Let R be the ring of integers in a number field and p an odd rational prime. In the paper "Higher class groups of orders" Math Zeitchrift 228, (1998) 229-246 , M. Kolster and R. Laubenbacher proved that p-torsion in odd-dimensional class groups of R-orders A can only occur for primes p which lie under the prime ideals of R at which A is not maximal --with consequences for group rings RG (G finite). The question arises if a similar result can be proved for even-dimensional class groups of arbitrary orders and grouprings. In the papers, "Higher class groups of generalized Eichler orders" , Communications in algebra, 33, 709-718 (2005); Higher class groups of locally triangular orders over number fields (Algebra Colloquim, 16, 1 (2009) 79 - 85, X Guo and I obtained similar characterization for p-torsion in even-dimensional higher class groups for generalized Eichler orders, e.g Eichler orders in quaternion algebras, hereditary orders as well as locally triangular orders.. Also, X Guo obtained further results along these lines in the paper " Even dimensional Higher class groups of orders" Math Zeitchrift ,261, 617-624 (2009). It is still open to obtain such a characterization(if it exists) for arbitrary R-orders and hence groups rings RG for even dimensional higher class groups.
Article Higher Class Groups Of Generalized Eichler Orders
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Algebra