The Cohen-Macaulay property of separating invariants of finite groups
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 actua...
| Main Authors: | , , |
|---|---|
| 格式: | Journal article |
| 出版: |
SP Birkhäuser Verlag Boston
2009
|
| _version_ | 1826267956395376640 |
|---|---|
| author | Dufresne, E Elmer, J Kohls, M |
| author_facet | Dufresne, E Elmer, J Kohls, M |
| author_sort | Dufresne, E |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T21:02:12Z |
| format | Journal article |
| id | oxford-uuid:3b4002ee-b8cf-4d9f-9385-a00bb1cfcdb0 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:02:12Z |
| publishDate | 2009 |
| publisher | SP Birkhäuser Verlag Boston |
| record_format | dspace |
| spelling | oxford-uuid:3b4002ee-b8cf-4d9f-9385-a00bb1cfcdb02022-03-26T14:06:30ZThe Cohen-Macaulay property of separating invariants of finite groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3b4002ee-b8cf-4d9f-9385-a00bb1cfcdb0Symplectic Elements at OxfordSP Birkhäuser Verlag Boston2009Dufresne, EElmer, JKohls, MIn 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. |
| spellingShingle | Dufresne, E Elmer, J Kohls, M The Cohen-Macaulay property of separating invariants of finite groups |
| title | The Cohen-Macaulay property of separating invariants of finite groups |
| title_full | The Cohen-Macaulay property of separating invariants of finite groups |
| title_fullStr | The Cohen-Macaulay property of separating invariants of finite groups |
| title_full_unstemmed | The Cohen-Macaulay property of separating invariants of finite groups |
| title_short | The Cohen-Macaulay property of separating invariants of finite groups |
| title_sort | cohen macaulay property of separating invariants of finite groups |
| work_keys_str_mv | AT dufresnee thecohenmacaulaypropertyofseparatinginvariantsoffinitegroups AT elmerj thecohenmacaulaypropertyofseparatinginvariantsoffinitegroups AT kohlsm thecohenmacaulaypropertyofseparatinginvariantsoffinitegroups AT dufresnee cohenmacaulaypropertyofseparatinginvariantsoffinitegroups AT elmerj cohenmacaulaypropertyofseparatinginvariantsoffinitegroups AT kohlsm cohenmacaulaypropertyofseparatinginvariantsoffinitegroups |