Let X be a locally Noetherian scheme and let f be a rational function on X (i.e. the equivalence class of a pair ( U , f ), where f ∈ O X ( U ) and U contains the associated points of X, under obvious equivalence relation).While reading Vakil's notes I wondered how could we define poles of such a rational function. After some thought I came up with the following definition: I'd say that a regular codimension one point p is a pole if it's not in the domain of definition of f. If X is also an integral scheme (or at least if all the stalks of O X are integral domains, in which case we can cover X with integral schemes), then this definition would coincide with the usual one, namely using the discrete valuation at p.But there is something unnatural about my definition, since I was not able to relate the rational function with the discrete valuation on O X , p and consequently was not able to determine the order of the pole. So I'd like to know if it's possible to define a meaningful notion of poles for rational functions on locally Noetherian schemes and how would it relate to my definition. By extension, consider the same question about zeros.

Question

Let   X be a locally Noetherian scheme and let   f be a rational function on   X (i.e. the equivalence class of a pair   (  U  ,  f  ), where   f  ∈            O        X    (  U  ) and   U contains the associated points of   X, under obvious equivalence relation).While reading Vakil&#039;s notes I wondered how could we define poles of such a rational function. After some thought I came up with the following definition: I&#039;d say that a regular codimension one point   p is a pole if it&#039;s not in the domain of definition of   f. If   X is also an integral scheme (or at least if all the stalks of             O        X   are integral domains, in which case we can cover   X with integral schemes), then this definition would coincide with the usual one, namely using the discrete valuation at   p.But there is something unnatural about my definition, since I was not able to relate the rational function with the discrete valuation on             O              X      ,      p       and consequently was not able to determine the order of the pole. So I&#039;d like to know if it&#039;s possible to define a meaningful notion of poles for rational functions on locally Noetherian schemes and how would it relate to my definition. By extension, consider the same question about zeros.

Bruce Bridges · Accepted Answer

Here&#039;s how I think this works. Take a codimension 1 point   p. The irreducible components containing   p correspond to minimal primes of             O              X      ,      p       and there&#039;s only one of these. Let&#039;s say the generic point for it is   η. Then there is an inclusion             O              X      ,      p        →            O              X      ,      η       which is just taking the fraction field. Your   U has to contain   η, so you can take the stalk of   f there and then find its valuation at   p.It&#039;s confusing. It&#039;s probably good to note that around any point you can find a Noetherian open subscheme, and then this is a finite disjoint union of Noetherian normal irreducible schemes, which is what you really like.

Let X be a locally Noetherian scheme and let f be a rational function on X (i.e. t

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