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For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

In this paper, we study abstract Burnside rings of essentially finite categories. Under unique epi-mono factorization property and the existence of coequalizers for some kind, we prove the existence of a fundamental exact sequence for ABR. Furthermore, an ABR can be embedded into a direct product of rational character rings.

Let $(H, R)$ be a finite dimensional quasitriangular Hopf algebra over a field $k$, and $_H\mathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center $^H_H\mathcal{YD}$ trivializable on $_H\mathcal{M}$. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra $_RH$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $\mathrm{BM}(k, H,R)$ in [18]. To this aim, we have to develop the braided bi-Galois theory initiated by Schauenburg in [14,15], which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category., Comment: 34 pages with figures

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.

We introduce the notion of relation type of an affine algebra and prove that it is well defined by using the Jacobi-Zariski exact sequence of Andre-Quillen homology. In particular, the relation type,is an invariant of an affine algebraic variety. Also as a consequence of the invariance, we show that in order to calculate the relation type of an ideal in a polynomial ring one can reduce the problem to trinomial ideals. When the relation type is at least two, the extreme equidimensional components play no role. This leads to the non-existence of affine algebras of embedding dimension three and relation type two. (C) 2015 Elsevier Inc. All rights reserved.

Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V-module V. Namely, if 0→Q→ϵP→ρV→0 is a short exact sequence of V-modules and W is a V-module, then 0→Q⊠W→ϵ⊠1WP⊠W→ρ⊠1WV⊠W→0 is also exact, where we view it as a sequence of g(V)-modules.

In the theory of canonical induction formulae for Mackey functors, Boltje [4] demonstrated that the plus constructions, together with the mark morphism, are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups. In this paper, we present a short exact sequence for the plus constructions derived from Cauchy–Frobenius lemma, and apply it to the proof of Boltje's integrality result for canonical induction formulae. The methods appearing in Boltje's theory, combined with the Dress construction for Mackey functors, are applicable to induction theorems on representations of the twisted quantum double of a finite group. As a sequel to such a research, we describe canonical versions of two induction theorems whose origins are Artin's induction theorem and Brauer's induction theorem on C -characters of a finite group.

Let I, I ′ be preinjective Kronecker modules (i.e. all their indecomposable components are preinjective). We describe the modules M for which there exists an exact sequence 0 → I ′ → M → I → 0 by explicit, easy to check numerical conditions, resulting in an algorithm (linear in the number of indecomposable components) for the decision problem. We also propose a method to generate all extensions of I ′ by I and we give a different proof for a theorem in [13] providing numerical criteria in terms of Kronecker invariants for the existence of a monomorphism f : I ′ → I . All these results apply dually to preprojective modules as well.

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.

Baeza showed that when char(F)=2 if E/F is the separable biquadratic extension E=F[℘−1(b1),℘−1(b2)], then ker[Wq(F)→Wq(E)]=WF⋅[1,b1]+WF⋅[1,b2]. Here we give the analogous result for the graded Witt group. Specifically we obtain an exact sequence νF(n,1)⊕νF(n,1)→H2n+1F→H2n+1E from which the result for GWqF follows by the isomorphisms of Kato. Applications to 2-algebras of exponent and index 4 are also given.

Let R be a discrete valuation ring with residue field of characteristic p > 0 . Let K be its fraction field. We prove that any finite and flat R -group scheme, isomorphic to μ p 2 , K on the generic fiber, is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models. In Appendix A X. Caruso shows how to classify models of μ p 2 , K , in the case of unequal characteristic, using the Breuil–Kisin theory.

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

A complex C is called Gorenstein injective if there exists an exact sequence of complexes ⋯ → I −1 → I 0 → I 1 → ⋯ such that each I i is injective, C = Ker ( I 0 → I 1 ) and the sequence remains exact when Hom ( E , − ) is applied to it for any injective complex E. We show that over a left Noetherian ring R, a complex C of left R-modules is Gorenstein injective if and only if C m is Gorenstein injective in R-Mod for all m ∈ Z . Also Gorenstein injective dimensions of complexes are considered.

The Witt group of a hyperelliptic curve over a field of characteristic different from two was determined by Parimala and Sujatha. Here, analogous results are obtained for the unramified Witt group in characteristic two using the analogue of Milnor's exact sequence for the Witt group of rational function fields developed earlier by the authors. In the elliptic case, if F is perfect and points of order two are rational, a generator and relation structure for the Witt group is given.

We give a unified treatment of the Baer sums in the context of efficiently homological categories which, on the one hand, contains any category of groups with multiple operators and more generally any semi-abelian variety and, on the other hand, the category of Hausdorff groups and more generally any category of semi-abelian Hausdorff algebras. This gives rise to a generalized “Euclide's Postulate” and a five terms exact sequence.

We show here that the Witt ring of the ring of regular functions is a direct summand of the Witt ring of a formally real algebraic function field over a real closed field. Next, we use this result to show that the Knebusch–Milnor exact sequence of these Witt rings fulfills a certain analogy of splitting.

Let $\Lambda$ be a quasi $k$-Gorenstein ring. For each $d$th syzygy module $M$ in mod $\Lambda$ (where $0 \leq d \leq k-1$), we obtain an exact sequence $0 \to B \to M \bigoplus P \to C \to 0$ in mod $\Lambda$ with the properties that it is dual exact, $P$ is projective, $C$ is a $(d+1)$st syzygy module, $B$ is a $d$th syzygy of Ext$_{\Lambda}^{d+1}(D(M), \Lambda)$ and the right projective dimension of $B^*$ is less than or equal to $d-1$. We then give some applications of such an exact sequence as follows. (1) We obtain a chain of epimorphisms concerning $M$, and by dualizing it we then get the spherical filtration of Auslander and Bridger for $M^*$. (2) We get Auslander and Bridger's Approximation Theorem for each reflexive module in mod $\Lambda ^{op}$. (3) We show that for any $0 \leq d \leq k-1$ each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if $\Lambda$ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer $d$, each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting., Comment: 13 pages

In this paper we study closely Yoneda’s correspondence between short exact sequences and the Ext 1 group. We prove a main theorem which gives conditions on the splitting of a short exact sequence after taking the tensor product with R=I, for any ideal I of R. As an application we prove a generalization of Miyata’s Theorem on the splitting of short exact sequences and we improve a proposition of Yoshino about ecien t systems of parameters. We introduce the notion of sparse module and we show that Ext 1 (M; N) is a sparse module provided that there are nitely many isomorphism classes of maximal Cohen-Macaulay modules having multiplicity the sum of the multiplicities of M and N. We prove that sparse modules are Artinian. We also give some information on the structure of certain Ext 1 modules.

The goal of this paper is to start to present a systematic presentation theory for pro-finite groups. Ž . Let G be a fixed finitely-generated pro-finite group and P a free presentation of it; i.e., P 1 R F G 1 Ž . is an exact sequence when F is a finitely generated free pro-finite group. Ž . Ž . We write d F for the minimal number of generators of F and d R for F the minimal number of generators of R as a normal subgroup of F. In G2 Gruenberg presented a systematic study of presentation theory Ž . of a discrete group G mainly for a finite group G and presented three basic problems: Ž . QUESTION 1. Given P 1 R F G 1 for i 1, 2, when F is i i Ž . Ž . a free group, is then d R d R ? F 1 F 2

The classical 3 × 3 lemma and snake lemma, valid in any abelian category, still hold in any quasi-pointed (the map 0 → 1 is a mono), regular, and protomodular category. Some applications are given, in this abstract context, concerning the denormalization of kernel maps and the normalization of internals groupoids (i.e., associated crossed modules).

We construct a ten-term exact sequence of low dimensional Leibniz homology associated to a short exact sequence of Leibniz algebras. As a consequence we obtain an eight-term exact sequence associated to a central extension of Leibniz algebras.

Previously we obtained a Picard-Brauer five term exact sequence for a symmetric monoidal functor between closed categories. Here we construct the corresponding sequence using the closed category of H-modules for a cocommutative Hopf algebra H. © 1996 Academic Press, Inc.

This paper is concerned with the structure of those space groups (or crystallographic groups) in n-dimensional Euclidean space for which the point groups are essential and generated by reflections. Recall [ 1 ] that a subgroup G in GL( V) is said to be essential if VG = (0). Those point groups are Weyl groups of root systems in the vector space [l]. The fundamental results are presented in Section 2. In a first step, we restrict our study to irreducible root systems. The principal result is stated in Theorem 2.6: in most cases the group H’(G, V/P(R)) is zero if the root system R is an irreducible one and the point group G is the Weyl group of R. Then, an easy consequence is given in Theorem 2.11: the same result holds even if the root system is a reducible one: we just have to forbid some types for its components. If /i is an invariant lattice such that Q(R) c /1 c P(R) the short exact sequence

and this mapping cone should, by rights, be a triangle. It is, for instance, true that any homological functor applied to (*) gives a long exact sequence. Unfortunately, the world of triangulated categories is a bad one, and (*) need not be a triangle. It is however true that, givenf and g, there exists an h for which (*) is a triangle (see Theorem 1.8); not all morphisms of triangles are equal. Some are better than others. It turns out that Theorem 1.8 is equivalent to the octahedral axiom. In the first two sections of this article, we study two possible notions of “good” morphisms between triangles, and we quickly decide that neither is satisfactory. The morphisms do not compose well; the composite of good morphisms need not be good. Worse still, the non-category of good morphisms is non-additive; the sum of two good morphisms need not be good. The problem goes right back to the definition of a triangulated category.

In this paper all modules are unital right modules over finite-dimensional algebras over algebraically closed fields, K. A submodule L of a module N is said to be a pure submodule of N if L is a direct sumand of M whenever L t M c N and M/L is finite-dimensional. A module L is pure-injective if whenever L is pure in M, it is a direct summand of M. A module N is pure-projective if every exact sequence 0 4 L + M 4 N -+ 0 with L pure in M splits. The following proposition is easy to prove

Here uniserial means with a unique composition series. In fact we choose these modules M from the indecomposable modules classified by Janusz [7] and use the canonical walk by Alperin and Janusz [I] to state this precise result (Theorem 3.1). An outline for the computation of Exti(S, M) is as follows: For a nonprojective indecomposable moduleM, we examine the property of B-‘M (Proposition 3.2) and then use Ext,“(S, M)Ext:(S, 8’-“M), where R is Heller’s loop-space operation [6]. We know that Exti(S, M) f 0 if and only if there is the exact sequence 0 + M--f W-t S + 0 with W indecomposable (see the proof of Proposition 3.3). We remark that, if A is the principal block of kG and S is the trivial kGmodule, then Extz(S, M) is the cohomology group H”(G, M), where G is a finite group with a cyclic Sylow p-subgroup and k has a characteristic p. Finally we refer to [4, Chap. VII; and 3] for blocks with cyclic defect groups and for the functor Ext, respectively.

The present work extends a number of earlier results; see [S, 7, 15-181. For example, the following fundamental exact sequence (essentially due to Bloch and Wigner in somewhat different form, but not published by them) can be found in DuPont and Sah [7]: o~o/z-tH3(SL(2,@))~~(@)~n:(a=~)~~,(a=)~o. (2.12) The group Y(C) is defined in (1.4))( 1.8) and is isomorphic to the group PC considered in DuPont and Sah [7]. Also, the study of the scissors congruence problem had led to two exact sequences in DuPont [S, Theorems 1.3 and 1.43 that roughly correspond to the f -eigenspaces of (2.12) under the action of complex conjugation: O~A~H,(SU(2))~~S3/L~[WO([W/Z)-*H,(SU(2))-tO; (0.1)

be an exact sequence of finitely generated free R-modules. We regard this as a free resolution of Coker d, , and say that the resolution is periodic of period n if for all i > 0, Fi+n = Fi and di+n = di . The resolution is minimal if Im di C mFi for all i > 0. Over commutatives rings, the only known periodic resolutions have period 2. Eisenbud [2, Theorem 4.11 proves that when R is a complete intersection, if [F is a minimal free resolution and {rankFi} is bounded, then [F becomes periodic of period 2 after at most 1 + dim R steps. He also notes that, in general, if a resolution has period 2 then (rankF,} is constant. Thus resolutions of period 2 are in 1-I correspondence with pairs (X, Y) of n by n matrices such that KerX=ImY and Ker Y = Im X. Eisenbud also gives a construction of resolutions of period 2 which is general enough to account for every periodic minimal resolution over a complete intersection [2, Theorem 5.21. The construction is this: let x E m be a non-zero-divisor on R, and let X and Y be n by n matrices such that XI’ = x1, = YX, where 1, is the n by n identity matrix. Let “-” denote reduction modulo xR. Then

Chevalley algebras associated with finite-dimensional simple Lie algebras L over the complex field were defined in 151, and their ideal structure was worked out there in terms of the ideal structure of the underlying commutative ring R with identity (See also ]lO].) Recently, Garland [2] has shown the existence of a Chevalley basis for infinite-dimensional Kac-Moody Lie algebras [6, 7 ] of affine type [8 J over the complex field. This makes possible the construction of Chevalley algebras by transfer of the scalars to a commutative ring R with identity, just as in the iinitedimensional case. We study here the ideal structure of these algebras, which we call aflne Chevalley algebras. In Section 1, we establish the notation we need to state results on the ideal structure of iinitc dimensional Chevalley algebras. In Section 2, we describe Kac-Moody Lie algebras and Garland’s Chevalley basis theorem for such algebras associated with afline generalized Cartan matrices. In Section 3, we study an important exact sequence for atline Chevalley algebras, extending a result of Kac and Moody [6,8] on the structure of Euclidean Kac-Moody Lie algebras. In Section 4, the ideal structure of the alline Chevalley algebras over Noctherian integral domains is studied, using the results of Section 3 and the known structure of finite-dimensional Chevalley algebras over Laurent polynomial rings. Our Main Theorem 4.7 takes the form of a sandwich result reminiscent of the situation in classical Chevalley algebras [ 5, Theorems 3.4-3.6; 10, Theorem 3.11. Our result constitutes a partial generalization of a theorem of Moody [7, Theorem 4] on the ideal structure of Euclidean Lie algebras over a field of characteristic zero. (see

Let R be a commutative ring with unit, H a finitely generated projective, commutative and cocommutative Hopf algebra over R. In [3], Long defined a Brauer group BM(R, H) whose elements are equivalence classes of R- Azumaya H-module algebras. In this note, we show that there is a split exact sequence 1 -+ R(R) + BM(R, H) + Gal(R, H) + 1, where Gal(R, H) is the group of Galois H-objects over R, defined in [l], and B(R) is the ordinary Brauer group of R. This sequence generalizes that obtained in [5] for graded algebras, i.e., for H = GR, the dual of the group ring RG.

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

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 Kaplansky's structural characterization of algebraically compact groups, which turn out to be those that have pure-injective dimension 1.

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.

Let R and S be arbitrary (not necessarily commutative) rings with unit and letfi R --f S be a ring morphism. By a module, we will mean left module unless otherwise stated. Mod(R) will denote the category of R-modules. Recall that S is of relative global dimension zero over R if every S-module M is an (S, R)-injective module, i.e., if 0 -+ M -+ A + B + 0 is an exact sequence of S-modules which splits when considered as an exact sequence of R-modules, then 0 + M -+ A + B -+ 0 splits over S also. See G. Hochschild [2] for general definitions. The basic result of this paper is to give a sufficient condition on

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

The notion of a covering morphism of groupoids has been developed by P. J. Higgins [4, 51 and shown to be a convenient tool in algebra, even for purely group theoretic results. That covering morphisms of groupoids model conveniently the covering maps of spaces is shown in [l]. If we weaken the conditions for a covering morphism we obtain what we shall call a fibration of groupoids, and our purpose is to explore this notion. The main results are that, even if we start in the category of groups, then certain constructions lead naturally to fibrations of groupoids; that for fibrations of groupoids we can obtain a family of exact sequences of a type familiar to homotopy theorists; and that these exact sequences include not only the bottom part of the usual exact sequence of a fibration of spaces, but also the well known six term exact sequences in the non-Abelian cohomolog!; of groups [6]. A further advantage of our procedure is that the same setup leads naturally to a definition of non-Abelian cohomology in dimensions 0 and 1 of a groupoid with coefficients in a groupoid. This cohomology (which will be dealt with elsewhere) generalises a non-Abelian cohomology of n group with coefficients in a groupoid which has been developed by A. Frohlich (unpublished) with a view to applications in Galois cohomology. Another question not touched on here is possible application of these methods to the non-bbelian Hz. There is some overlap of this paper with techniques used by J. Gray in 1131. However, the aims of that paper are quite different from ours, and so the theory is developed here from the beginning.

(1) Exta (T, T) = 0, (2) Ext: (T, ) = 0, (3) There is a short exact sequence 0 --+ A, + T> + T” + 0, with T’, T” being direct sums of summands of T. Then putting B= End T, we call (B, T, A) and Hom,(T, ): mod-A -+ mod-B a tilting triple and a tilting functor, respectively. Tilting functors have been introduced by Brenner and Butler [6] as a generalization of the Bernstein-Gelfand-Ponomarev’s reflection functors [S]. They and Happel and Ringel [S] have proved that we have (usually nonhereditary) torsion theories (F-, F) in mod-A and (X, g) in mod-B, where F = {Xe mod-A (Extj, (T, X) = 0} and X= { YEmod-BI YOB T=O), and the tilting functor and Ext: (T, -) give category-equivalences between F and g and between F and X, respectively. These equivalences give us, however, no information about indecomposable A-modules and B-modules which do not belong to the above subcategories F-, F”, !Z, and CV. The purpose of this paper is to point out that there is a method to enlarge our view by which we can supply the lack of information. Let us consider trivial extension algebras R = A K DA and S = B K DB of A and B by DA and DB respectively, where ,DA, = Hom,(A, k) and BDB, = Hom,(B, k) are injective cogenerator bimodules. In this case R and S are selfinjective (more precisely symmetric) algebras and mod-A and mod-B are naturally embedded into projectively stable categories m-R and m-S. Then our main theorem states that for any tilting triple

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 ]) → · · · .

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf–Pasquier–Roche quasi-Hopf algebra D ω ( Σ ), for some finite group Σ and some ω ∈ Z 3 ( Σ , k × ). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω ( Σ ), for an appropriate 3-cocycle ω coming from the Kac exact sequence.

Cataloged from PDF version of article. We show that there is an exact sequence of biset functors over p-groups0 → Cb over(→, j) B* over(→, Ψ) DΩ → 0 where Cb is the biset functor for the group of Borel-Smith functions, B* is the dual of the Burnside ring functor, DΩ is the functor for the subgroup of the Dade group generated by relative syzygies, and the natural transformation Ψ is the transformation recently introduced by the first author in [S. Bouc, A remark on the Dade group and the Burnside group, J. Algebra 279 (2004) 180-190]. We also show that the kernel of mod 2 reduction of Ψ is naturally equivalent to the functor B× of units of the Burnside ring and obtain exact sequences involving the torsion part of DΩ, mod 2 reduction of Cb, and B×. © 2006 Elsevier Inc. All rights reserved

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.

The orthogonal representations of a finite group over a Dedekind domain are studied. First, we study the equivariant Witt group W0(D, DG) of a finite nilpotent group G over a Dedekind domain D. Introducing a Morita correspondence on the set of orthogonal representations, we determine the structure of W0(D, DG) for a finite nilpotent group G of odd order. We next treat the exact sequence 0→W0( Z, Z G) → W0( Q, Q G) →∂ W0( Z, Z G), which was introduced by A. Dress (1975, Ann. of Math. (Z) 102, 291–325). We show that the boundary homomorphism δ is surjective when G is a finite group of odd order. Our last aim is to show that W0( Z, Z G) is sufficiently large to investigate the Witt group W0( Z G) in L-theory when G is a finite group of odd prime power order.

Let G be a finite group and k be a field whose characteristic p divides the order of G. In [3] we introduce the notion of Auslander-Reiten system of G on a symmetric interior G-algebra A (that is a symmetric k-algebra together with a homomorphism 4: G + A x ), as a generalization of the notion of almost split sequences of kG-modules. The language of interior G-algebras proves to be useful to study their restriction to certain subgroups such as the defect group (vertex) of their extremity. Furthermore we show in [3] that each non projective primitive idempotent i of AG (see Section 1) is the extremity of a unique Auslander-Reiten system, up to embedding of A into other symmetric interior G-algebras and to conjugacy (cf. [3, VI], recall embeddings are those one-to-one homomorphisms f: A + B which satisfy Im f = f (1) Af (1)). Thus in particular the middle term of “the” Auslander-Reiten system terminating in i is well defined (up to embedding and conjugacy), and we may look at its decomposition into primitive idempotents. Now if we set A [G] = A @ kG, we have a symmetric algebra again, which is similar to kG in many ways, and which enables us to view our systems of G over A ([3, I]) as short exact sequences of modules: to any idempotent i of AG, we associate the A-projective A[G]-module iA (with (b@ g) . iu = d(g) iub). This way any system Y = (i, i”, i’, d, d’) of G over A determines an exact sequence of A[G]-modules