THE BALANCED-PROJECTIVE DIMENSION OF UNITS IN COMMUTATIVE MODULAR GROUP-ALGEBRAS

Suppose F is a perfect field of characteristic p not equal 0 and G is a multiplicatively written abelian p-group. Write bpd(H) for the balan ced-projective dimension of an arbitrary p-group H. If V(G) is the gro up of normalized units of the group algebra F(G), it is shown that bpd (V(G)) = bpd(G). This was known previously only in the special case wh ere one of the dimensions is zero. Also, some partial results are obta ined concerning the conjecture that the functor G bar right arrow V(G) /G decreases balanced-projective dimension. Special cases of these res ults are related to the unresolved direct factor problem: When is G a direct factor of the group of units of F(G)? (C) 1998 Elsevier Science B.V. All rights reserved.