Cohomology of partially ordered sets and local cohomology of section rings

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings.     

Hochschild cohomology of brauer tree algebras

The aim of this paper is to study the structure of the Hochschild cohomology ring of Brauer tree algebras. We explicitly describe the even cohomology ring by generators and relations. This generalizes results of [11] on the Hochschild cohomology of blocks of modular group algebras with cyclic defect groups.     

On the push-forwards for motivic cohomology theories with invertible stable Hopf element

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.     

The Chen-Ruan cohomology of almost contact orbifolds

Comparing to the Ch-~Ruan cohomology theory for the almost complex orbifolds, we study the orbifold cohomology theory for almost contact orbifolds. We define the Chen-Ruan cohomology group of any almost contact orbifold. Using the methods for almost complex orbifolds, we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen~Ruan cup product for the Che-Ruan cohomology. We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring. Finally we calculate two examples.     

Transfer in Hochschild Cohomology of Blocks of Finite Groups

We develop the notion of a cohomology ring of blocks of finite groups and study its basic properties by means of transfer maps between the Hochschild cohomology rings of symmetric algebras associated with bounded complexes of finitely generated bimodules which are projective on either side.     

On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ _s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ _s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ _s is finitely generated as an algebra.     

The Weil-��tale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.     

Products in Hochschild cohomology and Grothendieck rings of group crossed products

We give a general construction of rings graded by the conjugacy classes of a finite group. Some examples of our construction are the Hochschild cohomology ring of a finite group algebra, the Grothendieck ring of the Drinfel'd double of a group, and the orbifold cohomology ring for a global quotient. We generalize the first two examples by deriving product formulas for the Hochschild cohomology ring of a group crossed product and for the Grothendieck ring of an abelian extension of Hopf algebras. Our results account for similarities in the product structures among these examples.     

Abelianization for hyperkähler quotients

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkähler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to quiver varieties, and compute as an example the ordinary and equivariant cohomology rings of a hyperpolygon space.     

Representation dimension and finitely generated cohomology

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.     

Equivariant cohomology distinguishes toric manifolds

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are weakly isomorphic. We also prove that quasitoric manifolds, which can be thought of as a topological counterpart to toric manifolds, are equivariantly homeomorphic if and only if their equivariant cohomology algebras are isomorphic.     

Zur Kohomologie von Gruppen mit abelschen Sylowuntergruppen

In 1962 Atiyah showed, with, the help of topological methods, the existence of a spectral sequence joining the cohomology ring of a finite group (with values in the rational integers) and the ring of generalized complex characters of the group. This paper undertakes to get that connection (at least in part) in a purely algebraic way. To that effect such ring filtrations of the character ring of a group are described that are functorially compatible with group morphisms and compatible with the induction of representations belonging to subgroups. The graded ring formed canonically by a filtration of that kind is to be compared with the cohomology ring of the same group, and it is shown, the group assumed to have abelian Sylow subgroups or to be a periodic group in a cohomological sense, that by taking a certain filtration its graded ring can be identified with a homomorphic part of the cohomology ring. (That corresponds to the infinite term of Atiyah's spectral sequence). The generalization of this result with respect to all finite groups has not yet been achieved; there is, however, a reduction theorem saying that it is sufficient to deal with groups of prime power order only. - As an application one gets lower bounds for the orders of the cohomology groups belonging to a finite group, especially, as a corollary, the theorem of its infinite cohomological dimension.     

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.

     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.     

量子外代数的Galois覆盖代数的Hochschild上同调

本文研究一类量子代数$\Lambda^n_q$的Hochschild上同调.量子代数$\Lambda^n_q$的极小投射双模分解被构造, $\Lambda^n_q$的各阶Hochschild上同调群的维数被清晰的给出.此外,对一些特殊的情况, $\Lambda^n_q$的上同调环也被清晰的刻画.     

On Carlson's depth conjecture in group cohomology

We establish a weak form of Carlson's conjecture on the depth of the mod-p cohomology ring of a p-group. In particular, Duflot's lower bound for the depth is tight if and only if the cohomology ring is not detected on a certain family of subgroups. The proofs use the structure of the cohomology ring as a comodule over the cohomology of the centre via the multiplication map. We demonstrate the existence of systems of parameters (so-called polarised systems) which are particularly well adapted to this comodule structure. Mathematics Subject Classification (2000): 20J06.     

一类量子Koszul代数的${\mathbb{Z}}_{2}$-Galois 覆盖的 Hochschild上同调

考虑一类量子Koszul代数的 ${\mathbb{Z}}_{2}$-Galois覆盖$\Lambda_{\q}$, 并计算 这类代数的各阶Hochschild上同调群的维数, 进而利用道路的语言, 刻画了 Hochschild上同调环的cup积. 作为应用, 给出了这类代数的Hochschild上同调环模掉幂零理想的 代数结构.     

Boundary manifolds of projective hypersurfaces

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge-theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring.