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 ).
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