Construction Of Cohomology Research Articles

Floquet topological phases with symmetry in all dimensions

Dynamical systems may host a number of remarkable symmetry-protected phases that are qualitatively different from their static analogs. In this work, we consider the phase space of symmetry-respecting unitary evolutions in detail and identify several distinct classes of evolution that host novel dynamical order. Using ideas from group cohomology, we construct a set of interacting Floquet drives that generate dynamical symmetry-protected topological order for each nontrivial cohomology class in every dimension, illustrating our construction with explicit two-dimensional examples. We also identify a set of symmetry-protected Floquet drives that lie outside of the group cohomology construction, and a further class of symmetry-respecting topological drives which host chiral edge modes. We use these special drives to define a notion of phase (stable to a class of local perturbations in the bulk) and the concepts of relative and absolute topological order, which can be applied to many different classes of unitary evolutions. These include fully many-body localized unitary evolutions and time crystals.

A flexible construction of equivariant Floer homology and applications

Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsv\'ath-Szab\'o, Lipshitz-Treumann, Szab\'o, Sarkar-Seed-Szab\'o, and others. In this paper we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith-type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar-Seed-Szab\'o's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

Syntomic cohomology and p-adic regulators for varieties over p-adic fields

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 \'etale 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 \'etale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul\'e's \'etale 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

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig's program. Precisely, let D be the quaternion algebra over a local non-Archimedean field K of positive characteristic, and let X be the p-adic Deligne–Lusztig ind-scheme associated to D×. There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified quadratic extension of K and representations of D× given by θ↦Hi(X)[θ]. We show that this correspondence is a bijection (after a mild restriction of the domain and target), and matches the bijection given by local Langlands and Jacquet–Langlands.

Integral Eisenstein cocycles on $\mathbf {GL}_n$, II: Shintani’s method

We define a cocycle on \mathbf {GL}_n(\mathbf Q) using Shintani's method. This construction is closely related to earlier work of Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q . As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintani's proof of the Klingen–Siegel rationality theorem on partial zeta functions of totally real fields. Next we relate the Shintani cocycle to the Sczech cocycle by showing that the two differ by the sum of an explicit coboundary and a simple "polar" cocycle. This generalizes a result of Sczech and Solomon in the case n=2 . Finally, we introduce an integral version of our cocycle by smoothing at an auxiliary prime \ell . This integral refinement has strong arithmetic consequences. We showed in previous work that certain specializations of the smoothed class yield the p -adic L -functions of totally real fields. Furthermore, combining our cohomological construction with a theorem of Spiess, one deduces that that the order of vanishing of these p -adic L -functions is at least as large as the expected one.

Monk’s rule and Giambelli’s formula for Peterson varieties of all Lie types

A Peterson variety is a subvariety of the flag variety $G/B$ which appears in the construction of the quantum cohomology of partial flag varieties. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type $A$ Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives Monk's rule and Giambelli's formula for all Lie types.

Chevalley-Monk and Giambelli formulas for Peterson Varieties

A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

Shintani cocycles and the order of vanishing of $$p$$ p -adic Hecke $$L$$ L -series at $$s=0$$ s = 0

Let \(\chi \) be a Hecke character of finite order of a totally real number field \(F\). By using Hill’s Shintani cocycle we provide a cohomological construction of the \(p\)-adic \(L\)-series \(L_p(\chi , s)\) associated to \(\chi \). This is used to show that \(L_p(\chi , s)\) has a trivial zero at \(s=0\) of order at least equal to the number of places of \(F\) above \(p\) where the local component of \(\chi \) is trivial.

Completed cohomology of Shimura curves and a p-adic Jacquet–Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet–Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.

Cyclic symmetry and adic convergence in Lagrangian Floer theory

In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infinity algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered A infinity structure associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudo-isotopy of cyclic (filtered) A infinity algebra.

The Face Semigroup Algebra of a Hyperplane Arrangement

Abstract.This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete systemof primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposablemodules, the Cartan invariants, projective resolutions of the simple modules, the Hochschild homology and cohomology, and the Koszul dual algebra. A new cohomology construction on posets is introduced, and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.

Comparison of relative cohomology theories with respect to semidualizing modules

We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology construction does in fact result in a different functor.

P-adic Superspaces and Frobenius

The notion of a p-adic superspace is introduced and used to give a transparent construction of the Frobenius map on p-adic cohomology of a smooth projective variety over \({\mathbb{Z}_p}\) (the ring of p-adic integers), as well as an alternative construction of the crystalline cohomology of a smooth projective variety over \({\mathbb{F}_p}\) (finite field with p elements).

On codihedral module for *-Hopf algebras

AbstractWe construct dihedral and reflexive cohomology theories for *-Hopf algebras. This generalizes the Connes–Moscovici construction of cyclic cohomology for Hopf algebras.

Axiomatic characterization of ordinary differential cohomology

Cheeger-Simons differential characters, Deligne cohomology in the smooth category, the Hopkins-Singer construction of ordinary differential cohomology and the recent Harvey-Lawson constructions are each in two distinct ways Abelian group extensions of known functors. In one desciption these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short exact sequences mesh with two interlocking long exact sequences (the Bockstein sequence and a DeRham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Theorem 1.1 shows that on the category of smooth manifolds and smooth maps any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as above is unique up to unique natural equivalence. Theorem 1.2 shows the natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of Theorem 1.1 couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

Cohomological construction of relative twists

Let g be a complex, semi-simple Lie algebra, h ⊂ g a Cartan subalgebra and D a subdiagram of the Dynkin diagram of g . Let g D ⊂ l D ⊆ g be the corresponding semi-simple and Levi subalgebras and consider two invariant solutions Φ ∈ ( U g ⊗ 3 〚 ℏ 〛 ) g and Φ D ∈ ( U g D ⊗ 3 〚 ℏ 〛 ) g D of the pentagon equation for g and g D respectively. Motivated by the theory of quasi-Coxeter quasitriangular quasibialgebras [V. Toledano Laredo, Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups, math.QA/0506529], we study in this paper the existence of a relative twist, that is an element F ∈ ( U g ⊗ 2 〚 ℏ 〛 ) l D such that the twist of Φ by F is Φ D . Adapting the method of Donin and Shnider [J. Donin, S. Shnider, Cohomological construction of quantized universal enveloping algebras, Trans. Amer. Math. Soc. 349 (1997) 1611–1632], who treated the case of an empty D, so that l D = h and Φ D = 1 ⊗ 3 , we give a cohomological construction of such an F under the assumption that Φ D is the image of Φ under the generalised Harish-Chandra homomorphism ( U g ⊗ 3 ) l D → ( U g D ⊗ 3 ) g D . We also show that F is unique up to a gauge transformation if l D is of corank 1 or F satisfies F Θ = F 21 where Θ ∈ Aut ( g ) is an involution acting as −1 on h .

The Generalized Cuspidal Cohomology Problem

AbstractLet Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\ℍ3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H1(Γ, ℍ41). Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

The stable derived category of a noetherian scheme

for a noetherian scheme, we introduce its unbounded stable derived category. this leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. some applications are included, for instance an analogue of maximal cohen–macaulay approximations, a construction of tate cohomology, and an extension of the classical grothendieck duality. in addition, the relevance of the stable derived category in modular representation theory is indicated.

Cohomologies bivariantes de type cyclique

In the article we propose a construction of bivariant cohomology of a couple of chain complexes with periodicities. In this way we obtain definitions of bivariant dihedral and bivariant reflective cohomology of an algebra A. Bivariant cyclic and quaternionic cohomologies appear as particular cases of this construction. The case of 2 invertible in the ground ring is studied particulary.

A cohomological construction of integrable hierarchies of hydrodynamic type

We explain how to use the theory of bidifferential ideals to construct integrable hierarchies of hydrodynamic type