Exact Cohomology Sequence Research Articles

Locally inner automorphisms of algebras

The connection between automorphisms of Azumaya algebras and the Picard group of the center has been noticed by Rosenberg-Zelinsky (RZ) [8], following a remark by Auslander-Goldman [l]. This connection has been generalized, using the Morita context, first by Bass [4] and more recently by Frohlich [5]. In my thesis I suggested another way of proving the result of [8]. This is done rather mechanically, by passing to the spectrum of the center of the given Azumaya algebra and taking the exact cohomology sequence resulting from an exact sequence of sheaves of units. This is described, in more detail, further on. However, this way of proving the RZ theorem shows two things:

A unified treatment of some basic problems in homotopy theory

0. Introduction. This note announces some new methods in algebraic topology, based on the results of [1], [2] and [3]. The relative lifting problem [5, p. 415] is fundamental to that subject. It includes the extension problem, the retraction problem, the lifting problem, the section problem, the relative section problem and the computation of [X, Y] problem, among its particular cases. These problems (excluding the last one) are usually taken to concern just the existence of extensions, retractions, etc. ; we use the terms in the wider sense, to include both the existence question and the homotopy classification question, for the respective extensions, retractions, etc. We will : (i) prove that many cases of the above problems (including the computation of the absolute and relative cohomology groups of the total space of a fibration, and the computation of certain homotopy groups) are equivalent to problems, involving the restricted fibered mapping projection (pq;a); (ii) give some examples of solutions of parallel problems. Our solutions include one lifting problem and two extension problems. The extension results have some immediate consequences; they give new derivations for the exact cohomology sequences of Serre and Wang. Gysin's sequence is derived separately. Future papers will discuss these and other applications in detail. Our argument is valid in several convenient categories, including the category of i-spaces [4], [2], [7] and the category of quasi-topological spaces [6], [1], [2]. This category of I-spaces, for a definition see [2, p. 276], contains the usual category of Hausdorff fc-spaces (= compactly generated spaces = Kelley spaces) as a subcategory.

On Inflation-Restriction Exact Sequences in Group and Amitsur Cohomology

On Inflation-Restriction Exact Sequences in Group and Amitsur Cohomology

On inflation-restriction exact sequences in group and Amitsur cohomology

is exact. In ?2 of this paper we show (under the assumption that n> 1) that this sequence is embedded in a considerably simpler exact sequence (for the precise statement see Theorem 1) and we are able to say something about the extent of deviation of the two sequences. Let C be a field, Kc F extension fields with [F: C] finite; a fundamental theorem of Amitsur cohomology [2], [6], [8] states: the sequence

On the differential geometry of homogeneous vector bundles

This paper falls roughly into three parts. The first section (??I-III) may be considered as an extension of the works of Nomizu and Wang [15; 19] to bundles over homogeneous complex manifolds. These results lay the groundwork for the succeeding paragraphs. The second section (??IV-VI) gives differential-geometric derivations of various properties of homogeneous complex manifolds. The last section (??VII-IX) gives some applications of differential geometry to homogeneous vector bundles in the sense of [4] and to the study of sheaf cohomology. To be more explicit, we let X be a homogeneous complex manifold which may be written as the coset space of complex Lie groups A, B (X = A/ B) and also as the coset space of compact Lie groups M, V (X = M/ V) where M is semi-simple. The first part treats the following question: If P -> X is any analytic principal bundle to which the action of A on X lifts, we ask for a algebraic description of those connexions x in P which are M-invariant and compatible with the complex structures involved. (It will be seen that the adjective computable is for us crucial.) Moreover, given such a x, we seek its form, and, in the case where P is the principal tangent bundle, we ask for the complex torsion of x. In the second part, we single out two types of connexions in such a P for further study. The first are those connexions arising from an invariant Hermitian metric in P. By examining these connexions, we are able to give a unified differentialgeometric treatment of several aspects in the theory of homogeneous complex manifolds. The complex torsion plays an important role here.The other connexion, which we call the canonical complex connexion, is the complex analogue of the Nomizu canonical affine connexion; it is important for applications to sheaf theory. In the third part, we first give a geometric realization of the curvature class, a sheaf cohomology class defined by Atiyah in [1]. Next, using a metric geometry, we discuss a vanishing theorem; the chief application here is the non-Kahler case. In the last paragraph, we give a new type of application of differential geometry to sheaf cohomology; this application arises from a connextion between the holonomy algebra of a linear connexion and the coboundary maps in an exact cohomology sequence.

The Eichler cohomology groups and automorphic forms

Introduction. In his papers [5; 6] Eichler demonstrated the significance for the study of automorphic forms of Bol 's discovery [3 ] of some remarkably simple differential operators taking automorphic forms into automorphic forms. In [6] in particular Eichler discussed a relation between the automorphic forms associated to a transformation group 9 on a Riemann surface D and some purely algebraic constructions involving the group 9, the first cohomology groups of 9 with certain modules of polynomials as coefficients; the cocycles appeared as the periods of the automorphic forms under iterated indefinite integration, generalizing the classical interpretation of the periods of the abelian integrals on D/g (which can of course be considered as automorphic forms on D) as cocycles of the group 9 or alternatively of the space ID/9. The object of studying such a relation is the development of tools for calculating the dimensions of spaces of automorphic forms and the traces of the Hecke operators on automorphic forms. The aim of the present paper is the study of a more general form of this relation in somewhat greater detail for one complex variable, but in such a manner that the results can be extended to several complex variables; the actual extension to several complex variables, as well as the application to the study of the Hecke operators, will be discussed elsewhere. As for the contents of this paper, ? 1 is devoted to an exposition of Bol's differential operators in a form more useful in the present context than that of [3]. In ?2 these differential operators are applied to give an exact cohomology sequence containing, in a rather more transparent form, the relation of Eichler discussed above. The interpretations of the terms appearing in this exact sequence are discussed in ??3 through 5; the only point of difficulty arises in ?4, Theorem 3 of that section really being a form of the Serre duality theorem [12] appropriate to the occasion. These results are combined in ?6 to give a formal statement of the fundamental result of the paper. 1. Differential operators preserving automorphic forms. Let SC be the group of 2 X 2 real matrices

Note on fundamental exact sequences in homology and cohomology for non-normal subgroups

Note on fundamental exact sequences in homology and cohomology for non-normal subgroups

A remark on fundamental exact sequences in cohomology of finite groups

A remark on fundamental exact sequences in cohomology of finite groups