====== 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.Solberg@math.ntnu.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 [[http://www.math.ntnu.no/~oyvinso/Courses/MA8202/2014/ma8202-syllabus.pdf|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) |