton's book "Intersection Theory". I believe your definition is (almost) what he calls a pseudo-divisor (Chapter 2). Jan 28, 2011 at 5:37
• EGA IV-4, (21.2.11)? Jan 28, 2011 at 8:27
• Since every Cartier divisor is a Weil divisor but not conversely, I would rather say that Weil divisors are a generalization of Cartier divisors... Jan 28, 2011 at 9:58
• I thought the point of pseudo-divisors is that the section $s$ can be zero (which allows you to pull-back pseudo-divisors without any problems). So at least the intention of their definition is rather orthogonal to Charles' intention. Jan 28, 2011 at 10:10
• If $K\otimes L$ is isomorphic to $K$, then $L$ injects into $K$ and then $L$ is associated to a Cartier divisor. This however is not true in general. There is an old counterexample by Kleiman (published later in Comm. in Algebra, 2000) and by Schröer (Arkiv der Math. 2000). Jan 28, 2011 at 10:32