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A Cartier divisor is usually defined to be a section of the sheaf $\mathscr{K}^\times/\mathscr{O}^\times$.

For an affine scheme, does a Cartier divisor on $\mathrm{Spec}(A)$ have a simple description in terms of commutative algebra, without mentioning sheaves?

Is it just a principal ideal generated by a non-zero-divisor, for instance?

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On an affine scheme $\operatorname{Spec} A$, one can represent (*) any Cartier divisor as a formal $\Bbb Z$-linear sum of locally principal ideals $I\subset A$ which are locally generated by non-zero-divisors. As there are locally principal ideals which are not principal (the most accessible example being $(2,1+\sqrt{-5})\subset \Bbb Z[\sqrt{-5}]$), your guess of "a principal ideal generated by a non-zero-divisor" is not correct.

(*) In general, one may find Cartier divisors which cannot be written as a difference of effective Cartier divisors, which is the meat of the claim here. But since we are on an affine scheme, it trivially has an ample invertible sheaf and as discussed in Sasha's comment here this is enough.

(A quick aside: I really would recommend learning to deal with the sheaf side of things at some point. It's worthwhile.)

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  • $\begingroup$ To make this feel more like commutative algebra, can we reformulate the "formal $\mathbf{Z}$-linear sum" as something like an actual sum of fractional ideals, to express our Cartier divisor as a single (fractional) ideal? $\endgroup$
    – Bun
    Apr 10, 2022 at 5:17
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    $\begingroup$ Fractional ideals are a good thing to think about here, but I didn't know that was something you wanted and they're a little far away from what I normally spend time on, so they didn't come to mind. If you like those and you agree to work with a domain, it turns out that fractional ideals can be identified with Cartier divisors. You might find Vakil's text helpful for a little more material about the relationship. $\endgroup$
    – KReiser
    Apr 10, 2022 at 6:04
  • $\begingroup$ I think this descrption in terms of fractional ideals is what I'm looking for. Can we extend this correspondence with fractional ideals beyond integral domains to include multiple irreducible pieces or nilpotents? Still affine, of course. $\endgroup$
    – Bun
    Apr 10, 2022 at 8:46

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