# Cohomology operations for moment-angle complexes
and resolutions of Stanley–Reisner rings

Steven Amelotte  Department of Mathematics, Western University, London, ON N6A
5B7, Canada<EMAIL_ADDRESS>and  Benjamin Briggs  Department of Mathematical
Sciences, University of Copenhagen, Universitetsparken 5 DK-2100 Copenhagen Ø
<EMAIL_ADDRESS>
(Date: 24 May 2023)

###### Abstract.

A fundamental result in toric topology identifies the cohomology ring of the
moment-angle complex $\mathcal{Z}_{K}$ associated to a simplicial complex $K$
with the Koszul homology of the Stanley–Reisner ring of $K$. By studying
cohomology operations induced by the standard torus action on the moment-angle
complex, we extend this to a topological interpretation of the minimal free
resolution of the Stanley–Reisner ring. The exterior algebra module structure
in cohomology induced by the torus action recovers the linear part of the
minimal free resolution, and we show that higher cohomology operations induced
by the action (in the sense of Goresky–Kottwitz–MacPherson) can be assembled
into an explicit differential on the resolution. Describing these operations
in terms of Hochster’s formula, we recover and extend a result due to Katthän.
We then apply all of this to study the equivariant formality of torus actions
on moment-angle complexes. For these spaces, we obtain complete algebraic and
combinatorial characterisations of which subtori of the naturally acting torus
act equivariantly formally.

###### Key words and phrases:

Moment-angle complex, equivariant formality, Stanley–Reisner ring, minimal
free resolution

###### 2020 Mathematics Subject Classification:

13F55, 57S12, 55U10

For part of this work, Amelotte was hosted by the Institute for Computational
and Experimental Research in Mathematics in Providence, RI, supported by the
National Science Foundation under Grant No. 1929284. Briggs was funded by the
European Union under the Grant Agreement no. 101064551 (Hochschild).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and notation
  3. 3 $\Lambda$-module models for $\mathcal{Z}_{K}$
    1. 3.1 Cellular decomposition of $\mathcal{Z}_{K}$
    2. 3.2 The reduced Koszul complex
    3. 3.3 Combinatorial model for $\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$
    4. 3.4 Higher cohomology operations
  4. 4 Hirsch–Brown models from perturbation theory
  5. 5 Equivariant formality and higher operations
    1. 5.1 $\mathcal{J}$-closed modules
    2. 5.2 Reduction to coordinate subtori
    3. 5.3 Equivariant formality in the flag case
    4. 5.4 $\mathcal{J}$-closed edge ideals
    5. 5.5 Dependence of equivariant formality on the characteristic
  6. 6 Combinatorial models for the higher operations
    1. 6.1 Secondary operations and the Mayer–Vietoris sequence
    2. 6.2 Higher operations and the Mayer–Vietoris spectral sequence
    3. 6.3 Equivariant formality from combinatorics

##  1\. Introduction

Equivariantly formal torus actions are among the richest and best understood
examples of group actions studied in geometry and topology. The action of a
torus $T$ on a space $X$ is called _equivariantly formal_ if the equivariant
cohomology $H^{*}_{T}(X)$ is free as a module over the polynomial ring
$H^{*}(BT)$. This condition makes the relationship between ordinary and
equivariant cohomology as simple as possible, and often allows $H^{*}_{T}(X)$
to be described in terms of fixed point data (as in the Borel localization
theorem and its variants) or low-dimensional orbits of the action (as in the
Chang–Skjelbred lemma [9] and its generalisations to the Atiyah–Bredon
sequence [16]). Examples of equivariantly formal torus actions include
Hamiltonian actions on compact symplectic manifolds, GKM manifolds and all
$T$-spaces with cohomology concentrated in even degrees.

An important source of torus actions is the _moment-angle complex_ , a central
construction in toric topology that functorially assigns a space
$\mathcal{Z}_{K}$ with a $T^{m}=(S^{1})^{m}$-action to each simplicial complex
$K$ on $m$ vertices. These $T^{m}$-spaces play a universal role in toric
topology. For example, every quasitoric manifold (including every smooth
projective toric variety) is diffeomorphic to the quotient
$\mathcal{Z}_{K}/T^{m-n}$ of a moment-angle complex by some restriction of its
standard $T^{m}$-action to a maximal freely acting subtorus $T^{m-n}\subseteq
T^{m}$ (cf. [7, 7.3]). On the other hand, the homotopy quotient of
$\mathcal{Z}_{K}$ by the entire $T^{m}$-action yields the _Davis–Januszkiewicz
space_ $DJ_{K}$, whose cohomology ring
$H^{*}(DJ_{K};k)=H^{*}_{T^{m}}(\mathcal{Z}_{K};k)$ is isomorphic to the
Stanley–Reisner ring

$k[K]=S/(v_{i_{1}}\\!\cdots v_{i_{q}}\,:\,\\{i_{1},\ldots,i_{q}\\}\notin K),$

where $S=H^{*}(BT^{m};k)=k[v_{1},\ldots,v_{m}]$ is a polynomial ring with
generators in degree $2$.

The (homotopy) quotients of moment-angle complexes by other subgroups
$H\subseteq T^{m}$ have recently been investigated by many authors [14, 17,
25, 26, 28, 30]. One motivation for this paper is to answer the following
question:

######  Question 1.

Let $K$ be a simplicial complex on the vertex set $[m]=\\{1,\ldots,m\\}$. For
which subtori $H\subseteq T^{m}$ is the $H$-action on the moment-angle complex
$\mathcal{Z}_{K}$ equivariantly formal?

Rather than directly computing the equivariant cohomology
$H^{*}_{H}(\mathcal{Z}_{K})$ as a module over a polynomial ring for every
subtorus $H\subseteq T^{m}$, we approach 1 from a Koszul dual perspective by
studying the ordinary cohomology $H^{*}(\mathcal{Z}_{K})$ as a module over an
exterior algebra.

Indeed, the $T^{m}$-action on $\mathcal{Z}_{K}$ equips
$H^{*}(\mathcal{Z}_{K})$ with a natural (in $K$) structure of a graded module
over $\Lambda=\Lambda(\iota_{1},\ldots,\iota_{m})$, with generators
$\iota_{j}$ acting by derivations of degree $-1$. In Section 3 we begin a
detailed study of this module structure. As a graded algebra, the ordinary
cohomology of $\mathcal{Z}_{K}$ is well understood in terms of the homological
algebra of the Stanley–Reisner ring (see [6]):

$H^{*}(\mathcal{Z}_{K};k)\cong\operatorname{Tor}^{S}(k[K],k).$

We describe the $\Lambda$-module structure on both sides of this isomorphism,
lifting it to a cochain-level isomorphism of differential graded
$\Lambda$-modules.

Hochster’s formula for the Betti numbers of a Stanley–Reisner ring yields
another description of the cohomology of a moment-angle complex, namely a
decomposition
$H^{*}(\mathcal{Z}_{K})\cong\bigoplus_{J\subseteq[m]}\widetilde{H}^{*}(K_{J})$
in terms of the reduced simplicial cohomology of full subcomplexes
$K_{J}\subseteq K$. We show that this can also be upgraded to an isomorphism
of $\Lambda$-modules, where the action of each $\iota_{j}$ on the components
of Hochster’s decomposition coincides, up to sign, with the maps

(1)
$\widetilde{H}^{*}(K_{J})\longrightarrow\widetilde{H}^{*}(K_{J\smallsetminus
j})$

induced by the inclusions of full subcomplexes $K_{J\smallsetminus
j}\hookrightarrow K_{J}$ (see Lemma 3.5). Together with A below, this recovers
a combinatorial description due to Katthän [24] of the linear part of the
minimal free resolution of $k[K]$.

The derivations $\iota_{j}\colon H^{*}(\mathcal{Z}_{K})\to
H^{*-1}(\mathcal{Z}_{K})$ extend to a family of higher operations

$\delta_{s}\colon H^{*}(\mathcal{Z}_{K})\longrightarrow
H^{*-2\deg(s)+1}(\mathcal{Z}_{K}),$

indexed by the monomials of $S$. In the context of equivariant cohomology, the
higher cohomology operations induced by a torus action were introduced by
Goresky, Kottwitz and MacPherson [18] as obstructions to equivariant
formality. As with other higher cohomology operations, $\delta_{s}$ is
generally only defined on the kernels of lower degree operations, taking
values with indeterminacy depending on these previous operations. For torus
actions on smooth manifolds, this indeterminacy can be avoided by identifying
de Rham cohomology with the space of harmonic forms, as in [1] and [8]. In our
case, since a moment-angle complex is not always a manifold and we wish to
work with coefficients in an arbitrary field $k$, we make use of the
homological perturbation lemma in Section 4 to obtain an explicit family of
higher operations which are well-defined endomorphisms of
$H^{*}(\mathcal{Z}_{K})$. The relevance to equivariant formality is made clear
by the main results of Section 4, which show that these operations assemble
into a differential yielding the minimal Hirsch–Brown model for the action of
any subtorus $H\subseteq T^{m}$ on $\mathcal{Z}_{K}$. In particular, for
$H=T^{m}$ we obtain the following.

######  Theorem A.

If $K$ is a simplicial complex on the vertex set $[m]$, then

$\big{(}S\otimes
H^{*}(\mathcal{Z}_{K};k),\delta\big{)},\qquad\delta=\sum_{J\subseteq[m]}v_{J}\otimes\delta_{J}$

is the minimal free resolution of the Stanley–Reisner ring $k[K]$, or
equivalently the minimal Hirsch–Brown model for the action of $T^{m}$ on
$\mathcal{Z}_{K}$. Here $v_{J}=v_{i_{1}}\\!\cdots v_{i_{q}}$ and $\delta_{J}$
is the cohomology operation indexed by $v_{J}$, for each
$J=\\{i_{1},\ldots,i_{q}\\}\subseteq[m]$.

Describing the linear part of the resolution of $k[K]$ amounts to
understanding the primary operations $\iota_{j}=\delta_{\\{j\\}}$, and these
are given in combinatorial terms by Katthän’s formula (1). In [24], Katthän
poses the question of describing the quadratic part of the resolution in
similar terms; in other words, how can the secondary operations
$\delta_{\\{ij\\}}$ be understood in terms of Hochster’s decomposition
$H^{*}(\mathcal{Z}_{K})\cong\bigoplus_{J\subseteq[m]}\widetilde{H}^{*}(K_{J})$?
More generally, one can ask

######  Question 2.

How can the higher operations $\delta_{s}$ on $H^{*}(\mathcal{Z}_{K};k)$—or
equivalently, the differentials in the minimal free resolution of $k[K]$—be
described in terms of the combinatorics of $K$?

In Section 6 we give combinatorial models for the higher operations by
identifying them with differentials in a Mayer–Vietoris spectral sequence
defined entirely in terms of simplicial cochains on full subcomplexes of $K$.
From the perspective of commutative algebra, this yields a description of the
minimal $S$-free resolution of the Stanley–Reisner ring, purely in terms of
Hochster’s decomposition and the combinatorics of $K$, up to the indeterminacy
in their definition, which remains an obstacle to a complete description. This
recovers and extends Katthän’s theorem.

Special attention is paid to the secondary operations (yielding the quadratic
part of the resolution of $k[K]$) in Section 6.1. We find that, just as the
primary operations $\delta_{\\{i\\}}$ are determined by the maps
$K_{J\smallsetminus i}\hookrightarrow K_{J}$ for all $J\subseteq[m]$, the
secondary operations $\delta_{\\{ij\\}}$ are essentially determined by the
natural inclusions

$K_{J\smallsetminus i}\cup K_{J\smallsetminus j}\hookrightarrow
K_{J}\quad\text{ and }\quad K_{J\smallsetminus i}\cup K_{J\smallsetminus
j}\hookrightarrow\Sigma K_{J\smallsetminus ij}.$

More generally, the behaviour of the higher operations is governed by
inclusions of _face deletions_ $K_{J}\smallsetminus F=\bigcup_{i\in
F}K_{J\smallsetminus i}$ for $F\in K_{J}$; see Section 6.3.

Returning to the question of equivariant formality, we find that this
condition can be read off from the minimal free resolution of the
Stanley–Reisner ring. We show in Section 5.1 that for any subtorus $H\subseteq
T^{m}$, the $H$-action on $\mathcal{Z}_{K}$ is equivariantly formal if and
only if $k[K]$ is $\mathcal{J}$-closed in the sense of Diethorn [11], where
$\mathcal{J}\subseteq S$ is a certain ideal generated by linear polynomials
associated to the torus $H$.

To obtain combinatorial characterisations of equivariant formality, we first
reduce 1 to the case of coordinate subtorus actions on $\mathcal{Z}_{K}$, that
is, actions of subtori of the form

$T^{I}=\big{\\{}(t_{1},\ldots,t_{m})\in T^{m}\,:\,t_{i}=1\text{ for }i\notin
I\big{\\}}$

for some $I\subseteq[m]$; see Section 5.2. The simplest case to consider is
that of a coordinate circle $S^{1}_{j}=T^{\\{j\\}}\subseteq T^{m}$. In this
case, equivariant formality is determined by the $\Lambda$-module structure on
$H^{*}(\mathcal{Z}_{K})$ alone: we show in Theorem 5.9 that the
$S^{1}_{j}$-action on $\mathcal{Z}_{K}$ is equivariantly formal if and only if
the derivation $\iota_{j}\colon H^{*}(\mathcal{Z}_{K})\to
H^{*-1}(\mathcal{Z}_{K})$ is trivial. By the combinatorial description (1) of
the primary operations, it is equivalent that the inclusion of full
subcomplexes $K_{J\smallsetminus j}\hookrightarrow K_{J}$ induces the trivial
map in reduced simplicial cohomology for all $J\subseteq[m]$ with $j\in J$.

For the $T^{I}$-actions on $\mathcal{Z}_{K}$ with $|I|>1$, equivariant
formality is not determined by the $\Lambda$-module $H^{*}(\mathcal{Z}_{K})$,
as the vanishing of higher operations is also necessary. Although the higher
operations are not simply induced by inclusions among the full subcomplexes
appearing in Hochster’s formula, it nonetheless turns out that the vanishing
of all $\delta_{J}$ with $J\subseteq I$ is detected by inclusions of the (not
necessarily full) face deletion subcomplexes.

######  Theorem B.

Let $K$ be a simplicial complex on the vertex set $[m]$ and let
$I\subseteq[m]$. Then the following conditions are equivalent:

  1. (a)

the coordinate $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal
over $k$;

  2. (b)

the cohomology operations $\delta_{J}$ vanish on $H^{*}(\mathcal{Z}_{K};k)$
for all $J\subseteq I$;

  3. (c)

the Stanley–Reisner ring $k[K]$ is $\mathcal{J}_{I}$-closed, where
$\mathcal{J}_{I}=(v_{i}\,:\,i\notin I)$;

  4. (d)

$K_{J}\smallsetminus(I\cap J)\hookrightarrow K_{J}$ induces the trivial map on
$\widetilde{H}^{\ast}(\;;k)$ for all $J\subseteq[m]$.

In case $K$ is flag (or equivalently, the Stanley–Reisner ring $k[K]$ is
quadratic), condition (d) above can be simplified considerably. In this
situation, we obtain the following characterisation of equivariant formality
purely in terms of the combinatorics of $K$.

######  Theorem C.

Let $K$ be a flag complex on $[m]$ and let $I\subseteq[m]$. The coordinate
$T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal if and only if
$I\in K$ and $K_{\\{i,j\\}}\ast K_{I\smallsetminus\\{i,j\\}}\subseteq K$ for
every missing edge $\\{i,j\\}\notin K$.

An interesting consequence is that the equivariant formality of these actions
is independent of the field $k$ in the flag case. We give examples in Section
5.5 illustrating that this is not true in general.

### Acknowledgements

The authors are very grateful to Rachel Diethorn and Matthias Franz for many
helpful comments on a draft of this work. The first author would also like to
thank Graham Denham for a helpful conversation regarding Macaulay2.

##  2\. Preliminaries and notation

Throughout this paper we fix a natural number $m$ and a simplicial complex $K$
on the vertex set $[m]=\\{1,\ldots,m\\}$. The _moment-angle complex_ over $K$
is the subspace of $(D^{2})^{m}$ defined by the polyhedral product

(2) $\mathcal{Z}_{K}=\bigcup_{\sigma\in
K}(D^{2},S^{1})^{\sigma},\qquad(D^{2},S^{1})^{\sigma}=\left\\{(z_{1},\ldots,z_{m})\in(D^{2})^{m}\,:\,z_{i}\in
S^{1}\text{ if }i\notin\sigma\right\\}.$

Note that the coordinatewise action of the torus $T^{m}=(S^{1})^{m}$ on
$(D^{2})^{m}$ restricts to an action of $T^{m}$ on $\mathcal{Z}_{K}$.

We also fix a field $k$, and we write

$S=k[v_{1},\ldots,v_{m}]\quad\text{and}\quad\Lambda=\Lambda(\iota_{1},\ldots,\iota_{m})$

for the polynomial algebra and exterior algebra over $k$, generated by
variables in bijection with $[m]$. We think of the exterior variables
$\iota_{i}$ as dual to the polynomial variables $v_{i}$. Both $S$ and
$\Lambda$ are multigraded by $\mathbb{N}^{m}$. In particular, when
$J=\\{j_{1},\ldots,j_{r}\\}\subseteq[m]$, we write $v_{J}=v_{j_{1}}\\!\cdots
v_{j_{r}}$ for the monomial with (squarefree) multidegree $J$.

The _Stanley–Reisner ring_ of $K$ is the multigraded algebra

$k[K]=S/(v_{i_{1}}\\!\cdots v_{i_{q}}\,:\,\\{i_{1},\ldots,i_{q}\\}\notin K).$

The Koszul complex of $k[K]$ is the multigraded dg algebra

$\big{(}k[K]\otimes\Lambda(u_{1},\ldots,u_{m}),d\big{)}$

with each $u_{i}$ given homological degree $1$ and multidegree
$(0,\ldots,1,\ldots,0)$ (a $1$ in the $i$th position), and with differential
determined by $d(u_{i})=v_{i}$ and the graded Leibniz rule. As before, for a
subset $J=\\{j_{1},\ldots,j_{r}\\}\subseteq[m]$ with $j_{1}<\cdots<j_{r}$, we
will use the notation $u_{J}=u_{j_{1}}\wedge\cdots\wedge u_{j_{r}}$.

For each $i$ there is a unique derivation $\iota_{i}$ on the Koszul complex
with (homological) degree $-1$, determined by

(3) $\iota_{i}(u_{i})=1,\ \iota_{i}(u_{j})=0\text{ if }i\neq j,\text{ and
}\iota_{i}(v_{j})=0\text{ for all }j.$

In other words $\iota_{i}=\frac{\partial}{\partial u_{i}}$. A computation
shows that $\iota_{i}^{2}=0$ and $\iota_{i}\iota_{j}+\iota_{j}\iota_{i}=0$,
and it follows that these derivations give the Koszul complex the structure of
a dg module over $\Lambda=\Lambda(\iota_{1},\ldots,\iota_{m})$.

According to [7, Corollary 4.3.3], the quotient $S\to k[K]$ provides an
algebraic model for the map

(4) $H^{*}(BT^{m};k)\longrightarrow H^{*}_{T^{m}}(\mathcal{Z}_{K};k)$

induced by the Borel fibration $ET^{m}\times_{T^{m}}\mathcal{Z}_{K}\to
BT^{m}$. Moreover, by [7, Theorem 4.5.4], the homology of the Koszul complex
of $k[K]$ computes the cohomology ring $H^{\ast}(\mathcal{Z}_{K};k)$ of the
corresponding moment-angle complex; see Lemma 3.1 below for a more precise
statement. In the next section we will use this to describe concretely a
(cochain-level) model for the $\Lambda$-module structure on
$H^{*}(\mathcal{Z}_{K};k)$, which is in some sense Koszul dual to (4).

Given two subsets $I,J\subseteq[m]$ we frequently use the notation

$\varepsilon(I,J)=|\\{(i,j)\in I\times J:i>j\\}|,$

and when $I=\\{i_{0},\ldots,i_{n}\\}$, we also use the short-hand
$\varepsilon(i_{0}\ldots i_{n},J)=\varepsilon(I,J)$. As we will see below, the
signs $(-1)^{\varepsilon(I,J)}$ occur often in the combinatorics of simplicial
complexes.

##  3\. $\Lambda$-module models for $\mathcal{Z}_{K}$

In this section we study exterior algebra module structures on the cochain
complex and cohomology of a moment-angle complex $\mathcal{Z}_{K}$ which are
induced by the standard torus action on $\mathcal{Z}_{K}$. We review two well-
known tools for computing the cohomology of a moment-angle complex: the Koszul
complex of the Stanley–Reisner ring $k[K]$, and Hochster’s formula. Regarding
these as algebraic and combinatorial models for $\mathcal{Z}_{K}$,
respectively, we show that both models can be equipped with a compatible
structure of a differential graded module over the exterior algebra $\Lambda$.

Throughout, $k$ continues to denote a field. All cochain and cohomology groups
are taken with coefficients in $k$, and all undecorated tensor products are
taken over $k$.

###  3.1. Cellular decomposition of $\mathcal{Z}_{K}$

We begin by recalling a convenient cellular decomposition of $\mathcal{Z}_{K}$
from [7, §4.4]. Consider the subdivision of the unit disk
$D^{2}=\\{z\in\mathbb{C}:|z|\leqslant 1\\}$ into three cells where the
$0$-skeleton is given by the point $z=1$, the $1$-skeleton is given by the
boundary circle $S^{1}\subset D^{2}$, and the $2$-skeleton is given by the
disk itself. Denoting the $0$-, $1$\- and $2$-cell by $1$, $\mathbb{S}$ and
$\mathbb{D}$, respectively, and taking products yields a cellular
decomposition of the polydisk $(D^{2})^{m}$ with cells parametrised by words
$\varkappa\in\\{1,\mathbb{S},\mathbb{D}\\}^{m}$ or, equivalently, pairs of
subsets $I,J\subseteq[m]$ with $I\cap J=\varnothing$. To each such pair is
associated the cell

$\displaystyle\varkappa(I,J)=\big{\\{}(z_{1},\ldots,z_{m})\in(D^{2})^{m}\,:\;\,$
$\displaystyle z_{i}\in\mathbb{D}\text{ for }i\in I,$ $\displaystyle
z_{j}\in\mathbb{S}\text{ for }j\in J,$ $\displaystyle z_{k}=1\text{ for
}k\notin I\cup J\big{\\}}.$

Then for any simplicial complex $K$ on the vertex set $[m]$, $\mathcal{Z}_{K}$
is the cellular subcomplex of $(D^{2})^{m}$ consisting of those cells
$\varkappa(I,J)$ with $I\in K$.

The cellular cochain complex $\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$
has a basis consisting of cochains $\varkappa(I,J)^{\vee}$ dual to the cells
above with $I\in K$, $J\subseteq[m]$ and $I\cap J=\varnothing$. Although the
cellular cochain complex in general does not come with a functorial
associative multiplication, Baskakov, Buchstaber and Panov [3, 6] have
constructed a cellular approximation $\widetilde{\Delta}_{K}$ to the diagonal
map $\Delta_{K}\colon\mathcal{Z}_{K}\to\mathcal{Z}_{K}\times\mathcal{Z}_{K}$
which is functorial in $K$ and defines a cup product

$\cup\colon\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})\otimes\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})\xrightarrow{\mathmakebox[2em]{\times}}\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K}\times\mathcal{Z}_{K})\xrightarrow{\mathmakebox[2em]{\widetilde{\Delta}_{K}}}\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$

giving $\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$ the structure of a
commutative differential graded algebra.

###  3.2. The reduced Koszul complex

The _reduced Koszul complex_ of $k[K]$ is the quotient of the Koszul complex
by the acyclic multigraded ideal of elements of non-squarefree multidegree:

(5) $R(K)=\bigl{(}\Lambda(u_{1},\ldots,u_{m})\otimes
k[K]\bigr{)}/(v_{i}^{2},v_{i}u_{i}),\quad d(u_{i})=v_{i}.$

The differential of the Koszul complex induces a well-defined differential on
$R(K)$, and moreover the quotient map

(6) $\bigl{(}\Lambda(u_{1},\ldots,u_{m})\otimes k[K],d\bigr{)}\xrightarrow{\
\simeq\ }R(K)$

is a quasi-isomorphism of dg algebras, see [7, Lemma 3.2.6]. In particular,

(7) $H_{*}\bigl{(}\Lambda(u_{1},\ldots,u_{m})\otimes k[K],d\bigr{)}\cong
H_{*}(R(K))\cong\operatorname{Tor}^{S}_{\ast}(k[K],k).$

The quotient map (6) is an isomorphism when restricted to any squarefree
multidegree. Making this identification, for any $U\subseteq[m]$ we write

$R(K)_{i,U}\coloneqq\bigl{(}k[K]\otimes\Lambda^{i}(u_{1},\ldots,u_{m})\bigr{)}_{U}.$

In particular, we may consider $R(K)$ as a subcomplex of the Koszul complex

(8) $R(K)\xhookrightarrow{\ \simeq\
}\bigl{(}\Lambda(u_{1},\ldots,u_{m})\otimes k[K],d\bigr{)},$

by including all elements with squarefree multidegree. Under this inclusion,
$R(K)$ is a dg $\Lambda$-submodule of the Koszul complex, using the action
$\iota_{i}=\frac{\partial}{\partial u_{i}}$ defined in (3). We note that this
inclusion is not a dg algebra homomorphism, and conversely the quotient (6) is
not a dg $\Lambda$-module homomorphism.

In the next lemma we use the total (cohomological) grading
$R^{n}(K)=\bigoplus_{n=2|U|-i}R(K)_{i,U}$.

######  Lemma 3.1 ([7, Lemma 4.5.3]).

There is an isomorphism of differential graded algebras

$\displaystyle g\colon R(K)$
$\displaystyle\longrightarrow\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$
$\displaystyle v_{I}u_{J}$ $\displaystyle\longmapsto\varkappa(I,J)^{\vee}$

which is functorial in $K$.

The action of $T^{m}$ on $\mathcal{Z}_{K}$ equips the cohomology ring
$H^{*}(\mathcal{Z}_{K})$ with the structure of a graded module over an
exterior algebra $\Lambda=\Lambda(\iota_{1},\ldots,\iota_{m})$ on $m$
generators of degree $-1$. This can be lifted to an action of $\Lambda$ on the
cellular cochains of $\mathcal{Z}_{K}$ as follows.

Regarding $S^{1}$ as a CW-complex with the same $0$-cell and $1$-cell as
$D^{2}$, we have identifications
$\mathcal{C}^{*}_{\mathrm{cw}}(S^{1})=H^{*}(S^{1})=\Lambda(u)$, $|u|=1$.
Taking products yields a cellular decomposition of the torus which makes
$T^{m}=(S^{1})^{m}$ a cellular subcomplex of $(D^{2})^{m}$. With respect to
these cell structures, it is easy to see that the standard action $\mu\colon
T^{m}\times\mathcal{Z}_{K}\to\mathcal{Z}_{K}$ is a cellular map and hence
induces a map on cellular cochains.

For each $j\in[m]$, let $\mu_{j}\colon
S^{1}_{j}\times\mathcal{Z}_{K}\to\mathcal{Z}_{K}$ be the coordinate circle
action obtained by restricting the $T^{m}$-action to the $j^{\mathrm{th}}$
factor. Identifying
$\mathcal{C}^{*}_{\mathrm{cw}}(S^{1}_{j}\times\mathcal{Z}_{K})$ with
$\Lambda(u_{j})\otimes\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$, each
coordinate circle action $\mu_{j}$ induces a map

(9)
$\displaystyle\mu_{j}^{*}\colon\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$
$\displaystyle\longrightarrow\Lambda(u_{j})\otimes\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$
$\displaystyle\alpha$
$\displaystyle\longmapsto\,1\otimes\alpha+u_{j}\otimes\iota_{j}\alpha,$

which defines a graded derivation

$\iota_{j}\colon\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})\longrightarrow\mathcal{C}^{*-1}_{\mathrm{cw}}(\mathcal{Z}_{K})$

for each $j\in[m]$. Since $[\iota_{j},d]=\iota_{j}d+d\iota_{j}=0$, these maps
induce graded derivations on the cohomology ring $H^{*}(\mathcal{Z}_{K})$
having degree $-1$, which we call _primary cohomology operations_ and also
denote by $\iota_{1},\ldots,\iota_{m}$. Moreover, since
$[\iota_{i},\iota_{j}]=\iota_{i}\iota_{j}+\iota_{j}\iota_{i}=0$ for all
$i,j\in[m]$, the actions of these derivations extend to graded
$\Lambda$-module structures on both
$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$ and $H^{*}(\mathcal{Z}_{K})$.

######  Remark 3.2.

Since $\Lambda$ acts on the cohomology of $\mathcal{Z}_{K}$ for every
simplicial complex $K$ on the vertex set $[m]$, and this action is functorial
in $K$, we think of $\Lambda$ as an algebra of cohomology operations for
moment-angle complexes. In Section 4 we will embed $\Lambda$ into a larger
algebra of _higher cohomology operations_.

######  Lemma 3.3.

The map $g\colon R(K)\to\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$ of
Lemma 3.1 is an isomorphism of differential graded
$\Lambda(\iota_{1},\ldots,\iota_{m})$-modules.

###### Proof.

Since the generators of $\Lambda$ act by derivations, it suffices by Lemma 3.1
to show that $g\circ\iota_{i}$ and $\iota_{i}\circ g$ agree on the generators
of $R(K)$ for each $i=1,\ldots,m$. By definition of the $\Lambda$-module
structure on $R(K)$, $g\circ\iota_{i}$ vanishes on all generators except
$u_{i}$, in which case
$(g\circ\iota_{i})(u_{i})=g(1)=\varkappa(\varnothing,\varnothing)^{\vee}$, the
dual of the $0$-cell
$(1,\ldots,1)\in\mathcal{C}_{0}^{\mathrm{cw}}(\mathcal{Z}_{K};k)$. On the
other hand, we have

$(\iota_{i}\circ
g)(v_{j})=\iota_{i}\varkappa(\\{j\\},\varnothing)^{\vee}\quad\text{and}\quad(\iota_{i}\circ
g)(u_{j})=\iota_{i}\varkappa(\varnothing,\\{j\\})^{\vee}.$

Since $v_{j}\in R^{2}(K)$ is closed for all $j$, so is
$\iota_{i}\varkappa(\\{j\\},\varnothing)^{\vee}$. It follows that
$\iota_{i}\varkappa(\\{j\\},\varnothing)^{\vee}=0$ since the only cocycle in
$\mathcal{C}^{1}_{\mathrm{cw}}(\mathcal{Z}_{K})$ is trivial by Lemma 3.1. To
see that $\iota_{i}\varkappa(\varnothing,\\{j\\})^{\vee}$ equals
$\varkappa(\varnothing,\varnothing)^{\vee}$ for $i=j$ and is trivial for
$i\neq j$, observe that the orbit of the $0$-cell $(1,\ldots,1)$ under the
action $\mu_{i}\colon S^{1}_{i}\times\mathcal{Z}_{K}\to\mathcal{Z}_{K}$ is
exactly the $1$-cell $\varkappa(\varnothing,\\{i\\})$. Dualizing, it follows
that

$\mu_{i}^{*}\bigl{(}\varkappa(\varnothing,\\{j\\})^{\vee}\bigr{)}=\begin{cases}1\otimes\varkappa(\varnothing,\\{i\\})^{\vee}+u_{i}\otimes\varkappa(\varnothing,\varnothing)^{\vee}&\text{if
}i=j\\\ 1\otimes\varkappa(\varnothing,\\{j\\})^{\vee}&\text{if }i\neq
j.\end{cases}$

Therefore by (9) we have

$\iota_{i}\varkappa(\varnothing,\\{j\\})^{\vee}=\begin{cases}\varkappa(\varnothing,\varnothing)^{\vee}&\text{if
}i=j\\\ 0&\text{if }i\neq j,\end{cases}$

which completes the proof. ∎

###  3.3. Combinatorial model for
$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$

The following celebrated result of Hochster interprets the Koszul homology of
a Stanley–Reisner ring $k[K]$ in terms of the simplicial cohomology groups of
full subcomplexes of $K$.

######  Theorem 3.4 (Hochster’s formula [23]).

For each squarefree multidegree $U\subseteq[m]$, there is an isomorphism of
cochain complexes

(10) $\displaystyle
h\colon\bigl{(}k[K]\otimes\Lambda^{*}(u_{1},\ldots,u_{m})\bigr{)}_{U}$
$\displaystyle\longrightarrow\widetilde{C}^{|U|-*-1}(K_{U})$ $\displaystyle
v_{I}u_{J}\;(\text{with }I\cup J=U)$
$\displaystyle\longmapsto(-1)^{\varepsilon(I,U)}I^{\vee}.$

Note that, summing over all $U\subseteq[m]$, the theorem above gives an
isomorphism of complexes
$R(K)\cong\bigoplus_{U\subseteq[m]}\widetilde{C}^{*}(K_{U})$. Taking
cohomology then yields the usual Hochster decomposition

$\operatorname{Tor}^{S}_{*}(k[K],k)\cong\bigoplus_{U\subseteq[m]}\widetilde{H}^{|U|-*-1}(K_{U}).$

The next result describes the $\Lambda$-module structures on
$R(K)\cong\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$ and
$H^{*}(\mathcal{Z}_{K})$ in terms of the decompositions above.

######  Lemma 3.5.

For each $j\in[m]$ and each squarefree multidegree $U\subseteq[m]$ containing
$j$, there are commutative diagrams

$\textstyle{R(K)_{i,U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\iota_{j}}$$\textstyle{\widetilde{C}^{|U|-i-1}(K_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R(K)_{i-1,U\smallsetminus
j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\widetilde{C}^{|U|-i-1}(K_{U\smallsetminus
j})}$$\textstyle{(-1)^{\varepsilon(j,U)+|U|-i}\omega|_{U\smallsetminus j}}$

where the horizontal maps are the isomorphisms of (10), and the right vertical
map is defined by the inclusion $K_{U\smallsetminus j}\hookrightarrow K_{U}$
(up to the indicated sign). In particular, there are commutative diagrams

$\textstyle{H^{n}(\mathcal{Z}_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{j}}$$\textstyle{\displaystyle\bigoplus_{U\subseteq[m]}\widetilde{H}^{n-|U|-1}(K_{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{n-1}(\mathcal{Z}_{K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\displaystyle\bigoplus_{U\subseteq[m]}\widetilde{H}^{n-|U|-2}(K_{U})}$

where the right vertical map is a sum of maps induced by inclusions
$K_{U\smallsetminus j}\hookrightarrow K_{U}$ for $j\in U$ (and is trivial on
summands indexed by $U\subseteq[m]$ with $j\notin U$).

###### Proof.

Let $v_{I}u_{J}\in R(K)_{i,U}$ (so $I\in K$, $J\in[m]$, $I\cap J=\varnothing$
and $I\cup J=U$) and assume $j\in U$. If $j\in I$, then
$\iota_{j}(v_{I}u_{J})=0$, and $h(v_{I}u_{J})=(-1)^{\varepsilon(I,U)}I^{\vee}$
is in the kernel of the restriction map
$\widetilde{C}^{*}(K_{U})\to\widetilde{C}^{*}(K_{U\smallsetminus j})$ since,
dually, $I\in\widetilde{C}_{*}(K_{U})$ is clearly not in the image of
$\widetilde{C}_{*}(K_{U\smallsetminus j})\to\widetilde{C}_{*}(K_{U})$ when
$j\in I$. On the other hand, if $j\in J$, then following anticlockwise around
the first diagram, we have

$\displaystyle h\iota_{j}(v_{I}u_{J})$
$\displaystyle=h\bigl{(}(-1)^{\varepsilon(j,J)}v_{I}u_{J\smallsetminus
j}\bigr{)}$ (11)
$\displaystyle=(-1)^{\varepsilon(j,J)}(-1)^{\varepsilon(I,U\smallsetminus
j)}I^{\vee}.$

Following clockwise around the diagram, the right vertical map sends
$h(v_{I}u_{J})=(-1)^{\varepsilon(I,U)}I^{\vee}$ to

(12)
$(-1)^{\varepsilon(I,U)}(-1)^{\varepsilon(j,U)+|U|-|J|}I^{\vee}=(-1)^{\varepsilon(I,U)}(-1)^{\varepsilon(j,U)+|I|}I^{\vee}.$

To see that (11) and (12) are equal, observe that

$\underbrace{\varepsilon(j,U)-\varepsilon(j,J)}_{\varepsilon(j,I)}+\underbrace{\varepsilon(I,U)-\varepsilon(I,U\smallsetminus
j)}_{\varepsilon(I,j)}+|I|=|I|+|I|\equiv 0\text{ mod }2,$

so the signs agree.

The second commutative diagram in the statement of the lemma is obtained from
the first by passing to cohomology, summing over all $i\in\mathbb{Z}$ and
$U\subseteq[m]$ with $2|U|-i=n$, and absorbing signs into the horizontal
isomorphisms. ∎

######  Remark 3.6.

By Lemma 3.5, $\bigoplus_{U\subseteq[m]}\widetilde{C}^{*}(K_{U})$ and hence
$\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U})$ can clearly be given
$\Lambda(\iota_{1},\ldots,\iota_{m})$-module structures by letting $\iota_{j}$
act on simplicial cochains by the right vertical map in the diagram. With
respect to these module structures, the Hochster decompositions

$R(K)\cong\bigoplus_{U\subseteq[m]}\widetilde{C}^{*}(K_{U})\quad\text{ and
}\quad\operatorname{Tor}^{S}(k[K],k)\cong\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U})$

become isomorphisms of differential graded
$\Lambda(\iota_{1},\ldots,\iota_{m})$-modules.

Combining Lemma 3.3, Lemma 3.5 and Remark 3.6, we obtain the following.

######  Corollary 3.7.

There is a zig-zag of isomorphisms of differential graded
$\Lambda(\iota_{1},\ldots,\iota_{m})$-modules

$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})\stackrel{{\scriptstyle\;g}}{{\longleftarrow}}R(K)\stackrel{{\scriptstyle
h\;}}{{\longrightarrow}}\bigoplus_{U\subseteq[m]}\widetilde{C}^{*}(K_{U}),$

inducing graded $\Lambda(\iota_{1},\ldots,\iota_{m})$-module isomorphisms

$H^{*}(\mathcal{Z}_{K})\cong\operatorname{Tor}^{S}(k[K],k)\cong\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U}).$

###  3.4. Higher cohomology operations

To any differential graded module $N$ over
$\Lambda=\Lambda(\iota_{1},\ldots,\iota_{m})$, one can associate a family of
_higher cohomology operations_ acting on $H^{*}(N)$, indexed by monomials:

$\delta_{s},\quad s=v_{1}^{n_{1}}\cdots v_{m}^{n_{m}}\in
S=k[v_{1},\ldots,v_{m}].$

In the context of Lie group actions and equivariant cohomology, these
operations were introduced by Goresky, Kottwitz and MacPherson as obstructions
to equivariant formality in [18]; see also Remarks 4.5 and 4.6 for other
contexts in which analogous operations arise. For any monomial
$s=v_{1}^{n_{1}}\cdots v_{m}^{n_{m}}$, the operation $\delta_{s}$ is of degree
$1-2\sum_{j=1}^{m}n_{j}$ and is well-defined as a map of the form

(13) $\bigcap_{t|s,t\neq
s}\operatorname{ker}\delta_{t}\longrightarrow\frac{H^{*}(N)}{\sum_{t|s,t\neq
s}\operatorname{im}\delta_{t}},$

see [18, Proposition 13.8]. In particular, the _primary operations_
$\delta_{v_{j}}$ are the degree $-1$ maps

$\displaystyle\delta_{v_{j}}\colon H^{*}(N)$ $\displaystyle\longrightarrow
H^{*-1}(N)$ $\displaystyle[\alpha]$
$\displaystyle\longmapsto[\iota_{j}\alpha]$

induced by the $\Lambda$-module structure on $N$.

If $[\alpha]\in H^{n}(N)$ and
$\delta_{v_{i}}[\alpha]=\delta_{v_{j}}[\alpha]=0$, then
$\iota_{i}\alpha=d\beta_{i}$ and $\iota_{j}\alpha=d\beta_{j}$ for some
$\beta_{i},\beta_{j}\in N^{n-2}$. Note that

$\displaystyle d(\iota_{i}\beta_{j}+\iota_{j}\beta_{i})$
$\displaystyle=-\iota_{i}d\beta_{j}-\iota_{j}d\beta_{i}$
$\displaystyle=-\iota_{i}\iota_{j}\alpha-\iota_{j}\iota_{i}\alpha$
$\displaystyle=0$

since $\iota_{i}\iota_{j}+\iota_{j}\iota_{i}=0$. In this case, the _secondary
operation_ $\delta_{v_{i}v_{j}}$, of degree $-3$, is defined by

(14) $\delta_{v_{i}v_{j}}[\alpha]=[\iota_{i}\beta_{j}+\iota_{j}\beta_{i}].$

Assuming the primary operations $\delta_{v_{i}}$, $\delta_{v_{j}}$,
$\delta_{v_{k}}$ and secondary operations $\delta_{v_{i}v_{j}}$,
$\delta_{v_{i}v_{k}}$, $\delta_{v_{j}v_{k}}$ act trivially on $H^{*}(N)$, the
_tertiary operation_ $\delta_{v_{i}v_{j}v_{k}}\colon H^{*}(N)\to H^{*-5}(N)$
is defined by

$\delta_{v_{i}v_{j}v_{k}}[\alpha]=[\iota_{i}\beta_{jk}+\iota_{j}\beta_{ik}+\iota_{k}\beta_{ij}],$

where

$\begin{aligned} d\beta_{ij}&=\iota_{i}\beta_{j}+\iota_{j}\beta_{i}\\\
d\beta_{ik}&=\iota_{i}\beta_{k}+\iota_{k}\beta_{i}\\\
d\beta_{jk}&=\iota_{j}\beta_{k}+\iota_{k}\beta_{j}\end{aligned}\qquad\text{
and }\qquad\begin{aligned} d\beta_{i}&=\iota_{i}\alpha\\\
d\beta_{j}&=\iota_{j}\alpha\\\ d\beta_{k}&=\iota_{k}\alpha\end{aligned}$

for some $\beta_{i},\beta_{j},\beta_{k}\in N^{n-2}$ and
$\beta_{ij},\beta_{ik},\beta_{jk}\in N^{n-4}$.

Similar cochain-level formulas can be given for all higher operations:
assuming representatives $\delta_{t}(\alpha)\in N$ of $\delta_{t}[\alpha]$
have been defined for all monomials $t$ dividing $s=v_{i_{1}}\cdots v_{i_{q}}$
($i_{1},\ldots,i_{q}$ not necessarily distinct) with $|t|<|s|$, and assuming
$\delta_{t}$ is trivial on $H^{*}(N)$ for all such $t$, then
$\delta_{s}[\alpha]$ is represented by

(15)
$\sum_{\ell=1}^{q}\iota_{i_{\ell}}d^{-1}\delta_{s/v_{i_{\ell}}}(\alpha)\in
N^{n-2q+1},$

where $d^{-1}\delta_{s/v_{i_{\ell}}}(\alpha)$ denotes a choice of preimage of
$\delta_{s/v_{i_{\ell}}}(\alpha)$ satisfying a certain system of equations
analogous to those above.

We will frequently make use of the fact that $\delta_{s}$ is a well-defined
map $H^{*}(N)\to H^{*-|s|+1}(N)$ with no indeterminacy when the previous
operations $\delta_{t}$ with $t|s$ all vanish. This follows from [18,
Proposition 13.8], cited above, which in turn follows from an identification
of the higher operations with differentials in a spectral sequence associated
to the Koszul dual dg $S$-module $S\otimes N$ with differential $1_{S}\otimes
d+\sum_{j=1}^{m}v_{j}\otimes\iota_{j}$. (For cohomology operations on
$H^{*}(\mathcal{Z}_{K})$, this fact will also follow from the proof of Lemma
6.6, which identifies the higher operations with differentials in a different
spectral sequence.) We include here a more direct proof for the special case
of a secondary operation, which we will need in Section 6.

######  Lemma 3.8.

Let $(N,d)$ be a differential graded
$\Lambda(\iota_{1},\ldots,\iota_{m})$-module. Suppose the primary operations
$\delta_{v_{i}}$ and $\delta_{v_{j}}$ vanish on $H^{*}(N)$ for some
$i,j\in[m]$. Then the secondary operation (14) is a well-defined map

$\delta_{v_{i}v_{j}}\colon H^{*}(N)\longrightarrow H^{*-3}(N).$

###### Proof.

Let $[\alpha]\in H^{n}(N)$. Since $\delta_{v_{i}}=\delta_{v_{j}}=0$, the
secondary operation $\delta_{v_{i}v_{j}}[\alpha]$ is represented by
$\iota_{i}\beta_{j}+\iota_{j}\beta_{i}$ where $\iota_{i}\alpha=d\beta_{i}$ and
$\iota_{j}\alpha=d\beta_{j}$. Suppose
$\beta_{i}^{\prime},\beta_{j}^{\prime}\in N^{n-2}$ also satisfy the equations
$\iota_{i}\alpha=d\beta_{i}^{\prime}$ and
$\iota_{j}\alpha=d\beta_{j}^{\prime}$. Then because
$\beta_{i}-\beta_{i}^{\prime}$ and $\beta_{j}-\beta_{j}^{\prime}$ are closed,
there must exist $\omega,\omega^{\prime}\in N^{n-4}$ with
$\iota_{i}(\beta_{j}-\beta_{j}^{\prime})=d\omega$ and
$\iota_{j}(\beta_{i}-\beta_{i}^{\prime})=d\omega^{\prime}$ since
$\delta_{v_{i}}[\beta_{j}-\beta_{j}^{\prime}]=\delta_{v_{j}}[\beta_{i}-\beta_{i}^{\prime}]=0$.
Therefore

$(\iota_{i}\beta_{j}+\iota_{j}\beta_{i})-(\iota_{i}\beta_{j}^{\prime}+\iota_{j}\beta_{i}^{\prime})=\iota_{i}(\beta_{j}-\beta_{j}^{\prime})+\iota_{j}(\beta_{i}-\beta_{i}^{\prime})=d(\omega+\omega^{\prime}),$

from which it follows that
$[\iota_{i}\beta_{j}+\iota_{j}\beta_{i}]=[\iota_{i}\beta_{j}^{\prime}+\iota_{j}\beta_{i}^{\prime}]\in
H^{n-3}(N)$ and $\delta_{v_{i}v_{j}}$ is well-defined. ∎

######  Remark 3.9.

Since the generators of $\Lambda$ act by derivations on the cochains of a
moment-angle complex $\mathcal{Z}_{K}$, it can be shown that
$\delta_{v_{i}v_{j}}$ is in fact a well-defined derivation of the cohomology
ring $H^{*}(\mathcal{Z}_{K})$ when the primary operations $\delta_{v_{i}}$ and
$\delta_{v_{j}}$ are trivial (although we will not need this). Analogous
computations establishing the well-definedness of tertiary and higher
operations when lower degree operations vanish are similarly straightforward
but tedious.

Taking $N$ to be $\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K})$, we next
describe how the higher operations interact with the multigrading on
$H^{*}(\mathcal{Z}_{K})$. Below we identify $H^{*}(\mathcal{Z}_{K})$ with
$\bigoplus_{J\subseteq[m]}\widetilde{H}^{*}(K_{J})$ via the equivalences of
Corollary 3.7 and denote the multidegree $J$ part of $H^{*}(\mathcal{Z}_{K})$
by $\widetilde{H}^{*}(K_{J})$.

######  Lemma 3.10.

Let $s=v_{i_{1}}\cdots v_{i_{q}}\in S$ be a monomial. Suppose the operation
$\delta_{t}$ acts trivially on
$H^{*}(\mathcal{Z}_{K})\cong\bigoplus_{J\subseteq[m]}\widetilde{H}^{*}(K_{J})$
for all $t|s$ with $|t|<|s|$, and consider the operation

$\delta_{s}\colon H^{*}(\mathcal{Z}_{K})\longrightarrow
H^{*-2q+1}(\mathcal{Z}_{K}).$

If $s$ is squarefree, then

$\delta_{s}\bigl{(}\widetilde{H}^{p}(K_{J})\bigr{)}\subseteq\widetilde{H}^{p-q+1}(K_{J\smallsetminus
i_{1}\cdots i_{q}})$

for all $J\subseteq[m]$ containing $i_{1},\ldots,i_{q}$. Otherwise,
$\delta_{s}=0$.

###### Proof.

The claim holds for primary operations $\delta_{v_{j}}=\iota_{j}$ by Lemma
3.5. Let $s=v_{i_{1}}\cdots v_{i_{q}}$ be a squarefree monomial and assume
inductively that for any $[\alpha]\in\widetilde{H}^{p}(K_{J})$ and
$1\leqslant\ell\leqslant q$, a representative
$\delta_{s/v_{i_{\ell}}}(\alpha)$ of the cohomology class
$\delta_{s/v_{i_{\ell}}}[\alpha]$ (as in (15)) can be chosen to lie in
$\widetilde{C}^{p-q+2}(K_{J\smallsetminus i_{1}\cdots\widehat{i_{\ell}}\cdots
i_{q}})$. Then for any $\alpha\in\widetilde{H}^{p}(K_{J})$,
$\delta_{s}[\alpha]$ is represented by a cochain of the form

$\sum_{\ell=1}^{q}\iota_{i_{\ell}}d^{-1}\delta_{s/v_{i_{\ell}}}(\alpha)\in\bigoplus_{J\subseteq[m]}\widetilde{C}^{*}(K_{J}),$

where each preimage $d^{-1}\delta_{s/v_{i_{\ell}}}(\alpha)$ can be chosen to
lie in $\widetilde{C}^{p-q+1}(K_{J\smallsetminus
i_{1}\cdots\widehat{i_{\ell}}\cdots i_{q}})$. Therefore each term
$\iota_{i_{\ell}}d^{-1}\delta_{s/v_{i_{\ell}}}(\alpha)$ is of multidegree
$J\smallsetminus i_{1}\cdots i_{q}$, and
$\delta_{s}[\alpha]\in\widetilde{H}^{p-q+1}(K_{J\smallsetminus i_{1}\cdots
i_{q}})$, as desired. Finally, if $s$ is not squarefree, the triviality of
$\delta_{s}$ follows from the fact that $\iota_{i_{\ell}}$ acts trivially on
$\widetilde{C}^{*}(K_{U})$ when $j\notin U$. ∎

By Lemma 3.10, among the higher cohomology operations induced by the standard
$T^{m}$-action on $\mathcal{Z}_{K}$, we may restrict attention to those
indexed by squarefree monomials in $S=k[v_{1},\ldots,v_{m}]$. (This is in
contrast to the case of a general $T^{m}$-space; see Examples 5.11 and 5.12).
Identifying squarefree monomials $s=v_{I}=v_{i_{1}}\\!\cdots v_{i_{q}}$ with
subsets $I=\\{i_{1},\ldots,i_{q}\\}\subseteq[m]$, we use the notation
$\delta_{I}$ or $\delta_{i_{1}\\!\cdots i_{q}}$ to denote $\delta_{s}$.

In the next section we take a different approach to the higher cohomology
operations for moment-angle complexes by rigidifying them into well-defined
endomorphisms of $H^{*}(\mathcal{Z}_{K})$. These operations can then be used
to describe explicit Hirsch–Brown models for these toric spaces, as has been
done in a different context in [8].

##  4\. Hirsch–Brown models from perturbation theory

In this section we explain how the minimal free resolution of the
Stanley–Reisner ring can be constructed from the reduced Koszul complex using
the homological perturbation lemma. As a consequence we obtain direct formulae
for the cohomology operations $\delta_{I}$ from Section 3.4, and we obtain an
explicit Hirsch–Brown model for the action of any subtorus $H\subseteq T^{m}$
on $\mathcal{Z}_{K}$.

We fix, as usual, a simplicial complex $K$ on a vertex set $[m]$, and we
consider the reduced Koszul complex $R(K)$ with its dg $\Lambda$-module
structure as in Section 3.2. We need to introduce some notation to be used in
the next lemma. The element

$\rho={\sum\limits_{i\in[m]}}v_{i}\otimes\iota_{i}\in S\otimes\Lambda$

defines by left multiplication a homological degree $-1$, multigraded
endomorphism

(16) $\rho\colon S\otimes R(K)\longrightarrow S\otimes R(K).$

A direct computation shows that $(S\otimes d+\rho)^{2}=0$, where $d$ is the
differential on $R(K)$. In other words, $\rho$ defines a perturbation of the
complex $S\otimes R(K)$.

We recall that $R(K)$ admits a basis of monomials $v_{V}u_{U}$ indexed by
disjoint subsets $U,V\subseteq[m]$ with $V\in K$, and we use this to define a
linear map

(17) $\theta\colon R(K)\to k[K],\quad v_{V}\mapsto v_{V}\ \text{ and }\
v_{V}u_{U}\mapsto 0\,\text{ if }\,U\neq\varnothing.$

In the next lemma $\theta_{S}\colon S\otimes R(K)\to k[K]$ is the unique
$S$-linear extension of $\theta$.

######  Lemma 4.1.

If $S\otimes R(K)$ is equipped with the perturbed differential $S\otimes
d+\rho$, then the map $\theta_{S}\colon S\otimes R(K)\to k[K]$ is a quasi-
isomorphism of dg $S$-modules.

###### Proof.

We use the quasi-isomorphism

$\theta^{\prime}\colon R(K)\xhookrightarrow{\ \simeq\
}\bigl{(}\Lambda(u_{1},\ldots,u_{m})\otimes k[K],d\bigr{)}$

of dg $\Lambda$-modules from (8) above. From this we obtain a chain map

$\theta^{\prime}_{S}\colon\big{(}S\otimes R(K),S\otimes
d+\rho\big{)}\to\big{(}S\otimes\Lambda(u_{1},\ldots,u_{m})\otimes
k[K],S\otimes d+\rho\big{)},$

giving both sides the differential perturbed by $\rho$. Since
$\theta^{\prime}$ is a quasi-isomorphism and
$\theta^{\prime}=k\otimes_{S}\theta^{\prime}_{S}$, it follows that
$\theta^{\prime}_{S}$ is a quasi-isomorphism by the (derived) Nakayama Lemma.

We also make use of the surjective chain map

$\theta^{\prime\prime}_{S}\colon\big{(}S\otimes\Lambda(u_{1},\ldots,u_{m})\otimes
k[K],S\otimes d+\rho\big{)}\to k[K],\quad f\otimes 1\otimes g\mapsto
fg,\,f\otimes u_{U}\otimes g\mapsto 0\text{ if }U\neq\varnothing.$

It is well known that

$\theta^{\prime\prime}_{S}\otimes_{k[K]}k\colon\big{(}S\otimes\Lambda(u_{1},\ldots,u_{m}),\rho\big{)}\to
k$

is a quasi-isomorphism, as the left-hand side is the standard Koszul complex
on the maximal homogeneous ideal of $S$. It follows that
$\theta^{\prime\prime}_{S}$ is a quasi-isomorphism, again by the (derived)
Nakayama Lemma. Finally, $\theta_{S}$ factors as

$S\otimes
R(K)\xrightarrow{\theta^{\prime}_{S}}S\otimes\Lambda(u_{1},\ldots,u_{m})\otimes
k[K]\xrightarrow{\theta^{\prime\prime}_{S}}k[K].$

From this we see that $\theta_{S}$ is a chain map and that it is a quasi-
isomorphism. ∎

######  Remark 4.2.

The perturbed differentials appearing above can be interpreted within the
framework of _twisted tensor products_ , using the _twisting cochain_
$\tau\colon\Lambda(u_{1},\ldots,u_{m})\to S$, with $u_{i}\mapsto v_{i}$ and
$u_{U}\mapsto 0$ if $|U|\neq 1$; consult [27] for more on this theory. Even
more classically, the perturbed (or twisted) tensor products $S\otimes-$ and
$\Lambda(u_{1},\ldots,u_{m})\otimes-$ that we have used are the well-known BGG
functors between dg $\Lambda$-modules and dg $S$-modules, realising the famous
Bernstein–Gel’fand–Gel’fand correspondence [4].

Recall from (7) that the reduced Koszul complex satisfies
$H(R(K))\cong\operatorname{Tor}^{S}(k[K],k)$. In order to construct cohomology
operations explicitly as well-defined endomorphisms of
$\operatorname{Tor}^{S}(k[K],k)$, we fix data that realises this isomorphism
at the chain level.

######  Definition 4.3.

A _multigraded deformation retraction_ for $R(K)$ is a diagram of maps,
homogeneous with respect to multidegree,

(18)
${\operatorname{Tor}^{S}(k[K],k)}$${R(K)}$$\scriptstyle{\sigma}$$\scriptstyle{\pi}$$\scriptstyle{h}$

such that $\pi$ and $\sigma$ are degree zero chain maps satisfying
$\pi\sigma=1$, and $h$ is a (homological) degree $1$ homotopy with
$dh+hd=1-\sigma\pi$.

One can readily construct multigraded deformation retraction data: Within
$R(K)$ denote $Z=\ker(d)$ and $B={\rm im}(d)$, and make the identification
$\operatorname{Tor}^{S}(k[K],k)=Z/B$. Next choose a splitting $s_{1}\colon
R(K)/Z\to R(K)$ of the surjection $p_{1}\colon R(K)\to R(K)/Z$, and a
splitting $s_{2}\colon Z/B\to Z$ of the surjection $p_{2}\colon Z\to Z/B$,
both as $\mathbb{Z}\times\mathbb{Z}^{m}$ graded vector spaces. Then we may
take $\sigma=s_{2}$ and $\pi=p_{2}(1-s_{1}p_{1})$, and since $ds_{1}\colon
R(K)/Z\to B$ is an isomorphism we may take $h=s_{1}(ds_{1})^{-1}$. These maps
all together satisfy the conditions of Definition 4.3.

######  Theorem 4.4.

Let $K$ be a simplicial complex on the vertex set $[m]$ and choose a
multigraded deformation retraction for the reduced Koszul complex $R(K)$, as
in (18). For each $I\subseteq[m]$ we define an operation on
$\operatorname{Tor}^{S}(k[K],k)$ having homological degree $-1$ and
multidegree $-I$ by the formula

$\partial_{I}=\sum_{I=\\{i_{1},...,i_{n}\\}}\pi\iota_{i_{1}}h\iota_{i_{2}}h\cdots
h\iota_{i_{n}}\sigma,$

where the sum is taken over all possible orderings of $I$ (therefore having
$|I|!$ summands). These operations are the coefficients of the differential in
the minimal (multigraded) free resolution of the Stanley–Reisner ring. In
other words, there is a quasi-isomorphism of dg $S$-modules

$\big{(}S\otimes\operatorname{Tor}^{S}(k[K],k),d\big{)}\xrightarrow{\ \simeq\
}k[K]\quad\text{ with }d=\sum_{I\subseteq[m]}v_{I}\otimes\partial_{I}.$

We have used the notation $\partial_{I}$ to distinguish these operations from
the operations $\delta_{I}$ defined in Section 3.4; however, we will show in
Proposition 4.9 that they are essentially the same.

###### Proof.

To begin with, we extend the chosen deformation retraction $S$-linearly:

${S\otimes\operatorname{Tor}^{S}(k[K],k)}$${S\otimes
R(K)}$$\scriptstyle{\sigma_{S}}$$\scriptstyle{\pi_{S}}$$\scriptstyle{h_{S}}$

where $\sigma_{S}=S\otimes\iota$, $\pi_{S}=S\otimes\pi$, $h_{S}=S\otimes h$,
and the complex $S\otimes R(K)$ has the differential $S\otimes d$. This is
again a deformation retraction.

We make use of the homological perturbation lemma, whose history goes back at
least to [21]; a convenient reference is [10]. Using the perturbation (16) and
[10, 2.4] we obtain the perturbed deformation retraction

${\big{(}S\otimes\operatorname{Tor}^{S}(k[K],k),d^{\prime}\big{)}}$${\big{(}S\otimes
R(K),S\otimes
d+\rho\big{)}}$$\scriptstyle{\sigma_{S}^{\prime}}$$\scriptstyle{\pi_{S}^{\prime}}$$\scriptstyle{h_{S}^{\prime}}$

where

${d^{\prime}=\sum_{n\geqslant
0}\pi_{S}\rho(h_{S}\rho)^{n}\sigma_{S},}$${\pi_{S}^{\prime}=\pi_{S}+\sum_{n\geqslant
0}\pi_{S}\rho(h_{S}\rho)^{n}h_{S},}$${\sigma_{S}^{\prime}=\sigma_{S}+\sum_{i\geqslant
0}h_{S}\rho(h_{S}\rho)^{n}\sigma_{S},}$${h_{S}^{\prime}=h_{S}+\sum_{n\geqslant
0}h_{S}\rho(h_{S}\rho)^{n}h_{S}.}$

In the given formula for $d^{\prime}$, each of $\pi_{S},\sigma_{S}$ and
$h_{S}$ are $S$-linear extensions of maps of vector spaces, and therefore all
non-constant $S$-coefficients in $d^{\prime}$ arise from
$\rho=\sum_{i\in[m]}v_{i}\otimes\iota_{i}$:

$d^{\prime}=\sum_{n\geqslant 0}\sum_{i_{1},\ldots,i_{n}\in[m]}(v_{i_{1}}\cdots
v_{i_{n}})\pi_{S}\iota_{i_{1}}h_{S}\iota_{i_{2}}h_{S}\cdots
h_{S}\iota_{i_{n}}\sigma_{S}.$

Since $R(K)$ is nonzero only in squarefree multidegree, and since $h_{S}$ has
multidegree zero while each $\iota_{i}$ has multidegree
$(0,\ldots,-1,\ldots,0)$, the summands in the above series can only be nonzero
when $i_{1},\ldots,i_{n}$ are distinct. In other words, the summands are
indexed by subsets $I=\\{i_{1},\ldots,i_{n}\\}$, with each ordering
contributing a different summand. This yields the expression
$d^{\prime}=\sum_{I\subseteq[m]}v_{I}\otimes\partial_{I}$.

Finally, using $\theta_{S}$ from (17), the natural projection

$\theta_{S}\sigma_{S}^{\prime}=\theta_{S}\sigma_{S}\colon\big{(}S\otimes\operatorname{Tor}^{S}(k[K],k),d^{\prime}\big{)}\longrightarrow
k[K]$

is a quasi-isomorphism by Lemma 4.1. ∎

######  Remark 4.5.

The system of cohomology operations on $\operatorname{Tor}^{S}(k[K],k)$
constructed in Theorem 4.4 contains exactly the data needed to recover the
derived tensor product $k[K]\otimes^{\rm L}_{S}k$ up to a quasi-isomorphism of
dg $\Lambda$-modules. In this sense, the cohomology operations play a similar
role to A∞-module structures. In fact, the operations $\\{\partial_{I}\\}$ can
be obtained by symmetrising the higher multiplications $\\{m_{n}\\}$ in a
choice of A∞-$\Lambda$-module structure on $\operatorname{Tor}^{S}(k[K],k)$.
Continuing along these lines, it is possible to consider the data
$\\{\partial_{I}\\}$ as a kind of $\infty$-$\Lambda$-module structure with
respect to the twisting cochain $\tau\colon S^{\vee}\to\Lambda$; cf. [27].

######  Remark 4.6.

The same operations are also closely related to the systems of higher
homotopies defined by Eisenbud, on resolutions of modules over local complete
intersection rings [12]. More precisely, if $A$ is a local ring with residue
field $k$, and $B=A/(f_{1},\ldots,f_{m})$ is the quotient by a regular
sequence, then for any $B$-module $M$, there is a natural
$\Lambda(x_{1},\ldots,x_{m})$-module structure on
$\operatorname{Tor}^{A}(M,k)$, with $x_{i}$ having homological degree $1$. A
system of higher homotopies on the minimal $A$-free resolution of $M$ (as in
[12]) induces a system of cohomology operators on
$\operatorname{Tor}^{A}(M,k)$ extending its
$\Lambda(x_{1},\ldots,x_{m})$-module structure, analogous to those in Theorem
4.4 but having different degrees.

These cohomology operations can be interpreted in terms of the equivariant
topology of the moment-angle complex $\mathcal{Z}_{K}$. This time we choose a
multigraded deformation retraction for the cellular cochain algebra

(19)
${H^{*}(\mathcal{Z}_{K};k)}$${\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K};k)}$$\scriptstyle{\sigma}$$\scriptstyle{\pi}$$\scriptstyle{h}$

satisfying the same conditions as in (18). Using the isomorphism
$R(K)\cong\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K};k)$ of dg
$\Lambda$-modules from Lemma 3.3, the statement of Theorem 4.4 translates to
the corollary below. We first recall a definition from rational homotopy
theory and the theory of transformation groups (generalised slightly to allow
for field coefficients other than $k=\mathbb{Q}$).

For a $T$-space $X$, although the $S=H^{*}(BT;k)$-module structure on
$H^{*}_{T}(X;k)=H^{*}(ET\times_{T}X;k)$ cannot in general be lifted to an
action of $S$ on $C^{*}(ET\times_{T}X;k)$, the existence of a dg algebra
quasi-isomorphism $f\colon C^{*}(BT;k)\to H^{*}(BT;k)$ identifies the derived
categories of dg $C^{*}(BT;k)$-modules and dg $H^{*}(BT;k)$-modules.
Explicitly, the functor

(20) $-\otimes_{C^{*}(BT^{m})}^{\rm L}H^{*}(BT)\colon
D(C^{*}(BT))\longrightarrow D(H^{*}(BT))$

is an equivalence of categories with quasi-inverse given by the restriction of
scalars along $f$. Below, we identify $C^{*}(ET\times_{T}X;k)$ with its image
under the equivalence above.

######  Definition 4.7.

The _minimal Hirsch–Brown model_ of the action of a torus $T$ on a space $X$
is the dg $S$-module minimal model of $C^{*}(ET\times_{T}X;k)$, where
$S=H^{*}(BT;k)$.

The minimal Hirsch–Brown model is therefore a semi-free dg $S$-module
$(S\otimes V,d)$ that satisfies the minimality condition
$\operatorname{im}(d)\subseteq S^{>0}\otimes V$ and is quasi-isomorphic to
(the image under (20) of) $C^{*}(ET\times_{T}X;k)$ in the derived category of
dg $S$-modules. When $k=\mathbb{Q}$, the minimal Hirsch–Brown model can be
defined as the dg $S$-module minimal model of a Sullivan model for the Borel
construction $ET\times_{T}X$. A convenient reference for the existence,
uniqueness and properties of these models is [2, Appendix A].

######  Corollary 4.8.

Let $K$ be a simplicial complex on the vertex set $[m]$ and choose a
multigraded deformation retraction for
$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K};k)$, as in (19). The cohomology
operations on $H^{*}(\mathcal{Z}_{K};k)$ defined by

$\partial_{I}=\sum_{I=\\{i_{1},...,i_{n}\\}}\pi\iota_{i_{1}}h\iota_{i_{2}}h\cdots
h\iota_{i_{n}}\sigma\quad\text{ for }I\subseteq[m]$

assemble to yield the minimal Hirsch–Brown model of the action of $T^{m}$ on
$\mathcal{Z}_{K}$:

$\big{(}S\otimes H^{*}(\mathcal{Z}_{K};k),d\big{)}\xrightarrow{\ \simeq\
}H^{*}_{T^{m}}(\mathcal{Z}_{K};k)\quad\text{ with
}d=\sum_{I\subseteq[m]}v_{I}\otimes\partial_{I}.$

###### Proof.

By Theorem 4.4, $(S\otimes H^{*}(\mathcal{Z}_{K};k),d)$ is quasi-isomorphic to
$H^{*}_{T^{m}}(\mathcal{Z}_{K};k)\cong k[K]$ as a dg $S$-module. It remains to
show that $(S\otimes H^{*}(\mathcal{Z}_{K};k),d)$ is quasi-isomorphic to the
image of $C^{*}(ET^{m}\times_{T^{m}}\mathcal{Z}_{K};k)$ under (20), and for
this we use a well-known formality result for
$ET^{m}\times_{T^{m}}\mathcal{Z}_{K}$. By [7, Theorem 4.3.2], the Borel
fibration $ET^{m}\times_{T^{m}}\mathcal{Z}_{K}\to BT^{m}$ can be identified up
to homotopy with the natural inclusion $DJ_{K}\to BT^{m}$ of the
Davis–Januszkiewicz space111Here $DJ_{K}$ is defined analogously to (2) as a
polyhedral product corresponding to the pair of spaces
$(\mathbb{C}P^{\infty},\ast)$ instead of $(D^{2},S^{1})$. associated to $K$
into $BT^{m}=(\mathbb{C}P^{\infty})^{m}$. By [13] or [29], there is a dg
algebra quasi-isomorphism $C^{*}(DJ_{K};k)\to H^{*}(DJ_{K};k)$ which is
functorial in $K$. Since $DJ_{\Delta^{m-1}}=BT^{m}$, the inclusion
$K\hookrightarrow\Delta^{m-1}$ gives rise to a commutative diagram

${C^{*}(BT^{m};k)}$${C^{*}(DJ_{K};k)}$${H^{*}(BT^{m};k)}$${H^{*}(DJ_{K};k).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Since the quasi-isomorphism $f$ defining (20) can be taken to be the left
vertical map in the diagram above, this shows that the $S$-module
$H^{*}_{T^{m}}(\mathcal{Z}_{K};k)\cong H^{*}(DJ_{K};k)$ is mapped under the
quasi-inverse of (20) (restricting scalars along $f$) to a
$C^{*}(BT^{m};k)$-module that is quasi-isomorphic to
$C^{*}(ET^{m}\times_{T^{m}}\mathcal{Z}_{K};k)\cong C^{*}(DJ_{K};k)$. We now
know that $(S\otimes H^{*}(\mathcal{Z}_{K};k),d)$ is quasi-isomorphic as a dg
$S$-module to $H^{*}_{T^{m}}(\mathcal{Z}_{K};k)$, which in turn is quasi-
isomorphic to the image under (20) of
$C^{*}(ET^{m}\times_{T^{m}}\mathcal{Z}_{K};k)$, and this completes the proof.
∎

The higher operations $\delta_{I}$ were defined in Section 3.4 by Massey-type
formulae, up to the indeterminacy explained in (13). The next result shows
that these are essentially equal to the operations $\partial_{I}$ defined in
this section. In particular, the indeterminacy of the $\delta_{I}$ can be
removed by fixing a deformation retraction for
$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K};k)$ as in (19), after which
they assemble to yield the minimal Hirsch–Brown model for the $T^{m}$-action
on $\mathcal{Z}_{K}$.

######  Proposition 4.9.

The operations $\partial_{I}$ provide representatives for the higher
operations $\delta_{I}$ where they are defined. More precisely, for an index
$I\subseteq[m]$ and an element $\alpha\in\operatorname{Tor}^{S}(k[K],k)$, if
$\delta_{J}(\alpha)=0$ for all $J\subsetneq I$, then
$\delta_{I}(\alpha)=\partial_{I}(\alpha)$ modulo $\sum_{J\subsetneq
I}\operatorname{im}(\delta_{J})$.

###### Proof.

If $I=\\{i\\}$ then $\partial_{i}=\delta_{i}=\iota_{i}$ by definition, and we
proceed inductively. Suppose that $\delta_{J}(\alpha)$ is defined and zero for
all $J\subsetneq I$. For each $i\in I$ we can assume by induction that

$\alpha_{I\smallsetminus i}=\sum_{I\smallsetminus
i=\\{i_{1},...,i_{n}\\}}\iota_{i_{1}}h\iota_{i_{2}}h\cdots
h\iota_{i_{n}}\sigma(\alpha)$

is a cycle representing $\delta_{I\smallsetminus i}(\alpha)$. Since we are
assuming $\delta_{I\smallsetminus i}(\alpha)=0$, each $\alpha_{I\smallsetminus
i}$ is a boundary. Then we choose preimages $d^{-1}\alpha_{I\smallsetminus i}$
under the differential, and by definition of the higher operations set
$\delta_{I}(\alpha)=[\sum_{i\in I}\iota_{i}d^{-1}\alpha_{I\smallsetminus i}]$.
We are free to use $h\alpha_{I\smallsetminus i}=d^{-1}\alpha_{I\smallsetminus
i}$ as our preimages, in which case

$\delta_{I}(\alpha)=\big{[}\sum_{i\in I}\iota_{i}h\alpha_{I\smallsetminus
i}\big{]}=\ \pi\\!\\!\\!\\!\sum_{\begin{subarray}{c}i\in I,\ I\smallsetminus
i=\\\
\\{i_{1},...,i_{n}\\}\end{subarray}}\iota_{i}h\iota_{i_{1}}h\iota_{i_{2}}h\cdots
h\iota_{i_{n}}\sigma(\alpha)=\sum_{I=\\{i,i_{1},...,i_{n}\\}}\\!\\!\\!\pi\iota_{i}h\iota_{i_{1}}h\iota_{i_{2}}h\cdots
h\iota_{i_{n}}\sigma(\alpha).$

This is the defining formula for $\partial_{I}(\alpha)$ given in Theorem 4.4,
so we are done. ∎

We finish this section by describing Hirsch–Brown models for the actions of
subtori $H\subseteq T^{m}$ (that is, compact connected abelian subgroups) on
moment-angle complexes.

Every one-parameter subgroup $S^{1}\to T^{m}$ of the $m$-torus is of the form
$t\mapsto(t^{a_{1}},\ldots,t^{a_{m}})$ for some
$a_{1},\ldots,a_{m}\in\mathbb{Z}$. Using this we assign to each subtorus
$H\subseteq T^{m}$ an ideal of $S$ that is generated by linear forms:

(21) $H\ \longmapsto\
\mathcal{J}_{H}=\bigg{(}{\textstyle\sum\limits_{i\in[m]}}b_{i}v_{i}\;:\;{\textstyle\sum\limits_{i\in[m]}}a_{i}b_{i}=0\text{
for all }S^{1}\to H,\ t\mapsto(t^{a_{1}},\ldots,t^{a_{m}})\bigg{)}.$

Recalling that the cohomology ring of the classifying space
$BT^{m}=(\mathbb{C}P^{\infty})^{m}$ is given by the polynomial ring
$H^{*}(BT^{m})=S=k[v_{1},\ldots,v_{m}]$ with generators in degree $2$, it is
straightforward to see that the map $H^{*}(BT^{m})\to H^{*}(BH)$ induced by
the inclusion $H\hookrightarrow T^{m}$ can be identified with the quotient map
$S\to S/{\mathcal{J}_{H}}$.

######  Theorem 4.10.

Let $K$ be a simplicial complex on the vertex set $[m]$ and let

$\big{(}S\otimes H^{*}(\mathcal{Z}_{K};k),d\big{)}\xrightarrow{\ \simeq\
}H^{*}_{T^{m}}(\mathcal{Z}_{K};k)$

be the minimal Hirsch–Brown model for the $T^{m}$-action on $\mathcal{Z}_{K}$.
Then for any subtorus $H\subseteq T^{m}$, taking the quotient by
$\mathcal{J}_{H}$ yields the minimal Hirsch–Brown model for the $H$-action on
$\mathcal{Z}_{K}$:

$\big{(}(S/\mathcal{J}_{H})\otimes
H^{*}(\mathcal{Z}_{K};k),d\big{)}\xrightarrow{\ \simeq\
}C^{*}(EH\times_{H}\mathcal{Z}_{K};k).$

###### Proof.

All cochain and cohomology groups are taken with coefficients in the field
$k$, which we suppress from the notation below. In the morphism of Borel
fibrations

${\mathcal{Z}_{K}}$${EH\times_{H}\mathcal{Z}_{K}}$${BH}$${\mathcal{Z}_{K}}$${ET^{m}\times_{T^{m}}\mathcal{Z}_{K}}$${BT^{m}}$

the right-hand square is a homotopy pullback. Taking cochains, the
Eilenberg–Moore model for the pullback yields a quasi-isomorphism

(22) $C^{*}(EH\times_{H}\mathcal{Z}_{K})\simeq
C^{*}(BH)\otimes_{C^{*}(BT^{m})}^{\rm
L}C^{*}(ET^{m}\times_{T^{m}}\mathcal{Z}_{K})$

in the homotopy category of dg $C^{*}(BH)$-modules. Since the given
Hirsch–Brown model is a semi-free dg $S$-module resolution of
$H^{*}_{T^{m}}(\mathcal{Z}_{K})$, taking the quotient by $\mathcal{J}_{H}$
computes the derived tensor product, giving a quasi-isomorphism of
$H^{*}(BH)=S/\mathcal{J}_{H}$-modules

(23) $\big{(}(S/\mathcal{J}_{H})\otimes H^{*}(\mathcal{Z}_{K}),d\big{)}\simeq
S/\mathcal{J}_{H}\otimes_{S}^{\rm L}H^{*}_{T^{m}}(\mathcal{Z}_{K}).$

We next construct a quasi-isomorphism between (22) and (23), implying the
collapse of the Eilenberg–Moore spectral sequence of the homotopy pullback
above. We identify $ET^{m}\times_{T^{m}}\mathcal{Z}_{K}$ with $DJ_{K}$ as was
done in the proof of Corollary 4.8. As reviewed in [15, Section 4.1], the
construction of the quasi-isomorphism $f\colon C^{*}(BT^{m})\to H^{*}(BT^{m})$
used in the proof of Corollary 4.8 depends on a choice of chain
representatives $c_{1},\ldots,c_{m}$ for a basis $x_{1},\ldots,x_{m}$ of
$H_{1}(T^{m})$ where each $c_{i}$ lies in the $i$th coordinate circle factor
of $T^{m}$. Given another basis $x^{\prime}_{1},\ldots,x^{\prime}_{m}\in
H_{1}(T^{m})$, write $x^{\prime}_{i}=\sum_{j}\alpha_{ij}x_{j}$ for each
$1\leqslant i\leqslant m$, and set $c^{\prime}_{i}=\sum_{j}\alpha_{ij}c_{j}$.
Then it follows from the construction in [15, Section 4.1] that the quasi-
isomorphism $f^{\prime}\colon C^{*}(BT^{m})\to H^{*}(BT^{m})$ corresponding to
the chains $c^{\prime}_{1},\ldots,c^{\prime}_{m}$ is equal to $f$, while any
other set of representatives $\tilde{c}_{1},\ldots,\tilde{c}_{m}$, with
$\tilde{c}_{i}$ homologous to $c^{\prime}_{i}$, leads to a quasi-isomorphism
$\tilde{f}$ which is homotopic to $f^{\prime}$ as a map of dg algebras by [15,
Proposition 4.4]. By fixing a decomposition into circles for each of the
factors of $T^{m}\cong H\times(T^{m}/H)$ and choosing a basis of
$H_{1}(T^{m})$ represented by chains $\tilde{c}_{i}$ that lie in the $i$th
circle factor for each $1\leqslant i\leqslant m$, we obtain a quasi-
isomorphism $\tilde{f}\colon C^{*}(BT^{m})\to H^{*}(BT^{m})$ that is natural
with respect to coordinatewise inclusions of subtori by [13, Theorem 5.3] and
is homotopic to $f$. We therefore have a diagram

${C^{*}(BH)}$${C^{*}(BT^{m})}$${C^{*}(DJ_{K})}$${H^{*}(BH)}$${H^{*}(BT^{m})}$${H^{*}(DJ_{K})}$$\scriptstyle{\simeq}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

where the right square strictly commutes, the left square commutes up to a dg
algebra homotopy, all vertical maps are quasi-isomorphisms of dg algebras and
the horizontal maps are induced by the evident inclusions. This diagram
induces the desired quasi-isomorphism between (22) and (23). Moreover, by
restricting scalars along the left vertical map, the derived tensor product
(23) becomes a $C^{*}(BH)$-module quasi-isomorphic to (22), so it follows that
$\big{(}(S/\mathcal{J}_{H})\otimes H^{*}(\mathcal{Z}_{K}),d\big{)}$ is the
minimal Hirsch–Brown model for the $H$-action on $\mathcal{Z}_{K}$. ∎

######  Remark 4.11.

A space $X$ with an $H$-action is said to be _MOD-formal_ (over $k$) if
$C^{*}(EH\times_{H}X;k)$ and $H^{*}_{H}(X;k)$ are isomorphic in the derived
category of dg $H^{*}(BH;k)$-modules; this condition was investigated by Amann
and Zoller in connection with the Toral Rank Conjecture [2]. The proof of
Corollary 4.8 shows that the $T^{m}$-action on $\mathcal{Z}_{K}$ is MOD-formal
(since $DJ_{K}\simeq ET^{m}\times_{T^{m}}\mathcal{Z}_{K}$ is a formal space).
However this property is not inherited by subtori; see [2, Example 7.1].

Given a subset $I\subseteq[m]$, we will write

(24) $T^{I}=\big{\\{}(t_{1},\ldots,t_{m})\,:\,t_{i}=1\text{ for }i\notin
I\big{\\}}\quad\text{and}\quad\mathcal{J}_{I}=\big{(}v_{i}\,:\,i\notin
I\big{)}$

for the _coordinate subtorus_ of $T^{m}$ specified by $I$, and the
corresponding ideal of $S$. The next result follows by combining Corollary 4.8
with Theorem 4.10.

######  Corollary 4.12.

Let $K$ be a simplicial complex on the vertex set $[m]$ and choose a
multigraded deformation retraction for
$\mathcal{C}^{*}_{\mathrm{cw}}(\mathcal{Z}_{K};k)$, resulting in cohomology
operations $\partial_{J}$ as in Corollary 4.8.

For any subset $I\subseteq[m]$, the minimal Hirsch–Brown model for the
$T^{I}$-action on $\mathcal{Z}_{K}$ is given by

$\big{(}(S/\mathcal{J}_{I})\otimes
H^{*}(\mathcal{Z}_{K};k),d\big{)}\xrightarrow{\ \simeq\
}C^{*}(ET^{I}\times_{T^{I}}\mathcal{Z}_{K};k)\quad\text{ with
}d=\sum_{J\subseteq I}v_{J}\otimes\partial_{J}.$

Small models for $H^{*}_{T^{I}}(\mathcal{Z}_{K})$ are described using Koszul
complexes in [30] (see also [28]). For our purposes, an advantage of the
Hirsch–Brown models above is that they make explicit the relationship between
the equivariant and ordinary cohomology of $\mathcal{Z}_{K}$. In particular,
the Hirsch–Brown model in Corollary 4.12 (together with Proposition 4.9) shows
that the freeness of $H^{*}_{T^{I}}(\mathcal{Z}_{K})$ over $H^{*}(BT^{I})$ is
equivalent to the vanishing of certain cohomology operations on
$H^{*}(\mathcal{Z}_{K})$ coming from the $T^{I}$-action. This will be used in
the next two sections to characterise equivariant formality for any (not
necessarily coordinate) subtorus action on $\mathcal{Z}_{K}$.

##  5\. Equivariant formality and higher operations

In the section we explain how equivariant formality of subtorus actions on a
moment angle complex can be understood from the minimal free resolution of the
corresponding Stanley-Reisner ring, and equivalently from the cohomology
operations $\delta_{I}$. The language of $\mathcal{J}$-closed ideals turns out
to be a convenient lens through which to view this relationship.

###  5.1. $\mathcal{J}$-closed modules

The class of $\mathcal{J}$-closed modules was introduced in the work of
Diethorn on free resolutions in local algebra, as a natural generalisation of
the notion of a weak complete intersection ideal [11]. As before, we write
$S=k[v_{1},\ldots,v_{m}]$, multigraded by $\mathbb{N}^{m}$.

######  Definition 5.1.

Let $\mathcal{J}\subseteq S$ be an ideal and let $M$ be a multigraded
$S$-module. Then $M$ is _$\mathcal{J}$ -closed_ if, in the minimal free
resolution

$\cdots\to F_{2}\xrightarrow{d_{2}}F_{1}\xrightarrow{d_{1}}F_{0}\to M\to 0,$

the differentials satisfy $d_{i}(F_{i})\subseteq\mathcal{J}F_{i-1}$ for all
$i$. A monomial ideal $\mathcal{I}\subseteq S$ is called _$\mathcal{J}$
-closed_ if $S/\mathcal{I}$ is a $\mathcal{J}$-closed module. (In [11] it is
assumed that $\mathcal{J}$ is generated by a regular sequence—this will be the
case in our main situation of interest, but we do not make that assumption
here.)

Let $\mathcal{I}(d_{i})$ denote the ideal generated by the entries of $d_{i}$,
when written as a matrix over $S$ (this is sometimes called the first Fitting
ideal of $d_{i}$, and it is independent of the matrix representing $d_{i}$).
If we also write $\mathcal{I}(M)=\sum_{i}\mathcal{I}(d_{i})$, then by
definition $M$ is $\mathcal{J}$-closed if and only if
$\mathcal{I}(M)\subseteq\mathcal{J}$.

######  Lemma 5.2.

Let $\mathcal{J}\subseteq S$ be an ideal (not necessarily multigraded) and let

$\mathcal{J}^{\prime}=\bigoplus_{a\in\mathbb{N}^{m}}\mathcal{J}\cap
S_{a}\subseteq S$

be the largest multigraded ideal contained in $\mathcal{J}$. Then a
multigraded $S$-module $M$ is $\mathcal{J}$-closed if and only if it is
$\mathcal{J}^{\prime}$-closed.

###### Proof.

Since $M$ is multigraded, it has a multigraded minimal resolution, and
therefore $\mathcal{I}(M)$ is a multigraded ideal. From this it follows that
$\mathcal{I}(M)\subseteq\mathcal{J}$ if and only if
$\mathcal{I}(M)\subseteq\mathcal{J}^{\prime}$. ∎

Equivariant formality is related to the $\mathcal{J}$-closed condition under
the correspondence (21).

######  Proposition 5.3.

Let $K$ be a simplicial complex on vertex set $[m]$ and let $H\subseteq T^{m}$
be a subtorus. Then the action of $H$ on $\mathcal{Z}_{K}$ is equivariantly
formal over $k$ if and only if the Stanley–Reisner ring $k[K]$ is
$\mathcal{J}_{H}$-closed as an $S$-module.

###### Proof.

Let $F$ be the minimal free resolution of $k[K]$ over $S$. By Theorem 4.10
there is an isomorphism $H^{*}(F/\mathcal{J}_{H}F)\cong
H^{*}_{T^{m}}(\mathcal{Z}_{K};k)$. Since $F/\mathcal{J}_{H}F$ is a minimal
complex of free $S/\mathcal{J}_{H}$-modules, the only way it can have free
homology is if its differentials vanish entirely. This happens exactly when
$d(F)\subseteq\mathcal{J}_{H}F$, that is, when $k[K]$ is
$\mathcal{J}_{H}$-closed. ∎

###  5.2. Reduction to coordinate subtori

In this section we show that, for the purpose of answering 1, it suffices to
study the coordinate subtorus actions on moment-angle complexes. For these
torus actions we give some characterisations of equivariant formality that
will be used and generalised in subsequent sections.

######  Definition 5.4.

Let $H\subseteq T^{m}$ be a subtorus. The _coordinate hull_ of $H$ is the
smallest coordinate subtorus of $T^{m}$ containing $H$,

$\operatorname{hull}(H)=\Big{\\{}(t_{1},\ldots,t_{m})\;:\;t_{i}=1\text{ if
}H\subseteq{\textstyle\prod\limits_{j<i}}S^{1}\times\\{1\\}\times{\textstyle\prod\limits_{j>i}}S^{1}\subseteq
T^{m}\Big{\\}}.$

In positive characteristic we will need to use the following slightly
different (possibly smaller) construction.

######  Definition 5.5.

Let $H\subseteq T^{m}$ be a subtorus. For any prime number $p$, the _$p$
-coordinate hull_ of $H$ is the coordinate subtorus of $T^{m}$ given by

$p\operatorname{-hull}(H)=\big{\\{}(t_{1},\ldots,t_{m})\;:\;t_{i}=1\text{ if
}p|a_{i}\text{ for any map }S^{1}\to H,\
t\mapsto(t^{a_{1}},\ldots,t^{a_{m}})\big{\\}}.$

######  Proposition 5.6.

Let $K$ be a simplicial complex on vertex set $[m]$ and let $H\subseteq T^{m}$
be a subtorus.

  1. (1)

If $k$ is a field of characteristic zero, then the action of $H$ on
$\mathcal{Z}_{K}$ is equivariantly formal over $k$ if and only if the action
of $\operatorname{hull}(H)$ is equivariantly formal over $k$.

  2. (2)

If $k$ is a field of characteristic $p>0$, then the action of $H$ on
$\mathcal{Z}_{K}$ is equivariantly formal over $k$ if and only if the action
of $p\operatorname{-hull}(G)$ is equivariantly formal over $k$.

###### Proof.

Taking one-parameter-subgroups, the inclusion $H\subseteq T^{m}$ induces an
inclusion of lattices

$L_{H}=\operatorname{Hom}(S^{1},H)\subseteq\operatorname{Hom}(S^{1},T^{m})=\mathbb{Z}^{m}.$

The smallest coordinate subspace of $\mathbb{Q}^{m}$ containing
$L_{H}\otimes_{\mathbb{Z}}\mathbb{Q}$ is exactly $L_{{\rm
hull}(H)}\otimes_{\mathbb{Z}}\mathbb{Q}$, and likewise the smallest coordinate
subspace of $\mathbb{F}_{p}^{m}$ containing
$L_{H}\otimes_{\mathbb{Z}}\mathbb{F}_{p}$ is exactly
$L_{p\operatorname{-hull}(H)}\otimes_{\mathbb{Z}}\mathbb{F}_{p}$.

The inclusion $L_{H}\subseteq\mathbb{Z}^{m}$ is dual to a surjection
$k^{m}\twoheadrightarrow\operatorname{Hom}(L_{H},k)$. This copy of $k^{m}$ can
be naturally identified with ${\rm span}_{k}\\{v_{1},\ldots,v_{m}\\}$, and we
define the subspace

$V_{H}=\ker\big{(}{\rm
span}_{k}\\{v_{1},\ldots,v_{m}\\}\twoheadrightarrow\operatorname{Hom}(L_{H},k)\big{)}.$

Unraveling all these definitions, the ideal $\mathcal{J}_{H}\subseteq S$
defined in (21) is the ideal generated by this subspace $V_{H}$. By duality,
when $k$ has characteristic zero the largest coordinate subspace contained in
$V_{H}$ is exactly $V_{\operatorname{hull}(H)}$, and likewise when $k$ has
characteristic $p$ the largest coordinate subspace contained in $V_{H}$ is
exactly $V_{p\operatorname{-hull}(H)}$. Combining these ingredients, the
statement now follows from Lemma 5.2 and Proposition 5.3. ∎

According to Proposition 5.6, the problem of determining which subtori
$H\subseteq T^{m}$ act equivariantly formally on $\mathcal{Z}_{K}$ can be
reduced to the case of the coordinate subtori $H=T^{I}$, where
$I\subseteq[m]$, as in (24). We therefore focus on these actions from now on.

######  Proposition 5.7.

Let $K$ be a simplicial complex on $[m]$ and let $I\subseteq[m]$. Then the
following conditions are equivalent:

  1. (a)

the $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal over $k$;

  2. (b)

the Stanley–Reisner ring $k[K]$ is $\mathcal{J}_{I}$-closed;

  3. (c)

the cohomology operations $\delta_{U}$ vanish on $H^{*}(\mathcal{Z}_{K};k)$
for all $U\subseteq I$.

###### Proof.

The equivalence of (a) and (b) is contained in Proposition 5.3. By Theorem 4.4
and Proposition 4.9, the minimal free resolution of $k[K]$ over $S$ is of the
form $F=S\otimes H^{*}(\mathcal{Z}_{K})$ with differential
$d=\sum_{U\subseteq[m]}v_{U}\otimes\delta_{U}$, where $v_{U}=\prod_{i\in
U}v_{i}$. This means $d(F)\subseteq\mathcal{J}_{I}F$ if and only if
$\delta_{U}=0$ whenever $v_{U}\notin\mathcal{J}_{I}$. Note that
$v_{U}\notin\mathcal{J}_{I}$ if and only if $U\subseteq I$, and altogether
this shows that (b) is equivalent to (c). (The equivalence of (a) and (c) also
follows immediately from Corollary 4.12 and Proposition 4.9.) ∎

######  Remark 5.8.

The equivalence of condition (a) above with the vanishing of an infinite
family of cohomology operations $\delta_{s}$, $s\in k[v_{i}:i\in I]$, is due
to Goresky, Kottwitz and MacPherson [18]. We remark that the proof of
Proposition 5.7 is quite different from the arguments in [18], and that in our
more restrictive setting the finite family of operations appearing in (c)
(indexed by squarefree monomials) form a complete set of obstructions for
equivariant formality.

Combining Proposition 5.7 with Lemma 3.5 (or Katthän’s combinatorial
description of the linear part of the minimal free resolution of $k[K]$ [24])
also yields the following characterisation of equivariant formality for
coordinate circle actions in terms of the homology of full subcomplexes of
$K$. It will be generalised to arbitrary coordinate subtori in Theorem 6.8.

######  Theorem 5.9.

Let $K$ be a simplicial complex on vertex set $[m]$ and let $j\in[m]$. Then
the following conditions are equivalent:

  1. (a)

the coordinate $S^{1}_{j}$-action on $\mathcal{Z}_{K}$ is equivariantly formal
over $k$;

  2. (b)

the derivation $\iota_{j}$ is trivial on $H^{\ast}(\mathcal{Z}_{K};k)$;

  3. (c)

$K_{J\smallsetminus j}\hookrightarrow K_{J}$ induces the trivial map on
$\widetilde{H}^{\ast}(\;;k)$ for all $J\subseteq[m]$ with $j\in J$.

######  Remark 5.10.

It follows that $H^{*}(\mathcal{Z}_{K})$ is a trivial
$\Lambda(\iota_{1},\ldots,\iota_{m})$-module if and only if the coordinate
$S^{1}_{j}$-action on $\mathcal{Z}_{K}$ is equivariantly formal for all
$j\in[m]$. This latter condition (the simultaneous equivariant formality of
every coordinate circle action) is considered in [30], where some
combinatorial characterisations are given in the case that $K$ is flag or
$1$-dimensional. (In the flag case, [30, Theorem 4.9] can readily be recovered
from the results of Section 5.3 below.)

We emphasise that the equivariant formality of a circle action on a space is
typically not equivalent to the vanishing of the primary cohomology operation
induced by the action as in Theorem 5.9; in general, the vanishing of an
infinite family of higher operations is also necessary (see [18, Section 13]).
In the case considered above, Lemma 3.10 automatically implies that
$\delta_{v_{j}^{n}}=0$ for all $n>1$, leaving the primary operation
$\delta_{v_{j}}=\iota_{j}$ as the only possible obstruction.

Below we give two examples of circle actions on manifolds inducing trivial
primary operations where equivariant formality is obstructed by associated
higher operations. The first example demonstrates that the vanishing of a
primary operation alone is not sufficient for the equivariant formality of
(non-coordinate) circle actions on moment-angle complexes.

######  Example 5.11.

Let $K=\partial\Delta^{1}$ and observe that $\mathcal{Z}_{K}=D^{2}\times
S^{1}\cup S^{1}\times D^{2}\cong S^{3}$, so the
$\Lambda(\iota_{1},\iota_{2})$-module structure on $H^{*}(\mathcal{Z}_{K})$ is
trivial for degree reasons. In this case, restricting the standard
$T^{2}$-action to the diagonal circle
$S^{1}_{\mathrm{diag}}=\\{(t,t)\\}\subseteq T^{2}$ yields the Hopf action on
$S^{3}$, and the primary operation induced by this circle action is
$\iota_{1}+\iota_{2}$. The fundamental class in $H^{3}(\mathcal{Z}_{K})$ is
represented by the cocycle $v_{1}u_{2}\in
R(K)=\bigl{(}k[v_{1},v_{2}]/(v_{1}v_{2})\otimes\Lambda(u_{1},u_{2})\bigr{)}/(v_{i}^{2},v_{i}u_{i})$.
Since there is no indeterminacy, the zig-zag

$v_{1}u_{2}\xmapsto{\mathmakebox[2.2em]{\iota_{1}+\iota_{2}}}v_{1}\xleftarrow{\mathmakebox[2.2em]{d}}\mapsfromchar
u_{1}\xmapsto{\mathmakebox[2.2em]{\iota_{1}+\iota_{2}}}1$

shows that the secondary operation $H^{3}(\mathcal{Z}_{K})\to
H^{0}(\mathcal{Z}_{K})$ induced by the $S^{1}_{\mathrm{diag}}$-action maps
$[v_{1}u_{2}]$ to $[1]$, obstructing equivariant formality.

More generally, for any $m\geqslant 2$, if $K=\partial\Delta^{m-1}$ on the
vertex set $[m]$, then the $m$-ary operation induced by the diagonal circle
action on $\mathcal{Z}_{K}\cong S^{2m-1}$ is given by the higher operation
$\delta_{[m]}$ and defines an isomorphism $H^{2m-1}(\mathcal{Z}_{K})\to
H^{0}(\mathcal{Z}_{K})$.

######  Example 5.12.

Consider the real solvable Lie algebra

$L=\langle W,X,Y,Z\;:\;[W,X]=X,\,[W,Y]=-Y,\,[X,Y]=Z\rangle.$

The corresponding simply connected solvable Lie group $G$ admits a lattice
$\Gamma$, and the de Rham cohomology of the solvmanifold $G/\Gamma$ is
computed by the Lie algebra cohomology of $L$ since the inclusion of left-
invariant differential forms $\Lambda
L^{\vee}=\Omega^{*}_{G}(G)\hookrightarrow\Omega^{*}_{\Gamma}(G)=\Omega^{*}(G/\Gamma)$
is a quasi-isomorphism by a theorem of Hattori [22]. Let $(\Lambda
L^{\vee},d)=(\Lambda(w,x,y,z),d)$ be the Chevalley–Eilenberg complex of $L$
with $|w|=|x|=|y|=|z|=1$ and differential determined by

$d(x)=wx,\quad d(y)=-wy,\quad d(z)=xy.$

It can be shown that $S^{1}$ acts smoothly on $G/\Gamma$ with fundamental
vector field $Z$ (see for example [5, Theorem 3.6]). The corresponding
cohomology operation is induced by the degree $-1$ derivation of the de Rham
complex given by interior multiplication by $Z$ (cf. [18, Section 10.5]). Let
$\iota_{Z}\colon\Lambda^{*}L^{\vee}\to\Lambda^{*-1}L^{\vee}$ denote the
derivation defined by interior multiplication by $Z$. Then a straightforward
computation with the dg $\Lambda(\iota_{Z})$-module $(\Lambda(w,x,y,z),d)$
shows that $H^{*}(G/\Gamma;\mathbb{R})$ together with all higher operations
induced by the $S^{1}$-action is given by:

${H^{4}\colon}$${\left[wxyz\right]}$${H^{3}\colon}$${\left[xyz\right]}$${H^{1}\colon}$${\left[w\right]}$${H^{0}\colon}$${\left[1\right]}$$\scriptstyle{-\delta_{Z^{2}}}$$\scriptstyle{\delta_{Z^{2}}}$

In particular, the primary operation $\delta_{Z}=\iota_{Z}$ acts trivially on
$H^{*}(G/\Gamma;\mathbb{R})$, but the $S^{1}$-action is not equivariantly
formal since the induced secondary operation $\delta_{Z^{2}}$ is nontrivial.

###  5.3. Equivariant formality in the flag case

For flag complexes we already have enough machinery to completely characterise
the subtori that act equivariantly formally on $\mathcal{Z}_{K}$ in terms of
the combinatorics of $K$. We find that in this case the answer does not depend
on the characteristic of the field involved.

A simplicial complex $K$ is called a _flag complex_ if every set of vertices
of $K$ which are pairwise connected by edges spans a simplex in $K$.
Equivalently, $K$ is flag if its corresponding Stanley–Reisner ideal is
quadratic.

The _join_ of two simplicial complexes $K$ and $L$ on disjoint vertex sets is
defined to be the simplicial complex

$K\ast L=\\{\sigma\cup\tau\;:\;\sigma\in K,\ \tau\in L\\}$

on the union of the vertex sets of $K$ and $L$. In the case that $L=\\{v\\}$
is a single vertex, the join $K\ast\\{v\\}$ is called the _cone_ over $K$.

######  Lemma 5.13.

Let $K$ be a flag complex on vertex set $[m]$ and let $v\in[m]$. If
$\\{i,v\\}\in K$ for every vertex $i\in[m]$, then $K=K_{[m]\smallsetminus
v}\ast\\{v\\}$.

###### Proof.

Since $K$ is flag, so is every full subcomplex of $K$. It follows that the
join of full subcomplexes $K_{[m]\smallsetminus v}\ast\\{v\\}$ is flag. By
definition, a flag complex is completely determined by its $1$-skeleton, so
the result follows from the fact that $K$ and $K_{[m]\smallsetminus
v}\ast\\{v\\}$ have the same $1$-skeleton by the assumption that $v$ is
connected by an edge to every vertex of $K$. ∎

######  Lemma 5.14.

Let $K$ be a flag complex on vertex set $[m]$ and let $v\in[m]$. Then the
following conditions are equivalent:

  1. (a)

$K_{J\smallsetminus v}\hookrightarrow K_{J}$ induces the trivial map on
$\widetilde{H}^{\ast}(\;;k)$ for all $J\subseteq[m]$ with $v\in J$;

  2. (b)

$K_{\\{i,j\\}}\ast\\{v\\}\subseteq K$ for every missing edge $\\{i,j\\}\notin
K$ with $v\notin\\{i,j\\}$.

###### Proof.

That condition (a) implies condition (b) is clear since for any missing edge
$\\{i,j\\}\notin K$, the inclusion $K_{\\{i,j\\}}\hookrightarrow
K_{\\{i,j,v\\}}$ is nontrivial in reduced cohomology unless the vertex $v$
cones over $K_{\\{i,j\\}}=\partial\Delta^{1}$.

Conversely, suppose $K_{J\smallsetminus v}\hookrightarrow K_{J}$ is nontrivial
on $\widetilde{H}^{\ast}(\;)$ for some $J\subseteq[m]$, $v\in J$. Then there
must exist a vertex $i\in J$ with $\\{i,v\\}\notin K$ since otherwise $K_{J}$
is a cone by Lemma 5.13 and hence contractible, contradicting the assumption.
Similarly, there must exist some $j\in J\smallsetminus v$ with
$\\{i,j\\}\notin K$ since otherwise $K_{J\smallsetminus v}$ is the cone
$K_{J\smallsetminus\\{v,i\\}}\ast\\{i\\}$ by Lemma 5.13. Now since
$\\{i,j\\}\notin K$ and $\\{i,v\\}\notin K$, we have
$K_{\\{i,j\\}}\ast\\{v\\}\not\subseteq K$. ∎

While condition (a) above apparently involves the coefficient ring $k$,
condition (b) does not. As we will see next, this will mean that in the flag
case the equivariant fomality of a coordinate torus action is independent of
$k$ as well. This is not true in general, see Section 5.5 for examples.

######  Theorem 5.15.

Let $K$ be a flag complex on vertex set $[m]$ and let $I\subseteq[m]$. The
coordinate $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal (over
$k$) if and only if $I\in K$ and $K_{\\{i,j\\}}\ast
K_{I\smallsetminus\\{i,j\\}}\subseteq K$ for every missing edge
$\\{i,j\\}\notin K$.

###### Proof.

Assume that the $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal.
Then $\delta_{U}=0$ for all $U\subseteq I$ by Proposition 5.7. In particular,
the secondary operation $\delta_{ij}$ is trivial on $H^{3}(\mathcal{Z}_{K})$
for all $i,j\in I$. Since $H^{3}(\mathcal{Z}_{K})\cong H^{3}(R(K))$ is spanned
by $\\{[v_{i}u_{j}]\,:\,\\{i,j\\}\notin K\\}$ and $\delta_{ij}[v_{i}u_{j}]=1$
for each missing edge $\\{i,j\\}\notin K$, it follows that $K_{I}$ has no
missing edges. Since $K_{I}$ is flag, this implies $K_{I}$ is a simplex, or
equivalently $I\in K$. Now to show that $K_{\\{i,j\\}}\ast
K_{I\smallsetminus\\{i,j\\}}\subseteq K$ for every $\\{i,j\\}\notin K$, it
suffices by flagness to show $K_{\\{i,j\\}}\ast\\{v\\}\subseteq K$ for all
$v\in I\smallsetminus\\{i,j\\}$. Since the primary operation
$\delta_{v}=\iota_{v}$ is trivial for all $v\in I$ by the assumption, this
follows from Theorem 5.9 and Lemma 5.14.

Conversely, suppose that $I\in K$ and $K_{\\{i,j\\}}\ast
K_{I\smallsetminus\\{i,j\\}}\subseteq K$ for every $\\{i,j\\}\notin K$. Assume
toward contradiction that the $T^{I}$-action on $\mathcal{Z}_{K}$ is not
equivariantly formal, and hence that $\delta_{U}\neq 0$ for some $U\subseteq
I$. Assuming without loss of generality that $U$ is a minimal such subset of
$I$, it follows from Lemma 3.10 that $\delta_{U}$ acts nontrivially on some
multidegree $J$ part of $H^{*}(\mathcal{Z}_{K})$ with $U\subseteq J$, defining
a nontrivial map

$\delta_{U}\colon\widetilde{H}^{p}(K_{J})\longrightarrow\widetilde{H}^{p-|U|+1}(K_{J\smallsetminus
U}).$

In particular, neither $K_{J}$ nor $K_{J\smallsetminus U}$ are contractible.
This implies that every $j\in J\smallsetminus U$ is contained in a missing
edge in $K_{J\smallsetminus U}$, since otherwise $K_{J\smallsetminus U}$ is a
cone by Lemma 5.13 and is thus contractible. But every $i\in U$ cones over
every missing edge in $J\smallsetminus U$ by the assumption. Therefore every
$i\in U$ is connected by an edge to every vertex $j\in J\smallsetminus U$.
Since every $i\in U$ is also connected by an edge to every other vertex in $U$
(as $U\subseteq I\in K$), it follows that $K_{J}$ is a cone by Lemma 5.13, a
contradiction. ∎

######  Remark 5.16.

For any simplicial complex $K$, the equivariant formality of the coordinate
$T^{I}$-action on $\mathcal{Z}_{K}$ implies that $I\in K$. As in the proof
above, this can be seen by observing that a missing face in $K_{I}$ would
imply that $1\in H^{0}(\mathcal{Z}_{K})$ is in the image of some $\delta_{U}$
with $U\subseteq I$. This also follows immediately from the well-known fact
that equivariantly formal actions have fixed points. (Note that the
$T^{I}$-action has fixed points precisely when $I\in K$ by definition (2) of
$\mathcal{Z}_{K}$).

An interesting consequence of Theorem 5.15 is that in the flag case the
equivariant formality of a subtorus action on $\mathcal{Z}_{K}$ is completely
determined by the action of primary and secondary cohomology operations on the
groups $H^{4}(\mathcal{Z}_{K})$ and $H^{3}(\mathcal{Z}_{K})$, respectively.

######  Corollary 5.17.

Let $K$ be a flag complex on vertex set $[m]$ and let $I\subseteq[m]$. If

$\delta_{v}\colon H^{4}(\mathcal{Z}_{K})\to H^{3}(\mathcal{Z}_{K})\quad\text{
and }\quad\delta_{ij}\colon H^{3}(\mathcal{Z}_{K})\to H^{0}(\mathcal{Z}_{K})$

are trivial for all $i,j,v\in I$, then $\delta_{U}$ vanishes everywhere for
all $U\subseteq I$. In particular, a coordinate $S^{1}_{v}$-action on
$\mathcal{Z}_{K}$ is equivariantly formal if and only if $\delta_{v}$ is
trivial on $H^{4}(\mathcal{Z}_{K})$.

###### Proof.

It is straightforward to check that a secondary operation $\delta_{ij}$ is
nontrivial on $H^{3}(\mathcal{Z}_{K})$ if and only if $\\{i,j\\}\notin K$.
Therefore if $\delta_{ij}$ is trivial on $H^{3}(\mathcal{Z}_{K})$ for all
$i,j\in I$, then $K_{I}$ is a simplex by flagness, so $I\in K$. If
additionally $\delta_{v}$ is trivial on $H^{4}(\mathcal{Z}_{K})$ for all $v\in
I$, then $K_{\\{i,j\\}}\hookrightarrow K_{\\{i,j,v\\}}$ is trivial in reduced
cohomology for every $\\{i,j\\}\notin K$ with $v\in I\smallsetminus\\{i,j\\}$
by Lemma 3.5. This implies $K_{\\{i,j\\}}\ast\\{v\\}\subseteq K$ for every
$\\{i,j\\}\notin K$ with $v\in I\smallsetminus\\{i,j\\}$. It follows that both
combinatorial conditions of Theorem 5.15 are satisfied, so the $T^{I}$-action
on $\mathcal{Z}_{K}$ is equivariantly formal. ∎

The equivariant formality of the action of any subtorus $H\subseteq T^{m}$ on
a moment-angle complex $\mathcal{Z}_{K}$ can be read off from the minimal free
resolution of the Stanley–Reisner ring $k[K]$ by Proposition 5.3. If $K$ is
flag, it follows from the above that the equivariant formality of the
$H$-action can be read off from the first two differentials in the resolution
alone.

###  5.4. $\mathcal{J}$-closed edge ideals

We obtain as another consequence of Theorem 5.15 a simple classification of
$\mathcal{J}$-closed edge ideals for any ideal $\mathcal{J}\subseteq
S=k[v_{1},\ldots,v_{m}]$ generated by linear forms.

Let $G$ be a simple graph with vertex set $[m]$ and edge set $E(G)$. The _edge
ideal_ of $G$ is the quadratic monomial ideal

$\mathcal{I}(G)=\big{(}v_{i}v_{j}\,:\,\\{i,j\\}\in E(G)\big{)}\subseteq S.$

The edge ideal of $G$ is the Stanley–Reisner ideal of a simplicial complex
associated to $G$ called the _independence complex_ ${\operatorname{Ind}(G)}$,
defined by

$\operatorname{Ind}(G)=\\{\sigma\subseteq[m]\,:\,\sigma\text{ is an
independent set of }G\\}.$

Equivalently, $\operatorname{Ind}(G)$ is the unique flag complex with
$1$-skeleton the graph complement $G^{c}$.

Next, let $\mathcal{J}\subseteq S$ be an ideal generated by linear forms. To
classify all graphs $G$ whose edge ideals are $\mathcal{J}$-closed, it
suffices by Proposition 5.3 and Proposition 5.6 to assume that $\mathcal{J}$
has the form $\mathcal{J}_{I}=(v_{i}:i\notin I)$ for some subset
$I\subseteq[m]$.

######  Corollary 5.18.

Let $G$ be a simple graph on vertex set $[m]$ and let $I\subseteq[m]$. Then
the following conditions are equivalent:

  1. (a)

the edge ideal $\mathcal{I}(G)$ is $\mathcal{J}_{I}$-closed;

  2. (b)

$d_{2}(F_{2})\subseteq\mathcal{J}_{I}F_{1}$ and
$d_{1}(F_{1})\subseteq\mathcal{J}_{I}F_{0}$ in the minimal free resolution
$(F,d)$ of $S/\mathcal{I}(G)$;

  3. (c)

$I$ is an independent set of $G$ and $\\{i,v\\},\\{j,v\\}\notin E(G)$ for
every edge $\\{i,j\\}\in E(G)$ and $v\in I\smallsetminus\\{i,j\\}$.

###### Proof.

Since $S/\mathcal{I}(G)$ is the Stanley–Reisner ring $k[K]$ of the
independence complex $K=\operatorname{Ind}(G)$ and since $S\otimes
H^{4}(\mathcal{Z}_{K})$ and $S\otimes H^{3}(\mathcal{Z}_{K})$ lie in
homological degree $2$ and $1$, respectively, in the minimal free resolution
$F=S\otimes H^{*}(\mathcal{Z}_{K})$ (see Theorem 4.4 and Proposition 4.9), it
follows from condition (b) that $\delta_{v}\colon H^{4}(\mathcal{Z}_{K})\to
H^{3}(\mathcal{Z}_{K})$ and $\delta_{ij}\colon H^{3}(\mathcal{Z}_{K})\to
H^{0}(\mathcal{Z}_{K})$ are trivial for $i,j,v\in I$. By Corollary 5.17, this
implies $\delta_{U}=0$ for all $U\subseteq I$, so $I(G)$ is
$\mathcal{J}_{I}$-closed by Proposition 5.7. This shows (b) implies (a). Since
(a) implies (b) by definition, (a) and (b) are equivalent.

To see that (a) and (c) are equivalent, first observe that (a) is equivalent
to the equivariant formality of the $T^{I}$-action on $\mathcal{Z}_{K}$ by
Proposition 5.7, where $K=\operatorname{Ind}(G)$. Since $K$ is flag, this in
turn is equivalent to the condition that $I\in K$ and
$K_{\\{i,j\\}}\ast\\{v\\}\subseteq K$ for every $\\{i,j\\}\notin K$ with $v\in
I\smallsetminus\\{i,j\\}$ by Theorem 5.15. These combinatorial conditions
translate precisely to condition (c) by definition of the the independence
complex of $G$. ∎

###  5.5. Dependence of equivariant formality on the characteristic

In contrast with the flag case, we give here some examples of torus actions on
moment-angle complexes where the equivariant formality of the action depends
on the characteristic of $k$. These examples show that a characterisation of
equivariant formality for $\mathcal{Z}_{K}$ purely in terms of the
combinatorics of $K$ (as in Theorem 5.15) cannot be expected in general.

######  Example 5.19.

Let $K=\partial\Delta^{1}$ so that $\mathcal{Z}_{K}\cong S^{3}$ (as in Example
5.11), and consider the subgroup $S^{1}\to T^{2}$, $t\mapsto(t,t^{p})$, where
$p$ is prime. The primary operation $\iota_{1}+p\iota_{2}$ induced by this
circle action is trivial in cohomology, but the cochain-level zig-zag

$v_{1}u_{2}\xmapsto{\mathmakebox[2.3em]{\iota_{1}+p\iota_{2}}}pv_{1}\xleftarrow{\mathmakebox[2.3em]{d}}\mapsfromchar
pu_{1}\xmapsto{\mathmakebox[2.3em]{\iota_{1}+p\iota_{2}}}p$

in the reduced Koszul complex $R(K)$ shows that the associated secondary
operation maps the fundamental class $[v_{1}u_{2}]\in
H^{3}(\mathcal{Z}_{K};k)$ to $[p]\in H^{0}(\mathcal{Z}_{K};k)$. Since all
tertiary and higher operations are trivial for degree reasons, it follows that
this circle action is equivariantly formal over $k=\mathbb{F}_{p}$ but not
over $k=\mathbb{Q}$.

######  Remark 5.20.

The example above illustrates the necessity of the $p$-coordinate hull
construction used in Proposition 5.6. If $H=S^{1}_{(1,2)}\subseteq T^{2}$
denotes the subtorus described above, then the coordinate hull of $H$ is
$\operatorname{hull}(H)=T^{2}$, which does not act equivariantly formally
(over any $k$) since the Stanley–Reisner ring
$H^{*}_{T^{2}}(\mathcal{Z}_{K};k)\cong k[v_{1},v_{2}]/(v_{1}v_{2})$ of
$K=\partial\Delta^{1}$ is not a free module over
$H^{*}(BT^{2};k)=k[v_{1},v_{2}]$. Thus, when working over $k=\mathbb{F}_{p}$,
the equivariant formality of the $H$-action is not detected by the smallest
coordinate subtorus $\operatorname{hull}(H)$ containing $H$. In this example
the $p$-coordinate hull is $p\operatorname{-hull}(H)=S^{1}_{1}$, the first
coordinate circle in $T^{2}$, which does act equivariantly formally on
$\mathcal{Z}_{K}\cong S^{3}$.

The minimal $6$-vertex triangulation of $\mathbb{R}P^{2}$ is the smallest
simplicial complex $K$ for which the associated moment-angle complex
$\mathcal{Z}_{K}$ has $2$-torsion in integral cohomology. (The homotopy type
of this moment-angle complex is worked out in [19].) Using Lemma 3.5, it is
straightforward to check that $\iota_{j}\colon H^{6}(\mathcal{Z}_{K};k)\to
H^{5}(\mathcal{Z}_{K};k)$ is nonzero for all $j=1,\ldots,6$ and all fields $k$
in this case. It follows that no coordinate $T^{I}$-action on
$\mathcal{Z}_{K}$ is equivariantly formal over any field $k$.

The complex $\hat{K}$ in the following example is obtained from the $6$-vertex
triangulation $K$ of $\mathbb{R}P^{2}$ by introducing a seventh vertex which
cones over all $10$ minimal non-faces of $K$. This introduces $10$ new minimal
non-faces containing the seventh vertex and has the effect that the primary
cohomology operation $\iota_{7}$ acts trivially on all cohomology groups
except for $H^{10}(\mathcal{Z}_{\hat{K}};k)$ when $\operatorname{char}(k)=2$.
According to the commutative diagrams of Lemma 3.5, $\iota_{7}\colon
H^{10}(\mathcal{Z}_{\hat{K}};k)\to H^{9}(\mathcal{Z}_{\hat{K}};k)$ is
determined by the map

$\widetilde{H}^{2}(\hat{K};k)\longrightarrow\widetilde{H}^{2}(\hat{K}_{\\{1,\ldots,6\\}};k)=\widetilde{H}^{2}(\mathbb{R}P^{2};k)$

induced by the inclusion of full subcomplexes
$\hat{K}_{\\{1,\ldots,6\\}}\hookrightarrow\hat{K}_{\\{1,\ldots,7\\}}=\hat{K}$.

######  Example 5.21.

Consider the simplicial complex $\hat{K}$ on $7$ vertices with Stanley–Reisner
ideal

$\displaystyle\big{(}v_{124},v_{126},$ $\displaystyle
v_{134},v_{135},v_{156},v_{235},v_{236},v_{245},v_{346},v_{456},$
$\displaystyle
v_{1237},v_{1257},v_{1367},v_{1457},v_{1467},v_{2347},v_{2467},v_{2567},v_{3457},v_{3567}\big{)},$

in $S=k[v_{1},\ldots,v_{7}]$, where $v_{i_{1}\cdots i_{q}}=v_{i_{1}}\\!\cdots
v_{i_{q}}$. One can verify with Macaulay2 [20] that $v_{7}$ appears as an
entry in the matrix representing the final differential in the minimal free
resolution of the Stanley–Reisner ring $\mathbb{F}_{2}[\hat{K}]$, whereas
$v_{7}$ does not appear as an entry in the differentials of the minimal free
resolution of $\mathbb{Q}[\hat{K}]$. In other words, for
$\mathcal{J}=(v_{1},\ldots,v_{6})$, the Stanley–Reisner ring $k[\hat{K}]$ is
$\mathcal{J}$-closed when $k=\mathbb{Q}$ but not when $k=\mathbb{F}_{2}$.
Consequently, by Proposition 5.7, the coordinate $S^{1}_{7}$-action on
$\mathcal{Z}_{\hat{K}}$ is equivariantly formal over $\mathbb{Q}$ but not over
$\mathbb{F}_{2}$.

##  6\. Combinatorial models for the higher operations

In this section we describe the action of the higher cohomology operations for
moment-angle complexes in terms of the Hochster decomposition
$H^{\ast}(\mathcal{Z}_{K})\cong\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U})$.
This has already been done for the primary operations $\delta_{i}=\iota_{i}$
(see Lemma 3.5). In general, an analogous description of the higher operations
$\delta_{U}$, for $U\subseteq[m]$, purely in terms of the combinatorics of
full subcomplexes of $K$ (and avoiding the issue of indeterminacy) is only
possible when the lower degree operations $\delta_{V}$ all vanish, for
$V\subsetneq U$. This will lead in Section 6.3 to a characterisation of
equivariantly formal torus actions in terms of the combinatorics of
subcomplexes of $K$, generalising Theorem 5.9.

###  6.1. Secondary operations and the Mayer–Vietoris sequence

We begin by describing the action of the secondary cohomology operations
$\delta_{ij}$, since these admit a particularly simple description in terms of
the Mayer–Vietoris long exact sequence. However, this will be generalised to
all cohomology operations in Section 6.2, and that section does not rely on
the results of this one.

Fix a multidegree $J\subseteq[m]$ and assume that $i,j\in J$ with $i<j$.
Consider the subcomplex $K_{J\smallsetminus i}\cup K_{J\smallsetminus j}$ of
$K_{J}$. Since $K_{J\smallsetminus i}\cap K_{J\smallsetminus
j}=K_{J\smallsetminus ij}$, the cover $\\{K_{J\smallsetminus
i},K_{J\smallsetminus j}\\}$ of $K_{J\smallsetminus i}\cup K_{J\smallsetminus
j}$ gives rise to a Mayer–Vietoris sequence

${\cdots}$${\widetilde{H}^{*}(K_{J\smallsetminus i}\cup K_{J\smallsetminus
j})}$${\widetilde{H}^{*}(K_{J\smallsetminus
i})\oplus\widetilde{H}^{*}(K_{J\smallsetminus
j})}$${\widetilde{H}^{*}(K_{J\smallsetminus
ij})}$${\cdots}$$\scriptstyle{\binom{-\mathrm{res}_{i}}{\mathrm{res}_{j}}}$$\scriptstyle{\left(\mathrm{res}_{j}\;\mathrm{res}_{i}\right)\;}$

where $\operatorname{res}_{i}$ and $\mathrm{res}_{j}$ are restriction maps
induced by the evident inclusions. We will often simply write $\alpha|_{i}$
for $\mathrm{res}_{i}(\alpha)$. Recall that on the summand
$\widetilde{H}^{*}(K_{J})$ of
$H^{*}(\mathcal{Z}_{K})\cong\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U})$,
the primary operations

$\iota_{i}\colon\widetilde{H}^{*}(K_{J})\to\widetilde{H}^{*}(K_{J\smallsetminus
i})\quad\text{ and
}\quad\iota_{j}\colon\widetilde{H}^{*}(K_{J})\to\widetilde{H}^{*}(K_{J\smallsetminus
j})$

are equal to the restriction maps up to a sign (see Lemma 3.5 and Remark 3.6).

If both $\iota_{i}$ and $\iota_{j}$ are zero on
$\bigoplus_{U\subseteq[m]}\widetilde{H}^{*}(K_{U})$, then this induces a
_unique_ lift $\ell$ through the connecting homomorphism of the Mayer–Vietoris
sequence:

(25)
${\widetilde{H}^{\ast}(K_{J})}$${\cdots}$${\widetilde{H}^{\ast-1}(K_{J\smallsetminus
ij})}$${\widetilde{H}^{\ast}(K_{J\smallsetminus i}\cup K_{J\smallsetminus
j})}$${\widetilde{H}^{\ast}(K_{J\smallsetminus
i})\oplus\widetilde{H}^{\ast}(K_{J\smallsetminus j})}$${\cdots\
}$$\scriptstyle{0}$$\scriptstyle{\ell}$$\scriptstyle{0}$

######  Proposition 6.1.

Under the assumptions above, $\delta_{ij}=(-1)^{\varepsilon(ij,J)}\ell$ on
$\widetilde{H}^{\ast}(K_{J})$.

###### Proof.

Let $[\alpha]\in\widetilde{H}^{\ast}(K_{J})$, and write
$[\bar{\alpha}]\in\widetilde{H}^{\ast}(K_{J\smallsetminus i}\cup
K_{J\smallsetminus j})$ for the restriction of $[\alpha]$ along the inclusion
$K_{J\smallsetminus i}\cup K_{J\smallsetminus j}\hookrightarrow K_{J}$. Then
$\ell([\alpha])$ is the unique preimage of $[\bar{\alpha}]$ under the
connecting homomorphism $\widetilde{H}^{\ast-1}(K_{J\smallsetminus
ij})\to\widetilde{H}^{\ast}(K_{J\smallsetminus i}\cup K_{J\smallsetminus j})$.
Consider the commutative diagram

${\widetilde{C}^{\ast-1}(K_{J\smallsetminus i}\cup K_{J\smallsetminus
j})}$${\widetilde{C}^{\ast-1}(K_{J\smallsetminus
i})\oplus\widetilde{C}^{\ast-1}(K_{J\smallsetminus
j})}$${\widetilde{C}^{\ast-1}(K_{J\smallsetminus
ij})}$${\widetilde{C}^{\ast}(K_{J\smallsetminus i}\cup K_{J\smallsetminus
j})}$${\widetilde{C}^{\ast}(K_{J\smallsetminus
i})\oplus\widetilde{C}^{\ast}(K_{J\smallsetminus
j})}$${\widetilde{C}^{\ast}(K_{J\smallsetminus ij}).\
}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$

According to the snake lemma, $\ell([\alpha])$ is represented by a cochain

(26)
$-\left(d^{-1}(\alpha|_{i})\right)|_{j}+\left(d^{-1}(\alpha|_{j})\right)|_{i}\in\widetilde{C}^{\ast-1}(K_{J\smallsetminus
ij})$

obtained by pushing $\bar{\alpha}$ through the zig-zag from the bottom left to
top right corner of the diagram above. Here we are writing
$d^{-1}(\alpha|_{i})$ for a choice of preimage of
$\alpha|_{i}\in\widetilde{C}^{\ast}(K_{J\smallsetminus i})$ under the
differential, and similarly for $d^{-1}(\alpha|_{j})$.

On the other hand,
$\delta_{ij}[\alpha]=[\iota_{i}\beta_{j}+\iota_{j}\beta_{i}]$ for any cochains
$\beta_{i},\beta_{j}$ satisfying $\iota_{i}\alpha=d\beta_{i}$ and
$\iota_{j}\alpha=d\beta_{j}$. Since

$\iota_{i}\alpha=(-1)^{\varepsilon(i,J)+|\alpha|+1}\alpha|_{i}\quad\text{ and
}\quad\iota_{j}\alpha=(-1)^{\varepsilon(j,J)+|\alpha|+1}\alpha|_{j}$

by definition of the $\Lambda$-module structure on
$\bigoplus_{J\subseteq[m]}\widetilde{C}^{*}(K_{J})$ (see Remark 3.6), it
follows that

$\displaystyle\delta_{ij}[\alpha]$
$\displaystyle=\left[(-1)^{\varepsilon(j,J)+|\alpha|+1}\iota_{i}\left(d^{-1}(\alpha|_{j})\right)+(-1)^{\varepsilon(i,J)+|\alpha|+1}\iota_{j}\left(d^{-1}(\alpha|_{i})\right)\right]$
$\displaystyle=\Big{[}(-1)^{\varepsilon(j,J)+|\alpha|+1}(-1)^{\varepsilon(i,J\smallsetminus
j)+|\alpha|}\left(d^{-1}(\alpha|_{j})\right)|_{i}$
$\displaystyle\qquad\qquad+(-1)^{\varepsilon(i,J)+|\alpha|+1}(-1)^{\varepsilon(j,J\smallsetminus
i)+|\alpha|}\left(d^{-1}(\alpha|_{i})\right)|_{j}\Big{]}$ (27)
$\displaystyle=\Big{[}(-1)^{\varepsilon(j,J)+\varepsilon(i,J\smallsetminus
j)+1}\left(d^{-1}(\alpha|_{j})\right)|_{i}+(-1)^{\varepsilon(i,J)+\varepsilon(j,J\smallsetminus
i)+1}\left(d^{-1}(\alpha|_{i})\right)|_{j}\Big{]}.$

Observe that since $i<j$, $\varepsilon(i,J\smallsetminus j)=\varepsilon(i,J)$
while $\varepsilon(j,J\smallsetminus i)=\varepsilon(j,J)-1$. Therefore the two
terms in (27) have opposite signs. Comparing with (26), we conclude that
$\ell([\alpha])$ and $\delta_{ij}[\alpha]$ are equal up to the indicated sign.
∎

######  Remark 6.2.

The combinatorial interpretation of the secondary operation
$\delta_{ij}[\alpha]$ given above holds just as well under the weaker
assumption that $\iota_{i}$ and $\iota_{j}$ vanish only on the summand
$\widetilde{H}^{|\alpha|}(K_{J})$ containing $[\alpha]$ and on
$\widetilde{H}^{|\alpha|-1}(K_{J\smallsetminus i})$ and
$\widetilde{H}^{|\alpha|-1}(K_{J\smallsetminus j})$. More generally, for any
cohomology class $[\alpha]\in\widetilde{H}^{\ast}(K_{J})$ in the kernel of the
primary operations $\iota_{i}$ and $\iota_{j}$, the secondary operation
$\delta_{ij}[\alpha]$ is defined up to indeterminacy analogous to the
indeterminacy of a triple Massey product. In this case, a statement analogous
to Proposition 6.1 still holds, with indeterminacy corresponding to the
nonuniqueness of a choice of lift $\ell$ in diagram (25).

It is well known that the connecting homomorphism in the Mayer–Vietoris
sequence for an excisive triad $(X;U,V)$ is induced by a map of spaces,
namely, the quotient map collapsing the ends of the double mapping cylinder
$U\cup\left((U\cap V)\times[0,1]\right)\cup V\simeq X$ to form $\Sigma(U\cap
V)$. For the Mayer–Vietoris sequence above, we can see the connecting
homomorphism $\widetilde{H}^{\ast-1}(K_{J\smallsetminus
ij})\to\widetilde{H}^{\ast}(K_{J\smallsetminus i}\cup K_{J\smallsetminus j})$
even more concretely, being induced by the inclusion $K_{J\smallsetminus
i}\cup K_{J\smallsetminus j}\hookrightarrow\Sigma K_{J\smallsetminus ij}$,
where $\Sigma K_{J\smallsetminus ij}$ is viewed as the union of the cones
$K_{J\smallsetminus ij}\ast\\{i\\}$ and $K_{J\smallsetminus ij}\ast\\{j\\}$
(cf. Figure 1).

Thus, just as the primary operations $\delta_{i}$ on $H^{*}(\mathcal{Z}_{K})$
are determined by the maps $K_{J\smallsetminus i}\hookrightarrow K_{J}$ for
all $J\subseteq[m]$, the secondary operations $\delta_{ij}$ are essentially
determined by the maps

(28) $K_{J\smallsetminus i}\cup K_{J\smallsetminus j}\hookrightarrow
K_{J}\quad\text{ and }\quad K_{J\smallsetminus i}\cup K_{J\smallsetminus
j}\hookrightarrow\Sigma K_{J\smallsetminus ij}.$

In particular, for each $J\subseteq[m]$ containing $i,j$, there is a
cofibration sequence

${K_{J\smallsetminus i}\vee K_{J\smallsetminus j}}$${K_{J\smallsetminus i}\cup
K_{J\smallsetminus j}}$${\Sigma K_{J\smallsetminus
ij}}$${K_{J}}$$\scriptstyle{\ell}$

inducing the Mayer–Vietoris sequence, and when the composite of the two
leftmost arrows is null homotopic, there exists an extension $\ell$ inducing
the lift in (25).

######  Example 6.3.

Let $K$ be the simplicial complex on the vertex set $[5]$ with minimal non-
faces $13$, $14$, $24$, $25$ and $345$. ($K$ is obtained from the boundary of
a pentagon by adding the edge $35$.) Take $J=[5]$ and consider the
Mayer–Vietoris sequence associated to the cover $\\{K_{J\smallsetminus
1},K_{J\smallsetminus 3}\\}$. In this case, $K_{J\smallsetminus 1}\cup
K_{J\smallsetminus 3}=K_{J}$, so the vertical map in (25) is an isomorphism.
The map $K_{J\smallsetminus 1}\cup K_{J\smallsetminus 3}\hookrightarrow\Sigma
K_{J\smallsetminus 13}$ inducing the connecting homomorphism (pictured in
Figure 1) is given up to homotopy by a map $S^{1}\vee
S^{1}\xrightarrow{1\vee\ast}S^{1}\vee\\{pt\\}$ collapsing the second wedge
summand to a point. If $\alpha\in\widetilde{H}^{1}(S^{1}\vee
S^{1})\cong\widetilde{H}^{1}(K_{J})\subset H^{*}(\mathcal{Z}_{K})$ is a
generator for the first circle summand, then $\alpha$ is in the kernel of the
restriction maps
$\widetilde{H}^{1}(K_{J})\to\widetilde{H}^{1}(K_{J\smallsetminus 1})$ and
$\widetilde{H}^{1}(K_{J})\to\widetilde{H}^{1}(K_{J\smallsetminus 3})$, and
hence $\iota_{1}\alpha=\iota_{3}\alpha=0$. Moreover, since $\alpha$ is clearly
in the image of the connecting homomorphism, it follows that the multidegree
$J$ part of $H^{*}(\mathcal{Z}_{K})$ supports a nontrivial secondary operation
$\delta_{13}\alpha\neq 0$.

$1$$2$$3$$4$$5$$K_{\\{1,2,4,5\\}}\cup
K_{\\{2,3,4,5\\}}$$K_{\\{1,2,4,5\\}}$$K_{\\{2,3,4,5\\}}$

$\quad\lhook\joinrel\longrightarrow\quad\;$ $1$$2$$3$$4$$5$$\Sigma
K_{\\{2,4,5\\}}$$K_{\\{2,4,5\\}}{\ast}\\{1\\}$$K_{\\{2,4,5\\}}{\ast}\\{3\\}$

Figure 1. The Mayer–Vietoris connecting map $K_{J\smallsetminus i}\cup
K_{J\smallsetminus j}\hookrightarrow\Sigma K_{J\smallsetminus ij}$
in Example 6.3

######  Remark 6.4.

In terms of the minimal free resolution $F$ of $k[K]$, the maps induced by the
inclusions of subcomplexes (28) yield a partial combinatorial interpretation
for the quadratic component of the differential. Together with the description
of the linear component of the differential given by [24, Theorem 1.1] (or
Lemma 3.5), this amounts to a combinatorial interpretation of the complex
$F/\mathfrak{m}^{3}F$ that partially answers Katthän’s Question 4.2 of [24].
One can remove the indeterminacy by fixing chain level data as in Section 4,
but to fully answer Katthän’s question one would need to understand how to
make these choices at the cohomology level, in terms of the inclusions (28)
and Hochster’s decomposition.

We also obtain the following characterisation of equivariant formality,
extending Theorem 5.9 to the case of coordinate $2$-torus actions. Since this
result will be further generalised in Theorem 6.8, we omit the proof. (See
Section 6.3 for the definition of the face deletion $K\smallsetminus F$.)

######  Theorem 6.5.

Let $K$ be a simplicial complex on vertex set $[m]$ and let
$I=\\{i,j\\}\subseteq[m]$ with $i\neq j$. Then the following conditions are
equivalent:

  1. (a)

the coordinate $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal
(over $k$);

  2. (b)

$\delta_{i}$, $\delta_{j}$ and $\delta_{ij}$ are trivial on
$H^{\ast}(\mathcal{Z}_{K};k)$;

  3. (c)

$K_{J}\smallsetminus(I\cap J)\hookrightarrow K_{J}$ induces the trivial map on
$\widetilde{H}^{\ast}(\;;k)$ for all $J\subseteq[m]$.

###  6.2. Higher operations and the Mayer–Vietoris spectral sequence

We generalise the results of the previous section to the case of an arbitrary
coordinate subtorus, that is, the $T^{I}$-action on $\mathcal{Z}_{K}$ for any
$I\subseteq[m]$. In place of the Mayer–Vietoris long exact sequence used to
describe secondary cohomology operations in the $|I|=2$ case, we will here
identify all higher operations with differentials in a Mayer–Vietoris spectral
sequence.

Fix $I\subseteq[m]$ and consider the $T^{I}$-action on $\mathcal{Z}_{K}$. To
describe the higher cohomology operations induced by this torus action in
terms of the Hochster decomposition
$H^{\ast}(\mathcal{Z}_{K})\cong\bigoplus_{J\subseteq[m]}\widetilde{H}^{*}(K_{J})$,
we fix a multidegree $J\subseteq[m]$ and, as before, consider the cover

$\mathcal{U}_{I\\!,J}=\\{K_{J\smallsetminus i}~{}:~{}i\in I\cap J\\}.$

The _(ordered) Čech complex_
$\check{C}^{*}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})$ of the cover
$\mathcal{U}_{I\\!,J}$ with coefficients in the presheaf $\widetilde{C}^{p}$
is given by

$\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})=\bigoplus_{\begin{subarray}{c}i_{0}<\cdots<i_{q}\\\
i_{0},\ldots,i_{q}\in I\cap
J\end{subarray}}\widetilde{C}^{p}(K_{J\smallsetminus i_{0}\cdots i_{q}}).$

For an element $\omega$ of
$\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})$, we write
$\omega_{i_{0}\cdots i_{q}}$ for its component in
$\widetilde{C}^{p}(K_{J\smallsetminus i_{0}\cdots i_{q}})$. The Čech
differential is then defined by

$\check{d}\colon\check{C}^{q-1}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})\to\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p}),\quad\quad(\check{d}\omega)_{i_{0}\cdots
i_{q}}=\sum_{\ell=0}^{q}(-1)^{\ell}\omega_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{q}}\big{|}_{K_{J\smallsetminus i_{0}\cdots i_{q}}}.$

We form the Čech double complex
$\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})$, whose vertical
differential is $(-1)^{p}\check{d}$, and whose horizontal differential
$d\colon\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})\to\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p+1})$
is induced by the simplicial cochain differential. The inclusions
$K_{J\smallsetminus i}\hookrightarrow K_{J}$ induce a morphism from
$\widetilde{C}^{*}(K_{J})$ to the Čech double complex, and with this we form
the augmented Čech double complex
$a\check{C}^{*}(\mathcal{U}_{I\\!,J},\widetilde{C}^{*})$:

(29)
${\vdots}$${\vdots}$${\vdots}$${\bigoplus\limits_{i<j}\widetilde{C}^{0}(K_{J\smallsetminus
ij})}$${\bigoplus\limits_{i<j}\widetilde{C}^{1}(K_{J\smallsetminus
ij})}$${\bigoplus\limits_{i<j}\widetilde{C}^{2}(K_{J\smallsetminus
ij})}$${\cdots}$${\bigoplus\limits_{i}\widetilde{C}^{0}(K_{J\smallsetminus
i})}$${\bigoplus\limits_{i}\widetilde{C}^{1}(K_{J\smallsetminus
i})}$${\bigoplus\limits_{i}\widetilde{C}^{2}(K_{J\smallsetminus
i})}$${\cdots}$${\widetilde{C}^{0}(K_{J})}$${\widetilde{C}^{1}(K_{J})}$${\widetilde{C}^{2}(K_{J})}$${\cdots}$$\scriptstyle{\check{d}}$$\scriptstyle{d}$$\scriptstyle{-\check{d}}$$\scriptstyle{d}$$\scriptstyle{\check{d}}$$\scriptstyle{d}$$\scriptstyle{\check{d}}$$\scriptstyle{d}$$\scriptstyle{-\check{d}}$$\scriptstyle{d}$$\scriptstyle{\check{d}}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{d}$

Taking cohomology with respect to the horizontal differential yields the first
page of the _augmented Mayer–Vietoris spectral sequence_ $(E^{p,q}_{r},d_{r})$
associated to $\mathcal{U}_{I\\!,J}$, with

(30) $E_{1}^{p,-1}=\widetilde{H}^{p}(K_{J}),\qquad
E_{1}^{p,q}=\bigoplus_{\begin{subarray}{c}i_{0}<\cdots<i_{q}\\\
i_{0},\ldots,i_{q}\in I\cap
J\end{subarray}}\widetilde{H}^{p}(K_{J\smallsetminus i_{0}\cdots
i_{q}})\;\text{ for }q\geqslant 0.$

(Equivalently, this is the spectral sequence associated to the filtration of
the total complex of (29) by column degree.) The differential $d_{1}$, being
induced by the vertical differential in (29), therefore has components given
up to sign by the primary cohomology operations $\iota_{i}=\delta_{i}$ for
$i\in I\cap J$. The next result identifies the higher differentials $d_{s}$ in
this spectral sequence with the higher cohomology operations $\delta_{U}$
indexed by subsets $U\subseteq I\cap J$ with $|U|=s$.

######  Lemma 6.6.

Let $K$ be a simplicial complex on the vertex set $[m]$, and fix
$I,J\subseteq[m]$. Suppose that $d_{r}=0$ for $1\leqslant r<s$ in the
augmented Mayer–Vietoris spectral sequence associated to
$\mathcal{U}_{I\\!,J}$. Then the differential $d_{s}\colon E_{s}^{p,-1}\to
E_{s}^{p-s+1,s-1}$ defines a map

$\widetilde{H}^{p}(K_{J})\longrightarrow\bigoplus_{i_{0}<\cdots<i_{s-1}}\widetilde{H}^{p-s+1}(K_{J\smallsetminus
i_{0}\cdots i_{s-1}})$

with each component given by $(-1)^{\varepsilon(i_{0}\ldots
i_{s-1},J)+p+s}\delta_{i_{0}\cdots i_{s-1}}$.

###### Proof.

Let $[\alpha]\in\widetilde{H}^{p}(K_{J})$. Observe that
$d_{1}[\alpha]=\bigl{(}[\alpha|_{i}]\bigr{)}_{i\in I\cap
J}\in\bigoplus_{i}\widetilde{H}^{p}(K_{J\smallsetminus i})$, where for each
$i\in I\cap J$ we have

$[\alpha|_{i}]=(-1)^{\varepsilon(i,J)+p+1}[\iota_{i}\alpha]=(-1)^{\varepsilon(i,J)+p+1}\delta_{i}[\alpha]$

by Remark 3.6. So $d_{1}[\alpha]=0$ implies that there exists an element
$(\beta_{i})_{i\in I\cap
J}\in\bigoplus_{i}\widetilde{C}^{p-1}(J_{K\smallsetminus i})$ with
$d(\beta_{i})=\alpha|_{i}$ for each $i\in I\cap J$, and $d_{2}[\alpha]$ is
then represented by $(-1)^{p-1}\check{d}\bigl{(}(\beta_{i})_{i\in I\cap
J}\bigr{)}$, as indicated in the cochain-level zig-zag below.

${\bigoplus\limits_{i<j}\widetilde{H}^{p-1}(K_{J\smallsetminus
ij})}$${\bigoplus\limits_{i<j}\widetilde{H}^{p}(K_{J\smallsetminus
ij})}$${(-1)^{p-1}\\!\left(\beta_{j}|_{i}-\beta_{i}|_{j}\right)_{i<j}}$${\bigoplus\limits_{i}\widetilde{H}^{p-1}(K_{J\smallsetminus
i})}$${\bigoplus\limits_{i}\widetilde{H}^{p}(K_{J\smallsetminus
i})}$${\left(\beta_{i}\right)_{i}}$${\left(\alpha|_{i}\right)_{i}}$${\widetilde{H}^{p-1}(K_{J})}$${\widetilde{H}^{p}(K_{J})}$${\alpha}$$\scriptstyle{(-1)^{p-1}\check{d}}$$\scriptstyle{d}$$\scriptstyle{d_{2}}$

Since the cohomology class of
$\left(\beta_{j}|_{i}-\beta_{i}|_{j}\right)_{i<j}\in\bigoplus_{i<j}\widetilde{C}^{p-1}(J_{K\smallsetminus
ij})$ does not depend on the choice of $d$-preimages $\beta_{i}$ of
$\alpha|_{i}$, we may assume for each $i\in I\cap J$ that
$\beta_{i}=(-1)^{\varepsilon(i,J)+p+1}d^{-1}\iota_{i}\alpha$ for some preimage
$d^{-1}\iota_{i}\alpha$ of $\iota_{i}\alpha$. Therefore, for each component of
$\left(\beta_{j}|_{i}-\beta_{i}|_{j}\right)_{i<j}$, we have

$\displaystyle\beta_{j}|_{i}-\beta_{i}|_{j}$
$\displaystyle=(-1)^{\varepsilon(i,J\smallsetminus
j)+p}\iota_{i}\beta_{j}-(-1)^{\varepsilon(j,J\smallsetminus
i)+p}\iota_{j}\beta_{i}$ $\displaystyle=(-1)^{\varepsilon(i,J\smallsetminus
j)+\varepsilon(j,J)+1}\iota_{i}d^{-1}\iota_{j}\alpha-(-1)^{\varepsilon(j,J\smallsetminus
i)+\varepsilon(i,J)+1}\iota_{j}d^{-1}\iota_{i}\alpha.$

Since $i<j$, it follows that $\varepsilon(i,J\smallsetminus
j)=\varepsilon(i,J)$ and $\varepsilon(j,J\smallsetminus
i)=\varepsilon(j,J)-1$, and hence

$\displaystyle\left[\beta_{j}|_{i}-\beta_{i}|_{j}\right]$
$\displaystyle=(-1)^{\varepsilon(i,J)+\varepsilon(j,J)+1}\left[\iota_{i}d^{-1}\iota_{j}\alpha+\iota_{j}d^{-1}\iota_{i}\alpha\right]$
$\displaystyle=(-1)^{\varepsilon(ij,J)+1}\delta_{ij}[\alpha].$

Finally, it follows that each component of $d_{2}[\alpha]$ is of the form

$(-1)^{p-1}\left[\beta_{j}|_{i}-\beta_{i}|_{j}\right]=(-1)^{\varepsilon(ij,J)+p}\delta_{ij}[\alpha],$

as claimed.

Proceeding inductively, we assume that $d_{r}=0$ for $1\leqslant r<s$ and that
the components of

$d_{s}[\alpha]\in
E_{s}^{p-s+1,s-1}\cong\bigoplus_{i_{0}<\cdots<i_{s-1}}\widetilde{H}^{p-s+1}(K_{J\smallsetminus
i_{0}\cdots i_{s-1}})$

are of the form $(-1)^{\varepsilon(i_{0}\ldots
i_{s-1},J)+p+s}\delta_{i_{0}\cdots i_{s-1}}[\alpha]$. Now suppose $d_{s}$ is
also trivial. To ease notation, we will write $\delta_{i_{0}\cdots
i_{s-1}}(\alpha)$ for a cochain representative of $\delta_{i_{0}\cdots
i_{s-1}}[\alpha]$ defined recursively by

$\delta_{i_{0}\cdots
i_{s-1}}(\alpha)=\sum_{\ell=0}^{s-1}\iota_{i_{\ell}}d^{-1}\delta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s-1}}(\alpha)\in\widetilde{C}^{p-s+1}(K_{J\smallsetminus i_{0}\cdots
i_{s-1}}),$

where $d^{-1}\delta_{i_{0}\cdots\widehat{i_{\ell}}\cdots i_{s-1}}(\alpha)$
denotes a choice of preimage of $\delta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s-1}}(\alpha)$. Then $d_{s}[\alpha]=0$ implies that there exists a zig-zag
in the double complex (29) of the form

${(-1)^{p-s}\left(\sum\limits_{\ell=0}^{s}(-1)^{\ell}\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}\big{|}_{i_{\ell}}\right)_{i_{0}<\cdots<i_{s}}}$${\left(\beta_{i_{0}\cdots
i_{s-1}}\right)_{i_{0}<\cdots<i_{s-1}}}$${\\!\left((-1)^{\varepsilon}\delta_{i_{0}\cdots
i_{s-1}}(\alpha)\right)_{i_{0}<\cdots<i_{s-1}}}$
${\left(\alpha|_{i_{0}}\right)_{i_{0}}}$${\hskip
44.0pt\alpha\in\widetilde{C}^{p}(K_{J})}$$\scriptstyle{(-1)^{p-s}\check{d}}$$\scriptstyle{d}$$\scriptstyle{(-1)^{p-s+1}\check{d}}$$\scriptstyle{\ddots}$$\scriptstyle{d}$

where $d(\beta_{i_{0}\cdots i_{s-1}})=(-1)^{\varepsilon}\delta_{i_{0}\cdots
i_{s-1}}(\alpha)$ with $\varepsilon=\varepsilon(i_{0}\ldots i_{s-1},J)+p+s$
for each strictly increasing sequence $i_{0},\ldots,i_{s-1}\in I\cap J$, and
the top-left cochain is a representative of $d_{s+1}[\alpha]$. Now for each
strictly increasing sequence $i_{0},\ldots,i_{s}\in I\cap J$ and for each
$0\leqslant\ell\leqslant s$, we have

$\displaystyle\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}\big{|}_{i_{\ell}}$
$\displaystyle=(-1)^{\varepsilon(i_{\ell},J\smallsetminus
i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s})+p-s+1}\iota_{i_{\ell}}\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}$
$\displaystyle=(-1)^{\varepsilon(i_{\ell},J)+\ell+p-s+1}\iota_{i_{\ell}}\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}.$

It follows that each component of $d_{s+1}[\alpha]$ is represented by a
cochain of the form

$\displaystyle(-1)^{p-s}\sum_{\ell=0}^{s}(-1)^{\ell}\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}\big{|}_{i_{\ell}}$
$\displaystyle=\sum_{\ell=0}^{s}(-1)^{\varepsilon(i_{\ell},J)+1}\iota_{i_{\ell}}\beta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}$
$\displaystyle=\sum_{\ell=0}^{s}(-1)^{\varepsilon(i_{\ell},J)+\varepsilon(i_{0}\ldots\widehat{i_{\ell}}\ldots
i_{s},J)+p+s+1}\iota_{i_{\ell}}d^{-1}\delta_{i_{0}\cdots\widehat{i_{\ell}}\cdots
i_{s}}(\alpha)$ $\displaystyle=(-1)^{\varepsilon(i_{0}\ldots
i_{s},J)+p+s+1}\delta_{i_{0}\cdots i_{s}}(\alpha),$

which closes the induction. ∎

Taking $J=[m]$ yields the augmented Mayer–Vietoris spectral sequence
associated to the cover $\mathcal{U}_{I}=\\{K_{[m]\smallsetminus i}~{}:~{}i\in
I\\}$, and we will see next that this spectral sequence contains all of the
higher operations as components in its differentials.

######  Theorem 6.7.

Suppose that $d_{r}=0$ for $1\leqslant r<s$ in the augmented Mayer–Vietoris
spectral sequence associated to $\mathcal{U}_{I}$. Then the differential
$d_{s}\colon E_{s}^{p,q}\to E_{s}^{p-s+1,s+q}$ defines a map

$\bigoplus_{U\subseteq I,\ |U|=m-q-1}\widetilde{H}^{p}(K_{[m]\smallsetminus
U})\longrightarrow\bigoplus_{V\subseteq I,\
|V|=m-q-s-1}\widetilde{H}^{p-s+1}(K_{[m]\smallsetminus V}),$

and the components are given by $(-1)^{\varepsilon(V\smallsetminus
U,[m])+p+s}\delta_{V\smallsetminus U}$ when $U\subseteq V$, and zero
otherwise.

###### Proof.

Writing $J=[m]\smallsetminus U$, there is a comparison map of augmented Čech
double complexes

$a\check{C}^{q}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})\longrightarrow
a\check{C}^{q+u}(\mathcal{U}_{I},\widetilde{C}^{p}),$

increasing vertical degree by $u=|U|$. On the component indexed by
$i_{0}\ldots i_{q}$ this is given by

$\widetilde{C}^{p}(K_{J\smallsetminus i_{0}\ldots
i_{q}})\xrightarrow{(-1)^{\varepsilon(i_{0}\ldots
i_{q},U)}}\widetilde{C}^{p}(K_{[m]\smallsetminus i_{0}\ldots i_{q},U}),$

using the equality $K_{J\smallsetminus i_{0}\ldots i_{q}}=K_{[m]\smallsetminus
i_{0}\ldots i_{q},U}$ to identify the two sides. On the first page of the
associated spectral sequences this map restricts to the isomorphism

$\widetilde{H}^{p}(K_{J\smallsetminus i_{0}\ldots
i_{q}})\xrightarrow{(-1)^{\varepsilon(i_{0}\ldots
i_{q},U)}}\widetilde{H}^{p}(K_{[m]\smallsetminus i_{0}\ldots i_{q},U}).$

Under the assumption that $d_{r}=0$ for $1\leqslant r<s$ in the spectral
sequence associated to $\mathcal{U}_{I}$, we may also assume by induction that
$d_{r}=0$ for $1\leqslant r<s$ in the spectral sequence associated to
$\mathcal{U}_{I\\!,J}$. Therefore on the $s$th page the comparison map induces
commutative squares

${\bigoplus\widetilde{H}^{p-s+1}(K_{J\smallsetminus i_{0}\ldots
i_{s+q}})}$${\bigoplus\widetilde{H}^{p-s+1}(K_{[m]\smallsetminus i_{0}\ldots
i_{s+q},U})}$${\widetilde{H}^{p}(K_{J\smallsetminus i_{0}\ldots
i_{q}})}$${\widetilde{H}^{p}(K_{[m]\smallsetminus i_{0}\ldots
i_{q},U}).}$$\scriptstyle{\oplus{(-1)^{\varepsilon(i_{0}\ldots
i_{s+q},U)}}}$$\scriptstyle{(-1)^{\varepsilon(i_{0}\ldots
i_{q},U)}}$$\scriptstyle{d_{s}}$$\scriptstyle{d_{s}}$

In particular, beginning on the $-1$st row and taking
$V=\\{i_{0},\ldots,i_{s-1}\\}\cup U$, the claimed formula follows from Lemma
6.6, with the sign $(-1)^{\varepsilon(V\smallsetminus
U,U)+\varepsilon(V\smallsetminus U,J)+p+s}=(-1)^{\varepsilon(V\smallsetminus
U,[m])+p+s}$. ∎

###  6.3. Equivariant formality from combinatorics

We are ready to prove our main result characterising the equivariant formality
of subtorus actions on moment-angle complexes $\mathcal{Z}_{K}$, purely in
terms of the cohomology of subcomplexes of $K$. By Proposition 5.6, it
suffices for us to treat the case of coordinate subtori.

The _face deletion_ of a simplicial complex $K$ at $F\in K$ is the largest
subcomplex $K\smallsetminus F$ of $K$ that does not contain $F$:

$K\smallsetminus F=\\{\sigma\in K\;:\;F\not\subseteq\sigma\\}.$

It follows from the definition that the face deletion can be written as the
union of full subcomplexes

$K\smallsetminus F=\bigcup_{i\in F}K_{[m]\smallsetminus i}.$

By convention, $K\smallsetminus\varnothing$ is the empty simplicial complex.
We also remind the reader of the notation

$\mathcal{U}_{I\\!,J}=\\{K_{J\smallsetminus i}~{}:~{}i\in I\cap J\\}$

and that this collection of subcomplexes of $K_{J}$ induces a Mayer–Vietoris
spectral sequence as in Section 6.2. The collection
$\mathcal{U}_{I}=\mathcal{U}_{I\\!,[m]}$ is of particular importance.

######  Theorem 6.8.

Let $K$ be a simplicial complex on vertex set $[m]$ and let $I\subseteq[m]$.
Then the following conditions are equivalent:

  1. (a)

the coordinate $T^{I}$-action on $\mathcal{Z}_{K}$ is equivariantly formal
over $k$;

  2. (b)

the cohomology operations $\delta_{J}$ vanish on $H^{*}(\mathcal{Z}_{K};k)$
for all $J\subseteq I$;

  3. (c)

the augmented Mayer–Vietoris spectral sequence associated to $\mathcal{U}_{I}$
degenerates at its first page (or equivalently, $\mathcal{U}_{I\\!,J}$ for all
$J$);

  4. (d)

$K_{J}\smallsetminus(I\cap J)\hookrightarrow K_{J}$ induces the trivial map on
$\widetilde{H}^{\ast}(\;;k)$ for all $J\subseteq[m]$.

###### Proof.

The equivalence of (a) and (b) is in Proposition 5.7. The extra equivalence
smuggled into (c) follows from the argument given for Theorem 6.7, since, for
any $J$, the differentials appearing augmented Mayer–Vietoris spectral
sequence associated to $\mathcal{U}_{I\\!,J}$ appear in that of
$\mathcal{U}_{I}$. After this, (b) is equivalent to (c) by Theorem 6.7. So it
is sufficient to show that (b) is equivalent to (d).

For brevity, we fix $J$ and write $\check{C}_{J}$ for the Čech double complex
associated to $\mathcal{U}_{I\\!,J}$, and $a\check{C}_{J}$ for the
corresponding augmented Čech double complex (29). There is then a short exact
sequence of complexes

(31)
$0\longrightarrow\operatorname{Tot}(\check{C}_{J})\longrightarrow\operatorname{Tot}(a\check{C}_{J})\longrightarrow\widetilde{C}^{*}(K_{J})[-1]\longrightarrow
0.$

It is well known that the Čech complex associated $\mathcal{U}_{I\\!,J}$ is
acyclic, that is, for each $p$ there is a quasi-isomorphism
$\widetilde{C}^{p}(\bigcup_{i\in I\cap J}K_{J\smallsetminus
i})\xrightarrow{\simeq}\check{C}^{*}(\mathcal{U}_{I\\!,J},\widetilde{C}^{p})$.
It follows that there is a quasi-isomorphism $\widetilde{C}^{*}(\bigcup_{i\in
I\cap J}K_{J\smallsetminus
i})\xrightarrow{\simeq}\operatorname{Tot}(\check{C}_{J})$. We also note that
$\bigcup_{i\in I\cap J}K_{J\smallsetminus i}=K_{J}\smallsetminus(I\cap J)$ is
the face deletion. Therefore, by taking cohomology (31) yields a long-exact
sequence

$\cdots\longrightarrow\widetilde{H}^{*}(K_{J}\smallsetminus(I\cap
J))\longrightarrow H^{*}(\operatorname{Tot}(a\check{C}_{J}))\xrightarrow{\ e\
}\widetilde{H}^{*-1}(K_{J})\xrightarrow{\ \delta\
}\widetilde{H}^{*-1}(K_{J}\smallsetminus(I\cap J))\longrightarrow\cdots$

By exactness, $e$ is surjective if and only if $\delta$ is zero; moreover the
connecting homomorphism $\delta$ is induced by the inclusion
$K_{J}\smallsetminus(I\cap J)\hookrightarrow K_{J}$, so this is equivalent to
(d). The edge map $e$ comes from the projection of $a\check{C}_{J}$ onto its
$-1$st row. Therefore $e$ is surjective if and only if, in the corresponding
spectral sequence (30), every differential $d_{s}$ leaving the $-1$st row is
zero, for $s\geqslant 1$. By Lemma 6.6 this is equivalent to (b). ∎

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