# Representations of $\mathbb{N}^{\infty}$-type combinatorial categories

Zhenxing Di  School of Mathematical Sciences, Huaqiao University, Quanzhou
362021, China<EMAIL_ADDRESS>,  Liping Li  LCSM(Ministry of Education),
School of Mathematics and Statistics, Hunan Normal University, Changsha
410081, China<EMAIL_ADDRESS>and  Li Liang  Department of Mathematics,
Lanzhou Jiaotong University, Lanzhou 730070, China<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider representations of certain combinatorial categories,
including the poset ${\mathscr{D}}$ of positive integers and division, the
Young lattice $\mathscr{Y}$ of partitions of finite sets, the opposite
category of the orbit category $\mathscr{Z}$ of $(\mathbb{Z},+)$ with respect
to nontrivial subgroups, and the category $\mathscr{CI}$ of finite cyclic
groups and injective homomorphisms. We describe explicit upper bounds for
homological degrees of their representations, and deduce that finitely
presented representations (resp., representations presented in finite degrees)
over a field form abelian subcategories of the representation categories. We
also give an explicit description for the category of sheaves over the ringed
atomic site $(\mathscr{Z},\,J_{at},\,\underline{\mathbb{C}})$, and show that
irreducible sheaves are parameterized by primitive roots of the unit.

Z.X. Di was partly supported by NSF of China (Grant No. 11971388), the
Scientific Research Funds of Huaqiao University (Grant No. 605-50Y22050) and
the Fujian Alliance of Mathematics (Grant No. 2024SXLMMS04); L.P. Li was
partly supported by NSF of China (Grant No. 12171146); L. Liang was partly
supported by NSF of China (Grant No. 12271230).

##  1\. Introduction

The poset ${\mathscr{D}}$ of positive integers and division is a well known
example of lattices, and is closely related to quite a few important posets in
combinatorics. For example, by the fundamental theorem of arithmetic, it is
isomorphism to the poset $\mathbb{N}^{\infty}$ whose elements are sequence
${\mathbf{x}}=(x_{1},x_{2},\ldots)$ of natural numbers with all but finitely
many entries being 0, and whose order is given by
${\mathbf{x}}\leqslant{\mathbf{y}}$ if $x_{i}\leqslant y_{i}$ for each $i$.
Furthermore, the infinite symmetric group $S_{\infty}=\varinjlim_{n}S_{n}$
acts on $\mathbb{N}^{\infty}$ in a natural way and this action respects the
partial order, so the quotient set consisting of orbits is again a poset, and
is isomorphic to the Young latttice $\mathscr{Y}$ whose elements are
partitions of $[n]=\\{1,2,\ldots,n\\}$ with $n$ ranging over all positive
integers and whose partial order is given by containment.

Although these posets and related variants have been widely considered in
various areas such as combinatorics, combinatorial topology, operad theory and
representation theory of symmetric groups or Lie algebras (see for instances
[2, 5, 15, 16, 17, 18, 19]), their representation theory are far from well
developed. One reason, as we guess, is that their representations are much
more complicated compared to their straightforward combinatorial structure.
However, we still think that this is a valuable topic, not only because of its
own interest, but also because of the deep relation between representations of
these posets and representations of other algebraic or combinatorial
structures. For instance, representations of ${\mathscr{D}}$ are precisely
multi-graded modules of the polynomial ring with countably many variables, and
correspondingly representations of $\mathscr{Y}$ are closely related to
$S_{\infty}$-symmetric modules of this polynomial ring, which have attracted
the attention of quite a few authors; see for instances [1, 4, 9, 11, 12] and
in particular the nice short survey paper [8]. Consequently, as a particular
application, a deep understanding on representations of these posets can shed
new light on studying asymptotic behavior of symmetric ideals of polynomial
rings.

In this paper we study representations of categories ${\mathscr{C}}$ sharing
the same combinatorial structure as ${\mathscr{D}}$, called combinatorial
categories of $\mathbb{N}^{\infty}$-type. Besides the poset ${\mathscr{D}}$,
typical examples include:

  * •

the opposite category of the orbit category $\mathscr{Z}$ of $(\mathbb{Z},+)$
with respect to the family of nontrivial subgroups;

  * •

the opposite category of the category $\mathscr{CS}$ of finite cyclic groups
and surjective group homomorphisms;

  * •

the category $\mathscr{CI}$ of finite cyclic groups and injective group
homomorphisms.

Let ${\mathscr{C}}$ be a combinatorial category of $\mathbb{N}^{\infty}$-type,
and $k$ a commutative ring. A representation of ${\mathscr{C}}$ (or called a
${\mathscr{C}}$-module) is a covariant functor from ${\mathscr{C}}$ to
$k\operatorname{-Mod}$, the category of all $k$-modules. The category
${\mathscr{C}}\operatorname{-Mod}$ is a rather big abelian category, in
practice one may work in relatively smaller abelian subcategories consisting
of ${\mathscr{C}}$-modules satisfying certain extra conditions. However, the
usual notion of finitely generated modules is not a proper choice. Therefore,
we want to find alternative conditions such that ${\mathscr{C}}$-modules
satisfying them form an abelian subcategory of
${\mathscr{C}}\operatorname{-Mod}$. The first result of this aspect is:

######  Theorem 1.1.

Let ${\mathscr{C}}$ be a combinatorial category of $\mathbb{N}^{\infty}$-type
and $k$ a field. Then the category algebra $k{\mathscr{C}}$ is locally left
coherent. Consequently, the category of finitely presented
${\mathscr{C}}$-modules is abelian.

There are many interesting finitely generated ${\mathscr{C}}$-modules (even
the simple modules) are not finitely presented. Moreover, for practical
purpose people frequently have to deal with infinitely generated
${\mathscr{C}}$-modules. In this case we introduce a few homological numerical
invariants and impose some finiteness conditions on them. It turns out that
${\mathscr{C}}$-modules satisfying these conditions also form an abelian
subcategory of ${\mathscr{C}}\operatorname{-Mod}$.

Let us give some details about these finiteness conditions. Since the object
set $\operatorname{Ob}({\mathscr{C}})$ can be identified with the set of
finite sequences ${\mathbf{x}}=(x_{1},\ldots,x_{n})$ of natural numbers, we
impose the sup norm on $\operatorname{Ob}({\mathscr{C}})$ by setting
$\|{\mathbf{x}}\|$ to be the maximal entry in the sequence. Given a
${\mathscr{C}}$-module $V$, we determine the smallest subset
$S\subseteq\operatorname{Ob}({\mathscr{C}})$ such that $V$ is generated by its
values on objects in $S$. The zeroth homological degree
$\operatorname{hd}_{0}(V)$ is defined to be the supremum of those
$\|{\mathbf{x}}\|$ for ${\mathbf{x}}\in S$. By a standard homological process
we can define $\operatorname{hd}_{i}(V)$ for all $i\geqslant 0$. We say that
$V$ is presented in finite degrees if both $\operatorname{hd}_{0}(V)$ and
$\operatorname{hd}_{1}(V)$ are finite.

######  Theorem 1.2.

Let ${\mathscr{C}}$ be a combinatorial category of type $\mathbb{N}^{\infty}$,
$k$ a field, $\|\bullet\|$ the sup norm on $\operatorname{Ob}({\mathscr{C}})$,
and $V$ a ${\mathscr{C}}$-module. Then

$\operatorname{hd}_{s}(V)\leqslant\max\\{\operatorname{hd}_{0}(V),\,\operatorname{hd}_{1}(V)\\}$

for $s\geqslant 0$. In particular, the category of ${\mathscr{C}}$-modules
presented in finite degrees is abelian.

At a first glance this result seems unexpectedly simple. This happens because
we define $\|{\mathbf{x}}\|$ to be the maximal entry in ${\mathbf{x}}$, which
is very weak in the sense that too many distinct objects share the same norm.
Instead, if we define homological degrees with respect to strong norms, for
instance, the sum norm by setting $\|{\mathbf{x}}\|$ to be the sum of all
entries in ${\mathbf{x}}$, then we cannot bound higher homological degrees in
terms of the first two; see Example 3.6. But for the Young lattice (which as a
quotient category of $\mathbb{N}^{\infty}$ is not a combinatorial category of
type $\mathbb{N}^{\infty}$), we do get an affirmative answer.

######  Theorem 1.3.

Let $\mathscr{Y}$ be the Young lattice, $k$ a field, and $\|\bullet\|$ the sum
norm on $\operatorname{Ob}(\mathscr{Y})$. If $s\geqslant 2$ and
$\operatorname{hd}_{s-1}(V)>0$, then

$\operatorname{hd}_{s}(V)\leqslant\operatorname{hd}_{s-1}(V)(\ln(\operatorname{hd}_{s-1}(V))+1).$

In particular, the category of $\mathscr{Y}$-modules presented in finite
degrees is abelian.

Another important subcategory of ${\mathscr{C}}\operatorname{-Mod}$ which has
a natural abelian structure (but not inherited from that of
${\mathscr{C}}\operatorname{-Mod}$) is the category
${\mathrm{Sh}}({\mathscr{C}}^{\operatorname{op}},\,J,\,\underline{k})$ of
sheaves over ${\mathscr{C}}^{\operatorname{op}}$ equipped with a certain
Grothendieck topology $J$ and the constant structure sheaf $\underline{k}$.
The main motivation for us to consider this category lies in three aspects:
the sheaf category over orbit categories equipped with the atomic Grothendieck
topology $J_{at}$ is closely related to continuous representations of
topological groups (see for instances [3, 7, 10]); the sheaf theoretic
approach can provide a novel viewpoint for representations, for instance, the
complicated shift functor constructed in Section 3 turns out to be the
sheafification functor; and finally, for some examples studied in this paper,
the sheaf category is much simpler than the representation category. For
instance, if ${\mathscr{C}}$ is a poset, then
${\mathrm{Sh}}({\mathscr{C}}^{\operatorname{op}},\,J_{at},\,\underline{k})$ is
equivalent to $k\operatorname{-Mod}$. Although the situation becomes more
complicated for other examples considered in this paper, we still obtain the
following result for a specific category.

######  Theorem 1.4.

Let $k$ be an algebraically closed field with characteristic 0, $\mathscr{Z}$
the orbit category of $(\mathbb{Z},+)$ with respect to the family of
nontrivial subgroups, and $J_{at}$ the atomic topology on $\mathscr{Z}$. Then
${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$ is a semisimple
category, and irreducible sheaves are parameterized by primitive roots of the
unit in $k$.

At this moment we have not obtained similar results for
${\mathrm{Sh}}(\mathscr{CS},\,J_{at},\,\underline{k})$ or
${\mathrm{Sh}}(\mathscr{CI}^{\operatorname{op}},\,J_{at},\,\underline{k})$.
The main obstacle is that an explicit description for automorphisms groups of
finite cyclic groups is not available.

This paper is organized as follows: in the second section we introduce
categories of type $\mathbb{N}^{\infty}$ and some homological invariants of
their representations. Section 3 is devoted to representations of
$\mathbb{N}^{\infty}$. We give an upper bound for homological degrees, and
investigate shift functors in details. In Section 4 we prove Theorems 1.1 and
1.2 for combinatorial categories ${\mathscr{C}}$ of type
$\mathbb{N}^{\infty}$, relying on the assumption that $\mathbb{N}^{\infty}$
can be viewed as a subcategory of ${\mathscr{C}}$ and the observation that the
restriction functor
${\mathscr{C}}\operatorname{-Mod}\to\mathbb{N}^{\infty}\operatorname{-Mod}$
has a nice property. Representations of Young lattice is considered in Section
5, where we prove Theorem 1.3. A sheaf theoretic approach as well as a proof
of Theorem 1.4 is given in the last section.

##  2\. Preliminaries

In this section we introduce combinatorial categories of type
$\mathbb{N}^{\infty}$, describe some examples, and propose a few important
invariants for their representations.

###  2.1. Combinatorial categories of type $\mathbb{N}^{\infty}$

Let ${\mathscr{C}}$ be a small skeletal EI category; that is, every
endomorphism in ${\mathscr{C}}$ is an isomorphism. Then
$G_{x}={\mathscr{C}}(x,x)$ is a group for every object $x$ in ${\mathscr{C}}$.
Furthermore, the binary relation $\preccurlyeq$ on
$\operatorname{Ob}({\mathscr{C}})$ given by $x\preccurlyeq y$ if
${\mathscr{C}}(x,y)\neq\emptyset$ is a partial order.

######  Definition 2.1.

A small skeletal EI category ${\mathscr{C}}$ is said to be a combinatorial
category of type $\mathbb{N}^{\infty}$ if the following conditions hold:

  1. (1)

there is a functor $\iota:\mathbb{N}^{\infty}\to{\mathscr{C}}$ which is a
bijection between the object sets;

  2. (2)

the group $G_{x}$ acts regularly (i.e., freely and transitively) on
${\mathscr{C}}(x,y)$ from the right side for
$x,y\in\operatorname{Ob}({\mathscr{C}})$;

  3. (3)

the group $G_{y}$ acts transitively on ${\mathscr{C}}(x,y)$ from the left side
for $x,y\in\operatorname{Ob}({\mathscr{C}})$

By the first condition, we can identify the underlying poset
$(\operatorname{Ob}({\mathscr{C}}),\preccurlyeq)$ of ${\mathscr{C}}$ and
$\mathbb{N}^{\infty}$, so in the rest of this paper by abuse of notation we
frequently denote objects in ${\mathscr{C}}$ by ${\mathbf{x}}$ with
${\mathbf{x}}\in\mathbb{N}^{\infty}$.

Recall the following definition of left Ore condition: given morphisms
$f:{\mathbf{x}}\to{\mathbf{y}}$ and $g:{\mathbf{x}}\to{\mathbf{z}}$ in
${\mathscr{C}}$, there exists an object ${\mathbf{w}}$ as well as morphisms
$f^{\prime}:{\mathbf{y}}\to{\mathbf{w}}$ and
$g^{\prime}:{\mathbf{z}}\to{\mathbf{w}}$ such that the diagram commutes

$\textstyle{{\mathbf{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{{\mathbf{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{{\mathbf{z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\textstyle{{\mathbf{w}}.}$

Dually, one can define right Ore condition.

The following facts can be easily deduced from definitions.

######  Lemma 2.2.

Let ${\mathscr{C}}$ be a combinatorial category of type $\mathbb{N}^{\infty}$.
Then one has:

  1. (1)

every morphism in ${\mathscr{C}}$ is a monomorphism;

  2. (2)

${\mathscr{C}}$ satisfies the left Ore condition.

###### Proof.

(1) Given two morphisms $\alpha$ and $\alpha^{\prime}$ from ${\mathbf{x}}$ to
${\mathbf{y}}$ and a morphism $\beta$ from ${\mathbf{y}}$ to ${\mathbf{z}}$
such that $\beta\alpha=\beta\alpha^{\prime}$, since $G_{{\mathbf{x}}}$ acts
regularly on ${\mathscr{C}}({\mathbf{x}},{\mathbf{y}})$, we can find an
element $g\in G_{{\mathbf{x}}}$ such that $\alpha^{\prime}=\alpha g$.
Consequently, $\beta\alpha=\beta\alpha g$. But $G_{{\mathbf{x}}}$ also acts
regularly on ${\mathscr{C}}({\mathbf{x}},{\mathbf{z}})$, this happens if and
only if $g$ is the identity in $G_{{\mathbf{x}}}$, so
$\alpha=\alpha^{\prime}$.

(2) Since ${\mathscr{C}}$ is of type $\mathbb{N}^{\infty}$, we can let
${\mathbf{w}}$ be a common upper bound of ${\mathbf{x}}$ and ${\mathbf{z}}$
and take $f^{\prime}:{\mathbf{y}}\to{\mathbf{w}}$ and
$g_{1}:{\mathbf{z}}\to{\mathbf{w}}$ to be arbitrary morphisms. Since
$G_{{\mathbf{w}}}$ acts transitively on
${\mathscr{C}}({\mathbf{x}},\,{\mathbf{w}})$, we can find a certain $\delta\in
G_{{\mathbf{w}}}$ such that $\delta\circ g_{1}\circ g=f^{\prime}\circ f$. Then
$\delta\circ g_{1}$ is the desired $g^{\prime}$. ∎

Here are some interesting examples of combinatorial categories of type
$\mathbb{N}^{\infty}$.

######  Example 2.3.

Let ${\mathscr{D}}$ be the poset of positive integers and division. This poset
is isomorphic to $\mathbb{N}^{\infty}$. Indeed, we can order all primes as
follows $p_{1}<p_{2}<p_{3}<\ldots$ and the fundamental theorem of arithmetic
asserts that every positive integer $n$ uniquely corresponds a sequence
$(x_{1},\,x_{2},\,\ldots)$ such that $n=p_{1}^{x_{1}}p_{2}^{x_{2}}\ldots$. It
is clear that the assignment $n\mapsto(x_{1},\,x_{2},\,\ldots)$ gives the
desired isomorphism of posets.

######  Example 2.4.

Let $G=(\mathbb{Z},+)$ and $\mathcal{H}$ the set of nontrivial subgroups of
$G$. Define $\mathscr{Z}$ to be the orbit category of $G$ with respect to
$\mathcal{H}$. Explicitly, objects are cosets
${\mathbf{n}}=\mathbb{Z}/n\mathbb{Z}$ with $n\in\mathbb{N}_{+}$, and morphisms
${\mathbf{m}}$ to ${\mathbf{n}}$ are $G$-equivariant maps. We check that
$\mathscr{Z}^{\operatorname{op}}$ is a combinatorial category of type
$\mathbb{N}^{\infty}$.

  1. (1)

$\mathscr{Z}({\mathbf{n}},\,{\mathbf{n}})$ is a cyclic group of order $n$, so
$\mathscr{Z}$ is a small skeletal EI category, and so is
$\mathscr{Z}^{\operatorname{op}}$.

  2. (2)

There is an embedding functor
$\iota:{\mathscr{D}}\to\mathscr{Z}^{\operatorname{op}}$ sending a morphism
$m\to n$ in ${\mathscr{D}}$ to the following $G$-equivariant map

$\mathbb{Z}/n\mathbb{Z}\to\mathbb{Z}/m\mathbb{Z},\quad 1+n\mathbb{Z}\mapsto
1+m\mathbb{Z}$

in $\mathscr{Z}$, which is a morphism in
$\mathscr{Z}^{\operatorname{op}}({\mathbf{m}},{\mathbf{n}})$. Clearly, this
functor gives a bijection on object sets.

  3. (3)

A direct computation shows that $\mathscr{Z}({\mathbf{n}},{\mathbf{n}})$
(resp., $\mathscr{Z}({\mathbf{m}},{\mathbf{m}})$) acts regularly (resp.,
transitively) on $\mathscr{Z}({\mathbf{m}},\,{\mathbf{n}})$ from the left
(resp., right) side when $n$ divides $m$, so
$\mathscr{Z}^{\operatorname{op}}({\mathbf{n}},{\mathbf{n}})$ (resp.,
$\mathscr{Z}^{\operatorname{op}}({\mathbf{m}},{\mathbf{m}})$) acts regularly
(resp., transitively) on
$\mathscr{Z}^{\operatorname{op}}({\mathbf{n}},\,{\mathbf{m}})$ from the right
(resp., left) side.

######  Example 2.5.

Let $\mathscr{CI}$ be the category whose objects are finite cyclic groups
${\mathbf{n}}=\mathbb{Z}/n\mathbb{Z}$ with $n\in\mathbb{N}_{+}$, and morphisms
are injective group homomorphisms. We check that $\mathscr{CI}$ is a
combinatorial category of type $\mathbb{N}^{\infty}$.

  1. (1)

There is a natural embedding functor from ${\mathscr{D}}$ to $\mathscr{CI}$
sending a morphism $m\to n$ in ${\mathscr{D}}$ to the following group
homomorphism

$\mathbb{Z}/m\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z},\quad
1+m\mathbb{Z}\mapsto\frac{n}{m}+n\mathbb{Z}.$

  2. (2)

When $n$ divides $m$, ${\mathbf{m}}$ contains a unique subgroup isomorphic to
${\mathbf{n}}$, so $\mathscr{CI}({\mathbf{n}},\,{\mathbf{n}})$ acts regularly
on $\mathscr{CI}({\mathbf{n}},{\mathbf{m}})$ from the right side.

  3. (3)

When $n$ divides $m$, every automorphism of ${\mathbf{n}}$ can extend to an
automorphism of ${\mathbf{m}}$, so $\mathscr{CI}({\mathbf{m}},\,{\mathbf{m}})$
acts transitively on $\mathscr{CI}({\mathbf{n}},{\mathbf{m}})$ from the left
side.

######  Example 2.6.

Let $\mathscr{CS}$ be the category whose objects are finite cyclic groups
${\mathbf{n}}=\mathbb{Z}/n\mathbb{Z}$ with $n\in\mathbb{N}_{+}$, and morphisms
are surjective group homomorphisms. The reader can check that
$\mathscr{CS}^{\operatorname{op}}$ is a combinatorial category of type
$\mathbb{N}^{\infty}$. Note that $\mathscr{CS}$ is different from the orbit
category $\mathscr{Z}$ in Example 2.4 since here we regard ${\mathbf{n}}$ as a
$\mathbb{Z}$-module, while in Example 2.4 we only regard ${\mathbf{n}}$ as a
$\mathbb{Z}$-set of the additive group $\mathbb{Z}$.

From now on let ${\mathscr{C}}$ be a combinatorial category of type
$\mathbb{N}^{\infty}$. A function
$\|\bullet\|:\operatorname{Ob}({\mathscr{C}})\to\mathbb{N}$ is called a norm
if it satisfies the following conditions:

  1. (1)

$\|{\mathbf{x}}\|\geqslant 0$ and the equality holds if and only if
${\mathbf{x}}=\boldsymbol{0}$, the sequence whose each entry is 0;

  2. (2)

$\|a{\mathbf{x}}\|=a\|{\mathbf{x}}\|$ for each $a\in\mathbb{N}$;

  3. (3)

$\|{\mathbf{x}}+{\mathbf{y}}\|\leqslant\|{\mathbf{x}}\|+\|{\mathbf{y}}\|$.

In this paper we mainly consider the following norms, called the sup norm and
the sum norm respectively:

${\mathbf{x}}\mapsto\sup\\{x_{i}\mid
i\in\mathbb{N}_{+}\\};\quad{\mathbf{x}}\mapsto\sum_{i\in\mathbb{N}_{+}}x_{i}.$

###  2.2. Representations and homological invariants

Let $k$ be a commutative ring. A representation of ${\mathscr{C}}$ over $k$,
or a ${\mathscr{C}}$-module, is a covariant functor from ${\mathscr{C}}$ to
$k\operatorname{-Mod}$. Denote the value of $V$ on an object ${\mathbf{x}}$ by
$V_{{\mathbf{x}}}$, and the support of $V$ is set to be

$\operatorname{supp}(V)=\\{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{C}})\mid
V_{{\mathbf{x}}}\neq 0\\}.$

The category of all ${\mathscr{C}}$-modules is a Grothendieck category
admitting enough projectives. For every object ${\mathbf{x}}$ in
${\mathscr{C}}$, denote by $P({\mathbf{x}})$ the $k$-linearization of the
representable functor ${\mathscr{C}}({\mathbf{x}},-)$.

Let $k{\mathscr{C}}$ be the category algebra, which is a free $k$-module with
a basis consisting of morphisms in ${\mathscr{C}}$, and whose multiplication
is defined by setting $f\cdot g=f\circ g$ if they can be composed, and
otherwise setting $f\cdot g=0$. The category algebra $k{\mathscr{C}}$ has a
two-sided ideal $\mathfrak{I}$ which as a free $k$-module is spanned by all
morphisms $f:{\mathbf{x}}\to{\mathbf{y}}$ such that
${\mathbf{x}}\neq{\mathbf{y}}$.

Given a ${\mathscr{C}}$-module $V$, we define the zeroth homology group of $V$
by

$H_{0}(V)=k{\mathscr{C}}/\mathfrak{I}\otimes_{k{\mathscr{C}}}V$

which is a ${\mathscr{C}}$-module, and its value on ${\mathbf{x}}$ is
isomorphic to

$V_{{\mathbf{x}}}/\sum_{{\mathbf{y}}<{\mathbf{x}}\atop
f\in{\mathscr{C}}({\mathbf{y}},{\mathbf{x}})}V_{f}(V_{{\mathbf{y}}}).$

Since $k{\mathscr{C}}/\mathfrak{I}\otimes_{k{\mathscr{C}}}-$ is right exact,
for each $i\geqslant 0$, we can define the $i$-th homology group of $V$ by

$H_{i}(V)=\operatorname{Tor}_{i}^{k{\mathscr{C}}}(k{\mathscr{C}}/\mathfrak{I},\,V).$

Correspondingly, the $i$-th homological degree with respect to a fixed norm
$\|\bullet\|$ is defined to be

$\operatorname{hd}_{i}(V)=\sup\\{\|{\mathbf{x}}\|\mid{\mathbf{x}}\in\operatorname{supp}(H_{i}(V))\\}$

which might be infinity. We set $\operatorname{hd}_{i}(V)$ to be $-1$ when the
above set is empty by convention. For brevity, we write $\operatorname{gd}(V)$
for $\operatorname{hd}_{0}(V)$ and $\operatorname{pd}(V)$ for
$\max\\{\operatorname{hd}_{0}(V),\,\operatorname{hd}_{1}(V)\\}$, and call them
the generation degree and presentation degree. The ${\mathscr{C}}$-module $V$
is said to be generated in finite degrees (resp., presented in finite degrees)
if $\operatorname{gd}(V)$ is finite (resp., $\operatorname{pd}(V)$ is finite).

######  Remark 2.7.

It is easy to see from the above definitions that if $V$ as a
${\mathscr{C}}$-module is generated by its values on objects in a subset
$S\subseteq\operatorname{Ob}({\mathscr{C}})$, then $S$ contains
$\operatorname{supp}(H_{0}(V))$ as a subset. It is also clear that every non-
invertible morphism in ${\mathscr{C}}$ acts on $H_{i}(V)$ as the zero map for
each $i\geqslant 0$.

Now we turn to torsion theory of ${\mathscr{C}}$-modules. Given a
${\mathscr{C}}$-module $V$ and an object
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{C}})$, an element $v\in
V_{{\mathbf{x}}}$ is called torsion if there is a morphism
$f:{\mathbf{x}}\to{\mathbf{y}}$ such that $V_{f}(v)=0\in V_{{\mathbf{y}}}$.
Since ${\mathscr{C}}$ satisfies the left Ore condition by Lemma 2.2, it
follows that all torsion elements in $V$ form a ${\mathscr{C}}$-submodule of
$V$, called the torsion part of $V$ and denoted by $V_{T}$. The quotient
module $V/V_{T}$ is called the torsion free part of $V$. We say that $V$ is
torsion if $V=V_{T}$, and torsion free if $V_{T}=0$. The torsion set of $V$
along the $i$-th direction is

$\operatorname{ts}_{i}(V)=\\{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{C}})\mid\exists\,0\neq
v\in V_{{\mathbf{x}}}\text{ such that there is a map
}f\in{\mathscr{C}}({\mathbf{x}},\,{\mathbf{x}}+{\mathbf{1}}_{i})\text{
satisfying }V_{f}(v)=0\\},$

where ${\mathbf{1}}_{i}$ is the sequence of natural numbers such that the
$i$-th entry is 1 and other entries are 0. Correspondingly, we define the
torsion height along the $i$-th direction of $V$ to be

$\operatorname{ht}_{i}(V)=\sup\\{x_{i}\mid{\mathbf{x}}\in\operatorname{ts}_{i}(V)\\}.$

Again, if $\operatorname{ts}_{i}(V)=\emptyset$, we set
$\operatorname{ht}_{i}(V)=-1$. The torsion height vector is set to be

$\operatorname{ht}(V)=(\operatorname{ht}_{1}(V),\,\operatorname{ht}_{2}(V),\,\ldots).$

######  Remark 2.8.

By definition, $\operatorname{ht}_{i}(V)=n$ if and only if for any morphism
$f:{\mathbf{x}}\to{\mathbf{x}}+{\mathbf{1}}_{i}$ with $x_{i}>n$, the map
$V_{f}:V_{{\mathbf{x}}}\to V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ is injective. We
remind the reader that in general $\operatorname{ht}(V)$ might contain
infinitely many nonzero entries.

##  3\. Representations of $\mathbb{N}^{\infty}$

In this section we let $\mathscr{N}$ be the poset $\mathbb{N}^{\infty}$ and
study its representations. As mentioned before, its representations are
closely related to modules of the polynomial ring
$A=k[X_{1},\,X_{2},\,\ldots]$ with countably many variables. Note that modules
over $A$ coincide with representations of the quiver with one vertex and
countably many loops such that every two loops commute. Since there is a
natural quotient functor from $\mathscr{N}$ to this quiver (viewed as a
category), every module over $A$ can be viewed as a representation of
$\mathscr{N}$. Explicitly, given an $A$-module $M$, one can construct a
representation $V$ of $\mathscr{N}$ by setting $V_{{\mathbf{x}}}=M$ for
${\mathbf{x}}\in\operatorname{Ob}(\mathscr{N})$, and the morphism
$V_{{\mathbf{x}}}\to V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ for each
$i\in\mathbb{N}_{+}$ can be defined by the action of $X_{i}$ on $M$.
Conversely, given a $\mathscr{N}$-module $V$, one can define a multi-graded
$A$-module

$M=\bigoplus_{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})}V_{{\mathbf{x}}},$

and the action of $X_{i}\in A$ on $M$ is given by maps $V_{{\mathbf{x}}}\to
V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ componentwise. These two procedures are
functorial, but are not inverse to each other even up to equivalence.

###  3.1. Bounds for invariants with respect to the sup norm

In this subsection we use the sup norm

$\|{\mathbf{x}}\|=\sup\\{x_{i}\mid i\in\mathbb{N}\\}.$

The following simple observation plays a key role for us to establish upper
bounds for homological invariants of ${\mathscr{N}}$-modules.

######  Lemma 3.1.

Let $V$ be an ${\mathscr{N}}$-module with $\operatorname{gd}(V)=n$, and let
${\mathbf{x}}$ be an object in ${\mathscr{N}}$. If $x_{i}\geqslant n$, then
the map $V_{h}:V_{{\mathbf{x}}}\to V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ is
surjective where $h$ is the unique morphism
${\mathbf{x}}\to{\mathbf{x}}+{\mathbf{1}}_{i}$.

###### Proof.

The conclusion holds trivially if $\operatorname{gd}(V)=-1$ since in this case
$V=0$. Suppose that $n$ is nonnegative. Since
$\|{\mathbf{x}}+{\mathbf{1}}_{i}\|\geqslant x_{i}+1>n$ and
$\operatorname{gd}(V)=n$, it follows that ${\mathbf{x}}+{\mathbf{1}}_{i}$ is
not contained in $\operatorname{supp}(H_{0}(V))$. Consequently, one has

$V_{{\mathbf{x}}+{\mathbf{1}}_{i}}=\sum_{{\mathbf{y}}<{\mathbf{x}}+{\mathbf{1}}_{i},\,\|{\mathbf{y}}\|\leqslant
n\atop
f:{\mathbf{y}}\to{\mathbf{x}}+{\mathbf{1}}_{i}}V_{f}(V_{{\mathbf{y}}}).$

However, ${\mathbf{y}}<{\mathbf{x}}+{\mathbf{1}}_{i}$ and
$\|{\mathbf{y}}\|\leqslant n$ imply that ${\mathbf{y}}\leqslant{\mathbf{x}}$.
It follows that any such $f$ can be expressed as a composite $f=hg$ depicted
as follows

$\textstyle{{\mathbf{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{\mathbf{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\mathbf{x}}+{\mathbf{1}}_{i}.}$

Thus $V_{f}(V_{{\mathbf{y}}})=V_{h}(V_{g}(V_{{\mathbf{y}}}))\subseteq
V_{h}(V_{{\mathbf{x}}})$, and hence
$V_{{\mathbf{x}}+{\mathbf{1}}_{i}}=V_{h}(V_{{\mathbf{x}}})$ as claimed. ∎

As the first application of this lemma, we can use the first two homological
degrees to bound torsion heights.

######  Proposition 3.2.

Let $V$ be an ${\mathscr{N}}$-module. Then for each $i\in\mathbb{N}_{+}$, one
has

$\operatorname{ht}_{i}(V)\leqslant\operatorname{pd}(V)-1.$

###### Proof.

Without loss of generality we can assume that $V$ is nonzero and
$\operatorname{pd}(V)$ is a finite integer $n$. If the conclusion is false,
then by the definition of $\operatorname{ht}_{i}(V)$, we can find an object
${\mathbf{x}}$ such that $x_{i}\geqslant n$ and the map
$V_{h}:V_{{\mathbf{x}}}\to V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ is not
injective.

Consider a short exact sequence $0\to W\to P\to V\to 0$ such that $P$ is a
projective ${\mathscr{N}}$-module with
$\operatorname{gd}(P)=\operatorname{gd}(V)$. It induces the following
commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{h}}$$\textstyle{P_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{h}}$$\textstyle{V_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{h}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{{\mathbf{x}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{{\mathbf{x}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By Lemma 2.2, every morphism in ${\mathscr{N}}$ is a monomorphism, so
$P({\mathbf{x}})=k{\mathscr{N}}({\mathbf{x}},-)$ is torsion free for every
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$. Thus we can assume that $P$
is torsion free, and hence $P_{h}$ is injective. But $V_{h}$ is not injective,
so applying the snake lemma we conclude that $W_{h}$ is not surjective. By
Lemma 3.1, we deduce that

$\operatorname{gd}(W)>x_{i}\geqslant n.$

On the other hand, a long exact sequence argument to the sequence $0\to W\to
P\to V\to 0$ tells us that

$\operatorname{gd}(W)\leqslant\max\\{\operatorname{gd}(P),\,\operatorname{hd}_{1}(V)\\}=\operatorname{pd}(V)=n.$

The conclusion follows by contradiction. ∎

The second application of Lemma 3.1 gives an extremely simple bound of
homological degrees for torsion free ${\mathscr{N}}$-modules.

######  Lemma 3.3.

Let $V$ be a torsion free ${\mathscr{N}}$-module. Then for every $s\geqslant
0$, one has $\operatorname{hd}_{s}(V)\leqslant\operatorname{gd}(V)$.

###### Proof.

Without loss of generality we can assume that $\operatorname{gd}(V)=n$ is a
natural number. It suffices to show the conclusion for $s=1$. Indeed, if this
is true, then take a short exact sequence $0\to W\to P\to V\to 0$ such that
$P$ is a torsion free projective ${\mathscr{N}}$-module with
$\operatorname{gd}(P)=\operatorname{gd}(V)$. Applying the long exact sequence
argument we deduce that

$\operatorname{gd}(W)\leqslant\max\\{\operatorname{gd}(P),\,\operatorname{hd}_{1}(V)\\}=\operatorname{gd}(V).$

Since $W$ is also torsion free, replacing $V$ by $W$ we conclude that

$\operatorname{hd}_{2}(V)=\operatorname{hd}_{1}(W)\leqslant\operatorname{gd}(W)\leqslant\operatorname{gd}(V).$

Then conclusion then follows by a recursive argument.

If the conclusion is false for $s=1$, then by the previous argument, we have
$\operatorname{gd}(W)\geqslant n+1$, so there exists an object
${\mathbf{x}}\in\operatorname{supp}(H_{0}(W))$ such that $x_{i}\geqslant n+1$
for a certain $i\in\mathbb{N}_{+}$. We consider the following commutative
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}-{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{h}}$$\textstyle{P_{{\mathbf{x}}-{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{P_{h}}$$\textstyle{V_{{\mathbf{x}}-{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{h}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that all vertical maps are injective since $V$ is torsion free.
Furthermore, since $({\mathbf{x}}-{\mathbf{1}}_{i})_{i}=x_{i}-1\geqslant n$,
Lemma 3.1 tells us that $P_{h}$ and $V_{h}$ are surjective. Consequently,
$W_{h}$ is surjective as well, so $(H_{0}(W))_{{\mathbf{x}}}=0$ and hence
${\mathbf{x}}\notin\operatorname{supp}(H_{0}(W))$. The conclusion follows by
contradiction. ∎

An application of the above result in commutative algebra is as follows.

######  Corollary 3.4.

Let $A=k[X_{1},\,X_{2},\,\ldots]$ be the polynomial ring with countably many
variables, and let $I\lhd A$ be an ideal generated by monomials such that the
degree of each $X_{i}$ in these monomials is bounded by $n$. Then $I$ has a
projective resolution $P^{\bullet}\to I\to 0$ such that each $P^{j}$ is a
direct sum of principal ideals of $A$ generated by a monomial in which the
degree of each $X_{i}$ is at most $n$.

###### Proof.

We can identify $I$ as a submodule $V$ of the ${\mathscr{N}}$-module
$k{\mathscr{N}}(\boldsymbol{0},\,-)$ in an obviously way. By the above lemma,
one can construct a projective resolution $Q^{\bullet}\to V\to 0$ of
${\mathscr{N}}$-modules such that $Q^{i}$ is isomorphic to a direct sum of
$k{\mathscr{N}}({\mathbf{x}},\,-)$ such that $\|{\mathbf{x}}\|\leqslant n$ for
each $i\in\mathbb{N}$. Note that each $k{\mathscr{N}}({\mathbf{x}},\,-)$ can
be identified with the principal ideal of $A$ generated by
$X_{1}^{x_{1}}X_{2}^{x_{2}}\ldots$ which is a monomial since $x_{i}=0$ for
$i\gg 0$. The conclusion follows from these observations. ∎

We generalize the conclusion of the above lemma to the general situation.
Recall that an ${\mathscr{N}}$-module $V$ is presented in finite degrees if
$\operatorname{pd}(V)$ is finite.

######  Proposition 3.5.

Let $V$ be an ${\mathscr{N}}$-module. Then
$\operatorname{hd}_{s}(V)\leqslant\operatorname{pd}(V)$ for $s\geqslant 0$. In
particular, the category of ${\mathscr{N}}$-modules presented in finite
degrees is abelian.

###### Proof.

The inequality holds trivially for $s\leqslant 1$. Take a short exact sequence
$0\to W\to P\to V\to 0$ such that $P$ is a torsion free projective
${\mathscr{N}}$-module and $\operatorname{gd}(P)=\operatorname{gd}(V)$. For
$s\geqslant 2$, by Lemma 3.3 one has

$\operatorname{hd}_{s}(V)=\operatorname{hd}_{s-1}(W)\leqslant\operatorname{gd}(W)\leqslant\max\\{\operatorname{gd}(V),\,\operatorname{hd}_{1}(V)\\}=\operatorname{pd}(V),$

so the first statement is valid.

Let $0\to U\to V\to W\to 0$ be a short exact sequence of
${\mathscr{N}}$-modules. By the induced long exact sequence of homology
groups, one has

$\operatorname{hd}_{s}(V)\leqslant\max\\{\operatorname{hd}_{s}(U),\,\operatorname{hd}_{s}(W)\\}$

for $s\geqslant 0$. Consequently, if both $U$ and $W$ are presented in finite
degrees, then their homological degrees are all finite. By the above
inequality, homological degrees of $V$ are also finite, and in particular $V$
is presented in finite degrees. Using the same argument, one can show that if
any two of the three terms in this sequence are presented in finite degrees,
so is the third one. The second statement follows from this observation. ∎

###  3.2. Bounds for invariants with respect to the sum norm

It might seem to the reader that some results established in previous
subsection with respect to the sup norm are too strong to be interesting, for
instance Proposition 3.5. The reason is that the sup norm is too weak, and
hence too many distinct objects share the same norm. For example, objects in
the following set

$\\{{\mathbf{1}}_{1}+{\mathbf{1}}_{2}+\ldots+{\mathbf{1}}_{s}\mid
s\in\mathbb{N}_{+}\\}$

all have norm 1, which seems quite unnatural. Indeed, this norm does not
respect the poset structure of ${\mathscr{N}}$; that is, one does not have
$\|{\mathbf{x}}\|<\|{\mathbf{y}}\|$ when ${\mathbf{x}}<{\mathbf{y}}$.

In this subsection we consider the more reasonable sum norm. Throughout this
subsection all homological degrees are defined with respect to the sum norm,
but we use the same notation. Hopefully this would not cause too much
confusion to the reader. Unfortunately, the situation becomes much more
complicated. For instance, in contrast to Proposition 3.5, the category of
${\mathscr{N}}$-modules presented in finite degrees with respect to the sum
norm is no longer abelian, as illustrated by the following example.

######  Example 3.6.

Let $k$ be a field and $P$ be the direct sum of countably many copies of
$P(\boldsymbol{0})$. Choose a basis $\\{b_{1},\,b_{2},\,\ldots\\}$ for
$P_{\boldsymbol{0}}$. For each
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$, $P_{{\mathbf{x}}}$ has a
basis consisting of elements $P_{f}(b_{i})$, $i\geqslant 0$, where $f$ is the
unique morphism from $\boldsymbol{0}$ to ${\mathbf{x}}$. By abuse of notation
we denote $V_{f}(b_{i})$ by $b_{i}$ again. Define a submodule $V$ of $P$ as
follows:

  * •

$V_{\boldsymbol{0}}=0$ and

$V_{{\mathbf{1}}_{s}}=\begin{cases}\langle b_{(i+1)/2}\rangle,&\text{if
$s=2i-1$ is odd};\\\ \langle
b_{p_{i}}+b_{{p_{i}}^{2}}+\ldots+b_{p_{i}^{i+1}}\rangle;&\text{if $s=2i$ is
even}\end{cases}$

where $p_{i}$ is the $i$-th prime number;

  * •

$V$ is generated in degree 1.

For instance, we have $V_{{\mathbf{1}}_{1}}=\langle b_{1}\rangle$,
$V_{{\mathbf{1}}_{2}}=\langle b_{2}+b_{4}\rangle$,
$V_{{\mathbf{1}}_{3}}=\langle b_{2}\rangle$, and $V_{{\mathbf{1}}_{4}}=\langle
b_{3}+b_{9}+b_{27}\rangle$.

Clearly, both $P$ and $P/V$ are presented in finite degrees. However, this is
not the case for the submodule $V$ of $P$. Indeed, we can construct a short
exact sequence

$0\to K\to\bigoplus_{s\in\mathbb{N}_{+}}P({\mathbf{1}}_{s})\to V\to 0.$

A direct computation shows that for any $s\in\mathbb{N}_{+}$, we can find an
object ${\mathbf{x}}$ with sum norm $p_{s}+1$ such that
${\mathbf{x}}\in\operatorname{supp}(H_{0}(K))$. For examples:

  * •

$s=1$, $p_{1}=2$, ${\mathbf{x}}=(0,\,1,\,1,\,0,\,0,\,0,\,1,\,0,\,\ldots)$,
$\dim_{k}(H_{0}(K))_{{\mathbf{x}}}=1$;

  * •

$s=2$, $p_{2}=3$,
${\mathbf{x}}={\mathbf{1}}_{4}+{\mathbf{1}}_{5}+{\mathbf{1}}_{17}+{\mathbf{1}}_{53}$,
$\dim_{k}(H_{0}(K))_{{\mathbf{x}}}=1$.

Thus $\operatorname{gd}(K)=\infty$, and hence $V$ is not presented in finite
degrees.

This example tells us that it is impossible to bound higher homological
degrees (even the second homological degree) of ${\mathscr{N}}$-modules $V$
presented in finite degrees in terms of its presentation degree. However, if
$k$ is a field and $V$ is moreover finitely generated111The condition that $V$
is finitely generated and presented in finite degrees does not imply that $V$
is finitely presented., then its second homological degree is finite. In the
rest of this section we give a proof for this fact.

######  Lemma 3.7.

Let $k$ be a field and let $V$ be a submodule of $P({\mathbf{x}})$ for a
certain ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$. Then

$\operatorname{hd}_{1}(V)\leqslant 2\operatorname{gd}(V).$

In particular, if $V$ is finitely generated, then it is finitely presented.

###### Proof.

Without loss of generality we assume that $V$ is nonzero,
$\operatorname{gd}(V)=n$ is finite, and ${\mathbf{x}}=\boldsymbol{0}$ since
$P({\mathbf{x}})$ can be identified with a submodule of $P(\boldsymbol{0})$.
Let $S=\operatorname{supp}(H_{0}(V))$, which is nonempty, and let
$V({\mathbf{x}})$ be the submodule of $V$ generated by $V_{{\mathbf{x}}}\cong
k$. Then

$V=\sum_{{\mathbf{x}}\in S}V({\mathbf{x}}).$

Since $V({\mathbf{x}})$ is torsion free, it is actually isomorphic to
$P({\mathbf{x}})$. Thus we obtain a surjection

$Q=\bigoplus_{{\mathbf{x}}\in
S}V({\mathbf{x}})\longrightarrow\sum_{{\mathbf{x}}\in S}V({\mathbf{x}})=V,$

and hence a short exact sequence $0\to K\to Q\to V\to 0$, from which we deduce
that $\operatorname{hd}_{1}(V)\leqslant\operatorname{gd}(K)$.

Now we investigate $K$. Fix ${\mathbf{z}}\in\operatorname{Ob}({\mathscr{N}})$
and let

$T=S\cap\\{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})\mid{\mathbf{x}}\leqslant{\mathbf{z}}\\}$

which is a finite set. We have

$K_{{\mathbf{z}}}=\\{(V_{{\mathbf{x}}\to{\mathbf{z}}}(v_{{\mathbf{x}}}))_{{\mathbf{x}}\in
T}\mid v_{{\mathbf{x}}}\in V_{{\mathbf{x}}},\,\sum_{{\mathbf{x}}\in
T}V_{{\mathbf{x}}\to{\mathbf{z}}}(v_{{\mathbf{x}}})=0\\},$

where ${\mathbf{x}}\to{\mathbf{z}}$ is the morphism from ${\mathbf{x}}$ to
${\mathbf{z}}$. Since $V$ is a submodule of $P=P(\boldsymbol{0})$, after
choosing a nonzero element $v\in P_{\boldsymbol{0}}$, we have
$v_{{\mathbf{x}}}=\lambda_{{\mathbf{x}}}P_{\boldsymbol{0}\to{\mathbf{x}}}(v)$
for a certain $\lambda_{{\mathbf{x}}}\in k$ and hence
$V_{{\mathbf{x}}\to{\mathbf{z}}}(v_{{\mathbf{x}}})=\lambda_{{\mathbf{x}}}P_{\boldsymbol{0}\to{\mathbf{z}}}(v)$.
Consequently,

$K_{{\mathbf{z}}}=\\{(\lambda_{{\mathbf{x}}}P_{\boldsymbol{0}\to{\mathbf{z}}}(v))_{{\mathbf{x}}\in
T}\mid\sum_{{\mathbf{x}}\in T}\lambda_{{\mathbf{x}}}=0\\}.$

Clearly, this space is spanned by elements of the following form:

$u=(0,\,\ldots,\,0,\,P_{\boldsymbol{0}\to{\mathbf{z}}}(v)=V_{{\mathbf{x}}\to{\mathbf{z}}}(\frac{v_{{\mathbf{x}}}}{\lambda_{{\mathbf{x}}}}),\,0,\,\ldots,\,0,\,-P_{\boldsymbol{0}\to{\mathbf{z}}}(v)=-V_{{\mathbf{y}}\to{\mathbf{z}}}(\frac{v_{{\mathbf{y}}}}{\lambda_{{\mathbf{y}}}}),\,0,\ldots,\,0)$

where ${\mathbf{x}}$ and ${\mathbf{y}}$ are two distinct elements in $T$ and
$\lambda_{{\mathbf{x}}}\neq 0\neq\lambda_{{\mathbf{y}}}$. However, by the
commutative diagram

$\textstyle{\boldsymbol{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbf{x}}\vee{\mathbf{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{z}}$

We know that

$w=(0,\,\ldots,\,0,\,V_{{\mathbf{x}}\to{\mathbf{x}}\vee{\mathbf{y}}}(\frac{v_{{\mathbf{x}}}}{\lambda_{{\mathbf{x}}}}),\,0,\,\ldots,\,0,\,-V_{{\mathbf{y}}\to{\mathbf{x}}\vee{\mathbf{y}}}(\frac{v_{{\mathbf{y}}}}{\lambda_{{\mathbf{y}}}}),\,0,\ldots,\,0)\in
K_{{\mathbf{x}}\vee{\mathbf{y}}},$

where ${\mathbf{x}}\vee{\mathbf{y}}$ is the least upper bound of
${\mathbf{x}}$ and ${\mathbf{y}}$. Furthermore,
$K_{{\mathbf{x}}\vee{\mathbf{y}}\to{\mathbf{z}}}$ sends $w$ to $u$.
Consequently, $K_{{\mathbf{z}}}$ can be generated by the values of $K$ on
objects ${\mathbf{x}}\vee{\mathbf{y}}$ with ${\mathbf{x}},\,{\mathbf{y}}\in
T$. Since
$\|{\mathbf{x}}\vee{\mathbf{y}}\|\leqslant\|{\mathbf{x}}\|+\|{\mathbf{y}}\|\leqslant
2n$, the desired inequality follows.

If $V$ is finitely generated (or equivalently, $V$ is generated by its values
on finitely many objects and the values on these objects are finite
dimensional), then $S$ is a finite set, and for each ${\mathbf{x}}\in S$, the
dimension of $V_{{\mathbf{x}}}$ is finite. Consequently, the dimensions of
$Q_{{\mathbf{x}}}$ and $K_{{\mathbf{x}}}$ for every
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$ are finite as well.
Furthermore, there are only finitely many ${\mathbf{x}}\vee{\mathbf{y}}$ with
${\mathbf{x}},{\mathbf{y}}\in S$. Since $K$ can be generated by its values on
these objects, we deduce that $K$ is finitely generated, and hence $V$ is
finitely presented. ∎

######  Remark 3.8.

The second statement of this lemma means that the category algebra
$k{\mathscr{N}}$ is locally left coherent. Consequently, finitely presented
${\mathscr{N}}$-modules coincide with coherent modules, and these modules form
an abelian subcategory of ${\mathscr{N}}\operatorname{-Mod}$. This result can
be deduced from the well known fact that the polynomial ring
$A=k[X_{1},\,X_{2},\,\ldots]$ is a coherent ring and
${\mathscr{N}}\operatorname{-Mod}$ can be viewed as a full subcategory of
$A\operatorname{-Mod}$. However, the inequality in this lemma holds in a more
general setup; that is, we do not need to impose the condition that $V$ is
generated by its values on finitely many objects, or the even stronger
condition that $V$ is finitely generated.

######  Proposition 3.9.

Let $k$ be a field, $V$ an ${\mathscr{N}}$-module and
$\omega=\dim_{k}(H_{0}(V))$. Then

$\operatorname{hd}_{2}(V)\leqslant 2^{\omega}\operatorname{hd}_{1}(V).$

###### Proof.

Without loss of generality one can assume that both $\omega$ and
$\operatorname{hd}_{1}(V)$ are finite. One can find a short exact sequence
$0\to W\to P\to V\to 0$ such that $P$ is a coproduct of $\omega$ copies of
${\mathscr{N}}$-modules of the form $P({\mathbf{x}})$. Furthermore, one can
assume that $H_{0}(P)\cong H_{0}(V)$, and hence
$\operatorname{gd}(W)=\operatorname{hd}_{1}(V)$.

When $\omega=1$, by the previous lemma, one has

$\operatorname{hd}_{2}(V)=\operatorname{hd}_{1}(W)\leqslant
2\operatorname{gd}(W)=2\operatorname{hd}_{1}(V),$

so the conclusion holds. Now assume that $\omega\geqslant 2$.

By taking a copy $P^{\prime}=P({\mathbf{x}})$ inside $P$ and letting
$W^{\prime}=P^{\prime}\cap W$, we get a commutative diagram of short exact
sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since
$\dim_{k}(H_{0}(V^{\prime\prime}))\leqslant\dim_{k}(H_{0}(P^{\prime\prime}))=\omega-1$
and
$\operatorname{hd}_{1}(V^{\prime\prime})\leqslant\operatorname{gd}(W^{\prime\prime})\leqslant\operatorname{gd}(W)=\operatorname{hd}_{1}(V)$,
the induction hypothesis tells us that

$\operatorname{hd}_{2}(V^{\prime\prime})\leqslant
2^{\omega-1}\operatorname{hd}_{1}(V^{\prime\prime})\leqslant
2^{\omega-1}\operatorname{gd}(W^{\prime\prime})\leqslant
2^{\omega-1}\operatorname{hd}_{1}(V).$

By looking at the first column, we deduce that

$\operatorname{hd}_{1}(V^{\prime})\leqslant\operatorname{gd}(W^{\prime})\leqslant\max\\{\operatorname{gd}(W),\,\operatorname{hd}_{1}(W^{\prime\prime})\\}\leqslant\max\\{\operatorname{gd}(W),\,2^{\omega-1}\operatorname{hd}_{1}(V)\\}=2^{\omega-1}\operatorname{hd}_{1}(V).$

Again, by the established conclusion for $\omega=1$, we have
$\operatorname{hd}_{2}(V^{\prime})\leqslant
2\operatorname{hd}_{1}(V^{\prime})\leqslant
2^{\omega}\operatorname{hd}_{1}(V)$. By looking at the last column, we get the
desired upper bound. ∎

###  3.3. Shift functors

In this subsection we introduce shift functors and describe some elementary
properties. Throughout this subsection all numerical invariants are defined
with respect to the sup norm.

Note that for each positive integer $i$ there is a functor
$\iota_{i}:{\mathscr{N}}\to{\mathscr{N}}$ sending
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$ to
${\mathbf{x}}+{\mathbf{1}}_{i}$, and the unique morphism
${\mathbf{x}}\to{\mathbf{y}}$ to the unique morphism
${\mathbf{x}}+{\mathbf{1}}_{i}\to{\mathbf{y}}+{\mathbf{1}}_{i}$. The shift
functor
$\Sigma_{i}:{\mathscr{N}}\operatorname{-Mod}\to{\mathscr{N}}\operatorname{-Mod}$
along the $i$-th direction is the pullback of $\iota_{i}$, mapping an
${\mathscr{N}}$-module $V$ to $V\circ\iota_{i}$. The following lemma collects
some elementary properties of $\Sigma_{i}$.

######  Lemma 3.10.

Let $\Sigma_{i}$ be the $i$-th shift functor. Then:

  1. (1)

$\Sigma_{i}$ is exact;

  2. (2)

$\Sigma_{i}\circ\Sigma_{j}=\Sigma_{j}\circ\Sigma_{i}$;

  3. (3)

there is a natural transformation $\phi^{i}$ between the identity functor on
${\mathscr{N}}\operatorname{-Mod}$ and $\Sigma_{i}$;

  4. (4)

the kernel $K_{i}V$ of the map $V\to\Sigma_{i}V$ has the following description

$K_{i}V=\bigoplus_{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})}\\{v\in
V_{{\mathbf{x}}}\mid v\text{ is sent to 0 by the map
}{\mathbf{x}}\to{\mathbf{x}}+{\mathbf{1}}_{i}\\}.$

  5. (5)

the natural maps $V\to\Sigma_{i}V$ is injective for each $i\in\mathbb{N}$ if
and only if $V$ is torsion free;

  6. (6)

for $P({\mathbf{x}})=k{\mathscr{N}}({\mathbf{x}},-)$, one has

$\Sigma_{i}P({\mathbf{x}})\cong\begin{cases}P({\mathbf{x}}-{\mathbf{1}}_{i}),&\text{if
}x_{i}\geqslant 1;\\\ P({\mathbf{x}}),&\text{else.}\end{cases}$

###### Proof.

The first two statements follow readily from the definition. The third
statement follows from the existence of a natural transformation between the
identity functor on ${\mathscr{N}}$ and $\iota_{i}$, illustrated by the
diagram

$\textstyle{{\mathbf{x}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{{\mathbf{y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\iota_{i}{\mathbf{x}}={\mathbf{x}}+{\mathbf{1}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{i}f}$$\textstyle{\iota_{i}{\mathbf{y}}={\mathbf{y}}+{\mathbf{1}}_{i}.}$

This diagram also implies the forth statement, which in turn applies the fifth
statement. The last statement holds by a very simple observation. ∎

Now we construct the shift functor along all directions, which intuitively can
be viewed as an infinite composite of $\Sigma_{i}$ for all $i\in\mathbb{N}$.
By statement (3) of Lemma 3.10, one can construct a sequence

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{V}^{1}}$$\textstyle{\Sigma_{1}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{2}_{\Sigma_{1}V}}$$\textstyle{\Sigma_{[2]}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{3}_{\Sigma_{[2]}V}}$$\textstyle{\Sigma_{[3]}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots,}$

where $\Sigma_{[n]}=\Sigma_{n}\circ\Sigma_{n-1}\circ\ldots\circ\Sigma_{1}$ for
$n\geqslant 1$. Taking the colimit of this diagram we obtain an
${\mathscr{N}}$-module. Clearly, the rule assigning $V$ to this colimit is
functorial, so induces an endofunctor in ${\mathscr{N}}\operatorname{-Mod}$
denoted by $\boldsymbol{\Sigma}$. Furthermore, there is a natural map from $V$
to $\boldsymbol{\Sigma}V$ such that the following diagram commutes

(3.1) |
---|---
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{1}_{V}}$$\scriptstyle{\psi}$$\textstyle{\Sigma_{1}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{2}_{\Sigma_{1}V}}$$\scriptstyle{\psi^{1}}$$\textstyle{\Sigma_{[2]}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{2}}$$\textstyle{\ldots}$$\textstyle{\boldsymbol{\Sigma}V.}$

We describe some useful facts about $\boldsymbol{\Sigma}$ in the following
lemma.

######  Lemma 3.11.

Suppose that $V$ is a nonzero ${\mathscr{N}}$-module. One has:

  1. (1)

$\boldsymbol{\Sigma}$ is an exact functor;

  2. (2)

$\operatorname{gd}(\boldsymbol{\Sigma}V)\leqslant\operatorname{gd}(V)$, and
$\operatorname{gd}(\boldsymbol{\Sigma}V)\leqslant\operatorname{gd}(V)-1$ when
$\operatorname{gd}(V)\geqslant 1$;

  3. (3)

for $i\in\mathbb{N}_{+}$, one has
$\operatorname{ht}_{i}(\boldsymbol{\Sigma}V)\leqslant\operatorname{ht}_{i}(V)-1$
whenever $\operatorname{ht}_{i}(V)\geqslant 0$;

  4. (4)

the map $\psi$ is injective if and only if $V$ is torsion free;

  5. (5)

if $V$ is torsion free, so is $\boldsymbol{\Sigma}V$.

  6. (6)

if $V$ is torsion, so is $\boldsymbol{\Sigma}V$.

###### Proof.

All facts follow from standard arguments about direct limits.

(1) This is clear since $\boldsymbol{\Sigma}$ is the direct limit of a
sequence of exact functors, and hence is exact.

(2) First we prove the conclusion for the special case that
$V=P({\mathbf{x}})$. Let $N$ be the minimal integer such that $x_{i}=0$ for
$i>N$. By (6) of Lemma 3.10, $\Sigma_{[i]}V\cong P({\mathbf{y}})$ for
$i\geqslant N$ where

$y_{j}=\begin{cases}x_{j},&\text{if }x_{j}=0;\\\ x_{j}-1,&\text{if
}x_{j}\geqslant 1.\end{cases}$

Furthermore, all but finitely many maps appearing in diagram (3.1) are
isomorphisms. Consequently, $\boldsymbol{\Sigma}P({\mathbf{x}})\cong
P({\mathbf{y}})$, and $\|{\mathbf{y}}\|=\|{\mathbf{x}}\|-1$ whenever
$\|{\mathbf{x}}\|\geqslant 1$, so the conclusion holds for this special case.

Now consider the general case. Take a surjection

$P=\bigoplus_{{\mathbf{x}}\in\operatorname{supp}(H_{0}(V))}P({\mathbf{x}})^{c_{{\mathbf{x}}}}\to
V$

where the multiplicity $c_{{\mathbf{x}}}$ might be infinity. It is clear that
$\operatorname{gd}(P)=\operatorname{gd}(V)$. Applying $\boldsymbol{\Sigma}$ to
it we obtain a surjection $\boldsymbol{\Sigma}P\to\boldsymbol{\Sigma}V$. If
$\operatorname{gd}(V)\geqslant 1$, the conclusion for the special case asserts
that $\operatorname{gd}(\boldsymbol{\Sigma}P)=\operatorname{gd}(P)-1$, so

$\operatorname{gd}(\boldsymbol{\Sigma}V)\leqslant\operatorname{gd}(\boldsymbol{\Sigma}P)=\operatorname{gd}(P)-1=\operatorname{gd}(V)-1$

as desired.

(3) Without loss of generality we can assume that
$\operatorname{ht}_{i}(V)=n\geqslant 0$ is finite. By Remark 2.8, for each
object ${\mathbf{x}}$ with $x_{i}\geqslant n+1$, the map
$V_{f}:V_{{\mathbf{x}}}\to V_{{\mathbf{x}}+{\mathbf{1}}_{i}}$ is injective.
Then when $s\geqslant i$, for any object ${\mathbf{y}}$ with $y_{i}\geqslant
n$, the map
$(\Sigma_{[s]}V)_{{\mathbf{y}}}\to(\Sigma_{[s]}V)_{{\mathbf{y}}+{\mathbf{1}}_{i}}$
is injective by Remark 2.8 since
$(\Sigma_{[s]}V)_{{\mathbf{y}}}=V_{{\mathbf{z}}}$ where

$z_{j}=\begin{cases}y_{j}+1,&j\leqslant s;\\\ y_{j},&j\geqslant
s+1,\end{cases}$

and in particular $z_{i}=y_{i}+1\geqslant n+1$. Taking direct limit for the
commutative diagram

$\textstyle{V_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Sigma_{1}V)_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Sigma_{[2]}V)_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{V_{{\mathbf{y}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Sigma_{1}V)_{{\mathbf{y}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Sigma_{[2]}V)_{{\mathbf{y}}+{\mathbf{1}}_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$

and noting that all but finitely many vertical maps are injective, we conclude
that the map
$(\boldsymbol{\Sigma}V)_{{\mathbf{y}}}\to(\boldsymbol{\Sigma}V)_{{\mathbf{y}}+{\mathbf{1}}_{i}}$
is also injective. Again, by Remark 2.8, it follows that
$\operatorname{ht}_{i}(\boldsymbol{\Sigma}V)\leqslant\operatorname{ht}_{i}(V)-1$.

(4) If $V$ is not torsion free, then by statement (5) of Lemma 3.10, there is
a certain $i\in\mathbb{N}_{+}$ such that the natural map $V\to\Sigma_{i}V$ is
not injective; that is, $\operatorname{ht}_{i}(V)\geqslant 0$. It is easy to
check that $\operatorname{ht}_{i}(\Sigma_{[i]}V)<\operatorname{ht}_{i}(V)$.
Consequently, in the top row of diagram (3.1), the composite map from $V$ to
$\Sigma_{[i]}V$ is not injective, and hence the map $\psi$ cannot be
injective.

Conversely, if $V$ is torsion free, then all maps in the top row of diagram
(3.1) are injective. If $\psi$ is not injective, then one may find an object
${\mathbf{x}}$ and a certain $0\neq v\in V_{{\mathbf{x}}}$ such that $v$
eventually becomes 0 by maps in the top row, which is impossible.

(5) This is clear. Indeed, if $V$ is torsion free, then by the previous
statement, the natural map $V\to\boldsymbol{\Sigma}V$ is injective. Since
$\boldsymbol{\Sigma}$ is exact, the natural map
$\boldsymbol{\Sigma}V\to\boldsymbol{\Sigma}(\boldsymbol{\Sigma}V)$ is also
injective. Again by the previous statement, $\boldsymbol{\Sigma}V$ is torsion
free.

(6) For each object ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$ as well
as an arbitrary element $v\in(\boldsymbol{\Sigma}V)_{{\mathbf{x}}}$, we want
to show that $v$ is torsion. By definition,
$(\boldsymbol{\Sigma}V)_{{\mathbf{x}}}$ is the direct limit of the following
sequence

$V_{{\mathbf{x}}}\to(\Sigma_{1}V)_{{\mathbf{x}}}\to(\Sigma_{[2]}V)_{{\mathbf{x}}}\to\ldots.$

Then for a sufficiently large $n$, we can find a representative
$\tilde{v}\in(\Sigma_{[n]}V)_{{\mathbf{x}}}$ of $v$. It is easy to see that
$\Sigma_{[n]}V$ is torsion, so we can find another object ${\mathbf{y}}$ such
that the map $f:{\mathbf{x}}\to{\mathbf{y}}$ sends $\tilde{v}$ to
$0\in(\Sigma_{[n]}V)_{{\mathbf{y}}}$. By looking at the commutative diagram

$\textstyle{(\Sigma_{[n]}V)_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\Sigma_{[n]}V)_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\boldsymbol{\Sigma}V)_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\boldsymbol{\Sigma}V)_{{\mathbf{y}}},}$

we conclude that $v$ is sent to 0 by $f$ as well. ∎

######  Remark 3.12.

In general one can not expect that
$\operatorname{gd}(\boldsymbol{\Sigma}V)=\operatorname{gd}(V)-1$ if
$\operatorname{gd}(V)\geqslant 1$. For example, let $V$ be a torsion free
module such that $V_{{\mathbf{x}}}=0$ if $\|{\mathbf{x}}\|\leqslant 1$, and
$V_{{\mathbf{x}}}=k$ otherwise. Then $\operatorname{gd}(V)\geqslant 2$ since
clearly the object $(2,\,0,\,0,\ldots)$ is contained in
$\operatorname{supp}(H_{0}(V))$. However, a direct computation shows that
$\boldsymbol{\Sigma}V\cong k{\mathscr{N}}(\boldsymbol{0},-)$, so
$\operatorname{gd}(\boldsymbol{\Sigma}V)=0$.

######  Remark 3.13.

Let $\boldsymbol{K}$ be the kernel functor induced by the natural
transformation $\psi$ in diagram (3.1). It is easy to see that an element
$v\in V_{{\mathbf{x}}}$ lies in $(\boldsymbol{K}V)_{{\mathbf{x}}}$ if there is
a non-invertible morphism $f:{\mathbf{x}}\to{\mathbf{y}}$ such that
$x_{i}\leqslant y_{i}\leqslant x_{i}+1$ for $i\in\mathbb{N}_{+}$ and
$V_{f}(v)=0$.

Now we introduce a type of special ${\mathscr{N}}$-modules, which is a
generalization of projective ${\mathscr{N}}$-modules.

######  Definition 3.14.

Let ${\mathscr{N}}_{{\mathbf{x}}}$ be the subcategory of ${\mathscr{N}}$
consisting of the single object ${\mathbf{x}}$. A ${\mathscr{N}}$-module $V$
is said to be induced at ${\mathbf{x}}$ if $V$ is isomorphic to the left Kan
extension of $V_{{\mathbf{x}}}$ with respect to the inclusion functor
${\mathscr{N}}_{{\mathbf{x}}}\to{\mathscr{N}}$; that is, $V\cong
k{\mathscr{N}}\otimes_{k{\mathscr{N}}_{{\mathbf{x}}}}V_{{\mathbf{x}}}$.

If $V$ is induced at ${\mathbf{x}}$, then for
${\mathbf{y}}\in\operatorname{Ob}({\mathscr{N}})$, $V_{{\mathbf{y}}}\cong
V_{{\mathbf{x}}}$ if ${\mathbf{y}}\geqslant{\mathbf{x}}$, and
$V_{{\mathbf{y}}}=0$ otherwise. Correspondingly, each map $V_{f}$ is either an
isomorphism or the zero map.

######  Proposition 3.15.

Let $V$ be an ${\mathscr{N}}$-module presented in finite degrees. Then for
$s\geqslant\operatorname{pd}(V)$, $\boldsymbol{\Sigma}^{s}V$ is either the
zero module or induced at $\boldsymbol{0}$, where $\boldsymbol{\Sigma}^{s}$ is
the composite of $s$ copies of $\boldsymbol{\Sigma}$.

###### Proof.

When $V$ is torsion, $\boldsymbol{\Sigma}^{s}V$ is also torsion by statement
(6) of Lemma 3.11. However, by statement (3) of this lemma as well as
Proposition 3.2, we know that
$\operatorname{ht}_{i}(\boldsymbol{\Sigma}^{s}V)=-1$ for all $i\geqslant 1$;
that is, $\boldsymbol{\Sigma}^{s}V$ is torsion free. This happens if and only
if $\boldsymbol{\Sigma}^{s}V$ is the zero module.

If $V$ is not torsion free, we consider the short exact sequence $0\to
V_{T}\to V\to V_{F}\to 0$ where $V_{T}$ is the torsion part and $V_{F}$ is the
torsion free part. Applying $\boldsymbol{\Sigma}^{s}$ to it and using the same
argument as in the previous paragraph we deduce that
$\boldsymbol{\Sigma}^{s}V_{T}=0$, so
$\boldsymbol{\Sigma}^{s}V\cong\boldsymbol{\Sigma}^{s}V_{F}$, which is torsion
free by statement (5) of Lemma 3.11, and whose generation degree is at most 0
by statement (2) of Lemma 3.11. Since $\boldsymbol{\Sigma}^{s}V_{F}$ cannot be
the zero module, it follows that it is a torsion free ${\mathscr{N}}$-module
with generation degree 0, which is clearly induced at $\boldsymbol{0}$. ∎

######  Corollary 3.16.

If $V$ is a torsion free ${\mathscr{N}}$-module presented in finite degrees,
then it can be embedded into an ${\mathscr{N}}$-module induced at
$\boldsymbol{0}$.

###### Proof.

Let $s=\operatorname{pd}(V)$. The composite of the following sequence is an
injective map

$V\to\boldsymbol{\Sigma}V\to\boldsymbol{\Sigma}^{2}V\to\ldots\to\boldsymbol{\Sigma}^{s}V$

with $\boldsymbol{\Sigma}^{s}V$ the desired ${\mathscr{N}}$-module induced at
$\boldsymbol{0}$. ∎

In the rest of this subsection we want to generalize the above results to
arbitrary ${\mathscr{N}}$-modules. For this purpose, we need to introduce the
“infinite product” of $\boldsymbol{\Sigma}$. Given an ${\mathscr{N}}$-module
$V$, we have a sequence of natural maps

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{V}}$$\textstyle{\boldsymbol{\Sigma}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\boldsymbol{\Sigma}V}}$$\textstyle{\boldsymbol{\Sigma}^{2}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots.}$

We denote the colimit of this sequence in ${\mathscr{N}}\operatorname{-Mod}$
by $\boldsymbol{\Sigma}^{\infty}V$. Clearly, this is a functorial
construction, and there is a natural map
$\boldsymbol{\psi}_{V}:V\to\boldsymbol{\Sigma}^{\infty}V$. Moreover,
$\boldsymbol{\Sigma}^{\infty}$ is an exact functor.

######  Remark 3.17.

We construct $\boldsymbol{\Sigma}^{\infty}$ as the direct limit of
$\boldsymbol{\Sigma}$, which itself is a direct limit. Actually, one can
define $\boldsymbol{\Sigma}^{\infty}$ in one step. Given an
${\mathscr{N}}$-module $V$, we define a functor
$\boldsymbol{V}:{\mathscr{N}}\to{\mathscr{N}}\operatorname{-Mod}$ as follows:
for
${\mathbf{x}}=(x_{1},\,\ldots\,x_{r},\,0,\,0,\,\ldots)\in\operatorname{Ob}({\mathscr{N}})$,
define
$\boldsymbol{V}_{{\mathbf{x}}}=\Sigma_{1}^{x_{1}}\ldots\Sigma_{r}^{x_{r}}V$.
When ${\mathbf{x}}\leqslant{\mathbf{y}}$, the morphism from
$\boldsymbol{V}_{{\mathbf{x}}}$ to $\boldsymbol{V}_{{\mathbf{y}}}$ is given by
natural transformations between the functor
$\Sigma_{1}^{x_{1}}\ldots\Sigma_{r}^{x_{r}}$ and the functor
$\Sigma_{1}^{y_{1}}\ldots\Sigma_{r}^{y_{s}}$, which is well defined by
statement (2) of Lemma 3.10. One can check that the colimit of
$\boldsymbol{V}$ in ${\mathscr{N}}\operatorname{-Mod}$ is precisely
$\boldsymbol{\Sigma}^{\infty}V$. We will give one more description for
$\boldsymbol{\Sigma}^{\infty}$ in Proposition 6.3.

######  Proposition 3.18.

Let $V$ be an ${\mathscr{N}}$-module. One has:

  1. (1)

If $V$ is torsion, then $\boldsymbol{\Sigma}^{\infty}V=0$; otherwise,
$\boldsymbol{\Sigma}^{\infty}V$ is induced at $\boldsymbol{0}$.

  2. (2)

If $V$ is torsion free, then it can be embedded into
$\boldsymbol{\Sigma}^{\infty}V$.

###### Proof.

(1) Suppose that $V$ is torsion. Then for any
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$ and any
$v\in(\boldsymbol{\Sigma}^{\infty}V)_{{\mathbf{x}}}$, one can find a
representative $\tilde{v}\in(\boldsymbol{\Sigma}^{r}V)_{{\mathbf{x}}}$ for a
large enough $r$. By statement (6) of Lemma 3.11, $\boldsymbol{\Sigma}^{r}V$
is also torsion. In particular, we can find a morphism
$f:{\mathbf{x}}\to{\mathbf{y}}$ such that $f$ sends $\tilde{v}$ to 0. Let

$s=\max\\{y_{i}-x_{i}\mid i\in\mathbb{N}_{+}\\}.$

By Remark 3.13, the kernel on ${\mathbf{x}}$ of the composite of the following
maps

$\boldsymbol{\Sigma}^{r}V\to\boldsymbol{\Sigma}^{r+1}V\to\ldots\to\boldsymbol{\Sigma}^{r+s}V$

consists of $u\in(\boldsymbol{\Sigma}^{r}V)_{x}$ satisfying the following
condition: there is an object
${\mathbf{z}}\in\operatorname{Ob}({\mathscr{N}})$ such that $z_{i}\leqslant
x_{i}+s$ for $i\geqslant 1$ and the morphism $g:{\mathbf{x}}\to{\mathbf{z}}$
sends $u$ to 0. Therefore, $\tilde{v}$ is contained in this kernel, so
$\tilde{v}$ becomes 0 in $(\boldsymbol{\Sigma}^{r+s}V)_{{\mathbf{x}}}$. Thus
$v=0$ and hence $\boldsymbol{\Sigma}^{\infty}V$ is the zero module.

In the case that $V$ is not torsion, it is easy to see that
$\boldsymbol{\Sigma}^{\infty}V\cong\boldsymbol{\Sigma}^{\infty}V_{F}$, where
$V_{F}$ is the torsion free part of $V$. Thus without loss of generality we
can assume that $V$ is a nonzero torsion free ${\mathscr{N}}$-module. In this
case, natural maps in the following sequence

$V\to\boldsymbol{\Sigma}V\to\boldsymbol{\Sigma}^{2}V\to\ldots$

are all injective and all terms in this sequence are nonzero torsion free
${\mathscr{N}}$-modules by statements (4) and (5) of Lemma 3.11. Consequently,
$\boldsymbol{\Sigma}^{\infty}V$ is a nonzero torsion free
${\mathscr{N}}$-module.

It remains to show that $\operatorname{gd}(\boldsymbol{\Sigma}^{\infty}V)=0$.
But this is clear. Indeed, for the special case that $V=P({\mathbf{x}})$, we
have $\boldsymbol{\Sigma}^{s}V\cong P(\boldsymbol{0})$ for
$s\geqslant\|{\mathbf{x}}\|$, so $\boldsymbol{\Sigma}^{\infty}V\cong
P(\boldsymbol{0})$. Taking a surjective homomorphism

$\bigoplus_{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})}P({\mathbf{x}})^{c_{{\mathbf{x}}}}\to
V$

and applying $\boldsymbol{\Sigma}^{\infty}$ we have a surjective homomorphism

$\boldsymbol{\Sigma}^{\infty}(\bigoplus_{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})}P({\mathbf{x}})^{c_{{\mathbf{x}}}})\cong\bigoplus_{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})}(\boldsymbol{\Sigma}^{\infty}P({\mathbf{x}}))^{c_{{\mathbf{x}}}}\to\boldsymbol{\Sigma}^{\infty}V.$

Consequently, $\operatorname{gd}(\boldsymbol{\Sigma}^{\infty}V)\leqslant 0$.
Since $\boldsymbol{\Sigma}^{\infty}V\neq 0$, it follows that
$\operatorname{gd}(\boldsymbol{\Sigma}^{\infty}V)=0$.

(2) Since all maps in the sequence
$V\to\boldsymbol{\Sigma}V\to\boldsymbol{\Sigma}^{2}V\to\ldots$ are injective,
the induced map $V\to\boldsymbol{\Sigma}^{\infty}V$ is injective as well and
gives the desired embedding. ∎

######  Remark 3.19.

It is easy to see that $\boldsymbol{\Sigma}^{\infty}V=0$ if and only if $V$ is
torsion. Moreover, the kernel of the map
$\boldsymbol{\psi}:V\to\boldsymbol{\Sigma}^{\infty}V$ coincides with the
torsion part $V_{T}$ of $V$.

##  4\. Homological degrees of representations of $\mathbb{N}^{\infty}$-type
combinatorial categories

Let ${\mathscr{C}}$ be an arbitrary combinatorial category of type
$\mathbb{N}^{\infty}$, and $k$ a commutative ring. In this section we give an
upper bound for homological degrees of ${\mathscr{C}}$-modules in terms of
their first two homological degrees, generalizing Proposition 3.5.

Denote by ${\mathscr{N}}$ the poset $\mathbb{N}^{\infty}$, and choose a
functor $\iota:{\mathscr{N}}\to{\mathscr{C}}$ such that $\iota$ is a bijection
restricted to object sets. The functor $\iota:{\mathscr{N}}\to{\mathscr{C}}$
induces a restriction functor
$\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}:{\mathscr{C}}\operatorname{-Mod}\to{\mathscr{N}}\operatorname{-Mod}$
via pullback. The following observation plays a key role for us to prove the
main result in this section.

######  Lemma 4.1.

Given ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{C}})$, let
$P({\mathbf{x}})=k{\mathscr{C}}({\mathbf{x}},-)$,
$Q({\mathbf{x}})=k{\mathscr{N}}({\mathbf{x}},-)$, and
$G_{{\mathbf{x}}}={\mathscr{C}}({\mathbf{x}},\,{\mathbf{x}})$. Then

$P({\mathbf{x}})\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\cong
Q({\mathbf{x}})^{|G_{{\mathbf{x}}}|},$

where $|G_{{\mathbf{x}}}|$ is the order of $G_{{\mathbf{x}}}$.

###### Proof.

Note that $P({\mathbf{x}})$ as a free $k$-module has a basis consisting of
morphisms in ${\mathscr{C}}({\mathbf{x}},-)$. Without loss of generality we
can view ${\mathscr{N}}$ as a subcategory of ${\mathscr{C}}$, so
${\mathscr{N}}({\mathbf{x}},-)$ is a subset of
${\mathscr{C}}({\mathbf{x}},-)$. For
$g\in{\mathscr{C}}({\mathbf{x}},{\mathbf{x}})$, define

${\mathscr{N}}({\mathbf{x}},-)g=\\{fg\mid
f\in{\mathscr{N}}({\mathbf{x}},-)\\}.$

Since $G_{{\mathbf{x}}}$ acts regularly on
${\mathscr{C}}({\mathbf{x}},{\mathbf{y}})$ for any
${\mathbf{y}}\in\operatorname{Ob}({\mathscr{C}})$, it follows that
${\mathscr{C}}({\mathbf{x}},-)$ is a disjoint union of
${\mathscr{N}}({\mathbf{x}},-)g$ with $g$ ranging over all elements in
$G_{{\mathbf{x}}}$. The conclusion then follows from this observation. ∎

######  Lemma 4.2.

Let $V$ be a ${\mathscr{C}}$-module, and $\|\bullet\|$ be either the sup norm
or the sum norm on $\operatorname{Ob}({\mathscr{C}})$. Then
$\operatorname{gd}(V)=\operatorname{gd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})$
and
$\operatorname{pd}(V)=\operatorname{pd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})$.

###### Proof.

Take a short exact sequence $0\to W\to P\to V\to 0$ such that $P$ is a
projective ${\mathscr{C}}$-module and
$\operatorname{gd}(P)=\operatorname{gd}(V)$. Applying the restriction functor
we get the following short exact sequence of ${\mathscr{N}}$-modules

$0\to W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to
P\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to
V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to 0,$

which implies that

$\operatorname{gd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\leqslant\operatorname{gd}(P\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})=\operatorname{gd}(P)=\operatorname{gd}(V)$

where the first equality comes from Lemma 4.1. On the other hand, for any
object ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{C}})$ such that
${\mathbf{x}}\in\operatorname{supp}(H_{0}(V))$, we know that

$\sum_{{\mathbf{y}}<{\mathbf{x}}\atop
f\in{\mathscr{C}}({\mathbf{y}},{\mathbf{x}})}V_{f}(V_{{\mathbf{y}}})\subsetneqq
V_{{\mathbf{x}}}.$

Since ${\mathscr{N}}$ can be viewed as a subcategory of ${\mathscr{C}}$, it
follows that

$\sum_{{\mathbf{y}}<{\mathbf{x}}\atop
f\in{\mathscr{N}}({\mathbf{y}},{\mathbf{x}})}V_{f}(V_{{\mathbf{y}}})\subsetneqq
V_{{\mathbf{x}}},$

that is,
${\mathbf{x}}\in\operatorname{supp}(H_{0}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}))$.
Consequently,
$\operatorname{gd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\geqslant\operatorname{gd}(V)$,
so the first equality follows.

Now we turn to the second equality. On one hand, we have

$\operatorname{hd}_{1}(V)\leqslant\operatorname{gd}(W)=\operatorname{gd}(W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\leqslant\max\\{\operatorname{gd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}),\,\operatorname{hd}_{1}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\\}=\operatorname{pd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})$

by considering the above two short exact sequences. By the same argument, we
also have

$\operatorname{hd}_{1}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\leqslant\operatorname{gd}(W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})=\operatorname{gd}(W)\leqslant\max\\{\operatorname{gd}(V),\,\operatorname{hd}_{1}(V)\\}=\operatorname{pd}(V).$

Consequently, one has
$\operatorname{pd}(V)\leqslant\operatorname{pd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})$
and
$\operatorname{pd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\leqslant\operatorname{pd}(V)$,
and the second equality holds. ∎

The main result of this section is:

######  Theorem 4.3.

Let $k$ be a field, $\|\bullet\|$ the sup norm on
$\operatorname{Ob}({\mathscr{C}})$, and $V$ a ${\mathscr{C}}$-module. Then the
category algebra $k{\mathscr{C}}$ is locally left coherent. Moreover, for
$s\geqslant 0$, one has

$\operatorname{hd}_{s}(V)\leqslant\operatorname{pd}(V),$

so the category of ${\mathscr{C}}$-modules presented in finite degrees is
abelian.

###### Proof.

The argument in the proof of Lemma 4.2 tells us that the
${\mathscr{C}}$-module $V$ is finitely generated if and only if so is the
${\mathscr{N}}$-module $V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}$. The
first statement then follows from this observation as well as Lemma 3.7 and
the remark after it.

Now we prove the inequality. Without loss of generality we can assume that $V$
is presented in finite degrees. The conclusion holds trivially for $s\leqslant
1$. As before, take a suitable short exact sequence $0\to W\to P\to V\to 0$ of
${\mathscr{C}}$-modules, which induces a short exact sequence of
${\mathscr{N}}$-modules:

$0\to W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to
P\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to
V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}\to 0,$

For $s\geqslant 2$, we have

$\displaystyle\operatorname{hd}_{s}(V)$
$\displaystyle=\operatorname{hd}_{s-1}(W)\leqslant\max\\{\operatorname{gd}(W),\,\operatorname{hd}_{1}(W)\\}$
by the induction hypothesis
$\displaystyle=\max\\{\operatorname{gd}(W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}),\,\operatorname{hd}_{1}(W\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\\}$
by Lemma 4.2
$\displaystyle=\max\\{\operatorname{gd}(W),\,\operatorname{hd}_{2}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\\}$
by Lemma 4.2
$\displaystyle\leqslant\max\\{\operatorname{gd}(W),\,\operatorname{gd}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}}),\,\operatorname{hd}_{1}(V\downarrow_{{\mathscr{N}}}^{{\mathscr{C}}})\\}$
by Proposition 3.5
$\displaystyle=\max\\{\operatorname{gd}(W),\,\operatorname{gd}(V),\,\operatorname{hd}_{1}(V)\\}$
by Lemma 4.2
$\displaystyle=\max\\{\operatorname{gd}(V),\,\operatorname{hd}_{1}(V)\\}$

as desired. ∎

##  5\. Representations of the Young lattice

Throughout this section let ${\mathscr{N}}$ be the poset
$\mathbb{N}^{\infty}$, and let ${\mathscr{Y}}$ be the Young lattice, which is
precisely the full subcategory of ${\mathscr{N}}$ consisting of objects
${\mathbf{x}}$ such that $x_{i}\geqslant x_{j}$ whenever $i\leqslant j$. We
obtain a natural inclusion functor $\iota:{\mathscr{Y}}\to{\mathscr{N}}$ and
hence a restriction functor
$\iota^{\ast}:{\mathscr{N}}\operatorname{-Mod}\to{\mathscr{Y}}\operatorname{-Mod}$.

The category ${\mathscr{Y}}$ can also be viewed as a quotient category of
${\mathscr{N}}$ by the action of the infinite symmetric group. Explicitly, let

$S_{\infty}=\varinjlim S_{n}=\\{\sigma:\mathbb{N}\to\mathbb{N}\mid\sigma\text{
is a bijection and }\sigma(n)=n\text{ for }n\gg 0\\}.$

This group naturally acts on ${\mathscr{N}}$ as follows:
$\sigma\cdot{\mathbf{x}}=(x_{\sigma(1)},\,x_{\sigma(2)},\,\ldots)$. Moreover,
${\mathbf{x}}\leqslant{\mathbf{y}}$ if and only if
$\sigma\cdot{\mathbf{x}}\leqslant\sigma\cdot{\mathbf{y}}$ for any $\sigma\in
S_{\infty}$. Therefore, one can obtain a quotient poset
${\mathscr{N}}/S_{\infty}$, whose objects are orbits $[{\mathbf{x}}]$ under
this action, and $[{\mathbf{x}}]\leqslant[{\mathbf{y}}]$ if and only if there
are ${\mathbf{x}}^{\prime}\in[{\mathbf{x}}]$ and
${\mathbf{y}}^{\prime}\in[{\mathbf{y}}]$ such that
${\mathbf{x}}\leqslant{\mathbf{y}}$. It is easy to see that ${\mathscr{Y}}$ is
isomorphic to this quotient poset. Via identifying ${\mathscr{Y}}$ with this
quotient poset, we obtain a quotient functor $q:{\mathscr{N}}\to{\mathscr{Y}}$
which induces a fully faithful lift functor
$q^{\ast}:{\mathscr{Y}}\operatorname{-Mod}\to{\mathscr{N}}\operatorname{-Mod}$.
Explicitly, given a ${\mathscr{Y}}$-module $W$, the corresponded
${\mathscr{N}}$-module $q^{\ast}W$ has the following structure: for a morphism
$f:{\mathbf{x}}\to{\mathbf{y}}$ in ${\mathscr{N}}$, one has
$(q^{\ast}W)_{{\mathbf{x}}}=W_{[{\mathbf{x}}]}$,
$(q^{\ast}W)_{{\mathbf{y}}}=W_{[{\mathbf{y}}]}$, and $(q^{\ast}W)_{f}$ is
defined to be $W_{[f]}:W_{[{\mathbf{x}}]}\to W_{[{\mathbf{y}}]}$.

Since the composite $q\iota$ is the identity functor on ${\mathscr{Y}}$, it
follows that $\iota^{\ast}q^{\ast}$ is isomorphic to the identity functor on
${\mathscr{Y}}\operatorname{-Mod}$. Furthermore, the image of $q^{\ast}$ is
the full subcategory of $S_{\infty}$-symmetric ${\mathscr{N}}$-modules. To
define this notion, let $V$ be an ${\mathscr{N}}$-module and $\sigma$ an
element in $S_{\infty}$. One can define another ${\mathscr{N}}$-module $\sigma
V$ as follows: for a morphism $f:{\mathbf{x}}\to{\mathbf{y}}$ in
${\mathscr{N}}$, one lets $(\sigma
V)_{{\mathbf{x}}}=V_{\sigma\cdot{\mathbf{x}}}$, $(\sigma
V)_{{\mathbf{y}}}=V_{\sigma\cdot{\mathbf{y}}}$, and $(\sigma V)_{f}$ is
defined to be $V_{\sigma\cdot f}:V_{\sigma\cdot{\mathbf{x}}}\to
V_{\sigma\cdot{\mathbf{y}}}$. We say that $V$ is $S_{\infty}$-symmetric if
$\sigma V$ is isomorphic to $V$ for every $\sigma\in S_{\infty}$.

######  Proposition 5.1.

Let $k$ be a field. Then the category of finitely generated
${\mathscr{Y}}$-modules over $k$ is abelian.

###### Proof.

It suffices to show that every submodule $V$ of
$k{\mathscr{Y}}({\mathbf{x}},-)$ is finitely generated for
${\mathbf{x}}\in\operatorname{Ob}({\mathscr{Y}})$. If $V$ is not finitely
generated, then since the dimension of $V_{{\mathbf{y}}}$ is either 1 or 0 for
${\mathbf{y}}\in\operatorname{Ob}({\mathscr{Y}})$, it follows that
$\operatorname{supp}(H_{0}(V))$ must be an infinite set. Furthermore, elements
in this set are pairwise incomparable. Therefore, we get an infinite anti-
chain in the poset ${\mathscr{Y}}$. This is impossible. Indeed, since
$\mathbb{N}$ is an well-ordered set, it follows from Higman’s lemma (see [14])
that the set of finite sequences over $\mathbb{N}$ is quasi-well-ordered with
respect to the following ordering:

${\mathbf{x}}=(x_{1},\,x_{2},\,\ldots,\,x_{m})\preccurlyeq{\mathbf{y}}=(y_{1},\,y_{2},\,\ldots,\,y_{n})$

if and only if there is a strictly increasing map $\sigma:[m]\to[n]$ such that
$x_{i}\leqslant y_{\sigma(i)}$ for each $i\in[m]$. But it is easy to see that
${\mathbf{x}}\preccurlyeq{\mathbf{y}}$ if and only if $x_{i}\leqslant y_{i}$
since entries in ${\mathbf{x}}$ and ${\mathbf{y}}$ are listed in a decreasing
way; that is, $\preccurlyeq$ coincides with the usual ordering $\leqslant$
used in this paper. Consequently, $\leqslant$ is a quasi-well-order, and hence
the poset ${\mathscr{Y}}$ cannot have an infinite anti-chain. ∎

######  Remark 5.2.

In other words, the incidence algebra $k{\mathscr{Y}}$ is locally Noetherian,
or the incidence algebra $k{\mathscr{N}}$ is $S_{\infty}$-symmetric
Noetherian: if an $S_{\infty}$-symmetric ${\mathscr{N}}$-module is generated
by its values on objects lying in only finitely many $S_{\infty}$-orbits and
each such value is finite dimensional, then its $S_{\infty}$-symmetric
submodules also have this property. Parallel results for
$A=k[X_{1},\,X_{2},\,\ldots]$ were established in the literature, see for
instances [1, 9].

Note that the sum norm on $\operatorname{Ob}({\mathscr{N}})$ induces the sum
norm on ${\mathscr{Y}}$, which is compatible with the poset structure on
$\operatorname{Ob}({\mathscr{Y}})$. One can also define homological degrees of
${\mathscr{Y}}$-modules with respect to this norm. For an
${\mathscr{N}}$-module, the finiteness of its zeroth and first homological
degrees cannot guarantee the finiteness of the second homological degree, as
shown in Example 3.6. But for ${\mathscr{Y}}$-modules, we have a different
situation.

######  Theorem 5.3.

Let $k$ be a field, $\|\bullet\|$ the sum norm on
$\operatorname{Ob}({\mathscr{Y}})$, and $V$ a ${\mathscr{Y}}$-module. If
${\mathscr{Y}}$ is presented in finite degrees, then
$\operatorname{hd}_{s}(V)$ is finite for all $s\geqslant 1$. In particular,
the category of ${\mathscr{Y}}$-modules presented in finite degrees is
abelian.

###### Proof.

We only prove the first statement since the second one follows immediately.
Take a short exact sequence $0\to W\to P\to V\to 0$ such that $P$ is a
projective ${\mathscr{Y}}$-module and $H_{0}(P)\cong H_{0}(V)$ (the reader can
easily check that this is possible). As in the proof of Proposition 3.5, it
suffices to show that $\operatorname{hd}_{2}(V)=\operatorname{hd}_{1}(W)$ is
finite. We give a proof similar to that of Lemma 3.7.

Without loss of generality suppose that $W$ is nonzero and
$\operatorname{gd}(W)=n$. Let $S=\operatorname{supp}(H_{0}(W))$, which is a
nonempty finite set, and let $W({\mathbf{x}})$ be the submodule of $W$
generated by $W_{{\mathbf{x}}}$. Then

$W=\sum_{{\mathbf{x}}\in S}W({\mathbf{x}}).$

Since $W({\mathbf{x}})$ is torsion free, it is actually a direct sum of
$k{\mathscr{Y}}({\mathbf{x}},-)$. Thus we obtain a surjection

$Q=\bigoplus_{{\mathbf{x}}\in
S}W({\mathbf{x}})\longrightarrow\sum_{{\mathbf{x}}\in S}W({\mathbf{x}})=W,$

and hence a short exact sequence $0\to K\to Q\to W\to 0$, from which we deduce
that $\operatorname{hd}_{1}(W)\leqslant\operatorname{gd}(K)$.

Now we investigate $K$. Fix ${\mathbf{z}}\in\operatorname{Ob}({\mathscr{Y}})$
such that $K_{{\mathbf{z}}}\neq 0$. Then ${\mathbf{z}}\geqslant{\mathbf{x}}$
for a certain ${\mathbf{x}}\in S$. Define

$T=S\cap\\{{\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})\mid{\mathbf{x}}\leqslant{\mathbf{z}}\\}$

which is again a nonempty finite set. We have

$K_{{\mathbf{z}}}=\\{(W_{{\mathbf{x}}\to{\mathbf{z}}}(w_{{\mathbf{x}}}))_{{\mathbf{x}}\in
T}\mid w_{{\mathbf{x}}}\in W_{{\mathbf{x}}},\,\sum_{{\mathbf{x}}\in
T}W_{{\mathbf{x}}\to{\mathbf{z}}}(w_{{\mathbf{x}}})=0\\},$

where ${\mathbf{x}}\to{\mathbf{z}}$ is the morphism from ${\mathbf{x}}$ to
${\mathbf{z}}$.

Note that ${\mathbf{z}}$ is an upper bound of elements in $T$. Let
$\tilde{{\mathbf{z}}}$ be the least upper bound of elements in $T$, which
always exists since $T$ is a finite set. Then
${\mathbf{z}}\geqslant\tilde{{\mathbf{z}}}$. However, if
${\mathbf{z}}>\tilde{{\mathbf{z}}}$, then

$W_{{\mathbf{x}}\to{\mathbf{z}}}(w_{{\mathbf{x}}})=W_{\tilde{{\mathbf{z}}}\to{\mathbf{z}}}(W_{{\mathbf{x}}\to\tilde{{\mathbf{z}}}}(w_{{\mathbf{x}}})).$

Note that the map $W_{\tilde{{\mathbf{z}}}\to{\mathbf{z}}}$ is injective since
$W$ as a submodule of $Q$ is torsion free. Consequently, we have

$\sum_{{\mathbf{x}}\in
T}W_{{\mathbf{x}}\to\tilde{{\mathbf{z}}}}(w_{{\mathbf{x}}})=0,$

so
$(W_{{\mathbf{x}}\to\tilde{{\mathbf{z}}}}(w_{{\mathbf{x}}}))_{{\mathbf{x}}\in
T}$ is contained $K_{\tilde{{\mathbf{z}}}}$. Furthermore, the map
$K_{\tilde{{\mathbf{z}}}\to{\mathbf{z}}}$ sends
$(W_{{\mathbf{x}}\to\tilde{{\mathbf{z}}}}(w_{{\mathbf{x}}}))_{{\mathbf{x}}\in
T}$ to $(W_{{\mathbf{x}}\to{\mathbf{z}}}(w_{{\mathbf{x}}}))_{{\mathbf{x}}\in
T}$. Consequently, the map $K_{\tilde{{\mathbf{z}}}\to{\mathbf{z}}}$ is
surjective, and hence ${\mathbf{z}}$ is not contained in
$\operatorname{supp}(H_{0}(K))$.

We have shown that ${\mathbf{z}}\in\operatorname{supp}(H_{0}(K))$ only if
there is a nonempty finite set $T$ of $S$ such that ${\mathbf{z}}$ is
precisely the least upper bound of elements in $T$. Since $S$ is a finite set,
there are only finitely many such least upper bounds. Consequently,
$\operatorname{supp}(H_{0}(K))$ is a finite set, so $\operatorname{gd}(K)$ is
finite as well. This finishes the proof. ∎

######  Remark 5.4.

By comparing the proof of this theorem and Example 3.6, the reason why the
above conclusion is not valid for ${\mathscr{N}}$-modules $V$ becomes
transparent: in that case $\operatorname{supp}(H_{0}(V))$ in general is an
infinite set, so it has infinitely many finite subsets and correspondingly one
can get infinitely many least upper bounds.

######  Remark 5.5.

Note that for each fixed $n\in\mathbb{N}$, there are only finitely many
objects ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{Y}})$ such that the sum
norm $\|{\mathbf{x}}\|=n$. Therefore, a ${\mathscr{Y}}$-module $V$ is
presented in finite degrees if and only if it is presented in finitely many
objects, an intrinsic property independent of norms on
$\operatorname{Ob}({\mathscr{Y}})$.

Finally we describe a recursive upper bound for homological degrees of
${\mathscr{Y}}$-modules.

######  Corollary 5.6.

Let $k$ be a field, $\|\bullet\|$ the sum norm on
$\operatorname{Ob}({\mathscr{Y}})$, and $V$ a ${\mathscr{Y}}$-module. If
$s\geqslant 2$ and $\operatorname{hd}_{s-1}(V)>0$, then one has

$\operatorname{hd}_{s}(V)\leqslant\operatorname{hd}_{s-1}(V)(\ln(\operatorname{hd}_{s-1}(V))+1).$

Note that $\operatorname{hd}_{s-1}(V)=-1$ if and only if $H_{s-1}(V)=0$, and
in this case $\operatorname{hd}_{s}(V)=-1$. If $\operatorname{hd}_{s-1}(V)=0$,
then there is a projective resolution $\ldots\to P^{1}\to V\to 0$ such that
the image $V^{s-1}$ of $P^{s}\to P^{s-1}$ is a torsion free
${\mathscr{Y}}$-module generated by the value of $V^{s-1}$ on the object
$\boldsymbol{0}$. But it follows that $V^{s-1}$ is projective, so
$\operatorname{hd}_{s}(V)=-1$ as well.

###### Proof.

It suffices to prove the inequality for $s=2$. Let $W$ and $K$ be as in the
proof of Theorem 5.3. Since we have assumed that $H_{0}(P)\cong H_{0}(V)$, it
follows that $\operatorname{hd}_{1}(V)=\operatorname{gd}(W)$. Without loss of
generality we can assume this number to be $n$. Let $T$ be the subset of
$\operatorname{Ob}({\mathscr{Y}})$ consisting of objects ${\mathbf{y}}$ with
$\|{\mathbf{y}}\|=n$, and let $S=\operatorname{supp}(H_{0}(W))$. For every
${\mathbf{x}}\in S$, since $\|{\mathbf{x}}\|\leqslant n$, we can find a
certain ${\mathbf{y}}\in T$ such that ${\mathbf{x}}\leqslant{\mathbf{y}}$.
Consequently, the least upper bound $\bigvee_{{\mathbf{x}}\in S}{\mathbf{x}}$
of $S$ is less than or equal to $\bigvee_{{\mathbf{x}}\in T}{\mathbf{x}}$.

It is not hard to see from the proof of Theorem 5.3 that
$\operatorname{gd}(K)$ is bounded by $\|\bigvee_{{\mathbf{x}}\in
S}{\mathbf{x}}\|$, so

$\operatorname{hd}_{2}(V)\leqslant\operatorname{gd}(K)\leqslant\|\bigvee_{{\mathbf{x}}\in
S}{\mathbf{x}}\|\leqslant\|\bigvee_{{\mathbf{x}}\in T}{\mathbf{x}}\|.$

But $T$ consists of all partitions of $[n]$ which have a common upper bound
$(\lambda_{1},\ldots,\lambda_{n})$ with $\lambda_{i}=\lfloor
n/i\rfloor\leqslant n/i$. Consequently, we have

$\operatorname{hd}_{2}(V)\leqslant\|\bigvee_{{\mathbf{x}}\in
T}{\mathbf{x}}\|\leqslant\sum_{i=1}^{n}\frac{n}{i}=n\sum_{i=1}^{n}\frac{1}{n}\leqslant
n(\ln n+1).$

∎

##  6\. A sheaf theoretic interpretation

In this section we use sheaf theory over ringed sites to give an
interpretation of results established in previous sections from another
viewpoint. Let $\mathcal{C}$ be a small category and $k$ a commutative ring.
Recall that a presheaf of $k$-modules over $\mathcal{C}^{\operatorname{op}}$
is precisely a $\mathcal{C}$-module. Suppose that $\mathcal{C}$ satisfies the
left Ore condition, so one can impose the atomic Grothendieck topology
$J_{at}$ on $\mathcal{C}^{\operatorname{op}}$. Let
$\boldsymbol{\mathcal{C}}^{\operatorname{op}}$ be the site
$(\mathcal{C}^{\operatorname{op}},\,J_{at})$, and let $\underline{k}$ be the
constant structure sheaf. For details about atomic topology and sheaves over
ringed sites, please refer to [3].

The following result is taken from [3].

######  Theorem 6.1.

A $\mathcal{C}$-module $V$ is a sheaf over the ringed site
$(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\underline{k})$ if and only if

$\operatorname{Hom}_{\mathcal{C}\operatorname{-Mod}}(T,V)=0=\operatorname{Ext}_{\mathcal{C}\operatorname{-Mod}}^{1}(T,V)$

for any torsion $\mathcal{C}$-module $T$. The category
${\mathrm{Sh}}(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})$
of sheaves of $k$-modules is equivalent to the Serre quotient
$\mathcal{C}\operatorname{-Mod}/\mathcal{C}\operatorname{-Mod}^{\operatorname{tor}}$,
where $\mathcal{C}\operatorname{-Mod}^{\operatorname{tor}}$ is the full
subcategory of $\mathcal{C}\operatorname{-Mod}$ consisting of torsion
${\mathscr{C}}$-modules.

When $\mathcal{C}$ is a poset satisfying the left Ore condition, we have a
very trivial description for
${\mathrm{Sh}}(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})$.

######  Corollary 6.2.

Every sheaf over
$(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})$ is a constant
sheaf, and hence
${\mathrm{Sh}}(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})\simeq
k\operatorname{-Mod}$.

###### Proof.

An explicit equivalence is given as follows. Firstly, since
$\mathcal{C}^{\operatorname{op}}$ is equipped with the atomic topology,
$\mathcal{C}$ is a directed set. Therefore, the direct limit functor

$\varinjlim:\mathcal{C}\operatorname{-Mod}\to k\operatorname{-Mod}$

is exact. It has a right adjoint, the diagonal functor
$\Delta:k\operatorname{-Mod}\to\mathcal{C}\operatorname{-Mod}$ sending each
$k$-module to the corresponded constant $\mathcal{C}$-module. Furthermore, a
$\mathcal{C}$-module $V$ is sent to 0 by $\varinjlim$ if and only if $V$ is
torsion. Therefore, by Theorem 6.1 and [6, Proposition III.2.5], we have

${\mathrm{Sh}}(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})\simeq\mathcal{C}\operatorname{-Mod}/\mathcal{C}\operatorname{-Mod}^{\operatorname{tor}}\simeq
k\operatorname{-Mod},$

and moreover each sheaf over
$(\boldsymbol{\mathcal{C}}^{\operatorname{op}},\,\underline{k})$ is an image
of $\Delta$, so must be a constant sheaf. ∎

###  6.1. $\boldsymbol{\Sigma}^{\infty}$ as the sheafification functor

Throughout this subsection let ${\mathscr{N}}$ be the poset
$\mathbb{N}^{\infty}$. Recall that the inclusion functor
$\textsl{inc}:{\mathrm{Sh}}(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k})\to{\mathscr{N}}\operatorname{-Mod}$
has a left adjoint $\sharp$, the sheafification functor, where
$\boldsymbol{{\mathscr{N}}}^{\operatorname{op}}=({\mathscr{N}}^{\operatorname{op}},J_{at})$.
It follows from the proof of Corollary 6.2 that $\sharp$ is isomorphic to the
following composite of functors

$\textstyle{{\mathscr{N}}\operatorname{-Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varinjlim}$$\textstyle{k\operatorname{-Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{{\mathrm{Sh}}(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k}).}$

Furthremore, by Proposition 3.18 and Corollary 6.2, the shift functor
$\boldsymbol{\Sigma}^{\infty}$ can be viewed as a functor from
${\mathscr{N}}\operatorname{-Mod}$ to
${\mathrm{Sh}}(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k})$.
Surprisingly, these two functors coincide up to isomorphism.

######  Proposition 6.3.

The functor
$\boldsymbol{\Sigma}^{\infty}:{\mathscr{N}}\operatorname{-Mod}\to{\mathrm{Sh}}(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k})$
is isomorphic to the sheafification functor.

###### Proof.

Take $V$ in ${\mathscr{N}}\operatorname{-Mod}$ and $W$ in
${\mathrm{Sh}}(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k})$.
We want to construct a natural isomorphism

$\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(\boldsymbol{\Sigma}^{\infty}V,\,W)\cong\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(V,\,W);$

that is, $\boldsymbol{\Sigma}^{\infty}$ is the left adjoint of inc, where on
the right side we write $W$ for $\textsl{inc}\,W$ for brevity.

Applying $\boldsymbol{\Sigma}^{\infty}$ to the short exact sequence $0\to
V_{T}\to V\to V_{F}\to 0$ we get
$\boldsymbol{\Sigma}^{\infty}V\cong\boldsymbol{\Sigma}^{\infty}V_{F}$ by
Proposition 3.18. Therefore, without loss of generality we can assume that $V$
is torsion free, so by Proposition 3.18 we have another short exact sequence

$0\to V\to\boldsymbol{\Sigma}^{\infty}V\to U\to 0,$

which induces a long exact sequence

$0\to\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(U,\,W)\to\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(\boldsymbol{\Sigma}^{\infty}V,\,W)\to\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(V,\,W)\to\operatorname{Ext}_{{\mathscr{N}}\operatorname{-Mod}}^{1}(U,\,W)\to\ldots.$

By Remark 3.19, $U$ is a torsion ${\mathscr{N}}$-module. Since $W$ is a sheaf
over $(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\,\underline{k})$,
Theorem 6.1 asserts that the first term and the last term vanish.
Consequently, we have

$\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(\boldsymbol{\Sigma}^{\infty}V,\,W)\cong\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(V,\,W).$

Furthermore, all constructions to establish this isomorphism are natural, so
the conclusion holds. ∎

######  Remark 6.4.

We give a more illustrative explanation that $\boldsymbol{\Sigma}^{\infty}$
indeed coincides with the sheafification functor. Since
$\boldsymbol{\Sigma}^{\infty}V$ is the colimit of the following sequence of
${\mathscr{N}}$-module homomorphisms

$V\to\boldsymbol{\Sigma}V\to\boldsymbol{\Sigma}^{2}V\to\boldsymbol{\Sigma}^{3}V\to\ldots,$

$\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(\boldsymbol{\Sigma}^{\infty}V,\,W)$
is isomorphic to the set of sequences
$(\phi^{0},\,\phi^{1},\,\phi^{2},\,\ldots)$ of ${\mathscr{N}}$-module
homomorphisms such that the following diagram commutes:

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{0}}$$\textstyle{\boldsymbol{\Sigma}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{1}}$$\textstyle{\boldsymbol{\Sigma}^{2}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{2}}$$\textstyle{\ldots}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$id$\textstyle{\ldots.}$

Clearly, this sequence determines an element in
$\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(V,\,W)$. The above
proposition asserts that the converse is also true: every element in
$\operatorname{Hom}_{{\mathscr{N}}\operatorname{-Mod}}(V,\,W)$ induces such a
sequence if $W$ is a sheaf over
$(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\underline{k})$, or
equivalently a constant ${\mathscr{N}}$-module.

To see the reason, let us construct $\phi^{1}:\boldsymbol{\Sigma}V\to W$ from
$\phi^{0}:V\to W$. For ${\mathbf{x}}\in\operatorname{Ob}({\mathscr{N}})$, we
need to define
$\phi^{1}_{{\mathbf{x}}}:(\boldsymbol{\Sigma}V)_{{\mathbf{x}}}\to
W_{{\mathbf{x}}}$. Since $(\boldsymbol{\Sigma}V)_{{\mathbf{x}}}$ is the
colimit of the following sequence

$V_{{\mathbf{x}}}\longrightarrow(\Sigma_{1}V)_{{\mathbf{x}}}=V_{{\mathbf{x}}+{\mathbf{1}}_{1}}\longrightarrow(\Sigma_{[2]}V)_{{\mathbf{x}}}=V_{{\mathbf{x}}+{\mathbf{1}}_{1}+{\mathbf{1}}_{2}}\longrightarrow\ldots,$

it suffices to define vertical maps to obtain a commutative diagram of
$k$-modules

$\textstyle{V_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{{\mathbf{x}}+{\mathbf{1}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{{\mathbf{x}}+{\mathbf{1}}_{1}+{\mathbf{1}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots.}$

Since $\phi^{0}$ gives a commutative diagram

$\textstyle{V_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{0}_{{\mathbf{x}}}}$$\textstyle{V_{{\mathbf{x}}+{\mathbf{1}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{0}_{{\mathbf{x}}+{\mathbf{1}}_{1}}}$$\textstyle{V_{{\mathbf{x}}+{\mathbf{1}}_{1}+{\mathbf{1}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{0}_{{\mathbf{x}}+{\mathbf{1}}_{1}+{\mathbf{1}}_{2}}}$$\textstyle{\ldots}$$\textstyle{W_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}+{\mathbf{1}}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{{\mathbf{x}}+{\mathbf{1}}_{1}+{\mathbf{1}}_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots,}$

and $W$ is a constant ${\mathscr{N}}$-module, we can successfully define
vertical maps in the first diagram.

######  Remark 6.5.

Given a torsion free ${\mathscr{N}}$-module $V$ (in other words, a separated
presheaf over
$(\boldsymbol{{\mathscr{N}}}^{\operatorname{op}},\underline{k})$), a standard
fact of sheaf theory tells us that the natural map $V\to
V^{\sharp}\cong\boldsymbol{\Sigma}^{\infty}V$ is injective, which is precisely
the second statement of Proposition 3.18.

###  6.2. Sheaves over the ringed site
$(\mathscr{Z},\,J_{at},\,\underline{k})$

In this subsection let $\mathscr{Z}$ be the category defined in Example 2.4.
Since $\mathscr{Z}$ satisfies the right Ore condition, we can impose the
atomic Grothendieck topology on $\mathscr{Z}$, and consider sheaves over the
ringed site $(\mathscr{Z},\,J_{at},\,\underline{k})$. Throughout this
subsection let $k$ be an algebraically closed field with characteristic 0.

Recall that every object in $\mathscr{Z}$ is the coset
${\mathbf{n}}=\mathbb{Z}/n\mathbb{Z}$, and morphisms
$f:{\mathbf{m}}\to{\mathbf{n}}$ are surjective $(\mathbb{Z},+)$-equivariant
maps. In particular, endomorphisms from ${\mathbf{n}}$ to itself form the
cyclic group $G_{n}=\langle g\mid g^{n}=1\rangle$. If $V$ is a presheaf of
$k$-modules over $\mathscr{Z}$, or equivalently, an
$\mathscr{Z}^{\operatorname{op}}$-module, since the canonical morphism
$f:{\mathbf{m}}\to{\mathbf{n}}$ in $\mathscr{Z}({\mathbf{m}},{\mathbf{n}})$
sending $\bar{1}\in{\mathbf{m}}$ to $\bar{1}\in{\mathbf{n}}$ induces an
embedding functor $kG_{n}\operatorname{-Mod}\to kG_{m}\operatorname{-Mod}$,
one can regard $V_{f}:V_{n}\to V_{m}$ as a $kG_{m}$-module homomorphism. In
particular, let $\xi$ be a $n$-th primitive root of 1, then the corresponded
irreducible $kG_{n}$-module $L_{\xi}$ can be regarded as an irreducible
$kG_{m}$-module while $n$ divides $m$. Denote by $\underline{L}_{\xi}$ the
$\mathscr{Z}^{\operatorname{op}}$-submodule (actually, an irreducible direct
summand) of $k\mathscr{Z}(-,{\mathbf{n}})$ generated by $L_{\xi}$ on
${\mathbf{n}}$. Explicitly, its value on ${\mathbf{m}}$ is $L_{\xi}$ if $n$
divides $m$ and $0$ otherwise.

The following result classifies all indecomposable injective objects in
${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$, which are precisely
indecomposable torsion free $\mathscr{Z}^{\operatorname{op}}$-modules by [3,
Corollary 3.9].

######  Theorem 6.6.

Let $k$ be an algebraically closed field with characteristic 0. Then
$\xi\mapsto\underline{L}_{\xi}$ gives a bijective correspondence between the
set of all primitive roots of 1 and the set of isomorphism classes of
indecomposable injective torsion free
$\mathscr{Z}^{\operatorname{op}}$-modules.

###### Proof.

Suppose that $\xi$ is a primitive $n$-th root of 1. Clearly,
$\underline{L}_{\xi}$ is torsion free and indecomposable. We show that it is
injective; that is, the functor
$\operatorname{Hom}_{\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}}(-,\,\underline{L}_{\xi})$
is exact. Given an $\mathscr{Z}^{\operatorname{op}}$-module $V$, let
$V^{\prime}$ be the submodule satisfying the following condition: for every
${\mathbf{m}}\in\operatorname{Ob}(\mathscr{Z}^{\operatorname{op}})$,
$V^{\prime}_{{\mathbf{m}}}$ is a direct sum of all copies of $L_{\xi}$
appearing in $V_{{\mathbf{m}}}$. This submodule is well defined by Schur’s
lemma. Moreover, by Schur’s lemma again, one has a decomposition
$V=V^{\prime}\oplus V^{\prime\prime}$. Clearly, one has

$\operatorname{Hom}_{\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}}(V,\,\underline{L}_{\xi})\cong\operatorname{Hom}_{\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}}(V^{\prime},\,\underline{L}_{\xi}).$

Since for each object ${\mathbf{m}}$ such that $n$ does not divide $m$,
$L_{\xi}$ cannot appear in $V_{{\mathbf{m}}}$, we can regard $V^{\prime}$ and
$\underline{L}_{\xi}$ as an
$\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}$-module, where
$\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}$ is the full
subcategory of $\mathscr{Z}^{\operatorname{op}}$ consisting of all objects
${\mathbf{m}}$ such that $n$ divides $m$. Clearly, one has

$\operatorname{Hom}_{\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}}(V^{\prime},\,\underline{L}_{\xi})\cong\operatorname{Hom}_{\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}\operatorname{-Mod}}(V^{\prime},\,\underline{L}_{\xi}).$

Note that $L_{\xi}$ is viewed as a $kG_{m}$-module via the natural quotient
group homomorphism $G_{m}\to G_{n}$ when $n$ divides $m$. Let $\mathscr{E}$ be
the following category:

  * •

$\operatorname{Ob}(\mathscr{E})=\operatorname{Ob}(\mathscr{Z}^{\operatorname{op}}_{\geqslant{\mathbf{n}}})$;

  * •

for ${\mathbf{m}}\in\operatorname{Ob}(\mathscr{E})$, set
$H_{{\mathbf{m}}}=\mathscr{E}({\mathbf{m}},\,{\mathbf{m}})$ be the unique
quotient group of $G_{m}$ which is isomorphic to $G_{n}$;

  * •

for ${\mathbf{m}},\,{\mathbf{x}}\in\operatorname{Ob}(\mathscr{E})$,
$\mathscr{E}({\mathbf{m}},\,{\mathbf{x}})$ are isomorphisms between
$H_{{\mathbf{m}}}$ and $H_{{\mathbf{x}}}$.

Then there is a natural quotient functor
$q:\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}\to\mathscr{E}$
which induces a fully faithful functor
$q^{\ast}:\mathscr{E}\operatorname{-Mod}\to\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}\operatorname{-Mod}$,
and one can view $V^{\prime}$ and $\underline{L}_{\xi}$ as
$\mathscr{E}$-modules. Furthermore, one has

$\operatorname{Hom}_{\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}\operatorname{-Mod}}(V^{\prime},\,\underline{L}_{\xi})\cong\operatorname{Hom}_{\mathscr{E}\operatorname{-Mod}}(V^{\prime},\,\underline{L}_{\xi}).$

We observe that each step in the following procedure

$V\in\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}\mapsto
V^{\prime}\in\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}\mapsto
V^{\prime}\in\mathscr{Z}_{\geqslant{\mathbf{n}}}^{\operatorname{op}}\operatorname{-Mod}\mapsto
V^{\prime}\in\mathscr{E}\operatorname{-Mod}$

is exact. Therefore, to show that $\underline{L}_{\xi}$ is injective in
$\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}$, it suffices to show that
$\underline{L}_{\xi}$ is injective in $\mathscr{E}\operatorname{-Mod}$.
However, using the same argument as in the proof of Corollary 6.2, one can
show that

${\mathrm{Sh}}(\mathcal{E}^{\operatorname{op}},\,J_{at},\,\underline{k})\simeq
kG_{n}\operatorname{-Mod}.$

In particular, $\underline{L}_{\xi}$ is an injective sheaf since it
corresponds to the $kG_{n}$-module $L_{\xi}$ which is injective in
$kG_{n}\operatorname{-Mod}$. Consequently, it is a torsion free injective
$\mathcal{E}$-module by [3, Corollary 3.9].

We have show that every $\underline{L}_{\xi}$ is a torsion free injective
$\mathscr{Z}^{\operatorname{op}}$-module. Now we prove the converse statement;
that is, every indecomposable torsion free injective
$\mathscr{Z}^{\operatorname{op}}$-module $I$ is isomorphic to
$\underline{L}_{\xi}$ for a certain primitive root $\xi$ of 1. Take a minimal
object ${\mathbf{n}}$ in $\mathscr{Z}^{\operatorname{op}}$ such that
$I_{{\mathbf{n}}}\neq 0$, and let $\underline{L}$ be the submodule of $I$
generated by a simple summand $L$ of $I_{{\mathbf{n}}}$. For any
$f:{\mathbf{x}}\to{\mathbf{y}}$ in $\mathscr{Z}^{\operatorname{op}}$, applying
the snake lemma to the commutative diagram of short exact sequences

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{L}_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{L}(f)}$$\textstyle{I_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{I(f)}$$\textstyle{(I/\underline{L})_{{\mathbf{x}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(I/\underline{L})(f)}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{L}_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(I/\underline{L})_{{\mathbf{y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

we know that the third vertical map is injective since the first one is an
isomorphism and the second one is injective. Consequently, $I/\underline{L}$
is a torsion free $\mathscr{Z}^{\operatorname{op}}$-module, so $\underline{L}$
is a sheaf over the ringed site
$(\mathscr{Z}^{\operatorname{op}},\,J_{at},\,\underline{k})$ by [3,
Proposition 3.7 and Theorem 3.8].

Note that $L$ corresponds to a unique $n$-th root $\xi$ of 1 which is an
$m$-th primitive root for a unique $m$ dividing $n$. Letting
$\underline{L}_{\xi}$ be the injective torsion free module corresponded to
$\xi$ constructed as before, we obtain a short exact sequence
$0\to\underline{L}\to\underline{L}_{\xi}\to\underline{L}_{\xi}/\underline{L}\to
0$. By [3, Proposition 3.7 and Theorem 3.8], the third term must be torsion
free. However, since its value on ${\mathbf{n}}$ is 0, this happens if and
only if $m=n$; that is, $\underline{L}\cong\underline{L}_{\xi}$. Since $I$ is
an indecomposable $\mathscr{Z}^{\operatorname{op}}$-module containing
$\underline{L}$ as a submodule, we must have $I\cong\underline{L}_{\xi}$. ∎

The above proof actually implies that indecomposable injective objects in
${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$ coincide with
irreducible objects. Consequently,
${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$, is a semisimple
category, as asserted in the following corollary.

######  Corollary 6.7.

Let $k$ be an algebraically closed field with characteristic 0. Then every
object in ${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$ is isomorphic
to a direct sum of irreducible sheaves of the form $\underline{L}_{\xi}$.
Consequently,

${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})\simeq\bigtimes_{\xi}k\operatorname{-Mod},$

the cartesian product of categories $k\operatorname{-Mod}$ indexed by all
primitive roots of 1.

###### Proof.

Let $V$ be an object in ${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$.
We can obtain a short exact sequence

$0\to W\to
P=k\mathscr{Z}^{\operatorname{op}}\otimes_{k\mathscr{Z}_{0}}H_{0}(V)\to V\to
0$

in $\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}$ where
$\mathscr{Z}_{0}$ is the subcategory of $\mathscr{Z}^{\operatorname{op}}$
consisting of all objects and all invertible morphisms in
$\mathscr{Z}^{\operatorname{op}}$. We claim that $P$ is a direct sum of
torsion free injective $\mathscr{Z}^{\operatorname{op}}$-modules. If this is
true, then the Grothendieck category
${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$ has a set of irreducible
generators

$\\{\underline{L}_{\xi}\mid\xi\text{ is an $n$-th primitive root for a certain
}n\in\mathbb{N}_{+}\\}.$

The conclusions then follow from this fact. Indeed, in this case $P$ is a
semismple object in ${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$.
Since a simple summand in $P$ becomes either 0 or a simple subobject of $V$
via the quotient map $P\to V$, and $V$ is the sum of these simple subobjects,
it follows that $V$ is semisimple by [13, Theorem 2.3].

Now we prove the claim. Note that every indecomposable direct summand
$\underline{L}$ of $P$ is generated by a certain $L_{\xi}$ in
$kG_{n}\operatorname{-Mod}$ with $\xi$ an $n$-th root of 1, so $L_{\xi}$ can
be embedded into $\underline{L}_{\xi}$ constructed before, and hence we obtain
a commutative diagram

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

where $\tilde{P}$ is a direct sum of those $\underline{L}_{\xi}$ and $\beta$
is the map induced by these embeddings. Clearly, $T$ is a torsion
$\mathscr{Z}^{\operatorname{op}}$-module (including the special case that
$T=0$). Since $V$ is a sheaf, by Theorem 6.1, the third column must split.
Applying the functor $H_{0}$ we obtain a commutative diagram

$\textstyle{H_{0}(P)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{H_{0}(\tilde{P})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(V)\oplus
H_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It turns out that the middle vertical arrow is also an isomorphism, and hence
the map $\alpha$ is a split monomorphism. It induces a split monomorphism

$P=k\mathscr{Z}^{\operatorname{op}}\otimes_{k\mathscr{Z}_{0}}H_{0}(V)\to\tilde{P}=k\mathscr{Z}^{\operatorname{op}}\otimes_{k\mathscr{Z}_{0}}H_{0}(\tilde{P}).$

But by our construction of $\tilde{P}$, this happens if and only if
$P\cong\tilde{P}$, so the claim is valid. ∎

######  Remark 6.8.

Note that $k_{\xi}$ is isomorphic to the endomorphism ring of
$\underline{L}_{\xi}$, so $\underline{L}_{\xi}$ is quipped with a
$(k\mathscr{Z}^{\operatorname{op}},\,k_{\xi})$-bimodule structure. With this
observation, the equivalence is given by the functors:

$\textstyle{\bigoplus_{\xi}\operatorname{Hom}_{\mathscr{Z}^{\operatorname{op}}\operatorname{-Mod}}(\underline{L}_{\xi},\,-):{\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigtimes_{\xi}(k_{\xi}\operatorname{-Mod})\simeq(\bigoplus_{\xi}k_{\xi})\operatorname{-Mod}:\bigoplus_{\xi}(\underline{L}_{\xi}\otimes_{k_{\xi}}-)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which both are exact and form an adjunction pair. Denote by $\mu$ the hom
functor and $\nu$ the tensor functor. Clearly, $\nu\mu$ is isomorphic to the
identity functor. Furthermore, since those $\underline{L}_{\xi}$’s form a set
of generators of ${\mathrm{Sh}}(\mathscr{Z},\,J_{at},\,\underline{k})$, the
kernel of $\mu$ is zero. Therefore, by [6, Proposition III.2.5], one obtains
the desired equivalence.

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