Cochain Complex Research Articles

Cohomology and products of real weight filtrations

We associate to each algebraic variety defined over $\mathbb{R}$ a filtered cochain complex, which computes the cohomology with compact supports and $\mathbb{Z}\_2$-coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusi\'nski, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive virtual Betti numbers. We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products in cohomology and homology are filtered with respect to the real weight filtrations.

Coefficients for higher order Hochschild cohomology

When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild (co)homology, presented by Pirashvili and Anderson, to study deformations seems only natural though the current definition allows only symmetric bi-module coefficients. In this paper we present an extended definition for higher order Hochschild cohomology which allows multi-module coefficients (when the simplicial sets $X_{\bullet}$ are accommodating) which agrees with the current definition. Furthermore we determine the types of modules that can be used as coefficients for the Hochschild cochain complexes based on the simplicial sets they are associated to.

Coxeter cochain complexes

We define the Coxeter cochain complex of a Coxeter group (G, S) with coefficients in a ℤ[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.

Crossed interval groups and operations on the Hochschild cohomology

We prove that the operad \mathcal{B} of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that \mathcal{B} is a certain crossed interval extension of an operad \mathcal{T} whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.

On a spectral sequence for twisted cohomologies

Let (Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex (Ω*(M), d + H ∧) and its associated twisted de Rham cohomology H*(M,H). The authors show that there exists a spectral sequence {E , d r } derived from the filtration $F_p (\Omega ^ * (M)) = \mathop \oplus \limits_{i \geqslant p} \Omega ^i (M)$ of Ω*M, which converges to the twisted de Rham cohomology H*(M,H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

Yang–Baxter deformations and rack cohomology

In his study of quantum groups, Drinfeld suggested considering set-theoretic solutions of the Yang–Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group or, more generally, every rack Q Q provides such a Yang–Baxter operator c Q : x ⊗ y ↦ y ⊗ x y c_Q \colon x \otimes y \mapsto y \otimes x^y . In this article we study deformations of c Q c_Q within the space of Yang–Baxter operators over some complete ring. Infinitesimally these deformations are classified by Yang–Baxter cohomology. We show that the Yang–Baxter cochain complex of c Q c_Q homotopy-retracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of c Q c_Q , including the modular case which had previously been left in suspense, by establishing that every deformation of c Q c_Q is gauge equivalent to a quasi-diagonal one. In a quasi-diagonal deformation only behaviourally equivalent elements of Q Q interact; if all elements of Q Q are behaviourally distinct, then the Yang–Baxter cohomology of c Q c_Q collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Gerstenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory.

De Rham and Dolbeault cohomology of solvmanifolds with local systems

Let G be a simply connected solvable Lie group with a lattice and the Lie algebra g and a representation � : G ! GL(V�) whose restriction on the nilradical is unipotent. Consider the flat bundle Egiven by �. By using many characters {�} of G and many flat line bundles {E�} over G/, we show that an isomorphism M {�} H � (g,VV�) � M {E�} H � (G/ , EE�) This isomorphism is a generalization of the well-known fact:If G is nilpotent andis unipotent then, the isomorphism H � (g, V�) � H � (G/ ,E�) holds. By this result, we construct an explicit finite dimensional cochain complex which compute the cohomology H � (G/ ,E�) of solvmanifolds even if the isomorphism H � (g,V�) � H � (G/ ,E�) does not hold. For Dolbeault co- homology of complex parallelizable solvmanifolds, we also prove an analogue of the above isomorphism result which is a generalization of computations of Dolbeault cohomology of complex parallelizable nilmanifolds. By this iso- morphism, we construct an explicit finite dimensional cochain complex which compute the Dolbeault cohomology of complex parallelizable solvmanifolds.

Pre-Lie systems and obstructions to $$A_\infty $$ A ∞ -structures over a ring

In this note we establish an obstruction theory for the existence of $$A_\infty $$ -algebra structures on a differential $$\mathbb Z $$ -graded $$R$$ -module $$A$$ equipped with a homotopy associative multiplication. This approach follows classical obstruction theory: the corresponding obstructions live in the Hochschild cohomology of the associative algebra $$H(A)$$ . Our theory makes use of the pre-Lie algebra structure on the Hochschild cochain complex of $$H(A)$$ . As a consequence, no additional assumptions on the commutative ring $$R$$ are needed.

Nekovár duality over p-adic Lie extensions of global fields

Tate duality is a Pontryagin duality between the i th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the (2-i) th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module T over a complete commutative local Noetherian ring R with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual T^* , which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of T and T^*(1) are then Grothendieck dual to each other in the derived category of finitely generated R -modules. Given a p -adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of T and T^*(1) in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with R -coefficients.

Exact Sequences of High Order Hardy Space Infinite Elements for Exterior Maxwell Problems

This paper describes the construction of a new family of prismatic infinite elements for electromagnetic scattering and resonance problems. The construction is based on the use of the pole condition to define the equivalent of an “outgoing wave.” Specifically, a function satisfies the pole condition if and only if a certain transformation of this function belongs to a Hardy space. We use tensor products of cochain complexes to obtain four different infinite element spaces which form an exact sequence corresponding to the deRham complex in the exterior domain. Numerical tests indicate superalgebraic convergence in the number of additional unknowns per degree of freedom on the coupling boundary.

Hochschild cohomology and string topology of global quotient orbifolds

Let M be a connected, simply connected, closed and oriented manifold, and G be a finite group acting on M by orientation-preserving diffeomorphisms. In this paper, we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M.

Modular isogeny complexes

We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products \({\mathbb{C}^{n}\ltimes_{\phi}N}\) of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product \({\mathbb{C}^{n}\times N}\) . As a corollary of this computation, we can compute the Dolbeault cohomology Hp,q(G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.

On the levels of maps and topological realization of objects in a triangulated category

The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover, we provide a method to compute the invariant for spaces over a K -formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander–Reiten theory for spaces due to Jørgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.

Compact generators in categories of matrix factorizations

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.

Calculus structure on the Lie conformal algebra complex and the variational complex

We construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a \documentclass[12pt]{minimal}\begin{document}${\mathfrak g}$\end{document}g-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009)10.1007/s00220-009-0886-1]. A special case of this construction is the variational calculus, for which we provide explicit formulas.

Formality Theorem for Hochschild Cochains via Transfer

We construct a 2-colored operad G^+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that Tamarkin's G-structure on the Hochschild cochain complex C(A) of an A-infinity algebra A extends naturally to a G^+ structure on the pair (C(A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this G^+ structure to the cohomology of the pair (C(A), A). We show that G^+ is a sub DG operad of the first sheet E^1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov's Swiss Cheese operad. Finally, we prove that the DG operads G^+ and E^1(SC) are non-formal.

Topological constructions for multigraded squarefree modules

Let R = k[x 1 , ..., x n ] and let M = R s /I be a multigraded squarefree module. We discuss the construction of cochain complexes associated to M and we show how to interpret homological invariants of M in terms of topological computations. This is a generalization of the well-studied case of squarefree monomial ideals.

Proof of Swiss Cheese Version of Deligne's Conjecture

For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss Cheese version of the Deligne conjecture formulated by M. Kontsevich in arXiv:math/9904055.

Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality $L_\infty$-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality $L_\infty$-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of {\bf Kontsevich eye}. In particular we clarify the r\^ole played by the {\bf I-cube} introduced in \cite{CR1}. Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular we prove that they naturally inherit the structure of an $A_\infty$-algebra with an $A_\infty$-(bi)module.