Is there a method for computing the generators of the cohomology of a chain complex over the ring $R=F[x,y]$?
When $R$ is a PID one can use Smith normal form and there are libraries in Sage that compute the kernel and cokernel.
If it helps working over the two-dimensional regular local rings $R=F[x,y]_{(x,y)}$ is enough for my purpose and I only need the generators for the torsion-free part of the cohomology.