Paper 1, Section II, I

Algebraic Geometry
Part II, 2017

Let kk be an algebraically closed field.

(a) Let XX and YY be varieties defined over kk. Given a function f:XYf: X \rightarrow Y, define what it means for ff to be a morphism of varieties.

(b) If XX is an affine variety, show that the coordinate ring A(X)A(X) coincides with the ring of regular functions on XX. [Hint: You may assume a form of the Hilbert Nullstellensatz.]

(c) Now suppose XX and YY are affine varieties. Show that if XX and YY are isomorphic, then there is an isomorphism of kk-algebras A(X)A(Y)A(X) \cong A(Y).

(d) Show that Z(x2y3)A2Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2} is not isomorphic to A1\mathbb{A}^{1}.