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

Chirvasitu, Alex; Kanda, Ryo; Smith, S.

doi:10.1090/tran/7672

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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