claimroutineinferred_dep
Unit ideal gives affine three-space fibre
Shows , so
applies the lifting lemma to get the three-space fibre
Claim·§3 The affine-space fibre and the critical point·inside thmNoncoordinate polynomial with affine three-space zero fibre
Statement (paper)
The elements generate the unit ideal in , i.e. is the zero ring. Consequently, for the algebra of Lemma one has and .
In plain English
In the elements and generate the unit ideal, so . Hence the lifted algebra is isomorphic to , and .
Why it's here
This checks the comaximality hypothesis of the lifting lemma, so . Since by the previous claim, is a polynomial ring in three variables. Together with this gives the isomorphism in the main theorem.
Classified routine:
The unit-ideal check is a calculation from the defining equations of B. The affine-three-space conclusion then follows directly from the lifting lemma and the preceding isomorphism B ≅ ℂ[X,Y].