Poincar\'e series of compressed local Artinian rings with odd top socle degree

TL;DR

This paper introduces a new class of compressed local Artinian rings without requiring a field, and proves that their modules have rational Poincaré series with a shared denominator, along with the existence of a Golod homomorphism.

Contribution

It defines compressed local Artinian rings without a field and establishes rationality of Poincaré series and Golod homomorphisms for these rings.

Findings

Poincaré series of modules over these rings are rational.

All modules share a common denominator in their Poincaré series.

Existence of a Golod homomorphism from a complete intersection to the ring.

Abstract

We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let $(R,\mathfrak m)$ be a compressed local Artinian ring with odd top socle degree $s$, at least five, and $\operatorname{socle}(R)\cap \mathfrak m^{s-1}=\mathfrak m^s$. We prove that the Poincar\'e series of all finitely generated modules over $R$ are rational, sharing a common denominator, and that there is a Golod homomorphism from a complete intersection onto $R$.