I. Yu. Limonchenko, T. E. Panov, “Monomial non-Golod face rings and Massey products”, Russian Math. Surveys, 77:4 (2022), 762

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;

(d) the Hilbert series satisfy the identity

$\begin{equation*} \operatorname{Hilb}(H_{*}(\Omega\mathcal Z_K;\Bbbk);t) =\frac1{1-\operatorname{Hilb}(\Sigma^{-1}\widetilde{H}^{*}(\mathcal Z_K;\Bbbk);t)}. \end{equation*} \notag$

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

$\begin{equation*} \operatorname{Hilb}(H_{*}(\Omega\mathcal Z_K;\Bbbk);t)= \operatorname{Hilb} (T\langle\Sigma^{-1}\widetilde{H}_{*}(\mathcal Z_K;\Bbbk)\rangle;t). \end{equation*} \notag$

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.

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

$\begin{equation*} \widetilde{H}^*(\mathcal K_{\{1,2,3,4,5,6\}})=\widetilde{H}^*(\mathcal K_{\{4,5,6,7,8,9\}})=0. \end{equation*} \notag$

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

$\begin{equation*} \widetilde{H}^*(\mathcal K_{[m]\setminus\{1,2,3\}})\cong \widetilde{H}^*(\mathcal K_{[m]\setminus\{4,5,6\}})\cong \widetilde{H}^*(\mathcal K_{[m]\setminus\{7,8,9\}})=0 \end{equation*} \notag$

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.

Bibliography