Grade (ring theory) explained

M over a Noetherian ring R is a cohomological invariant defined by vanishing of Ext-modules^[1] rm{grade}M= rm{grade} _{RM=inf\left\{i\inN} _{0:rm{Ext}} ^{i(M,R) ≠} ^{R} 0 \right\}. I\triangleleft R the grade is defined via the quotient ring viewed as a module over R rm{grade}I= rm{grade} _{RI=rm{grade}} _{RR/I=inf\left\{i\inN} _{0:rm{Ext}} ^{i(R/I,R) ≠} ^{R} 0 \right\}.

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

rm{grade} _{RI\leqrm{proj}\dim(R/I)}

where the projective dimension is another cohomological invariant.

The grade is tightly related to the depth, since

rm{grade} _{RI=rm{depth}} _{I} (R).

Notes and References

1. Book: Matsumura, Hideyuki . Hideyuki Matsumura . 1987 . Commutative Ring Theory . Cambridge . Cambridge University Press . 131 . 9781139171762.