In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative Gn+1-equivariant natural transformation Θ from some height n + 1 cohomology theory E∗(−) to the height n cohomology theory K∗(−)b ⊗FL, where K∗(−) is essentially the nth Morava K-theory. As a corollary, it is shown that the Gn-module K ∗(X) can be recovered from t...

The present paper can be considered as a natural extension the article [Ar7]. Fix root data (Y,X, . . . ) of the finite type (I, ·) and a positive integer number l. In [Ar7] we obtained a nice description for semiinfinite cohomology of the trivial module C over the small quantum group ul corresponding to the root data (Y,X, . . . ) in terms of local cohomology of the structure sheaf on the nilp...

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras. The classical First and Second Whitehead Lemmata state that the first, respectively second, cohomology group of a finite-dimensional semisimpl...

We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.

The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an R-module. The construction of the R-module is similar to the Alexander Wallace Spanier construction of the abelian group. In Chapter I, the algebraic and topological material necessary to construct the theor'y is lis...

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R = C[X1, . . . , Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a coro...

We present a recent improvement of the homotopy Conley index, the so-called Conley index over a base. We apply it to endow the homology and cohomology Conley indices with the structure of module over the cohomology ring of the phase space, and indicate how that structure provides information on continuation of cohomology classes of isolated invariant sets.

Let R be an irreducible root system with the Coxeter number h. Let l > h be an odd integer (we assume that l is not divisible by 3 if R is of type G2). Let U be the quantum group of type 1 with divided powers associated to these data, see [10] (of type 1 means that the elements K l i are equal to 1). Let u ⊂ U be the Frobenius kernel, see loc. cit. Let 1 be the trivial U−module. The cohomology ...