William Ullery

On commutative group algebras of mixed groups

Suppose F is a perfect field of characteristic p 0 and G is an abelian group whose torsion subgroup Gt is p-primary. If Gt is totally projective of countable length, it is shown that G is a direct factor of the group of normalized units V(G) of the group algebra F(G) and that V(G)/G is a totally projective p-group. The proof of this result is based on a new characterization of the class of totally projective p-groups of countable length. Li addition, the same result regarding V(G) is obtained if G has countable torsion-free rank and Gt is totally projective of length less than ω1 + ω0 . Finally, these results are applied to the question of whether the existence of an F-i pomorphism F(G) ≅ F(H) for some group H implies that G≅H.

Isotype Separable Subgroups of Totally Projective Groups

A p-primary abelian group G is called almost totally projective if it has a collection C of nice subgroups with the following properties: (0) 0 ∈ C, (1) C is closed with respect to unions of ascending chains, and (2) every countable subgroup of G is contained in a countable subgroup from C. Observe that this is a generalization of the Axiom 3 characterization of totally projective groups. In this paper, we show that the isotype subgroups of a totally projective group which are almost totally projective are precisely those that are separable. From this characterization it follows that every balanced subgroup of a totally projective group is almost totally projective. It is also shown that the class of almost totally projective groups is closed under the formation of countable extensions. Finally, in some special cases we settle the question of whether a direct summand of an almost totally projective group is again almost totally projective.

ISOTYPE SUBGROUPS OF LOCAL WARFIELD GROUPS

Necessary and sufficient conditions are given for an isotype subgroup H of a p-local Warfield group G to be, itself, Warfield.

Butler Modules Over Prüfer Domains

If R is an integral domain, let 𝒞 be the class of torsion free completely decomposable R-modules of finite rank. Denote by ℛ the class of those torsion-free R-modules A such that A is a homomorphic image of some C ∈ 𝒞, and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ∈ 𝒞. Further, let Q ℛ and Q 𝒫 be the respective closures of ℛ and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q ℛ = Q 𝒫, and ℛ = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ∈ 𝒞 has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then ℛ = Q ℛ = Q 𝒫 = 𝒫 if and only if R is almost maximal.

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

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 G↦V(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)?

Trim Extensions of Mixed Groups

If G is an extension of an Abelian group H with G/H torsion, call H ⊆ G a trim extension if, for each relevant prime p of G/H, the p-primary component of the torsion subgroup of G is torsion complete. In this article we give necessary and sufficient conditions for two trim extensions of a group H to be equivalent. We also characterize equivalent extension types (as defined by Baer). The class of trim extensions is more general than the class of slim extensions recently introduced by Hill and Megibben.

The sequentially pure projective dimension of global groups with decomposition bases

Abstract The sequentially pure projective dimension of a global group with a decomposition basis is computed in terms of a structural property that generalizes the earlier treatment of p -local torsion groups by Fuchs and Hill. As a corollary, it is established that a global group with a decomposition basis and cardinality ℵ n , for some nonnegative integer n , has sequentially pure projective dimension not exceeding n .