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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 ?

Thanks.

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

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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!

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