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:

    There is a mortifying experience in particular, which does not fail to wreak itself also in the general history; I mean “the foolish face of praise,” the forced smile which we put on in company where we do not feel at ease, in answer to conversation which does not interest us. The muscles, not spontaneously moved but moved, by a low usurping wilfulness, grow tight about the outline of the face, with the most disagreeable sensation.
    Ralph Waldo Emerson (1803–1882)

    There is not a more disgusting spectacle under the sun than our subserviency to British criticism. It is disgusting, first, because it is truckling, servile, pusillanimous—secondly, because of its gross irrationality. We know the British to bear us little but ill will—we know that, in no case do they utter unbiased opinions of American books ... we know all this, and yet, day after day, submit our necks to the degrading yoke of the crudest opinion that emanates from the fatherland.
    Edgar Allan Poe (1809–1845)