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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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New Artin-Schelter regular and Calabi-Yau algebras via normal extensions
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by Alex Chirvasitu, Ryo Kanda and S. Paul Smith PDF
Trans. Amer. Math. Soc. 372 (2019), 3947-3983 Request permission

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
  • MR Author ID: 868724
  • 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
  • MR Author ID: 990359
  • Email: ryo.kanda.math@gmail.com
  • S. Paul Smith
  • Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
  • MR Author ID: 190554
  • Email: smith@math.washington.edu
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 3947-3983
  • MSC (2010): Primary 14A22, 16S38, 16W50, 16W20
  • DOI: https://doi.org/10.1090/tran/7672
  • MathSciNet review: 4009424