Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group.
There are torsion free groups for which the analogous statement is false.
The Hochschild cohomology rings for self-injective algebras of tree classes $E_7$ and $E_8$ with finite representation type was described in terms of generators and relations.
We compute the multiplicative structure in the Hocshchild cohomology ring of a differential operators ring and the cap product of Hochschild cohomology on the Hochschild homology.
We introduce the Andre-Quillen cohomology of lambda-rings and Psi-rings, this is different to the lambda-ring cohomology defined by Yau in 2005. We show that there is a natural transformation connecting the cohomology of the K-theory of spheres to the homotopy groups of spheres.
It is proved that the third Mac Lane cohomology group of a ring R with coefficients in a bimodule B classifies categorical rings having R as the ring of isomorphism classes of objects and B as the bimodule of automorphisms of the neutral object.
We describe the Hochschild cohomology ring for a family of self-injective algebras of tree class $E_6$ in terms of generators and relations. Together with the results of the previous paper, this gives a complete description of the Hochschild cohomology ring for a self-injective algebras of tree class $E_6$.
In this thesis we develop the cohomology of diagrams of algebras and then apply this to the cases of the $λ$-rings and the $Ψ$-rings. A diagram of algebras is a functor from a small category to some category of algebras. For an appropriate category of algebras we get a diagram of groups, a diagram of Lie algebras, a diagram of commutative rings, etc. We define the cohomology of diagrams of algebras using comonads. The cohomology of diagrams of algebras classifies extensions in the category of functors. Our main result is that there is a spectral sequence connecting the cohomology of the diagra…
We realize explicitly the well-known additive decomposition of the Hochschild cohomology ring of a group algebra in the elements level. As a result, we describe the cup product, the Batalin-Vilkovisky operator and the Lie bracket in the Hochschild cohomology ring of a group algebra.
A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras (TQA's). As a direct application we describe explicitely the Yoneda product and derive several results on the structure of the cohomology ring of TQA's. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental examp…
We study the completion of a group relative to a Zariski dense representation in a reductive algebraic group over a field $k$. The characteristic zero case was worked out previously by R. Hain; we extend his results to arbitrary characteristic. The primary application is to the study of the cohomology of groups such as $SL_n(k[[T]])$.
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
The lambda-ring cohomology in dimensions 0 and 1 of certain truncated polynomial filtered lambda-rings are computed. The dimension 1 case is related to deformations of lambda-rings.
The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary q…
For a finite dimensional monomial algebra $Λ$ over a field $K$ we show that the Hochschild cohomology ring of $Λ$ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated $K$-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field by Snashall-Solberg.
This is a survey paper on cohomology theories for $A_\infty$ and $E_\infty$ ring spectra. Different constructions and main properties of topological André-Quillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of $A_\infty$ and $E_\infty$ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures.
We show that the Tate-Hochschild cohomology ring $HH^*(RG,RG)$ of a finite group algebra $RG$ is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of $G$. Moreover, our main result provides an explicit formula for the cup product in $HH^*(RG,RG)$ with respect to this decomposition. As an example, this formula helps us to compute the Tate-Hochschild cohomology ring of the symmetric group $S_3$ with coefficients in a field of characteristic 3.
Let $k$ be an algebraically closed field of characteristic two and let $Q_8$ be the quaternion group of order $8$. We determine the Gerstenhaber Lie algebra structure and the Batalin-Vilkovisky structure on the Hochschild cohomology ring of the group algebra $kQ_8$.
We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.
The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class.
We calculate the mod-two cohomology of all alternating groups together, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques is needed: an almost Hopf ring structure associated to the embeddings of products of alternating groups, the Gysin sequence relating the cohomology of alte…
We compute the Chow-Witt rings of split quadrics over a field of characteristic not two. We even determine the full bigraded I-cohomology and Milnor-Witt cohomology rings, including twists by line bundles. The results on I-cohomology corroborate the general philosophy that I-cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicit calculations confirm that the I-cohomology ring of a split quadric over the reals is isomorphic to the singular cohomology ring of the space of its real points.
This is a survey paper based on my talks at the 41st Symposium on Ring Theory and Representation Theory, held in Shizuoka University, Japan in September 2008, and will appear in the conference proceedings. The paper begins with a brief introduction to the use of Hochschild cohomology in developing the theory of support varieties for a module over an artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004), 705-732). The paper then describes the current status of research concerning the structure of the Hochschild cohomology ring modulo nilpotence.
Stefan and Guichardet have provided Lyndon-Hochschild-Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.
