Cohomology Of Complex Research Articles

Twisting theory, relative Rota-Baxter type operators and L∞-algebras on Lie conformal algebras

Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L∞-algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C[∂]-module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C[∂]-module homomorphism is a Maurer-Cartan element of the L∞-algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations.

Representation of relative sheaf cohomology

Abstract We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Čech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Čech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a “relative de Rham-type theorem” from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated. Particularly notable is the application of the Dolbeault complex case to the Sato hyperfunction theory and other problems in algebraic analysis.

Stable cohomology of graph complexes

We study three graph complexes related to the higher genus Grothendieck–Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus g tends to infty . As a byproduct, we find that the Malcev completion of the genus g mapping class group relative to the symplectic group is Koszul in the stable limit, partially answering a question of Hain.

A compact manifold with infinite-dimensional co-invariant cohomology

Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on the manifold $M$ by diffeomorphisms one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the differential complex $\Omega_c(M)_\Gamma=\mathrm{span}\{\omega-\gamma^*\omega,\;\omega\in\Omega_c(M),\;\gamma\in\Gamma\}.$ For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the complex $\mathcal{C}_{\mathcal{G}}(M)=\mathrm{span}\{L_X\omega,\;\omega\in\Omega_c(M),\;X\in\mathcal{G}\}.$ In this short paper we present a situation where these two cohomologies are infinite dimensional.

Index of the transversally elliptic complex in Pestunization

In this note we present a formula for the equivariant index of the cohomological complex obtained from localization of SYM on simply-connected compact four-manifolds with a T 2-action. Knowledge of said index is essential to compute the perturbative part of the partition function for the theory. In the topologically twisted case, the complex is elliptic and its index can be computed in a standard way using the Atiyah–Bott localization formula. Recently, a framework for more general types of twisting, so-called cohomological twisting, was introduced for which the complex turns out to be only transversally elliptic. While the index of such a complex has been computed for some cases where the manifold can be lifted to a Sasakian S 1-fibration in five dimensions, a general four-dimensional treatment was still lacking. We provide a formal, purely four-dimensional treatment of the cohomological complex, showing that the Laplacian part can be globally split off while the remaining part can be trivialized uniquely in the group-direction. This ultimately produces a simple formula for the index applicable for any compact simply-connected four-manifold. Finally, the index formula is applied to examples on S 4, and . For the latter, we use the result to compute the perturbative partition function.

On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve

We explore the cohomological structure for the (possibly singular) moduli of SL n $\mathrm{SL}_n$ -Higgs bundles for arbitrary degree on a genus g $g$ curve with respect to an effective divisor of degree > 2 g − 2 $>2g-2$ . We prove a support theorem for the SL n $\mathrm{SL}_n$ -Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel–Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder–Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ngô–type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.

Mod ℓ cohomology of some Deligne–Lusztig varieties for GLn(q)

In this article, we study the mod ℓ cohomology of some Deligne–Lusztig varieties for GLn(q). We prove that the cohomology groups of these varieties are torsion-free under some conditions on the characteristic. Under the torsion-free assumption we can compute the cohomology groups explicitly and we prove that the cohomology complex satisfies a partial-tilting condition, which is one of the necessary conditions in the geometric version of Broué's abelian defect group conjecture.

Equivariant perverse sheaves and quasi-hereditary algebras

Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field.Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field Fq, and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

On 3-Hom-Lie-Rinehart algebras

We introduce the notion of 3-Hom-Lie-Rinehart algebras and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an A-split abelian extension and the equivalence classes of A-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.

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.

Localization of IC-complexes on Kashiwara’s flag scheme and representations of Kac–Moody algebras

We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwara’s flag manifold. Using moment graph techniques we establish a link to the representation theory of Kac–Moody algebras and give a new proof of the Kazhdan–Lusztig conjecture for blocks containing an antidominant element.

Non-trivial higher Massey products in moment-angle complexes

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry.

Structure and cohomology of 3-Lie-Rinehart superalgebras

We introduce a concept of 3-Lie-Rinehart superalgebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we study the relationships between a Lie-Rinehart superalgebra and its induced 3-Lie-Rinehart superalgebra. The deformations of 3-Lie-Rinehart superalgebra are considered via a cohomology theory.

Lifting Chern classes by means of Ekedahl–Oort strata

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.

Recollement for perverse sheaves on real hyperplane arrangements

Consider the category of perverse sheaves on Cn smooth with respect to the stratification arising from a real hyperplane arrangement. We construct an algebra whose module category is equivalent to this category, building on work of Kapranov–Schechtman. We prove that the standard recollement of perverse sheaves is equivalent to a well-known recollement for algebras.As an application of our results, we give a new description of the category of representations of the fundamental group of the complement of such a hyperplane arrangement. We also identify the modules associated to all simple perverse sheaves, that is, the intersection cohomology complexes. Finally, we generalise our results to W-equivariant perverse sheaves for the reflection arrangement of a finite Coxeter group W, extending work of Weissman. As an application, we identify the modules associated to the equivariant simple perverse sheaves.

Local cohomology and local homology complexes with respect to a pair of ideals

We introduce local cohomology complexes and local homology complexes with respect to a pair of ideals $(I,J)$ of a commutative noetherian ring $R$, characterize them using Čech complexes and establish "MGM" equivalence between the local cohomology functor $\mathrm{R}\Gamma_{I,J}$ and local homology functor $\mathrm{L}\Lambda_{I,J}$ on $\mathrm{D}(R)$.

Relative cohomology of complexes II: Vanishing of relative cohomology

Let R be a ring such that (GP,GP⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R-modules. Recently, the first author defined for any complex X the relative cohomology functors ExtGP⁎(X,−) as H−⁎(Hom(G,−)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which ExtGPi(X,Q)=0 for all i>n and all R-modules Q∈GP⊥. This result gives a “Gorenstein” version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors ExtR⁎(−,−).

Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group

We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.