Buchsbaum–Eisenbud–Horrocks conjecture

Let $R$ be a commutative Noetherian ring such that $\operatorname{Spec}(R)$ is connected, and let $M$ be a non-zero, finitely generated $R$-module of finite projective dimension. For a finite projective resolution

$$0\longrightarrow P_d\longrightarrow\cdots\longrightarrow P_1\longrightarrow P_0\longrightarrow M\longrightarrow 0,$$

write $c=\operatorname{height}((0:_R M))$. Buchsbaum–Eisenbud–Horrocks conjecture. For every $0\leq i\leq d$, one has

$$\operatorname{rank}_R(P_i)\geq\binom{c}{i}.$$

This is a central lower-bound problem for the ranks of modules in finite projective resolutions; the paper's introduction points to survey literature for what is known and unknown, particularly over polynomial rings and for finite-length modules over local rings.