Generalised Witt algebras and idealizers

We construct Φ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let G be an arbitrary group and let A be a G-graded ring. A graded A-module M is an intermediate series module if Mg is one-dimensional for all g∈G. Given a shift-invariant family of intermediate series A-modules parametrised by a scheme X, we construct a homomorphism Φ from A to a skew extension of k[X]. The kernel of Φ consists of those elements which annihilate all modules in X.