The Fitting Ideal of Level 0 is Independent of Choice of Generators (15.1.36-37)

Let $M$ be a finitely generated module over $R$.

36. (a) Show that for all $p≥n$, the Fitting ideal of $M$ is also the ideal in $R$ generated by all the $n×n$ minors of all $p×n$ matrices.
(b) Let $A$ be a fixed $p×n$ matrix as in (a) and let $A'$ be a $p×n$ matrix obtained from $A$ by any elementary row or column operation. Show that the ideal in $R$ generated by all the $n×n$ minors of $A$ is the same as the ideal in $R$ generated by all the $n×n$ minors of $A'$.

37. Suppose $m_1,...,m_n$ and $m_1',...,m_{n'}'$ are two sets of generators for $M$. Let $\mathcal{F}$ be the Fitting ideal calculated with respect to $m_1,...,m_n$ and let $\mathcal{F}'$ be the Fitting ideal calculated with respect to $m_1,...,m_n,m_1',...,m_{n'}'$.

(a) Show that $m_s'=a_{s',1}m_1+...+a_{s',n}m_n$ for some $a_{s',i}∈R$ for all $1 ≤ s ≤ n'$, so that $(-a_{s',1},...,-a_{s',n},0,...,0,1,0,...,0)$ is a relation among $m_1,...,m_n,m_1',...,m_{n'}'$.
(b) If $A$ is an $n×n$ matrix whose rows are the coefficients of relations among $m_1,...,m_n$, show that $\text{det }A=\text{det }A'$ where $A'$ is an $(n+n')×(n+n')$ matrix whose rows are the coefficients of relations among $m_1,...,m_n,m_1',...,m_{n'}'$. Deduce $\mathcal{F}⊆\mathcal{F}'$.
(c) Prove $\mathcal{F}'⊆\mathcal{F}$ and conclude $\mathcal{F}=\mathcal{F}'$.
(d) Deduce from (c) that the Fitting ideal is independent of choice of generators for $M$.

Proof: (36)(a) $⊆$ is clear by simply fitting every $n×n$ matrix of relations into a $p×n$ relations matrix whose minor will be included in the latter ideal, and $⊇$ is also clear since every such minor is the determinant of an $n×n$ relations matrix.

(b) Let $a$ be an $n×n$ minor, the determinant of an $n×n$ matrix $N$. Then let $a'$ be the determinant of the matrix $N'$, which is identical to $N$ save for one of its rows or columns being replaced with another part of a row or column in the $p×n$ matrix. Then since determinants are $R$-bilinear on rows and columns, we have $a+r·a'$ is the $n×n$ minor under the column or row operation on $p×n$ by adding $r$ times the specified row or column to another affecting $N$. Since $N'$ is undisturbed, we can retrieve $a$ the original minor in the new ideal, and since this minor was arbitrary, this shows that the two ideals are equal.

(37)(a) Self-explanatory.

(b) Let $A'$ be the block matrix $$\begin{bmatrix} A & 0 \\ B & I \end{bmatrix}$$ where $I$ is the $n' × n'$ identity, and $B$ is the $n' × n$ matrix converting $n × 1$ vectors in terms of $m_i$ into $n' × 1$ vectors in terms of $m_i'$. Then $\text{det }A = \text{det }A'$ where $A'$ is a matrix of relations among $m_1,...,m_n,m_1',...,m_{n'}'$ as desired.

(c) Let $A$ be an $(n+n')×(n+n')$ relations matrix, and observe that its determinant ideal is contained in the $(n+n')×(n+n')$ minor-generated ideal of the $(n+2n')×(n+n')$ block matrix $A'$, where the top $(n+n')×(n+n')$ block is $A$ and the bottom $n'×(n+n')$ block is $[B~~I]$ as above. By 36(b) we may perform row operations on $A'$ to remain $A''$ a block matrix $$\begin{bmatrix} C & 0 \\ B & I \end{bmatrix}$$ where $C$ is an $(n+n')×n$ relations matrix in terms of $m_1,...,m_n$. The nontrivial minors of this matrix are those $n×n$ minors of $C$, which by 36(a) are all contained in $\mathcal{F}$. Since $\mathcal{F}'$ is the smallest ideal containing all these determinant ideals, we have $\mathcal{F}'⊆\mathcal{F}$ and $\mathcal{F}=\mathcal{F}'$.

(d) When $\mathcal{F}''$ is the Fitting ideal calculated with respect to $m_1',...,m_{n'}'$, we have $\mathcal{F}=\mathcal{F}'=\mathcal{F}''$.$~\square$