The multiplicity conjecture for finitely generated graded torsion modules

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $N$ be a finitely generated graded torsion $R$-module. Set $c=n-\dim N$ and, for a minimal graded free resolution of $N$, let $m_0$ be the minimum shift in homological degree zero and $M_i$ the maximum shift in homological degree $i$. Multiplicity conjecture for torsion modules. One has

$$e(N)\leq\frac{1}{c!}\prod_{i=1}^c(M_i-m_0),$$

with equality if and only if $N$ is Cohen–Macaulay and has a pure resolution. This extends the multiplicity conjecture from cyclic modules to finitely generated graded torsion modules; the paper proposes the bound and establishes it for torsion modules of codimension at most two, while the general statement remains open.