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

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.

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 $Fit_ k(M)$.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.)

Show that the following are equivalent:

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

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

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