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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 polynomial ring $k[x_{11},\ldots,x_{23}]$, I have no ideas to prove $I$ is a radical ideal(i.e. $I=\sqrt{I}$), could anyone give some hints?

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

But I have no ideas 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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In fact, I really want a direct algebraic (/elementary? )proof here. –  wxu Jul 29 '11 at 23:49

2 Answers 2

up vote 4 down vote accepted

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: http://books.google.com/books?id=Fm_yPgZBucMC&pg=PA107&lpg=PA107&dq=Eisenbud+determinantal&source=bl&ots=7Bi1FDR1Ip&sig=2a-d2oagWJgV8r0EhZq-EFusAIw&hl=en&ei=q2swTpTeBuj00gGN5M2GAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q=Eisenbud%20determinantal&f=false

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: http://www.amazon.com/Determinantal-Rings-Lecture-Notes-Mathematics/dp/3540194681/ref=ntt_at_ep_dpt_1

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

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