I want to prove the determinantal ideals over a field are prime ideals. To be concrete:

For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be an ideal of the polynomial ring $k[x_{11},\ldots,x_{23}]$. I have no idea how to prove that $I$ is a radical ideal (i.e. $I=\sqrt{I}$). Could anyone give some hints?

Generally, let $K$ be an algebraically closed field, then $\{A\mid\mathrm{Rank}(A)\leq r\}\subseteq K^{m\times n}$ is an irreducible algebraic set (I first saw this result from this question). And I tried to prove this by myself, then I have proved it (when I see the "Segre embedding").

But I have no idea how to show that the "determinantal ideals" are radical ideals (I hope this is true). BTW, is the statement that the determinantal ideals over a field are prime ideals true ?


  • $\begingroup$ In fact, I really want a direct algebraic (/elementary? )proof here. $\endgroup$ – wxu Jul 29 '11 at 23:49

There are several ways to prove that $I$ is radical. By the way, the statement that $I$ is prime is equivalent to $I$ being radical and the zero set of $I$ being an irreducible algebraic set.

An approach using Gröbner bases can be found in Chapter 16 of Miller-Sturmfels, Combinatorial Commutative Algebra

An approach using sheaf cohomology can be found in Sections 6.1-6.2 of Weyman, Cohomology of Vector Bundles and Syzygies. This requires a lot more background knowledge.

There is also the approach using induction on the size of the matrix and localization arguments in Chapter 2 of Bruns-Vetter, Determinantal Rings. Link to book: http://www.home.uni-osnabrueck.de/wbruns/brunsw/detrings.pdf


This isn't really an answer but I can point you to a reference that might be of some help:

For a discussion of this example for the ideal generated by 2x2 minors of a 3x3 matrix, see this nice discussion in Eisenbud's Commutative Algebra, p 107.

It seems that the general case is much more difficult. Eisenbud also mentions that Bruns and Vetter, Determinantal Rings [1988] is a nice reference for the general case.

I hope someone else can come along to tell you something more useful!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.