MA8202 Commutative Algebra (Fall 2014)

NOTE: The lecture on Friday September 5th is moved to Tuesday September 2nd 08:15-10:00, room 822, 8th floor, SBII.

Lecturer

Øyvind Solberg, room 854, Sentralbygg II, Oyvind [dot] Solberg [at] math [dot] ntnu [dot] no

Schedule

Lectures will take place on

Thursdays, 12:15–14:00, room 822, 8th floor, SBII.

Fridays, 12:15–14:00, room 734, 7th floor, SBII (except September 12th, then in 822).

Book

The course will follow the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.

Look at this link to see the syllabus for autumn 2014

Exercises

Extra exercises marked with -.

Date Problem sheets
04.09 Problem 1 = Exercise 1 (page 10)
Problem 2: Find the nilradical and radical of the rings \(k[x]\) and \(k[[x]]\).
Problem 3 = Exercise 10 (page 11)
Problem 4 = Exercise 1.15 (page 12)
25.09 Problem 5 (Nakayama lemma)
Let \(M\) be a finitely generated \(A\)-module. If \(\mathcal{J} M = M\) for an ideal \(J\) inside the Jacobson radical of \(A\), then \(M = 0\).
Problem 6 Show
- \(\mathbb{Z} / (2) \otimes_{\mathbb{Z}} \mathbb{Z} / (3) = 0\)
- \(\mathbb{Z}/(n) \otimes_{\mathbb{Z}} \mathbb{Z}/(m) = \mathbb{Z}/({\rm gcd}(m,n)) \)
- \( R/ \mathfrak{a} \otimes_R R/\mathfrak{b} = R/(\mathfrak{a} + \mathfrak{b}) \)
Problem 7 Exercise 2 (page 31)
Problem 8 Exercise 3 (page 31)
-Problem 9 Exerice 10 and 13 (page 32).
Problem 10 Exercise 1 (page 43).
Problem 11 Exercise 5 (page 44).
Problem 12 Exercise 19 i), ii), iii), v) and vi) (page 46).
09.10 Problem 13 = Exercise 23 (page 71)
Problem 14 = Exercise 24 (page 71)
Problem 15 = Exercise 31 (page 72)
Problem 16: Let \(k\) be an algebraically closed field, \(R = k[X,Y] / (XY) \).
- Determine \( {\rm MaxSpec}(R) \).
- For any \(\mathfrak{m} \in {\rm MaxSpec}(R)\), find the local ring \(R_{\mathfrak{m}}\).
23.10 Problem 17 = Exercise 1 (page 78)
Problem 18 = Exercise 3 (page 78)
Problem 19 = Exercise 4 (page 78)
Problem 20 Let \(R\) be a noetherian ring.
a) Show that any prime ideal is irreducible.
b) Show that for any irreducible ideal \(\mathfrak{a}\), the radical \(\sqrt{\mathfrak{a}}\) is prime.
Hint: Assume \( x,y \in R\) such that \(xy \in \sqrt{\mathfrak{a}} \), but \(y\not\in \sqrt{\mathfrak{a}}\).
Consider a chain of ideals formed by the \( (\mathfrak{a}:x^i) \).
Show that \(\mathfrak{a} = (\mathfrak{a} + (y^m)) \cap (\mathfrak{a} + (x^n)) \) for certain \(m, n\).
c) Show that any radical ideal (i.e. \(\mathfrak{a} = \sqrt{\mathfrak{a}} \)) can be written as a finite intersection of prime ideals.
Hint: Start with an irreducible decomposition and take radicals.
d) Show that the prime ideals in c) are unique, if non of them is a subset of another.
Prep time Problem 21 = Exercise 3 (page 99)
Problem 22 = Exercise 7 (page 99)
Problem 23 (We used this fact in the lectures)
Let \(M\) be an \(R\)-module. Show that \(M\) is flat if and only if for any finitely generated ideal \(\mathfrak{a} \triangleleft R\) the inclusion induces a monomorphism \( M \otimes_R \mathfrak{a} \to M \).
Strategy:
- May assume that we know that there are enough injective modules.
- Show: \(M\) flat ⇔ for every injective \(I\), \({\rm Hom}_R(M, I) \) is injective. (Use Hom-tensor-adjunction.)
- Show: A module \(X\) is injective if and only any map from an ideal to \(X\) can be extended to a map from \(R\) to \(X\). (Given any monomorphism \(A \to B\), and map \(A \to X\), use Zorn's lemma to extend as far as possible.)
Problem 24 = Exercise 3 (page 114)
Strategy:
- Let \(X = \bigcap_{\mathfrak{a} \subseteq \mathfrak{m} \in {\rm MaxSpec} R} {\rm Ker}[M \to M_{\mathfrak{m}}] \). Show that \(\mathfrak{a} X = X\). (Show that \(X_{\mathfrak{m}} = 0\) whenever \(\mathfrak{a} \subseteq \mathfrak{m} \in {\rm MaxSpec}R\).)
- Show that \(X = {\rm Ker}[M \to \widehat{M}_{\mathfrak{a}}]\). (Use the explicit description of this kernel.)
Problem 25
Determine, without using the Dimension theorem, \({\rm d}(R), \delta(R)\), and \(\dim R\) for
- \(R = k[[X]]\);
- \(R = k[[X,Y]]/(XY) \).
Problem 26 = Exercise 1 (page 125)
Problem 27 = Exercise 7 (page 126)
2014-11-21, Øyvind Solberg