Dear Andrew, your prescription for $\mathcal K'(U)$ does NOT yield a presheaf: you cannot restrict an element of that ring to a smaller open subset $V \subset U$ because non zerodivisors do not restrict to non zerodivisors. Don't feel bad about this error, you are in good company: Grothendieck, Kleiman and Hartshorne (among others) made the same mistake.Kleiman saw the light and wrote an article aptly named

Misconceptions about $K_X \quad$ (L'Enseignement Mathématique, 25(1979), 203-206)

where he gives a correct definition. He addresses your question by constructing a beautifully geometric (but sophisticated) example of an affine scheme X=Spec(A) where $\Gamma(X,\mathcal K)$ (with the correct definition of $\mathcal K$ !) is strictly bigger than Frac(A).

A gift from L'Enseignement Mathématique Our generous Swiss friends allow us to freely download all issues of their journal from 1899 to 2004 [click on Tome 1(1899)-50(2004), in the green column on the left]:

http://www.unige.ch/math/EnsMath/EM_fr/welcome.html

Kleiman's article in particular is here [take the line correponding to page 203 and click on the white PDF logo on the left]left]:

http://retro.seals.ch/digbib/fr/voltoc?rid=ensmat-001:1979:25&e=3&id=ssearch&id2=browse4&id3=#n3

Dear Andrew, your prescription for $\mathcal K'(U)$ does NOT yield a presheaf: you cannot restrict an element of that ring to a smaller open subset $V \subset U$ because non zerodivisors do not restrict to non zerodivisors. Don't feel bad about this error, you are in good company: Grothendieck, Kleiman and Hartshorne (among others) made the same mistake.Kleiman saw the light and wrote an article aptly named

Misconceptions about $K_X \quad$ Enseignement (L'Enseignement Mathématique, 25(1979), 203-206)

where he gives a correct definition. He addresses your question by constructing a beautifully geometric (but sophisticated) example of an affine scheme X=Spec(A) where $\Gamma(X,\mathcal K)$ (with the correct definition of $\mathcal K$ !) is strictly bigger than Frac(A).

A gift from L'Enseignement Mathématique Our generous Swiss friends allow us to freely download all issues of their journal from 1899 to 2004 [click on Tome 1(1899)-50(2004), in the green column on the left]:

http://www.unige.ch/math/EnsMath/EM_fr/welcome.html

Kleiman's article in particular is here [take the line correponding to page 203 and click on the white PDF logo on the left]

http://retro.seals.ch/digbib/fr/voltoc?rid=ensmat-001:1979:25&e=3&id=ssearch&id2=browse4&id3=#n3

1

Dear Andrew, your prescription for $\mathcal K'(U)$ does NOT yield a presheaf: you cannot restrict an element of that ring to a smaller open subset $V \subset U$ because non zerodivisors do not restrict to non zerodivisors. Don't feel bad about this error, you are in good company: Grothendieck, Kleiman and Hartshorne (among others) made the same mistake.Kleiman saw the light and wrote an article aptly named

Misconceptions about $K_X \quad$ Enseignement Mathématique, 25(1979), 203-206

where he gives a correct definition. He addresses your question by constructing a beautifully geometric (but sophisticated) example of an affine scheme X=Spec(A) where $\Gamma(X,\mathcal K)$ (with the correct definition of $\mathcal K$ !) is strictly bigger than Frac(A).