The Huneke–Srinivasan and Herzog–Srinivasan multiplicity bounds

Throughout, let $R=k[x_1,\dots,x_n]$ be a polynomial ring over a field, and let $I\subset R$ be a homogeneous ideal of codimension $c$. For the graded Betti numbers $\beta_{ij}(R/I)$, define

$$m_i=\min\{j:\beta_{ij}(R/I)\not=0\},\qquad M_i=\max\{j:\beta_{ij}(R/I)\not=0\}.$$

Here $m_i$ and $M_i$ are respectively the minimal and maximal degrees of a syzygy at step $i$ in the minimal graded free resolution of $R/I$, and $e(R/I)$ denotes its multiplicity.

Huneke–Srinivasan–Herzog–Srinivasan conjecture. If $R/I$ is Cohen–Macaulay, then

$$\frac{1}{c!}\prod_{i=1}^c m_i\le e(R/I)\le\prod_{i=1}^c M_i.$$

If $R/I$ is not Cohen–Macaulay, then only

$$e(R/I)\le\frac{1}{c!}\prod_{i=1}^c M_i.$$

These bounds seek to control the multiplicity of a homogeneous quotient from the degree shifts in its minimal graded free resolution. The source presents the assertion as a conjecture; no resolution status is supplied here.