On Gorenstein fiber products and applications

TL;DR

This paper characterizes when a fiber product of two local rings is Gorenstein, showing it occurs precisely when the fiber product is a one-dimensional hypersurface with both rings regular, and explores related applications.

Contribution

It provides a complete characterization of Gorenstein fiber products of local rings, linking them to hypersurfaces of dimension one and regularity of the factors.

Findings

Gorenstein fiber products are hypersurfaces of dimension 1.

Both rings in the fiber product are regular of dimension 1.

The paper discusses applications of this characterization.

Abstract

We show that a non-trivial fiber product $S\times_k T$ of commutative noetherian local rings $S,T$ with a common residue field $k$ is Gorenstein if and only if it is a hypersurface of dimension 1. In this case, both $S$ and $T$ are regular rings of dimension 1. We also give some applications of this result.