Regular sequence (algebra)

Regular sequence (algebra)

In commutative algebra, if "R" is a commutative ring and "M" an "R"-module, an element "r" in "R" is called M-regular if "r" is not a zerodivisor on "M", and "M/rM" is nonzero. An R-regular sequence on "M" is a "d"-tuple

:"r1, ..., rd" in "R"

such that for each "i ≤ d", "ri" is "Mi-1"-regular, where "Mi-1" is the quotient "R"-module

:"M/(r1, ..., ri-1)M".

Such a sequence is also called an "M"-sequence.

It may be that "r1, ..., rd" is an "M"-sequence, and yet some permutation of the sequence is not. It is, however, a theorem that if "R" is a local ring or if "R" is a graded ring and the "ri" are all homogeneous, then a sequence is an "R"-sequence only if every permutation of it is an "R"-sequence.

The depth of "R" is defined as the maximum length of a regular "R"-sequence on "R". More generally, the depth of an "R"-module "M" is the maximum length of an "R"-regular sequence on "M". The concept is inherently module-theoretic and so there is no harm in approaching it from this point of view.

The depth of a module is always at least "0" and no greater than the Krull dimension of the module.

Examples

# If "k" is a field, it possesses no non-zero non-unit elements so its depth as a "k"-module is "0".
# If "k" is a field and "X" is an indeterminate, then "X" is a nonzerodivisor on the formal power series ring "R = k""X", but "R/XR" is a field and has no further nonzerodivisors. Therefore "R" has depth 1.
# 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" ["X1, X2, ..., Xd"] 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.

References

* David Eisenbud, "Commutative Algebra with a View Toward Algebraic Geometry". Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
* Winfried Bruns; Jürgen Herzog, "Cohen-Macaulay rings". Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1


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