Betti numbers for connected sums of graded Gorenstein Artinian algebras

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

doi:10.1090/tran/9286

Betti numbers for connected sums of graded Gorenstein Artinian algebras
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by Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Sean Grate, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu and Junzo Watanabe;

Trans. Amer. Math. Soc. 378 (2025), 1055-1080

DOI: https://doi.org/10.1090/tran/9286

Published electronically: October 23, 2024

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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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