Deformations and generalized derivations of Hom-Lie conformal algebras

The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.     

Über die Cohomologie von BSO(2k+1) und Vn,2

If h^* is a multiplicative cohomology theory on the category of CW-pairs, then it is shown that h^*(BSO (2k+1)) is under certain conditions isomorphic to the ring of formal power series over h^*(*) in universal Pontrjagin-classes p₁,...,p_k. In the second part of the paper one finds a calculation of the cohomology of the Stiefel manifolds V_n,2=SO(n)/SO(n–2).     

Hyperbolic localization of intersection cohomology

For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = G_m, we consider the "hyperbolic localization" functor D^b(X) → D^b(X^T), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.     

A continuous analogue of Sperner's theorem

One of the best-known results of extremal combinatorics is Sperner's theorem, which asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient (n/(n/2)), that is, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present paper to propose yet another generalization that strikes in a different direction. We consider the lattice Mod(n) of linear subspaces (through the origin) of the vector space Rⁿ. Because this lattice is infinite, the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank k of the lattice Mod(n), that is, the set of all subspaces of dimension k of Rⁿ, or Grassmannian, possesses an invariant measure that is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner's theorem holds for Mod(n), with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice Mod(n) that is natural and unique in the sense defined below and for which an analogue of Sperner's theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice Mod(n) by similar reasoning. © 1997 John Wiley & Sons, Inc.     

Hochschild-Mitchell cohomology and Galois extensions

We define H-Galois extensions for k-linear categories and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology related to this situation. This spectral sequence is multiplicative and for a group algebra decomposes as a direct sum indexed by conjugacy classes of the group. We also compute some Hochschild-Mitchell cohomology groups of categories with infinite associated quivers.     

Homotopy Theory of A∞ Ring Spectra and Applications to MU-Modules

We give a definition of a derivation of an A ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A -ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of MU-algebras where MU is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.     

Cohomology of Products and Coproducts of Augmented Algebras

We show that the ordinary cohomology functor $\Lambda \mapsto {\operatorname{Ext}} ^* _\Lambda (k,k)$ from the category of augmented k-algebras to itself exchanges coproducts and products, then that Hochschild cohomology is close to sending coproducts to products if the factors are self-injective. We identify the multiplicative structure of the Hochschild cohomology of a product, modulo a certain ideal, in terms of the cohomology of the factors.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Restriction Between Cohomology Algebras of Blocks of Finite Groups

Let k be an algebraically closed field of characteristic p and G be a finite group. Let N be a normal subgroup of G and c be a G-stable block of kN. We shall discuss the cohomology algebra of the block c, defined by M. Linckelmann and, in this case a generalized block cohomology which can be defined using some generalized Brauer pairs, denoted (c, G)-Brauer pairs, which are introduced by R. Kessar and R. Stancu. We also analyze the restriction map between these two cohomology algebras associated to the block c through transfer maps between the Hochschild cohomology algebras of kGc and of the block c.     

Universal coefficient formulas for stable K-theory

The goal of this paper is to establish a Künneth spectral sequence and the connections between the stable complex K-functor k^* and other generalized cohomology theories for some classes of cell complexes (Theorems 1 and 3). In Theorem 2 criteria are formulated for a finite cell complex to admit a k^*-resolution. Analogous results were also obtained by Landweber [3].Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 53–60, January, 1972.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Duality for cochain DG algebras

This paper develops a duality theory for connected cochain DG algebras,with particular emphasis on the non-commutative aspects.One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology.As an application,it is proved that if the canonical module k=A/A≥1 has a semi-free resolution where the cohomological degree of the generators is bounded above,then the same is true for each DG module with finitely generated cohomology.     

Higher-order conservation laws for the nonlinear Poisson equation via characteristic cohomology

We study higher-order conservation laws of the nonlinearizable elliptic Poisson equation as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. We show that the \mathbbS¹{\mathbb{S}^1} -symmetry of the PDE leads to a normal form for the undifferentiated conservation laws. Zhiber and Shabat (in Sov Phys Dokl Akad 24(8):607–609, 1979) determine which potentials of nonlinearizable Poisson equations admit nontrivial Lie–B?cklund transformations. In the case that such transformations exist, they introduce a pseudo-differential operator that can be used to generate infinitely many such transformations. We obtain similar results using the theory of characteristic cohomology: we show that for higher-order conservation laws to exist, it is necessary that the potential satisfies a linear second-order ODE. In this case, at most two new conservation laws in normal form appear at each even prolongation. By using a recursion motivated by Killing fields, we show that, for the simplest class of potentials, this upper bound is attained. The recursion circumvents the use of pseudo-differential operators. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.     

Opération de la cohomologie elliptique définie par l'opérateur de Hecke T2

In this Note we show that isomorphism of formal group law classification, naturality of C-oriented multiplicative cohomology theories, and the strict isomorphism between the formal group law associated to Tate's curve and the multiplicative one, give a stable cohomology operation of degree 0 for elliptic cohomology of level 2, which induces the Hecke operator T₂ on coefficient group.     

Hochschild Cohomology and the Coradical Filtration of Pointed Coalgebras: Applications

In this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the Hochschild cohomology of that coalgebra with coefficients in some one-dimensional bicomodules. As an application, for a given prime numberpand an algebraically closed fieldkof characteristic 0, we classify all pointed Hopf algebras of dimensionp³overk.     

Hochschild and ordinary cohomology rings of small categories

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH^∗(kC)→H^∗(|C|,k) and prove it is split surjective, using the factorization category of Quillen [D. Quillen, Higher algebraic K-theory I, in: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85-147] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg [N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. 88 (3) (2004) 705-732].     

Stochastic cohomology of Chen–Souriau and line bundle over the Brownian bridge

We define a stochastic cohomology theory related to a stochastic diffeology for the Hoelder loop space. We show that the stochastic de Rham cohomology groups are equal to the deterministic de Rham cohomology groups of the Hoelder loop space. As an application, we show that a stochastic line bundle over the Brownian bridge (with fiber almost surely defined) is isomorphic to a true line bundle over the Hoelder loop space. Received: 9 November 1998 / Revised version: 14 July 2000 / Published online: 26 April 2001     

Varieties for cohomology with twisted coefficients

Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module Ext _kG ^* (M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples. Partially supported by grants from NSF and EPSRC     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.