Herzog–Huneke–Srinivasan multiplicity conjecture

Let $S={\bf k}[x_1,\ldots,x_n]$ be a polynomial ring over a field, and let $N$ be a finitely generated graded $S$-module with a minimal graded free resolution of length $l$. Write $\beta_{i,j}$ for its graded Betti numbers, set $c=\operatorname{codim}(N)$, and define

$$M_i=\max\{j:\beta_{i,j}\neq 0\},\qquad m_i=\min\{j:\beta_{i,j}\neq 0\}.$$

Let $I\subset S$ be a homogeneous ideal of codimension $c$, and let $e(N)$ denote the multiplicity of $N$. Herzog–Huneke–Srinivasan multiplicity conjecture. The multiplicity satisfies

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

Moreover, if $N$ is Cohen–Macaulay, then

$$e(N)\geq\frac{\prod_{i=1}^{c}m_i}{c!}.$$

The conjecture connects the multiplicity of a graded module with the extremal degree shifts in its minimal free resolution; the source reviews it as the central multiplicity bound motivating the paper. The supplied text does not state whether the conjecture is resolved in full generality.