Homotopy groups of a connective ring spectrum R form an Ngraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R repres...

We calculate Hochschild cohomology groups of the integers treated as an algebra over so-called ”field with one element”. We compare our results with calculation of the topological Hochschild cohomology groups of the integers this is the case when one considers integers as an algebra over the sphere spectrum.

Let G be a simple, simply connected algebraic group defined and split over the prime field F p > 3. Let F be a finite dimensional rational G-module. As shown in [1] , for d suitably large, the Eilenberg-Mac Lane cohomology groups H*(G(Fpd), V) achieve a stable value H*en(G, V)9 the so-called generic cohomology of V. This generic cohomology can in turn be determined from the rational cohomology ...

We apply constructions from equivariant topology to Benson-Carlson resolutions and hence prove in (2.1) that the group cohomology ring of a nite group enjoys remarkable duality properties based on its global geometry. This recovers and generalizes the result of Benson-Carlson stating that a Cohen-Macaulay cohomology ring is automatically Gorenstein. We give an alternative approach to Tate cohom...

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel-Kac construction from the latter.

For a truncated quiver algebra over a field of arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles. MSC(2000): 16E40, 16E10, 16G10

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection.

We prove that the normal cohomology groups H w(M, K(H)) of a von Neumann algebra M with coefficients in the algebra of compact operators are zero if M is atomic of type Ifin. In addition, the completely bounded normal cohomology groups H wcb(B(H), K(H)) are shown to be 0 as well.

From algebraic K-theory, we show that there exists a spectral sequence that has real cohomology of SL(n)(Z) as its E(1)-terms and converges to the tensor product of a polynomial algebra and an exterior algebra. On the basis of this spectral sequence, we discovered several families of real unstable cohomology classes of SL(n)(Z).

In theory, the bar construction suflices to calculate the homology groups of an augmented algebra. In practice, the bar construction is generally too large (has too many generators) to allow computation of higher dimensional homology groups. In this paper, we develop a procedure which simplifies the calculation of the homology and cohomology of Hopf algebras. Let A be a (graded) Hopf algebra ov...