Homework 2 – due 12/08/03
Math 603
Do as many as you can!
1. (a) Suppose Ais an integrally closed Noetherian domain with field of fractions
K. Suppose L/K is a finite separable extension, and let Bdenote the integral closure
of Ain L. Prove that Bis a finite A-module. Hint: Use Atiyah-Macdonald, Prop.
5.17.
(b) Let kbe a field and suppose Bis a domain which is also a finitely generated k-
algebra. Let L= Frac(B) and let ˜
Bdenote the integral closure of Bin L. Assume that
char(k) = 0 (for simplicity). Prove that ˜
Bis a finite B-module and a finitely generated
k-algebra. Hint: use the Noether Normalization lemma applied to Btogether with
part (a). (Remark: Spec( ˜
B) is called the normalization of the scheme Spec(B). This
exercise implies, for example, that the normalization of an irreducible variety is still
a variety, because it is still finite-type over the coefficient field.)
(c) Suppose Ais a domain which is also a finitely generated algebra over a field k
of characteristic zero. Let Lbe a finite extension of K= Frac(A), and let Bbe the
integral closure of Ain L. Prove that Bis a finite A-module and a finitely-generated
k-algebra. (Remark: This is essentially Theorem 3.9A in Hartshorne’s book Algebraic
Geometry, and is stated there –without proof – in the more general situation where
kneed not have characteristic zero.)
2. Let kbe a ring, Aand k′two k-algebras. Let A′=A⊗kk′. Show that ΩA′/k′=
ΩA/k ⊗kk′= ΩA/k ⊗AA′. If S⊂Ais multiplicative, show that ΩAS/k = ΩA/k ⊗AAS.
Hint: Use the fundamental exact sequences for differentials.
3. Let k=kbe an algebraically closed field. Suppose m1,...,mrare maximal
ideals in the ring A=k[X, Y ]. Is there necessarily a prime ideal of Awhich is
contained in every mi?
4. Atiyah-Macdonald, Chapter 11, # 2.
5. Atiyah-Macdonald, Chapter 11, # 3.
6. Atiyah-Macdonald, Chapter 11, # 4.
7. Atiyah-Macdonald, Chapter 10, # 11.
8. Exercise 22.2.4 from the notes.
9. Exercise 23.2.3 from the notes.