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
Classification MSC:  13A30, 13D40, 13H10
DOI:  10.21136/CMJ.2019.0259-18

[1] H. Ananthnarayan: Approximating Artinian Rings by Gorenstein Rings and Three-Standardness of the Maximal Ideal. Ph.D. Thesis, University of Kansas (2009).
[2] H. Ananthnarayan, L. L. Avramov, W. F. Moore: Connected sums of Gorenstein local rings. J. Reine Angew. Math. 667 (2012), 149-176. DOI 10.1515/CRELLE.2011.132 | MR 2929675 | Zbl 1271.13047
[3] H. Ananthnarayan, E. Celikbas, J. Laxmi, Z. Yang: Decomposing Gorenstein rings as connected sums. J. Algebra 527 (2019), 241-263. DOI 10.1016/j.jalgebra.2019.01.036 | MR 3924433 | Zbl 1410.13014
[4] L. L. Avramov, A. R. Kustin, M. Miller: Poincaré series of modules over local rings of small embedding codepth or small linking number. J. Algebra 118 (1988), 162-204. DOI 10.1016/0021-8693(88)90056-7 | MR 0961334 | Zbl 0648.13008
[5] W. Buczyńska, J. Buczyński, J. Kleppe, Z. Teitler: Apolarity and direct sum decomposability of polynomials. Mich. Math. J. 64 (2015), 675-719. DOI 10.1307/mmj/1447878029 | MR 3426613 | Zbl 1339.13012
[6] G. Casnati, J. Elias, R. Notari, M. E. Rossi: Poincaré series and deformations of Gorenstein local algebras. Commun. Algebra 41 (2013), 1049-1059. DOI 10.1080/00927872.2011.636643 | MR 3037178 | Zbl 1274.13032
[7] J. Elias, M. E. Rossi: Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system. Trans. Am. Math. Soc. 364 (2012), 4589-4604. DOI 10.1090/S0002-9947-2012-05430-4 | MR 2922602 | Zbl 1281.13015
[8] A. Iarrobino: The Hilbert function of a Gorenstein Artin algebra. Commutative Algebra. Mathematics Sciences Research Institute Publications 15, Springer, New York (1989), 347-364. DOI 10.1007/978-1-4612-3660-3_18 | MR 1015527 | Zbl 0733.13008
[9] J. Lescot: La série de Bass d'un produit fibré d'anneaux locaux. Sémin. d'Algèbre P. Dubreil et M.-P. Malliavin, 35ème Année, Proc. Lecture Notes in Math. 1029, Paris, Springer, Berlin (1983), 218-239. (In French.) DOI 10.1007/BFb0098933 | MR 0732477 | Zbl 0563.13007
[10] G. L. Levin, L. L. Avramov: Factoring out the socle of a Gorenstein ring. J. Algebra 55 (1978), 74-83. DOI 10.1016/0021-8693(78)90191-6 | MR 0515760 | Zbl 0407.13018
[11] J. D. Sally: Stretched Gorenstein rings. J. Lond. Math. Soc., II. Ser. 20 (1979), 19-26. DOI 10.1112/jlms/s2-20.1.19 | MR 0545198 | Zbl 0402.13018
[12] L. Smith, R. E. Stong: Projective bundle ideals and Poincaré duality algebras. J. Pure Appl. Algebra 215 (2011), 609-627. DOI 10.1016/j.jpaa.2010.06.011 | MR 2738376 | Zbl 1206.13004

Affiliations:   H. Ananthnarayan, Department of Mathematics, Indian Institute of Technology Bombay, Main Gate Rd, IIT Area, Powai, Mumbai, Maharashtra 400076, India, e-mail:; Ela Celikbas (corresponding author), Department of Mathematics, West Virginia University, 94 Beechurst Ave, Morgantown, WV 26505, USA, e-mail:; Jai Laxmi, University of Connecticut, 341 Mansfield Road U1009, Storrs, Connecticut, 06269-100, USA e-mail:; Zheng Yang, Sichuan University Pittsburgh Institute, Sichuan University, Jiang'an Campus, Chuanda Road, Shuangliu County, Chengdu, 610207 P. R. China e-mail:

PDF available at: