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
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Group theory, abstraction, and the 196,883-dimensional monster
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The One Ring Explained
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RPSC Assistant Professor | Ring Theory Paper Solution 2021 by GP Sir
Transcription
References
- ^ Matsumura, Hideyuki (1987). Commutative Ring Theory. Cambridge: Cambridge University Press. p. 131. ISBN 9781139171762.
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