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?

Question

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?

Keegan Barry · Accepted Answer

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.

Suppose f ∈ O p </msub> ( V ) is a rational func

Answered question

Answer & Explanation

New Questions in Algebra II