Grade (ring theory)

Template:One source Template:Short description 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

$\text{grade}M={\text{grade}}_{R}M=\inf \{i\in {\mathbb{N}}_0:{\text{Ext}}_R^i(M,R)\ne 0\}.$

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

$\text{grade}I={\text{grade}}_{R}I={\text{grade}}_{R}R/I=\inf \{i\in {\mathbb{N}}_0:{\text{Ext}}_R^i(R/I,R)\ne 0\}.$

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

${\text{grade}}_{R}I\le \text{proj}\dim (R/I)$

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

${\text{grade}}_{R}I={\text{depth}}_I(R).$

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

${\text{grade}}_{M}I=\inf \{i\in {\mathbb{N}}_0:{\text{Ext}}_R^i(R/I,M)\ne 0\}.$

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

References

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