Permutation of regular sequence is not necessarily regular

From Commalg
Jump to: navigation, search

Statement

Let R be a commutative unital ring, M a R-module, and x_1,x_2,\ldots,x_n a regular sequence in R, for the module M. Then, it is not necessarily true that every permutation of the x_is is regular.

Partial truth

The following are true:

  • If R is a Noetherian local ring, and the regular sequence comprises elements in the unique maximal ideal, then any permutation of it is regular.
  • If R is a graded ring, and all the elements in the regular sequence are homogeneous elements, then every permutation is regular.
  • Any permutation of a regular sequence of length one is regular (obviously)
  • For a unique factorization domain, and when the module is the ring itself, a sequence of length two is regular, if and only if the two elements are relatively prime. Thus, any permutation of a regular sequence of length two is regular.
  • For a principal ideal domain, when the module is the ring itself, there cannot exist regular sequences of length more than two, so any permutation of a regular sequence is regular.

Example

One example uses the polynomial ring k[x,y,z] over a field. Consider the sequences:

xy, xz, y - 1

versus:

xy, y - 1, xz

The first one of these is regular, while the second clearly isn't: in the second sequence, the term xz is actually equal to zero modulo the first two terms.