Let $k$ be an algebraically closed field. Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of $A$.

Is $k[x_1, \ldots, x_p]/I$ a Cohen-Macaulay ring?

If no, does the additional assumption that $I$ is a radical (or prime or any other nice property) ideal help?

If yes, is there a good reference for that?


No, the ring need not be Cohen-Macaulay. For instance, when $p$ equals $2$, $m$ equals $2$ and $n$ equals $3$, consider the matrix, $$ A = \left[ \begin{array}{rrr} x & 0 & 0 \\ 0 & x & y \end{array} \right].$$ Your ideal is $I = \langle x^2,xy \rangle$, so that $k[x,y]/I$ is not Cohen-Macaulay.

The positive results I know of all have a hypothesis on the height of $I$ relative to $(n-r)(m-r)$, where $r$ is the rank of the matrix. The authors (if I recall correctly) are Eagon, Northcott, Hochster, ..., probably others (it has been a while since I looked at this).

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    $\begingroup$ As far as I know, the mentioned authors deal with matrices of indeterminates and/or matrices whose entries are in a commutative ring. For the special case of matrices whose entries are linear forms I suggest this paper: msri.org/~de/papers/pdfs/1988-003.pdf, although there is no answer to the OP's question there. $\endgroup$
    – user26857
    Nov 27 '14 at 9:17

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