HOMEWORK 4 FOR 18.747, SPRING 2013
DUE FRIDAY, MARCH 8 BY 3PM.
(1) The projective linear group P GLn is the affine algebraic group whose ring of
functions is given by O(P GLn ) = O(GLn )Gm , where Gm acts via t(f )(x) =
f (tx).
Consider the natural map φ : SLn → P GLn .
(a) For which n is the map dφ : sln → pgln an isomorphism?
(b) Show that this happens exactly when the kernel of φ has n elements.
(c) (Optional) Describe the scheme theoretic kernel K of φ, and check
that O(K) is always n-dimensional.
(2) Let G be connected linear algebraic group and τ : G → G a homomorphism
such that dτ : g → g is nilpotent. Prove that the map x 7→ τ (x)x−1 is onto.
(3) Describe restricted Lie algebras for the following algebraic groups over a
field k of characteristic p > 0:
(a) The group in problem 1, pset 3.
(b) As an algebraic variety G = Gm × Ga with multiplcation given by
(t, x) · (s, y) = (ts, sx + y).
(c) As an algebraic variety G = Gm × Ga with multiplcation given by
(t, x) · (s, y) = (ts, sp x + y).
1