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Grothendieck Groups of Invariant Rings: Filtrations
Author(s) -
Brown Kenneth A.,
Lorenz Martin
Publication year - 1993
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-67.3.516
Subject(s) - mathematics , grothendieck group , surjective function , invariant (physics) , pure mathematics , combinatorics , krull dimension , noetherian ring , torsion (gastropod) , finitely generated abelian group , algebra over a field , noetherian , abelian group , mathematical physics , medicine , surgery
We investigate the Grothendieck group G 0 ( R ) of finitely generated modules over the ring of invariants R = S G of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G 0 ( R ) that is defined in terms of (Gabriel‐Rentschler) Krull dimension, properties of G 0 ( R ) are derived to a large extent from the connections between the sets of prime ideals of S and R . A crucial ingredient is an equivalence relation ∼ on Spec R that was introduced by Montgomery [ 25 ]. For example, we show that rank G 0 ( R ) ⩽ rank G 0 ( S ) G +∑ Ω( # Ω ‐ 1 )where Ω runs over the ∼‐equivalence classes in Spec R and (·) G denotes G ‐coinvariants. The torsion subgroup of G 0 ( R ) is also considered. We apply our results to group actions on the Weyl algebra in positive characteristics, the quantum plane, and the localized quantum plane for roots of unity.