Total Rank Conjecture

Establish that for a d-dimensional Noetherian local ring R and a finitely generated nonzero R-module M with finite length and finite projective dimension, the sum of Betti numbers across all homological degrees satisfies ∑_{i≥0} β_i(M) ≥ 2^d.

Background

Introduced by Avramov and Buchweitz as a weaker form of the BEH Conjecture, the Total Rank Conjecture claims a lower bound on the total Betti number count. It is known in several classes, e.g., complete intersections in characteristic not two (Walker) and in characteristic two (VandeBogert and Walker), but remains a conjecture in general.

The authors leverage their Betti number bounds over fiber product rings to provide positive results in particular cases, while explicitly stating the conjecture as an open problem.

References

Conjecture 4.13 (Total Rank Conjecture). Let (R,m,k) be a d-dimensional Noetherian local ring, and let M be a finitely generated nonzero R-module. If M has finite length and finite projective dimension, then ∑_{i≥0} β_i(M) ≥ 2d.

— On General fiber product rings, Poincaré series and their structure (2402.12125 - Freitas et al., 19 Feb 2024) in Section 4, Conjecture 4.13