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 P R (z) := ∞ p=0 dim k (Tor R p (k, k))z p be its Poincaré series. We compute P R when R belongs to a particular class defined in the Introduction, proving its rationality. As a by-product we prove the rationality of P R for all local, Gorenstein rings of multiplicity at most 10.

On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

“…The above theorems generalize the quoted results on stretched, almost-stretched and short algebras (see [16], [10], [6], [8]).…”

“…The rationality of the Poincaré series P A of every stretched ring A is proved in [16]. The proof has been generalized to rings with H A (2) = 2 in [10] and to rings with H A (2) = 3, H A (3) = 1 in [6] . The rationality of P A when A is a 2-stretched algebra has been studied in [8] with the restriction sdeg(A) = 3.…”

“…We now focus on the Poincaré series P A (z) of the algebra A defined in the introduction: we will generalize some classical results (see [16], [10], [6]). Out of the decomposition results proved in the previous section, the main tools we use are the following ones:…”

“…Nevertheless, several results show that large classes of local rings A have rational Poincaré series, e.g. complete intersections rings (see [19]), Gorenstein local rings with dim k (M/M 2 ) ≤ 4 (see [2] and [15]), Gorenstein local rings with dim k (M 2 /M 3 ) ≤ 2 (see [16], [10]), Gorenstein local rings of multiplicity at most 10 (see [6]), Gorenstein local algebras with dim k (M 2 /M 3 ) = 4 and M 4 = 0 (see [8]).…”

Poincaré series of compressed local Artinian rings with odd top socle degree

“…A list of applications of the hypothesis that a local ring is good may be found in [3]. The recent papers [26,20,15,14] all prove that a family of rings has rational Poincaré series.…”