Intersection Theory Class 9

I have one update from last time, and this is aimed more at the experts. Rob pointed out that there was no reason that we know that a Cartier divisor can be expressed as a difference (or quotient) of effective Cartier divisors. More precisely, a Cartier divisor can be described cohomologically as follows. Let X be a scheme. We have a sheaf O of invertible functions. There is another sheaf K that are things that locally look like quotients of a function by a nonzerodivisor. (If X is a variety, then K is the constant sheaf with K(U) = R(X) for all U.) Then I informally described Cartier divisors of X as determined by certain data: there is an open cover of X by open sets Ui; we have an element of K for each Ui; and on Ui ∩ Uj the quotient of the two elements of K ∗ corresponding to i and j is an element ofO X. We then mod out by an equivalence relation that I was careless about defining. This definition translates to themore compact notation: Cartier divisors are global sections of the (quotient) sheaf K/O X. (More generally, we get a sheaf of Cartier divisors K/O X.) The description I gave was the Cech description of a quotient sheaf. This drives home the point that any Cartier divisor is locally the quotient of two effective Cartier divisors, but not necessarily globally. I don’t know of any specific examples of a Cartier divisor that is not the quotient/difference of two effective Cartier divisors, and I would like to see one.