Consider an $1\times n$ matrix $$ \mathbf{A}=\begin{pmatrix} f_1 &f_2 & \dots & f_n \end{pmatrix} $$ over $R=\mathbb{C}[X_1,\dots,X_r]$. Let $M=\oplus_{i=1}^n R\mathbf{e}_i$ be the rank-$n$ free $R$-module. We have the following Koszul complex: $$ \wedge^2 M \xrightarrow{h} M\xrightarrow{g} R $$ where $g$ sends $\mathbf{e}_i$ to $\mathbf{A}_{1i}=f_i$ and $h$ sends $\mathbf{e}_i\wedge \mathbf{e}_j$ to $f_j\mathbf{e}_i-f_i\mathbf{e}_j$. In some cases ($f_i$'s form a regular sequence?) the complex is exact, and hence $f_j\mathbf{e}_i-f_i\mathbf{e}_j$'s form a set of generators for $\ker(g)\subseteq M$. I also learned from the basic theory of grobner basis that when $f_i$'s are all monomials, one can modify $h$ so that it sends $\mathbf{e}_i\wedge\mathbf{e}_j$ to $\frac{\mathrm{lcm}(f_i,f_j)}{f_i}\mathbf{e}_i-\frac{\mathrm{lcm}(f_i,f_j)}{f_j}\mathbf{e}_j$ and then the sequence is exact.

Question: do we have similar results in the case that $A$ is a $m\times n$ matrix?

Specifically, for $m\leq n$, consider an $m\times n$ matrix $$ \mathbf{A}=\begin{pmatrix} f_{11} &\dots & f_{1n} \\ \dots & \dots & \dots \\ f_{m1} &\dots & f_{mn} \end{pmatrix} $$ over $R=\mathbb{C}[X_1,\dots,X_r]$. Consider the map $g$ defined by $\mathbf{A}$: $$ g:\oplus_{i=1}^n R\mathbf{e}_i\to\oplus_{i=1}^m R\mathbf{e}'_i $$ that sends $\mathbf{e}_i$ to $\sum_{j=1}^m f_{ji}\mathbf{e}'_j$. Can we explicitly specify elements of $\ker(g)$? When do they generate $\ker(g)$? In the case that all $f_{ij}$'s are monomials, do we have a similar result as the one mentioned above?

Thank you very much!


1 Answer 1


The proper context of this question is how to generalize the Koszul complex (or at least its second differential) for a matrix with 1 row to a matrix with an arbitrary number of rows. The generalization is provided by the Buchsbaum-Rim complex.

I will give the kernel for the generic case (when all entries are separate variables; more generally, when the ideal generated by the maximal minors of the matrix has the expected depth, which is $n-m+1$ if $n \ge m$). In all cases these will give relations, though there could be more if the depth assumption is not satisfied. A treatment of this in more detail can be found in Appendix A2.6 of Eisenbud's Commutative Algebra, or in the paper "A generalized Koszul complex. I" by Buchsbaum.

First consider the case $n=m+1$. Let $g_i$ be the $(-1)^{i+1}$ times the determinant of the matrix obtained by deleting column $i$. Then I claim $(g_1, \dots, g_n)$ is a kernel element. (In fact it's everything in the generic case.) To see this, take the $j$th row of the matrix and add it to the top. Clearly this has determinant 0, and doing Laplace expansion along this row shows that the $j$th entry of $g \cdot (g_1, \dots, g_n)^T$ is 0.

For the general case, we can do the same thing, but we get one relation for every choice of $m+1$ columns. This gives all relations in the generic case, though more work is needed to show that. So the module of relations is naturally isomorphic to an exterior power $\wedge^{m+1} R^n$.


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