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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 4th 2014

should say that yesterday, right before my battery died, I had started a bare minimum at sheaf of rational functions, just so as to complete the corresponding entry at function field analogy – table

• CommentRowNumber2.
• CommentAuthorIngoBlechschmidt
• CommentTimeJul 7th 2014
• (edited Jul 7th 2014)

What is the relationship to the entry sheaf of meromorphic functions? There, a sheaf is defined by sheafification of the presheaf $U \quad\mapsto\quad \Gamma(U, \mathcal{O}_X)[\Gamma(U, \mathcal{S})^{-1}]$, where “$\Gamma(U, \mathcal{S})$ consists of only the regular sections of $\mathcal{O}_X$ over $U$, i.e. those elements of $\Gamma(U, \mathcal{O}_X)$ which are not zero divisors”.

Note that this definition is a bit unprecise: If by “not zero divisor” it is meant that the elements should not be zero divisors in $\Gamma(U, \mathcal{O}_X)$, then $\mathcal{S}$ fails to be a presheaf. If instead by “not zero divisor” it is meant that the elements should be such that their germs are not zero divisors in the rings $\mathcal{O}_{X,x}$, then we obtain exactly the definition in sheaf of rational functions.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeJul 7th 2014

@Ingo

For reference, the Stacks project has a good treatment. I suspect the current material was taken from PlanetMath.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 7th 2014

The stub material I had put in just to make the link work was just taken from Wikipedia. Should be improved.