The Poincaré series of a codimension four Gorenstein ring is rational (Q1064355)

Let R be a regular local ring over a field k of characteristic \(\neq 2\) and let \(S=R/I\) be a codimension 4 Gorenstein ring of the same embedding dimension as R. The Poincaré series \(P_ S(Z)=\sum \dim_ kTor^ S_ i(k,k)Z^ i\) is then shown to be a rational function. The authors use a classification of all possible \(Tor^ R(S,k)\) in this situation (made by Kustin and Miller), the existence of a DGA-structure on \(Tor^ R(S,k)\) (shown by Kustin and Miller) and a theorem of Avramov connecting the Poincaré series of S with that of \(Tor^ R(S,k)\). As a corollary of the classification of the Tor-algebras the authors show that S is a Golod factor of a complete intersection. This implies that the homotopy Lie algebra of S is either finite-dimensional (if S is a complete intersection) or the extension of a finite dimensional Lie algebra by a free Lie algebra.