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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

New Artin-Schelter regular and Calabi-Yau algebras via normal extensions


Authors: Alex Chirvasitu, Ryo Kanda and S. Paul Smith
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14A22, 16S38, 16W50, 16W20
DOI: https://doi.org/10.1090/tran/7672
Published electronically: May 30, 2019
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Abstract: We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by ``the missing one'' and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture in [J.Algebra 446 (2016), pp. 373-399] that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case [Trans. Amer. Math.Soc. 369 (2017), pp. 309-340, Cor. 5.4].


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Additional Information

Alex Chirvasitu
Affiliation: Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900
Email: achirvas@buffalo.edu

Ryo Kanda
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan; and Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: ryo.kanda.math@gmail.com

S. Paul Smith
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: smith@math.washington.edu

DOI: https://doi.org/10.1090/tran/7672
Keywords: Normal extension, Artin-Schelter regular algebra, Calabi-Yau algebra, superpotential algebra, non-commutative algebraic geometry
Received by editor(s): November 2, 2017
Received by editor(s) in revised form: July 8, 2018
Published electronically: May 30, 2019
Additional Notes: The first author was partially supported by NSF grants DMS-1565226 and DMS-1801011.
The second author was a JSPS Overseas Research Fellow and supported by JSPS KAKENHI Grant Numbers JP17K14164 and JP16H06337.
Article copyright: © Copyright 2019 American Mathematical Society