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...

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted int...

The purpose of this paper is to give a recursive description of the characters of the cohomology of the line bundles on the three dimensional flag variety over an algebraically closed field k of characteristic p > 0. In fact our recursive procedure also involves certain rank 2 bundles and we determine the characters of the cohomology of these bundles at the same time. The paper may be regarded ...

We explore the relationship between de Rham and Lie algebra cohomologies in the finite dimensional and affine settings. In particular, given a ĝκ-module that arises as the global sections of a twisted D-module on the affine flag manifold, we show how to compute its untwisted BRST reduction modulo n(K) using the de Rham cohomology of the restrictions to N(K) orbits. A similar relationship holds ...

We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C = R, this recovers the standard projective dimension. We show that three natural definitions of finite Cprojective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results th...

The exterior algebra over the centre of a Lie algebra acts on the cohomology of the Lie algebra in a natural way. Focusing on nilpotent Lie algebras, we explore the module structure afforded by this action. We show that for all two-step nilpotent Lie algebras, this module structure is non-trivial, which partially answers a conjecture of Cairns and Jessup [4]. The presence of free submodules ind...

Let S = k[x1, . . . , xn] be a polynomial ring over a field k with n variables x1, . . . , xn, m the irrelevant maximal ideal of S, I a monomial ideal in S and I ′ the polarization of I in the polynomial ring S′ with ρ variables. We show that each graded piece H i m (S/I)a, a ∈ Z , of the local cohomology module H i m (S/I) is isomorphic to a specific graded pieceH i+ρ−n m ′ (S′/I )α, α ∈ Z , o...

We define a transfer map in the setting of bounded cohomology with certain metric G-module coefficients. As an application, we extend a theorem on the comparison map from Borelbounded to Borel cohomology (cf. [1]), to cover the case of Lie groups with finitely many connected components.

We entirely compute the cohomology for a natural and large class of osp(1|2) modules M . We study the restriction to the sl(2) cohomology of M and apply our results to the module M = Dλ,μ of differential operators on the super circle, acting on densities.

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology a...