Grade (ring theory)

In commutative and homological algebra, the grade of a finitely generated module ${\displaystyle M}$ over a Noetherian ring ${\displaystyle R}$ is a cohomological invariant defined by vanishing of Ext-modules^[1]

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

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

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

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

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

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

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

Under the same conditions on ${\displaystyle R,I}$ and ${\displaystyle M}$ as above, one also defines the ${\displaystyle M}$-grade of ${\displaystyle I}$ as^[2]

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

This notion is tied to the existence of maximal ${\displaystyle M}$-sequences contained in ${\displaystyle I}$ of length ${\displaystyle {\textrm {grade}}_{M}\,I}$.