Grade (ring theory)
In commutative and homological algebra, the grade of a finitely generated module [math]\displaystyle{ M }[/math] over a Noetherian ring [math]\displaystyle{ R }[/math] is a cohomological invariant defined by vanishing of Ext-modules[1]
[math]\displaystyle{ \textrm{grade}\,M=\textrm{grade}_R\,M=\inf\left\{i\in\mathbb{N}_0:\textrm{Ext}_R^i(M,R)\neq 0\right\}. }[/math]
For an ideal [math]\displaystyle{ I\triangleleft R }[/math] the grade is defined via the quotient ring viewed as a module over [math]\displaystyle{ R }[/math]
[math]\displaystyle{ \textrm{grade}\,I=\textrm{grade}_R\,I=\textrm{grade}_R\,R/I=\inf\left\{i\in\mathbb{N}_0:\textrm{Ext}_R^i(R/I,R)\neq 0\right\}. }[/math]
The grade is used to define perfect ideals. In general we have the inequality
[math]\displaystyle{ \textrm{grade}_R\,I\leq\textrm{proj}\dim(R/I) }[/math]
where the projective dimension is another cohomological invariant.
The grade is tightly related to the depth, since
[math]\displaystyle{ \textrm{grade}_R\,I=\textrm{depth}_{I}(R). }[/math]
References
- ↑ Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 131. ISBN 9781139171762. https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1.
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