| Russian Mathematical Surveys |
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This article is cited in 1 scientific paper (total in 1 paper) Brief Communications Monomial non-Golod face rings and Massey products I. Yu. Limonchenkoa, T. E. Panovbac a National Research University Higher School of Economics b Lomonosov Moscow State University c Institute for Information Transmission Problems of the Russian Academy of Sciences
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     In this paper a criterion for the Golodness of the face ring $\Bbbk[K]$ of a simplicial complex $K$ over the field $\Bbbk$ is obtained. A similar criterion was proposed in [4], but one of the assertions there depended on the main result of [1], which was shown to be false in [5]. Our proof fills this gap. We also construct an example of a minimally non-Golod complex $K$ such that the cohomology of the corresponding moment-angle complex $\mathcal Z_K$ has a trivial cup product and a non-trivial triple Massey product. Let $K$ be a simplicial complex on the vertex set $[m]=\{1,2,\dots,m\}$. The face ring $\Bbbk[K]:=\Bbbk[v_1,\dots,v_m]/(v_{i_1}\cdots v_{i_r}\mid\{i_1,\dots,i_r\} \notin K)$ is said to be Golod (over $\Bbbk$) if the product and all higher Massey products in the Koszul complex $(\Lambda[u_1,\dots,u_m]\otimes\Bbbk[K],d)$ are trivial. By [3], $\Bbbk[K]$ is a Golod ring if and only if Serre’s inequality, relating the Hilbert series of $\operatorname{Ext}_{\Bbbk[K]}(\Bbbk,\Bbbk)$ and $\operatorname{Tor}_{\Bbbk[v_1,\dots,v_m]}(\Bbbk,\Bbbk[K])$, turns to equality. If $\Bbbk[K]$ is not Golod, but $\Bbbk[K_{[m]\setminus\{i\}}]$ is Golod for each $i\in [m]$, then $\Bbbk[K]$ is called minimally non-Golod (over $\Bbbk$). Given a topological pair $(X,A)$, its polyhedral product $(X,A)^K$ is defined as $\bigcup_{\sigma\in K}(X,A)^{\sigma}$ for $(X,A)^{\sigma}:=\prod_{i\in [m]}X_i$, where $X_i=X$ if $i\in\sigma$ and $X_i=A$ otherwise. Recall that $\mathcal Z_K:=(\mathbb D^2,\mathbb S^1)^K$ and $\mathit{DJ}(K):=(\mathbb{C}\mathbb P^\infty,\ast)^K$. The Koszul complex $(\Lambda[u_1,\dots,u_m]\otimes\Bbbk[K],d)$ is quasi-isomorphic to the cellular cochains of $\mathcal Z_K$ with an appropriate diagonal approximation ([2], Lemma 4.5.3); in particular, $H^*(\mathcal Z_K;\Bbbk)\cong \operatorname{Tor}_{\Bbbk[v_1,\dots,v_m]}(\Bbbk,\Bbbk[K])$. Theorem 1. Let $\Bbbk$ be a field. Then the following conditions are equivalent: (a) $\Bbbk[K]$ is a Golod ring over $\Bbbk$; (b) the cup product and all Massey products in $H^{+}(\mathcal Z_K;\Bbbk)$ are trivial; (c) $H_{*}(\Omega\mathcal Z_K;\Bbbk)$ is a graded free associative algebra; (d) the Hilbert series satisfy the identity Proof. The equivalence (a) $\Leftrightarrow$ (b) follows from [2], Theorem 4.5.4.
For (a) $\Leftrightarrow$ (d), a theorem of Golod [3] asserts that $\Bbbk[K]$ is a Golod ring if and only if the following identity holds for the Hilbert series:
$$
\begin{equation*}
\operatorname{Hilb}\bigl(\operatorname{Ext}_{\Bbbk[K]}(\Bbbk,\Bbbk);t\bigr) =\frac{(1+t)^m}{1-\sum_{i,j>0}\beta^{-i,2j}(\Bbbk[K])t^{-i+2j-1}}\,,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\beta^{-i,2j}(\Bbbk[K])=\dim\operatorname{Tor}^{-i,2j}_{\Bbbk[v_1,\dots,v_m]}(\Bbbk,\Bbbk[K]).
\end{equation*}
\notag
$$
By [2], Proposition 8.4.10, there is an algebra isomorphism $H_{*}(\Omega\mathit{DJ}(K);\Bbbk)\cong\operatorname{Ext}_{\Bbbk[K]}(\Bbbk,\Bbbk)$. The loop space decomposition
$$
\begin{equation*}
\Omega\mathit{DJ}(K)\simeq\Omega\mathcal Z_K\times\mathbb{T}^m
\end{equation*}
\notag
$$
([2], (8.15)) implies that
$$
\begin{equation*}
\operatorname{Hilb}(H_{*}(\Omega\mathit{DJ}(K);\Bbbk);t) =\operatorname{Hilb}(H_{*}(\Omega\mathcal Z_K;\Bbbk);t)\cdot (1+t)^m.
\end{equation*}
\notag
$$
Also,
$$
\begin{equation*}
\operatorname{Hilb}(\Sigma^{-1}\widetilde{H}^{*}(\mathcal Z_K;\Bbbk);t) =\sum_{i,j>0}\beta^{-i,2j}(\Bbbk[K])t^{-i+2j-1}
\end{equation*}
\notag
$$
by [2], Theorem 4.5.4. Substituting this in yields the identity in (d).
We prove that (c) $\Rightarrow$ (d). Let
$$
\begin{equation*}
Q=H_{>0}(\Omega\mathcal Z_K;\Bbbk)/(H_{>0}(\Omega\mathcal Z_K;\Bbbk)\cdot H_{>0}(\Omega\mathcal Z_K;\Bbbk))
\end{equation*}
\notag
$$
be the space of indecomposables. By assumption $H_{*}(\Omega\mathcal Z_K;\Bbbk)=T\langle Q\rangle$, where $T\langle Q\rangle$ is the free associative algebra on the graded $\Bbbk$-module $Q$. The Milnor–Moore (bar) spectral sequence has the $E_2$-term $E_{2}^{b}= \operatorname{Tor}_{H_{*}(\Omega\mathcal Z_K;\Bbbk)}(\Bbbk,\Bbbk)$ and converges to $\Sigma^{-1}H_{*}(\mathcal Z_K;\Bbbk)$. By assumption $E_{2}^{b}\cong\operatorname{Tor}_{H_{*}(T\langle Q\rangle)}(\Bbbk,\Bbbk) \cong\Bbbk\oplus Q$ (as $\Bbbk$-modules), so
$$
\begin{equation*}
\operatorname{Hilb}(\Sigma^{-1}\widetilde{H}_{*}(\mathcal Z_K;\Bbbk);t)= {\operatorname{Hilb}(E_{\infty}^{b};t)-1}\leqslant \operatorname{Hilb}(E_{2}^{b};t)-1=\operatorname{Hilb}(Q;t).
\end{equation*}
\notag
$$
In particular,
$$
\begin{equation*}
\operatorname{Hilb}(T\langle\Sigma^{-1} \widetilde{H}_{*}(\mathcal Z_K;\Bbbk)\rangle;t)\leqslant \operatorname{Hilb}(T\langle Q\rangle;t)= \operatorname{Hilb}(H_{*}(\Omega\mathcal Z_K;\Bbbk);t).
\end{equation*}
\notag
$$
The Adams (cobar) spectral sequence has $E_{2}^{c}=\operatorname{Cotor}_{H_{*}(\mathcal Z_K;\Bbbk)}(\Bbbk;\Bbbk)$ and converges to $H_{*}(\Omega\mathcal Z_K;\Bbbk)$. We have
$$
\begin{equation*}
\operatorname{Hilb}(H_{*}(\Omega\mathcal Z_K;\Bbbk);t)= \operatorname{Hilb}(E_{\infty}^{c};t)\leqslant \operatorname{Hilb}(E_{2}^{c};t)\leqslant \operatorname{Hilb}(T\langle\Sigma^{-1} \widetilde{H}_{*}(\mathcal Z_K;\Bbbk)\rangle;t),
\end{equation*}
\notag
$$
where the last inequality follows from the cobar construction (it turns to equality when all differentials in the cobar construction on $H_{*}(\mathcal Z_K;\Bbbk)$ vanish). By combining the two inequalities we obtain
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Hilb}\bigl(H_{*}(\Omega\mathcal Z_K;\Bbbk);t\bigr) &=\operatorname{Hilb}\bigl(T\langle\Sigma^{-1} \widetilde{H}_{*}(\mathcal Z_K;\Bbbk)\rangle;t\bigr) \\ &=\frac1{1-\operatorname{Hilb}(\Sigma^{-1}\widetilde{H}^{*}(\mathcal Z_K;\Bbbk);t)}\,. \end{aligned}
\end{equation*}
\notag
$$
To prove that (d) $\Rightarrow$ (c), observe that the identity in (d) is equivalent to This implies that all differentials in the Adams cobar construction on $H_{*}(\mathcal Z_K;\Bbbk)$ are trivial. Thus $H_{*}(\Omega\mathcal Z_K;\Bbbk)$ is a free associative algebra (on $\Sigma^{-1}\widetilde{H}_{*}(\mathcal Z_K;\Bbbk)$). $\Box$In the case of flag $K$ it was proved in [4] that $\Bbbk[K]$ is Golod if and only if $\operatorname{cup}(\mathcal Z_K)=1$, and if $\Bbbk[K]$ is minimally non-Golod, then $\operatorname{cup}(\mathcal Z_K)=2$. In general, for minimally non-Golod $\Bbbk[K]$ we have an upper bound $\operatorname{cup}(\mathcal Z_K)\leqslant2$, which follows easily from Baskakov’s description of the product in $H^*(\mathcal Z_K;\Bbbk)$: see [2], Theorem 4.5.4. Building upon a construction in [5] we give an example of a minimally non-Golod complex $\mathcal K$ such that $\mathrm{cup}(\mathcal Z_\mathcal K)=1$ and $H^*(\mathcal Z_\mathcal K)$ has a non-trivial triple Massey product. Theorem 2. Let $\mathcal K$ be given by its minimal non-faces $(1,2,3)$, $(4,5,6)$, $(7,8,9)$, $(1,4,7)$, $(1,2,4,5)$, $(5,6,7,8)$, $(2,3,7,8)$, $(2,3,5,6,7)$, $(1,2,4,6,8,9)$, $(1,3,4,5,8,9)$, $(1,3,5,6,7,9)$, $(2,3,4,5,7,9)$, $(2,3,4,5,8,9)$, $(2,3,4,6,7,9)$, $(2,3,5,6,8,9)$. Then $\mathcal K$ is a $4$-dimensional minimally non-Golod complex such that $\operatorname{cup}(\mathcal Z_{\mathcal K})=1$ and there exists a non-trivial indecomposable triple Massey product of $5$-dimensional Koszul cohomology classes in $H^{14}(\mathcal Z_\mathcal K)$: Proof. The description of the cup product for $\mathcal Z_\mathcal K$ ([2], Theorem 4.5.4) implies that $\operatorname{cup}(\mathcal Z_\mathcal K)=1$. The full subcomplex $\mathcal K_{[m]\setminus\{i\}}$ is Golod for any $i\in [m]$ by [5], Theorem 6.3, (5), so $\mathcal K$ is minimally non-Golod. It remains to show that the triple Massey product above is defined, non-trivial, and indecomposable. It is defined and single-valued due to [6], Lemma 3.3, since It is non-trivial since $[v_1v_2v_5v_7v_8u_3u_4u_6u_9]$ corresponds to a non-zero class in $H^4(\mathcal K)$ and $\dim\mathcal K=4$. It is indecomposable since and $\widetilde{H}^p(\mathcal K_{[m]\setminus\{1,4,7\}})\cong\Bbbk$ for $p=4$ and is zero otherwise, whereas $\widetilde{H}^q(\mathcal K_{\{1,4,7\}})\cong\Bbbk$ for $q=1$ and is zero otherwise. $\Box$
It follows directly from [5], Theorem 6.3, (5), (7), that that if $K$ is a minimally non-Golod simplicial complex such that $\operatorname{cup}(\mathcal Z_K)=1$ and there exists a non-trivial Massey product in $H^{*}(\mathcal Z_K)$, then $\dim(K)\geqslant\dim(\mathcal K)=4$ and $f_0(K)\geqslant f_0(\mathcal K)=9$. We are grateful to Victor Buchstaber for fruitful discussions and his interest to this work. 
Citation in format AMSBIB
\Bibitem{LimPan22} |
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