The Stacks project

111.1 Algebra

This first section just contains some assorted questions.

Exercise 111.1.1. Let $A$ be a ring, and ${\mathfrak m}$ a maximal ideal. In $A[X]$ let $\tilde{\mathfrak m}_1 = ({\mathfrak m}, X)$ and $\tilde{\mathfrak m}_2 = ({\mathfrak m}, X-1)$. Show that

\[ A[X]_{\tilde{\mathfrak m}_1} \cong A[X]_{\tilde{\mathfrak m}_2}. \]

Exercise 111.1.2. Find an example of a non Noetherian ring $R$ such that every finitely generated ideal of $R$ is finitely presented as an $R$-module. (A ring is said to be coherent if the last property holds.)

Exercise 111.1.3. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring. For any finite $A$-module $M$ define $r(M)$ to be the minimum number of generators of $M$ as an $A$-module. This number equals $\dim _ k M/{\mathfrak m} M = \dim _ k M \otimes _ A k$ by NAK.

  1. Show that $r(M \otimes _ A N) = r(M)r(N)$.

  2. Let $I\subset A $ be an ideal with $r(I) > 1$. Show that $r(I^2) < r(I)^2$.

  3. Conclude that if every ideal in $A$ is a flat module, then $A$ is a PID (or a field).

Exercise 111.1.4. Let $k$ be a field. Show that the following pairs of $k$-algebras are not isomorphic:

  1. $k[x_1, \ldots , x_ n]$ and $k[x_1, \ldots , x_{n + 1}]$ for any $n\geq 1$.

  2. $k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 5$.

  3. $k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 6$.

Remark 111.1.5. Of course the idea of this exercise is to find a simple argument in each case rather than applying a “big” theorem. Nonetheless it is good to be guided by general principles.

Exercise 111.1.6. Algebra. (Silly and should be easy.)

  1. Give an example of a ring $A$ and a nonsplit short exact sequence of $A$-modules

    \[ 0 \to M_1 \to M_2 \to M_3 \to 0. \]
  2. Give an example of a nonsplit sequence of $A$-modules as above and a faithfully flat $A \to B$ such that

    \[ 0 \to M_1\otimes _ A B \to M_2\otimes _ A B \to M_3\otimes _ A B \to 0. \]

    is split as a sequence of $B$-modules.

Exercise 111.1.7. Suppose that $k$ is a field having a primitive $n$th root of unity $\zeta $. This means that $\zeta ^ n = 1$, but $\zeta ^ m\not= 1$ for $0 < m < n$.

  1. Show that the characteristic of $k$ is prime to $n$.

  2. Suppose that $a \in k$ is an element of $k$ which is not an $d$th power in $k$ for any divisor $d$ of $n$ for $n \geq d > 1$. Show that $k[x]/(x^ n-a)$ is a field. (Hint: Consider a splitting field for $x^ n-a$ and use Galois theory.)

Exercise 111.1.8. Let $\nu : k[x]\setminus \{ 0\} \to {\mathbf Z}$ be a map with the following properties: $\nu (fg) = \nu (f) + \nu (g)$ whenever $f$, $g$ not zero, and $\nu (f + g) \geq min(\nu (f), \nu (g))$ whenever $f$, $g$, $f + g$ are not zero, and $\nu (c) = 0$ for all $c\in k^*$.

  1. Show that if $f$, $g$, and $f + g$ are nonzero and $\nu (f) \not= \nu (g)$ then we have equality $\nu (f + g) = min(\nu (f), \nu (g))$.

  2. Show that if $f = \sum a_ i x^ i$, $f\not= 0$, then $\nu (f) \geq min(\{ i\nu (x)\} _{a_ i\not= 0})$. When does equality hold?

  3. Show that if $\nu $ attains a negative value then $\nu (f) = -n \deg (f)$ for some $n\in {\mathbf N}$.

  4. Suppose $\nu (x) \geq 0$. Show that $\{ f \mid f = 0, \ or\ \nu (f) > 0\} $ is a prime ideal of $k[x]$.

  5. Describe all possible $\nu $.

Let $A$ be a ring. An idempotent is an element $e \in A$ such that $e^2 = e$. The elements $1$ and $0$ are always idempotent. A nontrivial idempotent is an idempotent which is not equal to zero. Two idempotents $e, e' \in A$ are called orthogonal if $ee' = 0$.

Exercise 111.1.9. Let $A$ be a ring. Show that $A$ is a product of two nonzero rings if and only if $A$ has a nontrivial idempotent.

Exercise 111.1.10. Let $A$ be a ring and let $I \subset A$ be a locally nilpotent ideal. Show that the map $A \to A/I$ induces a bijection on idempotents. (Hint: It may be easier to prove this when $I$ is nilpotent. Do this first. Then use “absolute Noetherian reduction” to reduce to the nilpotent case.)


Comments (2)

Comment #8727 by Jim Davis on

Exercise 078G. Replace idempotent by central idempotent


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0276. Beware of the difference between the letter 'O' and the digit '0'.