The Stacks project

111.25 Fitting ideals

Exercise 111.25.1. Let $R$ be a ring and let $M$ be a finite $R$-module. Choose a presentation

\[ \bigoplus \nolimits _{j \in J} R \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0. \]

of $M$. Note that the map $R^{\oplus n} \to M$ is given by a sequence of elements $x_1, \ldots , x_ n$ of $M$. The elements $x_ i$ are generators of $M$. The map $\bigoplus _{j \in J} R \to R^{\oplus n}$ is given by a $n \times J$ matrix $A$ with coefficients in $R$. In other words, $A = (a_{ij})_{i = 1, \ldots , n, j \in J}$. The columns $(a_{1j}, \ldots , a_{nj})$, $j \in J$ of $A$ are said to be the relations. Any vector $(r_ i) \in R^{\oplus n}$ such that $\sum r_ i x_ i = 0$ is a linear combination of the columns of $A$. Of course any finite $R$-module has a lot of different presentations.

  1. Show that the ideal generated by the $(n - k) \times (n - k)$ minors of $A$ is independent of the choice of the presentation. This ideal is the $k$th Fitting ideal of $M$. Notation $Fit_ k(M)$.

  2. Show that $Fit_0(M) \subset Fit_1(M) \subset Fit_2(M) \subset \ldots $. (Hint: Use that a determinant can be computed by expanding along a column.)

  3. Show that the following are equivalent:

    1. $Fit_{r - 1}(M) = (0)$ and $Fit_ r(M) = R$, and

    2. $M$ is locally free of rank $r$.

Comments (0)

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Your email address will not be published.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 027V. Beware of the difference between the letter 'O' and the digit '0'.