Connected sums of Gorenstein local rings

Abstract

A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\leftarrow S$ 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 $R{\times }_{T}S$. 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.

This version includes a new theorem (Theorem 1.8), in which results due to D'Anna and Shapiro are completed and strengthened. Other changes to the text are minor. To appear in Crelle's J

References in corpus (2)

• The Gorenstein Colength of an Artinian Local Ring
• Rings which are almost Gorenstein