Universal Galois algebras and cohomology of p-groups

Let G be a finite p-group and k a field of characteristic p>0. A universal Galois algebra of G is a weakly initial object in the category Ts of trace-surjective (commutative) k−G algebras. The objects in Ts are precisely the k−G-algebras that are Galois ring extensions over the ring of G-invariants. They are also characterized as k−G algebras which are free kG-modules. One example is Dk, the dehomogenized symmetric algebra of the regular representation, which is also an s-projective object in Ts (see the definition in the paper). In the previous work we proved that the polynomial ring Dk (of dimension ∣G∣−1) contains a polynomial retract U∈Ts of dimension , such that the invariant rings UG and are again polynomial rings. The G-action on U will in general be highly non-linear, but in special cases it can be chosen to be “almost linear”. In this paper we investigate such almost linear universal algebras, generalizing the construction of the algebras Dk and U. It is known that for k=Fp the minimal dimension of a polynomial universal algebra is n. Among other things we prove that such an algebra can be realized in an “almost linear” way, if and only if G is “crossed isomorphic” to an Fp-vector space. This is equivalent to the existence of a kG-module V such that there is 0≠[ρ]∈H1(G,V) with ρ∈Z1(G,V) being a bijective cocycle.