Connected sums of Gorenstein local rings (Q2898926)

Let \(\epsilon_R : R \to T\) and \(\epsilon_S : S \to T\) be surjective homomorphisms of commutative rings. Let \(V\) be a \(T\)-module with homomorphisms \(\iota_R: V \to R\) and \(\iota_S: V \to S\) of \(R\)- resp. \(S\)-modules. Suppose that \(\epsilon_R \iota_R = \epsilon_S \iota_S\). Then the connected sum \(R \#_T S\) is defined by \((R\times_T S)/ \{\iota_R(v), \iota_S(v) | v \in V\}\), where \(R\times_T S\) denotes the fibre product. That is, \(R \#_T S\) is obtained by factoring out the diagonal image of \(V\) in the fibre product \(R\times_T S\). Suppose that \(R, S, T\) have dimension \(d\), \(R,S\) are Gorenstein, \(T\) is Cohen-Macaulay, and \(V\) is a canonical module for \(T\), then one can choose \(\iota_R\) and \(\iota_S\) to be isomorphisms onto \(0: \text{Ker}(\epsilon_R)\) and \(0: \text{Ker}(\epsilon_S)\), respectively. Then as a Main Theorem it is proved that \(R \#_T S\) is Gorenstein of dimension \(d\) (provided \(\epsilon_R \iota_R = \epsilon_S \iota_S\)). This extends and generalizes partially results in the case of \(\epsilon_R = \epsilon_S\) shown by \textit{M. D'Anna} [J. Algebra 306, No. 2, 507--519 (2006; Zbl 1120.13022)]) and \textit{J. Shapiro} [J. Algebra 323, No. 4, 1155--1158 (2010; Zbl 1184.13069)]). The authors' results are used to study how closely an artinian ring can be approximated by a Gorenstein ring mapping onto it. This is done by the aid of the Gorenstein colength (see the first author's paper [J. Algebra 320, No. 9, 3438--3446 (2008; Zbl 1162.13008)]). Moreover, when \(T\) is regular, it is shown that \(R \#_T S\) is almost never a complete intersection ring. For these considerations the authors study the relations of the \(\text{Ext}\)-algebras \(\text{Ext}^{\star}_{X}(T,T)\) for \(X = R, S, R \#_T S\) respectively. In fact, when \(T\) is a field, then \(\text{Ext}^{\star}_{R \#_T S}(T,T)\) is presented as an amalgam of \(\text{Ext}^{\star}_{R}(T,T)\) and \(\text{Ext}^{\star}_{S}(T,T)\) over isomorphic polynomial subalgebras generated by one element of degree 2.