# Connected Sums of Graded Artinian Gorenstein Algebras and Lefschetz Properties

## TL;DR

This paper explores a connected sum construction for graded Artinian Gorenstein algebras, providing new descriptions and analyzing how this operation affects Lefschetz properties, thereby identifying new classes of rings with these properties.

## Contribution

It offers two alternative algebraic descriptions of the connected sum and studies its impact on Lefschetz properties in graded Artinian Gorenstein algebras.

## Key findings

- Connected sum can be described via Thom classes and Gysin homomorphisms.
- The construction can preserve weak and strong Lefschetz properties.
- New classes of rings with Lefschetz properties are identified.

## Abstract

A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.