Suppose f &#x2208;<!-- ∈ --> O p </msub> ( V ) is a rational func

Kolten Conrad 2022-07-09 Answered
Suppose f O p ( V ) is a rational function on V that has a value at p. Then write f = a / b = a / b where a , b , a , b Γ ( V ), the coordinate ring of V. Want to show the value of f at p is well-defined, i.e. a ( p ) / b ( p ) = a ( p ) / b ( p ).
So since a / b = a / b are the same equivalence class, there is some non-zero poly x Γ ( V ) such that x ( a b a b ) = 0. Then x ( p ) ( a ( p ) b ( p ) a ( p ) b ( p ) ) = 0. Then what? How do we know x ( p ) 0?
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Answers (1)

Keegan Barry
Answered 2022-07-10 Author has 18 answers
So x 0 ¯ Γ ( V ). Since Γ ( V ) is an integral domain, a b a b = 0 ¯ . Actually one can remove x from the definition of fraction field altogether.

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