# Wednesday April 20, 2016

• Read the section 6 on complete varieties (pages 45-46) and prepare to discuss Proposition 6.1.
• Read and prepare the proof that P^n is complete.
• Learn the definition of G_m, G_a, GL(n), SL(n), O(n),  SO(n), Sp(2m) as algebraic groups.  In each case, describe the morphism from A to A\otimes_k A that is dual to the multiplication mu:GxG->G, where G = Spec(A).
• 7.3. Let G be an affine algebraic group.  Show that G^0 is a normal subgroup of finite index.
• 7.4. Show that the Zariski closure of a subgroup is a subgroup.
• 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
• 8.1. State what an action is in the category of k-varieties.