### Homework Questions

#### Set II

(There are 10 questions.)
Let G=GL_{4}(K) and D=D_{4}(K) be the subgroup of G consisting of diagonal matrices. If possible, write 5 tori in D with non-isomorphic centralizers C_{G}(T).

Section 21: Questions 2, 4, 6, 9

Section 22: Questions 1, 6

Section 23: Questions 3, 4 (the first part only), 8

### Set I

(The homework is due on 31 October 2002 Thursday 12:40.)

K is an algebraically closed field.

1. Find an example of a subset X of K^{n} such that
X is different from V(I(X)).

2. Show that a point derivation is uniquely determined by its action on
K[G], where G is an algebraic group.

3. Find the Lie algebra of SL_{n}(K). Justify your claim.

4. Let X, Y, Z be affine varieties, P a point in X, f a morphism from Y to Z and g a morphism from X to Y. Show that d(f*g)_{P}=
df_{g(P)}*dg_{P}. [Here * stands for the composition of maps.]

5. If H is an abelian subgroup of an algebraic group G then show that the closure of H is also an abelian subgroup of G.

6. Exercise 1.11.

7. Exercise 7.4.

8. Exercise 7.11.

9. Exercise 8.1.

10. Exercise 10.2.

Good luck!