The balanced-projective dimension of units in commutative modular group algebras Original Research Article

Suppose F is a perfect field of characteristic p ≠ 0 and G is a multiplicatively written abelian p-group. Write bpd(H) for the balanced-projective dimension of an arbitrary p-group H. If V(G) is the group 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 where one of the dimensions is zero. Also, some partial results are obtained concerning the conjecture that the functor Gmaps toV(G)/G decreases balanced-projective dimension. Special cases of these results are related to the unresolved direct factor problem: When is G a direct factor of the group of units of F(G)?