Manfred Hartl. 
Polynomial properties of group extensions with torsion-free abelian kernel. 

Abstract : Polynomial 2-cocycles in the intrinsic and natural 
sense of Passi are defined also for non-trivial coefficient
modules; the corresponding notion of polynomial cohomology is studied as 
an approximation of ordinary cohomology in dimension 2.
It is proved that every extension of a finitely generated torsion-free 
nilpotent group $G$ by an additively torsion-free nilpotent
$G$-module $M$ is representable by a polynomial cocycle of the expected 
degree; also the minimal degree possible is determined.
This amounts to an identification of ordinary cohomology in this case 
with polynomial cohomology, and thus with the homology of a
cochain complex consisting of finitely generated free abelian groups 
(if $M$ is finitely generated, too). This complex can be explicitly
read off from a given free presentation of $G$ and allows to calculate 
the cohomology group $H^2(G,M)$ in terms of cocycles
given in the form of classical integer valued rational polynomial 
functions, by using only matrix calculus. Moreover, a functorial
equivalene is established between extensions of torsion-free nilpotent 
groups (finitely generated) and certain extensions of
additively torsion-free nilpotent modules (additively finitely generated).