# MA8202 Commutative Algebra (Fall 2012)

## Lecturer

Steffen Oppermann, room 844, Sentralbygg II, Steffen [dot] Oppermann [at] math [dot] ntnu [dot] no

## Schedule

Lectures will take place on Mondays, 10:15–12:00 and Wednesdays, 10:15–12:00 in room 822.

## Book

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

## Exercises

Date Problems
29.08. Exercise 1 (1.10 from the book)
Let $R$ be a ring. Show that the following are equivalent:
a) $R$ has exactly one prime ideal;
b) $R/\mathcal{R}$ is a field
c) every element of $R$ is either a unit or nilpotent
Exercise 2 (parts of 1.2 and 1.5)
Fint the Jacobson radicals of $\mathbb{C}[X]$ and $\mathbb{C}[[X]]$
Hints:
recall that for any $a \in \mathbb{C}$ we have an evaluation ringepimorphism $\mathbb{C}[X] \to \mathbb{C}$
show that $X$ lies in the Jacobson radical of $\mathbb{C}[[X]]$
Exercise 3 (1.15)
For a subset $E \subseteq R$ of a ring, we set $V(E) = \{ \mathfrak{p} \in {\rm Spec} R \mid E \subseteq \mathfrak{p} \}$.
Show that $\{V(E) \mid E \subseteq R\}$ are the closed set of a topology in ${\rm Spec} R$. This topology is called the Zariski topology.
05.09. Exercise 4 (Nakayama lemma)
Let $M$ be a finitely generated $R$-module. If $\mathcal{J} M = M$ then $M = 0$.
Exercise 5 Show
- $\mathbb{Z} / (2) \otimes_{\mathbb{Z}} \mathbb{Z} / (3) = 0$
- $\mathbb{Z}/(n) \otimes_{\mathbb{Z}} \mathbb{Z}/(m) = \mathbb{Z}/({\rm gcd}(m,n))$
- $R/ \mathfrak{a} \otimes_R R/\mathfrak{b} = R/(\mathfrak{a} + \mathfrak{b})$
Exercise 6 (2.3 from the book)
Let $R$ be a local ring, $M$ and $n$ finitely generated $R$-modules. If $M \otimes_R N = 0$ then $M = 0$ or $N = 0$.
12.09. Exercise 7 Determine ${\rm Spec} k[X]$ and ${\rm Spec} k[X, X^{-1}]$.
Exercise 8 Show that
"no nilpotent elements" is a local property;
"integral domain" is not a local property.
Exercise 9
For an $R$-module $M$ let ${\rm Supp}(M) = \{ \mathfrak{p} \in {\rm Spec}(R) \mid M_{\mathfrak{p}} \neq 0 \}$. Show
- ${\rm Supp}(M) = 0 \Leftrightarrow M = 0$
- ${\rm Supp}(R/\mathfrak{a}) = V(\mathfrak{a})$
- For any short exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ we have ${\rm Supp}(M_2) = {\rm Supp}(M_1) \cup {\rm Supp}(M_3)$.
- For finitely generated $M$ and $N$, we have ${\rm Supp}(M \otimes_R N) = {\rm Supp}(M) \cap {\rm Supp}(N)$.
19.09. Exercise 10 (5.31 from the book)
Let $\Gamma$ be a totally ordered abelian group, $K$ a field, and $v \colon K \setminus \{0\} \to \Gamma$ such that
- $v(k_1 k_2) = v(k_1) v(k_2)$ and
- $v(k_1 + k_2) \geq {\rm min}\{v(k_1), v(k_2)\}$.
Show that $V = \{ k \in K \setminus \{0\} \mid v(k) \geq 0 \} \cup \{0\}$ is a valuation ring of $K$.
Exercise 11
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 $R_{\mathfrak{m}}$.
26.09. Exercise 12 (6.1 from the book)
Let $M$ be an $R$-module, and $\varphi \colon M \to M$. Show
- If $M$ is noetherian, and $\varphi$ is epi, then $\varphi$ is iso;
- If $M$ is artinian, and $\varphi$ is mono, then $\varphi$ is iso;
Exercise 13 (6.4 from the book)
Let $M$ be a noetherian $R$-module. Show that $R / {\rm Ann}(M)$ is a noetherian ring.
Show that the corresponding statement for "artinian" does not hold.
03.10. Exercise 14
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 $r(\mathfrak{a})$ is prime.
Hint:
Assume $x,y \in R$ such that $xy \in r(\mathfrak{a})$, but $y\not\in r(\mathfrak{a})$.
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. $r(\mathfrak{a}) = \mathfrak{a}$) 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.
10.10. Exercise 15
Let $R$ be any ring. We consider the maps
$V(-) \colon \{\text{subsets of }R \} \to \{\text{subsets of } {\rm Spec} R\}$ - see Exercise 3 - and
$I(-) \colon \{\text{subsets of } {\rm Spec} R\} \to \{\text{subsets of }R \} \colon X \mapsto \bigcap_{\mathfrak{p} \in X} \mathfrak{p}$.
Show:
a) $I(V(S)) = r( (S) )$ for any $X \subseteq R$.
b) $V(I(X)) = \overline{X}$ for any $X \subseteq {\rm Spec} R$. (Here $\overline{X}$ denotes the closure of $X$ with respect to Zariski topology.)
Hint: First show that for closed $X$ one has $V(I(X)) = X$.
c) $V$ and $I$ induce mutually inverse bijections between $\{\text{radical ideals of } R\}$ and $\{ \text{closed subsets of } {\rm Spec} R\}$.
17.10. Exercise 16 (9.3 from the book)
Let $R$ be a valuation ring. Show that the following are equivalent:
- $R$ is notherian;
- $R$ is a discrete valuation ring.
Exercise 17 (9.7 from the book)
Let $R$ be a Dedekind domain. Show
a) For any non-zero $\mathfrak{a} \triangleleft R$ all ideals of $R / \mathfrak{a}$ are principal.
b) Any ideal of $R$ is generated by at most two elements.
24.10. Exercise 18
Let $R$ be an integral domain, $K$ its field of fractions (considered as an $R$-module). Let $\mathfrak{a}$ be a proper non-zero ideal of $R$. Show that $\widehat{R}_{\mathfrak{a}} \neq 0$, but $\widehat{K}_{\mathfrak{a}} = 0$. Conclude that taking $\mathfrak{a}$-adic completions is not exact.
Exercise 19
Let $R$ be noetherian, $\mathfrak{a} \triangleleft R$. Denote by $\widehat{\mathfrak{a}}$ the expansion of $\mathfrak{a}$ to $\widehat{R}_{\mathfrak{a}}$.
- Show that $\widehat{\mathfrak{a}}$ is contained in the Jacobson radical of $\widehat{R}_{\mathfrak{a}}$.
- Show that if $\mathfrak{a}$ is a maximal ideal then $\widehat{R}_{\mathfrak{a}}$ is a local ring.
31.10. Exercise 20
Let $M$ be an $R$-module. Show that $M$ is flat if and only if for any finitely generated ideal $\mathfrak{a} \triangleleft R$ inclusion induces a monomorphism $M \otimes_R \mathfrak{a} \to M$.
Strategy:
- May assume 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.)
Exercise 21 (10.4 from the book)
Let $\mathfrak{a} \triangleleft R$, $R$ noetherian, $M$ a finitely generated module. Show that $\widehat{M}_{\mathfrak{a}} = 0$ if and only if ${\rm Supp}(M) \cap V(\mathfrak{a}) = \emptyset$.
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 explicite description of this kernel.)
07.11. Exercise 22
Determine (independantly) ${\rm d}(R), \delta(R)$, and $\dim R$ for
- $R = k[[X]]$;
- $R = k[[X,Y]]/(XY)$.
14.11. Exercise 23
Let $k$ be an algebraically closed field. Determine $\dim k[X_1, \ldots, X_n]$.
Exercise 24
Let $R$ be a noetherian ring, and $\mathfrak{p}$ a prime ideal of height $h$.
a) Show that there is an ideal generated by $h$ elements $(r_1, \ldots, r_h )$ such that $\mathfrak{p}$ is minimal among prime ideals containing $(r_1, \ldots, r_h)$.
b) Show that if $\mathfrak{p}$ is minimal among prime ideals containing a given ideal $(r_1, \ldots, r_s)$, then $s \geq h$.
Hint: Consider the local ring.