Grade (ring theory)

In commutative and homological algebra, the grade of a finitely generated module

over a Noetherian ring

is a cohomological invariant defined by vanishing of Ext-modules[1]

= inf

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

the grade is defined via the quotient ring viewed as a module over

= inf

{\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

dim ⁡ (

{\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).}

{\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

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