### Abstract

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

Original language | English |
---|---|

Pages (from-to) | 771-785 |

Number of pages | 15 |

Journal | Transformation Groups |

Volume | 14 |

Issue number | 4 |

Early online date | 14 Nov 2009 |

DOIs | |

Publication status | Published - Dec 2009 |

### Keywords

- COMPUTING INVARIANTS
- REDUCTIVE GROUPS
- RINGS

### Cite this

*Transformation Groups*,

*14*(4), 771-785. https://doi.org/10.1007/s00031-009-9072-y

**The Cohen-Macaulay property of separating invariants of finite groups.** / Dufresne, Emilie; Elmer, Jonathan; Kohls, Martin.

Research output: Contribution to journal › Article

*Transformation Groups*, vol. 14, no. 4, pp. 771-785. https://doi.org/10.1007/s00031-009-9072-y

}

TY - JOUR

T1 - The Cohen-Macaulay property of separating invariants of finite groups

AU - Dufresne, Emilie

AU - Elmer, Jonathan

AU - Kohls, Martin

PY - 2009/12

Y1 - 2009/12

N2 - In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

AB - In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

KW - COMPUTING INVARIANTS

KW - REDUCTIVE GROUPS

KW - RINGS

U2 - 10.1007/s00031-009-9072-y

DO - 10.1007/s00031-009-9072-y

M3 - Article

VL - 14

SP - 771

EP - 785

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

IS - 4

ER -