Jeffery Clements
2022-06-21
Angelo Murray
Beginner2022-06-22Added 23 answers
You can define the "order" of a regular function at a point to be the length of the quotient: (which will be Artinian, so has finite length). The specifics are probably covered in Fulton's "Intersection theory" or Hartshorne's book.
Addition of details: Here is a sketch that clearly defines this: Let's say that your rational function may be described by and at the stalk, which I will call . is one-dimensional domain (you can assume less, but let's make it simple). We know and want to prove or .
Since , we are done by the following general fact:
Hint: look at the sequence
PS: One can also give a definition by using the normalization of , Although less attractive, I believe the above statement to be more practical and comprehensible.
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