Cohomology Of Algebras Research Articles

A basis of algebraic de Rham cohomology of complete intersections over a characteristic zero field

Let be a field of characteristic 0. Let X be a smooth complete intersection over of dimension n – k in the projective space for given positive integers n and k. When Terasoma and Konno provided an explicit representative (in terms of differential forms) of a basis for the primitive middle-dimensional algebraic de Rham cohomology Later Dimca constructed another explicit representative of a basis of . Moreover, he proved that his representative gives the same cohomology class as the previous representative of Terasoma and Konno. The goal of this article is to examine the above two different approaches without assuming that and provide a similar comparison result for any field Dimca’s argument depends heavily on the condition and our idea is to find appropriate Cech-de Rham complexes and spectral sequences corresponding to those two approaches, which work without restrictions on

On the BV structure of the Hochschild cohomology of finite group algebras

We give a simple algebraic recipe for calculating the components of the BV operator $\Delta$ on the Hochschild cohomology of a finite group algebra with respect to the centraliser decomposition. We use this to investigate the properties of $\Delta$ and to make some computations for some particular finite groups.

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov’s homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.

Hamiltonicity via cohomology of right-angled Artin groups

Let Γ be a finite graph and let A(Γ) be the corresponding right-angled Artin group. We characterize the Hamiltonicity of Γ via the structure of the cohomology algebra of A(Γ). In doing so, we define and develop a new canonical graph associated to a matrix, which as a consequence provides a novel perspective on the matrix determinant.

Classical and variational Poisson cohomology

We prove that, for a Poisson vertex algebra \({\cal V}\), the canonical injective homomorphism of the variational cohomology of \({\cal V}\) to its classical cohomology is an isomorphism, provided that \({\cal V}\), viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.

The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational or Ruled Types

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.

On the Cohomology of Lie Algebras with an Invariant Inner Product

In this work we consider the moduli space of all noncommutative metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \(\mathbb C\) and \(\mathbb R\) up to dimension 5 and all metric Lie algebras over \(\mathbb C\) in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.

The simplest cohomological invariants for vertex algebras

For the double complex structure of grading-restricted vertex algebra cohomology defined in [6,7], we introduce a multiplication of elements of double complex spaces. We show that the orthogonality and bi-grading conditions applied on double complex spaces, provide in relation among mappings and actions of co-boundary operators. Thus, we endow the double complex spaces with structure of bi-graded differential algebra. We then introduce the simplest cohomology classes for a grading-restricted vertex algebra, and show their independence on the choice of mappings from double complex spaces. We prove that its cohomology class does not depend on mappings representing of the double complex spaces. Finally, we show that the orthogonality relations together with the bi-grading condition bring about generators and commutation relations for a continual Lie algebra.

A note on Lie algebra cohomology

Given a finite dimensional Lie algebra $L$ let $I$ be the augmentation ideal in the universal enveloping algebra $U(L)$. We study the conditions on $L$ under which the $Ext$-groups $Ext (k,k)$ for the trivial $L$-module $k$ are the same when computed in the category of all $U(L)$-modules or in the category of $I$-torsion $U(L)$-modules. We also prove that the Rees algebra $\oplus _{n\geq 0}I^n$ is Noetherian if and only if $L$ is nilpotent. An application to cohomology of equivariant sheaves is given.

Cohomology Algebra of Orbit Spaces of Free Involutions on Some Wall Manifolds

In this paper, we investigate the existence of free involutions on some Wall manifolds and we compute the mod 2 cohomology algebra of the correspondent orbit space.

Algebraic elliptic cohomology and flops II: SL-cobordism

In this paper, we study the algebraic Thom spectrum MSL in Voevodsky's motivic stable homotopy category over an arbitrary perfect field k. Using the motivic Adams spectral sequence, we compute the geometric part of the η-completion of MSL. As an application, we study Krichever's elliptic genus with integral coefficients, restricted to MSL. We determine its image, and identify its kernel as the ideal generated by differences of SL-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.

A comparison between two de Rham complexes in diffeology

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech–de Rham spectral sequence by making use of the factor map which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_\theta$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over $T_\theta$.

Holonomic <i>D</i>-Modules Associated with a Simple Line Singularity and the Vertical Monodromy

Holonomic D-modules associated with certain kind of non-isolated hypersurface singularities, introduced by T. de Jong, are considered. Structures of these holonomic D-modules such as characteristic variety, monodromy are explicitly determined. The key of our approach is the use of the notion of algebraic local cohomology. An application to Betti numbers of local Milnor fibers is also discussed.

Hochschild Cohomology of the Cohomology Algebra of Closed Orientable Three- Manifolds

Let F be a field of characteristic other than 2. We show that the zeroth Hochschild cohomology vector space HH0(A) of a degree 3 graded commutative Frobenius F-algebra A = iAi, where we will always assume A0 = F, is isomorphic to the direct sum of the degree 0, 2 and 3 graded components and the kernel of a certain natural evaluation map ιμ : A1 Λ2(A1). In particular, this holds forA = H∗(M; F) the cohomology algebra of a closed orientable 3-manifoldM. In Theorem A of [1], Charette proves the vanishing of a certain discriminantΔassociated to a closed orientable 3-manifold L with vanishing cup product 3-form. It turns out that if we could show that HH2,−2(A) = 0for A = H∗(L;C), we would have found a more elementary proof of this part of Charette’s theorem. We show that for any β 3, the degree 3 graded commutative Frobenius algebra A with μA = 0and dim(A1) = β satisfiesHH2,−2(A) = 0. Thus Charette’s theorem is not simplified.

Codimension bounds for the Noether–Lefschetz components for toric varieties

For a quasi-smooth hypersurface X in a projective simplicial toric variety mathbb {P}_{Sigma }, the morphism i^*:H^p(mathbb {P}_{Sigma })rightarrow H^p(X) induced by the inclusion is injective for p=dim X and an isomorphism for p<dim X-1. This allows one to define the Noether–Lefschetz locus mathrm{NL}_{beta } as the locus of quasi-smooth hypersurfaces of degree beta such that i^* acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if dim mathbb {P}_{Sigma }=2k+1 and kbeta -beta _0=neta , nin mathbb {N}, where eta is the class of a 0-regular ample divisor, and beta _0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree beta satisfies the bounds n+1leqslant mathrm{codim},Z leqslant h^{k-1,,k+1}(X).

A new algorithm for computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals

A new algorithm is introduced for computing logarithmic vector fields along a hypersurface with an isolated singularity. The key ideas of the algorithm are computing an ideal quotient in a polynomial ring and the use of algebraic local cohomology. The use of these ideas allows us to compute a module, over a local ring, of germs logarithmic vector fields. The resulting algorithms are much faster than our previous algorithms in computation speed. As the applications, an effective method of computing Bruce-Roberts Milnor and Tjurina ideals (or numbers) is also introduced.

A motivic global Torelli theorem for isogenous K3 surfaces

We prove that the Chow motives of twisted derived equivalent K3 surfaces are isomorphic, not only as Chow motives (due to Huybrechts), but also as Frobenius algebra objects. Combined with a recent result of Huybrechts, we conclude that two complex projective K3 surfaces are isogenous (i.e. their second rational cohomology groups are Hodge isometric) if and only if their Chow motives are isomorphic as Frobenius algebra objects; this can be regarded as a motivic Torelli-type theorem. We ask whether, more generally, twisted derived equivalent hyper-Kähler varieties have isomorphic Chow motives as (Frobenius) algebra objects and in particular isomorphic graded rational cohomology algebras. In the appendix, we justify introducing the notion of “Frobenius algebra object” by showing the existence of an infinite family of K3 surfaces whose Chow motives are pairwise non-isomorphic as Frobenius algebra objects but isomorphic as algebra objects. In particular, K3 surfaces in that family are pairwise non-isogenous but have isomorphic rational Hodge algebras.

Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in $$M_{23}$$

Borcherds-Kac-Moody algebras generalise finite-dimensional, simple Lie algebras. Scheithauer showed that there are exactly ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of square-free level. These belong to a larger class of Borcherds-Kac-Moody (super)algebras Borcherds obtained by twisting the denominator identity of the Fake Monster Lie algebra. Borcherds asked whether these Lie (super)algebras admit natural constructions. For the ten Lie algebras from the classification we give a positive answer to this question, i.e. we prove that they can be realised uniformly as the BRST cohomology of suitable vertex algebras.

Tate–Hochschild cohomology for periodic algebras

This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.

Pro-$p$ groups with few relations and universal Koszulity

Let $p$ be a prime. We show that if a pro-$p$ group with at most $2$ defining relations has quadratic $\mathbb{F}_p$-cohomology algebra, then this algebra is universally Koszul. This proves the “Universal Koszulity Conjecture” formulated by J. Miná{č} et al. in the case of maximal pro-$p$ Galois groups of fields with at most $2$ defining relations.