I will explain a generalisation of the constructions Quillen used to prove that the $K$-groups of rings of integers are finitely generated. It takes the form of a ‘rank’ spectral sequence, converging to the homology of Quillen’s $Q$-construction on the category of coherent sheaves over a Noetherian integral scheme, and whose $E^1$ terms are given by homology of Steinberg modules. Computing its $d^1$ differentials is a challenge, which can be approached through the universal modular symbols of Ash-Rudolph.

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Last updated: 20 Mar 2023