We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H , and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra as...

We prove that the Hochschild homology (and cohomology) of a symmetric open Frobenius algebra A has a natural coBV and BV structure. The underlying coalgebra and algebra structure may not be resp. counital and unital. Moreover we prove that the product and coproduct satisfy the Frobenius compatibility condition i.e. the coproduct on HH∗(A) is a map of left and right HH∗(A)-modules. If A is commu...

Let k be a perfect field of characteristic p > 0, Wn = Wn(k). For separated k-schemes of finite type, we explain how rigid cohomology with compact supports can be computed as the cohomology of certain de Rham-Witt complexes with coefficients. This result generalizes the classical comparison theorem of Bloch-Illusie for proper and smooth schemes. In the proof, the key step is an extension of the...

Given a perfect field k of characteristic p > 0, a smooth proper k-scheme Y , a crystal E on Y relative to W (k) and a finite group G acting on Y and E, we show that, viewed as a virtual k[G]-module, the reduction modulo p of the crystalline cohomology of E is the de Rham cohomology of E modulo p. On the way we prove a base change theorem for the virtual Grepresentations associated with G-equiv...

In this paper the authors investigate infinite-dimensional representations L in blocks of the relative (parabolic) category OS for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the cohomology with coefficients in a finite-dimensional module, as in Kostant’s theorem. A complete classification of these “Kostant modules”...

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 main goal of this paper is to associate to each smooth quasi-projective scheme X over any field an E∞ differential graded algebra whose cohomology groups are the higher Chow groups of X. Modulo torsion, this provides a strictly commutative differential graded algebra. Such a construction is currently known only for the case X itself is a field. Needless to say there are several applications...

In a series of papers Grassi, Policastro, Porrati and van Nieuwenhuizen have introduced a new method to covariantly quantize the GS-superstring by constructing a resolution of the pure spinor constraint of Berkovits’ approach. Their latest version is based on a gauged WZNW model and a definition of physical states in terms of relative cohomology groups. We first put the off-shell formulation of...

For each group G having an infinite normal subgroup with the relative property (T) (e.g. G = H ×K, with H infinite with property (T) and K arbitrary) and each countable abelian group Λ we construct free ergodic measure-preserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H (σΛ, G), is equal to Char(G) × Λ. We deduce that G has uncountably many non stabl...

For each group G having an infinite normal subgroup with the relative property (T) (for instance groups of the form G = H ×K, where H is infinite with property (T) and K is arbitrary) and each countable abelian group Λ we construct free ergodic measure-preserving actions σΛ of G on the probability space such that the 1’st cohomology group of σΛ, H (σΛ, G), is equal to Char(G)×Λ. We deduce that ...