These compositions agree in general, though the proof is reasonably long. The idea is to expand the $*-!$ base change into a factorization, and prove it in the proper/etale cases separately. This is still formal, just using the triangle identities and unravelling base change. I'm writing up a general method for these kinds of problems, should be on arxiv in the new year. Let me know if this proof sketch doesn't suffice.

In your computations, may one always take $n$ to be the dimension of $\mathcal{A}$?

If one takes the view that the data of a spectral sequence is equivalent to the data of the filtered object, the following might be relevant. arxiv.org/abs/2109.01017

I'm not sure of a specific reference for this, but any source on Schubert varieties explains their stratification into affine spaces. For computing the cohomology, it's just the computation of cellular cohomology when all (CW) cells are even dimensional, this is explained in any algebraic topology text, eg, Hatcher's Algebraic topology.

As someone who is only an epsilon further than you down this path, I found the Gille, Szamuely book on CSAs and Galois cohomology to be incredibly readable, and it gives a good quick introduction to lots of ideas in this area. It doesn't use any homotopy theory though, it's more for the algebraic side.

In characteristic zero, the representation category is too simple to apply noncommutative geometric ideas (to my knowledge), there's just no homological algebra present. In the modular case though, there are definitely parallels, though I'll let someone more qualified explain them.

This is well answered here :) mathoverflow.net/questions/402623/…

Oh true, thanks for pointing that out!

Fantastic! Sorry for the dupe question.

A small point, but you need $\mathbb{Q}$ coefficients to have these generating sets agree in general, I think you only get $2e_2$ in this ideal with $\mathbb{Z}$ coefficients.

Yea having very few possibilities for maximal distance $k$ tuples would probably be more flexible. The examples I had in mind also had a kind of nondegeneracy, where one can complete $i$ tuples to $k$ tuples, but I’d be interested in either case. Even for say, graphs I’d be curious what this property resembles.