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Transformation Groups

, 14:771 | Cite as

The Cohen–Macaulay property of separating invariants of finite groups

  • Emilie DufresneEmail author
  • Jonathan Elmer
  • Martin Kohls
Article

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.

Keywords

Normal Subgroup Direct Summand Complete Intersection Cohomology Class Hilbert Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Mathematics Center Heidelberg (MATCH)Ruprecht-Karls Universität HeidelbergHeidelbergGermany
  2. 2.Lehrstuhl A für MathematikRWTH AachenAachenGermany
  3. 3.Technische Universität MünchenZentrum Mathematik-M11GarchingGermany

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