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Transactions of the American Mathematical Society

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

 
 

 

Quadratic Gorenstein rings and the Koszul property I


Authors: Matthew Mastroeni, Hal Schenck and Mike Stillman
Journal: Trans. Amer. Math. Soc. 374 (2021), 1077-1093
MSC (2020): Primary 13D02; Secondary 14H45, 14H50
DOI: https://doi.org/10.1090/tran/8214
Published electronically: November 18, 2020
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Abstract: Let $ R$ be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95-121], Conca-Rossi-Valla show that such a ring is Koszul if $ \mathrm {reg} R \leq 2$ or if $ \mathrm {reg} R = 3$ and $ c=\mathrm {codim} R \leq 4$, and they ask whether this is true for $ \mathrm {reg} R = 3$ in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring $ R$ that guarantee the Nagata idealization $ \tilde {R} = R \ltimes \omega _R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity $ 3$ satisfying our conditions for all $ c \ge 9$; this yields a negative answer to the question from the above-mentioned paper.


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Additional Information

Matthew Mastroeni
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: mmastro@okstate.edu

Hal Schenck
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: hschenck@iastate.edu

Mike Stillman
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850
Email: mike@math.cornell.edu

DOI: https://doi.org/10.1090/tran/8214
Keywords: Syzygy, Koszul algebra, Gorenstein algebra.
Received by editor(s): August 9, 2019
Received by editor(s) in revised form: May 2, 2020
Published electronically: November 18, 2020
Additional Notes: The second author was supported by NSF Grant 1818646.
The third author was supported by NSF Grant 1502294.
Article copyright: © Copyright 2020 American Mathematical Society