Mathematisches Institut SS 2002 der Universität München Set 2 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(5) Let X denote the plane curve y = x^{2}. Then X is isomorphic to the affine line.

(6) ^{∗} Let 𝕂 be an algebraically closed field. Let p be an irreducible square polynomial in 𝕂[x, y]. Let Z be the conic section defined by p with the affine algebra 𝕂[x, y]/(p). Show that Z is naturally isomorphic either to X or to U from problems (3) resp. (5).

(7) Let X be an affine scheme with affine algebra

A = 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}).

Define “coordinate functions” q_{i} : X (B) ⟶ B which describe the coordinates of B-points and identify these coordinate functions with elements of A.

(8) Let S_{3} be the symmetric group and A := 𝕂[S_{3}] be the group algebra on S_{3}. Describe the points of X (B) = 𝕂-Alg (A, B) as a subspace of 𝔸^{2}(B). What is the commutative part X_{c}(B) of X and what is the affine algebra of X_{c}?

Due date: Tuesday, 30.04.2002, 16:15 in Lecture Hall E41