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Grade (ring theory)

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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

Under the same conditions on and as above, one also defines the -grade of as[2]

This notion is tied to the existence of maximal -sequences contained in of length .

References

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  1. ^ Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 131. ISBN 9781139171762.
  2. ^ Brodmann, Markus P.; Sharp, Rodney Y. (2013). Local Cohomology (2nd ed.). Cambridge: Cambridge University Press. p. 113. ISBN 9780511629204.