Paper 1, Section II, I
Part II, 2017
Let be an algebraically closed field.
(a) Let and be varieties defined over . Given a function , define what it means for to be a morphism of varieties.
(b) If is an affine variety, show that the coordinate ring coincides with the ring of regular functions on . [Hint: You may assume a form of the Hilbert Nullstellensatz.]
(c) Now suppose and are affine varieties. Show that if and are isomorphic, then there is an isomorphism of -algebras .
(d) Show that is not isomorphic to .