Function Field (scheme Theory) - General Case

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)

Famous quotes containing the words general and/or case:

    A bill of rights is what the people are entitled to against every government on earth, general or particular, and what no just government should refuse, or rest on inference.
    Thomas Jefferson (1743–1826)

    The circumstances with which every thing in this world is begirt, give every thing in this world its size and shape;—and by tightening it, or relaxing it, this way or that, make the thing to be, what it is—great—little—good—bad—indifferent or not indifferent, just as the case happens.
    Laurence Sterne (1713–1768)