Regular Sequence - Examples

Examples

  1. If k is a field, it possesses no non-zero non-unit elements so its depth as a k-module is 0.
  2. If k is a field and X is an indeterminate, then X is a nonzerodivisor on the formal power series ring R = k], but R/XR is a field and has no further nonzerodivisors. Therefore R has depth 1.
  3. If k is a field and X1, X2, ..., Xd are indeterminates, then X1, X2, ..., Xd form a regular sequence of length d on the polynomial ring k and there are no longer R-sequences, so R has depth d, as does the formal power series ring in d indeterminates over any field.

An important case is when the depth of a ring equals its Krull dimension: the ring is then said to be a Cohen-Macaulay ring. The three examples shown are all Cohen-Macaulay rings. Similarly in the case of modules, the module M is said to be Cohen-Macaulay if its depth equals its dimension.

Read more about this topic:  Regular Sequence

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