Search

Let 1 → A → G → Q → 1 be an exact sequence of groups, where A is abelian, Q is polycyclic and ⨂ Q k ( A ⊗ Z Q ) is finitely generated as Q Q -module via the diagonal Q-action for k ≤ 2 m . Moreover we assume that if G is not metabelian, then it is of type FP 3 . Our main result is that sup U ∈ A ⁡ dim Q ⁡ H j ( U , Q ) ∞ for 0 ≤ j ≤ m , where A is the set of all subgroups of finite index in G.

It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the 3 × 3 lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties.

A duality of two categories is introduced. It generalizes the Malcev description of torsion free finite rank abelian groups. An equivalence of two categories is also introduced. It generalizes the Kurosh description of p -primitive groups. The composition of the duality and the equivalence is the duality earlier introduced by W. Wickless and the author. It is shown that the last duality preserves exactness for short exact sequences of homomorphisms.

If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras., 14 pages

Let G be a group. By an essential cover of G we shall mean here a short exact sequence of groups 1 -+ A -+ E -+ G -+ 1 in which A is abelian and with the additional property that A has no proper supplement in E: if E,A = E, then El = E. Similarly, if M is a module over some ring R, then the short exact sequence of R-modules 0 -+ A + V -+ M + 0 is an essential cover of M if no proper submodule VI of V satisfies Vr + A = V (i.e., A is an essential submodule of V). We have two aims in this paper. In the first place, we set up a framework for discussing group extensions and module extensions which is to be general enough to encompass all the possibilities normally encountered (though we must stress that in the case of group extensions we do limit the kernels to be abelian). Secondly, and this is really the core of the present paper, we show that in certain situations the essential covers can be classified in a very explicit way by means of projective geometries. For a given fixed group G, given commutative ring K and given subcategory 6 of KG-modules, we define the category (6 /I G) of all group extensions 13 A --j. E -+ G -+ 1, in which A E 6. Here CC is merely assumed to be a full additive subcategory closed under finite direct sums (cf. (l.l), below). Similarly, if R is a given ring, M is a given fixed R-module and 6 is now a subcategory of R-modules, we obtain the category (a Ij 44) of Rmo ueextensionsO-+A-+ V-+M+O. d 1 We shall handle both cases simultaneously: all proofs and definitions are given in such a way that they are valid in both categories. The fact that this can be done in a consistent way is not an accident: the category ((Xl] G) is actually equivalent to (CXllKg), where Kg is the augmentation ideal of KG. We shall discuss this fact in detail in a later paper (meanwhile, cf. [2, Sect. 10.51) but have

Let R be a commutative ring and G a finite abelian group. In [8], Long developed a Brauer group theory for G-dimodule algebras (i.e., algebras with a compatible G-grading and G-action) and constructed BD(R, G), the Brauer group of G-Azumaya algebras. Within BD(R, G) lies B(R, G), the set of classes of algebras which are R-Azumaya (i.e., central separable) as well as G-Azum$ya. B(R, G) is not always a group; we show that if every cocyle in H2(G, U(R)) is abelian, then it is. When B(R, G) is a group, we call it the Brauer group of central separable G-Azumaya algebras. If R is connected and Pit,(R) = 0 where m is the exponent of G, and if every cocycle in H2(G, U(R)) is abelian, then we show that there is a short exact sequence 1 + (BC(R, G)/B(R)) x (BM(R, G)/B(R)) -+ B(R, G)/B(R) -+ Aut(G) + 1, where B(R) is the usual Brauer group of R, BM(R, G) is the Brauer group of G-module algebras and BC(R, G) is the Brauer group of G-comodule algebras (cf. [S]). If either BM(R, G)/B(R) or BC(R, G)/B(R) is trivial, then the sequence splits. Using the above, we are able to describe BD(Z, G) for any cyclic G, and BD(IW, G) for any cyclic G of odd order. Our sequence provides a generalization of the results [9, Theorem 5.91 and [II, Theorem 4.41 and should be compared to the sequence in [6, Theorem 5.21 obtained under the assumptions that the order of G is a unit in R and R contains a primitive mth root of unity.

Suppose that R is any commutative ring with unity element and that M is a finitely generated R-module. The usual way in which one investigates the extent to which M deviates from being free is to map a free module of appropriate rank onto M and to examine the module of relations. This point of view was exploited by H. Fitting in [4] when he defined an ascending sequence of invariant ideals for M which generalized the classical invariant factors for a finite abelian group. For our purposes, the smallest ideal in this sequence is the one of greatest interest. This is the invariant which generalizes the order of a finite abelian group. Let us denote this invariant by F(M) (see Section 2 for definitions). The most important formal properties of the function F may be summarized as follows. If S is a multiplicatively closed set in R not containing 0, then F(M), =F(M,) where the subscript S denotes extension to the ring of quotients R, . If 0 + A -+ B + M + 0 is a short exact sequence of finitely generated R-modules and M satisfies the conditions (i) the annihilator of M contains a nondivisor of zero and (ii) the homological dimension of M is at most one, then F(A)F(M) = F(B). This is analogous to the fact that the order of a finite group divided by the order of an invariant subgroup is equal to the order of the quotient group. Finally, if M satisfies (i) and (ii) above and R is a noetherian ring, then F(M) is an invertible ideal of R. This generalizes the fact that the order of a finite abelian group is a nonzero integer. The details of these assertions are treated in Section 2. Now the multiplicative property of F on short exact sequences whose cokernel satisfies certain conditions shows that F bears a formal similarity to the notion of the rank of a module. This similarity turns out to be strong enough to define an Euler characteristic for long exact sequences

We prove that ifA→G→Qis a short exact sequence of groups whereGis finitely generated,AandQare abelian,Ais a2-torsion Krull dimension one3Q-module via conjugation then theFPm-Conjecture holds. In general ifGis of typeFPmand either the extension is split orAis4-torsion we show thatAism-tame as5Q-module.

The purpose of this note is to give another example of applications of the general results of an earlier paper. Here we apply the general localization theorem in K-theory of admissible abelian categories obtained there to twisted projective lines and obtain a long exact sequence relating the K-theory of twisted polynomial and twisted Laurent polynomial rings: · · · → K n+1 (R φ [t, t −1 ]) →, K n (R) ⊕ K n (R) → K n (R φ [t]) ⊕ K n (R φ [t −1 ]) → K n (R φ [t, t −1 ]) → · · · .

Let A and B be abelian p-groups with the property that there exists an isomorphism from A[pn] onto B[pn] that preserves heights (heights being computed in A and B, respectively) of all elements of height less than ω+ n. Let 0 → K → F → A → 0 be a pure exact sequence with F a direct sum of cyclic p-groups. Then there exists a pure subgroup L of F such that L[pn]= K[pn] and F L is isomorphic to B. This is used to show that the number of noncongruent pure subgroups of F supported by K[pn] is the same as the number of nonisomorphic p-groups G such that the valuated group (G[pn], vn), with an appropriate valuation vn, is isometric to the valuated group F[p n ] K[p n ]with the restriction of the height function of F K[p n ] as the valuation.

Perhaps the most significant difference between the theories of torsion and torsion-free abelian groups is the large number of splitting results in the former, and the lack of such in the latter. For instance, Szele’s theorem [F, Vol. 1, Proposition 27.11 has no obvious equivalent in the theory of torsion-free abelian groups, although it is one of the widely used results in the discussion of torsion and mixed groups: Let B g QJI Z/p’7 be a subgroup of an abelian group G. Then B is a direct summand of G if and only if B np”G = 0. The author was able to extend Szele’s theorem to valuated p-groups in [A2, Satz 6.81 using methods similar to those which proved useful in the discussion of Baer’s lemma in [AL], [A3], [A4]. It is the goal of this paper to prove a result which can be considered the equivalent of Szele’s theorem in the quasi-category of torsion-free abelian groups. Before we can describe the result, it is necessary to introduce some notation. We consider an abelian group A. The class of A-projective groups of finite A-rank is obtained by closing {A} with respect to direct summands and finite direct sums. If G is an abelian group, then the A-radical of G, which is denoted by R,(G), is R,(G) = n {ker 41 #E Hom(G, A)). Finally, consider modules B and C over a ring R. A homomorphism ~1: B + C quasi-splits if there are a non-zero integer n and a map /3~Hom,(C, B) with /?cr=n.id,, where id, denotes the identity map on B. If the additive group of B is torsion-free, then a quasi-splitting homomorphism is one-to-one [Jl, 52, Re]. We say that an abelian group A has the radical-splitting property if every exact sequence 0 + PA G of torsion-free abelian groups, in which P is a quasi-summand of an A-projective group of finite A-rank and u(P) n R,(G) = 0, quasi-splits. Szele’s original result guarantees the splitting of the sequence in the case in which A z Z/p’7 for some n < w. Moreover, if A z Z/p”Z, then R,(G) = p”G. Theorem 3.1 of this paper establishes that every strongly indecomposable group of finite rank has the