On the Golod property of Stanley-Reisner rings
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.
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