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

$\textrm\,M=\textrm_R\,M=\inf\left\.$

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

$\textrm\,I=\textrm_R\,I=\textrm_R\,R/I=\inf\left\.$

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

$\textrm_R\,I\leq\textrm\dim(R/I)$

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

$\textrm_R\,I=\textrm_(R).$

Under the same conditions on $R, I$ and $M$ as above, one also defines the $M$-grade of $I$ as

$\textrm_M\,I=\inf\left\.$

This notion is tied to the existence of maximal $M$-sequences contained in $I$ of length $\textrm_M\,I$.

References

{{abstract-algebra-stub

Ring theory
Homological algebra
Commutative algebra