MA8202 Commutative Algebra (Fall 2012)


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


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


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


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 \(\mathbb{C}[[X]]\)
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.
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
\(I(-) \colon \{\text{subsets of } {\rm Spec} R\} \to \{\text{subsets of }R \} \colon X \mapsto \bigcap_{\mathfrak{p} \in X} \mathfrak{p} \).
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 \( M \otimes_R \mathfrak{a} \to M \).
- 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 \( {\rm Supp}(M) \cap V(\mathfrak{a}) = \emptyset \).
- 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.
2012-12-06, Steffen Oppermann