BETTI NUMBERS FOR CONNECTED SUMS OF GRADED GORENSTEIN ARTINIAN ALGEBRAS

Nasrin Altafi; Roberta Di Gennaro; Federico Galetto; Sean Grate; Rosa M. Miró-Roig; Uwe Nagel; Alexandra Seceleanu; Junzo Watanabe

doi:10.1090/tran/9286

BETTI NUMBERS FOR CONNECTED SUMS OF GRADED GORENSTEIN ARTINIAN ALGEBRAS

Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Sean Grate, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, Junzo Watanabe

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring.

Original language	English
Pages (from-to)	1055-1080
Number of pages	26
Journal	Transactions of the American Mathematical Society
Volume	378
Issue number	2
DOIs	https://doi.org/10.1090/tran/9286
State	Published - Feb 2025

Bibliographical note

Publisher Copyright:
©c 2024 American Mathematical Society.

Funding

The first author was supported by Swedish Research Council grant VR2021-00472. The third author was supported by NSF DMS\u20132200844. The fifth author was partially supported by the grant PID2020-113674GB-I00. The sixth author was partially supported by Simons Foundation grant #636513. The seventh author was supported by NSF DMS\u20132101225. The project got started at the meeting \u201CWorkshop on Lefschetz Properties in Algebra, Geometry, Topology and Combinatorics\u201D, held at the Fields Institute in Toronto, Canada, May 15\u201319, 2023. The authors would like to thank the Fields Institute and the organizers for the invitation and financial support. Additionally, we thank Graham Denham for asking a question which motivated our work, and Mats Boij for useful discussions.

Funders	Funder number
Simons Foundation	636513, 2101225
Division of Mathematical Sciences	PID2020-113674GB-I00, 2200844
Vetenskapsrådet	VR2021-00472

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/tran/9286

Cite this

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abstract = "The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring.",

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AU - Altafi, Nasrin

AU - Di Gennaro, Roberta

AU - Galetto, Federico

AU - Grate, Sean

AU - Miró-Roig, Rosa M.

AU - Nagel, Uwe

AU - Seceleanu, Alexandra

AU - Watanabe, Junzo

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AB - The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring.

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BETTI NUMBERS FOR CONNECTED SUMS OF GRADED GORENSTEIN ARTINIAN ALGEBRAS

Abstract

Bibliographical note

Funding

ASJC Scopus subject areas

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