Semi-small type decomposition for the graded Möbius algebra in low degrees

Ascertain whether, for k ≤ d/2, the truncated graded Möbius algebra D^k H(M \ i) appears as a summand in the Krull–Schmidt decomposition of D^k H(M), where D^k denotes truncation to degree k and i ∈ E.

Background

Semi-small decompositions are known for matroid Chow-type rings and are central to inductive proofs of Hodge-theoretic properties. Extending analogous decompositions to the graded Möbius algebra, at least in low degrees, would provide new tools to understand its structure and its relation to A(M).