In commutative and homological algebra, the grade of a finitely generated module M {\displaystyle M} over a Noetherian ring R {\displaystyle R} is a cohomological invariant defined by vanishing of Ext-modules
grade M = grade R M = inf { i ∈ N 0 : Ext R i ( M , R ) ≠ 0 } . {\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 I ◃ R {\displaystyle I\triangleleft R} the grade is defined via the quotient ring viewed as a module over R {\displaystyle R}
grade I = grade R I = grade R R / I = inf { i ∈ N 0 : Ext R i ( R / I , R ) ≠ 0 } . {\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
grade R I ≤ proj dim ( R / I ) {\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
grade R I = depth I ( R ) . {\displaystyle {\textrm {grade}}_{R}\,I={\textrm {depth}}_{I}(R).}
Under the same conditions on R , I {\displaystyle R,I} and M {\displaystyle M} as above, one also defines the M {\displaystyle M} -grade of I {\displaystyle I} as
grade M I = inf { i ∈ N 0 : Ext R i ( R / I , M ) ≠ 0 } . {\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 M {\displaystyle M} -sequences contained in I {\displaystyle I} of length grade M I {\displaystyle {\textrm {grade}}_{M}\,I} .
References
