In Hartshorne's chapter on Weil divisors he fixes the following hypothesis:

$(*)$ Every scheme is Noetherian , integral, separated, and regular in codimension 1

I can understand why you would want the first three hypotheses, but the regularity in codimension 1 is a little mysterious to me. What is the motivation behind this hypothesis? Is there any geometric reason, or does this just make the algebra in this chapter easier? For reference, regular in codimension 1 means that any local ring of $X$ of dimension 1 is regular.

  • $\begingroup$ If you want the Weil divisors to form a group, then the hypothesis is necessary. Otherwise, for a divisor $D$, $nD$ will not make good sense. $\endgroup$ – Mohan Jun 16 '17 at 20:40
  • $\begingroup$ Can you give a non-example? $\endgroup$ – 54321user Jun 16 '17 at 22:30
  • 4
    $\begingroup$ If $P\subset A$ is a prime ideal in such a situation, defining $D$, try to see what $2D$ should be. The only reasonable one is to take $P^{(2)}$, but if $A_P$ is not a dvr, this will not have the right length. For a specific example, take $A=k[x^2,x^3]$ and $D$ defined by $(x^2,x^3)$. Show that $2D$ can be defined as two incomparable ideals and thus meaningless. $\endgroup$ – Mohan Jun 16 '17 at 22:52

It depends on what you want to do. If you just want to define the Weil divisor class group and the Weil divisor class associated to an invertible module, then it suffices to work with locally Noetherian integral schemes. See Tag 0BE0 and the next section in the Stacks project.

If you want to do other things, you may want to impose more conditions.


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