Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 4th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexshould 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]]_

   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

2. • CommentRowNumber2.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeJul 7th 2014
   • (edited Jul 7th 2014)
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexWhat 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]]_.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 7th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex@Ingo For reference, the Stacks project has a [good treatment](http://stacks.math.columbia.edu/tag/01X1). I suspect the current material was taken from PlanetMath.

   @Ingo

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 7th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe stub material I had put in just to make the link work was just taken from Wikipedia. Should be improved.

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