Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite sum. I remember from Arthur Ogus' lectures that the separated hypothesis isn't that necessary for things to make sense. Also, if we weaken "Noetherian" to "locally Noetherian", we can make a sheafy version and define Weil divisors to be global sections of that sheaf (then they'd only be "locally finite" sums).

So now I'm wondering, what's the most general "working" definition of Weil divisor around? What I'm looking for is a decent reference with a generalized definition and its consequences all worked out to convince me it's actually worthwhile.

(Among other things, this would help better understand the roles of Hartshorne's various hypotheses.)