On the Golod property of Stanley-Reisner rings
type: Item
instance of: scholarly article
title: On the Golod property of Stanley-Reisner rings
zbMATH Open document ID: 1129.13014
author: Alexander Berglund
author: Michael Jöllenbeck
published in: Journal of Algebra
publication date: 2007-10-17T00:00:00Z
review text: Let \(R = k[x_1,\ldots, x_n]\) be a polynomial ring over a field \(k\) and \(I\subset R\) a monomial ideal whose minimal set of generators is \(G(I)\). We consider the residue class field \(A = k[x_1,\ldots, x_n]/I\) and let \(F_\bullet\) be a multigraded minimal \(A\)-free resolution of \(k\) as an \(A\)-module. The authors show that there exist isomorphisms as \(k\)-vector spaces \[ F_i \cong \bigoplus_{\begin{matrix} J \subset \{1,\ldots, n\}\\ | J | = \ell \end{matrix}} \bigoplus_{\begin{matrix} u \in {\mathcal G}({\mathcal R})\\ | u| = i -\ell \end{matrix}} A(- (\alpha_J + \deg(u))) \] where \(\alpha_J\) is the characteristic vector of \(J\), defined by \((\alpha_J)_i=0\) if \(i\notin J\) and \(=1\) if \(i\in J\), and \({\mathcal G}({\mathcal R})\) is a monomial \(k\)-basis of the non-commutative associate algebra \({\mathcal R}\) constructed as follows: We first consider the free associate algebra \(\tilde{R} = k\langle \{Y_S\}\rangle\) where \(Y_S\) is the variable associated to a subset \(S\subset G(I)\) satisfying some condition coming from algebraic discrete Morse theory developed by \textit{E.~Sköldberg} [Trans. Am. Math. Soc. 358, No.~1, 115--129 (2006; Zbl 1150.16008)] and the second author. Then we set \({\mathcal R} = \tilde{\mathcal R}/J\) with \(J\) the ideal generated by skew symmetric relations of the variables \(Y_S\). This result was conjectured by \textit{M. Jöllenbeck} [J. Pure Appl. Algebra 207, No.~2, 261--298 (2006; Zbl 1101.13025)], in which the second author showed that \(A\) is Golod if and only if the first Massey operation on \(\text{Tor}_*^R(A,k)\) is trivial, under the assumption that the conjecture is correct. Moreover, the authors give a complete combinatorial condition of the Golod property of Stanley-Reisner rings for flag complexes.
reviewed by: Yukihide Takayama
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 5201685
zbMATH Keywords: Golod ring
zbMATH Keywords: Poincaré series
zbMATH Keywords: Stanley-Reisner ring
zbMATH Keywords: algebraic discrete Morse theory
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.04.018
OpenAlex ID: W2012044540
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DOI: 10.1016/J.JALGEBRA.2007.04.018
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description: scientific article