Abstract :
Consider the action of a group G ≤ Sn that permutes the n variables in a polynomial ring k[X1,...,Xn] over a field k. Two related properties, the Cohen-Macaulay property and F-rationality, are studied in the ring of invariants, and the following results are obtained. (1) The invariant ring k[X1,...,Xn]Cn produced by cyclic permutation of the variables is shown not to be Cohen-Macaulay in characteristics dividing n for n > 4. This completes the analysis of the characteristics in which this invariant ring is Cohen-Macaulay. (2) The non-F-rational locus of k[X1,...,Xn]An is found to have positive dimension for certain n and k, although this ring possesses many of the properties of F-rational rings.
