Homework for Week 11

Part 1

1. Let R be a ring with Spec R and closed sets defined as above.
   • Prove that V(f) ∪ V(g) = V(fg), meaning the closed sets of Spec R are closed under finite union.
   • Prove that ∩_{i ∈ I} V(f_i) = V(∪_{i ∈ I} f_i) for any set {f_i: i ∈ I}, meaning the closed sets of Spec R are closed under arbitrary intersection.
2. Describe the points and sheaf of functions of Spec C[x]/(x²-x). Describe the points and sheaf of functions of Spec C[x]/(x³-x²).
3. Describe the points and sheaf of functions of Spec R[x]/(x²+1).

Part 2

1. Explain why the points of Spec Z[x] are
   • (0)
   • (p) for p a prime integer
   • (f(x)) for f(x) a polynomial which is irreducible over Q and whose coefficients have no common denominator
   • (p,f(x)) where p is a prime integer and f(x) is a monic polynomial which is irreducible mod p.
   Which points are closed? What is the closure of the other points? Where do the closures (11) and (x-2) meet? Where do (11) and (4x+1) meet? Answer the same questions replacing 11 with 3.
2. Describe Spec R[x]. How does it compare topologically to R? What about to C?

• Part 1: Due Tuesday
• Part 2: Due Thursday