# 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) |