Rings of Functions

Theorem. A closed set in k" is irreducible iff its ring of functions is an integral domain. 

Let A be a Hopf algebra. We are going to study the /(-linear operators T A - A which are translation-invariant. As in the previous part, we begin by seeing what this means when A is the ring of functions on an algebraic matrix group S. An operator T on functions there is left-invariant iff it commutes with all the left-translation operators Tg defined by (Tgf)(x) — f(9x)- Now on A the map / h- f(gx) is (g, x) ° A and since Tg makes... 

Suppose for illustration that A and B are rings of functions on closed sets in k", with k = k. The maximal ideals P in A then correspond to points x in the set. If PB 4 B, some maximal ideal of B contains P, and the corresponding point maps to x. Thus when A - B is flat, the extra condition involved in faithful flatness is precisely surjectivity on the closed sets. Condition (2) is the generalization of that to arbitrary rings. 

Representable functor 5 Representation, see Linear representation Representation of a Lie algebra 96 Ring of functions on S 30 Root system 98... 

---