Lemma 10.15.5 (07DQ)—The Stacks project

Lemma 10.15.5. Let $R$ be a ring. Let $n \geq m$ . Let $A$ be an $n \times m$ matrix with coefficients in $R$ . Let $J \subset R$ be the ideal generated by the $m \times m$ minors of $A$ .

For any $f \in J$ there exists a $m \times n$ matrix $B$ such that $BA = f 1_{m \times m}$ .
If $f \in R$ and $BA = f 1_{m \times m}$ for some $m \times n$ matrix $B$ , then $f^ m \in J$ .

Proof. For $I \subset \{ 1, \ldots , n\}$ with $|I| = m$ , we denote by $E_ I$ the $m \times n$ matrix of the projection

$R^{\oplus n} = \bigoplus \nolimits _{i \in \{ 1, \ldots , n\} } R \longrightarrow \bigoplus \nolimits _{i \in I} R$

and set $A_ I = E_ I A$ , i.e., $A_ I$ is the $m \times m$ matrix whose rows are the rows of $A$ with indices in $I$ . Let $B_ I$ be the adjugate (transpose of cofactor) matrix to $A_ I$ , i.e., such that $A_ I B_ I = B_ I A_ I = \det (A_ I) 1_{m \times m}$ . The $m \times m$ minors of $A$ are the determinants $\det A_ I$ for all the $I \subset \{ 1, \ldots , n\}$ with $|I| = m$ . If $f \in J$ then we can write $f = \sum c_ I \det (A_ I)$ for some $c_ I \in R$ . Set $B = \sum c_ I B_ I E_ I$ to see that (1) holds.

If $f 1_{m \times m} = BA$ then by the Cauchy-Binet formula (72) we have $f^ m = \sum b_ I \det (A_ I)$ where $b_ I$ is the determinant of the $m \times m$ matrix whose columns are the columns of $B$ with indices in $I$ . $\square$

Comments (2)

Comment #3577 by Herman Rohrbach on September 17, 2018 at 10:18

If the philosophy of this section of not using any results other than those listed in the basic notions is to be followed, perhaps the Cauchy-Binet formula should either be included in the basic notions, or stated in this section? I know it's not very important, but I found it worth to at least mention it.

Comment #3701 by Johan on October 22, 2018 at 17:16

Thanks. I added the Cauchy-Binet formula to the list of basic notions here.

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