The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational

Abstract

Let R be a local, Gorenstein ring with algebraically closed residue field k of characteristic 0 and let PR(z):= Σp=0∞ dimk(TorpR(k, k))zp be its Poincare series. We compute PR when R belongs to a particular class defined in the Introduction, proving its rationality. As a by-product we prove the rationality of PR for all local, Gorenstein rings of multiplicity at most 10.