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On the growth of deviations


Authors: Adam Boocher, Alessio D’Alì, Eloísa Grifo, Jonathan Montaño and Alessio Sammartano
Journal: Proc. Amer. Math. Soc. 144 (2016), 5049-5060
MSC (2010): Primary 13D02; Secondary 16E45, 13D40, 16S37, 05C25, 05C38
DOI: https://doi.org/10.1090/proc/13132
Published electronically: August 18, 2016
MathSciNet review: 3556251
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Abstract: The deviations of a graded algebra are a sequence of integers that determine the Poincaré series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper, we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not increase when passing to an initial ideal and are maximized by the lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.


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Additional Information

Adam Boocher
Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ, Scotland
Email: adam.boocher@ed.ac.uk

Alessio D’Alì
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: dali@dima.unige.it

Eloísa Grifo
Affiliation: Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, Charlottesville, Virginia 22904
Email: er2eq@virginia.edu

Jonathan Montaño
Affiliation: Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Boulevard, Lawrence, Kansas 66045
Email: jmontano@ku.edu

Alessio Sammartano
Affiliation: Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, Indiana 47907
Email: asammart@purdue.edu

DOI: https://doi.org/10.1090/proc/13132
Received by editor(s): March 30, 2015
Received by editor(s) in revised form: January 27, 2016
Published electronically: August 18, 2016
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society