Paul Hill

A note on a theorem of May concerning commutative group algebras

Let G be a coproduct of p-primary abelian groups with each factor of cardinality not exceeding , , and let F be a perfect field of characteristic p . If V(G) is the group of normalized units of the group algebra F(G), it is shown that G is a direct factor of V(G) and that the complementary factor is simply presented. This generalizes a theorem of W. May, who proved the result in the case when G itself has cardinality not exceeding N, and length not exceeding co I Throughout let F denote a perfect field of characteristic p t 0 and let G denote a p-primary abelian group. In a recent paper, W. May [M2] proved the following result, where V(G) denotes the group of normalized units in the group algebra F(G); hence V(G) = { cigi E F(G): Ec = 1}. Theorem 1 ([M2]). If IGI < t1 and the length of G does not exceed wo1, then G is a direct factor of V(G) and the complementary factor is simply presented. Recall that a group is simply presented if it can be presented by generators and relations that involve at most two generators. The main purpose of this note is to show that the condition on the length of G in the above theorem can be omitted. In other words, the theorem is proved for a much more general class of groups (inasmuch as there exists an abundance of abelian p-groups of cardinality not exceeding t1 with arbitrary prescribed length i < wd2 ). Actually, we extend the theorem to any abelian p-group G which is a coproduct of groups with the cardinality of each factor not exceeding tj; of course such a G can have arbitrarily large cardinality. We refer to [MI] and [M3] as examples of the connection between the direct factor theorem and the isomorphism problem: When does F(G) F(G) imply that G G ? As is customary for group algebras, we use the multiplicative notation for G even though G is abelian. Consequently, for an ordinal a, G is the translation of the more familiar p7G; since there is only one relevant prime, Received by the editors November 28, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C)7; Secondary 20K10. Supported in part by NSF grant DMS 8800862. ? 1990 American Mathematical Society 0002-9939/90

On commutative group algebras of mixed groups

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Isotype Separable Subgroups of Totally Projective Groups

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The Classification Problem

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.

Units of commutative modular group algebras

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.

The projective dimension of valuated vector spaces

Having reached the full measure of a half century since countable abelian p-groups were classified, it seems appropriate that we consider this occasion (a survey talk on the classification of abelian groups presented to the Udihe Conference) as a kind of golden anniversary of that event, which still ranks as one of the great achievements in the history of abelian groups. We wish to honor here especially tne work of Prufer, Ulm, Zippin, Baer and other pioneer researchers who were able to determine completely the structure of important classes of abelian groups [1, 23, 27, 30]. It is proper and fitting that we review at this time what has been accomplished in classifying abelian groups over the last fifty years and that we reflect on what we have done to carry on and complete the work they started. Moreover, it may be beneficial for us to examine methods and techniques that have developed over this period and to analyse those in current use. Finally, we consider a few open problems and discuss briefly directions for future research.

The balance of Tor

Abstract Let F be a perfect field of characteristic p and let G be an arbitrary abelian p-group. The normalized units of the group algebra F(G) are denoted by V(G). The proposition that G is a direct factor of V(G) is a long-standing question, which we call the Direct Factor Problem. It is known that the Direct Factor Problem has an affirmative solution provided that V(G) G is simply presented. In this paper, we prove that V(G) G has a v-basis. It has recently been shown for an abelian p-group of cardinality not exceeding ℵ1 that having a v-basis implies the group is simply presented, but for larger groups this implication is unresolved.

ISOTYPE SUBGROUPS OF LOCAL WARFIELD GROUPS

This paper is primarily concerned with a projective dimension of vector spaces with valuations. Let Γ be a totally ordered set with suprema. The valuated vector spaces over Γ form a pre-abelian category V. A short exact sequence 0 → A → B → C → 0 in V is proper if each element of C has a preimage with the same value. The proper projectives in V are precisely the free objects in V. An object V ϵV is free if and only if it is the coproduct of one-dimensional spaces. The main purpose of this paper is to characterize those valuated spaces having proper projective dimension n for each positive integer n. Our characterization uses the notion of separability which is defined as follows. A subspace W of V is called ℵn-separable in V if for each x ϵ V there exists a subset S of W having cardinality not exceeding ℵn such that sup{¦x + w¦: w ϵ W} = sup{¦x + s¦: s ϵ S}. Roughly speaking, we show that proj. dim.(V) ⩽ n if V has enough ℵn − 1-separable subspaces, where we have abbreviated “proper projective dimension” to “proj. dim.” More precisely, we prove that proj. dim.(V) ⩽ n if and only if V is the union of a smooth ascending chain of ℵn − 1-separable subspaces, 0 = V0 ⊆ V1 ⊆ … ⊆ Vα ⊆ … such that dim(Vα + 1Vα) ⩽ ℵn. Many other results are required before this characterization can be obtained. After it is obtained, we are able to prove the existence of valuated vector spaces having projective dimension exactly n for each positive integer n. Also, it is shown that there exist spaces having infinite projective dimension. Although our main results are homological in nature, for the most part the techniques of the paper certainly are not. One might compare what we do, for example, with Kaplanskys structural characterization of algebraically compact groups, which turn out to be those that have pure-injective dimension 1.

On the structure of p-local abelian groups

Nunke has investigated Tor(A,B) in [10, 11] and 1-12]. Other studies of the torsion product of abelian groups A and B include [4, 8] and [-9]; accounts also appear in [,1] and [2]. Since Tor(A,B)=~Tor(Ap, Bp) p where Ap denotes the p-primary component of A, one specializes immediately to primary groups when studying the structure of the group Tor(A,B). In [10] Nunke computed the Ulm invariants of Tor(A,B) in terms of A and B. Therefore, if Tor(A,B) is totally projective, its structure is known. However, it is not known exactly when Tor(A, B) is totally projective. In fact, even when A and B have length co this question has only been partially resolved [-8]. In this paper, we show that Tor(A,B) is never totally projective when its length exceeds the first uncountable ordinal f2. This generalizes a recent result of Irwin, Snabb and Cellars [9] which states the same fact for groups A and B of unequal lengths. However, as [9, 10] and [12] all demonstrate, Tor(A,B) is more difficult to analyze when A and B have equal lengths. The notation and terminology is generally in agreement with [-2]. In particular, the length of a reduced p-primary group G is the smallest ordinal 2 such that pxG=O. Nunke [-10] has shown for any ordinal c~ that p~Tor(A,B) =Tor(p~A,p~B) and that the length of Tor(A,B) is the minimum of the lengths of A and B. An ascending chain {G~}~

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

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