Let $R= k[x\_{11} , x\_{12} \dotsc , x\_{nm}]$ denote the coordinate ring of a generic $n \times m$ matrix, $M$. It is well known that under the standard diagonal term order, the ideal of maximal minors of $M$ forms a Grobner basis (I am aware that it is a universal Grobner basis, but I don't need that result).


I am looking for a classical or elementary proof of this result, relying almost entirely on Buchberger's algorithm. I assume that this is written somewhere, but cannot find this result in any of the standard references. Thanks!



"Classical" proof that maximal minors form a Grobner basis under diagonal term order
Here are a few articles that might be relevant:


[Narasimhan, *The irreducibility of ladder determinantal varieties*](https://mathscinet.ams.org/mathscinet-getitem?mr=853237), from 1986. It takes about 20 pages to prove the result (that the minors are a standard basis for the ideal they generate). I confess, I don't want to try to read it; perhaps it can be streamlined.


I'm not sure about this one: [Caniglia, Guccione, Guccione, *Ideals of generic minors*](https://mathscinet.ams.org/mathscinet-getitem?mr=1074248), from 1990. I don't have access to the article, and the review doesn't describe the method of proof. So I don't know whether they stick to "elementary" methods.


[Herzog and Trung](https://mathscinet.ams.org/mathscinet-getitem?mr=1185786) and [Sturmfels](https://mathscinet.ams.org/mathscinet-getitem?mr=1069489) use Knuth-Robinson-Schensted correspondence and straightening laws.


