Relative Cohomology Groups Research Articles

Intersection numbers, polynomial division and relative cohomology

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential n-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural limiting mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotent orbit in several variables.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Quantisation of derived Lagrangians

We investigate quantisations of line bundles $\mathcal{L}$ on derived Lagrangians $X$ over $0$-shifted symplectic derived Artin $N$-stacks $Y$. In our derived setting, a deformation quantisation consists of a curved $A_{\infty}$ deformation of the structure sheaf $\mathcal{O}_{Y}$, equipped with a curved $A_{\infty}$ morphism to the ring of differential operators on $\mathcal{L}$; for line bundles on smooth Lagrangian subvarieties of smooth symplectic algebraic varieties, this simplifies to deforming $(\mathcal{L}, \mathcal{O}_{Y})$ to a DQ module over a DQ algebroid. For each choice of formality isomorphism between the $E_2$ and $P_2$ operads, we construct a map from the space of non-degenerate quantisations to power series with coefficients in relative cohomology groups of the respective de Rham complexes. When $\mathcal{L}$ is a square root of the dualising line bundle, this leads to an equivalence between even power series and certain anti-involutive quantisations, ensuring that the deformation quantisations always exist for such line bundles. This gives rise to a dg category of algebraic Lagrangians, an algebraic Fukaya category of the form envisaged by Behrend and Fantechi. We also sketch a generalisation of these quantisation results to Lagrangians on higher $n$-shifted symplectic derived stacks.

The Relative (Co)homology Theory through Operator Algebras

This paper introduces a new idea in the unital involutive Banach algebras and its closed subset. This paper aims to study the cohomology theory of operator algebra. We will study the Banach algebra as an applied example of operator algebra, and the Banach algebra will be denoted by <img src=image/13426521_01.gif>. The definitions of cyclic, simplicial, and dihedral cohomology group of <img src=image/13426521_01.gif> will be introduced. We presented the definition of <img src=image/13426521_14.gif>-relative dihedral cohomology group that is given by: <img src=image/13426521_02.gif>, and we will show that the relation between dihedral and <img src=image/13426521_14.gif>-relative dihedral cohomology group <img src=image/13426521_11.gif> <img src=image/13426521_12.gif> <img src=image/13426521_13.gif> can be obtained from the sequence <img src=image/13426521_03.gif>. Among the principal results that we will explain is the study of some theorems in the relative dihedral cohomology of Banach algebra as a Connes-Tsygan exact sequence, since the relation between the relative Banach dihedral and cyclic cohomology group (<img src=image/13426521_04.gif> and <img src=image/13426521_05.gif>) of <img src=image/13426521_01.gif> will be proved as the sequence <img src=image/13426521_06.gif>. Also, we studied and proved some basic notations in the relative cohomology of Banach algebra with unity and defined its properties. So, we showed the Morita invariance theorem in a relative case with maps <img src=image/13426521_07.gif> and <img src=image/13426521_08.gif>, and proved the Connes-Tsygan exact sequence that relates the relative cyclic and dihedral (co)homology of <img src=image/13426521_01.gif>. We proved the Mayer-Vietoris sequence of <img src=image/13426521_04.gif> in a new form in the Banach B-relative dihedral cohomology: <img src=image/13426521_09.gif> <img src=image/13426521_10.gif>. It should be borne in mind that the study of the cohomology theory of operator algebra is concerned with studying the spread of Covid 19.

Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

The relative Heller operator and relative cohomology for the Klein 4-group

Let G be the Klein four-group and let be an arbitrary field of characteristic 2. A classification of indecomposable -modules is known. We calculate the relative cohomology groups for every indecomposable -module N, where χ is the set of proper subgroups in G. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in are trivial.

Cohomology of the vector fields Lie algebras on acting on trilinear differential operators, vanishing on

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields with coefficients in the space of trilinear differential operators that acts on tensor densities, vanishing on the Lie algebra where and

A-hypergeometric systems and relative cohomology

We investigate the solution space to certain [Formula: see text]-hypergeometric [Formula: see text]-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by [Formula: see text], which generalizes the results of Huang, Lian, Yau and Zhu. As a corollary, we also prove the existence of rank one points for complete intersections in toric varieties.

Relative Čech–Dolbeault homology and applications

We define the relative Dolbeault homology of a complex manifold with currents via a Cech approach, and we prove its equivalence with the relative Cech–Dolbeault cohomology as defined by Suwa (in: Singularities–Niigata–Toyama 2007. Advanced studies in pure mathematics, vol 56. Mathematical Society of Japan, Tokyo, pp 321–340, 2009). This definition is then used to compare the relative Dolbeault cohomology groups of two complex manifolds of the same dimension related by a suitable proper surjective holomorphic map. Finally, an application to blow-ups is considered and a blow-up formula for the Dolbeault cohomology in terms of relative cohomology is presented.

Motivic spectral sequence for relative homotopy K-theory

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$.

Relative cohomology of complexes based on cotorsion pairs

Let [Formula: see text] be an associative ring with identity. The purpose of this paper is to establish relative cohomology theories based on cotorsion pairs in the setting of unbounded complexes of modules over [Formula: see text]. Let [Formula: see text] be a complete hereditary cotorsion pair in [Formula: see text]-Mod. Then [Formula: see text] and [Formula: see text] are complete hereditary cotorsion pairs in the category of [Formula: see text]-complexes. For any complexes [Formula: see text] and [Formula: see text] and any [Formula: see text], we define the [Formula: see text]th relative cohomology groups [Formula: see text] and [Formula: see text] by special [Formula: see text]-precovers of [Formula: see text] and by special [Formula: see text]-preenvelopes of [Formula: see text], respectively. They are common generalizations of absolute cohomology groups and Gorenstein cohomology groups of complexes. Some induced exact sequences concerning relative cohomology groups are considered. It is also shown that the relative cohomology functor of complexes we considered is balanced.

Balance for relative cohomology of complexes

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

Extended Hodge theory for fibred cusp manifolds

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted [Formula: see text] harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted [Formula: see text] harmonic forms are harmonic forms that are almost in the given weighted [Formula: see text] space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted [Formula: see text] case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.

Relative cohomology of complexes

Let R be an arbitrary ring and C a complex with finite Gorenstein projective dimension (that is, the supremum of Gorenstein projective dimension of all R-modules in C is finite). For each complex D, we define the nth relative cohomology group ExtGPn(C,D) by the equality ExtGPn(C,D)=HnHom(G,D), where G⟶C is a strict Gorenstein projective precover of C. If D is a complex with finite Gorenstein injective dimension (that is, the supremum of Gorenstein injective dimension of all R-modules in D is finite), then one can use a dual argument to define a notion of relative cohomology group ExtGIn(C,D). We show that if C is a complex with finite Gorenstein projective dimension and D a complex with finite Gorenstein injective dimension, then for each n∈Z there exists an isomorphism ExtGPn(C,D)≅ExtGIn(C,D). This shows that the relative cohomology functor of complexes is balanced. Some induced exact sequences concerning relative cohomology groups are considered.

Cohomology of the vector fields Lie algebras on ℝ acting on trilinear differential operators, vanishing on 𝔞𝔣𝔣(1)

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields [Formula: see text], with coefficients in the space of trilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra [Formula: see text].

Relative cohomology of bi-arrangements

A bi-arrangement of hyperplanes in a complex affine space is data of two sets of hyperplanes along with a coloring information on strata. To such a bi-arrangement, one naturally associates a relative cohomology group, that we call its motive. The motivation for studying such relative cohomology groups comes from notion of motivic period. More generally, we suggest systematic study of motive of a bi-arrangement of hypersurfaces in a complex manifold. We provide combinatorial and cohomological tools to compute structure of these motives. Our main object is Orlik-Solomon bi-complex of a bi-arrangement, which generalizes Orlik-Solomon algebra of an arrangement. Loosely speaking, our main result states that the motive of an exact bi-arrangement is computed by its Orlik-Solomon bi-complex, which generalizes classical facts involving Orlik-Solomon algebra of an arrangement. We show how this formalism allows us to explicitly compute motives arising from study of multiple zeta values and sketch a more general application to periods of mixed Tate motives.

Transporting cohomology in Lazard correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-[Formula: see text] groups of nilpotency class smaller than [Formula: see text] and finitely generated nilpotent [Formula: see text]-Lie algebras of nilpotency class smaller than [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for [Formula: see text], [Formula: see text] and [Formula: see text], and for a given category of modules the cohomology functors [Formula: see text] and [Formula: see text] are naturally equivalent. A similar result is proved for [Formula: see text] with the relative cohomology groups.

Computation of Relative 1-Cohomology Generators From a 1-Homology Basis for Eddy Currents Boundary Integral Formulations

Efficient boundary integral formulations based on stream functions for solving eddy current problems in thin conductors, which are modeled by the orientable combinatorial two-manifold with boundary, need generators of the first relative cohomology group to make the problem well defined. The state-of-the-art technique is to compute directly the relative cohomology generators with a combinatorial algorithm having linear worst-case complexity. In this paper, we propose to compute the relative cohomology generators from the homology generators, introducing a novel and general algorithm whose running time is again linear in the worst case. The advantage is that one may use an off-the-shelf software to compute the homology generators and implement only a simple and cheap procedure to obtain the required relative cohomology generators. Although the presented applications relate to ac power systems, the proposed technique is of general interest, and may be used for other applications in computational science and engineering.

Homogeneous generalized holomorphic structures

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds.