General Case
The trouble starts when X is no longer integral. Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, not only can this fail to give a sheaf, in general it does not even give a presheaf! The well-known article of Kleiman, listed in the bibliography, gives such an example.
The correct solution is to proceed as follows:
- For each open set U, let SU be the set of all elements in Γ(U, OX) that are not zero divisors in any stalk OX,x. Let KXpre be the presheaf whose sections on U are localizations SU-1Γ(U, OX) and whose restriction maps are induced from the restriction maps of OX by the universal property of localization. Then KX is the sheaf associated to the presheaf KXpre.
Read more about this topic: Function Field (scheme Theory)
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