Buchsbaum–Eisenbud–Horrocks 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 i-th Betti numbers satisfy β_i(M) ≥ (d choose i) for all i ≥ 0.

Background

The Buchsbaum–Eisenbud–Horrocks (BEH) Conjecture prescribes sharp lower bounds on individual Betti numbers of finite-length, finite projective dimension modules over local rings. It is proven in certain cases (e.g., regular rings for M = k; dimensions ≤ 4) but remains open in general for larger dimensions.

In this paper, the authors obtain lower bounds for Betti numbers over fiber product rings in various settings, giving positive answers for modules with infinite projective dimension, but they explicitly note that the general BEH Conjecture remains open for higher dimensions.

References

Conjecture 4.12 (Buchsbaum-Eisenbud-Horrocks). 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 for each i ≥ 0 the i-th Betti number of M over R satisfies the inequality β_i(M) ≥ (d choose i). This conjecture, from now on referred to as the (BEH) Conjecture, has a positive answer for local rings with dimension ≤ 4 (see [2]), but for larger dimensions the problem is still open.

— On General fiber product rings, Poincaré series and their structure (2402.12125 - Freitas et al., 19 Feb 2024) in Section 4, Conjecture 4.12