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
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
Under the same conditions on and as above, one also defines the -grade of as
This notion is tied to the existence of maximal -sequences contained in of length .