††footnotetext:  CS was partially supported by DFG grant SE 1114/5-2. UW was
supported in part by NSF Grant DMS-2100288 and by Simons Foundation
Collaboration Grant for Mathematicians #580839. UW was also supported by NSF
Grant DMS-1928930, while in residence at the Simons Laufer Mathematical
Sciences Institute (formerly MSRI) in Berkeley, California, during the Spring
2024 semester.
2020 _Mathematics Subject Classification._ 32C38, 14F10, 32S40
Keywords: GKZ-System, Euler–Koszul homology

# Hypergeometric systems from groups with torsion

Thomas Reichelt, Christian Sevenheck, and Uli Walther

###### Abstract

We consider $A$-hypergeometric (or GKZ-)systems in the case where the grading
(character) group is an arbitrary finitely generated Abelian group. Emulating
the approach taken for classical GKZ-systems in [MMW05] that allows for a
coefficient module, we show that these $D$-modules are holonomic systems. For
this purpose we formulate an Euler–Koszul complex in this context, built on an
extension of the category of $A$-toric modules. We derive that these new
systems are regular holonomic under circumstances that are similar to those
that lead to regular holonomic classical GKZ-systems.

For the appropriate coefficient module, our $D$-modules specialize to the
”better behaved GKZ-systems” introduced by Borisov and Horja. We certify the
corresponding $D$-modules as regular holonomic, and establish a holonomic
duality on the level of $D$-modules that was suggested on the level of
solutions by Borisov and Horja and later shown by Borisov and Han in a special
situation, [BH15, BH23].

###### Contents

  1. 1 Introduction
  2. 2 The $\mathcal{A}$-graded Category, Toric and Ttoric Modules
    1. 2.1 General Setup
    2. 2.2 Binomial Ideals
    3. 2.3 Toric Modules
  3. 3 Euler–Koszul Technology
  4. 4 $D$-modules from Modules over Semigroups
    1. 4.1 Semigroup Modules
    2. 4.2 Rank and Duality
  5. 5 Applications

##  1 Introduction

In a series of papers ([BH06, BPH13, BH15, BHW21, BH23]), Borisov together
with various collaborators has introduced a system of linear partial
differential equations called the _better behaved GKZ-system_. It has
considerable similarities with the members of a class of $D$-modules that
generalize the widely studied GKZ-systems, which themselves are far-reaching
generalizations of classical hypergeometric differential equations. We refer
to our survey [RSSW21] for a comprehensive overview on GKZ-systems.

A GKZ-system arises from a finite set of elements in a finite-dimensional
lattice (the character group), together with a complex parameter vector. In a
more general setup, an additional component enters in the form of a
coefficient module as introduced in [MMW05]. The systems studied by Borisov
_et al._ are special cases in a more general setup. The generalization arises
from replacing the lattice by an arbitrary finitely generated Abelian group.
This introduction of possible torsion creates interesting new phenomena: for
example, the associated toric ideal is now replaced by a smaller ideal that is
no longer prime. Within this more general situation, Borisov and his
collaborators consider two specific systems. One arises from taking the
coefficient module to be the normalization of the associated toric ring, while
the other comes from the canonical module of this normalization. In the
classical (torsion-free) situation, these choices for the coefficient module
assure that the rank of these systems is independent of the parameter vector
(which otherwise is not always true, even in the classical case without
coefficient module). This absence of rank jumps is why Borisov called these
system “better behaved”.

In the present paper, we study these systems in general: we allow torsion and
arbitrary coefficent modules. We begin with a study of the ideals that replace
of toric ideals, and in particular prove that they are still ”reasonably
close” to the toric ideal. This is necessary to bring in our main tool, the
_Euler–Koszul complex_ from [MMW05]. We show that these new systems are the
terminal homology group of an appropriate Euler–Koszul complex, but in order
to allow for torsion in the extended character group, we must extend the
category of toric modules (which serve as coefficient modules) from [MMW05] to
the category of _twisted toric module_ , or _ttoric modules_ for short, see
Subsection 2.3.

For this extended category we show that the homology groups of the
Euler–Koszul complex are holonomic, and we establish a criterion in terms of
the coefficient module and the parameter vector to determine (non-)vanishing.
As a consequence, we infer that all GKZ-systems with finitely generated
Abelian character group and with arbitrary coefficient module are holonomic,
and we characterize when those attached to the natural coefficient module are
regular holonomic. These statements apply in particular to Borisov’s systems.

Following the trail of ideas of [MMW05], we then investigate the behavior of
these systems under holonomic duality. Euler–Koszul functors are self-dual up
to a shift in the parameter vector, and thus holonomic duality reduces to
standard duality over the underlying ttoric ring. This allows us to give a
compact and conceptional proof of the duality statement in [BH23] regarding
the systems considered in [BPH13], but for an arbitrary character group
(allowing torsion) and without the assumption of normality of the underlying
toric ring.

##  2 The $\mathcal{A}$-graded Category, Toric and Ttoric Modules

###  2.1 General Setup

For any set $S$, and any ring ${\mathbb{K}}$ we write ${\mathbb{K}}^{S}$ for
the free ${\mathbb{K}}$-module $\bigoplus_{s\in S}{\mathbb{K}}$.

Let

$N=F\oplus{\mathbb{Z}}^{d}$

be a finitely generated Abelian group where $F$ is the torsion part of $N$.
The lattice

$N^{\vee}:=\operatorname{\textup{Hom}}_{\mathbb{Z}}(N,{\mathbb{Z}})$

embeds naturally into
$\operatorname{\textup{Hom}}_{\mathbb{Z}}(N,{\mathbb{C}})\simeq
N^{\vee}\otimes_{\mathbb{Z}}{\mathbb{C}}\simeq{\mathbb{C}}^{d}$. As in [BH06],
let

$\mathcal{A}=\\{{\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{n}\\}\subseteq N$

be a finite multi-set of elements of $N$ (_i.e._ , possibly with repetitions).
We set

$\ell:=|F|$

to be the torsion index, and choose

$\beta\in N\otimes_{\mathbb{Z}}{\mathbb{C}}.$

There is a morphism

$\displaystyle\pi:N$ $\displaystyle\rightarrow$ $\displaystyle
N/F=:\bar{N}={\mathbb{Z}}^{d},$ $\displaystyle{\mathbf{u}}$
$\displaystyle\mapsto$ $\displaystyle\bar{\mathbf{u}},$

and whenever it suits us, we identify $\pi(N)$ with its image under the
inclusion

$\pi(N)\hookrightarrow\pi(N)\otimes_{\mathbb{Z}}{\mathbb{R}}=:N_{\mathbb{R}}.$

We can identify non-canonically

$\bar{N}=\bigoplus_{1}^{d}{\mathbb{Z}}\cdot{\boldsymbol{\varepsilon}}_{i}$

where ${\boldsymbol{\varepsilon}}_{1},\ldots,{\boldsymbol{\varepsilon}}_{d}$
is a chosen basis for $N$. Denote by

$K_{\mathbb{R}}:={\mathbb{R}}_{\geq 0}\cdot\pi(\mathcal{A})\subseteq
N_{\mathbb{R}}$

the rational polyhedral cone generated by the images $\pi({\mathbf{a}}_{j})\in
N_{\mathbb{R}}$.

######  Hypothesis 2.0.

We shall work throughout under the following assumptions:

  1. 1.

The set $A:=\\{\pi({\mathbf{a}}_{j})\mid 1\leq j\leq n\\}$ spans the lattice
$\bar{N}=\pi(N)$ as a ${\mathbb{Z}}$-module; this implies in particular that
the index

$\delta:=[N:{\mathbb{Z}}\mathcal{A}]$

is a (possibly strict) divisor of $\ell=|F|$.

  2. 2.

The cone $K_{\mathbb{R}}$ is _pointed_ : the origin is the only unit of the
additive semigroup inherited from ${\mathbb{R}}^{d}$ on $K_{\mathbb{R}}$.

Among the set of all vector space morphisms $\\{\tau\colon
N_{\mathbb{R}}\to{\mathbb{R}}\\}$ choose a minimal set $\mathcal{F}_{K}$
defined such that

$K_{\mathbb{R}}=\bigcap_{\tau\in\mathcal{F}_{K}}(\tau^{-1}([0,\infty))$

(minimality means that the intersection is irredundant) and set

$K:=\pi^{-1}(K_{\mathbb{R}})\subseteq N,$ (2.0)

the preimage under $\pi$. Then $\mathcal{F}_{K}$ is finite, and
$\\{K_{\mathbb{R}}\cap\tau^{-1}(0)\\}_{\tau\in\mathcal{F}_{K}}$ are the
_facets of $\mathcal{A}$_. As is traditional, we identify a facet $\tau$ with
the intersection $\tau\cap\pi(A)$ as well as with the semigroup spanned by
this intersection, or with $\tau\cap K$.

######  Notation 2.0.

We will further denote by

$A:=\pi(\mathcal{A}),$

the finite multi-set consisting of the images in $\bar{N}={\mathbb{Z}}^{d}$ of
the elements of $\mathcal{A}$ under the morphism $\pi$. Fixing a lattice basis
$\\{{\mathbf{e}}_{1},\ldots,{\mathbf{e}}_{n}\\}$ for

$L:=\bigoplus{\mathbb{Z}}\cdot{\mathbf{e}}_{i}\simeq{\mathbb{Z}}^{\mathcal{A}}$

we will always allow ourselves to view $A$ as an integer $d\times n$ matrix,
with columns $\\{\bar{\mathbf{a}}_{j}\\}$ and entries $(a_{i,j})$; as $A$ is a
multi-set, this matrix might have repeated columns.

We define $N$\- resp. $\bar{N}$-graded polynomial rings

$R_{\mathcal{A}}:={\mathbb{C}}[\\{\partial_{j}\mid{\mathbf{a}}_{j}\in\mathcal{A}\\}],\qquad
R_{A}={\mathbb{C}}[\\{\partial_{j}\mid\bar{\mathbf{a}}_{j}\in\ A\\}],$

by grading them according to

$\deg_{\mathcal{A}}(\partial_{j})={\mathbf{a}}_{j},\qquad\deg_{A}(\partial_{j})=\bar{\mathbf{a}}_{j}.$

(If $\mathcal{A}$ is truly a multi-set, so
${\mathbf{a}}_{j}={\mathbf{a}}_{j^{\prime}}$ for some $j,j^{\prime}$ with
$j\neq j^{\prime}$ then $R_{\mathcal{A}}$ and $R_{A}$ have one generator each
for both copies).

We further introduce the non-commutative _Weyl algebras_

$D_{\mathcal{A}}:=R_{\mathcal{A}}[\\{x_{j}\mid{\mathbf{a}}_{j}\in\mathcal{A}\\}],\qquad
D_{A}:=R_{A}[\\{x_{j}\mid\bar{\mathbf{a}}_{j}\in\ A\\}],$

that we grade by by setting $\deg_{(-)}(x_{j})=-\deg_{(-)}(\partial_{j})$.

In $R_{A}$ there is the _toric_ (prime) _ideal_

$I_{A}:=R_{A}\cdot\\{\partial^{\mathbf{u}}-\partial^{\mathbf{v}}\mid
A\cdot({\mathbf{u}}-{\mathbf{v}})=0\\}.$

In analogy (by replacing $A$ with $\mathcal{A}$), we define the
$R_{\mathcal{A}}$-ideal

$I_{\mathcal{A}}:=R_{\mathcal{A}}\cdot\\{\partial^{\mathbf{u}}-\partial^{\mathbf{v}}\mid\mathcal{A}\cdot({\mathbf{u}}-{\mathbf{v}})=0\\},$
(2.0)

which we also call _toric_ ; the dot product here expresses the process of
forming ${\mathbb{Z}}$-linear combinations (inside $N$) of elements of
$\mathcal{A}$.

The quotient rings

$S_{A}:=R_{A}/I_{A}\quad\text{ and }\quad
S_{\mathcal{A}}:=R_{\mathcal{A}}/I_{\mathcal{A}}$

are the semigroup rings $S_{A}={\mathbb{K}}[{\mathbb{N}}A]$ and
$S_{\mathcal{A}}={\mathbb{K}}[{\mathbb{N}}\mathcal{A}]$ respectively.

There are graded ring morphisms

$\operatorname{\textup{id}}_{R}(\pi)\colon R_{\mathcal{A}}\to
R_{A},\qquad\operatorname{\textup{id}}_{D}(\pi)\colon D_{\mathcal{A}}\to
D_{A},$

induced by the identity map on the (ungraded) underlying rings, and by the
morphism of grading groups $N\to\bar{N}$. They induce functors (both denoted
$\iota_{\pi}$) from the categories of graded $R_{\mathcal{A}}$\- and
$D_{\mathcal{A}}$-modules to those of graded $R_{A}$\- and $D_{A}$-modules
that are the identity map on the underlying modules. They are, however, not
equivalences of categories, unless $F$ is trivial.

######  Remark 2.0.

  1. 1.

Note that syzygies between elements of $\mathcal{A}$ are also syzygies between
the corresponding elements of $A$, and so
$\iota_{\pi}(I_{\mathcal{A}})\subseteq I_{A}$.

  2. 2.

The ideal $I_{\mathcal{A}}$ is, in contrast to $I_{A}$, usually not prime. For
example, if $\mathcal{A}=\\{(1\mod 4{\mathbb{Z}},1),(2\mod
4{\mathbb{Z}},2)\\}\subseteq
N:=({\mathbb{Z}}/4{\mathbb{Z}})\oplus{\mathbb{Z}}$, then $R_{\mathcal{A}}$ and
$R_{A}$ are ${\mathbb{C}}[\partial_{1},\partial_{2}]$,
$I_{\mathcal{A}}=R_{\mathcal{A}}\cdot(\partial_{1}^{4}-\partial_{2}^{2})$,
$I_{A}=R_{A}\cdot(\partial_{1}^{2}-\partial_{2})$, and
${I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]}\subseteq R_{\mathcal{A}}$ (defined
in the proof of Lemma 2.3 below) is
$R_{\mathcal{A}}\cdot(\partial_{1}^{8}-\partial_{2}^{4})$.

###  2.2 Binomial Ideals

We review some facts from [ES96]. Let

$\rho\colon L_{\rho}\to{\mathbb{C}}^{*}$

be a character on a sublattice $L_{\rho}$ of $L$. We say that
$L_{\rho}\hookrightarrow L$ is _saturated_ (and then call $\rho$ a _saturated
partial character_) if the group $L/L_{\rho}$ is torsion-free—or,
equivalently, if $L_{\rho}$ is a direct summand of $L$.

######  Remark 2.0.

Suppose $\rho^{\prime}$ is a saturated partial character with associated
sublattice $L^{\prime}$. Let $L^{\prime\prime}$ be a complement,
$L=L^{\prime}\oplus L^{\prime\prime}$. Expand $\rho^{\prime}$ to a character
$\rho$ on all of $L$ by setting $\rho(L^{\prime\prime})=1$. Then there is a
monomial isomorphism $\rho^{*}$ of $R_{A}$ given by
$\rho^{*}(\partial^{{\mathbf{u}}^{\prime}+{\mathbf{u}}^{\prime\prime}})=\rho({\mathbf{u}}^{\prime})\cdot\partial^{{\mathbf{u}}^{\prime}+{\mathbf{u}}^{\prime\prime}}$
for any ${\mathbf{u}}^{\prime}\in L^{\prime},{\mathbf{u}}^{\prime\prime}\in
L^{\prime\prime}$.

From any partial character $\rho$ we define a binomial ideal

$I_{+}(\rho):=R_{A}\cdot\\{\partial^{\mathbf{u}}-\rho({\mathbf{u}}-{\mathbf{v}})\partial^{\mathbf{v}}\mid{\mathbf{u}},{\mathbf{v}}\in{\mathbb{N}}^{n},{\mathbf{u}}-{\mathbf{v}}\in
L_{\rho}\\}.$

This ideal is prime if and only if $\rho$ is saturated.

Let now $\mathfrak{p}$ be an arbitrary binomial prime ideal in $R_{A}$. There
is an induced partition
$\\{\partial_{1},\ldots,\partial_{n}\\}=\\{y_{1},\ldots,y_{s}\\}\cup\\{z_{1},\ldots,z_{n-s}\\}$
where
$\\{y_{1},\ldots,y_{s}\\}=\mathfrak{p}\cap\\{\partial_{1},\ldots,\partial_{n}\\}$
are the variables inside $\mathfrak{p}$. (Note that this perhaps not the
entire linear part of $\mathfrak{p}$; there might be linear binomials in
$\mathfrak{p}$). There is a corresponding splitting of lattices
${\mathbb{Z}}^{n}={\mathbb{Z}}^{s}\times{\mathbb{Z}}^{n-s}$. By [ES96, Cor.
2.4], $\mathfrak{p}$ has the form

$\mathfrak{p}=R_{A}\cdot\\{y_{1},\ldots,y_{s}\\}+I_{+}(\rho)$

for some partial saturated character $\rho$ in the lattice
${\mathbb{Z}}^{n-s}$.

A special case is when $L_{\rho}=\ker(A)$ and $\rho$ is the trivial character;
this results in $I_{+}(\rho)=I_{A}$. More generally, if $L_{\rho}=\ker(A)$ and
$\rho$ is an arbitrary character, we denote the prime ideal $I_{+}(\rho)$ by
$I_{A,\rho}$. Note that $I_{A,\rho}$ relates to $I_{A}$ via the
$R_{A}$-isomorphism from Remark 2.2.

For a face $\tau$ of the cone $K_{\mathbb{R}}={\mathbb{R}}_{\geq 0}A$, denote
by $I^{\tau}_{A}$ the prime ideal defining the toric ring of $\tau$ as a
quotient of $R_{A}$,

$R_{A}/I^{\tau}_{A}\simeq{\mathbb{C}}[\tau\cap{\mathbb{N}}A].$

Then $I^{\tau}_{A}$ is the ideal sum of $I_{A}$ with the ideal generated by
all $\partial_{j}$ with $\bar{\mathbf{a}}_{j}\not\in\tau$.

For a partial character $\rho$ with $L_{\rho}=\ker(A)$ we define similarly

$I^{\tau}_{A,\rho}:=R_{A}\cdot(\\{\partial_{j}\mid\bar{\mathbf{a}}_{j}\not\in\tau\\}\cup\underbrace{I_{+}(\rho)}_{=I_{A,\rho}}).$

We note that $I^{\tau}_{A,\rho}$ is a prime ideal. Indeed,
$R_{A}/I^{\tau}_{A,\rho}=R_{B}/I_{B,\rho^{\prime}}$ where $B=A\cap\tau$
corresponds to the variables that are not generators of $I^{\tau}_{A,\rho}$,
and where $\rho^{\prime}$ is the restriction of $\rho$ to ${\mathbb{Z}}B$.

We are going to need a statement of the following form.

######  Lemma 2.0.

If $\mathfrak{p}$ is a prime ideal in $R_{A}$ that is $A$-homogeneous,
contains $I_{A,\rho}$ for some character $\rho$ on ${\mathbb{Z}}A$, then it is
of the form $I^{\tau}_{A,\rho}$.

###### Proof.

In the $A$-graded module
$R_{A}/\mathfrak{p}=\bigoplus_{{\mathbf{u}}\in{\mathbb{Z}}A}(R_{A}/\mathfrak{p})_{\mathbf{u}}$,
each graded component is either zero or a 1-dimensional vector space. Indeed,
already the relations coming from $I_{A,\rho}$ identify (up to nonzero
scalars) monomials of $R_{A}$ if they have equal $A$-degree. Hence, any
$A$-homogeneous element of $\mathfrak{p}/I_{A,\rho}$ is the coset of a
monomial in $R_{A}$. But $\mathfrak{p}$ being prime implies that
$\mathfrak{p}$ contains a variable appearing in such monomial. Let
$V_{\mathfrak{p}}$ be the ideal generated by the variables that are in
$\mathfrak{p}$ and let $\tau$ be the face of $A$ defined by $[\partial_{j}\in
V_{\mathfrak{p}}]\Leftrightarrow[{\mathbf{a}}_{j}\not\in\tau]$; then
$\mathfrak{p}=I^{\tau}_{A,\rho}$. ∎

###  2.3 Toric Modules

We recall from [MMW05] the notion of a _toric $A$-module_, as an $A$-graded
$R_{A}$-module that has a finite filtration by $A$-graded modules such that
each composition factor is graded isomorphic to a shifted copy of some
$R_{A}/I_{A}^{\tau}$ where $I_{A}^{\tau}$ is the toric ideal of one of the
faces of the cone spanned by $A$ (the $\tau$ depending on the composition
factor). We modify this concept as follows.

######  Definition 2.0.

An _$\mathcal{A}$ -toric module_ is an $\mathcal{A}$-graded
$R_{\mathcal{A}}$-module that has a finite filtration by $\mathcal{A}$-graded
modules such that each composition factor is an $\mathcal{A}$-graded quotient
of $R_{\mathcal{A}}/I_{\mathcal{A}}$. Note that $\mathcal{A}$-toric modules
are necessarily finitely generated.

######  Lemma 2.0.

For any $\mathcal{A}$-toric module $M$, the $A$-graded module $\iota_{\pi}(M)$
has a finite filtration by $A$-graded submodules such that each composition
factor is graded isomorphic to a shifted copy of some
$R_{A}/I^{\tau}_{A,\rho}$ for suitable $\rho,\tau$ depending on the
composition factor.

###### Proof.

Recall that $\ell=|F|$ is the torsion index, and introduce the
$R_{\mathcal{A}}$-ideal

${I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]}:=R_{\mathcal{A}}\cdot\\{\partial^{\ell{\mathbf{u}}}-\partial^{\ell{\mathbf{v}}}\mid
A\cdot{\mathbf{u}}=A\cdot{\mathbf{v}}\\}.$

Note that this is an $R_{\mathcal{A}}$-ideal but the conditions defining it
come from $A$. Since $\ell{\mathbf{a}}\in 0\oplus{\mathbb{Z}}^{d}\subseteq
F\oplus{\mathbb{Z}}^{d}$ for any ${\mathbf{a}}\in N$, it follows that if
${\mathbf{a}}$ is a linear relation on $A$ then $\ell{\mathbf{a}}$ is a linear
relation on $\mathcal{A}$. In particular,
${I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]}\subseteq I_{\mathcal{A}}$, and so
${I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]}$ is $\mathcal{A}$-graded and there
is an $A$-graded surjection
$\iota_{\pi}(R_{\mathcal{A}}/{I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})\twoheadrightarrow\iota_{\pi}(R_{\mathcal{A}}/I_{\mathcal{A}})$.

By definition, $\iota_{\pi}({I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$ is an
$A$-graded binomial ideal in a polynomial ring, and by [ES96] all its
associated prime ideals $\\{\mathfrak{p}_{k}\\}$ are $A$-graded binomial
primes.

We study now the binomial $A$-graded prime ideals $\mathfrak{p}\subseteq
R_{A}$ containing $\iota_{\pi}({I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$.
For any $\partial^{\mathbf{u}}-\partial^{\mathbf{v}}\in I_{A}$ we have
$\partial^{\ell{\mathbf{u}}}-\partial^{\ell{\mathbf{v}}}=\prod(\partial^{\mathbf{u}}-\zeta_{\ell}^{i}\partial^{\mathbf{v}})\in\iota_{\pi}({I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$,
where $\zeta_{\ell}$ is a primitive $\ell$-th root of unity, and where $i$
runs through the elements of ${\mathbb{Z}}/\ell{\mathbb{Z}}$. A prime ideal
$\mathfrak{p}$ containing these products will contain at least one factor from
each such product, and so for each binomial in $I_{A}$ such prime ideal
$\mathfrak{p}$ contains that same binomial twisted by a power of
$\zeta_{\ell}$. On the other hand, by [ES96], $\mathfrak{p}$ is the sum of a
binomial prime ideal $I_{+}(\rho)$ with an ideal $V_{\mathfrak{p}}$ of
variables, and we can assume that $I_{+}(\rho)$ contains no monomial of degree
one. Suppose one of the binomials
$\partial^{\mathbf{u}}-\zeta_{\ell}^{i}\partial^{\mathbf{v}}\in\mathfrak{p}$
is not in $R_{A}\cdot V_{\mathfrak{p}}.$ Then neither of its two monomials can
be in $\mathfrak{p}$ since else they would have to be in $V_{\mathfrak{p}}$ as
$\mathfrak{p}$ is prime. Thus, such binomial is in $I_{+}(\rho)$, which then
forces ${\mathbf{u}}-{\mathbf{v}}\in L_{\rho}$. But then
$\partial^{\mathbf{u}}-\zeta_{\ell}^{i}\partial^{\mathbf{v}}$ and
$\partial^{\mathbf{u}}-\rho({\mathbf{u}}-{\mathbf{v}})\partial^{\mathbf{v}}$
need to agree up to a constant factor, since both are in $\mathfrak{p}$ (and
hence so are their linear combinations). It follows that for each
${\mathbf{u}}-{\mathbf{v}}\in\ker(A)$, either both
$\partial^{\mathbf{u}},\partial^{\mathbf{v}}$ are in $V_{\mathfrak{p}}$, or
$I_{+}(\rho)$ contains a
$\partial^{\mathbf{u}}-\zeta_{\ell}^{i}\partial^{\mathbf{v}}$ with $i$
depending on ${\mathbf{u}},{\mathbf{v}}$.

Pick an extension $\rho^{\prime}$ of $\rho$ to ${\mathbb{Z}}^{n}$, which
exists since $\mathfrak{p}$ is prime and hence $\rho$ is saturated. Then the
monomial automorphism of $R_{A}$ from Remark 2.2 that sends
$\partial^{\mathbf{u}}$ to
$\rho^{\prime}({\mathbf{u}})\cdot\partial^{\mathbf{u}}$ sends
$V_{\mathfrak{p}}$ to itself and $V_{\mathfrak{p}}+I_{+}(\rho)$ to an
$A$-homogeneous prime ideal generated by variables and binomials containing
$I_{A}$. The only such ideals are, by Lemma 2.2 the ideals $I^{\tau}_{A}$, and
so $\mathfrak{p}=I^{\tau}_{A,\rho^{\prime\prime}}$ for some character
$\rho^{\prime\prime}$ on $L$, and some face $\tau$ of ${\mathbb{N}}A$. In
particular, the associated primes of
$\iota_{\pi}(R_{\mathcal{A}}/{I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$ are
all of the form $I^{\tau}_{A,\rho^{\prime\prime}}$ for suitable
$\tau,\rho^{\prime\prime}$.

Suppose now that $M$ is an arbitrary $\mathcal{A}$-toric module, and choose
one of its composition factors $\tilde{M}_{i}:=M_{i}/M_{i-1}$, an
$\mathcal{A}$-graded quotient of $R_{\mathcal{A}}/I_{\mathcal{A}}$. Then
$\iota_{\pi}(\tilde{M}_{i})$ is an $A$-graded quotient of
$\iota_{\pi}(R_{\mathcal{A}}/{I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$.

It is well-known that any finitely generated ($\bar{N}$-graded) module
$\tilde{M}$ over any ($\bar{N}$-graded) Noetherian ring $R$ permits a finite
filtration by ($\bar{N}$-graded) modules such that each composition factor is,
up to a shift in the grading, of the form $R/\mathfrak{p}$ where
$\mathfrak{p}$ is an associated ($\bar{N}$-graded) prime ideal of $\tilde{M}$
over $R$. We apply this to
$R=\iota_{\pi}(R_{\mathcal{A}}/{I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$ and
$\tilde{M}=\iota_{\pi}(\tilde{M}_{i})$. This means that
$\iota_{\pi}(\tilde{M}_{i})$ has an $A$-graded filtration such that its
composition factors $\tilde{M}_{i,j}$ are, up to a shift in the grading, of
the form $R_{A}/\mathfrak{p}_{i,j}$ where each $\mathfrak{p}_{i,j}$ is an
$A$-graded prime containing
$\iota_{\pi}({I_{\mathcal{A}}}^{\\!\\!\\!\\!\\![\ell]})$. By what we have
proved earlier, each $\mathfrak{p}_{i,j}$ is thus of the form
$I^{\tau}_{A,\rho}$ for some face of ${\mathbb{N}}A$ and some character on
$L$, and hence each $\bar{M}_{i,j}$ has the claimed property.

It is now standard to assemble a composition chain for $M$ from these
composition chains of its composition factors. ∎

Inspired be this lemma, we make the following definition.

######  Definition 2.0.

An $A$-graded $R_{A}$-module $M$ is _$A$ -twisted-toric_, or short just _$A$
-ttoric_, if it has a finite filtration such that each composition factor is
$A$-graded isomorphic to a shifted copy of $R_{A}/I^{\tau}_{A,\rho}$ for some
face $\tau$ of $A$ and some character $\rho$ on the (saturated) lattice
${\mathbb{R}}\tau\cap N$.

The image under $\iota_{\pi}$ of any $\mathcal{A}$-toric module $M$ is
$A$-ttoric by Lemma 2.3.

##  3 Euler–Koszul Technology

We review from [MMW05] the following concepts and constructions. If $m\in M$
is an $A$-homogeneous element of $A$-degree ${\mathbf{u}}=\sum
u_{i}{\boldsymbol{\varepsilon}}_{i}$ inside an $A$-graded module $M$, set
$\deg_{A,i}(m):=u_{i}$. For $1\leq i\leq d$, let

$E_{i}=\sum_{j=1}^{n}a_{i,j}x_{j}\partial_{j}\in D_{A}$

be the $i$-th _Euler operator_. Then $\deg_{A}(E_{i})=0$ and we have the
commutator relation

$[E_{i},P]=-\deg_{A,i}(P)\cdot P$

for all $A$-homogeneous $P\in D_{A}$.

For $M$ any $A$-graded $R_{A}$-module, and for all $\beta\in
L\otimes_{\mathbb{Z}}{\mathbb{C}}=\bigoplus_{1}^{d}{\mathbb{C}}\cdot{\boldsymbol{\varepsilon}}_{i}$,
one can define an endomorphism (denoted $(E_{i}-\beta)\bullet$) of left
$D_{A}$-modules on $D_{A}\otimes_{R_{A}}M$ by setting

$(E_{i}-\beta)\bullet(P\otimes
m):=(E_{i}+\deg_{A,i}(P)+\deg_{A,i}(m)-\beta_{i}))(P\otimes m).$

If $M=R_{A}$ with $A$-grading that places $1\in R_{A}$ into degree zero, this
endomorphism on $D_{A}\otimes_{R_{A}}R_{A}=D_{A}$ of left $D_{A}$-modules is
right-multiplication by $E_{i}-\beta_{i}$ on $D_{A}$ as one checks from the
commutator relations.

The endomorphisms $(E_{i}-\beta_{i})\bullet$ and
$(E_{i^{\prime}}-\beta_{i^{\prime}})\bullet$ commute for all $i,i^{\prime}$
and on all $A$-graded modules $D_{A}\otimes_{R_{A}}M$, and thus one can define
the Koszul complex on the endomorphisms

$\mathopen{}\mathclose{{}\left(\leavevmode\hbox to172.38pt{\vbox
to17.57pt{\pgfpicture\makeatletter\hbox{\hskip 86.18817pt\lower-8.8351pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-86.18817pt}{-8.73526pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
31.0941pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.78856pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0\to
D_{A}\otimes_{R_{A}}M}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
31.0941pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 55.09407pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.78856pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${D_{A}\otimes_{R_{A}}M\to 0}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
31.0941pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07988pt}{2.39986pt}\pgfsys@curveto{-1.69989pt}{0.95992pt}{-0.85313pt}{0.27998pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85313pt}{-0.27998pt}{-1.69989pt}{-0.95992pt}{-2.07988pt}{-2.39986pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}{}{}{{}}\pgfsys@moveto{-23.79997pt}{-6.23526pt}\pgfsys@lineto{23.40001pt}{-6.23526pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.59999pt}{-6.23526pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.78204pt}{-2.13249pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{(E_{i}-\beta_{i})\bullet}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right).$

The resulting complex $K_{A,\bullet}(M;\beta)$ (well-defined up to ordering
the factors in the tensor product) is the _Euler–Koszul complex_ to $M$ and
$\beta$ and we view it as positioned in such a way that the terminal module
$D_{A}\otimes M$ of this complex is situated in homological degree zero. We
denote the $i$-th homology module by $H_{A,i}(M;\beta)$.

One of the main results in [MMW05] is that every homology group of
$K_{A,\bullet}(M,\beta)$ is a holonomic left $D_{A}$-module, provided that $M$
is an $A$-toric $R_{A}$-module. We prove next a generalization of this fact to
$A$-ttoric modules.

######  Theorem 3.0.

If $M$ is $A$-ttoric, then for every $\beta\in N_{\mathbb{C}}$, the
Euler–Koszul complex $K_{A,\bullet}(M,\beta)$ is holonomic (in the sense that
each of its homology groups is a holonomic $D_{A}$-module). Moreover,
$K_{A,\bullet}(M;\beta)$ is exact (has zero homology) if and only if
$\beta\notin\textup{qdeg}_{A}(M)$.

###### Proof.

We review the cornerstones of the proof in the $A$-toric case from [MMW05] and
indicate the necessary changes for the $A$-ttoric case.

If $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$ is an $A$-graded exact
sequence of $A$-toric modules, the Euler–Koszul functor induces a long exact
sequence of Euler–Koszul homology modules. Basic properties of holonomic
modules imply that $H_{A,\bullet}(M;\beta)$ is holonomic if and only if both
$H_{A,\bullet}(M^{\prime},\beta)$ and $H_{A,\bullet}(M^{\prime\prime};\beta)$
are holonomic. The definition of $A$-toric modules implies that holonomicity
of $K_{A,\bullet}(M;\beta)$ follows from holonomicity of all
$K_{A,\bullet}(R_{A}/I^{\tau}_{A};\beta)$.

The _true $A$-degrees_ $\operatorname{tdeg}_{A}(M)$ of an $A$-graded module
$M=\bigoplus_{{\mathbf{u}}\in\bar{N}}M_{\mathbf{u}}$ are the collection
$\\{{\mathbf{u}}\in\bar{N}\mid M_{\mathbf{u}}\neq 0\\}$. As a subset of
$\bar{N}_{\mathbb{C}}=\bigoplus{\mathbb{C}}\cdot{\boldsymbol{\varepsilon}}_{i}$,
the Zariski closure of $\operatorname{tdeg}_{A}(M)$ is the set of _$A$ -quasi-
degrees_ $\textup{qdeg}_{A}(M)$. For a toric $A$-module, the quasi-degrees are
a finite union of subspaces of the form
${\mathbb{C}}\cdot(A\cap\tau)+{\mathbf{u}}$ for faces $\tau$ of $A$ and
suitable shift vectors ${\mathbf{u}}\in\bar{N}$. The quasi-degrees of
$R_{A}/I^{\tau}_{A}$ are exactly ${\mathbb{C}}\cdot(A\cap\tau)$. One shows
next that $K_{A,\bullet}(R_{A}/I^{\tau}_{A};\beta)$ is an exact complex if and
only if $\beta$ is not a quasi-degree of $R_{A}/I^{\tau}_{A}$. Since the
construction of an Euler–Koszul complex is designed to work well with degree
shifts, it follows that $K_{A,\bullet}(M;\beta)$ is exact if and only if
$\beta\in\textup{qdeg}_{A}(M)$.

Let now
$\beta\in\textup{qdeg}_{A}(R_{A}/I^{\tau}_{A})={\mathbb{C}}\cdot(A\cap\tau)$.
The main (_i.e._ , terminal) homology group
$H_{A,0}(R_{A}/I^{\tau}_{A};\beta)$ is holonomic since it can be shown to be
isomorphic to the $D$-module inverse image under a projection, from
$\bar{N}_{\mathbb{C}}$ to a quotient space of $\bar{N}_{\mathbb{C}}$, of a
standard GKZ-system on that quotient space. (More precisely,
$H_{A,0}(R_{A}/I^{\tau}_{A};\beta)$ is of the form
$H_{B,0}(R_{B}/I_{B};\gamma)\otimes_{\mathbb{C}}{\mathbb{C}}[y_{1},\ldots,y_{k}]$
where $B$ is a full rank matrix with the same span as $\tau\cap A$ and $k$
fewer columns than $A$, compare the proof of [MMW05, Lemma 4.9]). This then
implies that all $H_{A,0}(M;\beta)$ are holonomic for toric $M$. The higher
Euler–Koszul homologies of $R_{A}/I^{\tau}_{A}$ are the cosets of
$A$-homogeneous elements $m$ in the kernel of the Euler–Koszul complex. Their
classes are annihilated by each $(E_{i}-\beta-\deg(m))\bullet$, and also by a
power of $I^{\tau}_{A}$ (compare the proof of [MMW05, Prop. 5.1]). Using a
filtration argument, they are a quotient of a finite direct sum of various
$H_{A,0}(R_{A}/I^{\tau}_{A};\gamma)$ and hence holonomic.

Inspecting this proof strategy shows that in order to adapt this proof to the
$A$-ttoric case, one just needs to know that
$H_{A,0}(R_{A}/I^{\tau}_{A,\rho};\beta)$ is holonomic, and that furthermore it
is nonzero if and only if $\beta$ is an $A$-quasi-degree of
$R_{A}/I^{\tau}_{A,\rho}$.

Let $M=R_{A}/I^{\tau}_{A,\rho}$. If $\beta$ lies outside
${\mathbb{C}}(A\cap\tau)=\textup{qdeg}_{A}(M)$, then the ${\mathbb{C}}$-linear
span of $E-\beta$ together with the products
$\\{x_{j}\partial_{j}\mid{\mathbf{a}}_{j}\notin\tau\\}$ is a subset of
$D_{A}\cdot I^{\tau}_{A,\rho}$ and contains a nonzero scalar (compare [MMW05,
Lemma 4.9]); hence,
$H_{A,0}(M;\beta)=D_{A}/D_{A}(I^{\tau}_{A,\rho},\\{E_{i}-\beta_{i}\\})$ is
zero. As in the untwisted case, this implies the vanishing of all higher
homology groups. If, on the other hand, $\beta$ is in the span of $A\cap\tau$
then choose $\dim(\tau)$ many rows of $A$ such that its restriction to the
columns in $\tau$ of these rows are linearly independent. Then write $B$ for
the corresponding $\operatorname{rank}(A\cap\tau)\times|A\cap\tau|$-submatrix,
denote $\beta^{\tau}$ the restriction of $\beta$ to the column span of $B$,
and set $E^{\tau}_{i}:=\sum_{j\in\tau}a_{i,j}x_{j}\partial_{j}$. If now
$\rho^{B}$ is the restriction of $\rho$ to $\ker(B)\subseteq\ker(A)$, it
follows that
$H_{A,0}(M;\beta)=(D_{B}/D_{B}\cdot(I_{B,\rho^{B}},\\{E_{i}^{\tau}-\beta_{i}^{\tau}\\}))\otimes_{\mathbb{C}}{\mathbb{C}}[\\{x_{j}\mid{\mathbf{a}}_{j}\notin\tau\\}]$.
The partial saturated character $\rho^{B}$ can be used to induce an
automorphism of ($R_{B}$ as in Remark 2.2, and hence also on) the Weyl algebra
$D_{B}$ that carries
$D_{B}/D_{B}\cdot(I_{B,\rho^{\rho}},\\{E_{i}^{\tau}-\beta_{i}^{\tau}\\})$ to
the classical GKZ-system $H_{B,0}(R_{B}/I_{B};\beta^{\tau})$. The latter is,
of course, nonzero. ∎

The _rank_ of a $D_{A}$-module $\mathscr{M}$ is the vector space dimension of
its classical solution space near a generic point of ${\mathbb{C}}^{n}$. The
Cauchy–Kovalevskaya–Kashiwara Theorem shows that this is also the dimension
over ${\mathbb{C}}(\boldsymbol{x})$ of the vector space
${\mathbb{C}}(\boldsymbol{x})\otimes_{{\mathbb{C}}[\boldsymbol{x}]}\mathscr{M}$.
For modules such as $H_{A,0}(M;\beta)$ that arise from toric modules $M$, it
is shown in [MMW05] that the rank is positive precisely when $\beta$ is a
quasi-degree of $M$. This generalizes to the $A$-ttoric case.

######  Lemma 3.0.

The parameter $\beta$ is a quasi-degree of the $A$-ttoric module $M$ precisely
if $H_{A,0}(M;\beta)$ has positive rank. More precisely, the rank function is
upper-semicontinuous in $\beta$.

###### Proof.

Since rank is additive in short exact sequences, it suffices to prove the
statement when $M=R_{A}/I^{\tau}_{A,\rho}$ for some face $\tau$ and some
character $\rho$ on $\ker(A)$. In that case, the automorphism on $R_{A}$
arising from Remark 2.2, and the induced automorphism of $D_{A}$ (that acts on
$x_{j}$ inversely to the action on $\partial_{j}$) reflects the issue from
${\mathbb{C}}(\boldsymbol{x})\otimes_{R_{A}}\mathscr{M}$ back to
$H_{A,0}(R_{A}/I^{\tau}_{A};\beta)$, in which case the result is known from
[MMW05].

The final claim follows the lines of [MMW05] since $A$-ttoric modules give
holonomic families. ∎

######  Corollary 3.0.

For any $A$-ttoric module $M$, there is a subspace arrangement
$\mathcal{E}_{M}$ in ${\mathbb{C}}^{d}$ such that $\beta\notin\mathcal{E}_{M}$
implies that the rank of $H_{A,0}(M;\beta)$ is independent of $\beta$.

###### Proof.

Following the train of the proof of Lemma 3, one needs to inspect the
statement of the lemma only when $M=R_{A}/I^{\tau}_{A,\rho}$. But because of
the isomorphism induced via Remark 2.2, the rank of
$H_{A,0}(R_{A}/I^{\tau}_{A,\rho};\beta)$ is the same as that of
$H_{A,0}(R_{A}/I_{A}^{\tau};\beta)$, and so one may once again refer to
[MMW05]. ∎

##  4 $D$-modules from Modules over Semigroups

In the next section we will review the definition of the systems introduced by
Borisov and Horja in [BH06, BPH13], show that they arise as terminal homology
groups of an Euler–Koszul complex on a suitable ttoric $A$-module, and deduce
finiteness conditions about it. Here, we lay some ground work regarding
modules over semigroups.

###  4.1 Semigroup Modules

######  Convention 4.0.

Throughout, we will assume that our semigroups are Abelian, have an identity,
and inject into their Grothendieck group.

If $S$ is any semigroup, then a set $T$ is an _$S$ -module_ if there is an
action

$\bullet\colon S\times T\to T$

that is additive in $S$. We call $T$ _finitely generated (over $S$)_ if
$T=\bigcup S\bullet t$ with $t$ running through some finite subset of $T$. Let
$S_{+}$ be the non-units in $S$ and remark that there are only finitely many
units in $S$ (since the torsion is finite and $\pi(S)$ is pointed).

We refer to

${T_{\mathrm{prim}}}:=T\smallsetminus(S_{+}\bullet T)$

as the _( $S$-)primitive elements of $T$_; this is the natural candidate set
for a “collection of minimal generators of $T$ over $S$”.

######  Lemma 4.0.

Suppose $S$ is a finitely generated semigroup and write $S_{+}$ for the non-
invertible elements of $S$. Suppose $T=\bigcup_{1}^{k}S\bullet t_{i}$ is a
finitely generated $S$-module, such that $s\bullet t=t$ only occurs when $s$
is a unit. Then the set ${T_{\mathrm{prim}}}:=T\smallsetminus S_{+}\bullet T$
is finite and generates $T$ over $S$.

###### Proof.

Let $U$ be the units in $S$; then one may form the factor semigroup
$\bar{S}:=S/U$ which acts on $T$ as well. The identification space $\bar{T}$
obtained from $T$ by identifying for all $t\in T$ the elements $\\{u\bullet
t\mid u\in U\\}$ is an $\bar{S}$-module and $\bar{s}\bullet\bar{t}=\bar{t}$ if
and only if $\bar{s}$ is the coset of the identity of $S$. Then
$S_{+}/U=(\bar{S})_{+}$ and cosets of primitive elements in $T$ consist
exclusively of primitive elements of $T$, forming primitive elements of
$\bar{T}$. Finiteness of $U$ now implies that for the purpose of the proof we
may assume that $U$ is trivial, $S=\bar{S}$ and $T=\bar{T}$. Note that this
reduction eliminates all torsion from $S$.

In general, let $t\in{T_{\mathrm{prim}}}$; according to the setup it is in
$\bigcup_{1}^{k}S\bullet t_{i}$ and so there is $i\in\\{1,\ldots,k\\}$ and
$s_{i}\in S$ with $t=s_{i}\bullet t_{i}$. But for a primitive element this can
only happen if $s_{i}$ is a unit (in the reduced setting, the identity,) of
$S$. Since $T$ is assumed to be finitely generated over $S$, we have shown
that $|{T_{\mathrm{prim}}}|$ is at most $k$ times the number of units of $S$.

We introduce a descending _level_ filtration on $T$ by
$\operatorname{level}(t)\geq h$ if
$t\in\underbrace{(S_{+}+S_{+}+\ldots+S_{+})}_{h\text{ copies}}\bullet T$. We
say that $t$ is of _height_ $h$ if $\operatorname{level}(t)\geq h$ but
$\operatorname{level}(t)\not\geq h+1$. It is convenient to say that every
element of $T$ is in level zero, and thus we attach to $t\in T\smallsetminus
S_{+}\bullet T$ the height zero.

Let now $g_{1},\ldots,g_{q}$ be nonzero semigroup generators for $S$ and let
$t_{1},\ldots,t_{k}$ be generators for $T$ over $S$. The reduction $S=\bar{S}$
forces $g_{i}\in S_{+}$, and every element of $S_{+}$ can be written as a sum
of the generators (perhaps in many ways).

We claim that every $t\in T$ has well-defined, finite, height. Indeed, suppose
$t$ is in every level. So there are equations
$t=(\sum_{i=1}^{h^{\prime}}g_{i,h})\bullet t_{i_{h}}$ for each
$h\in{\mathbb{N}}$, with $h^{\prime}\geq h$ and $g_{i,h}$ one of our chosen
generators.

Since there are only finitely many generators $t_{i_{h}}$ for $T$, but
infinitely many such expressions for $t$, we can choose a subsequence for
which always the same $t_{i_{h}}$ is used (and discard the other expressions
for $t$).

Let $\mu_{h,j}$ be the number of times that the generator $g_{j}$ appears in
the chosen $h$-th sum $t=(\sum_{i=1}^{h^{\prime}}g_{i,h})\bullet t_{i_{h}}$.
As $S$ is finitely generated, there is at least one generator $g_{1}$ of $S$
such that the set $\\{\mu_{h,1}\\}$ is finite. Choose a subsequence of the
infinitely many expressions for $t$ in which the number of times $g_{1}$
appears is strictly increasing, and discard all other expressions for $t$.

Now it can happen that for all other generators $g_{i}\in S_{+}$ of $S$ the
sets $\\{\mu_{i,h}\\}$ are finite. If so, move to the next paragraph.
Otherwise, take a generator $g_{2}$ that appears with unbounded multiplicity,
and choose a subsequence of expressions for $t$ in which now both $\mu_{h,1}$
and $\mu_{h,2}$ are strictly increasing. Repeat if necessary.

In this manner, we find a sequence of expressions for $t$ in which the
multiplicities for $g_{1},\ldots,g_{r}\in S_{+}$ are strictly increasing, and
the multiplicities for $g_{r+1},\ldots,g_{q}$ are uniformly bounded. The
pigeon hole principle now dictates that there are two expressions for $t$ in
which for all $g_{i}$ the multiplicity of $g_{i}$ in the second expression
strictly dominates the multiplicity in the first. So we have

$(a_{1}g_{1}+\ldots+a_{q}g_{q})\bullet t_{i}=t=(b_{1}g_{1}+\ldots
b_{q}g_{q})\bullet t_{i},$

where $a_{j}<b_{j}$ for $j=1,\ldots,q$, and so
$\sum_{j=1}^{q}(b_{j}-a_{j})g_{j}\in S_{+}$ acts on $t_{i}$ as identity. By
hypothesis, this implies that this sum is a unit (that is, the identity in the
reduced setting). By contradiction, $t$ must have a finite level, as claimed.

We are now ready finish the proof of the lemma. By definition,
${T_{\mathrm{prim}}}$ are the elements of height zero. Any generator $t_{i}$
of $T$ that is not height zero is of the form $s\bullet t$ for some $s\in
S_{+}$ and some $t\in T$. By definition of level and height, at least one such
rewriting $t_{i}=s\bullet t$ exists where the height of $t$ is smaller than
the height of $t_{i}$. Then
$\\{t_{1},\ldots,t_{k},t\\}\smallsetminus\\{t_{i}\\}$ is a generating set for
$T$ over $S$. By iteration, we can replace any element of positive height
among $\\{t_{1},\ldots,t_{k}\\}$ by one of height zero, while preserving the
fact that they are a generating set. It follows that ${T_{\mathrm{prim}}}$
generates $T$ over $S$. ∎

######  Notation 4.0.

On the semigroups ${\mathbb{N}}\mathcal{A}$ and ${\mathbb{N}}A$ there is a
tautological degree function, $\deg_{A}(-)$ with values in ${\mathbb{Z}}^{d}$
that sends an element to its natural image in ${\mathbb{N}}A$.

Suppose $T$ is an $S$-module with $S={\mathbb{N}}A$ or
${\mathbb{N}}\mathcal{A}$. We say $T$ _has an $A$-grading_ if there is a map
$(-)_{A}\colon T\to{\mathbb{Z}}A$ such that for all $s\in S,t\in T$ we have
$(s\bullet t)_{A}=\deg_{A}(s)+(t)_{A}$. All $S$-modules inside the
Grothendieck group ${\mathbb{Z}}S$ have a natural $A$-grading.

Define for an ${\mathbb{N}}\mathcal{A}$-module $T$ the
$S_{\mathcal{A}}$-module

$M_{T}:=\bigoplus_{t\in T}{\mathbb{K}}\cdot t$

via the rule $(c_{s}\cdot s)\bullet(c_{t}\cdot t):=(c_{s}c_{t})\cdot(s\bullet
t)$, for $c_{s},c_{t}\in{\mathbb{K}},s\in{\mathbb{N}}\mathcal{A},t\in T$. If
$T$ has an $A$-grading $(-)_{A}$, then $M_{T}$ becomes an $A$-graded
$R_{\mathcal{A}}$-module via $\deg_{A}(c_{t}\cdot t):=(t)_{A}$. If $T$ is
finitely generated over ${\mathbb{N}}\mathcal{A}$ then $M_{T}$ is a finitely
generated $A$-graded
$S_{\mathcal{A}}={\mathbb{K}}[{\mathbb{N}}\mathcal{A}]$-module, and hence
$\mathcal{A}$-toric.

######  Theorem 4.0.

For every finitely generated $A$-graded ${\mathbb{N}}\mathcal{A}$-module $T$
and all $\beta\in N_{\mathbb{C}}$, the $D_{A}$-module
$H_{A,0}(\iota_{\pi}(M_{T});\beta)$ is holonomic. In fact,
$H_{A,0}(\iota_{\pi}(M_{T});\beta)$ is the terminal homology group of an
Euler–Koszul complex in which every homology group is a holonomic
$D_{\mathcal{A}}$-module. This Euler–Koszul complex is exact if and only if
$\beta$ is not in the Zariski closure of $\deg_{A}(T)$.

###### Proof.

In view of Theorem 3.0, all we need to show is that $\iota_{\pi}(M_{T})$ is
$A$-ttoric. But that follows from Lemma 2.3. ∎

######  Corollary 4.0.

If there is a linear functional $h\colon N\to{\mathbb{R}}$ such that
$\mathcal{A}\subseteq h^{-1}(1)$ then for any finitely generated
${\mathbb{N}}\mathcal{A}$-module $T$ and for all
$\beta\in\operatorname{\textup{Hom}}(N,{\mathbb{C}})$, the Euler–Koszul
complex on $M_{T}$ has regular holonomic homology.

###### Proof.

By [SW08], every GKZ-system $H_{A}(\beta)$ is regular holonomic if a
functional $h$ as in the corollary exists. In the proof of Theorem 3.0 it is
shown that every homology group of the Euler–Koszul complex on $M_{T}$ is a
quotient of a finite sum of modules that are isomorphic to $A$-hypergeometric
systems in the sense of [MMW05]. The corollary follows. ∎

###  4.2 Rank and Duality

The transposition
$\tau(x^{\mathbf{a}}\partial^{\mathbf{b}})=(-\partial)^{\mathbf{b}}x^{\mathbf{a}}$
on $D_{\mathcal{A}}$ provides an equivalence of the categories of left and
right $D_{A}$-modules, that corresponds on the level of sheaves to the tensor
product with the canonical sheaf $\omega_{{\mathbb{C}}^{n}}$ in one direction,
and
$\operatorname{\mathscr{H}\mathit{om}}_{\mathcal{O}_{{\mathbb{C}}^{n}}}(\omega_{{\mathbb{C}}^{n}},-)$
in the other.

The automorphism $x\mapsto-x$ on ${\mathbb{C}}^{n}$ induces, via
$\partial\mapsto-\partial$, an auto-equivalence $(-)\mapsto(-)^{-}$ on the
categories of ${\mathbb{Z}}A$-graded $R_{A}$-modules and $D_{A}$-modules (but
not on the category of ${\mathbb{Z}}A$–graded $S_{A}$-modules, since $I_{A}$
is not preserved under $(-)^{-}$ unless it is projective). The formation of
Euler–Koszul complexes is equivariant under this sign change since
$E_{i}-\beta_{i}=(E_{i}-\beta_{i})^{-}$. Moreover, for ${\mathbb{Z}}A$-graded
$R_{A}$-modules $N$ we have
$D_{A}\otimes_{R_{A}}N\simeq\tau(N\otimes_{R_{A}}D_{A})^{-}$ as left
$D_{A}$-modules, where the tensor products exploit the two different
$R_{A}$-structures on $D_{A}$.

Let $M$ be a finitely generated ${\mathbb{Z}}A$-graded $R_{A}$-module and
choose a minimal free ${\mathbb{Z}}A$-graded resolution $F_{\bullet}$.
Denoting

$\varepsilon_{A}:=\sum_{{\mathbf{a}}\in\mathcal{A}}\pi({\mathbf{a}})$ (4.0)

the sum of all elements of the multi-set $A$, it is explained in [MMW05,
Section 6], that there is a natural identification of the complexes

$\tau\operatorname{\textup{Hom}}_{D_{A}}(K_{A,\bullet}(F_{\bullet};E-\beta),D_{A})^{-}\simeq
K_{,\bullet}(\operatorname{\textup{Hom}}_{D_{A}}(F_{\bullet},D_{A});-E-\beta-\varepsilon_{A}).$

The same construction works if $F_{\bullet}$ is any free finite
${\mathbb{Z}}A$-graded complex, and in this setting $F_{\bullet}$ having toric
homology assures the resulting homology groups of the Euler–Koszul complex to
be holonomic. In consequence, for a finite ${\mathbb{Z}}A$-graded complex
$F_{\bullet}$ with toric homology, the holonomic dual of the higher
Euler–Koszul homology $H_{A,i}(F_{\bullet};\beta)$ is, up to a degree shift by
$\varepsilon_{A}$, the Euler–Koszul homology
$H_{A,n-i}(\operatorname{\textup{Hom}}_{D_{A}}(F_{\bullet},D_{A});-\beta-\varepsilon_{A})^{-}$.
In particular, if $F_{\bullet}$ has only one homology module $M$, say in
cohomological position zero, then
${\mathbb{D}}H_{A,i}(M,\beta)=H_{A,n-i}(\operatorname{\textup{Hom}}_{D_{A}}(F_{\bullet},D_{A});-\beta-\varepsilon_{A})^{-}$,
and if $M$ is a Cohen–Macaulay $R_{A}$-module of dimension $d$ then (there is
no higher Euler–Koszul homology of $M$, and)

${\mathbb{D}}H_{A,0}(M,\beta)=H_{A,0}(\operatorname{{\textup{Ext}}}^{n-d}_{R_{A}}(M,R_{A});-\beta-\varepsilon_{A})^{-}.$
(4.0)

We want this isomorphism for finite free ${\mathbb{Z}}A$-graded
$D_{A}$-complexes with $A$-ttoric homology. Inspecting the proof of Thorem 6.3
in [MMW05] reveals that the only property of ”toric $F$” that is used is that
$A$-toric modules are $A$-graded and produce holonomic Euler–Koszul homology
modules. That, however is also the case for $A$-ttoric modules.

######  Corollary 4.0.

The duality statement (4.0) applies in particular to $\mathcal{A}$-toric
modules $M$.∎

##  5 Applications

######  Definition 5.0 (Differential systems on $\mathcal{A}$-toric modules).

Choose $r\in{\mathbb{N}}$, $\beta\in N\otimes_{\mathbb{Z}}{\mathbb{C}}$, and
let $T\subseteq N^{r}$ be a module over the semigroup
${\mathbb{N}}\mathcal{A}\subseteq N$. Let

$\Phi_{T}=(\phi_{\mathbf{u}})_{{\mathbf{u}}\in T}$

be an assignment of a (sufficiently differentiable) function in $x_{A}$ to
each element of $T$. The natural morphism
$N^{\vee}=\operatorname{\textup{Hom}}_{\mathbb{Z}}(N,{\mathbb{Z}})\to\operatorname{\textup{Hom}}_{\mathbb{C}}(N\otimes{\mathbb{C}},{\mathbb{C}})$
allows to define $\mu(\beta)$ for any $\mu\in N^{\vee}$.

Consider the following system of partial differential equations:

$\\{\partial_{j}\phi_{\mathbf{u}}=\phi_{{\mathbf{u}}+{\mathbf{a}}_{j}}\\}_{{\mathbf{u}}\in
T,{\mathbf{a}}_{j}\in\mathcal{A}}\qquad\cup\qquad\\{\sum_{{\mathbf{a}}_{j}\in\mathcal{A}}\mu({\mathbf{a}}_{j})x_{j}\partial_{j}\phi_{\mathbf{u}}=\mu(\beta-{\mathbf{u}})\phi_{\mathbf{u}}\\}_{{\mathbf{u}}\in
T,\mu\in N^{\vee}}.$

Let us write

$E_{\mu}:=\sum_{{\mathbf{a}}_{j}\in\mathcal{A}}\mu({\mathbf{a}}_{j})x_{j}\partial_{j}\quad\in
D_{\mathcal{A}}.$

A system of type $\Phi_{T}$ induces a $D_{\mathcal{A}}$-module

$\displaystyle\Phi_{\mathcal{A},0}(T;\beta):=D_{\mathcal{A}}^{T}/H_{\mathcal{A}}(T;\beta),$

where $H_{\mathcal{A}}(T;\beta)\subseteq D_{\mathcal{A}}$ is the left
$D_{\mathcal{A}}$-module

$\displaystyle
H_{\mathcal{A}}(T;\beta):=D_{\mathcal{A}}\cdot\mathopen{}\mathclose{{}\left(\\{\partial_{j}1_{\mathbf{u}}-1_{{\mathbf{u}}+{\mathbf{a}}_{j}}\\}_{{\mathbf{u}}\in
T,{\mathbf{a}}_{j}\in\mathcal{A}}\quad\cup\quad\\{(E_{\mu}-\mu(\beta-{\mathbf{u}}))\cdot
1_{{\mathbf{u}}}\\}_{{\mathbf{u}}\in T,\mu\in N^{\vee}}}\right)$ (5.0)

and $1_{\mathbf{u}}$ is the element of $\mathcal{D}^{T}$ that is 1 in
coordinate ${\mathbf{u}}$ and zero everywhere else.

The system $\Phi_{T}$ is the space of classical solutions to
$\Phi_{\mathcal{A}}(T;\beta)$. In particular, it can be identified with the
localization of $\Phi_{\mathcal{A}}(T;\beta)$ to a generic point.

Since ${T_{\mathrm{prim}}}$ generates $T$ as ${\mathbb{N}}\mathcal{A}$-module
by Lemma 4.1, the defining relations spelled out in (5.0) imply that
$\Phi_{\mathcal{A}}(T;\beta)$ is as $D_{\mathcal{A}}$-module generated by the
cosets of all $1_{\mathbf{u}}$ with ${\mathbf{u}}\in{T_{\mathrm{prim}}}$; it
is hence a finitely generated $D_{\mathcal{A}}$-module and we can replace in
the definition of $\Phi_{\mathcal{A},0}(T;\beta)$ the set $T$ by
${T_{\mathrm{prim}}}$:

$\Phi_{\mathcal{A},0}(T;\beta)\,\simeq_{D_{\mathcal{A}}}\,D_{\mathcal{A}}^{T_{\mathrm{prim}}}/{H_{\mathcal{A},\textrm{prim}}}(T;\beta)$

where

${H_{\mathcal{A},\textrm{prim}}}(T;\beta):=H_{\mathcal{A}}(T;\beta)\cap
D_{\mathcal{A}}^{T_{\mathrm{prim}}}.$

Note that if we set

${I_{\mathcal{A},\textrm{prim}}}(T):=D_{\mathcal{A}}\cdot\\{\partial_{\mathcal{A}}^{\mathbf{u}}\cdot
1_{{\mathbf{u}}^{\prime}}-\partial_{\mathcal{A}}^{\mathbf{v}}\cdot
1_{{\mathbf{v}}^{\prime}}\mid{\mathbf{u}}+{\mathbf{u}}^{\prime}={\mathbf{v}}+{\mathbf{v}}^{\prime}\\}_{{\mathbf{u}}^{\prime},{\mathbf{v}}^{\prime}\in{T_{\mathrm{prim}}}}$

then

${H_{\mathcal{A},\textrm{prim}}}(T;\beta)=D_{\mathcal{A}}\cdot({I_{\mathcal{A},\textrm{prim}}}(T),\\{(E-\mu(\beta-{\mathbf{u}}))\cdot
1_{\mathbf{u}}\\}_{\mu\in N^{\vee},\,{\mathbf{u}}\in{T_{\mathrm{prim}}}}).$
(5.0)

######  Example 5.0.

One example of a finitely generated ${\mathbb{N}}\mathcal{A}$-module is given
by $K$, as defined in Formula (2.0). Indeed, the image $\pi(K)$ is the
saturation semigroup of ${\mathbb{N}}A$, the set of all lattice points
${\mathbf{u}}\in{\mathbb{Z}}^{d}$ such that ${\mathbb{N}}{\mathbf{u}}$ meets
${\mathbb{N}}A$. That $\pi(K)$ is a finitely generated module over
${\mathbb{N}}A$ is Gordan’s Lemma (see, e.g., [CLS11, Proposition 1.2.17]).
That $K$ is finite over ${\mathbb{N}}\mathcal{A}$ then follows from the fact
that the fibers of $\pi$ are finite.

A second important finitely generated ${\mathbb{N}}\mathcal{A}$-module is the
set $K^{\circ}$, sitting over the interior points of $K$,

$K^{\circ}=\\{{\mathbf{u}}\in
K\mid\pi({\mathbf{u}})\notin\tau\,\,\forall\tau\in\mathcal{F}_{K}\\}.$ (5.0)

The shortest argument that $K^{\circ}$ is finitely generated is slightly
roundabout. Let ${\mathbb{K}}$ be any field, $S\subseteq{\mathbb{Z}}^{d}$ a
semigroup without non-trivial units, $T\subseteq{\mathbb{Z}}^{d}$ an
$S$-module, and consider the semigroup ring
${\mathbb{K}}[S]=\bigoplus_{S}{\mathbb{K}}\cdot s$ and the vector space
$\bigoplus_{T}{\mathbb{K}}\cdot t$, which in natural fashion has a
${\mathbb{K}}[S]$-module structure. If $S$ is finitely generated by $\tau$
elements, then $S$ is a quotient of ${\mathbb{N}}^{\tau}$ and so
${\mathbb{K}}[S]$ is the quotient of a polynomial ring in $\tau$ variables,
and in particular Noetherian. If then $T$ were an infinitely generated module
over $S$, one would have an infinite ascending chain of
${\mathbb{K}}[S]$-submodules of $\bigoplus_{T}{\mathbb{K}}\cdot t$, which
could then not be Noetherian. Now consider the ${\mathbb{N}}A$-module
$K^{\circ}$ and the corresponding ${\mathbb{K}}[{\mathbb{N}}A]$-module
${\mathbb{K}}[\pi(K^{\circ})]$. Since $\pi(K)$ is a finite
${\mathbb{N}}A$-module, ${\mathbb{K}}[\pi(K)]$ is module-finite over
${\mathbb{K}}[{\mathbb{N}}A]$. But ${\mathbb{K}}[\pi(K^{\circ})]$ is the
canonical module of the normal Cohen–Macaulay ring ${\mathbb{K}}[\pi(K)]$ (by
work of Hochster, Danilov and Stanley), and thus certainly module-finite over
${\mathbb{K}}[\pi(K)]$. Finiteness of fibers of $\pi$ then dictates that
$K^{\circ}$ is finitely generated over ${\mathbb{N}}\mathcal{A}$.

######  Remark 5.0.

It follows from the description (5.0) that for every $\mathcal{A}$-toric
module $T$,

$\iota_{\pi}(\Phi_{\mathcal{A},0}(T;\beta))=H_{A,0}(M_{T};\beta)$

is the distinguished (terminal) homology group of the Euler–Koszul complex to
$\beta$ on $M_{T}$.

The case $T=K$ inspired Borisov and Horja to write down the differential
system $\Phi_{K}$ and the corresponding $D_{A}$-module $\Phi_{A}(K;\beta)$
which they termed the _better behaved GKZ-system_.

In particular, the better behaved GKZ-system to $\beta$ in the torsion-free
case $F=0$ is precisely the $A$-hypergeometric module that corresponds to the
toric $S_{A}$-module $M_{K}=\tilde{S}_{A}$.

######  Example 5.0.

Let $N=({\mathbb{Z}}/2{\mathbb{Z}})\oplus{\mathbb{Z}}$, and assume
$\mathcal{A}$ is the singleton ${\mathbf{a}}_{1}=(\bar{1},1)$. Then
${\mathbb{N}}\mathcal{A}={\mathbb{N}}\cdot(\bar{1},1)\subseteq N$ is
isomorphic to ${\mathbb{N}}=\pi({\mathbb{N}}\mathcal{A})$ and so
$R_{\mathcal{A}}$ and $R_{A}$ are isomorphic to the polynomial ring
${\mathbb{K}}[\partial_{1}]$, with corresponding $\mathcal{A}$\- or
$A$-grading. The semigroup $K$ is
$({\mathbb{Z}}/2{\mathbb{Z}})\oplus{\mathbb{N}}$ and its semigroup ring is
isomorphic to
$({\mathbb{K}}[y_{0}]/(y_{0}^{2}-1))\otimes({\mathbb{K}}[y_{1}])$.

However, under this isomorphism, $\partial_{1}$ acts as the element
$y_{0}y_{1}$, and in particular does not act trivially on the $y_{0}$-factor.
Thus, (here and in general) there is an obvious abstract vector space
isomorphism between ${\mathbb{K}}[K]$ and a degree-$|F|$ ring extension of
$\tilde{S}_{A}$, but it is less clear (in general) that there is such an
isomorphism that is an isomorphism of $R_{A}$-algebras.

######  Corollary 5.0.

We consider the $D_{A}$-modules $H_{A,0}(M_{K};\beta)$ and
$H_{A,0}(M_{K^{\circ}};\beta)$. Both have rank equal to
$|F|\cdot\operatorname{\textup{vol}}(A\cup\\{0\\})$.

###### Proof.

By definition, $K$ is the preimage under $\pi$ of the lattice points in the
cone spanned by $A=\pi(\mathcal{A})$. In particular, as a semigroup,
$K=F\oplus\pi(K)$. As a semigroup ring, ${\mathbb{K}}[\pi(\mathcal{A})]$ is
the saturated semigroup ring to ${\mathbb{K}}[{\mathbb{N}}A]$, and

${\mathbb{K}}[K]\simeq\tilde{S}_{A}\otimes_{\mathbb{K}}{\mathbb{K}}[y_{1},\ldots,y_{k}]/(\\{y_{i}^{\ell_{i}}-1\\}_{1}^{k}),$

where $F=\bigoplus_{1}^{k}({\mathbb{Z}}/\ell_{i}{\mathbb{Z}})$ and
$\ell=\prod\ell_{i}$. While the $R_{A}$-structure on ${\mathbb{K}}[K]$ in the
above description is not compatible with the ”natural” subring
$\tilde{S}_{A}=\tilde{S}_{A}\otimes 1\hookrightarrow{\mathbb{K}}[K]$ (as seen
in the above Example 5), we show below that there _is_ such a direct sum
decomposition that is compatible with this natural structure; we are
interested in it because it certifies ${\mathbb{K}}[K]$ as a Cohen–Macaulay
$R_{A}$-module.

We start with considering $R_{\mathcal{A}}$-algebra morphisms from
${\mathbb{K}}[K]$ to $\tilde{S}_{A}$. For $1\leq i\leq k$, let $\zeta_{i}$ be
a primitive $\ell_{i}$-th root of unity, and choose
$\underline{r}:=(r_{i}\in{\mathbb{N}}\mid 0\leq
r_{i}\leq\ell_{i}-1)_{i=1}^{k}$. Then

$\displaystyle\phi_{\underline{r}}\colon{\mathbb{K}}[K]$ $\displaystyle\to$
$\displaystyle\tilde{S}_{A},$ $\displaystyle\\{y_{i}$ $\displaystyle\mapsto$
$\displaystyle(\zeta_{i})^{r_{i}}\\}_{1}^{k}$

is a surjective $R_{\mathcal{A}}$-algebra morphism. Now set
$\phi:=\bigoplus_{\underline{r}}\phi_{\underline{r}}\colon{\mathbb{K}}[K]\to\bigoplus_{\underline{r}}\tilde{S}_{A}$.

The map $\phi$ is an injective $R_{\mathcal{A}}$-module map. Indeed, if
$f\in{\mathbb{K}}[K]$ is in the kernel of $\phi_{\underline{r}}$, then $f$ is
in the ideal generated by $\\{y_{i}-\zeta_{i}^{r_{i}}\\}_{1}^{k}$. Thus, if
$f$ is in the kernel of $\phi$ then $f$ is in the ideal generated by
$\\{y_{i}^{\ell_{i}}-1\\}_{1}^{k}$ and hence zero in ${\mathbb{K}}[K]$. But
$\phi$ is also surjective, which becomes obvious if one considers one variable
at the time: for any ${\mathbb{K}}$-algebra $R$, the morphism that sends
$({\mathbb{K}}[y]/(y^{q}-1))\otimes_{\mathbb{K}}R$ to $\bigoplus_{0}^{q-1}R$
by evaluating $y$ to $\zeta_{q}^{i}$ on the $i$-th summand is onto since

$\begin{pmatrix}1&1&\ldots&1\\\
\zeta_{q}^{1}&\zeta_{q}^{2}&\ldots&\zeta_{q}^{q-1}\\\ \vdots&&&\vdots\\\
\zeta_{q}^{q-1}&\zeta_{q}^{2q-2}&\cdots&\zeta_{q}^{(q-1)(q-1)}\end{pmatrix}$

is a non-singular Vandermonde matrix.

It follows that ${\mathbb{K}}[K]$ is $R_{\mathcal{A}}$-isomorphic to
$\bigoplus_{|F|}\tilde{S}_{A}$ when $\tilde{S}_{A}$ is given its natural
$R_{\mathcal{A}}$-structure inherited from $\pi(K)$ being the saturation of
the semigroup $\pi({\mathbb{N}}\mathcal{A})={\mathbb{N}}A$.

Restriction of the map $\phi$ to ${\mathbb{K}}[K^{\circ}]$ shows that
${\mathbb{K}}[K^{\circ}]$ is $R_{\mathcal{A}}$-linearly isomorphic to the free
sum of $|F|$ copies of ${\mathbb{K}}[\pi(K^{\circ})]$. The decompositions show
that $H_{A,0}(M_{T};\beta)$ is isomorphic to $|F|$ copies of
$H_{A,0}(M_{\pi(T)};\beta)$, for $T\in\\{K,K^{\circ}\\}$.

It is well-known that $H_{A,0}(M_{\pi(K)};\beta)$ has rank equal to the
(simplicial) volume of the convex hull spanned by the origin and $A$,
independently of $\beta$. Moreover, ${\mathbb{K}}[\pi(K^{\circ})]$ is a
maximal Cohen–Macaulay $S_{A}$-module, and so the rank of
$H_{A,0}(M_{K^{\circ}};\beta)$ is independent of $\beta$ as well. It suffices
then to consider a generic $\beta$. But the quasi-degrees of
$\pi(K)/\pi(K^{\circ})$ are contained in the facets of $\mathcal{A}$, and thus
do not contain a generic $\beta$. By Theorem 3.0, applied to the long
Euler–Koszul homology sequence derived from
$0\to\pi(K^{\circ})\to\pi(K)\to\pi(K)/\pi(K^{\circ})\to 0$, one concludes that
the ranks of $H_{A,0}(M_{\pi K^{\circ}};\beta)$ and
$H_{A,0}(M_{\pi(K^{\circ})};\beta)$ are always equal. ∎

Finally, we apply the results on duality of Euler–Koszul homology groups to
the module $H_{A,0}(M_{K};\beta)$.

######  Theorem 5.0.

There is a natural non-degenerate pairing

$\mathopen{}\mathclose{{}\left({\mathbb{C}}({\mathbf{x}})\otimes_{{\mathbb{C}}[\boldsymbol{x}]}H_{A,0}(M_{K};\beta)}\right)\otimes\mathopen{}\mathclose{{}\left({\mathbb{C}}({\mathbf{x}})\otimes_{{\mathbb{C}}[\boldsymbol{x}]}H_{A,0}(M_{K^{\circ}};-\beta-\varepsilon_{A})^{-}}\right)\longrightarrow{\mathbb{C}}({\mathbf{x}}).$

In particular, if $\Gamma=H^{0}(\textup{Sol}(H_{A,0}(M_{K};\beta)_{\xi}))$
resp.
$\Gamma^{\prime}=H^{0}(\textup{Sol}(H_{A,0}(M_{K^{\circ}};-\beta-\varepsilon_{A})_{\xi})^{-})$
is the space of (classical) solutions of $H_{A,0}(M_{K};\beta)$ resp.
$H_{A,0}(M_{K^{\circ}};-\beta-\varepsilon_{A})^{-}$ near a generic point
$\xi$, then we have a non-degenerate pairing
$\Gamma\times\Gamma^{\prime}\rightarrow{\mathbb{C}}$.

###### Proof.

It follows from Formula (4.0) that

${\mathbb{D}}H_{A,0}(M_{K},\beta)\cong
H_{A,0}(M_{K^{\circ}};-\beta-\varepsilon_{A})^{-}(\varepsilon_{A}),$

since by what has been shown in the proof of Corollary 5, $M_{K}$ is a
Cohen–Macaulay $R_{A}$-module, and then its dualizing module is
$M_{K^{\circ}}$. Forgetting the graded structure, localize at the generic
point. Then we are looking at holonomically dual, smooth $D$-modules. These
are hence mutually dual bundles with mutually dual connections, as follows
from the Spencer resolution. (See, _e.g._ , [DS03, Lemma A.11] for the
argument in a slightly different but related situation). The theorem follows.
∎

######  Remark 5.0.

If $\bar{\mathbf{a}}_{1},\ldots,\bar{\mathbf{a}}_{n}$ lie on an affine
hyperplane in $\bar{N}$, the functor $(-)^{-}$ is an auto-equivalence of
graded $S_{A}$-modules, since $I_{A}$ is then projective. If in this case
$\beta$ is the zero vector, we obtain a duality result similar to [BH23,
Theorem 2.4.]. Formula (4.0) is more general, since the $D_{A}$-modules
$H_{A,0}(M_{K};\beta)$ and $H_{A,0}(M_{K^{\circ}};-\beta-\varepsilon_{A})^{-}$
cannot be reconstructed from their (classical) solutions, nor from their
restrictions to a generic point. On the other hand, holonomic duality of
modules always yields a pairing on the level of solutions at a generic point.

It is an interesting project to check whether the concrete construction in
_loc. cit._ agrees with our functorial pairing.

## References

  * [BH06] Lev A. Borisov and R. Paul Horja, _Mellin-Barnes integrals as Fourier-Mukai transforms_ , Adv. Math. 207 (2006), no. 2, 876–927.
  * [BH15] , _Applications of homological mirror symmetry to hypergeometric systems: duality conjectures_ , Adv. Math. 271 (2015), 153–187.
  * [BH23] Lev A. Borisov and Zengrui Han, _On hypergeometric duality conjecture_ , Preprint arXiv:2301.01374, 2023.
  * [BHW21] Lev Borisov, Zengrui Han, and Chengxi Wang, _On duality of certain GKZ hypergeometric systems_ , Asian J. Math. 25 (2021), no. 1, 65–88.
  * [BPH13] Lev A. Borisov and R. Paul Horja, _On the better behaved version of the GKZ hypergeometric system_ , Math. Ann. 357 (2013), no. 2, 585–603.
  * [CLS11] David A. Cox, John B. Little, and Henry K. Schenck, _Toric varieties_ , Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322
  * [DS03] Antoine Douai and Claude Sabbah, _Gauss-Manin systems, Brieskorn lattices and Frobenius structures. I_ , Ann. Inst. Fourier (Grenoble) 53 (2003), no. 4, 1055–1116.
  * [ES96] David Eisenbud and Bernd Sturmfels, _Binomial ideals_ , Duke Math. J. 84 (1996), no. 1, 1–45.
  * [MMW05] Laura Felicia Matusevich, Ezra Miller, and Uli Walther, _Homological methods for hypergeometric families_ , J. Amer. Math. Soc. 18 (2005), no. 4, 919–941 (electronic).
  * [RSSW21] Thomas Reichelt, Mathias Schulze, Christian Sevenheck, and Uli Walther, _Algebraic aspects of hypergeometric differential equations_ , Beitr. Algebra Geom. 62 (2021), no. 1, 137–203.
  * [SW08] Mathias Schulze and Uli Walther, _Irregularity of hypergeometric systems via slopes along coordinate subspaces_ , Duke Math. J. 142 (2008), no. 3, 465–509.

Thomas Reichelt
Lehrstuhl für Mathematik VI
Institut für Mathematik
Universität Mannheim, A 5, 6
68131 Mannheim
Germany
<EMAIL_ADDRESS>

Christian Sevenheck
Fakultät für Mathematik
Technische Universität Chemnitz
09107 Chemnitz
Germany
<EMAIL_ADDRESS>

Uli Walther
Purdue University
Dept. of Mathematics
150 N. University St.
West Lafayette, IN 47907
USA
<EMAIL_ADDRESS>