## 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].## References

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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