Here we give an example of a spectral sequence that takes n+2 steps to degenerate, where n is the embedding dimension of the ring. We present this when n = 2 but the user with computational power can easily do a bigger case.









Since this spectral sequence only consists of $k$ vector spaces, and all are generated in a single degree, for ease of presentation we may as well just look at the rank and degree which we can easily encode in a matrix with $rt^d$ encoding the rank $r$ and degree $d$ of each vector space $E_{i,j}$.


To see what we're going for, we compute the E_{infinity} page and also some earlier pages. Notice that it's clear that all terms except those in the top row of the matrix must eventually disappear, but for this to happen, there must a map of the right degree mapping to them.






The final two computations are meant to explain that the copy of $k^8$ in degree 3 that appears on the $E_1$ cancels in two steps via an $E_2$ map with $k^6$ and via an $E_3$ map with a $k^2$.