1

Definition: Let $A$ be a Noetherian ring and $M$ a non-zero finite $A$-module. Then the grade of $M$ is defined as $grade(M) = \inf_i \left\{Ext^i(M,A) \neq 0\right\}$.

Matsumura (CRT) p.132, says that if $I$ is an ideal of $A$, then by grade of $I$ we mean the grade of $A/I$, which turns out to be the $I$-depth of $A$. I find this confusing since $I$ itself is an $A$-module, so i would expect that $grade(I)$ should be just the quantity $\inf_i \left\{Ext^i(I,A) \neq 0\right\}$. Any comments?

Manos
  • 25,833

1 Answers1

3

Quote from Bruns and Herzog: "It is customary to set grade I = grade R/I = grade(I, R), for an ideal I" and "Of course grade I has two different meanings now but we will never use it to denote the grade of the module I."