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A locally compact contraction group is a pair ( G , α ) , where G is a locally compact group and α : G → G an automorphism such that α n ( x ) → e pointwise as n → ∞ . We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups ( G , α ) which are central extensions { 0 } → F p ( ( t ) ) → G → F p ( ( t ) ) → { 0 } of the additive group of the field of formal Laurent series over F p = Z / p Z by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.

In this text, we generalize Cech cohomology to sheaves F with values in blue B-modules where B is a blueprint with −1. If X is an object of the underlying site, then the cohomology sets H l ( X , F ) turn out to be blue B-modules. For a locally free O X -module F on a monoidal scheme X, we prove that H l ( X , F ) + = H l ( X + , F + ) where X + is the scheme associated with X and F + is the locally free O X + -module associated with F . In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over F 1 .

The cohomology of the configuration space of n points in R 3 is isomorphic to the regular representation of the symmetric group, which acts by permuting the points. We give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the Varchenko–Gelfand filtration on the cohomology of the configuration space of n points in R 1 . Along the way, we give a presentation of the equivariant cohomology ring of the R 3 configuration space with respect to a circle acting on R 3 via rotation around a fixed line. We extend our results to the settings of arbitrary real hyperplane arrangements (the aforementioned theorems correspond to the braid arrangement) as well as oriented matroids.

We compute the action of Hecke operators on tensor products of cohomology classes of lower congruence subgroups of SL ( n , Z ) in trivial weight. We use this computation to prove that if each representation in a collection of Galois representations is attached to a cohomology class of a lower congruence subgroup in trivial weight, then a sum of certain twists of the representations consistent with the main conjectures of [5] , [10] is also attached to such a cohomology class.

For J an integral domain and F its field of fractions, we construct a map from the 3-skeleton of the classifying space for Γ = S L 2 ( J [ t , t − 1 ] ) to a Euclidean building on which Γ acts. We then find an infinite family of independent cocycles in the building and lift them to the classifying space, thus proving that the cohomology group H 2 ( S L 2 ( J [ t , t − 1 ] ) ; F ) is infinite-dimensional.

We compute the group cohomology of 32 Γ 3 f , a certain group of order 32. For this we construct explicit cocycle representatives of the cohomology generators. We thus lay to rest a discrepancy between several published computations of this cohomology ring.

We observe that an n-dimensional crystallographic group G has periodic cohomology in degrees greater than n if it contains a torsion free finite index normal subgroup S ⊴ G whose quotient G / S has periodic cohomology. We then consider a different type of periodicity. Namely, we provide hypotheses on a crystallographic group G that imply isomorphisms H i ( G / γ c T , F ) ≅ H i ( G / γ c + d T , F ) for F the field of p elements and γ c T a term in the relative lower central series of the translation subgroup T ≤ G . The latter periodicity provides a means of calculating the mod-p homology of certain infinite families of finite p-groups using a finite (machine) computation.

Let k be an algebraically closed field of characteristic two and let Q 8 be the quaternion group of order 8. We determine the Gerstenhaber Lie algebra structure and the Batalin–Vilkovisky structure on the Hochschild cohomology ring of the group algebra k Q 8 .

This work studies the flag varieties of p-compact groups, principally through torus-equivariant cohomology, extending methods and tools of classical Schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups. In particular we give, for the infinite family of p-compact flag varieties corresponding to the complex reflection groups G ( r , 1 , n ) , a generalized GKM characterization (following Goresky–Kottwitz–MacPherson [8] ) of the torus-equivariant cohomology, building an explicit additive basis and showing its relationship with the polynomial or Borel presentation via the localization map.

Using Morel's weight truncations in categories of mixed sheaves, we attach to varieties over finite fields or the complex numbers a series of groups called the weightless cohomology groups. These lie between the usual cohomology and intersection cohomology, have natural ring structures, and are functorial for certain morphisms.

Recent research has demonstrated a connection between Weil-etale cohomology and special values of zeta functions. In particular, Lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a Weil-etale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology. We define Weil-etale cohomology for varieties over p-adic fields, and prove a duality theorem for the cohomology of G m on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a p-adic analogue of Lichtenbaum's Weil-etale duality theorem for curves over finite fields, as well as a Weil-etale analogue of his classical duality theorem for curves over p-adic fields. Finally, we show that our duality theorem implies this latter classical duality theorem for index 1 curves.

RO ( G ) -graded equivariant cohomology was originally defined by using equivariant spectra. In this paper we construct a sheaf version of RO ( G ) -graded equivariant cohomology theory. By introducing a complex of presheaves we obtain an isomorphism theorem of two different cohomology theories. We further apply the results to the study of real algebraic varieties.

In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin [1] is proved for varieties of cohomology algebras of fusion systems associated to block algebras of finite groups.

We introduce some topologies on the group cohomology of algebraic groups over complete valued fields and consider some applications.

For a field k of odd prime characteristic, we give an easy, but far-reaching sufficient combinatorial condition for a partition λ indicating that the first cohomology H 1 ( S n , S λ ) of the corresponding Specht module is trivial. Based on this condition, an infinite set of partitions λ is determined, for which the first cohomology groups H 1 ( S n , S λ ) are one-dimensional.

We introduce a criterion for the completeness of approximations of modular cohomology rings of finite groups that are not of prime-power order, and discuss how one can benefit from a combination of different completeness criteria in practical computations.

Local cohomology modules are, in general, not finitely generated over the ambient ring. Towards a better understanding of the structure of these modules, Huneke (1992) [Hu] asked if they have finitely many associated prime ideals. The answer is negative. Several authors have constructed a ring whose local cohomology module has infinitely many associated primes. In certain situations, one can prove that a local cohomology module has infinitely many associated primes by showing that Z/pZ embeds into the local cohomology module for each prime integer p. We show that in fact, each finitely generated abelian group embeds into a local cohomology module.

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions)., Comment: Updated version, new cocycle description added

We give an explicit affine algebraic variety whose coordinate ring is isomorphic (as an algebra with the action of the Weyl group) with the equivariant cohomology of some Springer fibers., Comment: 5 pages

The non-finiteness of the mod-p Schur multiplier of a finitely generated group G with trivial center implies the existence of uncountably many non-residually finite, non-isomorphic central extension groups with kernel C p (see Thm. A). This phenomenon is related to the comparison of the cohomology of the profinite completion G ˆ of a group G and the cohomology of the group G itself (see Thm. B); according to J-P. Serre [8] a group G is good if the cohomologies of G ˆ and G are naturally isomorphic on finite coefficients. It is shown that non-uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good (see Thm. C).

A solution to Linckelmannʼs gluing problem in all fusion systems of blocks yields an alternative formulation of Alperinʼs weight conjecture. Using Bredon equivariant cohomology techniques we solve the gluing problem in fusion systems (of blocks) which are isomorphic to the fusion systems of the symmetric groups or the alternating groups or GLd(q) or SLd(q) at the defining characteristic.

We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold $X$. Then we show that the stack of objects in the heart $\langle \Coh_{\leq 1}(X), \Coh_{\geq 2}(X)[1] \rangle$ satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart., 23 pages

We compute the Hochschild cohomology HH ⁎ ( B ) of an arbitrary block B of the Hecke algebra H = H ( n , q ) of the symmetric group S n at a primitive l th root of unity q , over a field of characteristic zero.

One main purpose of this article is to clarify, for any field, the relations between the infinitesimal dilogarithms, the extensions by itself of the additive group considered as a module over the multiplicative group, and the cohomology of the affine group.

We consider the orbit spaces of the products of complex projective spaces under a diagonal free involution and calculate the singular cohomology rings in terms of transfer homomorphism and characteristic classes.

We use a construction due to Fontaine to construct rings with almost vanishing local cohomology from rings of mixed characteristic and discuss the question of using this method to construct almost Cohen–Macaulay algebras over the original ring. We also show that the existence of almost Cohen–Macaulay algebras implies the Monomial Conjecture and give an example to show how this procedure can be carried out in a nontrivial case.

The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the case of nilpotent loops of order 2 q , q a prime), or lead to efficient classification computer programs. This allows us to enumerate all nilpotent loops of order less than 24.

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(\Sigma_d, S^\lambda). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a "generic cohomology" result for Specht modules equating cohomology of S^{p\lambda} with S^{p^2\lambda}. This is the first theorem we know relating Specht modules S^\lambda and S^{p\lambda}. The second result equates cohomology of S^\lambda with S^{\lambda + p^a\mu} for large a., Comment: Some substantial revisions from previous version

We introduce Δ-groups and show how they fit in the context of lattice field theory. To a topological space M we associate a Δ-group Γ(M). We define the symmetric cohomology HSn(G,A) of a group G with coefficients in a G-module A. The Δ-group Γ(M) is determined by the action of π1(M) on π2(M) and an element of HS3(π1(M),π2(M)).

In this paper we study the second Hochschild cohomology group ${HH}^2(\Lambda)$ of a finite dimensional algebra $\Lambda$. In particular, we determine ${HH}^2(\Lambda)$ where $\Lambda$ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field $K$ and show that this group is zero for most such $\Lambda$; we give a basis for ${HH}^2(\Lambda)$ in the few cases where it is not zero., Comment: Corrections to some calculations

It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the $\Ext$-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra., 10 pages

The membership of the local cohomology modules Han(M) of a module M in certain Serre subcategories of the category of modules is studied from below (in). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.

We compute the cohomology of crystallographic groups Γ=Zn⋊Z/p with holonomy of prime order by establishing the collapse at E2 of the spectral sequence associated to their defining extension. As an application we compute the group of gerbes associated to many six-dimensional toroidal orbifolds arising in string theory.

We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors.

We extend the notion of monogenic extension to the noncommutative setting, and we study the Hochschild cohomology ring of such an extension. As an aplication we complete the computation of the cohomology ring of the rank one Hopf algebras beggined in [S. M. Burciu and S. J. Witherspoon, Hochschild cohomology of smash products and rank one Hopf algebras]., 25 pages. Some examples added, and also the Gerstenhaber structure is computed in a big family of examples, including the rank one Hopf algebras

We show that the F-signature of a strongly F-regular local ring of characteristic p exists in the case that the non- Q -Gorenstein locus is dimension 1, given that a certain bound on zeroth local cohomology modules holds. This bound is shown to hold for rings essentially of finite type over a field. This is the first case in which hypotheses sufficient to prove the existence of the F-signature are not readily sufficient to prove the implication that weak F-regularity implies strong F-regularity.

We develop and study Tate and complete cohomology theory in the category of sheaves of O X -modules. Different approaches are included. We study the properties of these theories and show their power in reflecting the Gorensteinness of the underlying scheme. The connection of these two theories will be discussed.

Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin's theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds [P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004) 623–632] and consider the cohomology of trivial source modules.