An Isomorphism for the Grothendieck Ring of a Hopf Algebra Order

If G is a finite abelian group, R is a principal ideal domain with field of quotients an algebraic number field K which splits G, and if A is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated A-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules. The Grothendieck group a£ of the category of finitely generated modules over a ring A is analogous to the free group generated by the characters of a finite group. If H is a Hopf algebra, then h9 is a ringWe show if R is a principal ideal domain with field of quotients K which splits G, if G is a finite abelian group, and if A is a Hopf algebra order in KG, then rgQ — aQSwan [6] has shown that if G is a finite group, and if R is a semilocal Dedekind domain, then rg$ — kgQIt follows that if R is a semilocal Dedekind domain with field of quotients K, if G is a finite abelian group split by K, and if A is a Hopf algebra order in KG, then aQ — kgQSwan's result says that, factoring out relations induced by short exact sequences of representations, the representation theory of a group ring over a semilocal Dedekind domain is the same as the representation theory of the group algebra over the field of quotients. Our result says that, for abelian groups, the representation theory is the same for any Hopf algebra order in the group algebra as for the group ring. If C is an abelian category, the Grothendieck group of C is defined as follows: for each object M G C there is a generator [M]; for each short exact sequence 0 -» NT -> M -» M" -» 0 in C there is a relation [M] = [M'\ + [M"\. For the remainder of this paper R will be a principal ideal domain with field of quotients an algebraic number field K. If A is an fi-algebra which is a finitely generated projective fi-module, we will denote by aC the category of finitely generated A-modules, and by aC the category of finitely generated A-modules which are torsion free as fi-modules. We will denote the Grothendieck group of aC by aQ, and the Grothendieck group of ¿_C by aQThe embedding ¿_C —> aC induces an isomorphism ¿Q —> a9(See [4] for details.) If H is a Hopf algebra over R, and M,N G hC, then M ®ß N G hrhC. Pullback along the coproduct 8: H —> H (giß H gives an fi-module structure on M®rN. In this manner £§# gives rise to a ring structure on h$ with multiplication given by _ [M] \N] = [M®RN]. Received by the editors May 21, 1984. 1980 Mathematics Subject Classification. Primary 16A24, 16A54.