# 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)