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.)


1 Answer 1


The sheaf associated to a Weil divisor is reflexive, that is it is equal to its bidual. One can use this property as a definition to obtain a sheafy version of the notion of a Weil divisor. Hartshorne has written several articles about reflexive sheaves and their application. Take a look at the homepage of Karl Schwede (http://www-personal.umich.edu/~kschwede/), where you can find an introductory article on reflexivity and references to some of Hartshorne's publications on that matter.

In the 'classical case' treated in Hartshorne's book the relation between Weil divisors (defined as elements of the free abelian group generated by closed subschemes of codimension 1) and reflexive sheaves is based on the fact that the local rings in codimension 1 are discrete valuation rings. The whole setting can be generalized (at least) to separated, integral, quasi-compact schemes with coherent structure sheaf. You can find details in the article Hagen Knaf, Divisors on varieties over valuation domains, Israel Journal Math. 119.


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    $\begingroup$ A quick comment, the notes of mine referenced above, once had a mistake in them (this was fixed a while ago, but still). You see, the the notion of S2 Hartshorne uses in his papers is not the standard one (at least non-standard from the commutative algebra point of view). I confused the two notions. In some sense, it's not too important because for the sheaves considered, the ideas coincide, but it can lead to some incorrect statements if one is not careful. The current version of the notes fixes this (and explains it in more detail). $\endgroup$ Apr 16, 2010 at 14:05

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