Grade (ring theory)

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

$$\textrm{grade}\,M=\textrm{grade}_R\,M=\inf\left\{i\in\mathbb{N}_0:\textrm{Ext}_R^i(M,R)\neq 0\right\}.$$

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

$$\textrm{grade}\,I=\textrm{grade}_R\,I=\textrm{grade}_R\,R/I=\inf\left\{i\in\mathbb{N}_0:\textrm{Ext}_R^i(R/I,R)\neq 0\right\}.$$

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

$$\textrm{grade}_R\,I\leq\textrm{proj}\dim(R/I)$$

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

$$\textrm{grade}_R\,I=\textrm{depth}_{I}(R).$$