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# Hodge Theory for Polymatroids

Roberto Pagaria  Roberto Pagaria
Università di Bologna, Dipartimento di Matematica
Piazza di Porta San Donato 5 - 40126 Bologna
Italy<EMAIL_ADDRESS>and  Gian Marco Pezzoli  Gian Marco Pezzoli
Università di Bologna, Dipartimento di Matematica
Piazza di Porta San Donato 5 - 40126 Bologna
Italy<EMAIL_ADDRESS>
###### Abstract.

We construct a Leray model for a discrete polymatroid with arbitrary building
set and we prove a generalized Goresky-MacPherson formula. The first row of
the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard
Lefschetz, and Hodge-Riemann theorems for the Chow ring. Furthermore, we
provide a relative Lefschetz decomposition with respect to the deletion of an
element.

The first author is supported by PRIN 2017YRA3LK

###### Contents

  1. 1 Introduction
  2. 2 The Leray model
  3. 3 Generalized Goresky-MacPherson formula
  4. 4 Kähler package
  5. 5 The relative Lefschetz decomposition
  6. 6 Characteristic polynomial
  7. 7 An example

##  1\. Introduction

Recently, long standing conjectures about log-concavity of graphs and matroids
have been brilliantly solved by studying the Chow ring of matroids [Huh12,
HK12, Len13, HW17, AHK18, BHM+20, ADH20, BEST21]. A natural question is
whether the corresponding results hold for polymatroids [BEST21, Question
1.5].

_Discrete polymatroids_ generalize arrangements of subspaces in the same way
as matroids generalize hyperplane arrangements. Moreover polymatroids codify
invariants of hypergraphs as matroids do with graphs. Polymatroids have
application in linear programming [Edm70, Sch03], in code theory [GJLR20], and
in commutative algebra [HH02]. The base polytope of a polymatroids is a
_generalized permutahedron_ with a vertex in the origin, this yields a
criptmorphism between polymatroids and generalized permutahedra [Pos09, AA17].

The easiest definition of polymatroid is a pair $P=(E,\operatorname{cd})$
where $E$ is a finite ground set and $\operatorname{cd}\colon
2^{E}\to\mathbb{N}$ is a _increasing submodular function_. If $P$ is realized
by a subspace arrangement, then $\operatorname{cd}$ is the codimension of the
corresponding flat.

In the case of arrangements of subspaces, De Concini and Procesi [DCP95] have
constructed a _wonderful model_ , i.e. a smooth projective variety that
contains the complement of the arrangement as open subset. This wonderful
model $Y_{\mathcal{G}}$ is obtained from $\mathbb{P}^{r}$ by a sequence of
blowups along some linear subspaces; this collection of subspaces is called
_building set_ $\mathcal{G}$. The variety $Y_{\mathcal{G}}$ is used for
studying the complement of the subspace arrangement, by considering the Leray
spectral sequence for the inclusion of the complement in the wonderful model
$Y_{\mathcal{G}}$. The spectral sequence collapses at the third page yielding
a _Leray model_ (also known as Morgan algebra [Mor78]) for the rational
homotopy type.

Inspired by the realizable case, we provide a combinatorial definition of
_building set_ for polymatroids and we introduce a _Leray model_
$B(P,\mathcal{G})$ for a polymatroid with building set. In the case of
matroids the Leray model was recently studied by Bibby, Denham, and Feichtner
[BDF21]. The last combinatorial object that we need is the
$\mathcal{G}$_-nested set complex_ $n(P,\mathcal{G})$. In the realizable case
this complex remembers whether the intersection of the corresponding divisors
in $Y_{\mathcal{G}}$ is non-empty.

The problem of computing the cohomology of the complement of a subspace
arrangement was solved by Goresky and MacPherson [GM88] and by De Concini and
Procesi [DCP95] with different techniques. Goresky and MacPherson have used
the stratified Morse theory to describe the additive cohomology with integer
coefficients of the complement in terms of a poset. De Concini and Procesi
have provided the aforementioned Morgan algebra (alias the Leray model) that
describes the ring structure of the cohomology with rational coefficients.
These two results were connected by Yuzvinsky [Yuz02, Yuz99] constructing a
smaller rational model $\operatorname{CM}$, however this connection was found
only for the maximal building set.

We extend these results to the non-realizable setting and to arbitrary
building sets, see Theorems 3.14 and 3.15, by using the _critical monomial
algebra_ $\operatorname{CM}(P,\mathcal{G})$.

######  Theorem.

The inclusion $\operatorname{CM}(P,\mathcal{G})\hookrightarrow
B(P,\mathcal{G})$ is a quasi-isomorphism. Moreover

$H^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}(B(P,\mathcal{G}))\cong
H^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}(\operatorname{CM}(P,\mathcal{G}))\cong\bigoplus_{f\in
L}\bigotimes_{g\in
F}\tilde{H}_{2\operatorname{cd}(g)-2-{\mathbin{\mathchoice{\hbox
to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}\Bigl{(}n((\hat{0},g),\mathcal{G})\Bigr{)}.$

In the realizable case with maximal building set, the above decomposition
specializes to the Goresky-MacPherson formula.

The Leray model contains a subalgebra $\operatorname{DP}(P,\mathcal{G})$ as
the first row of the spectral sequence, we call this algebra the _Chow ring_
of the polymatroid. For subspace arrangements,
$\operatorname{DP}(P,\mathcal{G})$ is the cohomology (indeed the Chow ring) of
the wonderful model $Y_{\mathcal{G}}$. The combinatorial Chow ring for
matroids was studied by Feichtner and Yuzvinsky [FY04] and later by Huh, Katz,
and Adiprasito [Huh12, HK12, AHK18] and others.

We prove that the Chow ring $\operatorname{DP}(P,\mathcal{G})$ of a
polymatroid satisfies the Kähler package (see Theorems 4.7 and 4.21).

######  Theorem.

The algebra $\operatorname{DP}(P,\mathcal{G})$ satisfy the Poincaré duality
property. Moreover, there exists a simplicial cone $\Sigma_{P,\mathcal{G}}$
contained in $\operatorname{DP}^{1}(P,\mathcal{G})$ such that for each
$\ell\in\Sigma_{P,\mathcal{G}}$ the Hard-Lefschetz theorem and the Hodge-
Riemann relations hold.

We prove the above theorem using methods similar to ones in [AHK18]. A second
and easier proof of the Kähler package for matroid was given in [BHM+22] using
a semismall decomposition; the decomposition is the first step through the
singular Hodge theory [BHM+20]. In the realizable setting the decomposition is
induced by a map between wonderful models that is semismall (for semismall
maps in algebraic geometry see [dCM02, dCM09]). In the case of polymatroids
the corresponding map is not semismall, hence we cannot deduce the Kahler
package using this method. However, we obtain a relative Lefschetz
decomposition of the Chow ring (see Theorem 5.4).

######  Theorem.

Let $\operatorname{DP}_{(a)}$ be the Chow ring for the polymatroids
$P\setminus a$ where an element $a\in E$ is removed from the ground set. The
Chow ring $\operatorname{DP}(P,\mathcal{G})$ decomposes into irreducible
$\operatorname{DP}_{(a)}$-modules as

$\operatorname{DP}(P,\mathcal{G})=\operatorname{DP}_{(a)}\oplus\bigoplus_{f\in
S_{a}}\bigoplus_{k=1}^{n_{f}}x_{f}^{k}\operatorname{DP}_{(a)}.$

Moreover, these irreducibles are explicitly described by:

$x_{f}^{k}\operatorname{DP}_{(a)}\cong\operatorname{DP}((P\setminus
a)^{f\setminus a},(\mathcal{G}\setminus a)^{f\setminus
a})\otimes\operatorname{DP}(P_{f},\mathcal{G}_{f})[k].$

The _reduced characteristic polynomial_ of a polymatroid is defined by

(1) $\overline{\chi}_{P}(\lambda)=\frac{\sum_{A\subseteq E}(-1)^{\lvert
A\rvert}\lambda^{\operatorname{cd}(E)-\operatorname{cd}(A)}}{\lambda-1}.$

As final step we relate the coefficients of the reduced characteristic
polynomial to the Hodge-Riemann bilinear form (see Theorem 6.4). In order to
do that, we restrict to the case of the maximal building set and we fix an
isomorphism
$\deg\colon\operatorname{DP}^{r}(P,\mathcal{G}_{\max})\to\mathbb{Q}$.

######  Theorem.

There exist elements
$\alpha,\beta\in\operatorname{DP}^{1}(P,\mathcal{G}_{\max})$ such that

$\overline{\chi}_{P}(\lambda)=\sum_{i=0}^{r}(-1)^{i}\deg(\alpha^{i}\beta^{r-i})\lambda^{i}.$

The element $\alpha$ belongs to the closure of the $\sigma$-cone (morally it
is nef), but in general $\beta$ is not in the closure of the ample cone.
Hence, the coefficients of the reduced characteristic polynomial do not form a
log-concave sequence (see Remark 6.6). Indeed every finite sequence of non-
positive integers can appear as a substring of the coefficients.

### Techniques

The techniques used for our proof are various and inspired by [FY04, BDF21,
Yuz02, AHK18, BHM+22].

In Section 2 we make use of Gröbner basis to give two explicit additive bases
of the Leray model. In Section 3, by using algebraic Morse theory, we compute
the cohomology of the Leray model generalizing the Goresky-MacPherson formula.
In Section 4, we use an inductive procedure to prove the Kähler package. The
main difference with the previous methods is that we do not have partial
building sets as in [BDF21] nor order filters as in [AHK18]. Our induction is
based on the cardinality of the building set, and the inductive step involved
completely different polymatroids. Section 5 is devoted to the proof of the
relative Lefschetz decomposition using some lemmas from the previous sections.
The reduced characteristic polynomial is studied in Section 6. We prove the
claimed equality by showing that both polynomials satisfy the same recursion.
In this proof we use the properties of the Möbius function for posets.

Finally, Section 7 contains an explicit and exhaustive example that
illustrates our definitions and properties.

### Acknowledgements

The authors thank Corrado De Concini for holding a mini-course on log-
concavity results at the winter school “Geometry, Algebra and Combinatorics of
Moduli Spaces and Configurations IV”. The authors also thank the organizers of
the conference for having created this opportunity.

##  2\. The Leray model

For general references about polymatroids we suggest [Wel76].

######  Definition 2.1.

A polymatroid $P=(E,\operatorname{cd})$ on the ground set $E$ is a codimension
function $\operatorname{cd}\colon 2^{E}\to\mathbb{N}$ satisfying:

  1. (C1)

$\operatorname{cd}(\emptyset)=0$,

  2. (C2)

if $A\subseteq B$, then $\operatorname{cd}(A)\leq\operatorname{cd}(B)$, and

  3. (C3)

if $A,B\subseteq E$, then $\operatorname{cd}(A\cup B)+\operatorname{cd}(A\cap
B)\leq\operatorname{cd}(A)+\operatorname{cd}(B).$

A polymatroid is a matroid if the codimension of singletons are either zero or
one.

The closure of a subset $A\subseteq E$ is the subset

$\\{a\in E\ |\ \operatorname{cd}(A\cup\\{a\\})=\operatorname{cd}(A)\\}.$

A flat is a closed set and the collection of flats forms a lattice $L_{P}$,
that we call the poset of flats. We use the notation $\max(X)$ with
$X\subseteq L_{P}$ for the set of maximal elements of $X$.

######  Definition 2.2 (Geometric building set).

Let $P=(E,\operatorname{cd})$ be a polymatroid and let $L$ be its lattice of
flats. A subset $\mathcal{G}$ in $L\setminus\\{\hat{0}\\}$ is called a
geometric building set if for any $x\in L$ the morphism of lattices:

$\varphi_{x}\colon\prod_{y\in\max(\mathcal{G}_{\leq x})}\
[\hat{0},y]\to[\hat{0},x]$

induced by the inclusions is an isomorphism and the equality

$\operatorname{cd}(x)=\sum_{y\in\max(\mathcal{G}_{\leq
x})}\operatorname{cd}(y)$

holds.

We define $F(P,\mathcal{G},x)=\max(\mathcal{G}_{\leq x})$ the set of
$\mathcal{G}$-factors of $x$.

######  Definition 2.3 ($\mathcal{G}$-nested set complex).

A subset $S$ of $\mathcal{G}$ is called $\mathcal{G}$-nested if, for any set
of incomparable elements $x_{1},\dots,x_{t}$ in $S$ of cardinality at least
two, the join $x_{1}\vee\cdots\vee x_{t}$ is not contained in $\mathcal{G}$.
The $\mathcal{G}$-nested sets form an abstract simplicial complex
$n(P,\mathcal{G})$.

We suggest to visualize a (realizable) polymatroid $(E,\operatorname{cd})$ as
a collection of linear subspaces $S_{e}$ for $e\in E$ in a fixed complex
vector space $V$. For each $A\subseteq E$, the codimension
$\operatorname{cd}(A)$ is the codimension of the corresponding flat
$\cap_{a\in A}S_{a}$. The (geometric) building set $\mathcal{G}$ is a good
choice of some flats to blow up, in order to obtain a wonderful model
$Y_{\mathcal{G}}$ with some exceptional divisors $D_{g}\subset
Y_{\mathcal{G}}$, $g\in\mathcal{G}$ indexed by $\mathcal{G}$. A subset $S$ of
$\mathcal{G}$ is $\mathcal{G}$-nested if and only if the corresponding
divisors $\\{D_{W}\\}_{W\in S}$ have non empty intersection.

The following proposition summarizes the main properties of building and
nested sets.

######  Proposition 2.4.

Let $P$ be a polymatroid with poset of flats $L$ and $\mathcal{G}$ be a
building set. Then:

  1. (1)

For each $g\in\mathcal{G}$, $x\in L$ with $x\geq g$, there exists a unique
$\mathcal{G}$-factor $p$ of $x$ such that $p\geq g$.

  2. (2)

If $g,h\in\mathcal{G}$ and $g\wedge h>\hat{0}$, then $g\vee h\in\mathcal{G}$.

  3. (3)

If $S$ is a $\mathcal{G}$-nested set, then the $\mathcal{G}$-factors of
$\bigvee S$ are the maximal elements in $S$ (i.e. $F(P,\mathcal{G},\bigvee
S)=\max(S)$).

  4. (4)

Let $S$ be a $\mathcal{G}$-nested set, the Hasse diagram of $S$ (as subset of
$L$) is a forest.

###### Proof.

For $(1)$ see [FK04, Proposition 2.5(1)], for $(2)$ see [BDF21, Proposition
2.5.3(b)], and for $(3)$ see [FK04, Proposition 2.8]. In order to prove $(4)$
we suppose that the Hasse diagram $\Gamma_{S}$ of $S$ is not a forest. Thus
there exist two incomparable elements $g,h\in S$ and $t\in S$ such that
$g\wedge h\geq t\in S$; in particular $g\wedge h>\hat{0}$. By part $(2)$ we
get that $g\vee h\in\mathcal{G}$ but this contradicts the definition of nested
set. Therefore $\Gamma_{S}$ is a forest. ∎

Let $P=(E,\operatorname{cd})$ be a polymatroid, $L$ be its poset of flats, and
$\mathcal{G}$ be a building set in $L$. Let
$\mathcal{R}(\mathcal{G})=\mathbb{Q}[e_{g},x_{g}\mid g\in\mathcal{G}]$ be the
bigraded commutative algebra with exterior generators $e_{g}$ in bidegree
$(0,1)$ and commutative generators $x_{g}$ in bidegree $(2,0)$.

This algebra is equipped with a differential $d$ of bidegree $(2,-1)$ defined
on generators by $d(e_{g})=x_{g}$, $d(x_{g})=0$. Fix a linear extension
$\succ$ of the order on $\mathcal{G}$, this gives a reverse order among the
$e$-variables and among the $x$-variables, i.e. $x_{h}\prec x_{g}$ and
$e_{h}\prec e_{g}$ if and only if $h\succ g$. We also set $x_{g}\prec e_{h}$
for each $g,h$. The algebra $\mathcal{R}(\mathcal{G})$ has a monomial basis
given by:

$e_{T}x_{S}^{b}:=e_{g_{1}}\cdots e_{g_{t}}x_{h_{1}}^{b_{1}}\cdots
x_{h_{s}}^{b_{s}}$

where $T=\\{g_{1},\dots,g_{t}\\}$ with $g_{i}\in\mathcal{G}$ satisfying
$g_{1}\prec g_{2}\prec\cdots\prec g_{t}$, $S=\\{h_{1},\dots,h_{s}\\}$ with
$h_{i}\in\mathcal{G}$ and $b=(b_{1},\dots,b_{s})$ is a $s$-tuple of positive
integers. We define the element:

$c_{g}=\sum_{\begin{subarray}{c}h\in\mathcal{G}\\\ h\geq
g\end{subarray}}x_{h}.$

######  Definition 2.5 (The Leray model of a polymatroid).

Let $I(\mathcal{G})$ be the ideal of $\mathcal{R}(\mathcal{G})$ generated by

  1. (i)

$e_{T}x_{S}$ whenever $S\cup T\notin n(P,\mathcal{G})$,

  2. (ii)

$e_{T}x_{S}c_{g}^{b}$ whenever $S,T\subseteq\mathcal{G}$, $g\in\mathcal{G}$
and $b\geq\operatorname{cd}(g)-\operatorname{cd}(\bigvee(S\cup T)_{<g})$.

The ideal $I(\mathcal{G})$ is preserved by the differential
$\operatorname{d}$, so the quotient

$B(P,\mathcal{G})=\faktor{\mathcal{R}(\mathcal{G})}{I(\mathcal{G})}$

is a bigraded differential algebra, called the Leray model of the polymatroid.

In the realizable case, the Leray model $B(P,\mathcal{G})$ is the second page
of the Leray spectral sequence for the natural inclusion

$V\setminus\cup_{e\in E}S_{e}\cong
Y_{\mathcal{G}}\setminus\cup_{g\in\mathcal{G}}D_{g}\hookrightarrow
Y_{\mathcal{G}}.$

This spectral sequence collapses at the third page, hence it becomes a
differential bigraded algebra also known as the Morgan algebra [Mor78].

######  Remark 2.6.

Let $e_{T}x_{S}c_{g}^{b}$ be a monomial of type $(ii)$ and let

$S^{\prime}=S\cap\max(S\cup T)_{<g}\quad\hbox{and}\quad T^{\prime}=\max(S\cup
T)_{<g}\setminus S.$

The monomial $e_{T^{\prime}}x_{S^{\prime}}c_{g}^{b}$ divides
$e_{T}x_{S}c_{g}^{b}$. Thus, when we consider a monomial of type $(ii)$ we can
always assume that $S\cap T=\emptyset$, $S\cup T$ is an antichain and
$\bigvee(S\cup T)<g$.

######  Theorem 2.7.

The generators of type (i) and (ii) of the ideal $I(\mathcal{G})$ of
Definition 2.5 form a Gröbner basis with respect to the deg-lex order.

###### Proof.

We adapt the method used in [FY04, Theorem 2] and in [BDF21, Theorem 5.3.1].
We are fixing a linear extension of the order on $\mathcal{G}$ with
$x_{g}\prec e_{h}$ for each $g,h$. We consider the deg-lex monomial order on
$\mathcal{R}(\mathcal{G})$ and we explicitly compute $S$-polynomials.

Case (i)-(i):

Since relations type $(i)$ are monomials the $S$-polynomials is zero.

Case (i)-(ii):

Now we consider $f_{1}=e_{T}x_{S}$ of type $(i)$ and
$f_{2}=e_{A}x_{B}c_{g}^{b}$ of type $(ii)$. We assume that $\bigvee(A\cup
B)<g$ (see Remark 2.6). Let $U=T\cup A$, $V=B\cup S\smallsetminus\\{g\\}$,
therefore the $S$-polynomial is

$S(f_{1},f_{2})=e_{U}x_{V}x_{g}^{b}-e_{U}x_{V}c_{g}^{b}=e_{U}x_{V}(x_{g}^{b}-c_{g}^{b}).$

If $g\notin S$, we have that $S\subseteq V$ and therefore

$S(f_{1},f_{2})=\pm e_{A\setminus T}e_{T}x_{S}x_{V\setminus
S}(x_{g}^{b}-c_{g}^{b})$

is divisible by $e_{T}x_{S}$.

Then, assume $g\in S$, since $S\cup T$ is not nested we have that $U\cup
V\cup\\{g\\}$ is not nested. If $U\cup V$ is not nested, then $S(f_{1},f_{2})$
would be divisible by $e_{U}x_{V}$.

So assume $U\cup V$ is nested, (thus we have $g\notin U\cup V$) since $U\cup
V\cup\\{g\\}$ is not nested the $S$-polynomial modulo $e_{U}x_{V}x_{g}$ became

$S(f_{1},f_{2})\equiv e_{U}x_{V}\big{(}\sum_{f>g}x_{f}\big{)}^{b}.$

The set $U\cup V\cup\\{g\\}$ contains a non trivial antichain $Y$ whose join
$\bigvee Y=y$ is in $\mathcal{G}$ and $Y$ must contain $g$ since $U\cup V$ is
nested; let $y^{\prime}=\bigvee(Y\smallsetminus\\{g\\})$. We have

$\displaystyle b$
$\displaystyle=\operatorname{cd}(g)-\operatorname{cd}\Bigl{(}\bigvee_{\begin{subarray}{c}l\in
A\cup B\\\ l<g\end{subarray}}l\Bigr{)}$
$\displaystyle\geq\operatorname{cd}(g\vee
y^{\prime})-\operatorname{cd}\Bigl{(}\bigvee_{\begin{subarray}{c}l\in A\cup
B\\\ l<g\end{subarray}}l\vee y^{\prime}\Bigr{)}$
$\displaystyle\geq\operatorname{cd}(y)-\operatorname{cd}\Bigl{(}\bigvee_{\begin{subarray}{c}l\in
U\cup V\\\ l<y\end{subarray}}l\Bigr{)}=b^{\prime}$

and so $e_{U}x_{V}c_{y}^{b}$ is a relation of type $(ii)$. We claim that
modulo relations of type $(i)$

$S(f_{1},f_{2})\equiv e_{U}x_{V}c_{y}^{b}.$

To obtain this, we show that if $f\in\mathcal{G}$ with $f>g$ and $f\ngeq y$,
then $U\cup V\cup\\{f\\}$ is not nested. Suppose that $U\cup V\cup\\{f\\}$ is
nested and consider the antichain
$Y^{\prime}=\max(Y\smallsetminus\\{g\\}\cup\\{f\\})\subseteq U\cup
V\cup\\{f\\}$. The set $Y^{\prime}$ is nested and by Proposition 2.4 the
$\mathcal{G}$-factors of $\bigvee(Y\smallsetminus\\{g\\}\cup\\{f\\})$ are
exactly the elements of $Y^{\prime}=\\{y_{1},\dots,y_{k},f\\}$. We have

$\bigvee(Y\smallsetminus\\{g\\}\cup\\{f\\})=y^{\prime}\vee f\geq
y^{\prime}\vee g=y.$

By definition of $\mathcal{G}$-factor we have two cases:

  * •:

$y\leq y_{i}$ for a certain $i$. But this is impossible since $y_{i}<y$;

  * •:

$y\leq f$ contrary to the assumption $f\not\geq y$.

Thus, $U\cup V\cup\\{f\\}$ is not nested and $S(f_{1},f_{2})$ reduces to zero.

Case (ii)-(ii):

Let $f_{1}=e_{T}x_{S}c_{g}^{d}$ and $f_{2}=e_{A}x_{B}c_{h}^{f}$ be two
relations of type $(ii)$. We may assume $\bigvee(S\cup T)<g$ and
$\bigvee(A\cup B)<h$ (see Remark 2.6). We have the following cases:

First $g=h$ and $d\leq f$, then the $S$-polynomial is

$S(f_{1},f_{2})=e_{T\cup A}x_{S\cup B}c_{g}^{d}(x_{g}^{f-d}-c_{g}^{f-d});$

which is divisible by $e_{T}x_{S}c_{g}^{d}$.

Second $g\neq h$, $g\notin B$, $h\notin S$, we also assume that $h\succ g$.
The $S$-polynomial is

$S(f_{1},f_{2})=e_{T\cup A}x_{S\cup
B}(x_{h}^{f}c_{g}^{d}-x_{g}^{d}c_{h}^{f}).$

Let $y=e_{T\cup A}x_{S\cup B}c_{g}^{d}(c_{h}^{f}-x_{h}^{f})$, which is
divisible by $f_{1}=e_{T}x_{S}c_{g}^{d}$ and has a leading term smaller or
equal to that of $S(f_{1},f_{2})$. The remainder

$S(f_{1},f_{2})+y=e_{T\cup A}x_{S\cup B}(c_{g}^{d}-x_{g}^{d})c_{h}^{f},$

is divisible by $f_{2}=e_{A}x_{B}c_{h}^{f}$, and reduces to zero.

Finally, assume $g\neq h$ and $g\in B$, by Remark 2.6 we must have $g\prec h$
and $h\notin S$. Let $U=T\cup A$ and $V=S\cup B\smallsetminus\\{g\\}$, the
$S$-polynomial is

$S(f_{1},f_{2})=e_{U}x_{V}(x_{h}^{f}c_{g}^{d}-x_{g}^{d}c_{h}^{f}).$

Let $y=e_{U}x_{V}c_{g}^{d}(c_{h}^{f}-x_{h}^{f})$, which is divisible by
$f_{1}=e_{T}x_{S}c_{g}^{d}$ and has a leading term smaller or equal to that of
$S(f_{1},f_{2})$. It remains to verify that

$S(f_{1},f_{2})+y=e_{U}x_{V}(c_{g}^{d}-x_{g}^{d})c_{h}^{f}$

reduces to zero. First, through division by $f_{2}=e_{A}x_{B}c_{h}^{f}$, since
$g\in B$ we have

(2) $S(f_{1},f_{2})+y\equiv
e_{U}x_{V}\Big{(}\sum_{k>g}x_{k}\Big{)}^{d}c_{h}^{f}.$

We claim that for any $k>g$, $k\not\geq h$ we have

$e_{U}x_{V}x_{k}c_{h}^{f}\equiv e_{U}x_{V}x_{k}c_{h\vee k}^{f}\equiv 0$

modulo relations of type $(i)$ and $(ii)$. For the first claim, if $p\geq h$
but $p\ngeq h\vee k$ then $\\{p,k\\}$ is not nested by Proposition 2.4 and we
can divide by the relation $x_{h}x_{p}$ of type $(i)$. The last claim follows
since $h\vee k\in\mathcal{G}$ by Proposition 2.4 and

$\displaystyle f$
$\displaystyle\geq\operatorname{cd}(h)-\operatorname{cd}(\bigvee(A\cup B))$
$\displaystyle\geq\operatorname{cd}(h\vee k)-\operatorname{cd}(\bigvee(U\cup
V\cup\\{k\\})).$

Therefore, the element in eq. (2) reduces to

$S(f_{1},f_{2})+y\equiv e_{U}x_{V}c_{h}^{d+f}.$

Since $d+f\geq\operatorname{cd}(h)-\operatorname{cd}(\bigvee(U\cup
V\cup\\{k\\}))$ we may divide by $e_{U}x_{V}c_{h}^{d+f}$ and reduce to zero.

This completes the proof. ∎

######  Corollary 2.8.

The algebra $B(P,\mathcal{G})$ has an additive basis given by the monomials
$e_{T}x_{S}^{b}$ such that $S\cup T\in n(P,\mathcal{G})$ and
$0<b(s)<\operatorname{cd}(s)-\operatorname{cd}(\bigvee(S\cup T)_{<s})$ for all
$s\in S$.

###### Proof.

An additive basis for the algebra $B(P,\mathcal{G})$ is given by all the
monomials which are not divisible by the initial monomials of the Gröbner
basis provided by Theorem 2.7. The proof follows immediately. ∎

We now provide a second presentation for the algebra $B(P,\mathcal{G})$ using
a different set of generators ($\tau_{g}$ and $\sigma_{g}$ for
$g\in\mathcal{G}$). This second presentation is inspired by the work of
Yuzvinsky [Yuz99, Yuz02] and in the matroidal case coincides with the
simplicial presentation of Backman, Eur, and Simpson [BES20].

######  Theorem 2.9.

The morphism

$\varphi\colon\Lambda[\tau_{g}\mid
g\in\mathcal{G}]\otimes\mathbb{Q}[\sigma_{g}\mid g\in\mathcal{G}]\to
B(P,\mathcal{G})$

defined by $\varphi(\tau_{g})=\sum_{h\geq g}e_{h}$ and by
$\varphi(\sigma_{g})=\sum_{h\geq g}x_{h}$, is surjective with kernel generated
by:

  1. (i)

$\prod_{t\in T}(\tau_{t}-\tau_{g})\prod_{s\in S}(\sigma_{s}-\sigma_{g})$ for
$S\cup T$ a non-trivial antichain and $g=\bigvee(S\cup T)\in\mathcal{G}$,

  2. (ii)

$\prod_{t\in T}(\tau_{t}-\tau_{g})\prod_{s\in
S}(\sigma_{s}-\sigma_{g})\sigma_{g}^{b}$ for $g\in\mathcal{G}$ and
$b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee(S\cup T)_{<g})$.

We will identify the elements $\tau_{g},\sigma_{g}$ with their images in
$B(P,\mathcal{G})$. In the realizable case the element $\sigma_{g}$ is the
fundamental class of $D_{g}$, the total transform of the flat $g$.
Analogously, $\tau_{g}$ is the sum of irreducible components of the total
transform of the flat $g$. The elements $\sigma_{g}$ can be also seen in the
following way: consider the inclusion
$Y_{\mathcal{G}}\hookrightarrow\prod_{h\in\mathcal{G}}\mathbb{P}^{\operatorname{cd}(h)-1}$
of [DCP95], $\sigma_{g}$ is the pullback of the hyperplane class of the factor
$\mathbb{P}^{\operatorname{cd}(g)-1}$.

Before the proof of Theorem 2.9 we need a couple of technical lemmas.

######  Lemma 2.10.

Let $g\in\mathcal{G}$ and $S=\\{s_{1},\dots s_{n}\\}\subset\mathcal{G}$ such
that $\bigvee S\leq g$, set $b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee
S)$. Consider a set $A=\\{a_{1},\dots,a_{n}\\}\subset\mathcal{G}$ such that
$a_{i}\geq s_{i}$ and $a_{i}\not\geq g$ for all $i=1,\dots,n$. Then

$y_{A}c_{g}^{b}=0,$

where $y_{a_{i}}$ is equal to $e_{a_{i}}$ or $x_{a_{i}}$ and
$y_{A}=y_{a_{1}}\cdots y_{a_{n}}$.

###### Proof.

Define the element $h=\bigvee A\vee g$, we first prove the equality
$y_{A}\sigma_{g}^{b}=y_{A}\sigma_{h}^{b}$ and then $y_{A}\sigma_{h}^{b}=0$.

We show that $h\in\mathcal{G}$. Let $h^{\prime}\in\mathcal{G}$ be the unique
$\mathcal{G}$-factor of $h$ such that $h^{\prime}\geq g$. For each $a_{i}$ we
have $h^{\prime}\wedge a_{i}\geq s_{i}$ and so $a_{i}\vee
h^{\prime}\in\mathcal{G}$. By maximality of $h^{\prime}$ we have $a_{i}\leq
h^{\prime}$ for all $i$. Therefore $h=h^{\prime}\in\mathcal{G}$.

Firstly, let $g^{\prime}\in\mathcal{G}$ be any element such that
$g^{\prime}\geq g$ and $g^{\prime}\not\geq h$. Suppose that
$A\cup\\{g^{\prime}\\}$ is a $\mathcal{G}$-nested set. Then the
$\mathcal{G}$-factors of $h\vee g^{\prime}$ are the maximal elements of
$A\cup\\{g^{\prime}\\}$ by Proposition 2.4. So there exists an element in
$A\cup\\{g^{\prime}\\}$ bigger or equal to $h$, this is impossible since
$g^{\prime}\not\geq h$ and $a_{i}\not\geq g$. It follows that
$A\cup\\{g^{\prime}\\}$ is not $\mathcal{G}$-nested and
$y_{A}x_{g^{\prime}}=0$.

Finally, we show that $y_{A}\sigma_{h}^{b}=0$. Indeed,
$b\geq\operatorname{cd}(g)-\operatorname{cd}(g\wedge\bigvee A)$ which is
bigger than $\operatorname{cd}(h)-\operatorname{cd}(\bigvee A)$ by
submodularity of $\operatorname{cd}$. Applying the relations of type $(ii)$ in
Definition 2.5 we complete the proof. ∎

######  Lemma 2.11.

The elements $\prod_{t\in T}(\tau_{t}-\tau_{g})\prod_{s\in
S}(\sigma_{s}-\sigma_{g})\sigma_{g}^{b}$ for $g\in\mathcal{G}$, and
$b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee(S\cup T)_{<g})$ belong to
the kernel of $\varphi$.

###### Proof.

From the argument of Remark 2.6 we may assume that $S\cap T=\emptyset$,
$S\sqcup T$ is an antichain, and $\bigvee(S\cup T)\leq g$.

We have

$\varphi\Bigl{(}\prod_{t\in T}(\tau_{t}-\tau_{g})\prod_{s\in
S}(\sigma_{s}-\sigma_{g})\sigma_{g}^{b}\Bigr{)}=\sum_{A,B}e_{A}x_{B}\Bigl{(}\sum_{l\geq
g}x_{l}\Bigr{)}^{b},$

where the sum is taken over the sets $A=(a_{i})_{i}$ and $B=(b_{j})_{j}$ such
that $a_{i}\geq t_{i}$, $b_{j}\geq s_{j}$, $a_{i}\not\geq g$, and
$b_{j}\not\geq g$. Each term $e_{A}x_{B}\Bigl{(}\sum_{l\geq
g}x_{l}\Bigr{)}^{b}$ is zero by Lemma 2.10. ∎

###### Proof of Theorem 2.9.

Let $\prec$ be a reverse linear extension of the order on $\mathcal{G}$ with
$x_{g}\prec e_{h}$ and $\sigma_{g}\prec\tau_{h}$ for each $g,h$. Now we
consider the basis formed respectively by the $\sigma_{g}$, $\tau_{h}$ and by
the $x_{g}$,$e_{h}$ ordered with $\prec$; with respect of these two basis the
matrix associated to the morphism $\varphi$ is upper unitriangular and
therefore invertible. It follows that the map $\varphi$ is surjective.

We want to prove that $\ker\varphi$ is generated by relations of type $(i)$
and $(ii)$ of Theorem 2.9. From Lemma 2.11 we know that elements of the form
$(ii)$ belong to $\ker\varphi$. The relations $(i)$ are a particular case of
relations $(ii)$ with $b=0$. Let $J$ be the ideal generated by relations of
type $(i)$ and $(ii)$, we denote also by $\operatorname{in}(J)$ the initial
ideal of $J$. It suffices to prove that

$\dim\faktor{\mathcal{C}}{\operatorname{in}(J)}\leq\dim\faktor{\mathcal{R}(\mathcal{G})}{\operatorname{in}(I(\mathcal{G}))}$

where $\mathcal{C}=\Lambda[\tau_{g}\mid
g\in\mathcal{G}]\otimes\mathbb{Q}[\sigma_{g}\mid g\in\mathcal{G}]$. Let
$K\subseteq\operatorname{in}(I)$ be the ideal generated by the leading
monomial of relation of type $(i)$ and $(ii)$, since

$\dim\faktor{\mathcal{C}}{K}\geq\dim\faktor{\mathcal{C}}{\operatorname{in}(J)}$

it suffices to check that

(3)
$\dim\faktor{\mathcal{C}}{K}=\dim\faktor{\mathcal{R}(\mathcal{G})}{\operatorname{in}(I(\mathcal{G}))}.$

The leading monomials of relation type $(i)$ are of the form
$\tau_{T}\sigma_{S}$ where $S\cup T$ is not $\mathcal{G}$-nested; the leading
monomials of relation type $(ii)$ are of the form
$\tau_{T}\sigma_{S}\sigma_{g}^{b}$ whenever $S,T\subseteq\mathcal{G}$,
$g\in\mathcal{G}$ and $b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee(S\cup
T)_{<g})$. The monomials in $\mathcal{C}$, which are not divisible by the
these two type of leading monomials, are of the form $\tau_{T}\sigma_{S}^{m}$
with $S\cup T\in n(P,\mathcal{G})$ and
$0<m(s)<\operatorname{cd}(s)-\operatorname{cd}(\bigvee(S\cup T)_{<s})$ for all
$s\in S$. Hence eq. (3) follows. Since the map $\varphi$ is surjective it is
also injective; and the initial ideal $\operatorname{in}(J)$ is equal to $K$.
Therefore, relations of type $(i)$ and $(ii)$ form a Gröbner basis for
$\ker\varphi$. ∎

From the proof of Theorem 2.9 we obtain also the following corollary:

######  Corollary 2.12.

The set of monomials $\tau_{T}\sigma_{S}^{m}$ with $S\cup T\in
n(P,\mathcal{G})$ and, for each $s\in S$,
$0<m(s)<\operatorname{cd}(s)-\operatorname{cd}(\bigvee(S\cup T)_{<s})$ is an
additive basis of $B(P,\mathcal{G})$.

See Section 7 for an example of the application of Corollary 2.8 and Corollary
2.12.

##  3\. Generalized Goresky-MacPherson formula

In this section we generalize the Goresky-MacPherson formula (see [GM88]) to
the non-realizable case and to arbitrary building set. The choice of the
minimal building set yields a significantly smaller nested set complex and it
can be useful in practical computations. Other generalizations of this formula
can be found in [BLZ15, Des18, MP22].

###  3.1. Critical monomials

######  Definition 3.1.

A standard monomial $e_{T}x_{S}$ (resp. $\tau_{T}\sigma_{S}$) is a monomial
that appears in the basis given by Corollary 2.8 (resp. by Corollary 2.12).

For any standard monomial $\tau_{T}\sigma_{S}^{b}$ we extend the function $b$
by setting $b(g)=0$ for $g\not\in S$.

######  Definition 3.2.

Let $\tau_{T}\sigma_{S}^{b}$ be a standard monomial. An element
$g\in\mathcal{G}$ is called critical with respect to the monomial
$\tau_{T}\sigma_{S}^{b}$ if $g\in T$ and
$b(g)=\operatorname{cd}(g)-\operatorname{cd}(\bigvee(S\cup T)_{<g})-1$. If
every element of $S\cup T$ is critical with respect to
$\tau_{T}\sigma_{S}^{b}$ then the monomial $\tau_{T}\sigma_{S}^{b}$ is called
critical.

Notice that if the monomial $\tau_{T}\sigma_{S}^{b}$ is critical, then
$S\subseteq T$ and so the critical monomial is uniquely determined by $T$.

######  Definition 3.3.

The critical monomial associated with $T\in n(P,\mathcal{G})$ is

$c\mu(T)=\tau_{T}\sigma_{S}^{b},$

where $S=\\{t\in
T\mid\operatorname{cd}(t)-\operatorname{cd}(\bigvee(S_{<t}))>1\\}$ and
$b(s)=\operatorname{cd}(s)-\operatorname{cd}(\bigvee(T_{<s}))-1$ for all $s\in
S$.

In Theorem 3.14 we will prove that the linear span of critical monomials form
a subcomplex (indeed a subalgebra) of $B(P,\mathcal{G})$. Moreover, we will
show that this subalgebra is quasi-isomorphic to the Leray model. This first
lemma implies that the span of critical monomials is a sub-complex.

######  Lemma 3.4.

For every critical monomial $c\mu(T)$ we have

$\operatorname{d}(c\mu(T))=\sum_{t\in T\setminus\max(T)}(-1)^{\lvert T_{\prec
t}\rvert}c\mu(T\setminus\\{t\\}).$

###### Proof.

Let $c\mu(T)=\tau_{T}\sigma_{S}^{b}$, we have

$\displaystyle\operatorname{d}(c\mu(T))$ $\displaystyle=\sum_{t\in
T}(-1)^{\lvert T_{\prec
t}\rvert}\tau_{T\setminus\\{t\\}}\sigma_{S}^{b}\sigma_{t}$
$\displaystyle=\sum_{t\in T\setminus\min(T)}(-1)^{\lvert T_{\prec
t}\rvert}\tau_{T\setminus\\{t\\}}\sigma_{S}^{b}\sigma_{t},$

because if $t\in\min(T)$ then $b(t)=\operatorname{cd}(t)-1$ and so
$\sigma_{t}^{\operatorname{cd}(t)}=0$.

Fix $t\in T\setminus\min(T)$, the set $R=\max(T_{<t})$ is nonempty. By using
relation (ii) of Theorem 2.9 and the fact that $\tau_{t}^{2}=0$, we have

$\displaystyle\tau_{R}\sigma_{R}^{b}\sigma_{t}^{b(t)+1}$
$\displaystyle=\sum_{r\in R}(-1)^{\lvert R_{\prec
r}\rvert}\tau_{R\setminus\\{r\\}}\tau_{t}\sigma_{R}^{b}\sigma_{t}^{b(t)+1}$
$\displaystyle=\sum_{r\in R}(-1)^{\lvert R_{\prec
r}\rvert}\tau_{R\setminus\\{r\\}}\tau_{t}\sigma_{R\setminus\\{r\\}}^{b}\sigma_{t}^{b(t)+b(r)+1},$

where in the last equality we used

$0=\tau_{t}(\sigma_{r}-\sigma_{t})\sigma_{t}^{b(t)+1}\prod_{l\neq
r}(\tau_{l}-\tau_{t})=\tau_{t}\tau_{R\setminus\\{r\\}}(\sigma_{r}-\sigma_{t})\sigma_{t}^{b(t)+1}.$

Notice that $T\in n(P,\mathcal{G})$ implies $\operatorname{cd}(\bigvee
R)=\operatorname{cd}(\bigvee(R\setminus\\{r\\}))+\operatorname{cd}(r)$ and
$\operatorname{cd}(\bigvee(R\setminus\\{r\\})\vee\bigvee(T_{<r}))=\operatorname{cd}(\bigvee(R\setminus\\{r\\}))+\operatorname{cd}(\bigvee(T_{<r}))$;
so $b_{T\setminus\\{r\\}}(t)=b_{T}(t)+b_{T}(r)+1$. Therefore

$\tau_{R}\sigma_{R}^{b}\sigma_{t}^{b(t)+1}=\sum_{r\in R}(-1)^{\lvert R_{\prec
r}\rvert}c\mu((R\setminus\\{r\\})\cup\\{t\\})$

and finally:

$\displaystyle\operatorname{d}(c\mu(T))$ $\displaystyle=\sum_{t\in
T\setminus\min(T)}\sum_{r\in\max(T_{<t})}(-1)^{\lvert T_{\prec
r}\rvert}\mu(T\setminus\\{r\\})$ $\displaystyle=\sum_{r\in
T\setminus\max(T)}(-1)^{\lvert T_{\prec r}\rvert}\mu(T\setminus\\{r\\}),$

because $T$ is a forest by Proposition 2.4. This conclude the proof. ∎

We want to apply algebraic Morse theory to the complex $B(P,\mathcal{G})$. We
refer to [JW09] for basic definitions and properties of algebraic Morse
theory.

We define the following matching $\mathcal{M}$: for each non-critical monomial
$\tau_{T}\sigma_{S}^{b}$ let $g\in S\cup T$ be the smallest (with respect to
$\prec$) non-critical element. If $g$ belongs to $T$, then the pair
$(\tau_{T}\sigma_{S}^{b},\tau_{T\smallsetminus\\{g\\}}\sigma_{S}^{b}\sigma_{g})$
is in $\mathcal{M}$.

The algebraic Morse theory, together with Lemma 3.5 and Proposition 3.8,
implies that the complex of critical monomials is quasi-isomorphic to the
Leray model.

######  Lemma 3.5.

The set $\mathcal{M}$ is a matching. Moreover, a monomial is critical if and
only if it is critical for the matching $\mathcal{M}$.

###### Proof.

We check that each non-critical monomial appears exactly once in $\mathcal{M}$
and that all monomials in $\mathcal{M}$ are non-critical.

By definition if the monomial $\tau_{T}\sigma_{S}^{b}$ appears in the first
position in $\mathcal{M}$, it is non-critical. Moreover
$\tau_{T\smallsetminus\\{g\\}}\sigma_{S}^{b}\sigma_{g}$ is non-critical
because $S\cup\\{g\\}\not\subseteq T\smallsetminus\\{g\\}$. So every monomial
in the matching is non-critical.

Vice versa, if $\tau_{T}\sigma_{S}^{b}$ is a non-critical monomial, let $g$ be
the minimal non-critical element in $S\cup T$. If $g\in T$ then
$\tau_{T}\sigma_{S}^{b}$ appears in the matching (in the first position).
Otherwise, $g\in S\setminus T$ so the monomial
$\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$ is basic and non-critical.
Notice that an element $f\in\mathcal{G}$ is critical for
$\tau_{T}\sigma_{S}^{b}$ if and only if is critical for
$\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$. Therefore the pair
$(\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}},\tau_{T}\sigma_{S}^{b})$
is in $\mathcal{M}$. ∎

######  Definition 3.6.

Given a standard monomial $\tau_{T}\sigma_{S}^{b}$ we define $m(T,S,b)$ as the
multiset $\\{g^{a(g)}\mid g\in\mathcal{G}\\}$ where $a(g)$ is the sum of the
exponents of $\tau_{g}$ and $\sigma_{g}$ in the monomial
$\tau_{T}\sigma_{S}^{b}$. Moreover, we order these multisets lexicographically
using the reverse order on $\mathcal{G}$.

As an example, if $h<g$ then $h\prec g$ and $\\{h^{2}\\}\succ\\{h,g\\}$.

######  Definition 3.7.

Let $G$ be the directed graph whose vertices are the standard monomials with a
directed edge from $\tau_{T}\sigma_{S}^{b}$ to
$\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}}$ if the monomial
$\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}}$ appears with a nonzero
coefficient in $\operatorname{d}(\tau_{T}\sigma_{S}^{b})$.

Let $G_{\mathcal{M}}$ be the directed graph $G$ with all directed edges in
$\mathcal{M}$ reversed.

######  Proposition 3.8.

The matching $\mathcal{M}$ is a Morse matching.

###### Proof.

We show that the graph $G_{\mathcal{M}}$ is acyclic.

Although $m$ is not a term order (because $m(\tau_{g})=m(\sigma_{g})$) it has
the property that for any relation of Theorem 2.9

(4) $\prod_{t\in T}(\tau_{t}-\tau_{g})\prod_{s\in
S}(\sigma_{s}-\sigma_{g})\sigma_{g}^{b}$

with $\bigvee(S\cup T)\leq g$ the monomial $\tau_{T}\sigma_{S}\sigma_{g}^{b}$
has $m(T,S,b)$ strictly bigger than any other monomial in the expansion of eq.
(4). Moreover $m$ is multiplicative.

First notice that:

$\displaystyle d(\tau_{T}\sigma_{S}^{b})$ $\displaystyle=\sum_{g\in
T}(-1)^{\lvert T_{\prec g}\rvert}\tau_{T\setminus{g}}\sigma_{S}^{b}\sigma_{g}$
$\displaystyle=\sum_{\begin{subarray}{c}g\in T\\\ g\textnormal{ non-
critical}\end{subarray}}(-1)^{\lvert T_{\prec
g}\rvert}\tau_{T\setminus{g}}\sigma_{S}^{b}\sigma_{g}+\sum_{\begin{subarray}{c}g\in
T\\\ g\textnormal{ critical}\end{subarray}}(-1)^{\lvert T_{\prec
g}\rvert}\tau_{T\setminus{g}}\sigma_{S}^{b}\sigma_{g}$
$\displaystyle=\sum_{\begin{subarray}{c}g\in T\\\ g\textnormal{ non-
critical}\end{subarray}}(-1)^{\lvert T_{\prec
g}\rvert}\tau_{T\setminus{g}}\sigma_{S}^{b}\sigma_{g}+\sum_{\mathclap{\begin{subarray}{c}\textnormal{some
}T^{\prime},S^{\prime},b^{\prime}\\\ m(T^{\prime},S^{\prime},b^{\prime})\prec
m(T,S,b)\end{subarray}}}\alpha_{T^{\prime},S^{\prime},b^{\prime}}\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}},$

where $\alpha_{T^{\prime},S^{\prime},b^{\prime}}$ are some coefficients. In
the last equality we used the relations of Theorem 2.9 in order to write the
non-standard monomials $\tau_{T\setminus{g}}\sigma_{S}^{b}\sigma_{g}$ as
linear combination of standard ones. Notice also that if the pair
$(\tau_{T}\sigma_{S}^{b},\tau_{T\setminus{g}}\sigma_{S\cup{g}}^{b^{\prime}})$
is in $\mathcal{M}$ then $m(T,S,b)=m(T\setminus{g},S\cup\\{g\\},b^{\prime})$.
Hence the function $m$ is weakly decreasing on every direct path in
$G_{\mathcal{M}}$, so it is constant on every directed cycle.

It is enough to prove that there are no alternating directed cycles, i.e.
cycles such that for every pair of consecutive edges exactly one is in
$\mathcal{M}$. Suppose that there exists a directed cycle and consider two
consecutive edges. We can assume that the first one is in $\mathcal{M}$ and
the second one is not. The first edge is
$(\tau_{T}\sigma_{S}^{b},\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}})$
for some non-critical monomial $\tau_{T}\sigma_{S}^{b}$ with $g$ the smaller
non-critical element and $g\in S\setminus T$. The second edge is
$(\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}},\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}})$
for some standard monomial
$\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}}$. Since the value of $m$ is
constant on the cycle we have that
$\tau_{T^{\prime}}\sigma_{S^{\prime}}^{b^{\prime}}=\tau_{T\setminus\\{f\\}}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}\sigma_{f}$
for some $f\in T$ non-critical for the monomial
$\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$. These two edges are shown
below.

$\tau_{T}\sigma_{S}^{b}$$\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$$\tau_{T\setminus\\{f\\}}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}\sigma_{f}$

The sets of critical elements for $\tau_{T}\sigma_{S}^{b}$ and for
$\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$ coincide, so both $g$ and
$f$ are non-critical for $\tau_{T}\tau_{g}\frac{\sigma_{S}^{b}}{\sigma_{g}}$.
By minimality of $g$ we have $g\prec f$ and
$T\prec(T\setminus\\{f\\})\cup\\{g\\}=T^{\prime}$.

We have proved that in every alternating path after two steps the set indexing
the variable $\tau$ strictly increases. Therefore there are no alternating
cycles. ∎

###  3.2. Multiplicative structure

We want to describe the product of two critical monomials in
$B(P,\mathcal{G})$.

Let $(g_{1},g_{2},\dots,g_{k})$ be a list of elements in $\mathcal{G}$ and
recall that $\prec$ is a linear extension of the order on $\mathcal{G}$.
Define

$\tilde{\lambda}(g_{1},g_{2},\dots,g_{k})=(f_{1},f_{2},\dots,f_{k})$

where $f_{i}$ is the unique $\mathcal{G}$-factor of $g_{1}\vee
g_{2}\vee\dots\vee g_{i}$ bigger than $g_{i}$ guaranteed by Proposition
2.4(1). Define
$\lambda(g_{1},g_{2},\dots,g_{k})=\tilde{\lambda}(g_{1},g_{2},\dots,g_{k})$ if
$(f_{1},f_{2},\dots,f_{k})$ form a $\mathcal{G}$-nested set and $f_{i}\prec
f_{i+1}$ for $i=1,\dots,k-1$. Set $\lambda(g_{1},g_{2},\dots,g_{k})=0$
otherwise. We will use the convention that $c\mu(0)=0$ and
$c\mu(\emptyset)=1$. Let $\pi\in\mathfrak{S}_{k}$ be a permutation, we write
$\pi(g_{1},g_{2},\dots,g_{k})$ for the list
$(g_{\pi(1)},g_{\pi(2)},\dots,g_{\pi(k)})$ and we denote the concatenation of
two lists $T_{1}$ and $T_{2}$ by $T_{1}\cup T_{2}$.

######  Remark 3.9.

If $\tilde{\lambda}(g_{1},g_{2},\dots,g_{k})=(f_{1},f_{2},\dots,f_{k})$, then
$g_{1}\vee g_{2}\vee\dots\vee g_{j}=f_{1}\vee f_{2}\vee\dots\vee f_{j}$.
Moreover, $\lambda(g_{1},g_{2},\dots,g_{k})=0$ if there exist $i<j$ such that
$g_{j}\leq g_{i}$. Indeed, $f_{j}\leq f_{1}\vee f_{2}\vee\dots\vee f_{j-1}$
and $\\{f_{1},\dots,f_{j}\\}$ is $\mathcal{G}$-nested, hence $f_{j}=f_{c}$ for
some $c<j$ contradicting $f_{c}\prec f_{j}$.

Let $T_{1}$ and $T_{2}$ be two lists of elements in $\mathcal{G}$ and
$\pi\in\mathfrak{S}_{\lvert T_{1}\cup T_{2}\rvert}$ be a permutation. If
$\lambda(\pi(T_{1}\cup T_{2}))\neq 0$ then the last element of $\pi(T_{1}\cup
T_{2})$ belongs to $\max(T_{1}\cup T_{2})$.

In the particular case when $\mathcal{G}$ is the maximal building set and
$T_{1},T_{2}$ are chains in $\mathcal{G}$, $\lambda(\pi(T_{1}\cup T_{2}))$ is
zero if $\pi$ is not a $(\lvert T_{1}\rvert,\lvert T_{2}\rvert)$-shuffles.

The following proposition describes the multiplication of critical monomials
using shuffles.

######  Proposition 3.10.

Let $T_{1}$ and $T_{2}$ be $\mathcal{G}$-nested sets. If
$\operatorname{cd}(\bigvee(T_{1}\cup T_{2}))<\operatorname{cd}(\bigvee
T_{1})+\operatorname{cd}(\bigvee T_{2})$, then $c\mu(T_{1})c\mu(T_{2})=0$.
Otherwise

$c\mu(T_{1})c\mu(T_{2})=\sum_{\pi\in\mathfrak{S}_{\lvert T_{1}\cup
T_{2}\rvert}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T_{1}\cup T_{2})).$

Before the proof of Proposition 3.10 we need two technical lemmas.

######  Lemma 3.11.

Let $T_{1}$ and $T_{2}$ be nested sets such that $T_{1}\cup T_{2}$ is
$\mathcal{G}$-nested and $\operatorname{cd}(\bigvee
T_{1})+\operatorname{cd}(\bigvee T_{2})=\operatorname{cd}(\bigvee(T_{1}\cup
T_{2}))$. Then

$c\mu(T_{1})c\mu(T_{2})=(-1)^{s}c\mu(T_{1}\cup T_{2}),$

where $s$ is the length of the permutation that reorder $T_{1}$ and $T_{2}$.
Moreover:

$c\mu(T_{1})c\mu(T_{2})=\sum_{\pi\in\mathfrak{S}_{\lvert T_{1}\cup
T_{2}\rvert}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T_{1}\cup T_{2})).$

###### Proof.

Notice that $[\hat{0},\bigvee(T_{1}\cup T_{2})]=[\hat{0},\bigvee
T_{1}]\times[\hat{0},\bigvee T_{2}]$ with the same codimension, therefore

$c\mu(T_{1})c\mu(T_{2})=(-1)^{s}c\mu(T_{1}\cup T_{2}).$

Since each subset of $T_{1}\cup T_{2}$ is $\mathcal{G}$-nested, for each
$\pi\in\mathfrak{S}_{\lvert T_{1}\cup T_{2}\rvert}$ we have
$\tilde{\lambda}\pi(T_{1}\cup T_{2})=\pi(T_{1}\cup T_{2})$ by (3) of
Proposition 2.4. Hence $\lambda\pi(T_{1}\cup T_{2})$ is zero for all
permutations $\pi$ except for the unique permutation that reorders $T_{1}$ and
$T_{2}$. ∎

######  Lemma 3.12.

Suppose that $T$ is a $\mathcal{G}$-nested set and $g\in\mathcal{G}$ such that
$\operatorname{cd}(g\vee\bigvee T)-\operatorname{cd}(\bigvee
T)=\operatorname{cd}(g)-\operatorname{cd}(\bigvee T_{<g})$. Set
$b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee T_{<g})-1$, then

(5) $c\mu(T)\tau_{g}\sigma_{g}^{b}=\sum_{\pi\in\mathfrak{S}_{\lvert
T\rvert+1}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T\cup\\{g\\})),$

where the sum is taken over all permutations of $T\cup\\{g\\}$.

###### Proof.

We prove the statement by induction on $\lvert T\rvert$.

If $g\geq\bigvee T$, it follows directly from the definitions
$c\mu(T)\tau_{g}\sigma_{g}^{b}=c\mu(T\cup\\{g\\})=\sum_{\pi\in\mathfrak{S}_{\lvert
T\rvert+1}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T\cup\\{g\\}))$.

Let $f=g\vee\bigvee T$, $h$ be the unique $\mathcal{G}$-factor of $f$ bigger
than $g$ and set $T^{\prime}=T_{\leq h}\cup\\{g\\}$,
$T^{\prime\prime}=T_{\not\leq h}$. Notice that $T^{\prime\prime}=\emptyset$ if
and only if $f\in\mathcal{G}$. Define $t^{\prime}$ and $t^{\prime\prime}$ the
cardinality of $T^{\prime}$ and $T^{\prime\prime}$ respectively.

If $T^{\prime\prime}\neq\emptyset$, by using the inductive hypothesis and
Lemma 3.11, we have

$\displaystyle c\mu(T)\tau_{g}\sigma_{g}^{b}$
$\displaystyle=(-1)^{s}c\mu(T^{\prime\prime})c\mu(T^{\prime}\setminus\\{g\\})\tau_{g}\sigma_{g}^{b}$
$\displaystyle=(-1)^{s}c\mu(T^{\prime\prime})\sum_{\alpha\in\mathfrak{S}_{t^{\prime\prime}}}\operatorname{sgn}(\alpha)c\mu(\lambda\alpha(T^{\prime}))$
$\displaystyle=\sum_{\alpha\in\mathfrak{S}_{t^{\prime}}}(-1)^{s+s_{\alpha}}\operatorname{sgn}(\alpha)c\mu(T^{\prime\prime}\sqcup\lambda\alpha(T^{\prime}))$
$\displaystyle=\sum_{\pi\in\mathfrak{S}_{t^{\prime}+t^{\prime\prime}}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T\cup\\{g\\})),$

where $s$ corresponds to the permutation that reorders $T^{\prime\prime}$ and
$T^{\prime}\setminus\\{g\\}$, $s_{\alpha}$ to the permutation that reorders
$T^{\prime\prime}$ and $\alpha(T^{\prime})$. The set
$T^{\prime\prime}\cup\lambda\alpha(T^{\prime})$ is $\mathcal{G}$-nested for
all $\alpha\in\mathfrak{S}_{t^{\prime}}$ such that
$\lambda\alpha(T^{\prime})\neq 0$ and in this case $\operatorname{cd}(\bigvee
T^{\prime\prime})+\operatorname{cd}(\bigvee\lambda\alpha(T^{\prime}))=\operatorname{cd}(\bigvee(T^{\prime\prime}\cup\lambda\alpha(T^{\prime})))$.

Otherwise $T^{\prime\prime}=\emptyset$ and $f\in\mathcal{G}$. Let
$Y=\\{g_{1},g_{2},\dots,g_{k}\\}=\max(T\cup\\{g\\})$ numbered such that
$g=g_{k}$. We assume $g\succ t$ for all $t\in T$, the general case differs
only by a sign. We have $\prod_{i=1}^{k}(\tau_{g_{i}}-\tau_{f})=0$ and so

$\tau_{Y}=\sum_{i=1}^{k}(-1)^{k-i}\tau_{Y\setminus\\{g_{i}\\}\cup\\{f\\}}.$

Set $b(g_{i})=\operatorname{cd}(g_{i})-\operatorname{cd}(\bigvee
T_{<g_{i}})-1$, for all $i\leq k$ we have
$(\sigma_{g_{i}}-\sigma_{f})\prod_{j\neq i}(\tau_{g_{j}}-\tau_{f})=0$ and

$0=(\sigma_{g_{i}}^{b(g_{i})}-\sigma_{f}^{b(g_{i})})\tau_{f}\prod_{j\neq
i}(\tau_{g_{j}}-\tau_{f})=(\sigma_{g_{i}}^{b(g_{i})}-\sigma_{f}^{b(g_{i})})\tau_{f}\prod_{j\neq
i}\tau_{g_{j}},$

so
$\tau_{Y\setminus\\{g_{i}\\}\cup\\{f\\}}\sigma_{g_{i}}^{b(g_{i})}=\tau_{Y\setminus\\{g_{i}\\}\cup\\{f\\}}\sigma_{f}^{b(g_{i})}$.
Therefore we have

$\displaystyle c\mu(T)\tau_{g}\sigma_{g}^{b}$
$\displaystyle=(-1)^{s}c\mu(T\setminus
Y)\prod_{i=1}^{k}\tau_{g_{i}}\sigma_{g_{i}}^{b(g_{i})}$
$\displaystyle=(-1)^{s}c\mu(T\setminus Y)\sum_{i=1}^{k}(-1)^{k-i}\prod_{j\neq
i}\tau_{g_{j}}\sigma_{g_{j}}^{b(g_{j})}\tau_{f}\sigma_{f}^{b(g_{i})}$
$\displaystyle=\sum_{i=1}^{k-1}(-1)^{t_{i}+1}c\mu(T\setminus\\{g_{i}\\})\tau_{g}\sigma_{g}^{b}\tau_{f}\sigma_{f}^{b(g_{i})}+c\mu(T\cup\\{f\\}),$

where $s$ (and $t_{i}$) is the length of the permutation that reorder
$T\setminus Y$ and $Y\setminus\\{g\\}$ (respectively $T\setminus\\{g_{i}\\}$
and $\\{g_{i}\\}$). The last summand corresponds to the identity permutation.
Apply the inductive hypothesis on the terms
$c\mu(T\setminus\\{g_{i}\\})\tau_{g}\sigma_{g}^{b}$ so that

$(-1)^{t_{i}+1}c\mu(T\setminus\\{g_{i}\\})\tau_{g}\sigma_{g}^{b}\tau_{f}\sigma_{f}^{b(g_{i})}=\sum_{\pi}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T\cup\\{g\\}))$

where the sum is taken over all permutations $\pi$ in $\mathfrak{S}_{\lvert
T\rvert+1}$ that sends the element $g_{i}$ in the last position. Since every
$\pi$ such that $\lambda\pi(T\cup\\{g\\})\neq 0$ has in the last position an
element of $\max(T\cup\\{g\\})$, the result follows. ∎

###### Proof of Proposition 3.10.

For the first part notice that $c\mu(T_{i})$ is in bidegree
$(2(\operatorname{cd}(\bigvee T_{i})-\lvert T_{i}\lvert),\lvert T_{i}\rvert)$
for $i=1,2$. Let $f=\bigvee(T_{1}\cup T_{2})$, the product
$c\mu(T_{1})c\mu(T_{2})$ can be rewritten as sum of standard monomials using
only relations of type

$\prod_{s\in S}(\tau_{s}-\tau_{g})\prod_{t\in
T}(\sigma_{t}-\sigma_{g})\sigma_{g}^{b}$

for $\bigvee(S\cup T)\leq g\leq f$. The standard monomials
$\tau_{S}\sigma_{T}^{b}$ with $\bigvee(S\cup T)\leq f$ have bidegree at most
$(2(\operatorname{cd}(f)-\lvert S\rvert),\lvert S\rvert)$. Therefore, if
$\operatorname{cd}(f)<\operatorname{cd}(\bigvee
T_{1})+\operatorname{cd}(\bigvee T_{2})$ then $c\mu(T_{1})c\mu(T_{2})=0$ by
degree argument.

We prove the second statement by induction on $\lvert T_{2}\rvert$. The base
case $T_{2}=\emptyset$ is trivial. If $\bigvee T_{2}\notin\mathcal{G}$ then
there exist $T_{3}$ and $T_{4}$ nonempty $\mathcal{G}$-nested sets such that
$T_{2}=T_{3}\sqcup T_{4}$ and $[\hat{0},\bigvee T_{2}]=[\hat{0},\bigvee
T_{3}]\times[\hat{0},\bigvee T_{4}]$. Applying Lemma 3.11 and the inductive
step we have

$\displaystyle c\mu(T_{1})$ $\displaystyle
c\mu(T_{2})=(-1)^{s}c\mu(T_{1})c\mu(T_{3})c\mu(T_{4})$
$\displaystyle=(-1)^{s}\sum_{\alpha\in\mathfrak{S}_{t_{1}+t_{3}}}\operatorname{sgn}(\alpha)c\mu(\lambda\alpha(T_{1}\cup
T_{3}))c\mu(T_{4})$
$\displaystyle=(-1)^{s}\sum_{\alpha\in\mathfrak{S}_{t_{1}+t_{3}}}\operatorname{sgn}(\alpha)\sum_{\beta\in\mathfrak{S}_{t_{1}+t_{2}}}\operatorname{sgn}(\beta)c\mu(\lambda\beta(\alpha(T_{1}\cup
T_{3}),T_{4}))$
$\displaystyle=\sum_{\pi\in\mathfrak{S}_{t_{1}+t_{2}}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T_{1}\cup
T_{2})),$

where $t_{i}=\lvert T_{i}\rvert$.

Now we deal with the case $\bigvee T_{2}\in\mathcal{G}$. Let $g=\max
T_{2}\in\mathcal{G}$, $T^{\prime}_{2}=T_{2}\setminus\\{g\\}$, and
$m=\operatorname{cd}(g)-\operatorname{cd}(T^{\prime}_{2})-1$. We have

$\displaystyle c\mu(T_{1})$ $\displaystyle
c\mu(T_{2})=c\mu(T_{1})c\mu(T^{\prime}_{2})\tau_{g}\sigma_{g}^{m}$
$\displaystyle=\sum_{\alpha\in\mathfrak{S}_{t_{1}+t_{2}-1}}\operatorname{sgn}(\alpha)c\mu(\lambda\alpha(T_{1}\cup
T^{\prime}_{2}))\tau_{g}\sigma_{g}^{m}$
$\displaystyle=\sum_{\alpha\in\mathfrak{S}_{t_{1}+t_{2}-1}}\operatorname{sgn}(\alpha)\sum_{\beta\in\mathfrak{S}_{t_{1}+t_{2}}}\operatorname{sgn}(\beta)c\mu(\lambda\beta(\alpha(T_{1}\cup
T^{\prime}_{2})\cup\\{g\\}))$
$\displaystyle=\sum_{\pi\in\mathfrak{S}_{t_{1}+t_{2}}}\operatorname{sgn}(\pi)c\mu(\lambda\pi(T_{1}\cup
T_{2})),$

where we used the inductive hypothesis on $T_{1}$ and $T^{\prime}_{2}$ and
Lemma 3.12 on $\lambda\alpha(T_{1}\cup T^{\prime}_{2})$ and $\\{g\\}$. ∎

We define the algebra of critical monomials abstractly, by generators and
relations.

######  Definition 3.13.

Let $\operatorname{CM}(P,\mathcal{G})$ be the $\mathbb{Q}$-vector space
generated by all the $\mathcal{G}$-nested sets $T\in n(P,\mathcal{G})$ with
bidegree $(2(\operatorname{cd}(\bigvee T)-\lvert T\rvert),\lvert T\rvert)$.
The differential is defined on the base by

$\operatorname{d}(T)=\sum_{t\in T\setminus\max(T)}(-1)^{\lvert T_{\prec
t}\rvert}(T\setminus\\{t\\})$

and the product by $T\cdot S=0$ if $\operatorname{cd}(\bigvee(T\cup
S))<\operatorname{cd}(\bigvee T)+\operatorname{cd}(\bigvee S)$ and

$T\cdot S=\sum_{\pi\in\mathfrak{S}_{\lvert T\rvert+\lvert
S\rvert}}\operatorname{sgn}(\pi)\lambda(\pi(T\cup S))$

otherwise. This structure makes $\operatorname{CM}(P,\mathcal{G})$ a
differential bigraded algebra.

We summarize all the previous results of this section in the following
theorem.

######  Theorem 3.14.

The morphism $\xi\colon\operatorname{CM}(P,\mathcal{G})\to B(P,\mathcal{G})$
defined by $\xi(T)=c\mu(T)$ is an inclusion of differential algebras and a
quasi-isomorphism.

###### Proof.

The map $\xi$ is well defined as a morphism of $\mathbb{Q}$-vector spaces. It
is an inclusion since the monomials $c\mu(T)$ for $T\in n(P,\mathcal{G})$ are
standard monomials and are linearly independent by Corollary 2.12. The
equality $\operatorname{d}\xi=\xi\operatorname{d}$ follows from Lemma 3.4 and
the equality $\xi(S\cdot T)=\xi(S)\xi(T)$ from Proposition 3.10. This also
proves that $\operatorname{CM}(P,\mathcal{G})$ is a differential bigraded
algebra.

Finally, the algebraic Morse theory applied to $B(P,\mathcal{G})$ and the
matching $\mathcal{M}$ ensures that there exists a subcomplex
$N_{\mathcal{M}}$ such that the projection

$B(P,\mathcal{G})\twoheadrightarrow\faktor{B(P,\mathcal{G})}{N_{\mathcal{M}}}$

is a quasi-isomorphism and the quotient is freely generated by critical
monomials. The composition of $\xi$ with the projection gives an isomorphism
of chain complexes. Therefore $\xi$ is a quasi-isomorphism. ∎

See Section 7 for an explicit example of the construction of the algebra of
critical monomials.

Let $n((\hat{0},g),\mathcal{G})$ be the full subcomplex of $n(P,\mathcal{G})$
on the set of vertices $\\{h\in\mathcal{G}\mid h<g\\}$.

All the homology groups are taken with rational coefficients. We use the
standard convention for the reduced homology that
$\tilde{H}_{-1}(\emptyset)=\mathbb{Q}$.

This final theorem provides an explicit description of the cohomology of the
Leray model in term of cohomology of very small simplicial complexes.

######  Theorem 3.15.

Let $P$ be a polymatroid and $\mathcal{G}$ be a building set. Then

$H^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}(B(P,\mathcal{G}),\operatorname{d})\cong
H^{{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}(\operatorname{CM}(P,\mathcal{G}),\operatorname{d})\cong\bigoplus_{f\in
L}\bigotimes_{g\in
F}\tilde{H}_{2\operatorname{cd}(g)-2-{\mathbin{\mathchoice{\hbox
to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}\Bigl{(}n((\hat{0},g),\mathcal{G})\Bigr{)},$

where $F=F(P,\mathcal{G},f)$ is the set of $\mathcal{G}$-factors of $f$.

In particular the summand $\tilde{H}_{i}(n((\hat{0},g),\mathcal{G}))$
contributes in bidegree $(2(\operatorname{cd}(g)-2-i),2+i)$.

###### Proof.

Theorem 3.14 implies

$H(B(P,\mathcal{G}),\operatorname{d})\cong
H(\operatorname{CM}(P,\mathcal{G}),\operatorname{d}).$

For each flat $f$ let $\operatorname{CM}_{f}$ be the subcomplex of
$\operatorname{CM}(P,\mathcal{G})$ generated by all nested sets $T$ such that
$\max(T)=F(P,\mathcal{G},f)$. Moreover for each $g\in\mathcal{G}$ set
$\operatorname{CM}(g)$ to be the subcomplex of
$\operatorname{CM}(P,\mathcal{G})$ generated by all nested sets $T$ such that
$\\{g\\}=\max(T)$. We have

$\operatorname{CM}(P,\mathcal{G})=\bigoplus_{f\in L}\operatorname{CM}_{f}$

and

$\operatorname{CM}_{f}=\bigotimes_{g\in
F(P,\mathcal{G},f)}\operatorname{CM}(g)$

as complexes. It is enough to prove that

$H^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}(\operatorname{CM}(g),\operatorname{d})=\tilde{H}_{2\operatorname{cd}(g)-2-{\mathbin{\mathchoice{\hbox
to4.89995pt{\hfil\raise 0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}}\Bigl{(}n((\hat{0},g),\mathcal{G})\Bigr{)}.$

Indeed $\operatorname{CM}(g)$ coincides with the reduced simplicial chain
complex for $n((\hat{0},g),\mathcal{G})$, under the correspondence $T\mapsto
T\setminus\\{g\\}$. Notice that the bidegree of $T\in\operatorname{CM}(g)$ is
$(2(\operatorname{cd}(g)-\lvert T\rvert),\lvert T\rvert)$ and the degree of
$T\setminus\\{g\\}$ in the reduced chain complex is $\lvert
T\setminus\\{g\\}\rvert-1=\lvert T\rvert-2$. ∎

Definition 3.13 has a straightforward generalization to integer coefficients;
we left open the following question.

######  Conjecture 3.16.

Does Theorem 3.15 generalizes to integer coefficients?

The analogous statement in the realizable case with maximal building set was
proven in [DGM00, dLS01].

##  4\. Kähler package

Let $\operatorname{DP}^{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}}(P,\mathcal{G})$ be the
graded algebra $B^{2{\mathbin{\mathchoice{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\displaystyle\bullet$}}}\hfil}}{\hbox to4.89995pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\textstyle\bullet$}}}\hfil}}{\hbox to4.0833pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptstyle\bullet$}}}\hfil}}{\hbox to3.53883pt{\hfil\raise
0.0pt\hbox{\scalebox{0.75}{\lower
0.0pt\hbox{$\scriptscriptstyle\bullet$}}}\hfil}}}},0}(P,\mathcal{G})$. This
algebra, in the realizable case, is the Chow ring of the De Concini Procesi
wonderful model for the subspace arrangement. A presentation of
$\operatorname{DP}(P,\mathcal{G})$ is given by the generators $x_{g}$ for
$g\in\mathcal{G}$ with relations

$x_{S}c_{g}^{b}$

where $S\subseteq\mathcal{G}$, $g\in\mathcal{G}$ and
$b\geq\operatorname{cd}(g)-\operatorname{cd}(\bigvee S_{<g})$. The algebra
$\operatorname{DP}(P,\mathcal{G})$ has an additive basis given by

$x_{S}^{b}$

where $S\in n(P,\mathcal{G})$ and for each $s\in S$ we have that
$0<b(s)<\operatorname{cd}(s)-\operatorname{cd}(\bigvee(S)_{<s})$, see
Corollary 2.8.

A second presentation is given by the generators $\sigma_{g}$ for
$g\in\mathcal{G}$ with relations

$\sigma_{g}^{b}\prod_{s\in S}(\sigma_{s}-\sigma_{g})$

where $\bigvee S\leq g$ and $b=\operatorname{cd}(g)-\operatorname{cd}(\bigvee
S)$, see Theorem 2.9. The algebra $\operatorname{DP}(P,\mathcal{G})$ has an
additive basis given by

$\sigma_{S}^{b}$

where $S\in n(P,\mathcal{G})$ and for each $s\in S$ we have that
$0<b(s)<\operatorname{cd}(s)-\operatorname{cd}(\bigvee S_{<s})$, see Corollary
2.12.

######  Remark 4.1.

If $\hat{1}\notin\mathcal{G}$ then the polymatroid $P$ is direct sum of other
polymatroids. Indeed, let $a_{1},\dots,a_{k}$ be the $\mathcal{G}$-factors of
$\hat{1}$, the poset $L$ is a product $\prod_{i=1}^{k}[\hat{0},a_{k}]$. There
exist polymatroids $P^{a_{i}}$ (defined in the following, see Lemma 4.12) such
that $P=\oplus_{i=1}^{k}P^{a_{i}}$ and building sets
$\mathcal{G}^{a_{i}}=\mathcal{G}\cap[\hat{0},a_{i}]$. Moreover,
$\operatorname{DP}(P,\mathcal{G})=\otimes_{i=1}^{k}\operatorname{DP}(P^{a_{i}},\mathcal{G}^{a_{i}})$
and the dimension of $\operatorname{DP}(P,\mathcal{G})$ is
$\operatorname{cd}(\hat{1})-\lvert F(P,\mathcal{G},\hat{1})\rvert$ (where
$k=\lvert F(P,\mathcal{G},\hat{1})\rvert$).

For the clarity of exposition, we assume $\hat{1}\in\mathcal{G}$ in this
section. Consider the isomorphism
$\deg\colon\operatorname{DP}^{\operatorname{cd}(\hat{1})-1}(P,\mathcal{G})\to\mathbb{Q}$
defined by

$\deg(x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1})=(-1)^{\operatorname{cd}(\hat{1})-1}.$

######  Definition 4.2.

Let $A$ be a graded algebra with top degree $n$ and $\deg\colon
A^{n}\to\mathbb{Q}$ an isomorphism. We say that

  * •

the algebra $A$ satisfies Poincaré duality if the bilinear pairing

$A^{k}\times A^{n-k}\to\mathbb{Q}$

defined by $(a,b)\mapsto\deg(ab)$ is non-degenerate.

  * •

the element $\ell\in A^{1}$ satisfies the Hard Lefschetz property if the
multiplication map

$\cdot\ell^{n-2k}\colon A^{k}\to A^{n-k}$

is an isomorphism for all $k\leq\frac{n}{2}$.

  * •

the element $\ell\in A^{1}$ satisfies the Hodge-Riemann relations if the
bilinear form

$Q_{\ell}^{k}\colon A^{k}\times A^{k}\to\mathbb{Q}$

defined by $Q_{\ell}^{k}(a,b)=(-1)^{k}\deg(a\ell^{n-2k}b)$ (for
$k\leq\frac{n}{2}$) is positive definite on the subspace

$P_{k}=\ker(\cdot\ell^{n-2k+1}\colon A^{k}\to A^{n-k+1}).$

We will abbreviate these properties with $\operatorname{PD}_{A}$,
$\operatorname{HL}_{A}(\ell)$, and $\operatorname{HR}_{A}(\ell)$ respectively.

###  4.1. Poincaré duality

In this subsection we give a direct proof of the Poincaré duality property for
$\operatorname{DP}(P,\mathcal{G})$.

######  Definition 4.3.

Suppose that $\hat{1}\in\mathcal{G}$ and let $x_{S}^{b}$ be a standard
monomial. The element $\epsilon(x_{S}^{b})$ is

$\epsilon(x_{S}^{b})=(-1)^{\lvert
S\setminus\\{\hat{1}\\}\rvert}x_{S^{+}}^{c},$

where $S^{+}=S\cup\\{\hat{1}\\}$,
$c(\hat{1})=\operatorname{cd}(\hat{1})-\operatorname{cd}(\bigvee
S_{<\hat{1}})-b(\hat{1})-1$, and
$c(g)=\operatorname{cd}(g)-\operatorname{cd}(\bigvee S_{<g})-b(g)$ for $g\in
S\setminus\\{\hat{1}\\}$.

We will write $c_{S}$ instead of $c$ when we want to stress the dependency on
$S$ and $b$.

Recall the chosen monomial order with the property that if $h>g$ then $h\succ
g$ and $x_{h}\prec x_{g}$. We fix the basis of $\operatorname{DP}^{k}$
consisting in all standard monomials $x_{S}^{b}$ of degree $k$ ordered with
the aforementioned monomial order. In complementary degree
$\operatorname{DP}^{\operatorname{cd}(\hat{1})-k}$, we consider the basis
given by $\epsilon(x_{S}^{b})$ ordered using the monomial order on
$x_{S}^{b}$. In order to prove Poincaré duality we will show that the matrix
with entries $\deg(x_{S}^{b}\epsilon(x_{T}^{c}))$ is non-degenerate. Lemma 4.4
proves that the matrix has values $\pm 1$ on the diagonal and Lemma 4.6 shows
that the matrix is upper triangular.

######  Lemma 4.4.

If $\hat{1}\in\mathcal{G}$ then for all standard monomials we have

$x_{S}^{b}\epsilon(x_{S}^{b})=x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1}.$

###### Proof.

We prove the statement by induction on $\lvert S\setminus\\{\hat{1}\\}\rvert$.
The base case $S=\\{\hat{1}\\}$ is trivial. For the inductive step we choose
$g\in\max(S_{<\hat{1}})$ and set $T=S\setminus\\{g,\hat{1}\\}$. For the sake
of notation, let $n(h)=b(h)+c_{S}(h)$ for all $h\in S^{+}$ (where $c_{S}(h)$
is introduced in Definition 4.3). Notice that
$x_{T}x_{f}x_{\hat{1}}^{n(\hat{1})}=0$ for all
$f\in(g,\hat{1})\cap\mathcal{G}$, because $f\vee\bigvee T>g\vee\bigvee T$.
Since $x_{T}^{n}\sigma_{g}^{n(g)}=0$ by relation (ii), we have

$\displaystyle 0$
$\displaystyle=x_{T}^{n}\sigma_{g}^{n(g)}x_{\hat{1}}^{n(\hat{1})}$
$\displaystyle=x_{T}^{n}(x_{g}+x_{\hat{1}})^{n(g)}x_{\hat{1}}^{n(\hat{1})}$
$\displaystyle=x_{T}^{n}(x_{g}^{n(g)}+x_{\hat{1}}^{n(g)})x_{\hat{1}}^{n(\hat{1})},$

where in the last equality we used $x_{T}x_{g}x_{\hat{1}}^{n(\hat{1})+1}=0$.
Therefore,

$\displaystyle x_{S}^{b}\epsilon(x_{S}^{b})$ $\displaystyle=(-1)^{\lvert
S\setminus\\{\hat{1}\\}\rvert}x_{T}^{n}x_{g}^{n(g)}x_{\hat{1}}^{n(\hat{1})}$
$\displaystyle=(-1)^{\lvert
S\setminus\\{\hat{1}\\}\rvert-1}x_{T}^{n}x_{\hat{1}}^{n(g)}x_{\hat{1}}^{n(\hat{1})}$
$\displaystyle=(-1)^{\lvert
T\setminus\\{\hat{1}\\}\rvert}x_{T}^{n}x_{\hat{1}}^{n(g)+n(\hat{1})}$
$\displaystyle=x_{T}^{b}\epsilon(x_{T}^{b})=x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1},$

by inductive hypotheses on $T$. ∎

Let $d_{S}$ be the function defined by
$d_{S}(\hat{1})=\operatorname{cd}(\hat{1})-\operatorname{cd}(\bigvee
S_{<\hat{1}})-1$ and by
$d_{S}(g)=\operatorname{cd}(g)-\operatorname{cd}(\bigvee S_{<g})$ for
$g\neq\hat{1}$.

######  Lemma 4.5.

Let $S$ be a nested set, $g\in S$ and $x_{S}^{b}$ be a monomial such that for
all $h>g$ we have $b(h)\geq d_{S}(h)$ and $b(g)>d_{S}(h)$. Then $x_{S}^{b}=0$.

The proof of the lemma is the same of [BDF21, Lemma 5.4.1 (b)]. Recall the
chosen monomial order with the property that if $h>g$ then $h\succ g$ and
$x_{h}\prec x_{g}$. We need the following statement.

######  Lemma 4.6.

Let $x_{S}^{b}$ and $x_{T}^{c}$ be two standard monomials in
$\operatorname{DP}^{k}(P,\mathcal{G})$ such that
$x_{S}^{b}\prec_{revlex}x_{T}^{c}$. Then $x_{S}^{b}\epsilon(x_{T}^{c})=0$.

###### Proof.

Consider $T^{\prime}$ and $c^{\prime}$ such that
$x_{T^{\prime}}^{c^{\prime}}=\epsilon(x_{T}^{c})$ and notice that
$T^{\prime}\setminus\\{\hat{1}\\}=T\setminus\\{\hat{1}\\}$. Define
$g=\max_{\prec}\\{h\mid b(h)\neq c(h)\\}$ and, by hypothesis, $b(g)>c(g)$. If
$S\cup T^{\prime}$ is not $\mathcal{G}$-nested then we have
$x_{S}^{b}\epsilon(x_{T}^{c})=0$. Otherwise set $A=(S\cup T^{\prime})_{\geq
g}$, by (4) of Proposition 2.4 we have that $A$ is a chain
$(a_{1}<a_{2}<\dots<a_{l})$ with $a_{1}=g$. For $a_{i}\neq g,\hat{1}$ we have

$\displaystyle b(a_{i})+c^{\prime}(a_{i})$
$\displaystyle=b(a_{i})+\operatorname{cd}(a_{i})-\operatorname{cd}(\bigvee
T_{<a_{i}})-c(a_{i})$
$\displaystyle=\operatorname{cd}(a_{i})-\operatorname{cd}(\bigvee
T^{\prime}_{<a_{i}})$
$\displaystyle\geq\operatorname{cd}(a_{i})-\operatorname{cd}(\bigvee(S\cup
T^{\prime})_{<a_{i}})=d_{S\cup T^{\prime}}(a_{i}).$

The same holds for $\hat{1}$ (the proof has a minus one in the mid steps). For
$a_{1}$ we have $b(g)+c^{\prime}(g)>d_{S\cup T^{\prime}}(g)$ because
$b(g)>c(g)$. Therefore the monomial $x_{S}^{b}\epsilon(x_{T}^{c})=x_{S\cup
T^{\prime}}^{b+c^{\prime}}$ satisfies the hypothesis of Lemma 4.5 and we
obtain the claimed result $x_{S}^{b}\epsilon(x_{T}^{c})=0$. ∎

Finally we can prove the Poincaré duality property:

######  Theorem 4.7 (Poincaré duality).

If $\hat{1}\in\mathcal{G}$ then the algebra $\operatorname{DP}(P,\mathcal{G})$
is a Poincaré duality algebra of dimension $\operatorname{cd}(\hat{1})-1$.

More generally, $\operatorname{DP}(P,\mathcal{G})$ is a Poincaré duality
algebra of dimension $\operatorname{cd}(\hat{1})-\lvert
F(P,\mathcal{G},\hat{1})\rvert$.

###### Proof.

The function $\epsilon$ has the property $\epsilon^{2}=\operatorname{Id}$, and
gives a bijection between standard monomials in degree $k$ and in degree
$r-k$. This, together with Corollary 2.8, shows that
$\dim\operatorname{DP}^{k}(P,\mathcal{G})=\dim\operatorname{DP}^{r-k}(P,\mathcal{G})$.
We consider on standard monomials the reverse lexicographical order. Lemma 4.6
ensures that the matrix of the Poincaré pairing (in the chosen basis) is upper
triangular. From Lemma 4.4 we obtain that the entries on the diagonal are $\pm
1$ and so the Poincaré pairing is non degenerate. The last statement follows
from the first one together with Remark 4.1. ∎

We remark that the bases of standard monomials $\\{x_{S}^{b}\\}$ and
$\\{(-1)^{r}\epsilon(x_{S}^{b})\\}$ are not dual bases.

###  4.2. Tensor decomposition

This technical section is devoted to computing the annihilator
$\operatorname{Ann}(\sigma_{g})$ and $\operatorname{Ann}(x_{g})$ for
$g\in\mathcal{G}$. We describe it using the Chow ring of different
polymatroids: $\operatorname{tr}_{g}P$, $P^{g}$ and $P_{g}$. In the case of
matroids these operations are known as truncation, restriction, and
contraction.

The following proposition is needed for the proof of the main result of this
section.

######  Proposition 4.8.

Let $A$ and $B$ be Poincaré duality algebra of the same dimension $n$, then:

  * •

for each $x\in A^{k}$, $x\neq 0$, the ring $A/\operatorname{Ann}(x)$ is a
Poincaré duality algebra of dimension $n-k$,

  * •

each surjective homomorphism $f\colon A\to B$ is an isomorphism.

The proof of the above proposition can be found, for example, in [AHK18,
Proposition 7.2, Proposition 7.13].

Let $P=(E,\operatorname{cd})$ be a polymatroid with building set
$\mathcal{G}$. Consider $g\in\mathcal{G}$ such that $\operatorname{cd}(g)>1$
and let $\operatorname{tr}_{g}\operatorname{cd}\colon 2^{E}\to\mathbb{N}$ be
the function defined by:

$\operatorname{tr}_{g}\operatorname{cd}(h)=\begin{cases}\operatorname{cd}(h)-1&\textnormal{if
}\operatorname{cd}(h)=\operatorname{cd}(h\cup g),\\\
\operatorname{cd}(h)&\textnormal{otherwise.}\end{cases}$

We denote by $\operatorname{tr}_{g}L$ the poset of flats of
$\operatorname{tr}_{g}\operatorname{cd}$. Finally, define

$\operatorname{tr}_{g}\mathcal{G}=\\{\overline{h}\in\operatorname{tr}_{g}L\mid
h\in\mathcal{G}\\},$

where $\overline{h}$ is the closure with respect to
$\operatorname{tr}_{g}\operatorname{cd}$ of the flat $h$. Notice that
$\operatorname{tr}_{g}L$ is a subposet of $L$ but with a different codimension
function.

######  Lemma 4.9.

For all $g\in\mathcal{G}$ with $\operatorname{cd}(g)>1$, the pair
$\operatorname{tr}_{g}P=(E,\operatorname{tr}_{g}\operatorname{cd})$ is a
polymatroid and $\operatorname{tr}_{g}\mathcal{G}$ is a building set for the
poset of flats $\operatorname{tr}_{g}L$.

###### Proof.

It is easy to see that $(E,\operatorname{tr}_{g}\operatorname{cd})$ is a
polymatroid. Let $x\in\operatorname{tr}_{g}L$ and notice that, for all
$h\in\mathcal{G}$, $h\leq x$ in $L$ if and only if $\overline{h}\leq x$ in
$\operatorname{tr}_{g}L$. Thus, we have
$\max\operatorname{tr}_{g}\mathcal{G}_{\leq x}=\max\mathcal{G}_{\leq x}$ and
it follows that

$[\hat{0},x]\simeq\prod_{y\in\max(\mathcal{G}_{\leq
x})}[\hat{0},y]\simeq\prod_{y\in\max(\operatorname{tr}_{g}\mathcal{G}_{\leq
x})}[\hat{0},y].$

For the second part of the definition of a building set we have two cases. Let
$\\{y_{1},\dots,y_{n}\\}=\max\mathcal{G}_{\leq x}$ and assume $g\nleq x$,
which implies $g\nleq y_{i}$ for every $i$:

$\operatorname{tr}_{g}\operatorname{cd}(x)=\operatorname{cd}(x)=\sum_{y\in\max\mathcal{G}_{\leq
x}}\operatorname{cd}(y)=\sum_{y\in\max\operatorname{tr}_{g}\mathcal{G}_{\leq
x}}\operatorname{tr}_{g}\operatorname{cd}(y).$

Finally, let $g\leq x$ then by Proposition 2.4 there exists only one $h_{i}$
such that $g\leq h_{i}$. Thus, we have the following:

$\operatorname{tr}_{g}\operatorname{cd}(x)=\operatorname{cd}(x)-1=\Bigl{(}\sum_{y\in\max\mathcal{G}_{\leq
x}}\operatorname{cd}(y)\Bigr{)}-1=\sum_{y\in\max\operatorname{tr}_{g}\mathcal{G}_{\leq
x}}\operatorname{tr}_{g}\operatorname{cd}(y).$

This concludes the proof. ∎

Define the map

$\zeta_{g}\colon\operatorname{DP}(\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G})\to\operatorname{DP}(P,\mathcal{G})/\operatorname{Ann}(\sigma_{g})$

by $\zeta_{g}(\sigma_{k})=\sigma_{h}$ where $h$ is any element in
$\mathcal{G}$ such that $\overline{h}=k$.

######  Remark 4.10.

In the realizable case, this construction can be viewed geometrically:
consider a generic hyperplane $H$ containing the flat $g$. The intersection of
the subspace arrangement with $H$ describes a subspace arrangement in $H$
whose poset of intersection is $\operatorname{tr}_{g}L$. Moreover, the natural
closed inclusion between the two wonderful compactification induces a
surjective map
$\operatorname{DP}(P,\mathcal{G})\to\operatorname{DP}(\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G})$
with kernel $\operatorname{Ann}(\sigma_{g})$. The map $\zeta_{g}$ is its
pseudo-inverse.

######  Lemma 4.11.

For $g\in\mathcal{G}$ with $\operatorname{cd}(g)>1$, the map $\zeta_{g}$ is
well defined and an isomorphism. Moreover
$\deg(\alpha)=\deg(-\sigma_{g}\zeta_{g}(\alpha))$ for all
$\alpha\in\operatorname{DP}(\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G})$.

###### Proof.

We show that the map $\zeta_{g}$ does not depend on the choice of $h$: suppose
that exist $h,f\in\mathcal{G}$ such that $\overline{h}=\overline{f}$. By
symmetry we may assume $h\not\geq f$. Since $g\vee
h=\overline{h}=\overline{f}=g\vee f$, we have $\overline{h}\in\mathcal{G}$, so
replacing $f$ with $g\vee f$ we assume $f>h$. Notice that
$\operatorname{cd}(f)=\operatorname{cd}(h)+1$ and $f=g\vee h$ so

$\sigma_{g}(\sigma_{h}-\sigma_{f})=\sigma_{f}(\sigma_{h}-\sigma_{f})=0.$

We verify that the relations (i) and (ii) of Theorem 2.9 are send to zero.
Consider an antichain $A\subset\operatorname{tr}_{g}\mathcal{G}$ and
$k\in\operatorname{tr}_{g}\mathcal{G}$ such that $k\geq\bigvee A$, set
$n=\operatorname{tr}_{g}\operatorname{cd}(k)-\operatorname{tr}_{g}\operatorname{cd}(\bigvee
A)$. Let $h\in\mathcal{G}$ such that $\overline{h}=k$ and
$B\subset\mathcal{G}$ such that $\overline{b_{i}}=a_{i}$ for all $i$. We have

$\sigma_{g}\zeta_{g}\Bigl{(}\sigma_{k}^{n}\prod_{a\in
A}(\sigma_{a}-\sigma_{k})\Bigr{)}=\sigma_{g}\sigma_{h}^{n}\prod_{b\in
B}(\sigma_{b}-\sigma_{h}).$

Notice that $\operatorname{cd}(h)-\operatorname{cd}(\bigvee B)=n$ unless
$h\geq g$ and $\bigvee B\not\geq g$ in which case
$\operatorname{cd}(h)-\operatorname{cd}(\bigvee B)=n+1$. The non trivial case
is the latter. Notice also that $h=g\vee\bigvee B$. We use the relations to
obtain:

$\displaystyle\sigma_{g}\zeta_{g}\Bigl{(}\sigma_{k}^{n}\prod_{a\in
A}(\sigma_{a}-\sigma_{k})\Bigr{)}$
$\displaystyle=\sigma_{g}\sigma_{h}^{n}\prod_{b\in B}(\sigma_{b}-\sigma_{h})$
$\displaystyle=\sigma_{h}^{n+1}\prod_{b\in B}(\sigma_{b}-\sigma_{h})$
$\displaystyle=0.$

Hence $\zeta_{g}$ is well defined.

The map is surjective because for each $h\in\mathcal{G}$ we have
$\zeta_{g}(\sigma_{\overline{h}})=\sigma_{h}$. Finally applying Proposition
4.8 we obtain the sought isomorphism.

For the last statement it is enough to notice that
$\sigma_{g}\zeta_{g}(x_{\hat{1}}^{r-1})=x_{\hat{1}}^{r}$. ∎

Let $P=(E,\operatorname{cd})$ be a polymatroid, $\mathcal{G}$ be a building
set and $g\in\mathcal{G}$ any element. The restriction of the polymatroid to
the flat $g$ is $P^{g}=(E^{g},\operatorname{cd}^{g})$ where $E^{g}=\\{h\in
E\mid h\leq g\\}$ and $\operatorname{cd}^{g}=\operatorname{cd}_{|E^{g}}$. The
contraction of $P=(E,\operatorname{cd})$ to the flat $g$ is
$P_{g}=(E_{g},\operatorname{cd}_{g})$ where $E_{g}=E\setminus E^{g}$ and
$\operatorname{cd}_{g}(h)=\operatorname{cd}(h\vee g)-\operatorname{cd}(g)$.

Define $L^{g}=[\hat{0},g]$, $\mathcal{G}^{g}=\mathcal{G}\cap L^{g}$,
$L_{g}=[g,\hat{1}]$, and

$\mathcal{G}_{g}=\\{h\vee g\mid h\in\mathcal{G}\setminus[\hat{0},g]\\}.$

The proof of the following lemma is analogous to the one of Lemma 4.9, so we
omit it.

######  Lemma 4.12.

The restriction and the contraction at $g\in\mathcal{G}$ are polymatroids with
poset of flats $L^{g}$ (respectively $L_{g}$) and building set
$\mathcal{G}^{g}$ (resp. $\mathcal{G}_{g}$).

######  Remark 4.13.

In the case of matroids $M$, we have for every $e\in E$ that
$M_{e}=\operatorname{tr}_{e}M$ is the contraction of the matroid.

Define the map

$\psi_{g}\colon\operatorname{DP}(P^{g},\mathcal{G}^{g})\otimes\operatorname{DP}(P_{g},\mathcal{G}_{g})\to\faktor{\operatorname{DP}(P,\mathcal{G})}{\operatorname{Ann}(x_{g})}$

by $\psi_{g}(\sigma_{h}\otimes 1)=\sigma_{h}$ and
$\psi_{g}(1\otimes\sigma_{g\vee h})=\sigma_{h}$.

######  Lemma 4.14.

For all $g\in\mathcal{G}\setminus\\{\hat{1}\\}$ the map $\psi_{g}$ is well
defined and it is an isomorphism. Moreover,
$\deg(\alpha)\deg(\beta)=\deg(x_{g}\psi_{g}(\alpha\otimes\beta))$ for all
$\alpha\in\operatorname{DP}(P^{g},\mathcal{G}^{g})$ and
$\beta\in\operatorname{DP}(P_{g},\mathcal{G}_{g})$.

###### Proof.

We verify that $\psi_{g}(1\otimes\sigma_{g\vee h})$ does not depend on the
choice of the element $h$. Suppose that there exist $h,f\in\mathcal{G}$ such
that $g\vee h=g\vee f$ and $h,f\not\leq g$. By symmetry we may assume
$h\not\geq f$. Replacing $f$ with $g\vee f$ we assume $f>h$, then

$x_{g}(\sigma_{h}-\sigma_{f})=x_{g}\sum_{\begin{subarray}{c}l\geq h\\\
l\not\geq g\end{subarray}}x_{l}=0,$

because $\\{g,l\\}$ cannot be $\mathcal{G}$-nested since $g<f\leq g\vee l$ and
$l\not\geq g$.

We verify that all relations in the domain are mapped to zero. The ones in
$\operatorname{DP}(P^{g},\mathcal{G}^{g})$ hold also in
$\operatorname{DP}(P,\mathcal{G})$ trivially. Consider $h\in\mathcal{G}$ and
$S\subset\mathcal{G}$ an antichain such that $\bigvee S\leq h$ and $s\not\leq
g$ for all $s\in S$. Set $n=\operatorname{cd}(g\vee
h)-\operatorname{cd}(g\vee\bigvee S)$. There are two cases:

  * •

if $g\vee h\not\in\mathcal{G}$ then
$n=\operatorname{cd}(h)-\operatorname{cd}(\bigvee S)$ and

$x_{g}\psi_{g}\Bigl{(}1\otimes\sigma_{g\vee h}^{n}\prod_{a\in S}(\sigma_{g\vee
s}-\sigma_{g\vee h})\Bigr{)}=x_{g}\sigma_{h}^{n}\prod_{a\in
S}(\sigma_{s}-\sigma_{h})=0,$

  * •

if $g\vee h\in\mathcal{G}$ then

$\displaystyle x_{g}\psi_{g}\Bigl{(}1\otimes\sigma_{g\vee h}^{n}\prod_{s\in
S}(\sigma_{g\vee s}-\sigma_{g\vee h})\Bigr{)}$
$\displaystyle=x_{g}\sigma_{g\vee h}^{n}\prod_{s\in
S}(\sigma_{s}-\sigma_{g\vee h})$
$\displaystyle=\sum_{A}x_{g}x_{A}\sigma_{g\vee h}^{n},$

where the sum is taken over all sets $A=\\{a_{1},\dots a_{k}\\}$ such that
$a_{i}\geq s_{i}$ and $a_{i}\not\geq g\vee h$. Applying Lemma 2.10 to $g\vee
h$, $S\cup\\{g\\}$ and $A\cup\\{g\\}$ we obtain that each term
$x_{g}x_{A}\sigma_{g\vee h}^{n}$ is zero.

The map $\psi_{g}$ is surjective because either $h\in\mathcal{G}^{g}$ or
$g\vee h\in\mathcal{G}_{g}$ for all $h\in\mathcal{G}$. We apply Proposition
4.8, $\operatorname{DP}(L,\mathcal{G})/\operatorname{Ann}(x_{g})$ is a
Poincaré duality algebra of dimension $\operatorname{cd}(\hat{1})-2$. The
algebra
$\operatorname{DP}(P^{g},\mathcal{G}^{g})\otimes\operatorname{DP}(P_{g},\mathcal{G}_{g})$
is Poincaré duality of dimension
$(\operatorname{cd}(g)-1)+(\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1)$
(here is the only point were we use $g\neq\hat{1}$). Since $\psi_{g}$ is
surjective between Poincaré duality algebras of the same dimension, it is an
isomorphism.

For the last statement we have

$\displaystyle x_{g}\psi_{g}(x_{g}^{\operatorname{cd}(g)-1}\otimes
x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1})$
$\displaystyle=x_{g}\sigma_{g}^{\operatorname{cd}(g)-1}x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1}$
$\displaystyle=(x_{g}-\sigma_{g})\sigma_{g}^{\operatorname{cd}(g)-1}x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1}$
$\displaystyle=-x_{\hat{1}}\sigma_{g}^{\operatorname{cd}(g)-1}x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1}$
$\displaystyle=-x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1},$

so
$\deg(x_{g}^{\operatorname{cd}(g)-1})\deg(x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(g)-1})=(-1)^{\operatorname{cd}(\hat{1})}=\deg(-x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1})$.
∎

###  4.3. Hard Lefschetz and Hodge-Riemann

We define a simplicial cone
$\Sigma\subset\operatorname{DP}^{1}(P,\mathcal{G})$ and we will show that each
element $\ell\in\Sigma$ satisfies Hard Lefschetz and Hodge-Riemann relations.

######  Definition 4.15.

The $\sigma$-cone
$\Sigma_{P,\mathcal{G}}\subset\operatorname{DP}^{1}(P,\mathcal{G})$ is the
convex cone

$\Sigma_{P,\mathcal{G}}=\Bigl{\\{}-\sum_{g\in\mathcal{G}}d_{g}\sigma_{g}\mid
d_{g}>0\Bigr{\\}}.$

Let $a\in E$ be an atom in $L$, i.e. the interval $(\hat{0},a)$ is empty.
Consider the set

(6) $\\{g\in\mathcal{G}\setminus\\{a\\}\mid g\neq\overline{S}\textnormal{ for
all }S\subseteq E\setminus\\{a\\}\\},$

of all elements $g\in\mathcal{G}$ that cannot be written as the closure of
some subset $S\subset E$ not containing $a$. Define $E(a)$ as the disjoint
union of $E\setminus\\{a\\}$ and the minimal elements of the set in (6).
Define the pair $P(a)=(E(a),\operatorname{cd})$, where with a slight abuse of
notation

$\operatorname{cd}(\\{e_{1},\dots,e_{l},g_{1},\dots,g_{k}\\})=\operatorname{cd}(\\{e_{1},\dots,e_{l}\\}\cup
g_{1}\cup\dots\cup g_{k}\\}).$

We also define $\mathcal{G}(a)=\mathcal{G}\setminus\\{a\\}$. The polymatroid
$P(a)$ depends on $\mathcal{G}$ but we omit this dependency in our notation.

In the realizable case, this polymatroidal operation corresponds to remove
only the subspace $S_{a}$ from the building set $\mathcal{G}$ and from the
arrangement $\mathcal{A}$. Now, there are subspaces in the lattice of flats
$\mathcal{L}_{\mathcal{A}}$ that are not flats of $\mathcal{A}\setminus
S_{a}$. Among them we want to keep trace only of the ones blown up, i.e.
belonging to $\mathcal{G}$; so we add to the deleted arrangement
$\mathcal{A}\setminus S_{a}$ all the flats corresponding to minimal elements
in the set (6).

######  Lemma 4.16.

The pair $P(a)=(E(a),\operatorname{cd})$ is a polymatroid and $\mathcal{G}(a)$
is a building set for the poset of flats of $P(a)$.

###### Proof.

It is easy to see that $(E(a),\operatorname{cd})$ is a polymatroid and that
the lattice of flats $L_{P(a)}$ of $P(a)$ is a subposet of the lattice of
flats $L$ of $P$. We verify that $\mathcal{G}(a)$ is a building set. We check
the definition for all $x\in L_{P(a)}$: if $a$ is not a $\mathcal{G}$-factor
of $x$ then $\max(\mathcal{G}_{\leq x})=\max(\mathcal{G}(a)_{\leq x})$ and it
follows from the properties of $\mathcal{G}$. Otherwise, $a$ is a
$\mathcal{G}$-factor of $x$ and $x$ cannot lie in the lattice $L_{P(a)}$
generated by $\mathcal{G}\setminus\\{a\\}$. ∎

######  Lemma 4.17.

For an atom $a\in E$, $a\neq\hat{1}$, consider the element
$\mu_{0}=(x_{a}-\sigma_{a})^{\operatorname{cd}(a)}$. There exists an
isomorphism:

$p_{a}\colon\operatorname{DP}(P_{a},\mathcal{G}_{a})\to\faktor{\operatorname{DP}(P(a),\mathcal{G}(a))}{\operatorname{Ann}(\mu_{0})}.$

Moreover, $\deg(\alpha)=\deg(\mu_{0}p_{a}(\alpha))$ for all
$\alpha\in\operatorname{DP}(P_{a},\mathcal{G}_{a})$.

###### Proof.

Notice that $\mu_{0}=(x_{a}-\sigma_{a})^{\operatorname{cd}(a)}$ is a multiple
of $x_{a}$ because $\sigma_{a}^{\operatorname{cd}(a)}=0$, hence
$\operatorname{Ann}(x_{a})\subseteq\operatorname{Ann}(\mu_{0})$. Define the
morphism $p_{a}$ as the composition

$\operatorname{DP}(P_{a},\mathcal{G}_{a})\hookrightarrow\operatorname{DP}(P^{a},\mathcal{G}^{a})\otimes\operatorname{DP}(P_{a},\mathcal{G}_{a})\xrightarrow{\psi_{a}}\operatorname{DP}(P,\mathcal{G})/\operatorname{Ann}(x_{a})\twoheadrightarrow\operatorname{DP}(P,\mathcal{G})/\operatorname{Ann}(\mu_{0}),$

where the first map is the inclusion $x\mapsto 1\otimes x$. Explicitly
$p_{a}(\sigma_{a\vee h})=[\sigma_{h}]$ for all $h\neq a$. Since
$\mathcal{G}(a)$ is a subset of $\mathcal{G}$,
$\operatorname{DP}(P(a),\mathcal{G}(a))$ is a subalgebra of
$\operatorname{DP}(P,\mathcal{G})$. The range of the map $p_{a}$ is equal to
$\operatorname{DP}(P(a),\mathcal{G}(a))/\operatorname{Ann}(\mu_{0})$, so the
morphism in the statement is well defined and surjective. Since $a\neq\hat{1}$
we have $\mu_{0}\neq 0$ and by Proposition 4.8 the map $p_{a}$ is an
isomorphism, because both algebras satisfy Poincaré duality of dimension
$\operatorname{cd}(\hat{1})-\operatorname{cd}(a)-1$.

For the last statement we have
$\mu_{0}p_{a}(x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(a)-1})=(-1)^{\operatorname{cd}(a)}x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1}$
and so
$\deg(x_{\hat{1}}^{\operatorname{cd}(\hat{1})-\operatorname{cd}(a)-1})=(-1)^{\operatorname{cd}(\hat{1})-\operatorname{cd}(a)-1}=\deg((-1)^{\operatorname{cd}(a)}x_{\hat{1}}^{\operatorname{cd}(\hat{1})-1})$.
∎

######  Lemma 4.18.

Let $a\in E$, $a\neq\hat{1}$, be an atom and
$\mu_{0}=(x_{a}-\sigma_{a})^{\operatorname{cd}(a)}$. Consider the polynomial
$p(x)=\sum_{i=0}^{\operatorname{cd}(a)}\binom{\operatorname{cd}(a)}{i}x^{i}(x_{a}-\sigma_{a})^{\operatorname{cd}(a)-i}$,
then

$\faktor{\operatorname{DP}(P(a),\mathcal{G}(a))[x]}{(x\operatorname{Ann}(\mu_{0}),p(x))}\cong\operatorname{DP}(P,\mathcal{G}).$

###### Proof.

Define the morphism

$\operatorname{DP}(P(a),\mathcal{G}(a))[x]\to\operatorname{DP}(P,\mathcal{G})$

by $\sigma_{g}\mapsto\sigma_{g}$ and $x\mapsto-x_{a}$. By Lemmas 4.17 and 4.14
the elements of the form $x\operatorname{Ann}(\mu_{0})$ are in the kernel.
Also $p(x)$ is in the kernel because its image is
$(-\sigma_{a})^{\operatorname{cd}(a)}=0$. Clearly, the map is surjective.

Notice that if $A$ is a Poincaré duality algebra and $p(x)\in A[x]$ a monic
polynomial with constant term $\mu_{0}$ then
$A[x]/(x\operatorname{Ann}(\mu_{0}),p(x))$ is a Poincaré duality algebra.
Indeed, if a generic element $\sum_{i=0}^{j}a_{i}x^{i}$ (with
$a_{j}\not\in\operatorname{Ann}(\mu_{0})$ and $j<\deg(p)$) of degree $k$ is
orthogonal to all elements of degree $n-k$, then
$(\sum_{i=0}^{j}a_{i}x^{i})a^{\prime}=0$ for all $a^{\prime}\in A^{n-k}$. This
implies $a_{0}a^{\prime}=0$ and $a_{0}=0$. Moreover,
$(\sum_{i=1}^{j}a_{i}x^{i})a^{\prime}x^{\deg(p)-j}=0$ implies
$a_{j}a^{\prime}\mu_{0}=0$ and $a_{j}\mu_{0}=0$ by Poincaré duality in $A$,
contradicting the fact $a_{j}\not\in\operatorname{Ann}(\mu_{0})$. In
particular,
$\operatorname{DP}(P(a),\mathcal{G}(a))[x]/(x\operatorname{Ann}(\mu_{0}),p(x))$
is a Poincaré duality algebra of dimension $\operatorname{cd}(\hat{1})-1$.

The map
$\operatorname{DP}(P(a),\mathcal{G}(a))[x]\to\operatorname{DP}(P,\mathcal{G})$
is injective by Proposition 4.8 because domain and codomain are Poincaré
duality algebras of the same dimension equal to
$\operatorname{cd}(\hat{1})-1$. ∎

The following theorem provides an abstract procedure to prove the Hodge-
Riemann relations inductively.

######  Theorem 4.19.

Let $C$ be a Poincaré duality algebra and
$p(x)=x^{d}+\mu_{d-1}x^{d-1}+\dots+\mu_{0}=0\in C[x]$ be a homogeneous
polynomial with $\mu_{0}\neq 0$. Let $B=C/\operatorname{Ann}(\mu_{0})$ and
$A=C[x]/(x\operatorname{Ann}(\mu_{0}),p(x))$. Let $\ell\in C^{1}$ be an
element satisfying $\operatorname{HR}_{C}(\ell)$ and
$\operatorname{HR}_{B}(\ell)$. Then $\operatorname{HR}_{A}(\ell+\epsilon x)$
holds for sufficiently small positive $\epsilon$.

In the above theorem the degree function on $B$ is induced by $\mu_{0}$, i.e.
$\deg_{B}(\alpha)=\deg_{C}(\alpha\mu_{0})$. Since the top degrees coincide
$A^{top}=C^{top}$, we also implicitly assume that $\deg_{A}=\deg_{C}$.

The proof of Theorem 4.19 is the same of the proof of [AHK18, Proposition
8.2], so we omit it.

The following easy lemma shows that the maps introduced in Section 4.2
preserve the $\Sigma$-cone.

######  Lemma 4.20.

The following holds:

  1. (1)

For any $g\in\mathcal{G}$, $g\neq\hat{1}$ the natural map

$\operatorname{DP}^{1}(P,\mathcal{G})\to\operatorname{DP}^{1}(P^{g},\mathcal{G}^{g})\oplus\operatorname{DP}^{1}(P_{g},\mathcal{G}_{g})$

induced by the quotient by $\operatorname{Ann}(x_{g})$ composed with
$\psi_{g}^{-1}$, maps $\Sigma_{P,\mathcal{G}}$ into
$\Sigma_{P^{g},\mathcal{G}^{g}}\times\Sigma_{P_{g},\mathcal{G}_{g}}$.

  2. (2)

For any $g\in\mathcal{G}$ the morphism

$\operatorname{DP}^{1}(P,\mathcal{G})\to\operatorname{DP}^{1}(\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G})$

induced by the quotient by $\operatorname{Ann}(\sigma_{g})$ composed with
$\zeta_{g}^{-1}$, maps $\Sigma_{P,\mathcal{G}}$ into
$\Sigma_{\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G}}$.

  3. (3)

For any atom $a\in E,a\neq\hat{1}$ the natural map

$\operatorname{DP}^{1}(P(a),\mathcal{G}(a))\to\operatorname{DP}^{1}(P_{a},\mathcal{G}_{a})$

induced by the quotient by $\operatorname{Ann}(\mu_{0})$ composed with
$p_{a}^{-1}$, maps $\Sigma_{P(a),\mathcal{G}(a)}$ into
$\Sigma_{P_{a},\mathcal{G}_{a}}$.

###### Proof.

  1. (1)

Let $l=-\sum_{h\in\mathcal{G}}d_{h}\sigma_{h}$ be an element of the
$\sigma$-cone, we have

$\psi_{g}^{-1}([l])=-\sum_{h\leq g}d_{h}\sigma_{h}\otimes 1-\sum_{h\nleq
g}d_{h}\otimes\sigma_{g\vee h}.$

It may occur that there are two different $h,h^{\prime}\in\mathcal{G}$ such
that $g\vee h=g\vee h^{\prime}$ but, also in this case, the coefficient of
$1\otimes\sigma_{g\vee h}$ is still negative. It follows that
$\psi_{g}^{-1}([l])\in\Sigma_{P^{g},\mathcal{G}^{g}}\times\Sigma_{P_{g},\mathcal{G}_{g}}$.

  2. (2)

Let $l=-\sum_{h\in\mathcal{G}}d_{h}\sigma_{h}$ be an element of the
$\sigma$-cone, we have

$\zeta_{g}^{-1}([l])=-\sum_{h\in\mathcal{G}}d_{h}\sigma_{\bar{h}}.$

It may occur that there are two different $h,h^{\prime}\in\mathcal{G}$ such
that $\bar{h}=\bar{h^{\prime}}$ but, also in this case, the coefficient of
$\sigma_{\bar{h}}$ is still negative. Thus,
$\zeta_{g}^{-1}([l])\in\Sigma_{\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G}}$.

  3. (3)

Let $l=-\sum_{h\in\mathcal{G}}d_{h}\sigma_{h}$ be an element of the
$\sigma$-cone, we have

$p_{a}^{-1}([l])=-\sum_{h\in\mathcal{G}}d_{h}\sigma_{a\vee h}.$

It may occur that there are two different $h,h^{\prime}\in\mathcal{G}$ such
that $a\vee h=a\vee h^{\prime}$ but, also in this case, the coefficient of
$\sigma_{a\vee h}$ is still negative. It follows that
$p_{a}^{-1}([l])\in\Sigma_{P_{a},\mathcal{G}_{a}}$.

∎

Now we are ready to prove the main theorem.

######  Theorem 4.21.

For every element $\ell$ in the $\sigma$-cone $\Sigma_{P,\mathcal{G}}$ the
conditions $\operatorname{HL}_{\operatorname{DP}(P,\mathcal{G})}(\ell)$ and
$\operatorname{HR}_{\operatorname{DP}(P,\mathcal{G})}(\ell)$ hold.

###### Proof.

We prove the statement by induction on $\lvert\mathcal{G}\rvert$ and
$\operatorname{cd}(\hat{1})$. The base case is $\lvert\mathcal{G}\rvert=1$, so
$\operatorname{DP}(P,\mathcal{G})=\mathbb{Q}[x_{\hat{1}}]/(x_{\hat{1}}^{\operatorname{cd}(\hat{1})})$.
In this case, it is known that $-\lambda x_{\hat{1}}$ satisfies Hard Lefschetz
and Hodge-Riemann for all positive $\lambda$.

For the inductive step consider a polymatroid $P$, a building set
$\mathcal{G}$, and an element $\ell\in\Sigma_{P,\mathcal{G}}$. Under the
morphisms of Lemma 4.20 Item 2 $\ell$ is mapped in
$\Sigma_{\operatorname{tr}_{g}P,\operatorname{tr}_{g}\mathcal{G}}$ for all
$g\in\mathcal{G}$. Therefore by the inductive hypothesis the image of $\ell$
in $\operatorname{DP}(P,\mathcal{G})/\operatorname{Ann}(\sigma_{g})$ satisfies
Hodge-Riemann relations for all $g\in\mathcal{G}$. Notice also that $\ell$ is
a sum of $-\sigma_{g}$ with positive coefficients. By [AHK18, Proposition
7.15], $\operatorname{HL}_{\operatorname{DP}(P,\mathcal{G})}(\ell)$ holds.

We want to prove that the Hodge-Riemann relations hold for all
$\ell\in\Sigma_{P,\mathcal{G}}$. By [AHK18, Proposition 7.16] it is enough to
prove $\operatorname{HR}_{\operatorname{DP}(P,\mathcal{G})}(\ell)$ for some
$\ell\in\Sigma_{P,\mathcal{G}}$. We apply Theorem 4.19: consider any atom
$a\in E$, since $\lvert\mathcal{G}\rvert>1$ then $a\neq\hat{1}$. Set
$C=\operatorname{DP}(P(a),\mathcal{G}(a))$ and
$p(x)=\sum_{i=0}^{\operatorname{cd}(a)}\binom{\operatorname{cd}(a)}{i}x^{i}(x_{a}-\sigma_{a})^{\operatorname{cd}(a)-i}$;
Lemma 4.17 ensures that $B=\operatorname{DP}(P_{a},\mathcal{G}_{a})$ and Lemma
4.18 that $A=\operatorname{DP}(P,\mathcal{G})$. Let
$\ell\in\Sigma_{P(a),\mathcal{G}(a)}$, then under the morphism $C\to B$ (Lemma
4.20 Item 3) the class $\ell$ is mapped in $\Sigma_{P_{a},\mathcal{G}_{a}}$.
By the inductive hypothesis we have
$\operatorname{HR}_{\operatorname{DP}(P(a),\mathcal{G}(a))}(\ell)$ and
$\operatorname{HR}_{\operatorname{DP}(P_{a},\mathcal{G}_{a})}(\ell)$, hence by
Theorem 4.19
$\operatorname{HR}_{\operatorname{DP}(P,\mathcal{G})}(\ell-\epsilon x_{a})$
holds for sufficiently small $\epsilon>0$.

Moreover if $\epsilon$ is small enough then $\ell-\epsilon x_{a}$ belongs to
$\Sigma_{P,\mathcal{G}}$. Indeed using the Möbius inversion formula we have

$x_{a}=\sum_{g\geq a}\mu_{\mathcal{G}}(a,g)\sigma_{g}$

(where we consider $\mathcal{G}$ as a sub-poset of $L$). Let
$\ell=-\sum_{g\in\mathcal{G}}d_{g}\sigma_{g}$, taking $\epsilon$ smaller than

$\min_{g\geq
a}\left\\{\Big{\lvert}\frac{d_{g}}{\mu_{\mathcal{G}}(a,g)}\Big{\rvert}\right\\},$

then $\ell-\epsilon x_{a}\in\Sigma_{P,\mathcal{G}}$. This concludes the proof.
∎

######  Remark 4.22.

The ample cone depends on the geometric realization, however our $\sigma$-cone
is contained in the ample cone of every realization. Indeed, consider $3$
distinct lines in $\mathbb{C}^{3}$ and let $P$ be the polymatroid realized by
this subspace arrangement. The projective wonderful model is the blowup of
$\mathbb{P}^{2}$ in $3$ distinct points; there are two cases. If the three
points are collinear the ample cone coincides with the $\sigma$-cone.
Otherwise the three points are in general position and the ample cone is

$\\{-d_{\hat{1}}x_{\hat{1}}-d_{a}x_{a}-d_{b}x_{b}-d_{c}x_{c}\mid
d_{\hat{1}}>d_{a}+d_{b},d_{\hat{1}}>d_{a}+d_{c},d_{\hat{1}}>d_{b}+d_{c}\\}$

which strictly contains the $\sigma$-cone.

######  Remark 4.23.

If we restrict to the case of matroids with arbitrary building sets, the
generator $x_{\hat{1}}$ can be eliminated using the relation
$x_{\hat{1}}=-\sum_{g\geq e,\,g\neq\hat{1}}x_{g}$ for any $e\in E$. Thus the
Hard Lefschetz theorem (and so the Hodge-Riemann relations) can be proven for
the entire ample cone using as generators $\\{x_{g}\\}_{g\neq\hat{1}}$ instead
of $\\{\sigma_{g}\\}_{g\in\mathcal{G}}$ and Lemma 4.14 instead of Lemma 4.11.

##  5\. The relative Lefschetz decomposition

In this section we provide a decomposition of
$\operatorname{DP}(P,\mathcal{G})$ as $\operatorname{DP}(P\setminus
a,\mathcal{G}\setminus a)$-module. This is analogous to the semi-small
decomposition of [BHM+22], but in this more general setting the corresponding
map is not always semi-small.

Indeed, consider an arrangement of hyperplanes $\mathcal{A}$ and the deleted
arrangement $\mathcal{A}^{\prime}=\mathcal{A}\setminus\\{H\\}$ for some
hyperplane $H\in\mathcal{A}$. There is a projection map between the wonderful
models $Y_{\mathcal{A}}\to Y_{\mathcal{A}^{\prime}}$ (constructed using the
maximal building sets). This map is semi-small and induces the semi-small
decomposition of the Chow ring.

In the case of subspace arrangements, the projection between the wonderful
models exists but is not semi-small, because the dimension of the fiber of the
blow up is too big. Therefore, the proof of the Kähler package done in
[BHM+22] for matroids cannot be adapted to polymatroids.

Recall that for a polymatroid $P=(E,\operatorname{cd})$ an atom $a\in E$ is an
element such that the interval $(\hat{0},\overline{a})\subset L$ is empty
(where $\overline{a}$ is the closure of $a$).

######  Definition 5.1.

For an atom $a$ define the polymatroid $P\setminus a$ on the ground set
$E\setminus\\{a\\}$ with the restricted codimension function
$\operatorname{cd}$. The building set $\mathcal{G}\setminus a$ is the
intersection of $\mathcal{G}$ with the poset of flats of $P\setminus a$.

Define a map

$\theta_{a}\colon\operatorname{DP}(P\setminus a,\mathcal{G}\setminus
a)\to\operatorname{DP}(P,\mathcal{G})$

by $\theta_{a}(\sigma_{h})=\sigma_{\overline{h}}$ where $\overline{h}$ is the
closure of $h$ in $P$. Define the subalgebra
$\operatorname{DP}_{(a)}=\operatorname{Im}(\theta_{a})$.

######  Lemma 5.2.

The map $\theta_{a}$ is injective.

###### Proof.

Consider a standard monomial $\sigma_{S}^{b}\in\operatorname{DP}(P\setminus
a,\mathcal{G}\setminus a)$ and let $\overline{S}=\\{\overline{h}\mid h\in
S\\}$. We have $\theta_{a}(\sigma_{S}^{b})=\sigma_{\overline{S}}^{b}$ and it
is enough to prove that $\sigma_{\overline{S}}^{b}$ is a standard monomial.
Notice that $\overline{h}\vee\overline{g}=\overline{h\vee g}$ and the map
between the two poset of flats is an inclusion. Therefore $\overline{S}$ is
$\mathcal{G}$-nested. Since
$\operatorname{cd}(h)=\operatorname{cd}(\overline{h})$, then
$\sigma_{\overline{S}}^{b}$ is a standard monomial. ∎

Let $S_{a}=\\{g\in\mathcal{G}\mid a\in g\textnormal{ and }g\setminus\\{a\\}\in
L\\}$ be the set of all flats such that $a$ is a coloop for that flat.

######  Remark 5.3.

Notice that $\theta_{a}(x_{g})=x_{g}+x_{g\cup\\{a\\}}$, where we use the
convention that $x_{h}=0$ if $h$ is not a flat of $P$. Moreover
$\operatorname{DP}_{(a)}$ is generated as an algebra by $\sigma_{g}$ with
$g\notin S_{a}$ and as vector space by the monomials $\sigma_{S}^{b}$ with
$S\cap S_{a}=\emptyset$.

For $f\in S_{a}$ define $\operatorname{DP}_{f}$ as the
$\operatorname{DP}_{(a)}$-submodule of $\operatorname{DP}(P,\mathcal{G})$
generated by $x_{f},x_{f}^{2},\dots x_{f}^{n_{f}}$, where

$n_{f}=\operatorname{cd}(f)-\operatorname{cd}(f\setminus\\{a\\})-1+\lvert
F(P,\mathcal{G},f\setminus\\{a\\})\rvert.$

For a graded module $M=\oplus_{i}M^{i}$ we define $M[k]$ to be the graded
module such that $(M[k])^{i}=M^{i+k}$.

######  Theorem 5.4.

Let $a$ be an atom, then:

(7) $\displaystyle
x_{f}^{k}\operatorname{DP}_{(a)}[-k]\cong\operatorname{DP}((P\setminus
a)^{f\setminus a},(\mathcal{G}\setminus a)^{f\setminus
a})\otimes\operatorname{DP}(P_{f},\mathcal{G}_{f}),$ (8)
$\displaystyle\operatorname{DP}_{f}=\bigoplus_{k=1}^{n_{f}}x_{f}^{k}\operatorname{DP}_{(a)},$
(9)
$\displaystyle\operatorname{DP}(P,\mathcal{G})=\operatorname{DP}_{(a)}\oplus\bigoplus_{f\in
S_{a}}\operatorname{DP}_{f}.$

as $\operatorname{DP}_{(a)}$-modules. Moreover, the last decomposition is
orthogonal with respect to the Poincaré pairing, with the exception of the
summand $\operatorname{DP}_{(a)}$ and $\operatorname{DP}_{\hat{1}}$ (if $a$ is
a coloop).

For an example of the application of Theorem 5.4 see Section 7. Before the
proof of the above theorem we need some lemmas.

######  Lemma 5.5.

For all $f\in S_{a}$ and $k\leq n_{f}$ we have

(10)
$x_{f}\sigma_{f}^{k-1}\operatorname{DP}_{(a)}[-k]\cong\operatorname{DP}((P\setminus
a)^{f\setminus a},(\mathcal{G}\setminus a)^{f\setminus
a})\otimes\operatorname{DP}(P_{f},\mathcal{G}_{f}),$

and these modules are in direct sum in $\operatorname{DP}(P,\mathcal{G})$.

###### Proof.

Notice that for $k\leq n_{f}$

$\operatorname{DP}((\operatorname{tr}_{f}^{k-1}(P^{f}))\setminus
a,(\operatorname{tr}_{f}^{k-1}(\mathcal{G}^{f}))\setminus
a)=\operatorname{DP}((P\setminus a)^{f\setminus a},(\mathcal{G}\setminus
a)^{f\setminus a}).$

Using Lemmas 4.14 and 4.11 we obtain the isomorphism

$x_{f}\sigma_{f}^{k-1}\operatorname{DP}(P,\mathcal{G})[-k]\simeq\operatorname{DP}(\operatorname{tr}_{f}^{k-1}(P^{f}),\operatorname{tr}_{f}^{k-1}(\mathcal{G}^{f}))\otimes\operatorname{DP}(P_{f},\mathcal{G}_{f}).$

It is easy to check that the above isomorphism restricts to the one in eq.
(10). For the second claim suppose that there exists a linear combination

$\sum_{k=l}^{n_{f}}\sigma_{f}^{k-1}p_{k}=0,$

for some $p_{k}\in\operatorname{DP}((P\setminus a)^{f\setminus
a},(\mathcal{G}\setminus a)^{f\setminus a})$ with $p_{l}\neq 0$. The above
equality implies

$\sum_{k=l}^{n_{f}}\sigma_{f}^{k-l}p_{k}=0$

in
$\operatorname{DP}(\operatorname{tr}_{f}^{l-1}P^{f},\operatorname{tr}_{f}^{l-1}\mathcal{G}^{f})$.
Therefore $p_{l}$ belongs to the ideal generated by $\sigma_{f}$ (where $f$ is
the top element in the poset of flats of $\operatorname{tr}_{f}^{l-1}P^{f}$).
The ideal $(\sigma_{f})$ is linearly generated by all monomials
$\sigma_{T}^{b}$ with $f\in T$. This yields a contradiction since $p_{l}$ lies
in $\operatorname{DP}((P\setminus a)^{f\setminus a},(\mathcal{G}\setminus
a)^{f\setminus a})$, which does not contain the generator $\sigma_{f}$. ∎

######  Lemma 5.6.

For all elements $f,g\in S_{a}$ such that $f\not\geq g$ we have
$x_{f}\sigma_{g}=x_{f}\sigma_{g\setminus\\{a\\}}$. Moreover, we have

$\operatorname{DP}_{f}=\bigoplus_{k=1}^{n_{f}}x_{f}\sigma_{f}^{k-1}\operatorname{DP}_{(a)}.$

###### Proof.

Consider $h\in\mathcal{G}$ such that $h\geq g\setminus\\{a\\}$ and $h\not\geq
g$, we need to prove that $x_{f}x_{h}=0$. Notice that $\\{f,h\\}$ is an
antichain, $f\vee g\in\mathcal{G}$ and so $(f\vee g)\vee h\in\mathcal{G}$
because $g\setminus\\{a\\}\neq\hat{0}$. Therefore

$f\vee h=(f\vee a)\vee((g\setminus\\{a\\})\vee h)=f\vee g\vee
h\in\mathcal{G},$

$\\{f,h\\}$ is not $\mathcal{G}$-nested, and $x_{f}x_{h}=0$.

For the second statement it is sufficient to prove that
$x_{f}\sigma_{f}^{k-1}=x_{f}^{k}+z$ with some
$z\in\sum_{j=1}^{k-1}x_{f}^{j}\operatorname{DP}_{(a)}$. Write
$\sigma_{f}=x_{f}+\sum_{g>f}b_{g}\sigma_{g}$ for some coefficients
$b_{g}\in\mathbb{Z}$, then

$x_{f}\sigma_{f}=x_{f}^{2}+x_{f}\sum_{\begin{subarray}{c}g>f\\\ g\notin
S_{a}\end{subarray}}b_{g}\sigma_{g}+x_{f}\sum_{\begin{subarray}{c}g>f\\\ g\in
S_{a}\end{subarray}}b_{g}\sigma_{g\setminus\\{a\\}},$

and all the summands (except $x_{f}^{2}$) belong to
$x_{f}\operatorname{DP}_{(a)}$. An inductive argument on the exponent $k$
concludes the proof. ∎

######  Lemma 5.7.

The submodules $\operatorname{DP}_{(a)}$ and $\operatorname{DP}_{f}$ for all
$f\in S_{a}$ generate $\operatorname{DP}(P,\mathcal{G})$.

###### Proof.

We prove that each monomial $\sigma_{S}^{b}$ belongs to the submodule
$M:=\operatorname{DP}_{(a)}+\sum_{F\in S_{a}}\operatorname{DP}_{F}$ by
complete induction on $f=\min(S\cap S_{a})$ and on $b(f)$.

The base case is $S\cap S_{a}=\emptyset$ and so
$\sigma_{S}^{b}\in\operatorname{DP}_{(a)}$. For the inductive step notice that
$S\cap S_{a}$ is $\mathcal{G}$-nested, so it is a chain. Call $f=\min(S\cap
S_{a})$ and suppose that all monomials $\sigma_{S}^{b^{\prime}}$ with
$b^{\prime}(f)<b(f)$ and all monomials $\sigma_{S^{\prime}}^{b^{\prime}}$ with
$\min(S^{\prime}\cap S_{a})>f$ belong to $M$.

Let $\\{g_{1},\dots,g_{l}\\}=F(P,\mathcal{G},f\setminus\\{a\\})$ be the set of
$\mathcal{G}$-factors of $f\setminus\\{a\\}$. The relation

$\sigma_{f}^{\operatorname{cd}(f)-\operatorname{cd}(f\setminus\\{a\\})}\prod_{i=1}^{k}(\sigma_{g_{i}}-\sigma_{f})=0,$

holds and in the case $b(f)>n_{f}$ we can rewrite
$\sigma_{f}^{b(f)}\sigma_{T\setminus\\{f\\}}$ as sum of monomials with
$b^{\prime}(f)<b(f)$ using the above relation and the fact that $g_{i}\not\in
S_{a}$.

In the case $b(f)\leq n_{f}$ we have

$\sigma_{S}^{b}=x_{f}\sigma_{f}^{b(f)-1}\sigma_{S\setminus\\{f\\}}^{b}+(\sigma_{f}-x_{f})\sigma_{f}^{b(f)-1}\sigma_{S\setminus\\{f\\}}^{b}.$

Using the first assertion of Lemma 5.6, it follows that the element
$x_{f}\sigma_{f}^{b(f)-1}\sigma_{S\setminus\\{f\\}}^{b}$ belongs to
$x_{f}\sigma_{f}^{b(f)-1}\operatorname{DP}_{(a)}\subset M$. The second summand
$(\sigma_{f}-x_{f})\sigma_{f}^{b(f)-1}\sigma_{S\setminus\\{f\\}}^{b}$ is a
linear combination of monomials
$\sigma_{h}\sigma_{f}^{b(f)-1}\sigma_{S\setminus\\{f\\}}^{b}$ with $h>f$ and
so belongs to $M$ by the inductive hypothesis. ∎

###### Proof of Theorem 5.4.

As in Remark 4.1, we may assume that $\hat{1}\in\mathcal{G}$. By Lemmas 5.6
and 5.5, $\operatorname{DP}_{f}$ is a free $\operatorname{DP}((P\setminus
a)^{f\setminus a},(\mathcal{G}\setminus a)^{f\setminus
a})\otimes\operatorname{DP}(P_{f},\mathcal{G}_{f})$-module with basis
$x_{f}\sigma_{f}^{k-1}$ for $k=1,\dots,n_{f}$. The elements
$\\{x_{f}^{k}\\}_{k}$ written in the basis $\\{x_{f}\sigma_{f}^{k-1}\\}_{k}$
form an upper triangular matrix with ones on the diagonal (the inverse of the
one given in Lemma 5.6). Eq. (7) and eq. (8) follow.

In order to prove eq. (9) we first prove the orthogonality. Let
$f\neq\hat{1}$; the elements $\operatorname{DP}_{f}$ and
$\operatorname{DP}_{(a)}$ are orthogonal because the product is contained in
$\operatorname{DP}_{f}$ which is zero in degree
$\operatorname{cd}(\hat{1})-1$. Indeed from eq. (7) and eq. (8), it follows
that the top degree of $\operatorname{DP}_{f}$ is
$\operatorname{cd}(\hat{1})-2$.

Consider generic elements $x_{f}^{b}y\in\operatorname{DP}_{f}$ and
$x_{g}^{c}z\in\operatorname{DP}_{g}$ in complementary degrees (with
$y,z\in\operatorname{DP}_{(a)}$). The product is zero if $f$ and $g$ are
incomparable. Otherwise, by symmetry we may assume $g>f$, hence

$x_{f}x_{g}^{c}=x_{f}(x_{g}+x_{g\setminus\\{a\\}})^{c}.$

Since $x_{g}+x_{g\setminus\\{a\\}}\in\operatorname{DP}_{(a)}$, we obtain that
the product lie in $\operatorname{DP}_{f}$. Again the top degree is zero since
$f\neq\hat{1}$.

We prove that if $a$ is a coloop then
$\operatorname{DP}_{(a)}\cap\operatorname{DP}_{\hat{1}}=0$. In that case
$\operatorname{DP}_{\hat{1}}$ is the ideal generated by $\sigma_{\hat{1}}$.
This ideal is linearly generated by all standard monomials $\sigma_{S}^{b}$
with $\hat{1}\in S$. Since $\hat{1}\in S_{a}$ then
$\operatorname{DP}_{(a)}\cap\operatorname{DP}_{\hat{1}}=0$. The direct sum of
eq. (9) follows from the orthogonality of all other summands together with
Lemma 5.7 and Theorem 4.7. ∎

##  6\. Characteristic polynomial

In this section we study the coefficients of the (reduced) characteristic
polynomial of a polymatroid.

We consider only maximal building sets, so we omit the building set from the
notations. Moreover we suppose that the polymatroid is without loops, i.e.
$\operatorname{cd}(\\{e\\})>0$ for all $e\in E$.

Let $\alpha=\alpha_{P}=-x_{\hat{1}}$ and
$\beta=\beta_{P}=\sum_{g\in\mathcal{G}_{\max}}x_{g}$ be two elements in
$\operatorname{DP}^{1}(P)$. We denote by $\mu_{L}(a,b)$ the Möbius function of
$L$.

######  Lemma 6.1.

For any polymatroid $P$ with $\operatorname{cd}(E)>0$ and
$r=\operatorname{cd}(E)-1$ we have

$\deg(\beta_{P}^{r})=(-1)^{r}+\sum_{g\in
L\setminus\\{\hat{0},\hat{1}\\}}(-1)^{\operatorname{cd}(g)-1}\deg(\beta_{P_{g}}^{r-\operatorname{cd}(g)}).$

###### Proof.

A flag with repetition is $\mathcal{F}=(F_{1}^{a_{1}}\subsetneq
F_{2}^{a_{2}}\subsetneq\dots\subsetneq F_{l}^{a_{l}})$ where $a_{i}>0$ are the
multiplicity of the flats $F_{i}\in L$. We also require that
$\sum_{i=1}^{l}a_{i}=r$. Define
$x_{\mathcal{F}}=\prod_{i=1}^{\lvert\mathcal{F}\rvert}x_{F_{i}}^{a_{i}}$, we
will prove that $x_{\mathcal{F}}=0$ if $\operatorname{cd}(F_{1})>a_{1}$. More
generally we have $x_{\mathcal{F}}=0$ if
$\operatorname{cd}(F_{i})>\sum_{j=1}^{i}a_{j}$ for some $i$, but we prove and
use the implication only for $i=1$. From the isomorphism $\psi_{g}$ of Lemma
4.14 we obtain

$x_{\mathcal{F}}=x_{F_{1}}\psi_{F_{1}}((x_{F_{1}}\otimes
1-1\otimes\beta_{P_{g}})^{a_{1}-1}(1\otimes x_{\mathcal{F}^{\prime}})),$

where $\mathcal{F}^{\prime}=(F_{2}^{a_{2}}\subsetneq\dots\subsetneq
F_{l}^{a_{l}})$. Notice that the degree of $x_{\mathcal{F}^{\prime}}$ is
$r-a_{1}$, which is greater than $r-\operatorname{cd}(F_{1})$, the top degree
of $\operatorname{DP}(P_{g})$.

Let $\binom{r}{a}$ be the multinomial coefficient where
$a=(a_{1},\dots,a_{l})$ and $\sum_{i=1}^{l}a_{i}=r$. Since $x_{f}x_{g}=0$ if
$f$ and $g$ are incomparable, we have

$\displaystyle\beta_{P}^{r}$ $\displaystyle=\sum_{\mathcal{F}\textnormal{ flag
of }P}\binom{r}{a}x_{\mathcal{F}}$ $\displaystyle=\sum_{F\in
L\setminus\\{\hat{0}\\}}\sum_{\begin{subarray}{c}\mathcal{F}\textnormal{ flag
of }P\\\ F_{1}=F\end{subarray}}\binom{r}{a}x_{\mathcal{F}}$
$\displaystyle=\sum_{F\in
L\setminus\\{\hat{0}\\}}\sum_{k=\operatorname{cd}(F)}^{r}\binom{r}{k}x_{F}^{k}\sum_{\mathcal{F}^{\prime}\textnormal{
flag of }P_{F}}\binom{r-k}{a^{\prime}}x_{\mathcal{F}^{\prime}}$
$\displaystyle=\sum_{F\in
L\setminus\\{\hat{0}\\}}\sum_{k=\operatorname{cd}(F)}^{r}\binom{r}{k}x_{F}^{k}\beta_{P_{F}}^{r-k}.$

The summand relative to $F=\hat{1}$ is exactly $x_{\hat{1}}^{r}$ and
contributes $(-1)^{r}$ to $\deg(\beta_{P}^{r})$. It is enough to prove

$\deg\Bigl{(}\sum_{k=\operatorname{cd}(g)}^{r}\binom{r}{k}x_{g}^{k}\beta_{P_{g}}^{r-k}\Bigr{)}=\deg(\beta_{P_{g}}^{r-\operatorname{cd}(g)})$

for every $g\in L\setminus\\{\hat{0},\hat{1}\\}$. We use Lemma 4.14 to obtain:

$\displaystyle\deg$
$\displaystyle\Bigl{(}\sum_{k=\operatorname{cd}(g)}^{r}\binom{r}{k}x_{g}^{k}\beta_{P_{g}}^{r-k}\Bigr{)}=$
$\displaystyle=\sum_{k=\operatorname{cd}(g)}^{r}\binom{r}{k}\deg((x_{g}\otimes
1-1\otimes\beta_{P_{g}})^{k-1}(1\otimes\beta_{P_{g}}^{r-k}))$
$\displaystyle=\sum_{k=\operatorname{cd}(g)}^{r}(-1)^{k-\operatorname{cd}(g)}\binom{r}{k}\binom{k-1}{\operatorname{cd}(g)-1}\deg(x_{g}^{\operatorname{cd}(g)-1}\otimes\beta_{P_{g}}^{r-\operatorname{cd}(g)})$
$\displaystyle=\sum_{k=\operatorname{cd}(g)}^{r}(-1)^{k-1}\binom{r}{k}\binom{k-1}{\operatorname{cd}(g)-1}\deg(\beta_{P_{g}}^{r-\operatorname{cd}(g)})$
$\displaystyle=(-1)^{\operatorname{cd}(g)-1}\deg(\beta_{P_{g}}^{r-\operatorname{cd}(g)}),$

where in the last equality we used the identity

$\sum_{k=\operatorname{cd}(g)}^{r}(-1)^{k}\binom{r}{k}\binom{k-1}{\operatorname{cd}(g)-1}=(-1)^{\operatorname{cd}(g)}$

which follows from [GKP94, eq. 5.24] with $l=r$, $m=0$,
$n=\operatorname{cd}(g)-1$, and $s=-1$. ∎

######  Lemma 6.2.

For every polymatroid $P$ with poset of flats $L$ and
$r=\operatorname{cd}(E)-1$ with $\operatorname{cd}(E)>0$ we have

$\deg(\beta_{P}^{r})=(-1)^{\operatorname{cd}(E)}\mu_{L}(\hat{0},\hat{1}).$

###### Proof.

It is known that
$\mu_{L}(\hat{0},\hat{1})=\tilde{\chi}(\Delta(\hat{0},\hat{1}))$, i.e. the
Möbius function coincides with the reduced Euler characteristic of the order
complex of the poset $L\setminus\\{\hat{0},\hat{1}\\}$ (e.g. see [Rot64]). Let
$L^{\operatorname{op}}$ be the opposite (dual) lattice of $L$ which is defined
on the same set of $L$ but with reversed order, i.e., $x\leq y$ in
$L^{\operatorname{op}}$ if and only if $y\leq x$ in $L$. Since the order
complexes of $L$ and $L^{\operatorname{op}}$ are the same simplicial complex,
we have

$\mu_{L}(\hat{0},\hat{1})=\mu_{L^{\operatorname{op}}}(\hat{0},\hat{1}).$

Define $\deg(\beta_{P}^{0})=1$ for rank zero polymatroids $P$. Therefore, the
functions $(-1)^{\operatorname{cd}(E)}\deg(\beta_{P}^{r})$ and
$\mu_{L^{\operatorname{op}}}(\hat{0},\hat{1})$ satisfy the same recurrence
relation. One is given by the definition of
$\mu_{L^{\operatorname{op}}}(\hat{0},\hat{1})$ and the other by Lemma 6.1.
This concludes the proof. ∎

Notice that if $L$ is a geometric lattice (i.e. the poset of flats of a
matroid), then the Möbius function has alternating sign, hence in this case
$\deg(\beta^{r})\in\mathbb{N}_{0}$.

######  Definition 6.3.

The characteristic polynomial of a polymatroid $P$ is

$\chi_{P}(\lambda)=\sum_{g\in L}\mu_{L}(\hat{0},g)\lambda^{\dim(g)},$

where $\dim(g)=\operatorname{cd}(\hat{1})-\operatorname{cd}(g)$. Since
$\chi_{P}(1)=0$ by the definition of Möbius function, we define the reduced
characteristic polynomial as

$\overline{\chi}_{P}(\lambda)=\frac{\chi_{P}(\lambda)}{\lambda-1}.$

This definition of reduced characteristic polynomial coincides with the one in
the introduction eq. (1), as stated in [Whi93].

######  Theorem 6.4.

For every polymatroid $P$, we have

$\overline{\chi}_{P}(\lambda)=\sum_{i=0}^{r}(-1)^{i}\deg_{P}(\alpha_{P}^{i}\beta_{P}^{r-i})\lambda^{i}$

where $r=\operatorname{cd}(E)-1$.

###### Proof.

We show that $\overline{\chi}_{P}(\lambda)$ and the right hand side satisfy
the same recurrence:

$q_{P}(\lambda)-\lambda
q_{\operatorname{tr}_{\hat{1}}P}(\lambda)=-\mu_{L}(\hat{0},\hat{1})$

where $L$ is the poset of flats of $P$.

Let $\operatorname{tr}_{\hat{1}}L$ be the poset of flats of
$\operatorname{tr}_{\hat{1}}P$ and notice that
$\mu_{L}(\hat{0},g)=\mu_{\operatorname{tr}_{\hat{1}}L}(\hat{0},g)$ for all $g$
such that $\dim(g)>1$. Therefore
$\chi_{P}(\lambda)-\lambda\chi_{\operatorname{tr}P}(\lambda)$ is a polynomial
of degree one divisible by $\lambda-1$. Hence
$\overline{\chi}_{P}(\lambda)-\lambda\overline{\chi}_{\operatorname{tr}P}(\lambda)$
is constant and equal to $\overline{\chi}_{P}(0)=-\mu_{L}(\hat{0},\hat{1})$.
This proves that $\overline{\chi}_{P}(\lambda)$ satisfies the recurrence.

Now observe that for $i>0$
$\deg_{P}(\alpha_{P}^{i}\beta_{P}^{r-i})=\deg_{\operatorname{tr}P}(\alpha_{\operatorname{tr}P}^{i-1}\beta_{\operatorname{tr}P}^{r-i})$
by Lemma 4.11. This proves that

$\begin{split}\sum_{i=0}^{r}(-1)^{i}\deg_{P}(\alpha_{P}^{i}\beta_{P}^{r-i})\lambda^{i}-\lambda\sum_{i=0}^{r-1}(-1)^{i}\deg_{\operatorname{tr}P}(\alpha_{\operatorname{tr}P}^{i-1}\beta_{\operatorname{tr}P}^{r-i})\lambda^{i}=\\\
=(-1)^{r}\deg_{P}(\beta^{r}),\end{split}$

and so Lemma 6.2 proves the recurrence.

The base case $\operatorname{cd}(E)=1$ is trivial, so the proof follows. ∎

For an explicit example check Section 7.

######  Corollary 6.5.

The coefficient of $\lambda^{i}$ of the reduced characteristic polynomial
$\overline{\chi}_{P}(\lambda)$ is (up to the sign) the reduced Euler
characteristic of the order complex of the poset
$(\operatorname{tr}_{\hat{1}}^{i}L)\setminus\\{\hat{0},\hat{1}\\}$:

$[\lambda^{i}]\overline{\chi}_{P}(\lambda)=(-1)^{\operatorname{cd}(E)}\tilde{\chi}(\Delta((\operatorname{tr}_{\hat{1}}^{i}L)\setminus\\{\hat{0},\hat{1}\\})).$

###### Proof.

It follows from Theorem 6.4 and Lemma 6.2. ∎

######  Remark 6.6.

The coefficients of the characteristic polynomial $\chi_{P}$ and of the
reduced characteristic polynomial $\overline{\chi}_{P}$ do not form a log-
concave sequence. Indeed if $P_{1}$ is the polymatroid associated to $4$
subspaces of codimension $2,3,4,4$ in $\mathbb{C}^{5}$ in general position,
then

$\chi_{P_{1}}(\lambda)=\lambda^{5}-\lambda^{3}-\lambda^{2}-2\lambda+3,$

which is not log-concave. Let $P_{2}$ be the polymatroid on
$E=\\{a,b,c,d,e\\}$ with rank defined by $\operatorname{cd}(a)=2$,
$\operatorname{cd}(b)=3$, $\operatorname{cd}(c)=4$, $\operatorname{cd}(d)=4$,
$\operatorname{cd}(e)=1$, by $\operatorname{cd}(A)=6$ if $\lvert A\rvert\geq
3$ and
$\operatorname{cd}(\\{x,y\\})=\min\\{5,\operatorname{cd}(x)+\operatorname{cd}(y)\\}$.
The reduced characteristic polynomial is not log-concave because

$\overline{\chi}_{P_{2}}(\lambda)=\lambda^{5}-\lambda^{3}-\lambda^{2}-2\lambda+6.$

##  7\. An example

Let $E=\\{a,b,c\\}$ and $\operatorname{cd}\colon 2^{E}\to\mathbb{N}$ the
function defined by

$\displaystyle\operatorname{cd}(a)=\operatorname{cd}(b)=2,\quad\quad\operatorname{cd}(ab)=\operatorname{cd}(c)=4,$
$\displaystyle\operatorname{cd}(ac)=\operatorname{cd}(bc)=\operatorname{cd}(abc)=5.$

This function defines a polymatroid $P$ with poset of flats $L$ shown in
Figure 1. Near every cover relation, the relative codimension of the two flats
is shown.

$\hat{1}$$\hat{0}$$a$$b$$c$$ab$$\scriptstyle 2$$\scriptstyle 2$$\scriptstyle
2$$\scriptstyle 2$$\scriptstyle 1$$\scriptstyle 4$$\scriptstyle 1$ Figure 1.
The Hasse diagram of the poset of flats $L$ of Section 7

This polymatroid is realizable: a realization is the collection in
$\mathbb{C}^{5}$ of two subspace of dimension $3$ and a line in general
position.

Consider the (minimal) building set $\mathcal{G}=\\{a,b,c,\hat{1}\\}$; the
nested set complex $n(P,\mathcal{G})$ is shown in Figure 2.

$\hat{1}$$a$$b$$c$ Figure 2. The nested set complex $n(P,\mathcal{G})$.

The algebra $B(P,\mathcal{G})$ is generated by
$x_{a},x_{b},x_{c},x_{\hat{1}},e_{a},e_{b},e_{c},e_{\hat{1}}$ with relations:

$\displaystyle e_{a}e_{c}=e_{b}e_{c}=0$ $\displaystyle
x_{a}x_{c}=x_{b}x_{c}=0$ $\displaystyle x_{a}e_{c}=x_{b}e_{c}=0$
$\displaystyle e_{a}x_{c}=e_{b}x_{c}=0$
$\displaystyle(x_{a}+x_{\hat{1}})^{2}=(x_{b}+x_{\hat{1}})^{2}=0$
$\displaystyle(x_{c}+x_{\hat{1}})^{4}=0$ $\displaystyle x_{\hat{1}}^{5}=0$
$\displaystyle x_{c}x_{\hat{1}}=e_{c}x_{\hat{1}}=0$ $\displaystyle
x_{a}x_{\hat{1}}^{3}=e_{a}x_{\hat{1}}^{3}=0$ $\displaystyle
x_{b}x_{\hat{1}}^{3}=e_{b}x_{\hat{1}}^{3}=0$ $\displaystyle
x_{a}x_{b}x_{\hat{1}}=e_{a}x_{b}x_{\hat{1}}=0$ $\displaystyle
e_{a}e_{b}x_{\hat{1}}=x_{a}e_{b}x_{\hat{1}}=0$

The homogeneous component $B^{4,1}(P,\mathcal{G})$ has dimension $12$ and the
additive basis provided by Corollary 2.8 is:

$\displaystyle
e_{\hat{1}}x_{a}x_{b},\,e_{\hat{1}}x_{a}x_{\hat{1}},\,e_{\hat{1}}x_{b}x_{\hat{1}},\,e_{\hat{1}}x_{c}^{2},\,e_{\hat{1}}x_{\hat{1}}^{2},$
$\displaystyle
e_{a}x_{a}x_{b},\,e_{a}x_{a}x_{\hat{1}},\,e_{a}x_{\hat{1}}^{2},\,e_{b}x_{a}x_{b},\,e_{b}x_{a}x_{\hat{1}},\,e_{b}x_{\hat{1}}^{2},\,e_{c}x_{c}^{2}.$

2 | 1 | 2 | 1 |  |
---|---|---|---|---|---
1 | 3 | 7 | 7 | 3 |
0 | 1 | 4 | 5 | 4 | 1
| 0 | 1 | 2 | 3 | 4
Table 1. The dimensions of $B^{2p,q}(P,\mathcal{G})/(e_{1})$ in position $(p,q).$ 3 | 1 | 2 | 1 |  |
---|---|---|---|---|---
2 | 4 | 9 | 8 | 3 |
1 | 4 | 11 | 12 | 7 | 1
0 | 1 | 4 | 5 | 4 | 1
| 0 | 1 | 2 | 3 | 4
Table 2. The dimensions of $B^{2p,q}(P,\mathcal{G})$ in position $(p,q).$

Notice that $B(P,\mathcal{G})=B(P,\mathcal{G})/(e_{\hat{1}})\otimes\langle
1,e_{\hat{1}}\rangle$ and their dimensions are reported in Tables 1 and 2.

The other presentation of $B(P,\mathcal{G})$ is given by generators
$\sigma_{a},\sigma_{b},\sigma_{c},\sigma_{\hat{1}}$,
$\tau_{a},\tau_{b},\tau_{c},\tau_{\hat{1}}$ and relations:

$\displaystyle(\tau_{a}-\tau_{\hat{1}})(\tau_{c}-\tau_{\hat{1}})=0$
$\displaystyle(\tau_{b}-\tau_{\hat{1}})(\tau_{c}-\tau_{\hat{1}})=0$
$\displaystyle\sigma_{c}^{4}=0$
$\displaystyle(\sigma_{a}-\sigma_{\hat{1}})(\sigma_{c}-\sigma_{\hat{1}})=0$
$\displaystyle(\sigma_{b}-\sigma_{\hat{1}})(\sigma_{c}-\sigma_{\hat{1}})=0$
$\displaystyle\sigma_{a}^{2}=0$
$\displaystyle(\sigma_{a}-\sigma_{\hat{1}})(\tau_{c}-\tau_{\hat{1}})=0$
$\displaystyle(\sigma_{b}-\sigma_{\hat{1}})(\tau_{c}-\tau_{\hat{1}})=0$
$\displaystyle\sigma_{b}^{2}=0$
$\displaystyle(\tau_{a}-\tau_{\hat{1}})(\sigma_{c}-\sigma_{\hat{1}})=0$
$\displaystyle(\tau_{b}-\tau_{\hat{1}})(\sigma_{c}-\sigma_{\hat{1}})=0$
$\displaystyle\sigma_{\hat{1}}^{5}=0$
$\displaystyle(\sigma_{c}-\sigma_{\hat{1}})\sigma_{\hat{1}}=0$
$\displaystyle(\tau_{c}-\tau_{\hat{1}})\sigma_{\hat{1}}=0$
$\displaystyle(\sigma_{a}-\sigma_{\hat{1}})\sigma_{\hat{1}}^{3}=0$
$\displaystyle(\tau_{a}-\tau_{\hat{1}})\sigma_{\hat{1}}^{3}=0$
$\displaystyle(\sigma_{b}-\sigma_{\hat{1}})\sigma_{\hat{1}}^{3}=0$
$\displaystyle(\tau_{b}-\tau_{\hat{1}})\sigma_{\hat{1}}^{3}=0$
$\displaystyle(\sigma_{a}-\sigma_{\hat{1}})(\sigma_{b}-\sigma_{\hat{1}})\sigma_{\hat{1}}=0$
$\displaystyle(\tau_{a}-\tau_{\hat{1}})(\sigma_{b}-\sigma_{\hat{1}})\sigma_{\hat{1}}=0$
$\displaystyle(\tau_{a}-\tau_{\hat{1}})(\tau_{b}-\tau_{\hat{1}})\sigma_{\hat{1}}=0$
$\displaystyle(\sigma_{a}-\sigma_{\hat{1}})(\tau_{b}-\tau_{\hat{1}})\sigma_{\hat{1}}=0$

The homogeneous component $B^{4,1}(P,\mathcal{G})$ has dimension $12$ and the
additive basis provided by Corollary 2.12 is:

$\displaystyle\tau_{\hat{1}}\sigma_{a}\sigma_{b},\,\tau_{\hat{1}}\sigma_{a}\sigma_{\hat{1}},\,\tau_{\hat{1}}\sigma_{b}\sigma_{\hat{1}},\,\tau_{\hat{1}}\sigma_{c}^{2},\,\tau_{\hat{1}}\sigma_{\hat{1}}^{2},$
$\displaystyle\tau_{a}\sigma_{a}\sigma_{b},\,\tau_{a}\sigma_{a}\sigma_{\hat{1}},\,\tau_{a}\sigma_{\hat{1}}^{2},\,\tau_{b}\sigma_{a}\sigma_{b},\,\tau_{b}\sigma_{a}\sigma_{\hat{1}},\,\tau_{b}\sigma_{\hat{1}}^{2},\,\tau_{c}\sigma_{c}^{2}.$

The set of critical monomials is:

$\displaystyle
1,\tau_{a}\sigma_{a},\,\tau_{b}\sigma_{b},\,\tau_{c}\sigma_{c}^{3},\,\tau_{\hat{1}}\sigma_{\hat{1}}^{4},\,\tau_{a}\tau_{b}\sigma_{a}\sigma_{b},\,\tau_{a}\tau_{\hat{1}}\sigma_{a}\sigma_{\hat{1}}^{2},\,\tau_{b}\tau_{\hat{1}}\sigma_{b}\sigma_{\hat{1}}^{2},\tau_{c}\tau_{\hat{1}}\sigma_{c}^{3},\,\tau_{a}\tau_{b}\tau_{\hat{1}}\sigma_{a}\sigma_{b},$

and the dimensions of $\operatorname{CM}^{2p,q}(P,\mathcal{G})$ are given in
Table 3. The rank of the cohomology group of
$(B(P,\mathcal{G}),\operatorname{d})$ are given in Table 4

3 | 0 | 0 | 1 |  |
---|---|---|---|---|---
2 | 0 | 0 | 1 | 3 |
1 | 0 | 2 | 0 | 1 | 1
0 | 1 | 0 | 0 | 0 | 0
| 0 | 1 | 2 | 3 | 4
Table 3. The dimensions of $\operatorname{CM}(P,\mathcal{G})$ in position $(p,q).$ 3 | 0 | 0 | 0 |  |
---|---|---|---|---|---
2 | 0 | 0 | 1 | 1 |
1 | 0 | 2 | 0 | 1 | 0
0 | 1 | 0 | 0 | 0 | 0
| 0 | 1 | 2 | 3 | 4
Table 4. The dimensions of $H^{2p,q}(B(P,\mathcal{G}),\operatorname{d})$ in
position $(p,q).$

As an example we have

$\displaystyle\operatorname{d}(c\mu(ab\hat{1}))$
$\displaystyle=\operatorname{d}(\tau_{a}\tau_{b}\tau_{\hat{1}}\sigma_{a}\sigma_{b})=\tau_{a}\tau_{b}\sigma_{\hat{1}}\sigma_{a}\sigma_{b}$
$\displaystyle=\tau_{b}\tau_{\hat{1}}\sigma_{b}\sigma_{\hat{1}}^{2}-\tau_{a}\tau_{\hat{1}}\sigma_{a}\sigma_{\hat{1}}^{2}=c\mu(b\hat{1})-c\mu(a\hat{1}),$

that coincides with $\operatorname{d}((a,b,\hat{1}))=(b,\hat{1})-(a,\hat{1})$
in the differential algebra $\operatorname{CM}(P,\mathcal{G})$. Moreover, in
$\operatorname{CM}(P,\mathcal{G})$ we have

$(a)\cdot(b)=\lambda(a,b)-\lambda(b,a)=(a,b),$

because $a\prec b$ and it corresponds to the equality

$c\mu(a)c\mu(b)=\tau_{a}\sigma_{a}\tau_{b}\sigma_{b}=\tau_{a}\tau_{b}\sigma_{a}\sigma_{b}=c\mu(ab).$

The posets related to $P$ and $a$ are shown in Figure 3. The polymatroids
$P(a)$ and $P\setminus a$ are equal by coincidence; see below for the poset
$P(a)$ relative to the maximal building set.

$\hat{1}$$ab$$\hat{0}$$a$$b$$\scriptstyle 2$$\scriptstyle 1$$\scriptstyle
2$$\scriptstyle 1$$\scriptstyle 1$ (a) The poset of flats of
$\operatorname{tr}_{a}P$.

$a$$\hat{0}$$\scriptstyle 2$ (b) The poset of flats of $P^{a}$.

$\hat{1}$$a$$ab$$\scriptstyle 1$$\scriptstyle 2$ (c) The poset of flats of
$P_{a}$.

$\hat{1}$$\hat{0}$$b$$c$$\scriptstyle 4$$\scriptstyle 1$$\scriptstyle
3$$\scriptstyle 2$ (d) The poset of flats of $P(a)$.

$\hat{1}$$\hat{0}$$b$$c$$\scriptstyle 4$$\scriptstyle 1$$\scriptstyle
3$$\scriptstyle 2$ (e) The poset of flats of $P\setminus a$.

Figure 3. The Hasse diagram of some posets related to $a$. The circled nodes
are in the corresponding building sets.

The $\sigma$-cone $\Sigma_{P,\mathcal{G}}$ is given by the linear combinations
$-d_{a}\sigma_{a}-d_{b}\sigma_{b}-d_{c}\sigma_{c}-d_{\hat{1}}\sigma_{\hat{1}}$
with positive coefficients $d_{g}>0$.

We have
$\operatorname{Ann}(x_{a})=(x_{c},x_{b}\sigma_{\hat{1}},\sigma_{\hat{1}}^{3})$
and so in $\operatorname{DP}(P,\mathcal{G})/\operatorname{Ann}(x_{a})$ we have
$\sigma_{c}=\sigma_{\hat{1}}$,
$(\sigma_{b}-\sigma_{\hat{1}})\sigma_{\hat{1}}=0$, and
$\sigma_{\hat{1}}^{3}=0$. The last two equations correspond to the defining
relation for $\operatorname{DP}(P_{a},\mathcal{G}_{a})$. Similarly,
$\operatorname{Ann}(\sigma_{a})=(\sigma_{c}-\sigma_{\hat{1}},\sigma_{a},\sigma_{\hat{1}}^{4},(\sigma_{b}-\sigma_{\hat{1}})\sigma_{\hat{1}}^{2})$
and these are exactly the equations defining
$\operatorname{DP}(\operatorname{tr}_{a}P,\operatorname{tr}_{a}\mathcal{G})$
that do not appear in $\operatorname{DP}(P,\mathcal{G})$.

The relative Lefschetz decomposition with respect to the atom $a$ is

$\operatorname{DP}(P,\mathcal{G})=\operatorname{DP}_{(a)}\oplus
x_{a}\operatorname{DP}_{(a)},$

where

$\operatorname{DP}_{(a)}=\langle
1,\sigma_{b},\sigma_{c},\sigma_{\hat{1}},\sigma_{b}\sigma_{\hat{1}},\sigma_{c}^{2},\sigma_{\hat{1}}^{2},\sigma_{b}\sigma_{\hat{1}}^{2},\sigma_{c}^{3},\sigma_{\hat{1}}^{3},\sigma_{\hat{1}}^{4},\rangle$

and

$\operatorname{DP}_{a}=x_{a}\operatorname{DP}_{(a)}=\langle
x_{a},x_{a}\sigma_{b},x_{a}\sigma_{\hat{1}},x_{a}\sigma_{\hat{1}}^{2}\rangle\simeq\operatorname{DP}(P_{a},\mathcal{G}_{a})[1].$

The relative Lefschetz decomposition with respect to the atom $c$ is

$\operatorname{DP}(P,\mathcal{G})=\operatorname{DP}_{(c)}\oplus\operatorname{DP}_{\hat{1}}\oplus\operatorname{DP}_{c},$

where $\operatorname{DP}_{(c)}=\langle
1,\sigma_{a},\sigma_{b},\sigma_{a}\sigma_{b}\rangle$ and the other
$\operatorname{DP}_{(c)}$-modules are $\operatorname{DP}_{c}=\langle
x_{c},x_{c}^{2},x_{c}^{3}\rangle$ and
$\operatorname{DP}_{\hat{1}}=x_{\hat{1}}\operatorname{DP}_{(c)}\oplus
x_{\hat{1}}^{2}\operatorname{DP}_{(c)}$. Moreover we have
$x_{\hat{1}}\operatorname{DP}_{(c)}[-1]\simeq
x_{\hat{1}}^{2}\operatorname{DP}_{(c)}[-2]\simeq\operatorname{DP}((P\setminus
c)^{ab},(\mathcal{G}\setminus c)^{ab})$.

### Maximal building set

Now consider the same polymatroid $P$ with the maximal building set
$\mathcal{G}_{\max}=\\{a,b,c,ab,\hat{1}\\}$. The polymatroid $P(a)$ relative
to the maximal building set is shown in Figure 4 and the groundset $E(a)$ is
$\\{b,c,ab\\}$. This polymatroid $P(a)$ associated with $\mathcal{G}_{\max}$
is different from the polymatroid $P(a)$ defined from the minimal building set
$\mathcal{G}$ (shown in Figure 3(d)).

$\hat{1}$$\hat{0}$$b$$c$$\\{b,ab\\}$$\scriptstyle 2$$\scriptstyle
2$$\scriptstyle 1$$\scriptstyle 4$$\scriptstyle 1$ Figure 4. The Hasse diagram
of the poset of flats of $P(a)$ with maximal building set on the groundset
$\\{b,c,ab\\}$.

The characteristic polynomial is
$\chi_{P}(\lambda)=\lambda^{5}-2\lambda^{3}+1$ and the reduced one is

$\overline{\chi}_{P}(\lambda)=\lambda^{4}+\lambda^{3}-\lambda^{2}-\lambda-1.$

We have $\alpha=-x_{\hat{1}}$, $\beta=x_{a}+x_{b}+x_{c}+x_{ab}+x_{\hat{1}}$
and $\deg(\alpha^{4})=1$, $\deg(\alpha^{3}\beta)=-1$,
$\deg(\alpha^{2}\beta^{2})=-1$, $\deg(\alpha\beta^{3})=1$, and
$\deg(\beta^{4})=-1$.

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