Associated Graded Rings and Connected Sums

TL;DR

This paper explores conditions under which the associated graded ring of a Gorenstein Artin local ring can be expressed as a connected sum, leading to new insights on the structure and Poincare series of such rings.

Contribution

It characterizes when a Gorenstein Artin local ring's associated graded ring is a connected sum, extending previous results on short and stretched Gorenstein rings.

Findings

Conditions identified for associated graded rings to be connected sums

Revealed structure of short and stretched Gorenstein Artin rings

Results on the rationality of Poincare series

Abstract

In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincare series of Q.