Connected sums of Gorenstein local rings

Abstract

Given surjective homomorphisms R → T ← S of local rings, and ideals in R and S that are isomorphic to some T-module V, the connected sumR⋕TS is defined to be the ring obtained by factoring out the diagonal image of V in the fiber product R ×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 a Gorenstein ring mapping onto it. When T is regular, it is shown that R⋕TS almost never is a complete intersection ring. The proof uses a presentation of the cohomology algebra as an amalgam of the algebras and over isomorphic polynomial subalgebras generated by one element of degree 2.