Grade (ring theory)

Short description: An invariant for finitely generated modules over a Noetherian rings

In commutative and homological algebra, the grade of a finitely generated module $M$ over a Noetherian ring $R$ is a cohomological invariant defined by vanishing of Ext-modules

$\text{<mrow data-mjx-texclass="ORD">grade</mrow>}M={\text{<mrow data-mjx-texclass="ORD">grade</mrow>}}_{R}M=\inf \{i\in {\mathbb{\mathbb{N}}}_0:{\text{<mrow data-mjx-texclass="ORD">Ext</mrow>}}_R^i(M,R)\ne 0\}.$

For an ideal $I◃R$ the grade is defined via the quotient ring viewed as a module over $R$

$\text{<mrow data-mjx-texclass="ORD">grade</mrow>}I={\text{<mrow data-mjx-texclass="ORD">grade</mrow>}}_{R}I={\text{<mrow data-mjx-texclass="ORD">grade</mrow>}}_{R}R/I=\inf \{i\in {\mathbb{\mathbb{N}}}_0:{\text{<mrow data-mjx-texclass="ORD">Ext</mrow>}}_R^i(R/I,R)\ne 0\}.$

The grade is used to define perfect ideals. In general we have the inequality

${\text{<mrow data-mjx-texclass="ORD">grade</mrow>}}_{R}I\le \text{<mrow data-mjx-texclass="ORD">proj</mrow>}\dim (R/I)$

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

${\text{<mrow data-mjx-texclass="ORD">grade</mrow>}}_{R}I={\text{<mrow data-mjx-texclass="ORD">depth</mrow>}}_I(R).$