Let X,Y be affine varieties, we know that the coordinate ring of the product variety X × Y satisfies k [ X × Y ] ≅ k [ X ] ⊗ k k [ Y ].My question is is it true that for rational function field, we also have k ( X × Y ) ≅ k ( X ) ⊗ k k ( Y )? If not, is there a way to relate k ( X × Y ) with k ( X ), and k ( Y )?

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Let X,Y be affine varieties, we know that the coordinate ring of the product variety   X  ×  Y satisfies   k  [  X  ×  Y  ]  ≅  k  [  X  ]      ⊗    k    k  [  Y  ].My question is is it true that for rational function field, we also have   k  (  X  ×  Y  )  ≅  k  (  X  )      ⊗    k    k  (  Y  )? If not, is there a way to relate   k  (  X  ×  Y  ) with   k  (  X  ), and   k  (  Y  )?

Alessandro Schmidt · Accepted Answer

I just thought I would expand on Alex&#039;s answer a bit. First I want to make sure we&#039;re assuming that   k  =            k      ¯      . Then we can always think of   k  (  X  )      ⊗    k    k  (  Y  ) as a subset of   k  (  X  ×  Y  ) by the map extending   ϕ  ⊗  ψ  ↦  ϕ  ψ (see note below), but you can see that in this way,   k  (  X  )      ⊗    k    k  (  Y  ) only contains rational functions   f  (      x    ,      y    )      /    g  (      x    ,      y    )  ∈  k  (  X  ×  Y  ) where   g  (      x    ,      y    ) is a product       g    1    (      x    )      g    2    (      y    ) of polynomials in just       x   and       y   separately, and this is not all possible rational functions -- e.g.       1          x      y      −      1       for   X  =  Y  =            A        k    1  . Of course, you may write   k  (  X  ×  Y  )  ≅      frac    (  k  (  X  )      ⊗    k    k  (  Y  )  )  ≅      frac    (  k  [  X  ]      ⊗    k    k  [  Y  ]  ), or you may just localize   k  (  X  )      ⊗    k    k  (  Y  ) at the set of polynomials which are not of the above form, but these are probably not super helpful.Note. It is non-trivial that, when   k  =            k      ¯       and A,B are k-algebras and also domains, then   A      ⊗    k    B is a domain. It is not always true when   k  ≠            k      ¯      .

Let X,Y be affine varieties, we know that the coordinate ring of the product variety X &#x00D

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