The multiplicity conjecture for standard graded algebras

Let $A=T/I$ be a standard graded $k$-algebra, where $T=k[x_1,\ldots,x_n]$, and let

$$0\longrightarrow\bigoplus_{j\geq 0}T(-j)^{\beta_{r,j}}\longrightarrow\cdots\longrightarrow\bigoplus_{j\geq 0}T(-j)^{\beta_{1,j}}\longrightarrow T\longrightarrow A\longrightarrow 0$$

be the minimal free resolution of $A$ as a $T$-module. Let $e(A)$ be the multiplicity of $A$, let $h=\operatorname{height}(I)$ be the height, or codimension, of $I$, and define

$$M_i=\max\{j\mid j\geq 0\text{ and }\beta_{i,j}\neq 0\},\qquad m_i=\min\{j\mid j\geq 0\text{ and }\beta_{i,j}\neq 0\}.$$

Multiplicity conjecture. The inequalities

$$e(A)\leq\frac{1}{h!}\prod_{i=1}^h M_i$$

and, if $A$ is Cohen--Macaulay,

$$e(A)\geq\frac{1}{h!}\prod_{i=1}^h m_i$$

hold. If $A$ is Cohen--Macaulay, equality holds if and only if $A$ has a pure resolution over $T$. The conjecture relates the multiplicity of a standard graded algebra to the shifts in its minimal free resolution up to the height of its defining ideal; the paper verifies it for Stanley--Reisner rings of barycentric subdivisions, while the general statement is attributed to Huneke and Herzog--Srinivasan.