The lifting problem in group cohomology

Group actions on curves and the lifting problem

1.1 The lifting problem The problem we are concerned with in our lectures and which we shall refer to as the lifting problem was originally formulated by Frans Oort in [17]. To state it, we fix an algebraically closed field κ of positive characteristic p. Let W (κ) be the ring of Witt vectors over κ. Throughout our notes, o will denote a finite local ring extension of W (κ) and k = Frac(o) the ...

Module cohomology group of inverse semigroup algebras

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

Lifting Problem in Codimension 2 and Initial Ideals

Isotropy in Group Cohomology

The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N⊳G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N . This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation f...

Cohomology of the Lie Algebras of Differential Operators: Lifting Formulas

(i) Tr(DiA) = 0 for any A ∈ A and any Di ∈ D (ii) [Di,Dj ] = ad(Qij) — inner derivation (Qij ∈ A) for any Di,Dj ∈ D (iii) Alt i,j,k Dk(Qij) = 0 for all i, j, k. The main example of such a situation is the Lie algebra ΨDifn(S 1) of the formal pseudodifferential operators on (S1)n (see [A]). The trace Tr in this example is the “noncommutative residue”, Tr(D) = the coefficient of the term x 1 · x ...