# Chow rings of matroids are Koszul

Matthew Mastroeni  Iowa State University, Department of Mathematics, Ames, IA,
USA<EMAIL_ADDRESS>and  Jason McCullough  Iowa State University,
Department of Mathematics, Ames, IA, USA<EMAIL_ADDRESS>
###### Abstract.

Chow rings of matroids were instrumental in the resolution of the Heron-Rota-
Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the
Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow
ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with
linear and quadratic relations defined by the matroid. Dotsenko conjectured
that the Chow ring of any matroid is Koszul. The purpose of this paper is to
prove Dotsenko’s conjecture. We also show that the augmented Chow ring of a
matroid is Koszul. As a corollary, we show that the Chow rings and augmented
Chow rings of matroids have rational Poincaré series.

###### Key words and phrases:

Koszul algebra, Chow ring, matroid, lattice, Gorenstein ring

###### 2020 Mathematics Subject Classification:

Primary: 16S37, 13E10, 05B35; Secondary: 13H10, 05E40

##  1\. Introduction

Let $M$ be a simple matroid on a finite set $E$. The Chow ring of $M$ is the
following quotient ring

$\operatorname{\underline{CH}}(M):=\underline{S}_{M}/(\underline{I}_{M}+\underline{J}_{M}),$

where

$S_{M}=\mathbb{Q}[x_{F}\mid F\text{ is a nonempty flat of }M],$

and $\underline{I}_{M}$ and $\underline{J}_{M}$ are the ideals:

$\displaystyle\underline{I}_{M}$ $\displaystyle=({\textstyle\sum_{i\in
F}}\,x_{F}\mid i\in E)$ $\displaystyle\underline{J}_{M}$
$\displaystyle=\left(x_{F}x_{G}\mid F,G\text{ are incomparable, nonempty flats
of }M\right).$

These algebras were employed in [AHK18] in the proof of the Heron-Rota-Welsh
Conjecture and more recently in [BHM+20b] in the proof of the Top-Heavy
Conjecture. It was shown that $\operatorname{\underline{CH}}(M)$ is a graded,
Artinian $\mathbb{Q}$-algebra with Poincaré duality and versions of the hard
Lefschetz and Hodge-Riemann conditions. In particular,
$\operatorname{\underline{CH}}(M)$ is a Gorenstein algebra with a presentation
by quadratic relations. It is natural to ask whether it is a Koszul algebra;
in fact, Dotsenko explicitly conjectured that
$\operatorname{\underline{CH}}(M)$ is always Koszul [Dot21, Conjecture 5.3].
Our first main result is to confirm Dotsenko’s conjecture.

######  Theorem 1.1.

For any simple matroid $M$, $\operatorname{\underline{CH}}(M)$ is Koszul.

It is well-known that all Koszul algebras have quadratic presentations; we
will call such algebras quadratic. However, not all quadratic algebras are
Koszul. In fact, it is not even guaranteed that a quadratic algebra will be
Koszul if it is also Artinian and Gorenstein; the first non-Koszul such
algebra was constructed by Matsuda [Mat18]. A thorough study of the Koszul
properties of quadratic, Gorenstein algebras was conducted by Schenck,
Stillman, and the first author in [MSS21a, MSS21b]. In particular, they
determined (with only 3 exceptions) the ordered pairs $(r,c)$ for which there
is a non-Koszul, quadratic, Gorenstein algebra of characteristic zero with
regularity $r$ and codimension $c$; see also [MS20].

In order for a quadratic algebra to be Koszul, it is sufficient for the
defining ideal of relations to have a quadratic Gröbner basis, yet there are
Koszul algebras whose defining ideals have no quadratic Gröbner bases with
respect to any monomial order or any linear change of variables. Previously,
Feichtner and Yuzvinsky [FY04] gave a non-quadratic Gröbner basis for a more
general construction of the Chow ring of an atomic lattice (see Section 2). An
argument similar to [Dot21, Proposition 3.1] shows that there is no possible
quadratic Gröbner basis for $\underline{I}_{M}+\underline{J}_{M}$ in the given
presentation.

Our approach to showing that $\operatorname{\underline{CH}}(M)$ is always
Koszul involves the construction of a Koszul filtration (see Theorem 4.12). A
key ingredient in the construction is the study of lattices that admit what we
call a total coatom ordering. This is an ordering on the elements of the
lattice satisfying certain properties similar to that of a shellable
simplicial complex. This aspect of the paper was inspired by and is similar to
the notion of a strongly shellable simplicial complex of Guo, Shen, and Wu
[GSW19]. A very similar notion is implicit in the work of Delucchi [Del08]. In
Section 3, we show that the lattice of flats of a matroid always admits a
total coatom ordering thereby making it possible to always construct a Koszul
filtration on $\operatorname{\underline{CH}}(M)$.

It is also natural to study the Poincaré series of
$\operatorname{\underline{CH}}(M)$. The Poincaré series of a
$\mathbb{K}$-algebra $R$, denoted $\operatorname{P}_{\mathbb{K}}^{R}(t)$, is
the generating function of the total Betti numbers of $\mathbb{K}$ over $R$:

$\operatorname{P}_{\mathbb{K}}^{R}(t):=\sum_{t\geq
0}\dim_{\mathbb{K}}\operatorname{Tor}^{R}_{i}(\mathbb{K},\mathbb{K})t^{i}.$

For many years, it was expected that all such Poincaré series were rational;
this expectation became known as the Serre-Kaplansky Problem. In [Ani82,
Example 7.1], Anick gave the first example of a commutative
$\mathbb{K}$-algebra with irrational Poincaré series. Later examples of
Artinian, Gorenstein algebras with irrational Poincaré series included one by
Bøgvad [Bøg83]. We observe that this algebra is quadratic (see Example 6.1).
Thus, not all quadratic, Artinian, Gorenstein $\mathbb{K}$-algebras have
rational Poincaré series. Our second main result, a corollary to the first, is
the following.

######  Theorem 1.2.

For any simple matroid $M$, $\operatorname{\underline{CH}}(M)$ has a rational
Poincaré series.

We also consider the augmented Chow ring $\operatorname{CH}(M)$ of a matroid
$M$. These algebras were introduced in [BHM+20a] as an analog of the Chow ring
of the augmented wonderful variety associated to a hyperplane arrangement in
the realizable case. Like the Chow ring of a matroid, these are quadratic,
Artinian, Gorenstein algebras satisfying the Kähler package. We show (Theorem
5.10) that the augmented Chow ring of a matroid is also Koszul.

The rest of the paper is organized as follows. Section 2 collects necessary
definitions and background results on lattices, matroids, Chow rings, and
Koszul algebras. Section 3 contains a proof that all geometric lattices admit
a total coatom ordering. Section 4 contains the proofs of our main results for
Chow rings of matroids, while Section 5 contains the analogous results for
augmented Chow rings of matroids. A final Section 6 collects some examples and
questions.

##  2\. Background

In this section, we collect some necessary notation and background for our
main results. The reader can safely skip to Section 3 and review this section
as necessary.

######  Notation.

Throughout this section, $R=\oplus_{i\geq 0}R_{i}$ will denote a commutative
graded $\mathbb{K}$-algebra finitely generated as an algebra over
$R_{0}=\mathbb{K}$ by elements of degree one. We call such an algebra standard
graded and set $R_{+}=\oplus_{i\geq 1}R_{i}$ to be the graded maximal ideal.

###  2.1. Hilbert functions and free resolutions

Let $M=\oplus_{i\in\mathbb{Z}}M_{i}$ be a finitely generated, graded
$R$-module. The Hilbert function of $M$ is defined as
$\operatorname{\mathrm{HF}}_{M}(i):=\dim_{\mathbb{K}}M_{i}$, and its
generating function is the Hilbert series of $M$, denoted
$\operatorname{\mathrm{H}}_{M}(t):=\sum_{i\geq
0}\operatorname{\mathrm{HF}}_{M}(i)t^{i}$. The module $M$ has a minimal graded
free resolution $\mathbf{F}_{\bullet}$ over $R$, which is an exact sequence of
the form

$\mathbf{F}_{\bullet}:\cdots\xrightarrow{\partial_{i+1}}F_{i}\xrightarrow{\partial_{i}}F_{i-1}\xrightarrow{\partial_{i-1}}\cdots\xrightarrow{\partial_{2}}F_{1}\xrightarrow{\partial_{1}}F_{0},$

where $M\cong\operatorname{Coker}(\partial_{1})$, each
$F_{i}=\oplus_{j}R(-j)^{\beta^{R}_{i,j}(M)}$ is a finite-rank, graded, free
$R$-module, and all maps are graded homomorphisms. Here, $R(-j)$ is the rank-
one free $R$-module with graded components $R(-j)_{i}=R_{i-j}$. The numbers
$\beta^{R}_{i,j}(M)$ are the graded Betti numbers of $M$ over $R$. The total
Betti numbers are denoted

$\beta_{i}^{R}(M):=\sum_{j}\beta_{i,j}^{R}(M),$

and their generating function

$\operatorname{P}_{M}^{R}(t)=\sum_{i\geq 0}\beta_{i}^{R}(M)t^{i}$

is called the Poincaré series of $M$. The regularity of $M$ is

$\operatorname{reg}_{R}(M)=\max\\{j-i\mid\beta_{i,j}^{R}(M)\neq 0\text{ for
some $i$}\\}.$

When no maximum exists, we say that $M$ has infinite regularity. When $M$ is
Artinian and $R$ is a polynomial ring, $\operatorname{reg}_{R}(M)=\max\\{i\mid
M_{i}\neq 0\\}$. See [Eis05] for connections between regularity, free
resolutions, and local cohomology.

###  2.2. Koszul algebras

A standard graded $\mathbb{K}$-algebra $R$ is called Koszul if $\mathbb{K}$
has a linear free resolution over $R$; equivalently, $R$ is Koszul if and only
if $\operatorname{reg}_{R}(\mathbb{K})=0$. This definition extends to the
noncommutative case, but all rings we consider in this paper are commutative.
Koszul algebras are ubiquitous in algebraic geometry and topology; they appear
as coordinate rings of canonical curves [VF93], quotients of polynomial rings
by quadratic monomial ideals [Frö75], and all sufficiently high Veronese
subalgebras of standard graded $\mathbb{K}$-algebras [Bac86]. There are
several equivalent conditions to being Koszul. Notably, Fröberg [Frö99] showed
that $R$ is Koszul if and only if

$\displaystyle\operatorname{P}_{\mathbb{K}}^{\,R}(t)\operatorname{\mathrm{H}}_{R}(-t)$
$\displaystyle=1.$ (2.1)

Moreover, it follows from work of Avramov, Eisenbud, and Peeva [AE92, AP01]
that $R$ is Koszul if and only if all finite, graded $R$-modules have finite
regularity. See [Con14] and [Frö99] for surveys of the theory of Koszul
algebras.

Any commutative Koszul algebra has a presentation $S/I$, where
$S=\mathbb{K}[x_{1},\ldots,x_{n}]$ is a standard graded polynomial ring over
$\mathbb{K}$ and $I$ is an ideal generated by quadrics (homogeneous
polynomials of degree $2$); we call such rings quadratic. Not all quadratic
$\mathbb{K}$-algebras are Koszul. It is sufficient for $I$ to have a quadratic
Gröbner basis with respect to some monomial order, perhaps after a change of
coordinates; such algebras are called G-quadratic. Yet not all Koszul algebras
are G-quadratic; see [Con14, Remark 1.14]. If $\ell\in R$ is a nonzerodivisor,
then $R$ is Koszul if and only if $R/\ell R$ is Koszul; see e.g. [CDNR13,
Theorem 3.1].

When considering the Chow ring of a matroid in the introduction, there are two
approaches one might take to prove the Koszul property. Since quadratic
monomial ideals, like $\underline{J}_{M}$, are Koszul, if the linear forms in
$\underline{I}_{M}$ formed a regular sequence on $S_{M}/\underline{J}_{M}$,
one could conclude Koszulness for $\operatorname{\underline{CH}}(M)$
immediately; however, in general, these linear forms do not form a regular
sequence. Another natural approach is to look for a quadratic Gröbner basis
for $\underline{I}_{M}+\underline{J}_{M}$. Yet, as Dotsenko has noted [Dot21],
in the given presentation there are obstructions that prevent the existence of
a quadratic Gröbner basis. It is not clear if a change of basis would resolve
this obstruction. We address this issue in Section 6.

In the absence of a quadratic Gröbner basis, one way to prove that a
$\mathbb{K}$-algebra has the Koszul property is to construct a Koszul
filtration, a concept first formally defined by Conca, Trung, and Valla.

######  Definition 2.1.

Let $R$ be a standard graded $\mathbb{K}$-algebra. A family $\mathcal{F}$ of
ideals of $R$ is said to be a Koszul filtration of $R$ if the following
conditions hold.

  1. (1)

Every ideal $I\in\mathcal{F}$ is generated by linear forms.

  2. (2)

The ideals $(0)$ and $R_{+}$ are in $\mathcal{F}$.

  3. (3)

For every nonzero $I\in\mathcal{F}$, there exists an ideal $J\in\mathcal{F}$
such that $J\subsetneq I$, $I/J$ is cyclic, and $(J:I)\in\mathcal{F}$.

If $R$ has a Koszul filtration, it follows from an inductive argument on the
ideals in the filtration that $R$ is Koszul [CTV01, Proposition 1.2].

###  2.3. Lattices

Recall that a (finite) lattice is a finite partially ordered set
$(\operatorname{\mathcal{L}},\leq)$ in which every two elements
$a,b\in\operatorname{\mathcal{L}}$ have a unique least upper bound called the
join, denoted $a\vee b$, and a unique greatest lower bound called the meet,
denoted $a\wedge b$. Consequently, all nonempty lattices have a unique minimum
element and maximum element, denoted $\widehat{0}$ and $\widehat{1}$
respectively.

Given $a,b\in\operatorname{\mathcal{L}}$ with $a\leq b$, the intervals $[a,b]$
and $(a,b]$ are the sets:

$\displaystyle[a,b]$ $\displaystyle=\\{c\in\operatorname{\mathcal{L}}\mid
a\leq c\leq b\\}$ $\displaystyle(a,b]$
$\displaystyle=\\{c\in\operatorname{\mathcal{L}}\mid a<c\leq b\\}.$

Clearly, every interval $[a,b]$ is also a lattice. If $a<b$ and there are no
elements $c\in\operatorname{\mathcal{L}}$ with $a<c<b$, then we say that $b$
covers $a$ or $b$ is a cover of $a$, and we write $a\lessdot b$. The Hasse
diagram of $\operatorname{\mathcal{L}}$ is a graph with vertices labeled by
the elements of $\operatorname{\mathcal{L}}$ positioned in such a way that if
$a<b$ then vertex $a$ is lower than vertex $b$; an edge from $a$ to $b$ is
drawn exactly when $b$ covers $a$. See Figure 2 for an example.

The elements $a\in\operatorname{\mathcal{L}}$ that cover $\widehat{0}$ are
called the atoms of $\operatorname{\mathcal{L}}$, collectively denoted
$\operatorname{at}(\operatorname{\mathcal{L}})$. The elements
$b\in\operatorname{\mathcal{L}}$ covered by $\widehat{1}$ are called the
coatoms of $\operatorname{\mathcal{L}}$, collectively denoted
$\operatorname{coat}(\operatorname{\mathcal{L}})$. For each element
$\widehat{0}\neq a\in\operatorname{\mathcal{L}}$, we also let
$\operatorname{coat}(a)$ denote the set of elements of
$\operatorname{\mathcal{L}}$ covered by $a$, which are precisely the coatoms
of the lattice $[\widehat{0},a]$.

The lattice $\operatorname{\mathcal{L}}$ is atomic if every element of
$\operatorname{\mathcal{L}}$ is a join of finitely many atoms. We say that
$\operatorname{\mathcal{L}}$ is graded (or ranked) if there is a function
$\operatorname{rk}:\operatorname{\mathcal{L}}\to\mathbb{Z}_{\geq 0}$ such that
$\operatorname{rk}\widehat{0}=0$ and if $a,b\in\operatorname{\mathcal{L}}$ and
$a\lessdot b$, then $\operatorname{rk}b=\operatorname{rk}a+1$, in which case
the number
$\operatorname{rk}\operatorname{\mathcal{L}}=\max\\{\operatorname{rk}a\mid
a\in\operatorname{\mathcal{L}}\\}$ is called the rank of
$\operatorname{\mathcal{L}}$. The lattice $\operatorname{\mathcal{L}}$ is
semimodular if for all $a,b\in\operatorname{\mathcal{L}}$ such that $a\wedge
b\lessdot a$ and $a\wedge b\lessdot b$, then $a\lessdot a\vee b$ and
$b\lessdot a\vee b$; equivalently, $\operatorname{\mathcal{L}}$ is semimodular
if it is graded and its rank function satisfies

$\operatorname{rk}a+\operatorname{rk}b\geq\operatorname{rk}(a\wedge
b)+\operatorname{rk}(a\vee b),$

for all $a,b\in\operatorname{\mathcal{L}}$. Finally, a lattice is called
geometric if it is finite, atomic, and semimodular. For basic results on
lattices, we refer the reader to [Sta12].

###  2.4. The lattice of flats of a matroid

A matroid $M$ is a pair $(E,\mathcal{I})$ consisting of a finite set $E$,
called the ground set of $M$, and a collection $\mathcal{I}$ of subsets of $E$
satisfying three properties:

  1. (1)

$\varnothing\in\mathcal{I}$.

  2. (2)

If $I\in\mathcal{I}$ and $I^{\prime}\subseteq I$, then
$I^{\prime}\in\mathcal{I}$.

  3. (3)

If $I_{1},I_{2}\in\mathcal{I}$ and $|I_{1}|<|I_{2}|$, then there exists an
element $e\in I_{2}\smallsetminus I_{1}$ such that $I_{1}\cup
e\in\mathcal{I}$.

######  Notation.

In the above definition and the subsequent sections, we abuse notation and
identify elements $e\in E$ with the corresponding set $\\{e\\}$. Thus, if
$F\subseteq E$, we will frequently write $F\cup e$ in place of $F\cup\\{e\\}$.

The members of $\mathcal{I}$ are called independent sets of $M$. A subset of
$E$ that is not in $\mathcal{I}$ is called dependent. A maximal independent
set is called a basis. All bases of a matroid have the same cardinality,
called the rank of $M$. Given a subset $X\subseteq E$, the rank of $X$,
denoted $\operatorname{rk}_{M}X$, is the cardinality of the largest
independent set contained in $X$; we drop the subscript when the matroid is
clear from context.

The closure of a subset $X\subset E$ in $M$ is

$\operatorname{cl}(X)=\\{e\in E\mid\operatorname{rk}(X\cup
e)=\operatorname{rk}X\\}.$

A subset $F$ is called a flat of $M$ if $F=\operatorname{cl}(F)$. A hyperplane
of $M$ is a flat $H$ of rank $\operatorname{rk}H=\operatorname{rk}M-1$ is
called. One can give an equivalent definition of matroid as a pair
$(E,\operatorname{\mathcal{L}})$ consisting of a finite set $E$ and a
collection $\operatorname{\mathcal{L}}$ of subsets of $E$, called flats,
satisfying:

  1. (1)

If $F,G\in\operatorname{\mathcal{L}}$, then $F\cap
G\in\operatorname{\mathcal{L}}$.

  2. (2)

If $F\in\operatorname{\mathcal{L}}$ and $e\in E\smallsetminus F$, then there
is a unique flat $G\in\operatorname{\mathcal{L}}$ that minimally contains
$F\cup e$.

Matroids can also be characterized by their rank functions, by their bases, or
by their minimal dependent sets.

The set of flats of $M$, denoted $\operatorname{\mathcal{L}}(M)$, has the
structure of a lattice; for any two flats
$F,G\in\operatorname{\mathcal{L}}(M)$, the meet is the intersection, $F\wedge
G=F\cap G$, and the join is the closure of the union, $F\vee
G=\operatorname{cl}(F\cup G)$. The atoms of $\operatorname{\mathcal{L}}(M)$
are precisely the rank-one flats, and the coatoms of
$\operatorname{\mathcal{L}}(M)$ are precisely the hyperplanes of $M$. A
lattice is geometric if and only if it is isomorphic to the lattice of flats
of some matroid [Oxl11, Theorem 1.7.5]. However, this correspondence is not
quite bijective, as different matroids can have isomorphic lattices of flats.

An element $e\in E$ is a loop of $M$ if the set $\\{e\\}$ is dependent. If
$e,f\in E$ are not loops, then $e$ and $f$ are parallel if $\\{e,f\\}$ is
dependent. A matroid is simple if it has no loops and no pairs of parallel
elements. For any matroid $M$, there is a unique simple matroid (up to
isomorphism) whose lattice of flats is isomorphic to
$\operatorname{\mathcal{L}}(M)$, called the simplification of a matroid $M$.
It can be constructed as the matroid on the set of rank-one flats of $M$ such
that a set of flats $\\{Y_{1},\dots,Y_{t}\\}$ is independent if and only if
$\operatorname{rk}_{M}(Y_{1}\vee\cdots\vee Y_{t})=t$.

######  Remark 2.2.

Since it is clear from the definition of the Chow ring of a matroid that any
two matroids with isomorphic lattices of flats will have isomorphic Chow
rings, we can always replace a matroid $M$ with its simplification, and so, we
may henceforth assume that all matroids under consideration are simple without
any loss of generality.

There are two constructions for producing new matroids from a given matroid
that will play an important role in the subsequent sections. Given a matroid
$M$ on a ground set $E$ and a subset $S\subseteq E$, the restriction of $M$ to
$S$ is the matroid $M|S$ on the ground set $S$ whose flats are of the form
$F\cap S$ for some flat $F$ of $M$. In particular, when $H$ is a flat of $M$,
every flat of $M|H$ is also a flat of $M$ so that $M|H$ is a matroid quotient
of $M$, and the lattice of flats of $M|H$ is just the interval
$[\varnothing,H]$ in $\operatorname{\mathcal{L}}(M)$. The truncation of $M$ is
the matroid $T(M)$ on the ground set $E$ with rank function
$\operatorname{rk}_{T(M)}X=\max\\{\operatorname{rk}_{M}X,\operatorname{rk}M-1\\}$.
The lattice of flats of the truncation $T(M)$ is obtained by removing all the
hyperplanes from the lattice of flats of $M$. We refer the reader to [Oxl11]
for further details about these constructions as necessary.

###  2.5. Chow rings of atomic lattices

We now recall the more general definition of the Chow ring of an atomic
lattice from [FY04]. Let $\operatorname{\mathcal{L}}$ be an atomic lattice. A
subset
$\mathcal{G}\subseteq\operatorname{\mathcal{L}}\smallsetminus\\{\widehat{0}\\}$
is a building set if for any
$G\in\operatorname{\mathcal{L}}\smallsetminus\\{\widehat{0}\\}$, there is a
poset isomorphism

$\varphi_{G}:\prod_{i=1}^{t}\,[\widehat{0},G_{i}]\to[\widehat{0},G],$

where $\max\mathcal{G}_{\leq G}=\\{G_{1},\ldots,G_{k}\\}$ and
$\varphi_{G}(\widehat{0},\ldots,\widehat{0},G_{i},\widehat{0},\ldots,\widehat{0})=G_{i}$
for $1\leq i\leq t$. Here, $\max\mathcal{G}_{\leq G}$ denotes the set of
maximal elements among all elements in $\mathcal{G}$ that are less than or
equal to $G$. Every atomic lattice $\operatorname{\mathcal{L}}$ has a unique
maximal building set
$\mathcal{G}_{\max}=\operatorname{\mathcal{L}}\smallsetminus\\{\widehat{0}\\}$
and a unique minimal building set $\mathcal{G}_{\min}$ consisting of the
irreducible elements in
$\operatorname{\mathcal{L}}\smallsetminus\\{\widehat{0}\\}$. (The atoms of
$\operatorname{\mathcal{L}}$ are always irreducible, but in general, there can
be more irreducibles than just the atoms; see [FK04, Section 2].) Given any
building set $\mathcal{G}$ of $\operatorname{\mathcal{L}}$, a subset
$\mathcal{S}\subseteq\mathcal{G}$ is called nested if for any pairwise
incomparable elements $G_{1},\ldots,G_{t}\in\mathcal{S}$, either $t\leq 1$ or
$G_{1}\vee\cdots\vee G_{t}\notin\mathcal{G}$. The set of nested subsets of
$\mathcal{G}$ is an abstract simplicial complex denoted by
$\mathcal{N}(\operatorname{\mathcal{L}},\mathcal{G})$.

######  Definition 2.3.

Let $\operatorname{\mathcal{L}}$ be a finite, atomic lattice and building set
$\mathcal{G}$ for $\operatorname{\mathcal{L}}$. Then the Chow ring of
$\operatorname{\mathcal{L}}$ with respect to $\mathcal{G}$ is the algebra

$D(\operatorname{\mathcal{L}},\mathcal{G}):=\mathbb{Z}[x_{G}\mid
G\in\mathcal{G}]/\mathcal{I},$

where $\mathcal{I}$ is generated by

$\prod_{i=1}^{t}G_{i}\qquad\text{for}\qquad\\{G_{1},\ldots,G_{t}\\}\notin\mathcal{N}(\operatorname{\mathcal{L}},\mathcal{G})$

and

$\sum_{G\geq A}x_{G}\qquad\text{for}\qquad
A\in\operatorname{at}(\operatorname{\mathcal{L}}).$

Note that $D(\operatorname{\mathcal{L}},\mathcal{G})$ is a quotient of the
Stanley-Reisner ring associated to the simplicial complex
$\mathcal{N}(\operatorname{\mathcal{L}},\mathcal{G})$.

For any building set, Feichtner and Yuzvinsky showed that
$D(\operatorname{\mathcal{L}},\mathcal{G})$ is the Chow ring of a certain
smooth, affine, toric variety [FY04, Theorem 3]. They also show that for any
essential complex hyperplane arrangement $\mathcal{A}$ and any building set
$\mathcal{G}$ for the intersection lattice
$\operatorname{\mathcal{L}}(\mathcal{A})$ containing $\\{0\\}$, the cohomology
ring of the De Concini and Procesi wonderful compactification
$\overline{Y}_{\mathcal{A},\mathcal{G}}$ of the arrangement [DCP95a, DCP95b]
is isomorphic to $D(\operatorname{\mathcal{L}}(\mathcal{A}),\mathcal{G})$.

######  Remark 2.4.

While reading [BES20] we discovered the following discrepancy: It is claimed
in [BES20, Remark 3.2.2] that the Chow ring of the graphic matroid $M(K_{n})$
of a complete graph on $n$ vertices is the cohomology ring of the Deligne-
Mumford space $\overline{\mathcal{M}_{0,n+1}}$, which was shown to be Koszul
by Dotsenko. This claim may be somewhat misleading. Although Dotsenko proves
the cohomology ring of $\overline{\mathcal{M}_{0,n+1}}$ is Koszul [Dot21] and
the cohomology ring is a Chow ring associated to the lattice of flats of
$M(K_{n})$, it is not _the_ Chow ring of $M(K_{n})$. The lattice of flats of
$M(K_{n})$ is isomorphic to the partition lattice $\Pi_{n}$ of partitions of
$\\{1,\dots,n\\}$ ordered by refinement. The distinction between the two rings
comes down to the choice of building set. The Chow ring of $M(K_{n})$ is the
Chow ring $D(\Pi_{n},\mathcal{G}_{\max})$ with respect to the maximal building
set $\mathcal{G}_{\max}=\Pi_{n}\smallsetminus\\{\widehat{0}\\}$. On the other
hand, as pointed out in [FY04, §7], the cohomology ring of
$\overline{\mathcal{M}_{0,n+1}}$ is isomorphic to
$D(\Pi_{n},\mathcal{G}_{\min})$ with respect to the minimal building set.
These two rings are not isomorphic, which can easily be seen for $n=4$ from
the fact that both rings are Artinian with nondegenerate presentations of
different codimensions. Thus, to the best of our knowledge, there is not any
large class of matroids whose Chow rings were previously known to be Koszul.

At the other extreme, the Chow ring of a matroid $M$ is the special case
corresponding to the maximal building set of the lattice of flats of $M$; that
is,

$\operatorname{\underline{CH}}(M)\cong
D(\operatorname{\mathcal{L}}(M),\mathcal{G}_{\mathrm{max}})\otimes_{\mathbb{Z}}\mathbb{Q}.$

##  3\. Lattices with total coatom orderings

As Björner writes in [Bjö92, p. 232], _“A number of remarkable properties of
matroids are revealed by, but not dependent on, assigning a linear order to
the underlying point set.”_ In this section, we follow Björner’s lead and
define the notion of a total coatom ordering on a lattice. The definition is
related to and inspired by the definition of a strongly shellable simplicial
complex of Guo, Shen, and Wu in [GSW19], where they showed that the
independence complex of a matroid is strongly shellable. We prove a parallel
result (Theorem 3.3) that the lattice of flats of a matroid admits a total
coatom ordering. This result is essential to creating the Koszul filtration on
the Chow ring of a matroid in Theorem 4.12.

######  Definition 3.1.

Let $\operatorname{\mathcal{L}}$ be a lattice, and let $\prec$ be any total
order on $\operatorname{\mathcal{L}}$. Given an element
$F\in\operatorname{\mathcal{L}}$, a set
$\mathcal{G}\subseteq\operatorname{coat}(F)$ is called an initial segment
covered by $F$ if for all $G,G^{\prime}\in\operatorname{coat}(F)$ such that
$G^{\prime}\in\mathcal{G}$ and $G\prec G^{\prime}$, then $G\in\mathcal{G}$.

######  Definition 3.2.

We say that a finite lattice $\operatorname{\mathcal{L}}$ admits total coatom
ordering if there is a total order $\prec$ on $\operatorname{\mathcal{L}}$
with the following properties for all $F\in\operatorname{\mathcal{L}}$:

  1. (i)

For all $G,G^{\prime}\in\operatorname{coat}(F)$, if $G\prec G^{\prime}$, then
there exists $G^{\prime\prime}\in\operatorname{coat}(F)$ such that:

    1. (i.a)

$G^{\prime\prime}\prec G^{\prime}$,

    2. (i.b)

$G^{\prime}\wedge G^{\prime\prime}\in\operatorname{coat}(G^{\prime})$, and

    3. (i.c)

$G\wedge G^{\prime}\leq G^{\prime\prime}$.

  2. (ii)

If $\mathcal{G}\subseteq\operatorname{\mathcal{L}}$ is an initial segment
covered by $F$ and $G^{\prime}$ is the largest element in $\mathcal{G}$ with
respect to $\prec$, then the set

$\operatorname{coat}_{\mathcal{G}}(G^{\prime}):=\\{G\wedge G^{\prime}\mid
G\in\mathcal{G}\text{ and }G\wedge
G^{\prime}\in\operatorname{coat}(G^{\prime})\\}$

is an initial segment covered by $G^{\prime}$.

The first defining property of a total coatom order can be visualized in the
Hasse diagram of $\operatorname{\mathcal{L}}$ as shown in Figure 1; the dashed
lines represent relations in the lattice that are not necessarily covering
relations.

$F$$G^{\prime}$$G^{\prime\prime}$$G$$G^{\prime\prime}\wedge
G^{\prime}$$G\wedge G^{\prime}$$\prec$$\prec$ Figure 1. A total coatom order

First, we show that geometric lattices have the above property.

######  Theorem 3.3.

Geometric lattices admit total coatom orderings.

###### Proof.

Let $\operatorname{\mathcal{L}}$ be a geometric lattice. We may assume that
$\operatorname{\mathcal{L}}$ is the lattice of flats of some simple matroid
$M$ on the ground set $E=[n]=\\{1,2,\dots,n\\}$. Given flats
$F^{\prime}=\\{i_{1}>\cdots>i_{r}\\}$ and $F=\\{j_{1}>\cdots>j_{s}\\}$ of $M$,
we will say that $F^{\prime}\succ F$ if
$\operatorname{rk}F^{\prime}>\operatorname{rk}F$ or if
$\operatorname{rk}F^{\prime}=\operatorname{rk}F$ and for some
$\ell\leq\min\\{r,s\\}$ we have $i_{p}=j_{p}$ for all $p<\ell$ and
$i_{\ell}<j_{\ell}$. We claim that $\prec$ is the desired total coatom order
of $\operatorname{\mathcal{L}}$.

First, we observe $\prec$ is compatible with restriction to flats of $M$ in
the sense that the induced order on the lattice of flats of $M|F$ is the
analogously defined order on $M|F$ for any flat
$F\in\operatorname{\mathcal{L}}$. Hence, it suffices to prove that properties
(i) and (ii) of Definition 3.2 are satisfied when $F=E$ so that $G$ and
$G^{\prime}$ are hyperplanes of $M$. Since $\operatorname{\mathcal{L}}$ is a
geometric lattice, we also note that saying a flat $F$ covers a flat $G$ is
equivalent to saying that $G\subseteq F$ and
$\operatorname{rk}G=\operatorname{rk}F-1$.

(i) Let $H^{\prime}$ and $H$ be hyperplanes of $M$ such that $H\prec
H^{\prime}$. Let $X$ be a basis for $H\cap H^{\prime}$ (in other words, a
basis for the matroid $M|(H\cap H^{\prime})$). Then
$\operatorname{cl}(X)=H\cap H^{\prime}$ so that
$\left|X\right|=\operatorname{rk}(X)=\operatorname{rk}(H\cap H^{\prime})$. We
can then find a set $Y\subseteq E$ with $X\cap Y=\varnothing$ such that $X\cup
Y$ is a basis for $H^{\prime}$.

Suppose that $H^{\prime}=\\{i_{1}>\cdots>i_{r}\\}$ and
$H=\\{j_{1}>\cdots>j_{s}\\}$. By assumption, there is an
$\ell\leq\min\\{r,s\\}$ such that $i_{\ell}<j_{\ell}<j_{\ell-1}=i_{\ell-1}$ so
that $j_{\ell}\notin H^{\prime}$. Hence, $X\cup Y\cup j_{\ell}$ is a basis for
$M$. We note that
$\left|Y\right|=\operatorname{rk}H^{\prime}-\operatorname{rk}(H\cap
H^{\prime})\geq 1$. Consider the hyperplane
$H^{\prime\prime}=\operatorname{cl}(X\cup(Y\smallsetminus y)\cup j_{\ell})$
for some $y\in Y$, and suppose that
$H^{\prime\prime}=\\{k_{1}>\cdots>k_{t}\\}$. Since $X\cup(Y\smallsetminus
y)\subseteq H\cap H^{\prime\prime}$, we clearly have $\operatorname{rk}(H\cap
H^{\prime\prime})=\operatorname{rk}H-1$ by construction, and
$\\{j_{1},\dots,j_{\ell-1}\\}\subseteq H\cap
H^{\prime}=\operatorname{cl}(X)\subseteq H^{\prime\prime}$ so that $k_{i}\geq
j_{i}$ for all $i\leq\ell$. It follows that for some $h\leq\ell$ we have
$k_{h}>i_{h}$ and $k_{p}=i_{p}$ for all $p<h$ so that $H^{\prime\prime}\prec
H^{\prime}$.

(ii) Let $\mathcal{H}$ be an initial segment of hyperplanes of $M$ with
largest element $H^{\prime}=\\{i_{1}>i_{2}>\cdots>i_{r}\\}$. We must prove
that the set $\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ of induced
hyperplanes in $M|H^{\prime}$ is also an initial segment. Let
$F^{\prime\prime}\prec F$ be hyperplanes of $M|H^{\prime}$ with
$F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$. We must show that
$F^{\prime\prime}\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$. As
$F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$, we know there is a
hyperplane $H\in\mathcal{H}\smallsetminus\\{H^{\prime}\\}$ with $F=H\cap
H^{\prime}$. Since $H\prec H^{\prime}$ by assumption, we know that there is an
index $\ell$ and element $j>i_{\ell}$ of the ground set of $M$ such that

$H=\\{i_{1}>\cdots>i_{\ell-1}>j>j_{\ell+1}>\cdots>j_{s}\\}.$

In particular, we note that $j\notin H^{\prime}$ and
$F=\\{i_{1}>\cdots>i_{\ell-1}>i_{h_{\ell}}>\cdots>i_{h_{t}}\\}$ for some
indices $h_{k}$. Since $F^{\prime\prime}\prec F$, we similarly know that
$F^{\prime\prime}$ contains an element $e$ not in $F$. As
$F^{\prime\prime}\smallsetminus F\subseteq H^{\prime}\smallsetminus F$, it
follows that $i_{\ell}\geq e$ so that
$F^{\prime\prime}=\\{i_{1}>\cdots>i_{\ell-1}>i_{h_{\ell}}>\cdots>i_{h_{p-1}}>e>e_{p+1}>\cdots>e_{q}\\}$.

Consider the hyperplane
$H^{\prime\prime}=\operatorname{cl}(F^{\prime\prime}\cup j)$. On the one hand,
we have $F^{\prime\prime}\subseteq H^{\prime}\cap H^{\prime\prime}$ so that
$\operatorname{rk}H^{\prime}-1=\operatorname{rk}F^{\prime\prime}\leq\operatorname{rk}(H^{\prime}\cap
H^{\prime\prime})$. On the other hand, we note that $H^{\prime\prime}\neq
H^{\prime}$ since $j\notin H^{\prime}$, and so, it follows that
$\operatorname{rk}(H^{\prime}\cap H^{\prime\prime})\leq\operatorname{rk}M-1$
since the lattice of flats of $M$ is semimodular. Hence,
$\operatorname{rk}F^{\prime\prime}=\operatorname{rk}(H^{\prime}\cap
H^{\prime\prime})$ so that $F^{\prime\prime}=H^{\prime}\cap H^{\prime\prime}$,
and so, we are done if we can show that $H^{\prime\prime}\prec H^{\prime}$. If
this is not the case, then $H^{\prime}\prec H^{\prime\prime}$ so that there is
an index $m$ such that
$H^{\prime\prime}=\\{i_{1}>\cdots>i_{m-1}>k_{m}>\cdots>k_{b}\\}$, where
$i_{m}>k_{m}$. In particular, $i_{m}\notin H^{\prime\prime}$ so that
$m\geq\ell$. If $m>\ell$, then $j\in H^{\prime\prime}$ and $j>i_{\ell}\geq
i_{m-1}$, which is impossible. Thus, we must have $m=\ell$ and $k_{m}\geq
j>i_{\ell}$, which implies that $H^{\prime\prime}\prec H^{\prime}$, a
contradiction. Hence, $H^{\prime\prime}\prec H^{\prime}$ as wanted. ∎

######  Example 3.4.

Consider the uniform matroid $U_{5,5}$ on the ground set $E=\\{1,2,3,4,5\\}$.
For simplicity of notation, we will write subsets of $E$ as strings of their
elements. Since every subset of $E$ is independent, the hyperplanes of
$U_{5,5}$ are precisely the subsets of size 4. Take
$M=T_{H^{\prime}}(U_{5,5})$ to be the principal truncation of $U_{5,5}$ with
respect to the hyperplane $H^{\prime}=1234$. (See [Oxl11, p.279] for a
definition of principal truncation.) Equivalently, $M$ can be viewed as just
the matroid of linearly independent columns of the matrix

$A=\begin{pmatrix}1&0&0&1&0\\\ 0&1&0&1&0\\\ 0&0&1&1&0\\\
0&0&0&0&1\end{pmatrix},\vspace{0.5 em}$

where the columns are indexed $1$ to $5$. The lattice of flats of $M$ is as
shown in Figure 2. The flats of $M$ have been arranged in descending order
according to the total coatom order of the preceding theorem when read from
top to bottom and left to right.

1234512341251352351452453451213231424341525354512345$\varnothing$ Figure 2.
The lattice $\operatorname{\mathcal{L}}(M)$

A finite lattice $\operatorname{\mathcal{L}}$ with
$\widehat{0}\neq\widehat{1}$ admits a recursive coatom ordering if there is a
total order $\prec=\prec_{\widehat{1}}$ on the set of coatoms of
$\operatorname{\mathcal{L}}$ such that:

  1. (i)

For all $H,H^{\prime}\in\operatorname{coat}(\operatorname{\mathcal{L}})$, if
$H\prec H^{\prime}$ and $G\in\operatorname{\mathcal{L}}$ such that
$G<H,H^{\prime}$, then there exists an
$H^{\prime\prime}\in\operatorname{coat}(\operatorname{\mathcal{L}})$ such
that:

    1. (i.a)

$H^{\prime\prime}\prec H^{\prime}$,

    2. (i.b)

There exists $F\in\operatorname{coat}(H^{\prime})$ such that $G\leq
F<H^{\prime\prime}$

  2. (ii)

For each $H^{\prime}\in\operatorname{coat}(\operatorname{\mathcal{L}})$, the
interval $[\widehat{0},H^{\prime}]$ admits a recursive coatom ordering
$\prec_{H^{\prime}}$ such that if
$F\in\operatorname{coat}(H)\cap\operatorname{coat}(H^{\prime})$ for some
$H\prec H^{\prime}$ and
$F^{\prime}\in\operatorname{coat}(H^{\prime})\smallsetminus\bigcup_{H^{\prime\prime}\prec
H}\operatorname{coat}(H^{\prime\prime})$, then
$F\prec_{H^{\prime}}F^{\prime}$.

By convention, we also consider the lattice with $\widehat{0}=\widehat{1}$ to
admit a recursive coatom order.

It is known that geometric lattices admit recursive atom orderings [WW86,
Theorem 7.2], the dual notion to recursive coatom orderings. We could not find
a reference in the literature for the following result and include a proof for
completeness. The following argument was communicated to us by Vic Reiner.

######  Proposition 3.5.

If $\operatorname{\mathcal{L}}$ is a geometric lattice, then
$\operatorname{\mathcal{L}}$ admits a recursive coatom ordering.

###### Proof.

If $\operatorname{\mathcal{L}}$ is geometric, then
$\operatorname{\mathcal{L}}$ is SL-shellable by [Bjö80, Theorem 3.7]. The dual
$\operatorname{\mathcal{L}}^{\ast}$ of the SL-shellable lattice
$\operatorname{\mathcal{L}}$ is also SL-shellable by [Bjö80, Proposition 3.5].
By definition, $\operatorname{\mathcal{L}}^{\ast}$ is then EL-shellable
[Bjö80, Definition 3.4] and hence CL-shellable by [BW83, Proposition 2.3].
Since $\operatorname{\mathcal{L}}^{\ast}$ is CL-shellable,
$\operatorname{\mathcal{L}}^{\ast}$ admits a recursive atom ordering by [BW96,
Theorem 5.11], which is equivalent to $\operatorname{\mathcal{L}}$ admitting a
recursive coatom ordering. ∎

We refer the reader to the cited references for precise definitions of the
variants of shellability; we will not need them for the remainder of this
paper.

It is clear from the definition that a total recursive coatom ordering induces
a recursive coatom ordering. That the reverse also holds is essentially the
content of [Del08, Lemma 2.10] restricted to the case of a graded lattice.

######  Proposition 3.6.

Let $\operatorname{\mathcal{L}}$ be a graded lattice. Then
$\operatorname{\mathcal{L}}$ admits a recursive coatom orderings if and only
if $\operatorname{\mathcal{L}}$ admits a total coatom ordering.

We refer the reader to [Del08] for a proof. Thus one could replace Theorem 3.3
by appealing to Propositions 3.5 and 3.6; however, we find our argument more
constructive and direct. By [BW82, Theorem 3.3], CL-shellable lattices (or
more generally, graded posets) are shellable. Thus, graded lattices with a
total coatom order are shellable.

##  4\. The Koszul property of the Chow ring of a matroid

The aim of this section is to show that the Chow ring of any matroid is Koszul
by building a suitable Koszul filtration. First, we describe a presentation of
$\operatorname{\underline{CH}}(M)$ and a corresponding Gröbner basis better
suited to our purposes.

###  4.1. The Atom-free presentation of the Chow Ring of a Matroid

Let $M$ be a simple matroid with finite ground set $E$ and lattice of flats
$\operatorname{\mathcal{L}}=\operatorname{\mathcal{L}}(M)$. Abusing notation
slightly, we identify elements $i\in E$ with the rank-one flats $\\{i\\}$ of
$M$. There are now several presentations of the Chow ring of a matroid. The
Feichtner-Yuzvinsky presentation of the Chow ring of $M$ [FY04] is the ring

$\operatorname{\underline{CH}}_{FY}(M)=\mathbb{Q}\left[x_{F}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}\right]/I_{FY}(M),$

where

$I_{FY}(M)=\left(x_{F}x_{F^{\prime}}\mid
F,F^{\prime}\;\text{incomparable}\;\right)+\left(\textstyle\sum_{i\in
F}\,x_{F}\mid i\in E\right).$

In several papers [AHK18, BHM+20a, BHM+20b], the variable $x_{E}$ is
eliminated so that variables are associated to nonempty, proper flats.
Instead, we eliminate the variables associated to the atoms of
$\operatorname{\mathcal{L}}$ to define a quadratic presentation.

######  Definition 4.1.

Let $\operatorname{\mathcal{L}}_{\geq 2}$ denote the set of flats of $M$ of
rank at least 2. We define the atom-free presentation of the Chow ring of $M$
to be the ring

$\operatorname{\underline{CH}}_{\operatorname{af}}(M)=\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}_{\geq 2}]/I_{\operatorname{af}}(M),$

where

$\displaystyle I_{\operatorname{af}}(M)$
$\displaystyle=\left(x_{F}x_{F^{\prime}}\mid
F,F^{\prime}\in\operatorname{\mathcal{L}}_{\geq
2}\;\text{incomparable}\;\right)+\left(\textstyle
x_{F}\sum_{F^{\prime}\supseteq F\vee
i}\,x_{F^{\prime}}\,\bigg{\rvert}\,F\in\operatorname{\mathcal{L}}_{\geq
2},i\in E\smallsetminus F\right)$
$\displaystyle+\left(\textstyle\sum_{F\supseteq i\vee
j}\,x_{F}^{2}+\sum_{F^{\prime}\supsetneq F\supseteq i\vee
j}\,2x_{F}x_{F^{\prime}}\,\bigg{\rvert}\,i,j\in E,i\neq j\right).$

After a change variables $\varphi$ on the ring $\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}]$ sending
$x_{i}\mapsto x_{i}-\sum_{F\supsetneq i}x_{F}$ for each $i\in E$, we see that

$\displaystyle\begin{split}\varphi(I_{FY}(M))&=\left(x_{i}\mid i\in
E)+(x_{F}x_{F^{\prime}}\mid F,F^{\prime}\in\operatorname{\mathcal{L}}_{\geq
2}\;\text{incomparable}\;\right)\\\ &+\left(\textstyle
x_{F}\sum_{F^{\prime}\supsetneq
i}\,x_{F^{\prime}}\,\bigg{\rvert}\,F\in\operatorname{\mathcal{L}}_{\geq
2},i\in E\smallsetminus F\right)+\left(\textstyle\sum_{F\supsetneq
i,F^{\prime}\supsetneq j}\,x_{F}x_{F^{\prime}}\,\bigg{\rvert}\,i,j\in E,i\neq
j\right)\\\ &=\left(x_{i}\mid i\in
E\right)+I_{\operatorname{af}}(M),\end{split}$ (4.1)

so that
$\operatorname{\underline{CH}}_{FY}(M)\cong\mathbb{Q}[x_{F}\mid\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}]/((x_{i}\mid
i\in
E)+I_{\operatorname{af}}(M))\cong\operatorname{\underline{CH}}_{\operatorname{af}}(M)$.
There is also a simplicial presentation of the Chow ring of a matroid [BES20],
but we will not need it here.

######  Remark 4.2.

Although we use $\mathbb{Q}$ as our coefficient field in this paper to keep
with the terminology established in [BHM+20b], our constructions work equally
well over any coefficient field.

While a quadratic Gröbner basis for $I_{FY}(M)$ or
$I_{{\operatorname{af}}}(M)$ in not known, a non-quadratic Gröbner basis for
$I_{FY}(M)$ was computed by Feichtner and Yuzvinsky, which we recall here.

######  Theorem 4.3 ([FY04, Theorem 2]).

With respect to any lexicographic order $>$ such that $x_{F}>x_{G}$ implies
$F\nsupseteq G$, the ideal $I_{FY}(M)$ has a Gröbner basis consisting of the
following polynomials for all
$F,F^{\prime}\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}$:

$\displaystyle x_{F^{\prime}}x_{F}$ $\displaystyle
F,F^{\prime}\text{incomparable}$ (4.2) $\displaystyle
x_{F^{\prime}}\left(\textstyle\sum_{G\supseteq
F}\,x_{G}\right)^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}$
$\displaystyle F^{\prime}\subsetneq F$ (4.3)
$\displaystyle\left(\textstyle\sum_{G\supseteq
F}\,x_{G}\right)^{\operatorname{rk}F}$ (4.4)

A slight modification yields a non-quadratic Gröbner basis in the atom-free
setting.

######  Corollary 4.4.

With respect to any lexicographic order $>$ such that $x_{F}>x_{G}$ implies
$F\nsupseteq G$, the ideal $I_{\mathrm{af}}(M)$ has a Gröbner basis consisting
of the following polynomials for all
$F,F^{\prime}\in\operatorname{\mathcal{L}}_{\geq 2}$:

$\displaystyle x_{F}x_{F^{\prime}}$ $\displaystyle
F,F^{\prime}\text{incomparable}$ (4.5) $\displaystyle
x_{F^{\prime}}\left(\textstyle\sum_{G\supseteq
F}\,x_{G}\right)^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}$
$\displaystyle F^{\prime}\subsetneq F$ (4.6)
$\displaystyle\left(\textstyle\sum_{G\supseteq
F}\,x_{G}\right)^{\operatorname{rk}F}$ (4.7)

###### Proof.

As in the proof of [BES20, 3.2.2], we note that the automorphism $\varphi$ of
$\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}]$ of (4.1)
satisfies $\operatorname{in}_{>}\varphi(x_{F})=x_{F}$ for all flats $F$;
hence, $\operatorname{in}_{>}\varphi(f)=\operatorname{in}_{>}f$ for all
polynomials $f$ so that $I_{FY}(M)$ and $I_{\mathrm{af}}(M)$ have the same
initial ideal and the images under $\varphi$ of the Gröbner basis for
$I_{FY}(M)$ form a Gröbner basis for $\varphi(I_{FY}(M))=(x_{i}\mid i\in
E)+I_{\mathrm{af}}(M)$. This Gröbner basis includes the polynomials listed
above and the polynomials:

$\displaystyle x_{F}\left(\textstyle x_{i}-\sum_{F^{\prime}\supsetneq
i}\,x_{F^{\prime}}\right)$ $\displaystyle F\in\operatorname{\mathcal{L}}_{\geq
2},i\in E,F\nsupseteq i$ (4.8) $\displaystyle\left(\textstyle
x_{i}-\sum_{F^{\prime}\supsetneq
a}\,x_{F^{\prime}}\right)\left(\textstyle\sum_{G\supseteq
F}x_{G}\right)^{\operatorname{rk}F-1}$ $\displaystyle i\in E,i\subsetneq F$
(4.9) $\displaystyle x_{i}$ $\displaystyle i\in E$ (4.10)

From this, we see that

$\displaystyle\begin{split}\operatorname{in}_{>}(\varphi(I_{FY}(M))&=\left(x_{i}\mid
i\in E\right)+\left(x_{F}x_{F^{\prime}}\mid
F,F^{\prime}\in\operatorname{\mathcal{L}}_{\geq
2}\;\text{incomparable}\right)\\\
&+\left(x_{F^{\prime}}x_{F}^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}\mid
F,F^{\prime}\in\operatorname{\mathcal{L}}_{\geq 2},F^{\prime}\subsetneq
F\right)+\left(x_{F}^{\operatorname{rk}F}\mid
F\in\operatorname{\mathcal{L}}_{\geq 2}\right),\end{split}$ (4.11)

which shows that omitting the above polynomials in (4.8)–(4.9) still leaves a
Gröbner basis for $\varphi(I_{FY}(M))$.

Since $<$ is an elimination order for the variables $x_{i}$ with $i\in E$, it
follows that the polynomials listed in the statement of the corollary are a
Gröbner basis for the elimination ideal
$\varphi(I_{FY}(M))\cap\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}_{\geq 2}]$ by [EH12, 3.3]. It then suffices to
note that $\varphi(I_{FY}(M))\cap\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}_{\geq 2}]=\pi((x_{i}\mid i\in
E)+I_{\operatorname{af}}(M))=I_{\operatorname{af}}(M)$, where
$\pi:\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}]\to\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}_{\geq 2}]$ is the natural algebra retract
sending $x_{i}\mapsto 0$ for all $i\in E$. ∎

Combining (4.11) with Macaulay’s Theorem yields the following.

######  Corollary 4.5.

The ring $\operatorname{\underline{CH}}_{\mathrm{af}}(M)$ has a
$\mathbb{Q}$-basis consisting of all monomials

$x_{F}^{\alpha}=x_{F_{1}}^{\alpha_{1}}\cdots x_{F_{r}}^{\alpha_{r}}$

for all chains of flats $F=\\{F_{1}\supsetneq F_{2}\supsetneq\cdots\supsetneq
F_{r}\supsetneq F_{r+1}=\varnothing\\}$ for some $r\geq 0$ with
$F_{i}\in\operatorname{\mathcal{L}}_{\geq 2}$ for all $i\leq r$ and all
$\alpha\in\mathbb{Z}_{\geq 0}^{r+1}$ such that
$\sum_{i}\alpha_{i}=\operatorname{rk}F_{1}$ and
$1\leq\alpha_{i}<\operatorname{rk}F_{i}-\operatorname{rk}F_{i+1}$ if $i\leq
r$.

Following [FY04] and [BES20], we call the monomials of Corollary 4.5 the
nested monomials of $\operatorname{\underline{CH}}_{\operatorname{af}}(M)$. In
this notation, we note that $x_{\\{\varnothing\\}}^{\mathbf{0}}=1$.

Adiprasito, Huh, and Katz [AHK18] proved that
$\operatorname{\underline{CH}}(M)$ has the Kähler package; that is,
$\operatorname{\underline{CH}}(M)$ is an Artinian, Gorenstein
$\mathbb{Q}$-algebra of regularity
$\operatorname{reg}(\operatorname{\underline{CH}}(M))=\operatorname{rk}(M)-1$
with versions of the hard Lefschetz and Hodge-Riemann conditions. Other proofs
of these properties have now been given in [BES20] and [BHM+20a]. It is well
known that an Artinian, graded $\mathbb{K}$-algebra $A$ is a Poincaré duality
algebra if and only if $A$ is Gorenstein; see e.g. [MW09, Proposition 2.1].

######  Remark 4.6.

As a consequence of Corollary 4.4, we note that $I_{\mathrm{af}}(M)$ contains
the polynomials

$x_{H}x_{E},\qquad
x_{F}x_{H}^{\operatorname{rk}H-\operatorname{rk}F}+x_{F}x_{E}^{\operatorname{rk}M-\operatorname{rk}F-1},\qquad
x_{H}^{\operatorname{rk}H}+x_{E}^{\operatorname{rk}M-1}$ (4.12)

for any $F\in\operatorname{\mathcal{L}}_{\geq 2}$ and hyperplane $H\supsetneq
F$. Moreover, we have $x_{F}x_{E}^{\operatorname{rk}M-1}=0$ in
$\operatorname{\underline{CH}}_{\operatorname{af}}(M)$ for all
$F\in\operatorname{\mathcal{L}}_{\geq 2}$. Since
$\operatorname{\underline{CH}}_{\operatorname{af}}(M)$ is an Artinian
Gorenstein ring, it follows that $x_{E}^{\operatorname{rk}M-1}$ is a socle
generator of $\operatorname{\underline{CH}}_{\operatorname{af}}(M)$. These
observations will be useful in the computations below.

###  4.2. A Koszul filtration for the Chow ring of a matroid

We are now in position to describe the Koszul filtration. When $M$ is a
matroid of rank 1 or 2, we note that
$\operatorname{\underline{CH}}(M)\cong\mathbb{Q}$ or
$\operatorname{\underline{CH}}(M)\cong\mathbb{Q}[x_{E}]/(x_{E}^{2})$,
respectively. In either case, the Chow ring is clearly Koszul, and moreover,
it is has a Koszul filtration. In the former case, the filtration consists of
only the zero ideal, and in the latter case, the filtration consists of the
zero ideal and the graded, maximal ideal. Hence, in what follows, we may
assume the following without any loss of generality.

######  Notation.

Throughout the remainder of this section, $M$ denotes a simple matroid with
$\operatorname{rk}M\geq 3$ and $A=\operatorname{\underline{CH}}(M)$ with
respect to the atom-free presentation.

Although we will not exactly present the proof this way, the idea behind the
construction of the desired Koszul filtration is that we can use natural
matroid operations such as truncation and restriction to reduce the rank of
our matroid down to the low-rank cases mentioned above and inductively lift a
Koszul filtration to the Chow ring of our arbitrary matroid $M$. Recall that a
flat $H$ of $M$ is called a hyperplane if
$\operatorname{rk}H=\operatorname{rk}M-1$, and the hyperplanes are precisely
the coatoms in the lattice of flats
$\operatorname{\mathcal{L}}=\operatorname{\mathcal{L}}(M)$. Since the
hyperplanes are exactly the flats omitted when truncating the matroid $M$, we
are naturally led to study ideals generated by hyperplane variables.

######  Lemma 4.7.

Let $\mathcal{H}$ be a set of hyperplanes of $M$. Then the ideal $(x_{H}\mid
H\in\mathcal{H})$ has a $\mathbb{Q}$-basis consisting of all nested monomials
$x_{F}^{\alpha}$ with $F=\\{F_{1}\supsetneq F_{2}\supsetneq\cdots\supsetneq
F_{r}\supsetneq F_{r+1}=\varnothing\\}$ for some $r\geq 0$ such that either:

  1. (i)

$F_{1}\in\mathcal{H}$, or

  2. (ii)

$F_{1}=E$, $H\supsetneq F_{2}$ for some $H\in\mathcal{H}$, and
$\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1$.

###### Proof.

Consider the case of a single hyperplane $H$. The general case follows easily
from the principal case. If $F_{1}=H$, then clearly
$x_{F}^{\alpha}\in(x_{H})$. If we have $F_{1}=E$, $H\supsetneq F_{2}$, and
$\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1$, then by (4.12) we
have $x_{E}^{\alpha_{1}}x_{F_{2}}=-x_{H}^{\alpha_{1}}x_{F_{2}}$ provided
$F_{2}\neq\varnothing$ or $x_{E}^{\alpha_{1}}=-x_{H}^{\alpha_{1}}$ if
$F_{2}=\varnothing$; either way, it follows that $x_{F}^{\alpha}\in(x_{H})$.
Hence, $(x_{H})$ contains all monomials in the statement of the lemma.
Conversely, if $f\in(x_{H})$, then $f=x_{H}g$ for some
$g\in\operatorname{\underline{CH}}(M)$. Write $g=\sum
c_{F,\alpha}x_{F}^{\alpha}$ as a $\mathbb{Q}$-linear combination of nested
monomials. For any monomial $x_{F}^{\alpha}$ in the support of $g$, we see
that $x_{H}x_{F}^{\alpha}$ is nonzero only if $F_{1}\subseteq H$. Indeed, if
$F_{1}\nsubseteq H$, then either $F_{1}$ and $H$ are incomparable or
$F_{1}=E$. Either way, the defining relations of
$\operatorname{\underline{CH}}(M)$ or (4.12) imply that $x_{F_{1}}x_{H}=0$. If
$F_{1}=H$, then $x_{H}x_{F}^{\alpha}$ is a scalar multiple of a nested
monomial, even if
$\alpha_{1}=\operatorname{rk}H-\operatorname{rk}F_{2}-1=\operatorname{rk}M-\operatorname{rk}F_{2}-2$
since then
$x_{H}x_{F}^{\alpha}=-x_{E}^{\operatorname{rk}M-\operatorname{rk}F_{2}-1}x_{F_{2}}^{\alpha_{2}}\cdots
x_{F_{r}}^{\alpha_{r}}$. If $H\supsetneq F_{1}$ and
$\operatorname{rk}H-\operatorname{rk}F_{1}\geq 2$, then $x_{H}x_{F}^{\alpha}$
is also a nested monomial. If $H\supsetneq F_{1}$ and
$\operatorname{rk}H-\operatorname{rk}F_{1}=1$, then
$x_{H}x_{F}^{\alpha}=-x_{E}x_{F_{1}}^{\alpha_{1}}\cdots
x_{F_{r}}^{\alpha_{r}}$ is also a nested monomial. Thus, we can write

$f=\sum_{G_{1}=H}c_{G,\beta}x_{G}^{\beta}+\sum_{H\supsetneq
G_{2}}c_{G,\beta}x_{E}^{\operatorname{rk}M-\operatorname{rk}G_{2}-1}x_{G_{2}}^{\beta_{2}}\cdots
x_{G_{r}}^{\beta_{r}}.\qed$

Since $A=\operatorname{\underline{CH}}(M)$ is an Artinian, Gorenstein ring, it
follows by linkage [HU87, Remark 2.7] that the quotient ring $A/(0:m)$ is also
Gorenstein for any monomial $m$, and one might hope that this ring is the Chow
ring of a matroid quotient of $M$. We identify the matroids associated to such
quotients in some simple but important cases.

######  Proposition 4.8.

For a matroid $M$, we have $(0:x_{E})=(x_{H}\mid H\in\operatorname{coat}(E))$
and $A/(0:x_{E})\cong\operatorname{\underline{CH}}(T(M))$.

###### Proof.

Clearly, the latter ideal is contained in the former by (4.12). Suppose that
$f\in A$ with $x_{E}f=0$, and write $f=\sum c_{F,\alpha}x_{F}^{\alpha}$ as a
$\mathbb{Q}$-linear combination of nested monomials. For each nested monomial
$x_{F}^{\alpha}$ in $A$, we note that $x_{E}x_{F}^{\alpha}=0$ only if $F_{1}$
is a hyperplane or $F_{1}=E$ and
$\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1$ (including the
possibility that $F_{2}=\varnothing$). Otherwise, $x_{E}x_{F}^{\alpha}$ is
still a nested monomial. It then follows from the equality $x_{E}f=0$ that

$f=\sum_{F_{1}\in\operatorname{coat}(E)}c_{F,\alpha}x_{F}^{\alpha}+\sum_{F_{1}=E,\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1}c_{F,\alpha}x^{\alpha}_{F}.$

For each monomial in the latter sum, if $F_{2}\neq\varnothing$, we note that
$\operatorname{rk}F_{2}\leq\operatorname{rk}M-2$ so that there is a hyperplane
$H\supsetneq F_{2}$ as the lattice of flats of $M$ is graded, and
$x_{E}^{\operatorname{rk}M-\operatorname{rk}F_{2}-1}x_{F_{2}}=-x_{H}^{\operatorname{rk}H-\operatorname{rk}F_{2}}x_{F_{2}}$.
Otherwise, if $F_{2}=\varnothing$, we have
$x_{E}^{\operatorname{rk}M-1}=-x_{H}^{\operatorname{rk}H}$ for any hyperplane
$H$. This shows that $f\in(x_{H}\mid H\in\operatorname{coat}(E))$.

It is easily checked that there is a well-defined surjective
$\mathbb{Q}$-algebra map $\pi:A\to\operatorname{\underline{CH}}(T(M))$ defined
by $x_{F}\mapsto x_{F}$ for $F\notin\operatorname{coat}(E)$ and $x_{F}\mapsto
0$ otherwise. We will show that $\operatorname{Ker}\pi=(x_{H}\mid
H\in\operatorname{coat}(E))$. Again, it is clear that the latter ideal is
contained in the former. If $f\in\operatorname{Ker}\pi$ and we write
$f=\sum_{\alpha}c_{F,\alpha}x_{F}^{\alpha}$ as a $\mathbb{Q}$-linear
combination of nested monomials, then

$0=\pi(f)=\sum_{F_{1}\notin\operatorname{coat}(E)}c_{F,\alpha}x_{F}^{\alpha}=\sum_{\operatorname{rk}F_{1}\leq\operatorname{rk}M-2}c_{F,\alpha}x_{F}^{\alpha}+\sum_{F_{1}=E,\alpha_{1}<\operatorname{rk}M-\operatorname{rk}F_{2}-1}c_{F,\alpha}x_{F}^{\alpha}$

expresses $\pi(f)$ as a linear combination of nested monomials in
$\operatorname{\underline{CH}}(T(M))$. In particular, note that the latter sum
does not include any nested monomials with
$\operatorname{rk}F_{2}=\operatorname{rk}M-2$ since $F_{2}$ is a hyperplane in
$T(M)$ so that $x_{F_{2}}x_{E}=0$ by (4.12). Hence, the coefficients of all
the above monomials in $f$ must be zero so that

$f=\sum_{F_{1}\in\operatorname{coat}(E)}c_{F,\alpha}x_{F}^{\alpha}+\sum_{F_{1}=E,\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1}c_{F,\alpha}x_{F}^{\alpha}.$

However, for every nonempty flat $F_{2}$ we note that
$x_{F_{2}}x_{E}^{\operatorname{rk}M-\operatorname{rk}F_{2}-1}=-x_{F_{2}}x_{H}^{\operatorname{rk}H-\operatorname{rk}F_{2}}$
for any hyperplane $H\supsetneq F_{2}$, and
$x_{E}^{\operatorname{rk}M-1}=-x_{H}^{\operatorname{rk}H}$ for any hyperplane
$H$. Thus, we see that $f\in(x_{H}\mid H\in\operatorname{coat}(E))$ as wanted,
and combining this with the preceding paragraph yields
$A/(0:x_{E})\cong\operatorname{\underline{CH}}(T(M))$. ∎

######  Corollary 4.9.

Let $F$ be a flat of $M$ with $\operatorname{rk}F\geq 2$.

  1. (a)

The kernel of the natural surjective homomorphism
$\pi_{M,F}:A\to\operatorname{\underline{CH}}(M|F)$ sending $x_{G}\mapsto
x_{G}$ if $G\subseteq F$ and $x_{G}\mapsto 0$ otherwise is $(x_{G}\mid
G\nsubseteq F)$.

  2. (b)

Suppose that $\mathcal{G}$ is a set of flats of $M$ such that
$(F,E]\subseteq\mathcal{G}$ and $\mathcal{G}\cap[\varnothing,F]=\varnothing$.
If $\operatorname{rk}F\geq 3$, then

$(x_{G}\mid G\in\mathcal{G}):x_{F}=(x_{G}\mid G\nsubseteq F)+(x_{G}\mid
G\in\operatorname{coat}(F)).$

Otherwise, $(x_{G}\mid G\in\mathcal{G}):x_{F}=A_{+}$.

###### Proof.

(a) It is easily checked that there is a well-defined surjective
$\mathbb{Q}$-algebra map $\pi:A\to\operatorname{\underline{CH}}(M|F)$ defined
by the given assignments. We will show that $\operatorname{Ker}\pi=(x_{G}\mid
G\nsubseteq F)$. Again, it is clear that the latter ideal is contained in the
former. If $f\in\operatorname{Ker}\pi$ and we write $f=\sum
c_{G,\alpha}x_{G}^{\alpha}$ as a $\mathbb{Q}$-linear combination of nested
monomials, then $0=\pi(f)=\sum_{G_{1}\subseteq F}c_{G,\alpha}x_{G}^{\alpha}$
expresses $\pi(f)$ as a linear combination of nested monomials in
$\operatorname{\underline{CH}}(M|F)$. Hence, the coefficients of all the
preceding monomials in $f$ must be zero so that $f=\sum_{G_{1}\nsubseteq
F}c_{G,\alpha}x_{G}^{\alpha}\in(x_{G}\mid G\nsubseteq F)$ as wanted.

(b) First, we note that the colon ideal contains $(x_{G}\mid G\nsubseteq F)$.
Indeed, if $G^{\prime}$ is flat such that $G^{\prime}\nsubseteq F$, then
either $F\subsetneq G^{\prime}$ so that $x_{G^{\prime}}\in(x_{G}\mid
G\in\mathcal{G})$, or $G^{\prime}$ is incomparable with $F$ so that
$x_{G^{\prime}}x_{F}=0$. Since both ideals in question contain $(x_{G}\mid
G\nsubseteq F)$, it suffices by part (a) to show that their images agree in
$\operatorname{\underline{CH}}(M|F)$. Hence, it suffices to assume $F=E$ and
to prove that $(0:x_{E})=(x_{H}\mid H\in\operatorname{coat}(E))$ when
$\operatorname{rk}M\geq 3$ and $(0:x_{E})=A_{+}$ when $\operatorname{rk}M=2$,
which follow from the preceding proposition and the discussion at the
beginning of this section respectively. ∎

######  Proposition 4.10.

Let $\mathcal{H}$ be a nonempty set of hyperplanes of $M$ and $H^{\prime}$ be
any hyperplane. Then:

  1. (a)

$(0:x_{H^{\prime}})=(x_{F}\mid F\nsubseteq H^{\prime})$ and
$A/(0:x_{H^{\prime}})\cong\operatorname{\underline{CH}}(M|H^{\prime})$.

  2. (b)

$(0:(x_{H}\mid H\in\mathcal{H}))=(x_{F}\mid F\nsubseteq H\;\text{for
all}\;H\in\mathcal{H})$.

  3. (c)

Suppose that $H^{\prime}\notin\mathcal{H}$. If
$\operatorname{rk}H^{\prime}\geq 3$ and for every $H\in\mathcal{H}$ there
exists an $F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ such that $H\cap
H^{\prime}\subseteq F$, then

$(x_{H}\mid H\in\mathcal{H}):x_{H^{\prime}}=(0:x_{H^{\prime}})+(x_{F}\mid
F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime}))$

Otherwise, if $\operatorname{rk}H^{\prime}=2$, then $(x_{H}\mid
H\in\mathcal{H}):x_{H^{\prime}}=A_{+}$.

###### Proof.

(a) We have already seen in the proof of Lemma 4.7 that for any nested
monomial $x_{F}^{\alpha}$ in $A$ we have $x_{H^{\prime}}x_{F}^{\alpha}=0$ if
$F_{1}\nsubseteq H^{\prime}$ and $x_{H^{\prime}}x_{F}^{\alpha}$ is a scalar
multiple of a nested monomial otherwise. Furthermore, multiplication by
$x_{H^{\prime}}$ takes distinct nested monomials with $F_{1}\subseteq
H^{\prime}$ to scalar multiples of distinct nested monomials. In particular,
it is immediate that the latter ideal is contained in the former. Moreover, if
$f\in A$ with $x_{H^{\prime}}f=0$ and we write $f=\sum
c_{F,\alpha}x_{F}^{\alpha}$ as a $\mathbb{Q}$-linear combination of nested
monomials, it follows from the equality $x_{H^{\prime}}f=0$ that
$f=\sum_{F_{1}\nsubseteq H^{\prime}}c_{F,\alpha}x_{F}^{\alpha}$, which shows
that $f\in(x_{F}\mid F\nsubseteq H^{\prime})$. The isomorphism
$A/(0:x_{H^{\prime}})\cong\operatorname{\underline{CH}}(M|H^{\prime})$ then
follows from part (a) of the preceding corollary.

(b) Consider the homomorphism
$\eta:A\to\bigoplus_{H\in\mathcal{H}}\operatorname{\underline{CH}}(M|H)$
sending each $f\in A$ to the tuple of its images under the natural maps
$A\to\operatorname{\underline{CH}}(M|H)$. By part (a), we know that
$\operatorname{Ker}\eta=\bigcap_{H\in\mathcal{H}}(0:x_{H})=(0:(x_{H}\mid
H\in\mathcal{H}))$. Clearly, this ideal contains $x_{F}$ for every flat $F$
such that $F\nsubseteq H$ for all $H\in\mathcal{H}$. Conversely, if
$f\in\operatorname{Ker}\eta$, then the proof of the previous part shows that
$f$ has an expansion in terms of nested monomials of the form
$f=\sum_{F_{1}\nsubseteq
H,\;\text{all}\;H\in\mathcal{H}}c_{F,\alpha}x_{F}^{\alpha}$. Hence,
$f\in(x_{F}\mid F\nsubseteq H\;\text{for all}\;H\in\mathcal{H})$ as wanted.

(c) The statement is clear if $\operatorname{rk}H^{\prime}=2$, since
$x_{H^{\prime}}x_{F}=0$ for every $F\neq H^{\prime}$ and
$x_{H^{\prime}}^{2}=-x_{E}^{2}=x_{H}^{2}\in(x_{H}\mid H\in\mathcal{H})$ for
any $H\in\mathcal{H}$ by (4.12). So, we may assume that
$\operatorname{rk}H^{\prime}\geq 3$. In that case, the latter ideal is again
clearly contained in the former since
$x_{F}x_{H^{\prime}}=-x_{F}x_{E}=x_{F}x_{H}$ if $F=H\cap H^{\prime}$ and
$\operatorname{rk}F=\operatorname{rk}M-2$. Suppose that $f\in A$ with
$x_{H^{\prime}}f\in(x_{H}\mid H\in\mathcal{H})$, and write $f=\sum
c_{F,\alpha}x_{F}^{\alpha}$ as a $\mathbb{Q}$-linear combination of nested
monomials. Then:

$\displaystyle x_{H^{\prime}}f=\sum_{F_{1}\subseteq
H^{\prime}}c_{F,\alpha}x_{H^{\prime}}x_{F}^{\alpha}$
$\displaystyle=\sum_{\operatorname{rk}H^{\prime}-\operatorname{rk}F_{1}\geq
2}c_{F,\alpha}x_{H^{\prime}}x_{F}^{\alpha}+\sum_{\begin{subarray}{c}F_{1}=H^{\prime}\\\
\alpha_{1}<\operatorname{rk}H^{\prime}-\operatorname{rk}F_{2}-1\end{subarray}}c_{F,\alpha}x_{H^{\prime}}^{\alpha_{1}+1}x_{F_{2}}^{\alpha_{2}}\cdots
x_{F_{r}}^{\alpha_{r}}$
$\displaystyle-\sum_{\operatorname{rk}H^{\prime}-\operatorname{rk}F_{1}=1}c_{F,\alpha}x_{E}x_{F}^{\alpha}-\sum_{\begin{subarray}{c}F_{1}=H^{\prime}\\\
\alpha_{1}=\operatorname{rk}H^{\prime}-\operatorname{rk}F_{2}-1\end{subarray}}c_{F,\alpha}x_{E}^{\operatorname{rk}M-\operatorname{rk}F_{2}-1}x_{F_{2}}^{\alpha_{2}}\cdots
x_{F_{r}}^{\alpha_{r}}$

By Lemma 4.7, the coefficients of the monomials in the first two sums must all
be zero, and the coefficient of a monomial in the last two sums is also zero
unless $F_{1}\subsetneq H$ for some $H\in\mathcal{H}$ or $F_{1}=H^{\prime}$
and $F_{2}\subsetneq H$ for some $H\in\mathcal{H}$. Hence, we have

$\displaystyle f$ $\displaystyle=\sum_{F_{1}\nsubseteq
H^{\prime}}c_{F,\alpha}x_{F}^{\alpha}+\sum_{\begin{subarray}{c}\operatorname{rk}H^{\prime}-\operatorname{rk}F_{1}=1\\\
F_{1}\subsetneq
H\in\mathcal{H}\end{subarray}}c_{F,\alpha}x_{F}^{\alpha}+\sum_{F_{2}\subsetneq
H\in\mathcal{H}}c_{F,\alpha}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F_{2}-1}x_{F_{2}}^{\alpha_{2}}\cdots
x_{F_{r}}^{\alpha_{r}}.$

It follows that
$f\in(0:x_{H^{\prime}})+(x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1},x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-1}\mid
F\subseteq H\cap H^{\prime}\;\text{for some}\;H\in\mathcal{H})$. It remains to
show that

$(0:x_{H^{\prime}})+(x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1},x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-1}\mid
F\subseteq H\cap H^{\prime}\;\text{for some}\;H\in\mathcal{H})$

is contained in the ideal $(0:x_{H^{\prime}})+(x_{F}\mid
F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime}))$. After quotienting by
$(0:x_{H^{\prime}})$, it suffices by part (a) to show that

$(x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1},x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-1}\mid
F\subseteq H\cap H^{\prime}\;\text{for some}\;H\in\mathcal{H})$

is contained in the ideal $J=(x_{F}\mid
F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime}))$ in
$\operatorname{\underline{CH}}(M|H^{\prime})$.

Since $\mathcal{H}\neq\varnothing$, we have
$\operatorname{coat}_{\mathcal{H}}(H^{\prime})\neq\varnothing$ by assumption
so that $J\neq 0$. By Remark 4.6, we know
$x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-1}$ generates the socle of
$\operatorname{\underline{CH}}(M|H^{\prime})$ and is, therefore, contained in
$J$. We must also show that
$x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1}\in J$
if $F\subseteq H\cap H^{\prime}$ for some $H\in\mathcal{H}$ and
$\operatorname{rk}H^{\prime}\geq\operatorname{rk}F+2$. For this, since the
elements $\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ are hyperplanes of
the matroid $M|H^{\prime}$, we note that

$(0:J)=(x_{G}\mid G\subseteq H^{\prime},G\nsubseteq F^{\prime}\;\text{for
all}\;F^{\prime}\in\operatorname{coat}_{\mathcal{H}}(H^{\prime}))$

by part (b). Because $\operatorname{\underline{CH}}(M|H^{\prime})$ is
Gorenstein, it follows by linkage [Eis95, Theorem 21.23] that $(0:(0:J))=J$,
and so, it suffices to note that each such $x_{G}$ annihilates
$x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1}$ to
prove that
$x_{F}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1}\in J$.
Given a flat $G$ with $G\subseteq H^{\prime}$ and $G\nsubseteq F^{\prime}$ for
all $F^{\prime}\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$, we note that
$G\nsubseteq F$ since otherwise we would have $G\subseteq F\subseteq H\cap
H^{\prime}\subseteq F^{\prime}$ for some
$F^{\prime}\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ by assumption.
Consequently, either $G$ and $F$ are incomparable so that $x_{F}x_{G}=0$ or
$F\subsetneq G$ so that
$x_{G}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1}=0$ by
Corollary 4.4 as
$\operatorname{rk}H^{\prime}-\operatorname{rk}F-1\geq\operatorname{rk}H^{\prime}-\operatorname{rk}G$.
And so, either way we have
$x_{F}x_{G}x_{H^{\prime}}^{\operatorname{rk}H^{\prime}-\operatorname{rk}F-1}=0$
as wanted. ∎

The condition in part (c) of the above proposition that for every
$H\in\mathcal{H}$ there exists an
$F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ such that $H\cap
H^{\prime}\subseteq F$ is essential for the colon ideal to be generated by
linear forms. We give some examples below showing how the colon ideal can fail
to be generated by linear forms if this condition is not satisfied.

######  Example 4.11.

Consider the uniform matroid $U_{5,6}$ on the ground set
$E=\\{1,2,3,4,5,6\\}$. Since every subset of $E$ of size at most 5 is
independent, the hyperplanes of $U_{5,6}$ are precisely the subsets of size 4.
For $H^{\prime}=1256$, the proof of the proposition shows that in
$\operatorname{\underline{CH}}(U_{5,6})$ we have

$(x_{1234}):x_{1256}=(0:x_{1256})+(x_{12}x_{1256},x_{1256}^{3})=(0:x_{1256})+(x_{12}x_{1256})$

where $x_{12}x_{1256}\notin(0:x_{1256})$ since it is a nested monomial of
$\operatorname{\underline{CH}}(U_{5,6}|H^{\prime})$.

If we take $M=T_{H^{\prime}}(U_{5,6})$ to be the principal truncation
$U_{5,6}$ with respect to $H^{\prime}$, then $M$ is a rank 4 matroid whose
hyperplanes are $H^{\prime}$ and all subsets of $E$ of size 3 that are not
contained in $H^{\prime}$. Even in this case, we still have

$(x_{234}):x_{1256}=(0:x_{1256})+(x_{1256}^{2}),$

where $x_{1256}^{2}\notin(0:x_{1256})$ since it is a nested monomial of
$\operatorname{\underline{CH}}(M|H^{\prime})$.

In the first example above, the issue is that the two hyperplanes in question
have rank 4 but their meet $1234\wedge 1256=12$ has rank $2$. One might hope
that by choosing only sets of hyperplanes that are locally connected (in the
sense that between any two hyperplanes there is a sequence of hyperplanes in
the set such each pair of consecutive hyperplanes has a meet of rank
$\operatorname{rk}M-2$), we might avoid quadratic terms in these colon ideals.
The following example shows even this is not enough.

Again, consider $\operatorname{\underline{CH}}(U_{5,6})$. Although the set of
hyperplanes $\\{1234,2345,3456,1456,1256,1236\\}$ in $U_{5,6}$ is locally
connected (the corresponding graph is a $6$-cycle), one computes that

$(x_{1234},x_{2345},x_{3456},x_{1456},x_{1236}):x_{1256}=(0:x_{1256})+(x_{125},x_{126},x_{12}x_{1256}).$

where the quadratic monomial $x_{12}x_{1256}$ is a minimal generator of this
colon ideal.

######  Theorem 4.12.

The Chow ring of a matroid has a Koszul filtration.

###### Proof.

Let $M$ be a matroid with lattice of flats $\operatorname{\mathcal{L}}$. As
previously noted at the beginning of this subsection, we may assume without
loss of generality that $M$ is simple and $\operatorname{rk}M\geq 3$. By
Theorem 3.3, we also know that the lattice $\operatorname{\mathcal{L}}$ admits
a total coatom order $\prec$. In what follows, all ideals are considered with
respect to the atom-free presentation of $A=\operatorname{\underline{CH}}(M)$,
and all initial segments of flats are with respect to the chosen total coatom
ordering. However, all up-sets refer to the natural partial order of
$\operatorname{\mathcal{L}}$. Consider the collections of ideals:

$\displaystyle\mathcal{F}_{0}$ $\displaystyle=\\{(x_{G}\mid
G\in\mathcal{G})\mid\mathcal{G}\subseteq\operatorname{\mathcal{L}}_{\geq
2}\;\text{an up-set}\\}$ $\displaystyle\mathcal{F}_{1}$
$\displaystyle=\\{(x_{G}\mid G\nsubseteq F)+(x_{G}\mid G\in\mathcal{G})\mid
F\in\operatorname{\mathcal{L}}_{\geq 2},\;\mathcal{G}\;\text{an initial
segment covered by}\;F\\},$

and set $\mathcal{F}=\mathcal{F}_{0}\cup\mathcal{F}_{1}$. Note that
$(0)\in\mathcal{F}_{0}$ when $\mathcal{G}=\varnothing$ and that
$A_{+}\in\mathcal{F}_{0}$ when $\mathcal{G}=\operatorname{\mathcal{L}}_{\geq
2}$. We claim that $\mathcal{F}$ is the desired Koszul filtration.

Let $0\neq I\in\mathcal{F}_{0}$ so that $I=(x_{G}\mid G\in\mathcal{G})$ for
some nonempty up-set $\mathcal{G}$. We then choose any flat
$G^{\prime}\in\mathcal{G}$ minimal in $\mathcal{G}$ with respect to the
partial order of $\operatorname{\mathcal{L}}$ and set
$\mathcal{G}^{\prime}=\mathcal{G}\smallsetminus\\{G^{\prime}\\}$. It is easy
to see that $\mathcal{G}^{\prime}$ is also an up-set since we are removing a
minimal element of $\mathcal{G}$. Consider the ideal $J=(x_{G}\mid
G\in\mathcal{G}^{\prime})\in\mathcal{F}_{0}$. Clearly $J\subseteq I$ and $I/J$
is cyclic. Since $\mathcal{G}^{\prime}$ contains all flats properly containing
$G^{\prime}$ and none of the flats contained in $G^{\prime}$, it follows from
Corollary 4.9 that

$(J:I)=(J:x_{G^{\prime}})=(x_{G}\mid G\nsubseteq G^{\prime})+(x_{G}\mid
G\in\operatorname{coat}(G^{\prime}))$

if $\operatorname{rk}G^{\prime}\geq 3$ and $(J:I)=A_{+}$ if
$\operatorname{rk}G^{\prime}=2$. In the former case, it follows that
$(J:I)\in\mathcal{F}_{1}$, and we have $(J:I)\in\mathcal{F}_{0}$ in the latter
case.

Let $0\neq I\in\mathcal{F}_{1}$ so that $I=(x_{G}\mid G\nsubseteq
F)+(x_{G}\mid G\in\mathcal{G})$ for some flat $F$ and $\mathcal{G}$ an initial
segment of flats covered by $F$. If $\mathcal{G}\neq\varnothing$, then
$\operatorname{rk}G\geq 3$, and we can take $G^{\prime}$ to be the largest
element of $\mathcal{G}$ with respect to the total coatom order and
$\mathcal{G}^{\prime}=\mathcal{G}\smallsetminus\\{G^{\prime}\\}$. Then
$J=(x_{G}\mid G\nsubseteq F)+(x_{G}\mid
G\in\mathcal{G}^{\prime})\in\mathcal{F}_{1}$. Again, $J\subseteq I$ with $I/J$
cyclic. Moreover

$(J:I)=(J:x_{G^{\prime}})=(x_{G}\mid G\nsubseteq G^{\prime})+(x_{G}\mid
G\in\operatorname{coat}_{\mathcal{G}}(G^{\prime}))$

by Corollary 4.9 and part (c) of Proposition 4.10 applied to $M|F$. By the
definition of a total coatom ordering,
$\operatorname{coat}_{\mathcal{G}}(G^{\prime})$ is an initial segment of flats
covered by $G^{\prime}$ so that $(J:I)\in\mathcal{F}_{1}$. On the other hand,
if $\mathcal{G}=\varnothing$, then $I=(x_{G}\mid G\nsubseteq
F)\in\mathcal{F}_{0}$, and so, we have already seen in the preceding paragraph
how to choose an ideal contained in $I$ with the desired properties. Thus,
$\mathcal{F}$ is a Koszul filtration for $A$. ∎

We get the following corollary to our main theorem.

######  Corollary 4.13.

The Chow ring of a matroid has a rational Poincaré series.

###### Proof.

It is well-known that any commutative Koszul $\mathbb{K}$-algebra $A$ has a
rational Poincaré series; indeed, the Hilbert Series
$\operatorname{\mathrm{H}}_{A}(t)$ is always rational so that the Poincare
series
$\operatorname{P}_{\mathbb{K}}^{A}(t)=\frac{1}{\operatorname{\mathrm{H}}_{A}(-t)}$
is as well by (2.1). ∎

##  5\. The Koszul property of the augmented Chow ring of a matroid

###  5.1. The augmented Chow ring of a Matroid

The augmented Chow ring of a simple matroid $M$ with lattice of flats
$\operatorname{\mathcal{L}}=\operatorname{\mathcal{L}}(M)$ is the ring

$\operatorname{CH}(M):=S_{M}/(I_{M}+J_{M}),$

where

$S_{M}=\mathbb{Q}[y_{i},x_{F}\mid i\in
E,F\in\operatorname{\mathcal{L}}\smallsetminus\\{E\\}],$

and $I_{M}$ and $J_{M}$ are the ideals

$\displaystyle I_{M}$ $\displaystyle=(y_{i}-\textstyle\sum_{i\notin
F}\,x_{F}\mid i\in E),$ $\displaystyle J_{M}$ $\displaystyle=(x_{F}x_{G}\mid
F,G\;\text{incomparable}\;)+(y_{i}x_{F}\mid i\in E,i\notin F).$

Augmented Chow rings of matroids were introduced in [BHM+20a] and employed in
[BHM+20b] as a key ingredient in the resolution of the Top-Heavy Conjecture
that interpolates between the combinatorics of the lattice of flats of a
matroid (as encoded by its graded Möbius algebra) and the good algebraic
properties of the Chow ring of the matroid. In particular, it is shown
[BHM+20b, Theorem 2.19] that the augmented Chow ring of a matroid is also a
quadratic Artinian Gorenstein graded algebra. It was pointed out to us by
Chris Eur that the augmented Chow ring of $M$ could be viewed as the
Feichtner-Yuzvinsky Chow ring $D(\operatorname{\mathcal{L}},\mathcal{G})$ of
the lattice of flats of a related matroid, the free coextension of $M$, with
respect to a certain building set. Using this observation, we show how to
obtain an atom-free presentation for the augmented Chow ring of a matroid and
that all of the results of the preceding section carry over with only minor
modifications to show that the augmented Chow ring of a matroid is also
Koszul.

We begin by sketching the details of Eur’s observation about the augmented
Chow ring. Let $M$ be a simple matroid on the ground set $E$. The free
coextension of $M$ is the matroid $(M^{*}+e)^{*}$; it is the dual of the free
single element extension of the dual matroid of $M$. We refer the interested
reader to [Oxl11, Sections 2.1, 7.2] for further details on duality and free
extensions of matroids. For our purposes, it suffices to note that the free
coextension $(M^{*}+e)^{*}$ is a matroid on the ground set $E\cup\\{e\\}$,
where $e$ is any element not in $E$, with rank function defined as follows for
any set $F\subseteq E$:

$\displaystyle\operatorname{rk}_{(M^{*}+e)^{*}}F$
$\displaystyle=\left\\{\begin{array}[]{ll}\operatorname{rk}_{M}F+1,&\text{if}\;\operatorname{cl}_{M^{*}}(E\smallsetminus
F)\neq E\\\\[5.0pt]
\operatorname{rk}_{M}F,&\text{if}\;\operatorname{cl}_{M^{*}}(E\smallsetminus
F)=E\end{array}\right.$ $\displaystyle\operatorname{rk}_{(M^{*}+e)^{*}}(F\cup
e)$ $\displaystyle=\operatorname{rk}_{M}F+1$

Using the description of the rank function of the dual of $M$, it is easily
checked that the condition $\operatorname{cl}_{M^{*}}(E\smallsetminus F)=E$ is
satisfied if and only if $\operatorname{rk}_{M}F=\left|F\right|$, which is
equivalent to $F$ being an independent set of $M$. From this, it can be seen
that a set of the form $F\cup e$ is a flat of the free coextension of $M$ if
and only if $F$ is a flat of $M$, and a set $F$ is a flat of the free
coextension if and only if $F$ is an independent set of $M$. Thus, in a way,
the free coextension treats independent sets and flats of $M$ on equal
footing. By identifying the flat $F$ of $M$ with the flat $F\cup e$ of the
free coextension, we see that the lattice of flats of the free coextension
contains an isomorphic copy of the lattice of flats of $M$, just shifted
upward in rank by one, in addition to all independent sets of $M$ with their
usual rank.

Consider the subset
$\mathcal{G}_{\operatorname{aug}}\subseteq\operatorname{\mathcal{L}}((M^{*}+e)^{*})$
defined by

$\mathcal{G}_{\operatorname{aug}}=\\{i\mid i\in E\\}\cup\\{F\cup e\mid
F\in\operatorname{\mathcal{L}}(M)\\}.$

We claim that $\mathcal{G}_{\operatorname{aug}}$ is a building set for the
lattice of flats of the free coextension. By the preceding paragraph, we know
that every non-minimal element of the lattice of flats of the free
coextensions is either a nonempty independent set of $M$ or of the form $F\cup
e$ for some flat $F$ of $M$. When $F$ is a flat of $M$, $F\cup
e\in\mathcal{G}_{\operatorname{aug}}$ so that the required poset isomorphism
for $\mathcal{G}_{\operatorname{aug}}$ to be building is just the identity map
$[\varnothing,F\cup e]\to[\varnothing,F\cup e]$. On the other hand, when $F$
is an independent set of $M$, $\max(\mathcal{G}_{\operatorname{aug}})_{\leq
F}=\\{i\mid i\in F\\}$. Since the interval $[\varnothing,F]$ is just the
Boolean lattice of all subsets of $F$, it is easily seen that the map
$\varphi_{F}:\prod_{i\in F}[\varnothing,i]\to[\varnothing,F]$ given by
$\varphi_{F}(A_{1},\dots,A_{r})=\bigcup_{j=1}^{r}A_{j}$ is the required poset
isomorphism for $\mathcal{G}_{\operatorname{aug}}$ to be building.

The Feichtner-Yuzvinsky presentation of the augmented Chow ring of a simple
matroid $M$ is the ring

$\operatorname{CH}_{FY}(M)=\mathbb{Q}\left[y_{i},y_{F\cup e}\mid i\in
E,F\in\operatorname{\mathcal{L}}\right]/I_{FY}(M),$

where

$\displaystyle I_{FY}(M)$ $\displaystyle=\left(y_{F\cup e}y_{F^{\prime}\cup
e}\mid F,F^{\prime}\;\text{incomparable}\;\right)+(y_{i}y_{F\cup e}\mid i\in
E,i\notin F)$ $\displaystyle\qquad\qquad+(\textstyle y_{i}+\sum_{i\in
F}\,y_{F\cup e},\sum_{F}\,y_{F\cup e}\mid i\in E).$

Setting $x_{F}=y_{F\cup e}$ and using the last linear form above to eliminate
$x_{E}$ recovers the usual presentation of the augmented Chow ring of $M$.

######  Definition 5.1.

The atom-free presentation of the augmented Chow ring of $M$ is the ring

$\operatorname{CH}_{\operatorname{af}}(M)=\mathbb{Q}[x_{F}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}]/I_{\operatorname{af}}(M),$

where

$\displaystyle I_{\operatorname{af}}(M)$
$\displaystyle=(x_{F}x_{F^{\prime}}\mid
F,F^{\prime}\;\text{incomparable}\;)+(\textstyle x_{F}\sum_{i\in
F^{\prime}}\,x_{F^{\prime}}\mid i\in E,i\notin F)+((\textstyle\sum_{i\in
F^{\prime}}\,x_{F^{\prime}})^{2}\mid i\in E)$

After a change variables $\varphi$ on the ring $\mathbb{Q}[y_{i},x_{F}\mid
i\in E,F\in\operatorname{\mathcal{L}}]$ sending $y_{i}\mapsto y_{i}-\sum_{i\in
F}x_{F}$ for each $i\in E$ and sending $x_{\varnothing}\mapsto
x_{\varnothing}-\sum_{F\neq\varnothing}x_{F}$, we see that

$\displaystyle\begin{split}\varphi(I_{FY}(M))&=\left(x_{\varnothing},y_{i}\mid
i\in E)+(x_{F}x_{F^{\prime}}\mid
F,F^{\prime}\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}\;\text{incomparable}\;\right)\\\
&+(\textstyle x_{F}\sum_{i\in F^{\prime}}\,x_{F^{\prime}}\mid
F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\},i\in E,i\notin
F)\\\ &+(\textstyle(\sum_{i\in
F^{\prime}}\,x_{F^{\prime}})(\sum_{F\neq\varnothing}\,x_{F})\mid i\in E)\\\
&=(x_{\varnothing},y_{i}\mid i\in E)+I_{\operatorname{af}}(M),\end{split}$
(5.1)

where the last equality follows after a suitable change of generators from the
fact that

$\textstyle(\sum_{i\in
F^{\prime}}\,x_{F^{\prime}})(\sum_{F\neq\varnothing}\,x_{F})=(\sum_{i\in
F^{\prime}}\,x_{F^{\prime}})^{2}+\sum_{F\neq\varnothing,i\notin
F}x_{F}(\sum_{i\in F^{\prime}}\,x_{F^{\prime}}).$

And so, we see that

$\operatorname{CH}(M)\cong\operatorname{CH}_{FY}(M)\cong\frac{\mathbb{Q}[y_{i},x_{F}\mid
i\in E,F\in\operatorname{\mathcal{L}}]}{(x_{\varnothing},y_{i}\mid i\in
E)+I_{\operatorname{af}}(M)}\cong\operatorname{CH}_{\operatorname{af}}(M).$

######  Theorem 5.2 ([FY04, Theorem 2]).

With respect to any lexicographic order $>$ such that $x_{F}>x_{G}$ implies
$F\nsupseteq G$ and $y_{i}>x_{F}$ for all $i\in E$ and all flats $F$, the
defining ideal $I_{FY}(M)$ of the augmented Chow ring of $M$ has a Gröbner
basis consisting of the following polynomials for all $i\in E$ and
$F,F^{\prime}\in\operatorname{\mathcal{L}}$:

$\displaystyle x_{F^{\prime}}x_{F}$ $\displaystyle
F,F^{\prime}\text{incomparable}$ (5.2) $\displaystyle y_{i}x_{F}$
$\displaystyle i\notin F$ (5.3) $\displaystyle
x_{F^{\prime}}(\textstyle\sum_{G\supseteq
F}\,x_{G})^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}$ $\displaystyle
F^{\prime}\subsetneq F$ (5.4) $\displaystyle y_{i}(\textstyle\sum_{G\supseteq
F}\,x_{G})^{\operatorname{rk}F}$ $\displaystyle i\in F$ (5.5)
$\displaystyle(\textstyle\sum_{G\supseteq F}\,x_{G})^{\operatorname{rk}F+1}$
(5.6) $\displaystyle\textstyle y_{i}+\sum_{i\in F}\,x_{F}$ $\displaystyle i\in
E$ (5.7)

A slight modification yields a Gröbner basis in the atom-free setting. We omit
the proof since it is completely analogous to the proof of Corollary 4.4.

######  Corollary 5.3.

With respect to any lexicographic order $>$ such that $x_{F}>x_{G}$ implies
$F\nsupseteq G$, the defining ideal $I_{\operatorname{af}}(M)$ of the
augmented Chow ring of $M$ has a Gröbner basis consisting of the following
polynomials for all
$F,F^{\prime}\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}$:

$\displaystyle x_{F}x_{F^{\prime}}$ $\displaystyle
F,F^{\prime}\;\text{incomparable}$ (5.8) $\displaystyle
x_{F^{\prime}}(\textstyle\sum_{G\supseteq
F}\,x_{G})^{\operatorname{rk}F-\operatorname{rk}F^{\prime}}$ $\displaystyle
F^{\prime}\subsetneq F$ (5.9) $\displaystyle(\textstyle\sum_{G\supseteq
F}\,x_{G})^{\operatorname{rk}F+1}$ (5.10)

######  Corollary 5.4.

The ring $\operatorname{CH}_{\operatorname{af}}(M)$ has a $\mathbb{Q}$-basis
consisting of all monomials

$x_{F}^{\alpha}=x_{F_{1}}^{\alpha_{1}}\cdots x_{F_{r}}^{\alpha_{r}}$

for all chains of flats $F=\\{F_{1}\supsetneq F_{2}\supsetneq\cdots\supsetneq
F_{r}\supsetneq F_{r+1}=\varnothing\\}$ for some $r\geq 0$ and all
$\alpha\in\mathbb{Z}_{\geq 0}^{r+1}$ such that
$\sum_{i}\alpha_{i}=\operatorname{rk}F_{1}+1$,
$1\leq\alpha_{i}<\operatorname{rk}F_{i}-\operatorname{rk}F_{i+1}$ if $i<r$,
and $\alpha_{r}\leq\operatorname{rk}F_{r}$.

As before, we call the monomials of Corollary 5.4 the nested monomials of
$\operatorname{CH}_{\mathrm{af}}(M)$.

###  5.2. A Koszul filtration for the augmented Chow ring of a matroid

When $M$ is a matroid of rank 1, we note that
$\operatorname{CH}(M)\cong\mathbb{Q}[x_{E}]/(x_{E}^{2})$, which we have
previously noted has a Koszul filtration. We may, therefore, assume the
following without any loss of generality.

######  Notation.

Throughout the remainder of this section, $M$ denotes a simple matroid with
$\operatorname{rk}M\geq 2$ and $A=\operatorname{CH}(M)$ with respect to the
atom-free presentation.

######  Remark 5.5.

As a consequence of Corollary 5.3, we note that $I_{\mathrm{af}}(M)$ contains
the polynomials

$x_{H}x_{E},\qquad
x_{F}x_{H}^{\operatorname{rk}H-\operatorname{rk}F}+x_{F}x_{E}^{\operatorname{rk}M-\operatorname{rk}F-1},\qquad
x_{H}^{\operatorname{rk}H+1}+x_{E}^{\operatorname{rk}M}$ (5.11)

for any $F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}$ and
hyperplane $H\supsetneq F$. In particular, we note that the last type of
relation is the primary difference when compared with the ordinary Chow ring
of $M$. Since $x_{F}x_{E}^{\operatorname{rk}M}=0$ in $A$ for all
$F\in\operatorname{\mathcal{L}}\smallsetminus\\{\varnothing\\}$ and $A$ is an
Artinian Gorenstein ring, it follows that $x_{E}^{\operatorname{rk}M}$ is a
socle generator of $A$.

The proofs of the following results are completely analogous to those of
Section 4 aside from some minor modifications based on the above remark.

######  Lemma 5.6.

Let $\mathcal{H}$ be a set of hyperplanes of $M$. Then the ideal $(x_{H}\mid
H\in\mathcal{H})$ in $A=\operatorname{CH}(M)$ has a $\mathbb{Q}$-basis
consisting of all nested monomials $x_{F}^{\alpha}$ with $F=\\{F_{1}\supsetneq
F_{2}\supsetneq\cdots\supsetneq F_{r}\supsetneq F_{r+1}=\varnothing\\}$ for
some $r\geq 0$ such that either:

  1. (i)

$F_{1}\in\mathcal{H}$, or

  2. (ii)

$F_{1}=E$, $H\supsetneq F_{2}\neq\varnothing$ for some $H\in\mathcal{H}$ and
$\alpha_{1}=\operatorname{rk}M-\operatorname{rk}F_{2}-1$, or

  3. (iii)

$F_{1}=E$, $F_{2}=\varnothing$, and $\alpha_{1}=\operatorname{rk}M$.

######  Proposition 5.7.

For any matroid $M$, we have $(0:x_{E})=(x_{H}\mid
H\in\operatorname{coat}(E))$ and $A/(0:x_{E})\cong\operatorname{CH}(T(M))$.

######  Corollary 5.8.

Let $F$ be a flat of $M$ with $\operatorname{rk}F\geq 1$.

  1. (a)

The kernel of the natural surjective homomorphism
$\pi_{M,F}:A\to\operatorname{CH}(M|F)$ sending $x_{G}\mapsto x_{G}$ if
$G\subseteq F$ and $x_{G}\mapsto 0$ otherwise is $(x_{G}\mid G\nsubseteq F)$.

  2. (b)

Suppose that $\mathcal{G}$ is a set of flats of $M$ such that
$(F,E]\subseteq\mathcal{G}$ and $\mathcal{G}\cap[\varnothing,F]=\varnothing$.
If $\operatorname{rk}F\geq 2$, then

$(x_{G}\mid G\in\mathcal{G}):x_{F}=(x_{G}\mid G\nsubseteq F)+(x_{G}\mid
G\in\operatorname{coat}(F)).$

Otherwise, $(x_{G}\mid G\in\mathcal{G}):x_{F}=A_{+}$.

######  Proposition 5.9.

Let $\mathcal{H}$ be a nonempty set of hyperplanes of $M$ and $H^{\prime}$ be
any hyperplane. Then:

  1. (a)

$(0:x_{H^{\prime}})=(x_{F}\mid F\nsubseteq H^{\prime})$ and
$A/(0:x_{H^{\prime}})\cong\operatorname{CH}(M|H^{\prime})$.

  2. (b)

$(0:(x_{H}\mid H\in\mathcal{H}))=(x_{F}\mid F\nsubseteq H\;\text{for
all}\;H\in\mathcal{H})$.

  3. (c)

Suppose that $H^{\prime}\notin\mathcal{H}$. If
$\operatorname{rk}H^{\prime}\geq 2$ and for every $H\in\mathcal{H}$ there
exists an $F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime})$ such that $H\cap
H^{\prime}\subseteq F$, then

$(x_{H}\mid H\in\mathcal{H}):x_{H^{\prime}}=(0:x_{H^{\prime}})+(x_{F}\mid
F\in\operatorname{coat}_{\mathcal{H}}(H^{\prime}))$

Otherwise, if $\operatorname{rk}H^{\prime}=1$, then $(x_{H}\mid
H\in\mathcal{H}):x_{H^{\prime}}=A_{+}$.

######  Theorem 5.10.

The augmented Chow ring of a matroid has a Koszul filtration. In particular,
such rings have rational Poincaré series.

It is tempting to seek a common framework for the Koszul property of Chow
rings and augmented Chow rings of matroids. The first natural approach is to
consider atomic Chow rings of geometric lattices with respect to arbitrary
building sets. Examples 6.2–6.3 show that there is no such result in this
level of generality.

##  6\. Examples and questions

In this section, we collect some relevant examples and questions. We begin
with Bøgvad’s algebra with irrational Poincaré series.

######  Example 6.1 (A quadratic Artinian Gorenstein algebra with irrational
Poincaré series).

This example is due to Bøgvad [Bøg83] and is based on Anick’s [Ani82]
construction; however, it is not obvious that this construction leads to a
quadratic algebra. Let $\mathbb{K}$ denote a field of characteristic $2$.
(Computation suggests that any field will do.) Set

$R=\mathbb{K}[x_{1},x_{2},x_{3},x_{4},x_{5}]/(x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{4}^{2},x_{5}^{2},x_{1}x_{2},x_{4}x_{5},x_{1}x_{3}+x_{3}x_{4}+x_{2}x_{5}).$

Anick proved that $R$ has irrational Poincaré series. One can easily check
that $R$ is superlevel in the language of Mastroeni, Schenck, and Stillman
[MSS21a]. Therefore, the idealization $\widetilde{R}=R\ltimes\omega_{R}(-3)$
is quadratic and Gorenstein; here $\omega_{R}$ denotes the canonical module of
$R$. The Poincaré series of $\widetilde{R}$ is rationally related to that of
$R$ [Gul72], and so $\widetilde{R}$ also has irrational Poincaré series. (In
both papers, the ideal $(x_{1},x_{2},x_{3},x_{4},x_{5})^{3}$ is included among
the relations but is redundant.)

One might hope for a more general result about the Koszul property of Chow
rings of atomic lattices and arbitrary building sets. Even for geometric
lattices, the following examples show that this is not possible.

######  Example 6.2 (A geometric lattice with Gorenstein but not Koszul Chow
ring).

In general, the Chow rings $D(\operatorname{\mathcal{L}},\mathcal{G})$ of
Feichtner and Yuzvinsky need not be Koszul for all building sets $\mathcal{G}$
even when $\operatorname{\mathcal{L}}$ is a geometric lattice. For example,
suppose $\operatorname{\mathcal{L}}=B_{3}$ is the Boolean lattice on 3
elements, which is the lattice of flats of the uniform matroid $U_{3,3}$ on
the ground set $\\{1,2,3\\}$. Then
$\mathcal{G}=\\{\\{1\\},\\{2\\},\\{3\\},\\{1,2,3\\}\\}$ is easily checked to
be a building set for $\operatorname{\mathcal{L}}$, but (after suppressing
commas and braces in the variable indices) the Chow ring is

$D(\operatorname{\mathcal{L}},\mathcal{G})=\frac{\mathbb{Q}[x_{1},x_{2},x_{3},x_{123}]}{(x_{1}x_{2}x_{3},x_{1}+x_{123},x_{2}+x_{123},x_{3}+x_{123})}\cong\frac{\mathbb{Q}[x_{123}]}{(x_{123}^{3})},$

which is evidently not Koszul. Interestingly, one can check using Macaulay2
that this is the only building set for the lattice $B_{3}$ for which the Chow
ring is not Koszul.

######  Example 6.3 (A geometric lattice with non-Koszul Chow ring for the
minimal building set).

Even when $\mathcal{G}=\mathcal{G}_{\min}$ is the minimal building set of
geometric lattice $\operatorname{\mathcal{L}}$,
$D(\operatorname{\mathcal{L}},\mathcal{G})$ can fail to be Koszul. Let $C_{4}$
denote the $4$-cycle and $M=M(C_{4})$ its graphic matroid. The minimal
building set of $\operatorname{\mathcal{L}}(M)$ is
$\mathcal{G}=\\{\\{1\\},\\{2\\},\\{3\\},\\{4\\},\\{1,2,3,4\\}\\}$. The Chow
ring is

$\frac{\mathbb{Q}[x_{1},x_{2},x_{3},x_{4},x_{1234}]}{(x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{3}x_{4},x_{2}x_{3}x_{4},x_{1}+x_{1234},x_{2}+x_{1234},x_{3}+x_{1234},x_{4}+x_{1234})}\cong\frac{\mathbb{Q}[x_{1234}]}{(x_{1234}^{3})},$

which is also not Koszul.

The Gorenstein property of the (augmented) Chow ring of a matroid seems to
play an important role in our methods. It was already clear from the work of
Feichtner and Yuzvinsky that not all Chow rings of atomic lattices were
Gorenstein; see [FY04, pp. 22-23]. We present a simple example.

######  Example 6.4 (A non-geometric lattice with Koszul but not Gorenstein
Chow ring).

Consider the lattice $\operatorname{\mathcal{L}}$ with Hasse diagram in Figure
3. It is easy to check that $\operatorname{\mathcal{L}}$ is atomic but not
semimodular. The Chow ring of $\operatorname{\mathcal{L}}$ with respect to the
maximal building set
$\mathcal{G}_{\max}=\operatorname{\mathcal{L}}\smallsetminus\\{\widehat{0}\\}$
is

$D(\operatorname{\mathcal{L}},\mathcal{G}_{\max})\cong\frac{\mathbb{Q}[x_{e},x_{f},x_{\widehat{1}}]}{(x_{e},x_{f},x_{\widehat{1}})^{2}},$

which is Koszul but not Gorenstein.

$e$$f$$\widehat{1}$$a$$b$$c$$d$$\widehat{0}$ Figure 3. An atomic lattice
$\operatorname{\mathcal{L}}$ with non-Koszul Chow ring
$\operatorname{\underline{CH}}(\operatorname{\mathcal{L}},\mathcal{G}_{\max})$.

The preceding examples naturally lead to the following questions.

######  Question 6.5.

Under what combinatorial conditions on the building set $\mathcal{G}$ and the
lattice $\operatorname{\mathcal{L}}$ is the Chow ring
$D(\operatorname{\mathcal{L}},\mathcal{G})$ Koszul? Is there a Chow ring of a
geometric lattice that fails to be Koszul with respect to some building set
while still being quadratic?

######  Question 6.6.

Is the Chow ring of any graded, atomic lattice with respect to its maximal
building set always Koszul?

While Dotsenko showed that $D(\Pi_{n},\mathcal{G}_{\mathrm{min}})$ has a
quadratic Gröbner basis, a similar monomial order on
$\operatorname{\underline{CH}}(M)=D(\operatorname{\mathcal{L}}(M),\mathcal{G}_{\mathrm{max}})$
does not seem to produce a quadratic Gröbner basis. On the other hand, there
does not appear to be a Hilbert function obstruction to one, leaving open the
following question.

######  Question 6.7.

Is the Chow ring of a matroid G-quadratic?

We close with a potential generalization of our results.

######  Question 6.8.

Does every quadratic, Artinian, Gorenstein $\mathbb{K}$-algebra with the
Kähler package have the Koszul property?

It follows from [MS20, Theorem 4.3] that not all quadratic, Gorenstein
$\mathbb{K}$-algebras have the Kähler package; in particular, there are
quadratic Gorenstein $\mathbb{K}$-algebras with non-unimodal $h$-vectors.

## Acknowledgements

The authors thank Emanuele Delucchi, Vladimir Dotsenko, Chris Eur, Vic Reiner,
and Botong Wang for many helpful conversations. Computations with Macaulay2
[GS], especially with Justin Chen’s matroid package [Che19], were very helpful
while working on this project. Mastroeni was supported by an AMS-Simons Travel
Grant. McCullough was supported by NSF grant DMS–1900792.

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