In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q , which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincaré series of Q .

中文翻译：

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相关分级环和连接和

2012 年，Ananthnarayan、Avramov 和 Moore 用两个 Gorenstein 局部环重新构造了 Gorenstein 环，称为它们的连通和。在本文中，我们研究了 Gorenstein Artin 局部环 Q 的相关分级环上的条件，这迫使它成为其残差域上的连通和。特别是，我们恢复了一些关于短的和拉伸的 Gorenstein Artin 环的结果。最后，使用这些分解，我们得到关于 Q 的 Poincaré 级数的合理性的结果。