Advanced Algebra II

Advanced Algebra II

Homework 4 due on Apr. 13, 2007

(1) * Complete the exercises and incomplete proofs in the note.

(2) Let M be a Noetherian R-module, and let aC be the annihilator of M. Prove that M is a Noetherian R/a-module. How about if we replace Noetherian by Artinian?

(3) * Let R be a Noetherian local ring and M be a finitely generated R-module. Show that M is free if and only if M is flat.

(4) Let R be an Noetherian ring, and q be a p-primary ideal. Show that there exists n ≥ 1 such that pⁿ ⊂ q.

Is it still true if R is not necessarily Noetherian?

(5) Let f : A → B be a homomorphism between local rings (A, m_A), (B, m_B).

We say that f is local if f⁻¹(m_B) = m_A.

If we start with a homomorphism f : A → B of rings. For any q ∈ Spec(B), we have p := f⁻¹q ∈ Spec(A). Show that the induced map A_p → B_q is local.

(6) Let k be an algebraically closed field. Consider the ring homo- morphism f : A := k[x] → B := k[x, y]/(y² − x) which sends f (x) = x.

(a) Show that B is integral over A.

(b) For each prime ideal p ∈ Spec(A), determine the prime ideals of B lying over p.

(c) Show that for each prime ideal q ∈ Spec(B), lying over p, we have a local homomorphism (A_p, m_p) → (B_q, m_q).

Moreover, a k-vector space homomorphism f_q : m_p/(m_p)² → m_q/(m_q)².

(d) Show that for q 6= 0, all the above vector space m_p/(m_p)², m_q/(m_q)² has dimension 1. And also determine when f_q is not iso-

morphism.

(7) Consider B = k[x, y]/(xy − 1).

(a) Let A₁ be the subring generated by x, show that B is not integral over A₁.

(b) Let A2 be the subring generated by x + y, show that B is integral over A2.

(c) Show that dim k[x, y]/(xy − 1) = 1.

(8) Let R be a local Noetherian domain of dim R = 1. Show that R is integrally closed if and only if the maximal ideal is principal and every ideal is of the form mⁿ.

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