Commutative Algebra

Exercises 11

Let A be a ring. Recall that a finite locally free A-module M is a module such that for every 𝔭 ∈ Spec A there exists an f ∈ A, f ∉ 𝔭 such that M_{f} is a finite free A_{f}-module.

1. Let A be a ring.

(a) Suppose that M is a finite locally free A-module, and suppose that φ : M → M is an endomorphism. Define/construct the trace and determinant of φ and prove that your construction is “functorial in the triple (A, M, φ)”.

(b) Show that if M, N are finite locally free A-modules, and if φ : M → N and ψ : N → M then T race(φ ∘ ψ) = T race(ψ ∘ φ) and Det(φ ∘ ψ) = Det(ψ ∘ φ).

(c) In case M is finite locally free show that Det defines a multiplicative map End_{A}(M) → A.

2. Now suppose that B is an A-algebra which is finite locally free as an A-module, in other words B is a finite locally free A-algebra.

(a) Define T race_{B/A} and Norm_{B/A} using T race and Det as defined above.

(b) Let b ∈ B and let π : Spec B → Spec A be the induced morphism. Show that π(V (b)) = V (Norm_{B/A}(b)). (Recall that V (f) = {𝔭 | f ∈ 𝔭}.)

(c) (Base change.) Suppose that i : A → A′ is a ring map. Set B′ = B ⊗_{A} A′. Indicate why i(Norm_{B/A}(b)) equals Norm_{B′/A′}(b ⊗ 1).

(d) Compute Norm_{B/A}(b) when B = A × A × A × … × A and b = (a_{1}, …, a_{n}).

(e) Compute the norm of y − y^{3} under the finite flat map ℚ[x] → ℚ[y], x → y^{n}. (Hint: use the “base change” A = ℚ[x] ⊂ A′ = ℚ(ζ_{n})(x^{1/n}).)

Remark. Let h ∈ ℤ[y] be a monic polynomial of degree d. Then:

(a) The map A = ℤ[x] → B = ℤ[y], x ↦ h is finite locally free of rank d.

(b) For all primes p the map A_{p} = 𝔽_{p}[x] → B_{p} = 𝔽_{p}[y], y ↦ h(y) mod p is finite locally free of rank d.

3. Let h, A, B, A_{p}, B_{p} be as in the remark. For f ∈ ℤ[x, u] we define f_{p}(x) = f(x, x^{p}) mod p ∈ 𝔽_{p}[x]. For g ∈ ℤ[y, v] we define g_{p}(y) = g(y, y^{p}) mod p ∈ 𝔽_{p}[y].

(a) Give an example of a h and g such that there does not exist a f with the property

f_{p} = Norm_{B_{p}/A_{p}}(g_{p}).

(b) Show that for any choice of h and g as above there exists a nonzero f such that for all p we have

Norm_{B_{p}/A_{p}}(g_{p}) divides f_{p}.

If you want you can restrict to the case h = y^{n}, even with n = 2, but it is true in general.

(c) Discuss the relevance of this to Exercises 6 & 7 of the previous set.

4. Unsolved problems. They may be really hard or they may be easy. I don’t know.

(a) Is there any f ∈ ℤ[x, u] such that f_{p} is irreducible for an inifinite number of p?

(b) Let f ∈ ℤ[x, u] nonzero, and suppose deg_{x}(f_{p}) = dp + d′ for all large p. (In other words deg_{u}(f) = d and the coefficient c of u^{d} in f has deg_{x}(c) = d′.) Suppose we can write d = d_{1} + d_{2} and d′ = d′_{1} + d′_{2} with d_{1}, d_{2} > 0 and d′_{1}, d′_{2} ≥ 0 such that for all sufficiently large p there exists a factorization

f_{p} = f_{1,p}f_{2,p}

with deg_{x}(f_{1,p}) = d_{1}p + d′_{1}. Is it true that f comes about via a norm construction as in Exercise 4? (More precisely, are there a h and g such that Norm_{B_{p}/A_{p}}(g_{p}) divides f_{p} for all p >> 0.)

(c) Analogous question to the one in (b) but now with f ∈ ℤ[x_{1}, x_{2}, u_{1}, u_{2}] irreducible and just assuming that f_{p}(x_{1}, x_{2}) = f(x_{1}, x_{2}, x_{1}^{p}, x_{2}^{p}) mod p factors for all p >> 0.