Answer Happy

Let K be a field and let A C B be an inclusion of finitely generated K-algebras. Let m C B be a maximal ideal and write P = m n A. (a) Use Noether Normalization Theorem to show that B/m is module finite over K. (2 marks) (b) Use the fact that A/P injects into B/m to deduce that A/P is module finite over K. (1 mark) (c) Use the previous exercise to deduce that P is a maximal ideal. (1 mark) Following is some useful hints (Only need prove above, do not need prove below) Theorem 30.4 (Noether's Normalization Theorem). Let R be a finitely generated algebra over a field K. There exist elements u₁,..., ud ER which are algebraically independent and such that R is module-finite over Klu₁,..., ua]. Definition 29.1. Let R be a commutative ring and let S be an R-algebra. (a) We say that S is module finite over R if it is a finitely generated R-module. (b) If RC S, we say that an element s ES is integral over R if there exists a monic polynomial f(X) = Xd + ad-1X1+...+a₁X + a0 € R[X] such that f(s) = 0. Proposition 29.2. Let R be a commutative ring. Let S be an R algebra and T be an S algebra. If S is module finite over R and T is module finite over S then T is an R-algebra which is module finite over R. Theorem 29.3. Let R be a commutative ring and let S be an R-algebra containing R. The following are equivalent: (a) S is module finite over R, (b) there exist 01,..., On ES which are integral over R such that S = R[0₁,...,0], (c) S is a finitely generated R-algebra and every element in S is integral over R.