sentence1: we show that , in many instances , conditions which ensure that the ring of invariants is non cohen-macaulay , in fact imply that no graded geometric separating algebra is cohen-macaulay .
sentence2: 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 .