Elementary ideals of a finitely presented module

Let $R$ be a commutative ring (Commutative ring) and let $M$ be a finitely presented $R$-module (Finitely presented modules and finitely presented algebras), so that there are $m,n\in \mathbb{N}$ and a presentation

$R^m\stackrel{A}{\to }{R}^n\stackrel{\beta }{\to }M⟶0.$

Fix the standard bases $e_1,\dots ,e_m$ of $R^m$ and $f_1,\dots ,f_n$ of $R^n$ (Finitely presented modules and finitely presented algebras) and let $A=(a_{ij})$ be the $n\times m$ matrix of the $R$-linear map $\alpha$ with respect to them: the $j$-th column of $A$ is the coordinate vector of $\alpha (e_j)$, so $\alpha (e_j)={\sum }_{i=1}^n{a}_{ij}{f}_i$. Such a matrix is called a presentation matrix of $M$ (for the chosen presentation). A minor of $A$ of size $r\times r$ is the determinant $\det (a_{i_s{j}_t}{)}_{s,t=1}^r$ (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix) of the square submatrix obtained by choosing rows $i_1<\cdots <i_r$ and columns $j_1<\cdots <j_r$; determinants of $1\times 1$ and $0\times 0$ submatrices are allowed, with the empty determinant equal to $1$.

For $k\ge 0$ the $k$-th elementary ideal $E_k(M)$ of the presented module $M$ is the ideal of $R$ generated by the determinants of all $(n-k)\times (n-k)$ minors of $A$ (The ideal generated by a subset and principal ideals, Left, right and two-sided ideals): if $n-k>n$ or $n-k>m$ there is no such minor, and the ideal generated by the empty set is $0$. In particular no minor is nonzero-by-convention: the conventions are $E_k(M)=R\text{for }k\ge n,E_k(M)=0\text{for }k<n-m\mathrm{.}$ For $0\le k\le \min (n,m)$ the ideal $E_k(M)$ is the ideal generated by the $(n-k)\times (n-k)$ minors; e.g. $E_1(M)$ is generated by the $(n-1)\times (n-1)$ minors of $A$, and $E_n(M)=R$ because the $0\times 0$ minor is $1$. This is the $k$-th Fitting ideal ${\mathrm{Fit}}_k(M)$ of the Stacks Project (Tag 07Z6, Lemma 15.8.2 and Definition 15.8.3), which is indexed by the size $n-k$ of the complementary minors; the two conventions $0=E_{-1}\subset E_0\subset \cdots$ agree on the range $0\le k\le \min (n,m)$.

Caveats. (i) The notation $E_k(M)$ suppresses the chosen presentation; the ideal defined here is a priori attached to the presentation and it is the content of Elementary ideals are independent of the presentation that it is in fact an invariant of $M$. (ii) The conventions in the degenerate ranges are conventions, not theorems; they are the ones compatible with the Stacks numbering and with independence of the presentation.