Multiplication in Grothendieck ring and a ring lifting

Abstract

For a ring A, the Grothendieck group G,(A) is defined by generators [Ml, one for each finitely generated left A-module M, with relations [M] = [M’] + [M”] for each short exact sequence 0 -+ M’ -+ M + M” -+ 0 of finitely generated left A-modules. In particular, let G be a finite group, Q be the rational number field, and Z be the ring of rational integers. Denote by ZG the integral group ring of G over Z and QG the group algebra of G over Q. Then the Grothendieck group G,(ZG) may be given a ring structure as follows: for all ZG-lattices M and N, set [M] [IN] = CM@ N], where MQ N is a ZG-module with the action of G given by g(m 0 n) = gm Q gn, for all g E G, m E M, n E N. We can define a product similarly on G,(QG). Then Swan [13] has shown that this makes G,(ZG) and G,(QG) into commutative rings. Since there is a natural epimorphism 0: G,(ZG) + G,(QG) defined by 0( [M] ) = [Q Oz M] and G,( QG) is Z-free, there is a linear map o: G,(QG) + G,(ZG) with oB= IcoCpc) and so we have G,(ZG) = ker 8 0 o(G,(QG)) as abelian groups. Such a map o is called a lifting for G,(ZG) (see [12]). Furthermore, Swan [13] has showed that Ker 0 is a square-nilpotent ideal. Therefore, in order to get an explicit formula of the ring structure, we suffice to determine the multiplications of elements of o(GO(QG)) in G,(ZG) and then the action of G,(QG) on Ker 8. The Grothendieck ring G,(ZG) has been studied by Heller and Reiner [S, 61, Swan [13, 143, Stancl [12], and Obayashi [S]. Swan first gave a formula for multiplication in G,(ZG) when G is cyclic of prime power order and Stancl generalized this to the case of an arbitrary cyclic group and an elementary abelian group, and then Obayashi treated abelian p-groups. In these cases, they settled the first part by finding a special lifting w, which is just a ring homomorphism. Then the structure of w(G,(QG)) is easy to 291 0021-8693/87 $3.00