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DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

Part of: Rings and algebras arising under various constructions Commutative algebra: Homological methods

Published online by Cambridge University Press:  18 June 2018

AMANDA CROLL ,
ROGER DELLACA ,
ANJAN GUPTA ,
JUSTIN HOFFMEIER ,
VIVEK MUKUNDAN ,
LIANA M. ŞEGA ,
GABRIEL SOSA ,
PEDER THOMPSON  and
DENISE RANGEL TRACY
AMANDA CROLL
Affiliation:
Concordia University Irvine, Irvine, CA 92612, USA email amanda.croll@cui.edu
ROGER DELLACA
Affiliation:
Department of Mathematics, University of California–Irvine, Irvine, CA 92697, USA email rdellaca@uci.edu
ANJAN GUPTA
Affiliation:
Dipartimento di Matematica, Università degli studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy email agmath@gmail.com
JUSTIN HOFFMEIER
Affiliation:
Department of Mathematics & Statistics, Northwest Missouri State University, Maryville, MO 64468, USA email jhoff@nwmissouri.edu
VIVEK MUKUNDAN
Affiliation:
Purdue University & School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India email vivekm85@gmail.com
LIANA M. ŞEGA
Affiliation:
Department of Mathematics and Statistics, University of Missouri, Kansas City, MO 64110, USA email segal@umkc.edu
GABRIEL SOSA
Affiliation:
Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002, USA email gsosa@amherst.edu
PEDER THOMPSON
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA email peder.thompson@ttu.edu
DENISE RANGEL TRACY
Affiliation:
Department of Mathematics, Central Connecticut State University, New Britain, CT 06050, USA email d.a.rangeltracy@ccsu.edu
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Abstract

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.) 13D02: Syzygies, resolutions, complexes 16S37: Quadratic and Koszul algebras
Type
Article
Information
Nagoya Mathematical Journal , Volume 238 , June 2020 , pp. 47 - 85
Copyright
© 2018 Foundation Nagoya Mathematical Journal

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Footnotes

This material is based upon work supported by the National Science Foundation under Grant #1321794, as part of the Mathematical Research Communities 2015 program in Snowbird, Utah, and by a grant from the Simons Foundation (#354594, Liana Şega). Anjan Gupta was supported by the Istituto Nazionale di Alta Matematica “Francesco Severi” fellowship.

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