"Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree.

For example if we take the quotient of the polynomial ring in three variables by five random cubics:

$S = \mathbb{Q}[x1,x2,x3]$

$M$ = coker random( S^1, S^{5:-3} )

then Macaulay2 "always" (e.g. 1000 out of 1000 times) gives the following Betti table

total: 1 5 9 5

0: 1 . . .

1: . . . .

2: . 5 . .

3: . . 9 5

It seems that the behavior can be explained by the fact that we can resolve the cokernel of a generic matrix of the given form and this resolution remains exact when specializing to any point in a Zariski open subset of some affine space. My question is whether anyone knows a slick proof of this fact.

To elaborate: we can adjoin a new variable to our original ring for each coefficient appearing in each entry of the matrix. So in the above example we would adjoin 10*5 = 50 new variables to $S$, say $y1..y50$. Call the new ring $T$. Consider the $1x3$ matrix $N$ over $T$ whose entries are cubic in the $x_i$ and linear in the $y_i$. Resolve the cokernel of $N$ over $T$ to get a complex $F$. We can then substitute any point in $\mathbb{Q}^{50}$ into the maps of $F$ to get a complex over the original ring $S$. The claim is that this complex is exact on a Zariski open set of the affine space $\mathbb{Q}^{50}$.

It seems like this must be well known but I'm having trouble finding references.