Scarf resolution conjecture for powers of extremal ideals

Prove that for every pair of positive integers r and q, the Scarf complex of the rth power of the q-extremal ideal E_q supports a minimal free resolution of E_q^r.

Background

The paper studies powers of extremal ideals E_q because they provide universal upper bounds for Betti numbers and related invariants of powers of square-free monomial ideals with q generators. The Scarf complex is the smallest canonical subcomplex that every resolution must contain, and a Scarf resolution would therefore give sharp combinatorial control of the minimal free resolution.

The conjecture is known for r less than or equal to 2 or q less than or equal to 4, and the paper proves the first unresolved power case r=3 for every q. The cases with r greater than or equal to 4 remain covered by the conjecture.

References

Our guiding conjecture is that all powers of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for r ≤ 2 or q ≤ 4.

— Realizing resolutions of powers of extremal ideals  (2502.09585 - Chau et al., 13 Feb 2025) in Conjecture 1.1 in Section 1, Introduction