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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 UΓ(U,𝒪 X)[Γ(U,𝒮) 1]U \quad\mapsto\quad \Gamma(U, \mathcal{O}_X)[\Gamma(U, \mathcal{S})^{-1}], where “Γ(U,𝒮)\Gamma(U, \mathcal{S}) consists of only the regular sections of 𝒪 X\mathcal{O}_X over UU, i.e. those elements of Γ(U,𝒪 X)\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 Γ(U,𝒪 X)\Gamma(U, \mathcal{O}_X), then 𝒮\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 𝒪 X,x\mathcal{O}_{X,x}, then we obtain exactly the definition in sheaf of rational functions.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 7th 2014


    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.