Connected sums of Gorenstein local rings

TL;DR

This paper introduces a new construction called the connected sum of Gorenstein local rings, explores its properties, and investigates how it relates to Cohen-Macaulay and Gorenstein conditions, with implications for ring approximation and cohomology algebra structure.

Contribution

It defines the connected sum of local rings via fiber products and ideals, proves Gorenstein properties are preserved under certain conditions, and analyzes the cohomology algebra structure of these sums.

Findings

Connected sums preserve Gorenstein properties under specified conditions.

The cohomology algebra of connected sums is an amalgam of the original algebras.

Connected sums with regular T are rarely complete intersections.

Abstract

A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\gets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the \emph{connected sum} $R#_TS$ is defined to be the local ring obtained by factoring out the diagonal image of $V$ in the fiber product $R\times_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R#_TS$. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra $\Ext^*_{R#_kS}(k,k)$ is an amalgam of the algebras $\Ext^*_{R}(k,k)$ and $\Ext^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring $R#_TS$ almost never is complete intersection.