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

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