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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.