Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 261-279, 2020
Associated graded rings and connected sums
H. Ananthnarayan, Ela Celikbas, Jai Laxmi, Zheng Yang
Received May 22, 2018. Published online December 18, 2019.
Abstract: 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 $Q$.
Keywords: associated graded ring; fibre product; connected sum; short Gorenstein ring; stretched Gorenstein ring; Poincaré series
Affiliations: H. Ananthnarayan, Department of Mathematics, Indian Institute of Technology Bombay, Main Gate Rd, IIT Area, Powai, Mumbai, Maharashtra 400076, India, e-mail: firstname.lastname@example.org; Ela Celikbas (corresponding author), Department of Mathematics, West Virginia University, 94 Beechurst Ave, Morgantown, WV 26505, USA, e-mail: email@example.com; Jai Laxmi, University of Connecticut, 341 Mansfield Road U1009, Storrs, Connecticut, 06269-100, USA e-mail: firstname.lastname@example.org; Zheng Yang, Sichuan University Pittsburgh Institute, Sichuan University, Jiang'an Campus, Chuanda Road, Shuangliu County, Chengdu, 610207 P. R. China e-mail: email@example.com