Invariant for finitely generated modules over a Noetherian ring
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]
For an ideal the grade is defined via the quotient ring viewed as a module over
The grade is used to define perfect ideals. In general we have the inequality
where the projective dimension is another cohomological invariant.
The grade is tightly related to the depth, since