Math 902
Homework # 5
Due: Wednesday, April 6th
Throughout R denotes a commutative ring with identity.
1. Let R ⊆ S ⊆ T be commutative rings. Prove that T is integral over R if and only if T
is integral over S and S is integral over R.
2. Let S be a finitely generated (commutative) R-algebra which is integral over R. Prove
that for each prime p of R there are only finitely many primes of S lying over p.
3. Let R be a domain and K its field of fractions. For each maximal ideal m of R, we
can consider Rm as a subring of K containing R. Prove that
\
R=
Rm .
m∈maxSpec R
4. Let R be a domain. Prove that the following are equivalent:
(a) R is integrally closed.
(b) Rp is integrally closed for every p ∈ Spec R.
(c) Rm is integrally closed for every m ∈ maxSpec R.
5. Let k be a field. A nonempty variety V ⊆ Ank is irreducible if V is not the union of two
proper subvarieties. Prove that V is irreducible if and only if I(V ) is prime.
6. Let k be a field and V a nonempty affine k-variety. Prove that there exists unique
irreducible subvarieties W1 , . . . , Wr such that Wi 6⊂ Wj for all i 6= j and V = W1 ∪ · · · ∪
Wr . The Wi are called the irreducible components of V . (Hint: For existence, note
that Ank satisfies DCC on varieties. [You should say why.] By choosing a nonempty
variety minimal with respect to not being the finite union of irreducible subvarieties,
one gets a contradiction. Now throw out superfluous subvarieties in this union. For
uniqueness, use Problem # 6.)
7. Let k be a field and V ⊆ Ank a nonempty affine variety. Define the dimension of V
to be the supremum of integers ` ≥ 0 such that there exists a chain of irreducible
subvarieties of V :
W0 ( W1 ( · · · ( W` .
Prove that if k is algebraically closed, then dim V = dim k[x1 , . . . , xn ]/ I(V ).