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!