Commutative Algebra

Exercises 10

A Noetherian ring A is said to be catenary if for any triple of prime ideals 𝔭_{1} ⊂ 𝔭_{2} ⊂ 𝔭_{3} we have

ht(𝔭_{3}/𝔭_{1}) = ht(𝔭_{3}/𝔭_{2}) + ht(𝔭_{2}/𝔭_{1}).

1. Show that a Noetherian local domain of dimension 2 is catenary.

2. Let A be a ring, and 𝔪 a maximal ideal. In A[X] let 𝔪̃_{1} = (𝔪, X) and 𝔪̃_{2} = (𝔪, X − 1). Show that

A[X]𝔪̃_{1} ≅ A[X]𝔪̃_{2}.

3. Find an example of a non Noetherian ring R such that every finitely generated ideal of R is finitely presented as an R-module. (A ring is said to be coherent if the last property holds.)

4. Consider the domain

ℚ[r, s, t]/(s^{2} − (r − 1)(r − 2)(r − 3), t^{2} − (r + 1)(r + 2)(r + 3)).

Find a domain of the form ℚ[x, y]/(f) with isomorphic field of fractions.

5. Give an example of a finite type ℤ-algebra R with the following two properties:

(a) There is no ring map R → ℚ.

(b) For every prime p there exists a maximal ideal 𝔪 ⊂ R such that R/𝔪 =≅ 𝔽_{p}.

6. For f ∈ ℤ[x, u] we define f_{p}(x) = f(x, x^{p}) mod p ∈ 𝔽_{p}[x]. Give an example of an f ∈ ℤ[x, u] such that the following two properties hold:

(a) There exist infinitely many p such that f_{p} does not have a zero in 𝔽_{p}.

(b) For all p >> 0 the polynomial f_{p} either has a linear or a quadratic factor.

7. For f ∈ ℤ[x, y, u, v] we define f_{p}(x, y) = f(x, y, x^{p}, y^{p}) mod p ∈ 𝔽_{p}[x, y]. Give an “interesting” example of an f such that f_{p} is reducible for all p >> 0. For example, f = xv − yu with f_{p} = xy^{p} − x^{p}y = xy(x^{p−1} − y^{p−1}) is “uninteresting”; any f depeding only on x, u is “uninteresting”, etc.