Cohomology Theory Research Articles

Twisted iterated algebraic K-theory and topological T-duality for sphere bundles

We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory.

Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of D\'eglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.

On Hom-Lie antialgebra

In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.Communicated by Dr. Pavel Kolesnikov

THE HIERARCHY OF ASSOCIATIVITY EQUATIONS FOR n=3 CASE WITH AN METRIC ƞ11≠0

This paper describes the hierarchy for N = 2 and n=3 case with an metric ƞ11≠0 when V0 = 0 of associativity equations. The equation of associativity arose from the 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. We give a description of nonlinear partial differential equations of associativity in 2D topological field theories as integrable nondiagonalizable weakly nonlinear homogeneous system of hydrodynamic type. The article discusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x, t. In this work we consider the associativity equation for n=3 case with an a metric 0 11   . The solution of some cases of hierarchy when N = 2 and V0 = 0 equations of associativity illustrated

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial

Automorphism groups of superspecial curves of genus 4 over [formula omitted

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus 4 over F11. Our algorithm determining automorphism groups works for any nonhyperelliptic curve of genus 4 over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus 4 over F11 obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.

Operations and poly-operations in algebraic cobordism

In the case of a field of characteristic zero, we describe all operations (including non-additive ones) from a theory A⁎ obtained from Algebraic Cobordism Ω⁎ of M. Levine-F. Morel by change of coefficients to any oriented cohomology theory B⁎ (in the sense of Definition 2.1). We prove that such an operation can be reconstructed out of it's action on the products of projective spaces. This reduces the construction of operations to algebra and extends the additive case done in [24], as well as the topological one obtained by T. Kashiwabara - see [6]. The key new ingredients which permit us to treat the non-additive operations are: the use of poly-operations and the “Discrete Taylor expansion”. As an application we construct the only missing, the 0-th (non-additive) Symmetric operation, for arbitrary p - see [23], which permits to sharpen results on the structure of Algebraic Cobordism - see [22]. We also prove the general Riemann-Roch theorem for arbitrary (even non-additive) operations (over an arbitrary field). This extends the case of multiplicative operations proved by I. Panin in [13].

A counter-example by Yagita

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

Non-abelian Galois cohomology via descent cohomology

It is shown that the descent cohomology theory of monads includes Serre’s non-abelian Galois cohomology theory as a special case.

Representations and cohomology of a family of finite supergroup schemes

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form (Ga−×Ga−)⋊(Ga(r)×(Z/p)s). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.We also completely determine the cohomology ring in the smallest cases, namely (Ga−×Ga−)⋊Ga(1) and (Ga−×Ga−)⋊Z/p. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes.

Equivariant $K$-theory of divisive torus orbifolds

The category of torus orbifolds is a generalization of the category of toric orbifolds which contains projective toric varieties associated to complete simplicial fans. We introduce the concept of “divisive” torus orbifolds following divisive weighted projective spaces. The divisive condition may ensure an invariant cell structure on a locally standard torus orbifold. We give a combinatorial description of equivariant K-theory, equivariant cobordism theory and equivariant cohomology theory of divisive torus orbifolds. In particular, we get a combinatorial description of these generalize cohomology theories for torus manifolds over acyclic polytopes.

Mixed cohomology of Lie superalgebras

Supermanifolds are known to admit both differential forms and integral forms, thus any appropriate super analogue of the de Rham theory should take both types of forms into account. However, the cohomology of Lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de Rham theory for Lie supergroups. Thus a new cohomology theory of Lie superalgebras is needed to fully incorporate differential-integral forms, and we investigate such a theory here. This new cohomology is defined by a BRST complex of Lie superalgebra modules, and includes the standard Lie superalgebra cohomology as a special case. General properties expected of a cohomology theory are established for the new cohomology, and examples of the new cohomology groups are computed.

Treatment of Operator Algebras through Cohomology Theory

Treatment of Operator Algebras through Cohomology Theory

Hodge Laplacians on Graphs

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. T...

On the Morse–Novikov Cohomology of blowing up complex manifolds

Inspiré par les récents travaux de S. Rao, S. Yang, X.-D. Yang et L. Meng sur les formules donnant le comportement des groupes de cohomologie de de Rham et Morse-Novikov dans les éclatements, nous donnons une nouvelle preuve simple de la formule pour la cohomologie de Morse-Novikov en introduisant le groupe de cohomologie de Morse-Novikov relatif via la cohomologie des faisceaux et en explicitant l’isomorphisme de la formule.

On the axiomatic systems of singular cohomology theory

On the category of pairs of topological spaces having a homotopy type of CW complexes the singular (co)homology theory was axiomatically studied by J. Milnor [8]. In particular, Milnor gave additivity axiom for a (co)homology theory and proved that any additive (co)homology theory on the given category is isomophic to the singular (co)homology. On the other hand, the singular homology is a homology with compact support [3]. In the paper [6], L. Mdzinarishvili proposed partially compact support property for a cohomology theory and gave another axiomatic characterization of the singular cohomology theory. In this paper, we will give additional different axiomatic characterizations of the singular cohomology theory. Moreover, we will study connections of the mentioned axiomatic systems (cf. [2]).

Separability of Nonassociative Algebras with Metagroup Relations

This article is devoted to a class of nonassociative algebras with metagroup relations. This class includes, in particular, generalized Cayley–Dickson algebras. The separability of the nonassociative algebras with metagroup relations is investigated. For this purpose the cohomology theory is utilized. Conditions are found under which such algebras are separable. Algebras satisfying these conditions are described.

Cohomology structure for a Poisson algebra: II

We introduce for any Poisson algebra a bicomplex of free Poisson modules, and use it to show that the Poisson cohomology theory introduced in the paper "[M. Flato, M. Gerstenhaber and A. A. Voronov, Cohomology and Deformation of Leibniz Pairs, Lett. Math. Phys. 34 (1995) 77--90]" is given by certain derived functor. Moreover, by constructing a long exact sequence connecting Poisson cohomology groups and Yoneda-extension groups of certain quasi-Poisson modules, we provide a way to compute this Poisson cohomology via the Lie algebra cohomology and the Hochschild cohomology.

Local cohomology in Grothendieck categories

Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

On the Relation of Symplectic Algebraic Cobordism to Hermitian K-Theory

We reconstruct hermitian K-theory via algebraic symplectic cobordism. In the motivic stable homotopy category SH(S), there is a unique morphism ϕ: MSp → BO of commutative ring T-spectra which sends the Thom class thMSp to the Thom class thBO. Using ϕ we construct an isomorphism of bigraded ring cohomology theories on the category $${\mathop{\rm Sm}\nolimits} {\mathcal O}p/S,\bar \varphi :{{\mathop{\rm MSp}\nolimits} ^{*,*}}(X,U){ \otimes _{{\rm{MS}}{{\rm{p}}^{4*,0*}}({\rm{pt}})}}{\rm{B}}{{\rm{O}}^{4*,2*}}({\rm{pt}}) \cong {\rm{B}}{{\rm{O}}^{*,*}}(X,U)$$. The result is an algebraic version of the theorem of Conner and Floyd reconstructing real K-theory using symplectic cobordism. Rewriting the bigrading as MSpp,q = MSp1[q], we have an isomorphism $$\bar \varphi :{{\mathop{\rm MSp}\nolimits} _*}^{[*]}(X,U){ \otimes _{{\rm{MSp}}_0^{[2*]}({\rm{pt}})}}{\rm{KO}}_0^{[2*]}({\rm{pt}}) \cong {\rm{K}}{{\rm{O}}_*}^{[*]}(X,U)$$, where the KOi[](X,U) are Schlichting’s hermitian K-theory groups.