---
tag: SXD7X0WQH1
kind: concept
name: Determinantal ideal
msc: [13C40, 14M12]
sources:
  - url: https://en.wikipedia.org/wiki/Determinantal_variety
---

## Summary

A determinantal ideal is the ideal generated by all minors of a fixed size of a matrix with entries in a commutative ring.
Take the generic matrix, whose entries are independent variables.
Then the zero set of the ideal is the determinantal variety of matrices of rank less than the size of the minors.
These ideals are among the best understood classes of ideals.
For the generic matrix they are prime, and their quotient rings are Cohen–Macaulay with a known height.
They appear in the study of resolutions, of rank conditions, and of secant varieties.

## Explanations

### L1 human

A matrix has rank less than $t$ exactly when all of its $t \times t$ minors vanish.
So the condition "rank less than $t$" is a system of polynomial equations in the entries.
The determinantal ideal is the ideal generated by those minors.
For a $2 \times 2$ matrix of variables $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $t = 2$, it is the principal ideal $(ad - bc)$, and its zero set is the set of singular $2 \times 2$ matrices.