Cohomology of Semidirect Product Groups

Characteristic classes for cohomology of split Hopf algebra extensions

On the cohomology of the finite special linear groups, I

It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops.A lemma about Tor(Af, TV) as a module over R S is proved.Two spectral sequences of Hochschild and Serre are shown to be the same.1. Introduction.Let G be a finite group, 77 a normal subgroup, and A a finitely generated G-module.We show here that the two spectral sequences ^(G/H, FP(H, A)) * Hp+q(G, A) and Hp(G/H,Hq(H,A))=>Hp+q(G,A) stop in a finite number of steps.Historical note.In [E], I proved a special case of the result for cohomology.Subsequently, I was able to handle the general case along the lines presented here.(At about the same time Atiyah also disposed of the case of cohomology at least for coefficients in Z by another method.)Since that time I have been asked repeatedly about this fact, and it seems appropriate to provide a published proof.Moreover, there is some added interest to this fact given a recent result of James Blowers which shows that the corresponding spectral sequence for nonnormal subgroup 77 need not stop.Finally, whUe I am at it, I take the trouble to deal also with the homology spectral sequence.The situation there is almost, but not quite, dual to that of cohomology.2. Cohomology.All modules in this section are left modules-but of course this makes no real difference.Theorem 1.Let G be a finite group, H a normal subgroup, A a finitely

A class of resolutions of objects of an abelian category determines a theory of derived functors if each morphism between objects extends to a morphism, unique to within homotopies, between their resolutions. This paper is primarily concerned with resolutions canonically associated with certain natural classes of extensions (E-functors), and the known examples are constructed by using pairs of adjoint functors. An inclusion between two E-functors on the same category induces natural transformations between functors derived from their associated resolutions, and other relations exist in the form of invariant exact couples. The relations simplify for the special and frequently occurring class of ‘central’ inclusions of E-functors; in particular the operations of forming satellites of a functor on the two resolutions commute. Amongst various applications the general theory provides generalizations of: results on groups of extensions of modules over Dedekind domains; the Hochschild—Serre spectral sequences in the homology theory of groups; the spectral sequences for coherent algebraic sheaves that determine Ext by means of vector bundle resolutions and affine coverings.

We prove standard results of group cohomology – namely, existence of a long exact sequence, classification of torsors via the first cohomology group, Shapiro’s lemma, the Hochschild-Serre spectral sequence, a decomposition of the cochain complex in the direct product case, and Jannsen’s result on the recovery problem – for cohomology theories such as continuous, analytic, bounded, and pro-analytic cohomology. We also prove these results for certain monoids, as the applications we have in mind concern $$(\varphi ,\Gamma )$$ -modules. The cohomology groups considered here all have very concrete interpretations by means of a cochain complex. Therefore, we do not use methods of homological algebra, but explicit calculations on the level of cochains, using techniques dating back to Hochschild and Serre.

Abstract. We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology- syntomic cohomology- is a Bloch-Ogus cohomology theory, admits period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the étale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé’s étale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on p-adic comparison theorems.

ABSTRACTWe construct and study the map from Leibniz homology HL∗(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH∗−1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.

This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical $\operatorname {mod} p$ coalgebra and shows them to be a complete list of all such possible Steenrod operations. These operations are compatible with the differentials and with Steenrod operations on the total complex. A general rule is given for computing the operations on ${E_2}$. A generalization of the Kudo transgression theorem is also proved, placing it in a larger and more natural setting.

Conventions The spectral sequence of a bisimplicial coalgebra Bialgebra actions on the cohomology of algebras Extensions of Hopf algebras Steenrod operations in the change-of-rings spectral sequence The Eilenberg-Moore spectral sequence Steenrod operations in the Eilenberg-Moore spectral sequence Bibliography Index.

If G is a finite group, and A a G-ring, then generators and relations for H*(G,A), the cohomology of G with coefficients in A, may be obtained via the Hochschild-Serre spectral sequence. Examples include. Prerequisite to such a calculation is the existence of a proper normal subgroup of G, so that for simple groups this method is not available. Perhaps, in such cases, a more appropriate method is to extract the p-part of the cohomology ([H*(G,A)]p) from the cohomology ring of the Sylow subgroup for each prime p dividing ∣G∣, the order of G ([1, p.259]). In general, this requires a knowledge of the intersection of the Sylow p-subgroups, but has the following two simplifications. Let Gp be the Sylow p-subgroup of G, and Φp the group of automorphisms of Gp induced by inner automorphisms of G.

We set up a Grothendieck spectral sequence which generalizes the Lyndon–Hochschild–Serre spectral sequence for a group extension K ? G ? Q by allowing the normal subgroup K to be replaced by a subgroup, or family of subgroups which satisfy a weaker condition than normality. This is applied to establish a decomposition theorem for certain groups as fundamental groups of graphs of Poincare duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group F.

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

Let H be a finite group having a fixed point free au tomorph ism c~ of order p". Consider the semidirect product G = (c~)H. It is well known that (eh) v" = 1 if h 9 H (see [3], p. 334). Put K = ( ev ) H. Then G # K and the elements in G K are p-elements. This last si tuation was considered by Kurzweil in [7]. It includes as a special case the groups having a proper generalized Hughes subgroup, i.e. those verifying G + Hr, (G) where Hp, (G) = ( x 9 G I xl" Je 1). A classical result of Hughes-Thompson and Kegel assures that if G :# H v (G) then H v (G) is ni lpotent (see [5] and [6]). Assuming that G is solvable Kurzweil showed that the Fi t t ing length of Hr, (G) (and hence that of G) is bounded by a function of n (see [7]). His bound for exceptional primes (in the Hal l -Higman sense) was improved by Har t ley and Rae as a product of their work in [4]. More recently Meixner obtained a l inear bound in [8]. Finally, in [2], the best possible bound f (Hr, (G)) < n was obtained for p odd. The case p = 2 is open. The purpose of this note is to consider the general problem. We may assume that G = ( x ) K, G K consists of p-elements and the order of x is, say, p". Assuming that G is solvable, what can be said about its Fi t t ing length? In [7] Kurzweil considered the case n = I and showed that f (K) < 2. Here we prove that f (K) < n + 1 if p is odd and the bound is best possible. The result is false for p = 2 even in the case n = 2. Our theorem is a new appl icat ion of the non-coprime Shult type theorems stated in [2]. There is another problem connected to this. Let G be a finite group having a proper subgroup H and a proper normal subgroup N of H such that H c~ H ~ < N if g 9 G H. Then G is said to be a Frobenius-Wie landt group (see [1] for more details and notation). We write (G, H, N) to indicate this situation. A theorem of Wielandt (see [1] for example) assures that, in such conditions, there exists a normal subgroup K of G such that G K = ~) (H -N) o, G = H K and H c~ K = N. Assume that H is a p-group. Then osG G K consists of p-elements. Thus we are in the above situation. Conversely, if G is p-solvable and K is a normal subgroup of G such that G K consists of p-elements then taking P 9 $1, (G) we have that (G, P, P c~ K) is an F W group. To show this observe that if x 9 G K then x acts f.p.f, on every x-invariant p '-section of K. Suppose that y 9 P c~ Po where g is a nontrivial p ' -element of G. As K is p-solvable we have a p '-section A/B of K where A and B are normal in G and g 9 A B. Then [y, g 1] 9 p c~ A < B. Thus y 9 P c~ K.

Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit

This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical mod p coalgebra and shows them to be a complete list of all such possible Steenrod operations.These operations are compatible with the differentials and with Steenrod operations on the total complex.A general rule is given for computing the operations on E2.A generalization of the Kudo transgression theorem is also proved, placing it in a larger and more natural setting.