Decomposing Gorenstein Rings as Connected Sums

TL;DR

This paper investigates when Gorenstein Artinian local rings can be decomposed into connected sums, providing criteria for decomposition, uniqueness of components, and implications for their algebraic invariants.

Contribution

It offers a characterization of decomposable Gorenstein Artin rings, criteria for indecomposability, and establishes the uniqueness of components in graded cases.

Findings

Characterization of decomposable Gorenstein Artin rings

Conditions for indecomposability as connected sums

Uniqueness of indecomposable components in graded cases

Abstract

In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know whether it decomposes as a connected sum and if so, what are its components. We answer these questions in the Artinian case and investigate conditions on the ring which force it to be indecomposable as a connected sum. We further give a characterization for Gorenstein Artin local rings to be decomposable as connected sums, and as a consequence, obtain results about its Poincare series and minimal number of generators of its defining ideal. Finally, in the graded case, we show that the indecomposable components appearing in the connected sum decomposition are unique up to isomorphism.