Sequence For Cohomology Research Articles

Monodromy and local-global compatibility forl=p

We strengthen the compatibility between local and global Langlands correspondences for GL_{n} when n is even and l=p. Let L be a CM field and \Pi\ a cuspidal automorphic representation of GL_{n}(\mathbb{A}_{L}) which is conjugate self-dual and regular algebraic. In this case, there is an l-adic Galois representation associated to \Pi, which is known to be compatible with local Langlands in almost all cases when l=p by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless \Pi\ has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane's weight spectral sequence for log crystalline cohomology.

Equivariant Poincaré–Alexander–Lefschetz duality and the Cohen–Macaulay property

We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.

On the motive of the group of units of a division algebra

AbstractWe consider the algebraic group GL1 (A), where A is a division algebra of prime degree over a field F, and the associated motive in the Voevodsky category of motivic complexes (F). We relate the motive of GL1 (A) to the motive of the Čech simplicial scheme χ, associated to the Severi-Brauer variety of A, and compute the second differential in the resulting spectral sequence for motivic cohomology.

The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product

We consider the Belkale-Kumar cup product ⊙ t \odot _t on H ∗ ( G / P ) H^*(G/P) for a generalized flag variety G / P G/P with parameter t ∈ C m t \in \mathbb {C}^m , where m = dim ⁡ ( H 2 ( G / P ) ) m=\dim (H^2(G/P)) . For each t ∈ C m t\in \mathbb {C}^m , we define an associated parabolic subgroup P K ⊃ P P_K \supset P . We show that the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) contains a graded subalgebra A A isomorphic to H ∗ ( P K / P ) H^*(P_K/P) with the usual cup product, where P K P_K is a parabolic subgroup associated to the parameter t t . Further, we prove that ( H ∗ ( G / P K ) , ⊙ 0 ) (H^*(G/P_K), \odot _0) is the quotient of the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) with respect to the ideal generated by elements of positive degree of A A . We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.

Lifting problems and transgression for non-abelian gerbes

We discuss lifting and reduction problems for bundles and gerbes in the context of a Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen’s long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces–in case of a Lie 2-group with a single object–to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structures on the loop space.

On Hochschild–Serre spectral sequence of Lie algebras

Using non-abelian exterior product and free presentation of a Lie algebra the Hochschild–Serre spectral sequence for cohomology of Lie algebras will be extended a step further. Also, some results about this sequence are obtained.

Low-dimensional non-abelian Leibniz cohomology

We construct the zero and first non-abelian cohomologies of Leibniz algebras with coefficients in crossed modules, which differ from those of Gnedbaye and generalize the zero and first Leibniz cohomologies of Loday and Pirashvili. We also introduce the second non-abelian Leibniz cohomology and describe its relationship with extensions of Leibniz algebras by crossed modules. We obtain a nine-term exact non-abelian cohomology sequence. For Lie algebras we compare the non-abelian Leibniz and Lie cohomologies.

Recursion for Poincaré polynomials of subspace arrangements

A subspace arrangement $\mathcal{A}$ in $\mathbb{C}^m$ is a finite set $\{x_0, \dots, x_n \}$ of vector subspaces. The complement space $M(\mathcal{A})$ is $\mathbb{C}^m \setminus \bigcup_{x \in \mathcal{A}} x$. When each subspace is an hyperplane, it is also known as an arrangement of hyperplanes. In that case, it is known that the Poincare polynomials of $M(\mathcal{A})$ is connected to the Poincare polynomials of the complements of the deleted arrangement $\mathcal{A}' = \mathcal{A} \setminus \{x_0\}$ and of the restricted arrangement $\mathcal{A}'' = \{ x_0 \cap y \st y \in \mathcal{A}' \}$ by the nice formula \[ Poin(M(\mathcal{A}),t) = Poin(M(\mathcal{A}'),t) + tPoin(M(\mathcal{A}''),t). \] In this paper, we prove that for a subspace arrangement, there is a long exact sequence in cohomology which connects $M(\mathcal{A})$ to $M(\mathcal{A}')$ and $M(\mathcal{A}'')$. Using it, we can extend the above formula to arrangements with a geometric lattice, and to some other specific arrangements.

A generalization of the Wang sequence

The classical Wang sequence (for cohomology of fiber bundles over a sphere) is extended to a more generalized setting, given by gluing together two disc bundles over manifolds along their boundaries.

Seidel’s long exact sequence on Calabi-Yau manifolds

In this paper, we generalize construction of Seidel's long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of anchored Lagrangian submanifolds developed in \cite{fooo:anchor} and some compactness theorem of \emph{smooth} $J$-holomorphic sections of Lefschetz Hamiltonian fibration for a generic choice of $J$. The proof of the latter compactness theorem involves a study of proper pseudoholomorphic curves in the setting of noncompact symplectic manifolds with cylindrical ends.

A spectral sequence for de Rham cohomology

A spectral sequence for de Rham cohomology

Bloch–Ogus Sequence in Degree Two

We prove exactness of the Bloch–Ogus type étale cohomology sequence with initial term of degree two with -coefficients, over any 2-dimensional excellent regular local ring of residue characteristic prime-to-n. Parts of the sequence are exact for arbitrarily twisted μ n -coefficients and excellent regular local rings of any dimension.

Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism, the scattering matrix at low energy may be regarded as an operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud–Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.

Group cohomology with coefficients in a crossed module

AbstractWe compare three different ways of defining group cohomology with coefficients in a crossed module: (1) explicit approach via cocycles; (2) geometric approach via gerbes; (3) group theoretic approach via butterflies. We discuss the case where the crossed module is braided and the case where the braiding is symmetric. We prove the functoriality of the cohomologies with respect to weak morphisms of crossed modules and also prove the ‘long’ exact cohomology sequence associated to a short exact sequence of crossed modules and weak morphisms.

Butterflies I: Morphisms of 2-group stacks

Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a complete description of the resulting bicategory of crossed modules, which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As a consequence we obtain a complete characterization of the non-abelian derived category of complexes of length 2. Deligne's analogous theorem in the case of Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity laws on 2-group stacks are also analyzed in terms of butterflies, yielding new characterizations of braided, symmetric, and Picard 2-group stacks. Furthermore, the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non-abelian cohomology, removing a preexisting fibration condition on the coefficients short exact sequence.

A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If D is a restriction functor for a finite group G , then the mark morphism φ : D + → D + is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G ) after composing with a suitable isomorphism of D + . As a consequence, we obtain an exact sequence of Mackey functors 0 → Ext ̂ γ − 1 ( ρ , D ) → D + ⟶ φ D + → Ext ̂ γ 0 ( ρ , D ) → 0 where ρ denotes the restriction algebra and γ denotes the conjugation algebra for G . Then, we show how one can calculate these Tate groups explicitly using group cohomology and give some applications to integrality conditions.

Directions for the Long Exact Cohomology Sequence in Moore Categories

A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced by Bourn (Cahiers Topologie Géom Différentielle Catég XL:297–316, 1999; J Pure Appl Algebra 168:133–146, 2002). The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good context for non-abelian cohomology, in particular for the Baer Extension Theory.

Exact cohomology sequences with integral coefficients for torus actions

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free equivariant cohomology over the cohomology of BT. Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.

A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space Λ X . We introduce two spectral sequences converging towards H * ( Λ X ; Z / p ) and H * ( ( Λ X ) h T ; Z / p ) . The E 2 -terms are certain non-Abelian-derived functors applied to H * ( X ; Z / p ) . When H * ( X ; Z / p ) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p = 2 .

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups