---
tag: M4606WJDYG
kind: definition
name: Determinantal ideal
msc: [13C40, 14M12]
defines: SXD7X0WQH1
uses: [0310REJJY8, 0VD9ZSWWJG, 0BT9BYWVRY, 096Z9MP2ME, 0SDC934R52, 0ZC9YTXZRY, 00MHHZ9SNE, 87WH65FJB8, N1QJQEGBFC]
sources:
  - url: https://en.wikipedia.org/wiki/Determinantal_variety
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]], let $M$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] with entries in $R$, and let $t$ be an integer with $1 \le t \le \min(m, n)$.
The *determinantal ideal* $I_t(M)$ is the [[ideal|096Z9MP2ME]] of $R$ generated by all $t \times t$ [[minors|0BT9BYWVRY]] of $M$.
When $k$ is a field, $R = k[x_{ij} : 1 \le i \le m, \; 1 \le j \le n]$ is the [[polynomial ring|0SDC934R52]] and $X = (x_{ij})$ is the generic matrix, $I_t(X)$ is the *generic determinantal ideal*.

## Remarks

When $k$ is algebraically closed, the zero set of $I_t(X)$ in $k^{mn}$ is the set of $m \times n$ matrices of [[rank|0ZC9YTXZRY]] less than $t$.
For the generic matrix, $I_t(X)$ is a [[prime ideal|00MHHZ9SNE]] of [[height|87WH65FJB8]] $(m - t + 1)(n - t + 1)$, and $R/I_t(X)$ is [[Cohen–Macaulay|N1QJQEGBFC]].

Example: for $m = 2$, $n = 3$ and $t = 2$, the ideal $I_2(X)$ is generated by the three minors $x_{11} x_{22} - x_{12} x_{21}$, $x_{11} x_{23} - x_{13} x_{21}$ and $x_{12} x_{23} - x_{13} x_{22}$.
Its height is $(2 - 2 + 1)(3 - 2 + 1) = 2$.

Non-example: in the same ring, the principal ideal $(x_{11} x_{22} - x_{12} x_{21})$ is generated by one $2 \times 2$ minor only.
It has height $1$ by Krull's principal ideal theorem, so it is not $I_2(X)$, which has height $2$.