# Bergman algebras
The graded universal algebra constructions

R. Hazrat  Roozbeh Hazrat: Centre for Research in Mathematics and Data Science
Western Sydney University, Australia<EMAIL_ADDRESS>,  H. Li
Huanhuan Li: Center for Pure Mathematics
School of Mathematical Sciences
Anhui University, Hefei, China<EMAIL_ADDRESS>and  R. Preusser  Raimund
Preusser: School of Mathematics and Statistics
Nanjing University of Information Science and Technology, Nanjing, China
<EMAIL_ADDRESS>
###### Abstract.

A half a century ago, George Bergman introduced stunning machinery which would
realise any commutative conical monoid as the non-stable $K$-theory of a ring.
The ring constructed is “minimal” or “universal”. Given the success of graded
$K$-theory in classification of algebras and its connections to dynamics and
operator algebras, the realisation of $\Gamma$-monoids (monoids with an action
of an abelian group $\Gamma$ on them) as non-stable graded $K$-theory of
graded rings becomes vital. In this paper, we revisit Bergman’s work and
develop the graded version of this universal construction. For an abelian
group $\Gamma$, a $\Gamma$-graded ring $R$, and non-zero graded finitely
generated projective (left) $R$-modules $P$ and $Q$, we construct a universal
$\Gamma$-graded ring extension $S$ such that $S\otimes_{R}P\cong
S\otimes_{R}Q$ as graded $S$-modules. This makes it possible to bring the
graded techniques, such as smash products and Zhang twists into Bergman’s
machinery. Given a commutative conical $\Gamma$-monoid $M$, we construct a
$\Gamma$-graded ring $S$ such that $\mathcal{V}^{\operatorname{gr}}(S)$ is
$\Gamma$-isomorphic to $M$. In fact we show that any finitely generated
$\Gamma$-monoid can be realised as the non-stable graded $K$-theory of a hyper
Leavitt path algebra. Here $\mathcal{V}^{\operatorname{gr}}(S)$ is the monoid
of isomorphism classes of graded finitely generated projective $S$-modules and
the action of $\Gamma$ on $\mathcal{V}^{\operatorname{gr}}(S)$ is by shift of
degrees. Thus the group completion of $M$ can be realised as the graded
Grothendieck group $K^{\operatorname{gr}}_{0}(S)$. We use this machinery to
provide a short proof to the fullness of the graded Grothendieck functor
$K^{\operatorname{gr}}_{0}$ for the class of Leavitt path algebras (i.e.,
Graded Classification Conjecture II).

###### Key words and phrases:

Bergman algebra, Leavitt path algebra, graded ring, $\Gamma$-monoid, graded
Grothendieck group

_In memory of Nikolai Vavilov who conveyed the joy of mathematics to us in
Chinese, German and Persian_

###### Contents

  1. 1 Introduction
  2. 2 $\Gamma$-monoids and graded algebras
  3. 3 Leavitt path algebras
  4. 4 Bergman algebras
  5. 5 Graded Bergman universal construction
  6. 6 Smash products of graded Bergman algebras
  7. 7 The graded $\mathcal{V}$-monoid of a graded Bergman algebra
  8. 8 Realisation of $\Gamma$-monoids as non-stable $K$-theory of graded rings
  9. 9 Application to graph algebras

##  1\. Introduction

In 1974 in the seminal paper [7] George Bergman introduced machinery whereby
by adding sufficient generators and relations to an algebra, one can lift any
pair of non-zero finitely generated projective modules to become isomorphic.
To be precise, let $R$ be an algebra over a field $\mathsf{k}$, and let
$(P_{i},Q_{i})_{i\in I}$ be pairs of non-zero finitely generated projective
left $R$-modules. Bergman’s machinery adjoins certain generators and relations
to $R$, so that the resulted extension algebra $S$ obtained provides universal
isomorphisms of modules, $S\otimes_{R}P_{i}\cong S\otimes_{R}Q_{i}$, where
$i\in I$. Furthermore, the non-stable $K$-theory of the algebra $S$ can be
obtained from that of $R$ by identifying the pairs $(P_{i},Q_{i})_{i\in I}$ in
the structure.

Starting from a field $\mathsf{k}$ and choosing the pair of
$\mathsf{k}$-modules $(\mathsf{k}^{n},\mathsf{k}^{m})$,
$n,m\in\mathbb{N}^{+}$, Bergman observed that his machinery retrieves the
celebrated Leavitt algebra $R=L_{\mathsf{k}}(n,m)$, where
$R^{n}\cong\mathsf{k}^{n}\otimes_{\mathsf{k}}R\cong\mathsf{k}^{m}\otimes_{\mathsf{k}}R\cong
R^{m}$ as $R$-modules. Many combinatorial algebras constructed in the last 50
years, such as Leavitt path algebras and their generalisations, can be
obtained from Bergman’s machinery.

Two natural questions immediately arise:

  1. (1)

The structural properties: To describe the structure of the extension algebra
$S$ obtained from $R$ relative to the pairs $(P_{i},Q_{i})_{i\in I}$.

  2. (2)

The classification: Given pairs $(P_{i},Q_{i})_{i\in I}$ of $R$-modules and
$(P^{\prime}_{i},Q^{\prime}_{i})_{i\in I}$ of $R^{\prime}$-modules, for
$\mathsf{k}$-algebras $R$ and $R^{\prime}$, respectively, to describe how the
extension $\mathsf{k}$-algebras $S$ and $S^{\prime}$ obtained from the Bergman
machinery, respectively, compare.

We highlighted Leavitt path algebras as substantial activity is currently
taking place on both questions (1) and (2). In fact, there are several
conjectures concering the classification problems and what could be the right
invariant for Leavitt path algebras [1]. Leavitt path algebras associated to
graphs have rich $\mathbb{Z}$-graded structures (arising by assigning degree
$1$ to edges) which play a vital role in their studies. Their non-stable
_graded_ $K$-theory is related to invariants of symbolic dynamics and the
equivariant $K$-theory of operator algebras and the current open conjecture is
that the graded $K$-theory could be a complete invariant for these algebras
(see [13, 15] and §9). This is one of the main motivations of this project: to
study the Bergman machinery in the graded setting.

Let $\mathsf{k}$ be a commutative ring and let $R$ be a unital
$\mathsf{k}$-algebra. Further let $N$ be an $R$-module and $P$ a finitely
generated projective $R$-module. Let $R\text{-rings}_{\mathsf{k}}$ be the
category of $\mathsf{k}$-algebras $T$ equipped with a
$\mathsf{k}$-homomorphism $R\rightarrow T$. Bergman’s work started by showing
that the following functor is representable:

$\begin{split}\mathcal{F}_{N,P}:R\text{-rings}_{\mathsf{k}}&\longrightarrow{}\operatorname{Sets}\\\
T&\longmapsto\operatorname{Hom}(T\otimes_{R}N,T\otimes_{R}P).\end{split}$
(1.1)

That is, there exists a universal algebra $S$ (or the initial object
$R\rightarrow S$ in an appropriate category), denoted by $R\langle
f:\overline{N}\rightarrow\overline{P}\rangle$, which provides the
$S$-homomorphism $S\otimes_{R}N\rightarrow S\otimes_{R}P$. Bergman further
showed that there exists a universal algebra $S$, denoted by
$R\langle\overline{f}=0\rangle$, such that the extension of a given $R$-module
homomorphim $f:N\rightarrow P$ vanishes.

Combining these two constructions, allows for forming very interesting
universal rings. For example, given a homomorphism $f:P\rightarrow Q$, one can
construct a universal ring $R\langle\overline{f}^{-1}\rangle$, where the
extension of $f$ becomes invertible. The two important universal constructions
needed to realise any conical monoid are the following: Let $P$ and $Q$ be
non-zero finitely generated projective left $R$-modules. Constructing the
universal homomorphisms $h:\overline{P}\rightarrow\overline{Q}$ and
$h^{\prime}:\overline{\overline{Q}}\rightarrow\overline{\overline{P}}$ in
subsequent extensions and setting $h^{\prime}\overline{h}-1$ and
$\overline{h}h^{\prime}-1$ to zero, we obtain the universal ring

$S:=R\big{\langle}h,h^{-1}:\overline{P}\cong\overline{Q}\big{\rangle},$ (1.2)

where $S\otimes_{R}P\cong S\otimes_{R}Q$ as $S$-modules.

On the other hand, for $P$ a non-zero finitely generated projective left
$R$-module, by constructing the universal homomorphism $e:P\rightarrow P$ and
setting the homomorphism $e^{2}-e$ to zero we form a universal ring

$T:=R\big{\langle}e:\overline{P}\to\overline{P};~{}e^{2}=e\big{\rangle},$
(1.3)

such that the extension of $e$ in $T$ is an idempotent endomorphism.

Bergman then proved [7, Theorems 5.1, 5.2] that
$\mathcal{V}(S)\cong\mathcal{V}(R)/\langle[P]=[Q]\rangle$ and
$\mathcal{V}(T)\cong\langle\mathcal{V}(R),[P_{1}],[P_{2}]\rangle/\langle[P]=[P_{1}]+[P_{2}]\rangle$,
for the universal rings $S$ and $T$ of (1.2) and (1.3), respectively. Here
$\mathcal{V}(S)$ is the monoid of isomorphism classes of finitely generated
projective left $S$-modules. These constructions allowed Bergman to build up a
machinery to realise any (finitely generated) conical commutative monoid as
the non-stable $K$-theory of a ring.

Graded algebras frequently appear when there is a group acting on an algebraic
structure. A graded ring can be a model of an algebraic structure that
captures and reflects the time evolution and the dynamics. A ring $R$ is
_graded_ by a group $\Gamma$ when, roughly, $R$ can be partitioned by $\Gamma$
in a way that is compatible with the structure of $R$. (See §2.2 for more
details.) Consequently, three categories play a prominent role in this
setting: the category of left $R$-modules $R$-$\operatorname{{\bf Mod}}$, the
category of graded left $R$-modules $R$-$\operatorname{{\bf Gr}}$, and the
category of left $R_{0}$-modules $R_{0}$-$\operatorname{{\bf Mod}}$, where
$R_{0}$ is the subring of $R$ consisting of the elements in the partition
corresponding to the identity element $0$ of $\Gamma$. A substantial portion
of the theory of graded rings concerns the relationships between these
categories. While the applications of this theory are numerous, one prominent
example is the fundamental theorem of $K$-theory, proved by Quillen [22],
using the category of graded modules in a crucial way.

Our starting point is to consider a $\Gamma$-graded $\mathsf{k}$-algebra $R$
over a $\Gamma$-graded commutative ring $\mathsf{k}$, a $\Gamma$-graded (left)
$R$-module $N$ and a $\Gamma$-graded finitely generated projective (left)
$R$-module $P$, where $\Gamma$ is an abelian group, and consider the
corresponding functors in the graded setting:

$\begin{split}\mathcal{F}^{\operatorname{gr}}_{N,P}:R\text{-gr-
rings}_{\mathsf{k}}&\longrightarrow{}\operatorname{Sets}\\\
T&\longmapsto\operatorname{Hom}_{T\text{-}\operatorname{{\bf
Gr}}}(T\otimes_{R}N,T\otimes_{R}P)\end{split}$ (1.4)

$\begin{split}\mathcal{F}_{N_{0},P_{0}}:R_{0}\text{-rings}_{\mathsf{k}_{0}}&\longrightarrow{}\operatorname{Sets}\\\
T&\longmapsto\operatorname{Hom}(T\otimes_{R_{0}}N_{0},T\otimes_{R_{0}}P_{0})\end{split}$
(1.5)

Here $R\text{-gr-rings}_{\mathsf{k}}$ is the category of $\Gamma$-graded
$\mathsf{k}$-algebras $T$ equipped with a graded $\mathsf{k}$-homomorphism
$R\rightarrow T$. In this paper we show that
$\mathcal{F}^{\operatorname{gr}}_{N,P}$ is representable, i.e., there is a
$\Gamma$-graded algebra $S$ such that
$\operatorname{Hom}(S,-)\cong\mathcal{F}^{\operatorname{gr}}_{N,P}$.
Furthermore, if $R$ is strongly graded, then $S\cong R\ast_{R_{0}}S_{0}$,
where $S_{0}$ is the representation for $\mathcal{F}_{N_{0},P_{0}}$ and $\ast$
is the coproduct. Forgetting the grading on $S$ gives us the representation
for $\mathcal{F}_{N,P}$ of (1.1).

Having these graded universal constructions, we can realise any conical
commutative $\Gamma$-monoid as the nonstable graded $K$-theory of a suitable
graded ring. Namely, for such a $\Gamma$-monoid $M$, there is a
$\Gamma$-graded $\mathsf{k}$-algebra $R$ such that
$M\cong\mathcal{V}^{\operatorname{gr}}(R)$, as $\Gamma$-monoids. Here
$\mathcal{V}^{\operatorname{gr}}(R)$ is the monoid of isomorphism classes of
graded finitely generated projective $R$-modules and the action of $\Gamma$ on
$\mathcal{V}^{\operatorname{gr}}(R)$ is by shift of degrees (Theorem 8.1). In
order to carry out our $\Gamma$-monoid realisations, we had two options;
either to develop the entire theory of monoids for co-products rings ([8]) in
the graded setting, or to start with a graded ring, and use the fact that the
category of graded modules is equivalent to the category of modules over its
corresponding smash product (i.e., to pass from the graded setting to the non-
graded case and use the available techniques). We opted for the second option;
we pass from the graded structure via smash product to the category of modules
and use the established results in the non-graded setting. The price we had to
pay for such a shortcut, was to drop some generality, such as working with
graded $\mathsf{k}$-algebras, where $\mathsf{k}$ is a field concentrated in
degree zero, rather than with $\mathsf{k}$-algebras, where $\mathsf{k}$ is a
graded field.

Our realisations of $\Gamma$-monoids allow us to provide a short proof to one
of the Graded Classification Conjectures (See Conjecture 9.1(1)): Let $A$ be a
$\mathbb{Z}$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is concentrated in
degree zero and $\mathcal{V}^{\operatorname{gr}}(A)$ is cancellative. Then any
order preserving $\mathbb{Z}[x,x^{-1}]$-module homomorphism
$\phi:K_{0}^{\operatorname{gr}}(L_{\mathsf{k}}(E))\rightarrow
K_{0}^{\operatorname{gr}}(A)$ with $\phi([L_{\mathsf{k}}(E)])=[A]$ is induced
by a unital $\mathbb{Z}$-graded $\mathsf{k}$-homomorphism
$\psi:L_{\mathsf{k}}(E)\rightarrow A$, i.e.,
$K^{\operatorname{gr}}_{0}(\psi)=\phi$. Here $L_{\mathsf{k}}(E)$ is the
Leavitt path algebra associated to the graph $E$. Replacing the algebra $A$
with a Leavitt path algebra $L_{\mathsf{k}}(F)$ provides a positive answer to
the fullness conjecture.

In general given two finitely generated projective modules, it is not
straightforward to describe Bergman’s (localisation) algebras explicitly.
Given finitely generated $R$-modules $P$ and $Q$, and a map $f:P\rightarrow
Q$, there are few instances where the localisation rings $R\langle
f:\overline{P}\rightarrow\overline{Q}\rangle$, or
$R\langle\overline{f}^{-1}\rangle$ are explicitly described. We refer to the
paper of Sheiham [26] where the localisation of triangular matrix rings has
been described. In this paper, by representing the finitely generated
projective modules by the corresponding idempotent matrices, we first describe
the Bergman’s constructions in terms of generators and relations. Thus given
idempotent matrices $e\in\mathbb{M}_{m}(R)$ and $f\in\mathbb{M}_{n}(R)$, we
define the Bergman algebra $B_{R}(e,f)$ via generators and relations such that
$e$ and $f$ become equivalent idempotents in $B_{R}(e,f)$ in a universal way.
The construction can be extended to families of pairs of idempotents
$(e,f):=\\{(e_{i},f_{i})\\}_{i\in I}$. In fact this construction will be
carried out for graded rings and homogeneous idempotents in Section 4. The
algebra $B_{R}(e,f)$ obtained this way, coincides with the universal
localisation ring $R\langle f,f^{-1}:\overline{P}_{i}\cong\overline{Q}_{i}\mid
i\in I\rangle$, for pairs of finitely generated projective modules
$(P_{i},Q_{i}),i\in I$ which represent $(e_{i},f_{i}),{i\in I}$.

Returning to the graded setting, we show that when $R$ is a $\Gamma$-graded
commutative $\mathsf{k}$-semisimple ring which is concentrated in degree zero,
then the universal localisation ring $R\langle
f,f^{-1}:\overline{P}_{i}\cong\overline{Q}_{i}\mid i\in I\rangle$, for any
family of pairs of finitely generated projective modules $(P_{i},Q_{i}),i\in
I$, is graded isomorphic to a Leavitt path algebra of a hypergraph and
conversely any such hyper Leavitt path algebra can be realised as a graded
Bergman’s construction over a $\Gamma$-graded commutative
$\mathsf{k}$-semisimple ring (Theorem 4.10). Built on this, we show that any
finitely generated conical $\Gamma$-monoid can be realised as the non-stable
graded $K$-theory of a hyper Leavitt path algebra (Theorem 8.2).

As mentioned, starting with a field $\mathsf{k}$, and projective modules
$\mathsf{k}$ and $\mathsf{k}^{n}$, Bergman’s machinery would give the Leavitt
algebra $L_{\mathsf{k}}(1,n)$. This ring can be graded by assigning $1$ and
$-1$ to the suitable generators. However, we can start with a field
$\mathsf{k}$ concentrated in degree zero, and consider the $\mathbb{Z}$-graded
projective modules $\mathsf{k}$ and $\mathsf{k}(-1)^{n}$, where
$\mathsf{k}(-1)$ is the projective module $\mathsf{k}$ shifted by $-1$.
Applying our graded Bergman construction, we obtain the algebra
$L_{\mathsf{k}}(1,n)$ which is naturally $\mathbb{Z}$-graded from the outset.
This allows us to build the graded invariants such as non-stable graded
$K$-theory from the machinery we develop.

The paper is organised as follows: Section 2 provides a background needed for
the concepts of $\Gamma$-monoids and graded algebras, as well as the notion of
smash products. In Section 3, we recall the notion of Leavitt path algebras of
hypergraphs, which generalises several extensions of Leavitt path algebras.
This is the generalisation which covers all Bergman algebras constructed from
commutative semisimple rings as a base ring. In Section 4, we introduce
Bergman algebras, via generators and relations, both the non-graded version
and the graded version. Section 5 which is the core part of this paper,
carries out Bergman’s localisation construction [7] in the setting of graded
rings. Section 6 investigates how the graded Bergman localisation behaves
under the smash product. This is needed in Section 7 to compute the non-stable
graded $K$-theory of graded Bergman algebras. Section 8 develops many
consequences of the graded Bergman machinery: we realise any conical
$\Gamma$-monoid as the non-stable graded $K$-theory of an appropriate graded
algebra. In fact we show that any finitely generated $\Gamma$-monoid can be
obtained as the non-stable graded $K$-theory of a hyper Leavitt path algebra.
Using our result, we provide a short positive answer to one of the Graded
Classification Conjectures (Conjecture 9.1(1)) for Leavitt path algebras in
Section 9.

In this paper, all rings are unital and all modules are considered as left
modules unless otherwise stated. $\mathbb{N}$ denotes the natural numbers with
$0$ and $\mathbb{N}^{+}=\mathbb{N}\backslash\\{0\\}$.

##  2\. $\Gamma$-monoids and graded algebras

###  2.1. $\Gamma$-monoids

Recall that a monoid is a semigroup with an identity element. Throughout this
paper monoids are commutative, written additively, with the identity element
denoted by $0$. A monoid homomorphism $\phi:M\rightarrow N$ is a map between
monoids $M$ and $N$ which respects the structures and satisfies $\phi(0)=0$.
Every monoid $M$ is equipped with a natural preordering: $n\leq m$ if $n+p=m$
for some $p\in M$. We write

$n\propto m\text{ if }n\leq km,\text{ for some }k\in\mathbb{N}^{+},$

and

$n\asymp m\text{ if }n\propto m\text{ and }m\propto n.$

Thus $n\asymp m$ if there is a $k\in\mathbb{N}^{+}$ such that $m\leq kn$ and
$n\leq km$. We say $i\in M$ is _an order unit_ if $m\propto i$ for any $m\in
M$. We say $M$ is _conical_ if $m+n=0$ implies that $m=n=0$, where $m,n\in M$.
We say $M$ is _cancellative_ if $m_{1}+n=m_{2}+n$ implies $m_{1}=m_{2}$, for
all $m_{1},m_{2},n\in M$.

Given a group $\Gamma$, a _$\Gamma$ -monoid_ consists of a monoid $M$ equipped
with an action of $\Gamma$ on $M$ (by monoid automorphisms). We denote the
action of $\alpha\in\Gamma$ on $m\in M$ by ${}^{\alpha}m$. Throughout this
paper $\Gamma$ will be an abelian group and $M$ a commutative monoid. A monoid
homomorphism $\phi\colon M\rightarrow N$ between two $\Gamma$-monoids is
called $\Gamma$-_monoid homomorphism_ if $\phi$ respects the actions of
$\Gamma$, i.e., $\phi({}^{\alpha}m)={}^{\alpha}\phi(m)$ for all $m\in M$. The
natural preordering is respected by the action of $\Gamma$.

An order unit of a $\Gamma$-monoid $M$ is an element $i\in M$ such that for
any $m\in M$, there are $\gamma_{1},\dots,\gamma_{t}\in\Gamma$ such that

$m\leq\sum_{k=1}^{t}{}^{\gamma_{k}}i.$ (2.1)

We denote the $\Gamma$-monoid $M$ along with the order unit $i$ by $(M,i)$ and
call it a _pointed $\Gamma$-monoid_. A morphism $f:(M,i)\rightarrow(N,j)$ in
the category of pointed $\Gamma$-monoids, is a $\Gamma$-monoid morphism $f$
such that $f(i)=j$.

The order unit $i$ is called a _strong order unit_ if for any $m\in M$, we can
choose all $\gamma_{k}=0$ in (2.1). We say $i$ is a _invariant order unit_ if
${}^{\gamma}i=i$ for any $\gamma\in\Gamma$.

A congruence on a $\Gamma$-monoid $M$ is an equivalence relation $\sim$ on $M$
such that $m\sim m^{\prime}$ and $n\sim n^{\prime}$ implies $m+n\sim
m^{\prime}+n^{\prime}$, and $m\sim m^{\prime}$ implies
${}^{\gamma}m\sim{}^{\gamma}m^{\prime}$, for any $\gamma\in\Gamma$ and
$m,m^{\prime},n,n^{\prime}\in M$. If $\sim$ is a congruence on $M$, then
$M/\sim$ is a $\Gamma$-monoid in the obvious way. We call $M/\sim$ a quotient
of $M$.

If $M$ is a $\Gamma$-monoid and $(m_{i},n_{i})$, where $i\in I$, a family of
pairs of elements of $M$, then _the congruence on the $\Gamma$-monoid $M$
generated by the relations $m_{i}=n_{i}$_ will be the congruence generated on
$M$ as a monoid by the relations ${}^{\gamma}m_{i}={}^{\gamma}n_{i}$, where
$i\in I$ and $\gamma\in\Gamma$.

The action of the abelian group $\Gamma$ respects the relations $\propto$ and
$\asymp$ and one can see that $\asymp$ is an equivalence relation.

######  Lemma 2.1.

Let $M$ be a $\Gamma$-monoid and $i\in M$ a $\Gamma$-order unit. Then the
following are equivalent.

  1. (1)

The element $i$ is a strong order unit;

  2. (2)

For any $\gamma\in\Gamma$ we have $i\asymp{}^{\gamma}i$;

  3. (3)

The equivalence class $[i]$ under $\asymp$, which is clearly a sub-semigroup
of $M$, is in fact a $\Gamma$-subsemigroup of $M$.

  4. (4)

If $M$ has a strong order-unit, then all such elements form an equivalence
class under $\asymp$.

###### Proof.

The proofs are straightforward. ∎

Clearly the image of a homomorphism $f:M\to N$, denoted by
$\operatorname{im}f$, is a $\Gamma$-submonoid of $N$. The kernel $\ker f$ of
$f$ is the congruence $\sim$ on $M$ defined by $m\sim m^{\prime}$ if
$f(m)=f(m^{\prime})$, so that $M/\ker f$ is a $\Gamma$-monoid and the
canonical homomorphism $M/\ker f\rightarrow N$ is injective (see Lemma 2.3).

Let $X$ be a set. The free $\Gamma$-monoid on $X$, denoted by $F\langle
X\rangle$, is constructed as the free monoid on the set $X\times\Gamma$, with
the action of $\delta\in\Gamma$ on $(x,\gamma)$ defined by
$(x,\gamma+\delta)$. To be precise,

$F\langle X\rangle=\Big{\\{}\sum_{x\in
X,\gamma\in\Gamma}n_{x,\gamma}(x,\gamma)\mid n_{x,\gamma}\geq 0\text{, for any
}x\in X\text{ and }\gamma\in\Gamma;\text{ almost all }n_{x,\gamma}\text{ are
zero}\Big{\\}}$

which becomes a $\Gamma$-monoid with addition

$\sum_{x\in X,\gamma\in\Gamma}n_{x,\gamma}(x,\gamma)+\sum_{x\in
X,\gamma\in\Gamma}n^{\prime}_{x,\gamma}(x,\gamma)=\sum_{x\in
X,\gamma\in\Gamma}(n_{x,\gamma}+n^{\prime}_{x,\gamma})(x,\gamma)$

and $\Gamma$-action

${}^{\delta}\Big{(}\sum_{x\in
X,\gamma\in\Gamma}n_{x,\gamma}(x,\gamma)\Big{)}=\sum_{x\in
X,\gamma\in\Gamma}n_{x,\gamma}x(\gamma+\delta).$

One checks easily that $F\langle X\rangle$ has the following universal
property.

######  Lemma 2.2.

Let $X$ be a set, $M$ a $\Gamma$-monoid and $f:X\to M$ a map. Let $g:X\to
F\langle X\rangle$ be the map defined by $x\to x(0)$. Then there is a unique
homomorphism $h:F\langle X\rangle\to M$ of $\Gamma$-monoids such that $f=hg$.

######  Lemma 2.3 (Homomorphism theorem).

Let $f:M\to N$ be a homomorphism of $\Gamma$-monoids. Then $M/\ker
f\cong\operatorname{im}f$.

###### Proof.

Define a map $\bar{f}:M/\ker f\to\operatorname{im}f$ by $\bar{f}([m])=f(m)$
for any $m\in M$. Clearly $\bar{f}$ is well-defined and bijective. One checks
easily that $f$ is a homomorphism of $\Gamma$-monoids. ∎

######  Proposition 2.4.

Any $\Gamma$-monoid is isomorphic to a quotient of a free $\Gamma$-monoid.

###### Proof.

Let $M$ be a $\Gamma$-monoid. It follows from Lemma 2.2 that there is a
surjective homomorphism $\gamma:F\langle X\rangle\to M$ of $\Gamma$-monoids.
Thus $F\langle X\rangle/\ker\gamma\cong\operatorname{im}\gamma=M$ by Lemma
2.3. ∎

Let $X$ be a set and $R$ a set of relations $m_{i}=n_{i}$, where
$m_{i},n_{i}\in F\langle X\rangle$, for any $i\in I$. We write $M=\langle
X\mid R\rangle$ if $M\cong F\langle X\rangle/\sim$, where $\sim$ is the
congruence on $F\langle X\rangle$ generated by $R$. In this case we say that
$M$ is a $\Gamma$-monoid presented by the generating set $X$ and the relations
$R$.

###  2.2. Graded algebras

In this section we collect basic facts that we need about graded rings and
graded algebras. We refer the reader to [17, 21] for the theory of graded
rings.

Let $\Gamma$ be an abelian group with identity denoted by $0$. A ring $R$
(possibly without unit) is called a _$\Gamma$ -graded ring_ if
$R=\bigoplus_{\gamma\in\Gamma}R_{\gamma}$ such that each $R_{\gamma}$ is an
additive subgroup of $R$ and $R_{\gamma}R_{\delta}\subseteq R_{\gamma+\delta}$
for all $\gamma,\delta\in\Gamma$. The group $R_{\gamma}$ is called the
$\gamma$-_homogeneous component_ of $R.$ When it is clear from context that a
ring $R$ is graded by the group $\Gamma,$ we simply say that $R$ is a _graded
ring_. We denote the set of all homogeneous elements of the graded ring $R$,
by $R^{h}$.

A unital $\Gamma$-graded ring $R=\bigoplus_{\gamma\in\Gamma}R_{\gamma}$ is
_strongly graded_ if $R_{\alpha}R_{\beta}=R_{\alpha+\beta}$ for any
$\alpha,\beta\in\Gamma$ and it is _crossed-product_ if each component
$R_{\alpha}$, $\alpha\in\Gamma$ contains an invertible element. Furthermore,
we say $R$ is a _graded field_ if it is commutative and all non-zero
homogeneous elements are invertible. The notion of strongly graded, in
particular, is an important case as it widely generalises the case of group
rings, and it has been gradually established that many results on group rings
can be extended to strongly graded rings.

Let $\mathsf{k}$ be a $\Gamma$-graded commutative ring and $R$ a unital
$\Gamma$-graded ring. We say $R$ is a _graded $\mathsf{k}$-algebra_ if it is
equipped with a graded ring homomorphism $\phi:\mathsf{k}\rightarrow Z(R)$,
where $Z(R)$ is the centre of $R$. Note that since $\Gamma$ is abelian, $Z(R)$
is a graded ring. Let $S$ be a graded $\mathsf{k}$-algebra. Then we say $S$ is
a _$\Gamma$ -graded $R\text{-ring}_{\mathsf{k}}$_ if it is equipped with a
graded $\mathsf{k}$-algebra homomorphism $R\rightarrow S$. We denote the
category of $\Gamma$-graded $R\text{-rings}_{\mathsf{k}}$ by $R\text{-gr-
rings}_{\mathsf{k}}$. Considering $\Gamma$ to be trivial we obtain the
category of $R\text{-rings}_{\mathsf{k}}$.

For a graded $A$-module $M$, we define the $\alpha$-_shifted_ graded
$A$-module $M(\alpha)$ as

$M(\alpha)=\bigoplus_{\gamma\in\Gamma}M(\alpha)_{\gamma},$

where $M(\alpha)_{\gamma}=M_{\alpha+\gamma}$. That is, as an ungraded module,
$M(\alpha)$ is a copy of $M$, but the grading is shifted by $\alpha$.

For a $\Gamma$-graded ring $R$, the category of graded $R$-modules is denoted
by $R$-$\operatorname{{\bf Gr}}$ and the category of $R$-modules by
$R$-$\operatorname{{\bf Mod}}$. For $\alpha\in\Gamma$, the _shift functor_

$\mathcal{T}_{\alpha}:R\text{-}\operatorname{{\bf Gr}}\longrightarrow
R\text{-}\operatorname{{\bf Gr}},\quad M\longmapsto M(\alpha)$

is an isomorphism with the property
$\mathcal{T}_{\alpha}\mathcal{T}_{\beta}=\mathcal{T}_{\alpha+\beta}$ for
$\alpha,\beta\in\Gamma$.

Let $R$ be a strongly $\Gamma$-graded $\mathsf{k}$-algebra. For any
$R_{0}$-module $N$ and any $\gamma\in\Gamma$, we identify the $R_{0}$-module
$R_{\gamma}\otimes_{R_{0}}N$ with its image in $R\otimes_{R_{0}}N$. Since
$R=\bigoplus_{\gamma\in\Gamma}R_{\gamma}$ and $R_{\gamma}$ are
$R_{0}$-bimodules, $R\otimes_{R_{0}}N$ is a $\Gamma$-graded $R$-module, with

$R\otimes_{R_{0}}N=\bigoplus_{\gamma\in\Gamma}R_{\gamma}\otimes_{R_{0}}N.$

Recall that we have the _restriction functor_

$(-)_{0}:R\text{-}\operatorname{{\bf Gr}}\longrightarrow
R_{0}\text{-}\operatorname{{\bf Mod}}$ (2.2)

sending $M$ to $M_{0}$ and $\phi$ to $\phi_{|_{M_{0}}}$ with $M$ a graded
$R$-module and $\phi$ a graded module homomorphism from $M$, and the
_induction functor_

$\begin{split}R\otimes_{R_{0}}-:R_{0}\text{-}\operatorname{{\bf
Mod}}&\longrightarrow R\text{-}\operatorname{{\bf Gr}}\\\ N&\longmapsto
R\otimes_{R_{0}}N\\\ \psi&\longmapsto R\otimes\psi.\end{split}$ (2.3)

When $R$ is strongly graded, Dade’s Theorem [17, Theorem 1.5.1] (see also [21,
Theorem 3.1.1]), guarantees that the functors $(-)_{0}$ and
$R\otimes_{R_{0}}-$ form mutually inverse equivalences of categories.

Note that in general $(-)_{0}:R\text{-}\operatorname{{\bf
Gr}}\xrightarrow{}R_{0}\text{-}\operatorname{{\bf Mod}}$ does not induce a
functor $(-)_{0}:R\text{-}\operatorname{{\bf
Gr}}_{\operatorname{proj}}\xrightarrow[]{}R_{0}\text{-}\operatorname{{\bf
Mod}}_{\operatorname{proj}}$. But $R$ is strongly graded if and only if
$R\text{-}\operatorname{{\bf Gr}}_{\operatorname{proj}}$ is equivalent to
$R\text{-}\operatorname{{\bf Mod}}_{\operatorname{proj}}$, if and only if
$R\text{-}{\bf gr}$ is equivalent $R_{0}\text{-}{\bf mod}$, both under the
functors (2.2) and (2.3). Here $R\text{-}{\bf gr}$ denotes the category of
graded finitely generated $R$-modules and $R_{0}\text{-}{\bf mod}$ is the
category of finitely generated $R_{0}$-modules.

Let $R$, $S$ and $A$ be $\Gamma$-graded $\mathsf{k}$-algebras and let
$i:A\rightarrow R$ and $j:A\rightarrow S$ be $\mathsf{k}$-graded algebra
monomorphisms (so we identify $A$ as a graded subring of $R$ and $S$). The
coproduct $R*_{A}S$ can be given a natural $\Gamma$-grading, so that this
coproduct is indeed the pushout for the graded maps $i$ and $j$.

###  2.3. Matrix form of graded homomorphisms

Let $\Gamma$ be an abelian group, $\mathsf{k}$ a $\Gamma$-graded commutative
ring with identity and $R$ a $\Gamma$-graded $\mathsf{k}$-algebra.

A graded $R$-module is called a graded free module, if it is a free module
with homogeneous basis. Clearly a graded free module is a free module. A
graded free $R$-module can be presented as $\bigoplus_{i\in I}R(\gamma_{i})$,
where $\gamma_{i}\in\Gamma$.

A graded $R$-module $P$ is called a graded projective module if $P$ is a
projective $R$-module. One can check that $P$ is graded projective if and only
if the functor $\operatorname{Hom}_{R\text{-}\operatorname{{\bf Gr}}}(P,-)$ is
an exact functor in the category $R$-$\operatorname{{\bf Gr}}$ if and only if
$P$ is graded isomorphic to a direct summand of a graded free $R$-module ([17,
Proposition 1.2.15]). So a module is graded projective module if and only if
it is graded and projective. This is not the case for graded free module. A
module could be graded and free, but not graded free (see [17, § 1.2.4]).

In particular, if $P$ is a graded finitely generated projective $R$-module,
then there is a graded finitely generated projective $R$-module $Q$ such that

$P\bigoplus Q\cong_{\operatorname{gr}}\bigoplus_{i=1}^{n}R(\alpha_{i}),$ (2.4)

with $\alpha_{i}\in\Gamma$. We denote $\bigoplus_{i=1}^{n}R(\alpha_{i})$ by
$R^{n}(\overline{\alpha})$ with
$\overline{\alpha}=(\alpha_{1},\cdots,\alpha_{n})$.

One observes that
${\operatorname{Hom}}_{R}(R(\delta_{j}),R(\delta_{i}))\cong_{\operatorname{gr}}R(\delta_{i}-\delta_{j})$
as graded $R$-modules. Hence we have the following isomorphisms:

$\begin{split}\operatorname{Hom}_{R}(\bigoplus_{i\in
I}R(\alpha_{i}),\bigoplus_{j\in J}R(\beta_{j}))&\cong\prod_{i\in
I}\operatorname{Hom}_{R}(R(\alpha_{i}),\bigoplus_{j\in J}R(\beta_{j}))\\\
&\cong\prod_{i\in I}\bigoplus_{j\in
J}\operatorname{Hom}_{R}(R(\alpha_{i}),R(\beta_{j}))\\\ &\cong\prod_{i\in
I}\bigoplus_{j\in J}R(\beta_{j}-\alpha_{i}).\end{split}$ (2.5)

For a $\Gamma$-graded $\mathsf{k}$-algebra $R$,
$\overline{\alpha}=(\alpha_{i})_{i\in I}$, $\overline{\beta}=(\beta_{j})_{j\in
J}$ with $\alpha_{i},\beta_{j}\in\Gamma$, set

$\mathbb{M}_{I\times
J}(R)[\overline{\alpha}][\overline{\beta}]:={\begin{pmatrix}R_{\beta_{j}-\alpha_{i}}\end{pmatrix}}_{i\in
I,j\in J},$ (2.6)

where for each $i\in I$, only a finite number of entries in
$(R_{\beta_{j}-\alpha_{i}})$ are non-zero. Thus $\mathbb{M}_{I\times
J}(R)[\overline{\alpha}][\overline{\beta}]$ consists of matrices with the
$ij$-entry in $R_{\beta_{j}-\alpha_{i}}$, with a finite number of non-zero
elements in each row. One observes that $\mathbb{M}_{I\times
J}(R)[\overline{\alpha}][\overline{\beta}]$ represents the set
$\operatorname{Hom}_{R\text{-}\operatorname{{\bf Gr}}}(\bigoplus_{i\in
I}R(\alpha_{i}),\bigoplus_{j\in J}R(\beta_{j}))$ as follows: Suppose that
$u:\bigoplus_{i\in I}R(\alpha_{i})\xrightarrow{}\bigoplus_{j\in
J}R(\beta_{j})$ is a graded $R$-module homomorphism. Then for each $i\in I$,
we can write $u(1_{i})=(u_{ij})_{j\in J}$, where $1_{i}$ is the element of
$\bigoplus_{i\in I}R(\alpha_{i})$ whose $i$-th component is $1$ and whose
other components are zero (note that $\deg(1_{i})=-\alpha_{i})$), and
$u_{ij}\in R(\beta_{j})_{-\alpha_{i}}=R_{\beta_{j}-\alpha_{i}}$ are all zero
except for a finite number of $j$ in $J$. Then the graded homomorphism
$u:\bigoplus_{i\in I}R(\alpha_{i})\xrightarrow{}\bigoplus_{j\in
J}R(\beta_{j})$ is given by multiplying the matrix $(u_{ij})_{i\in I,j\in
J}\in\mathbb{M}_{I\times J}(R)[\overline{\alpha}][\overline{\beta}]$ from the
right. Suppose that $v:\bigoplus_{j\in
J}R(\beta_{j})\xrightarrow{}\bigoplus_{k\in K}R(\gamma_{k})$ is given by
multiplying the matrix $(v_{jk})_{j\in J,k\in K}\in\mathbb{M}_{J\times
K}(R)[\overline{\beta}][\overline{\gamma}]$ from the right. Then the
composition $v\circ u$ is given by multiplying the matrix $(u_{ij})_{i\in
I,j\in J}\cdot(v_{jk})_{j\in J,k\in K}$ from the right. Here,

$(u_{ij})_{i\in I,j\in J}\cdot(v_{jk})_{j\in J,k\in K}$ (2.7)

is the multiplication of matrices. Notice that the matrix representation of
the composition of functions $v\circ u$ is obtained by swapping the orders
(2.7).

The above notion of _mixed-shift_ was developed for graded right $R$-modules
in [17, §1.3.4].

In fact, any element in $\prod_{i\in I}\bigoplus_{j\in
J}R(\beta_{j}-\alpha_{i})$ in the last equation of (2.5) can be written as
$(x_{i})_{i\in I}$, where $x_{i}\in\bigoplus_{j\in J}R(\beta_{j}-\alpha_{i})$
is a row vector indexed by $J$ ($x_{i}$ having finitely many non-zero entries
for each $i\in I$).

For a $\Gamma$-graded $\mathsf{k}$-algebra $R$,
$\overline{\alpha}=(\alpha_{1},\cdots,\alpha_{m})\in\Gamma^{m}$,
$\overline{\beta}=(\beta_{1},\cdots,\beta_{n})\in\Gamma^{n}$, (2.6) takes the
form

$\mathbb{M}_{m\times
n}(R)[\overline{\alpha}][\overline{\beta}]={\begin{pmatrix}R_{\beta_{1}-\alpha_{1}}&R_{\beta_{2}-\alpha_{1}}&\cdots&R_{\beta_{n}-\alpha_{1}}\\\
R_{\beta_{1}-\alpha_{2}}&R_{\beta_{2}-\alpha_{2}}&\cdots&R_{\beta_{n}-\alpha_{2}}\\\
\vdots&\vdots&\ddots&\vdots\\\
R_{\beta_{1}-\alpha_{m}}&R_{\beta_{2}-\alpha_{m}}&\cdots&R_{\beta_{n}-\alpha_{m}}\\\
\end{pmatrix}},$ (2.8)

and $\mathbb{M}_{m\times n}(R)[\overline{\alpha}][\overline{\beta}]$
represents the set $\operatorname{Hom}_{R\text{-}\operatorname{{\bf
Gr}}}(R^{m}(\overline{\alpha}),R^{n}(\overline{\beta}))$. When $m=n$, we
simply write $\mathbb{M}_{m\times n}(R)[\overline{\alpha}][\overline{\beta}]$
as $\mathbb{M}_{n}(R)[\overline{\alpha}][\overline{\beta}]$.

By (2.5) we have the graded matrix ring
$\operatorname{End}_{R}(R^{n}(\overline{\alpha}))\cong\bigoplus_{i=1}^{n}\bigoplus_{j=1}^{n}R(\alpha_{j}-\alpha_{i})$,
denoted by $\mathbb{M}_{n}(R)(\overline{\alpha})$. More precisely

$\mathbb{M}_{n}(R)(\overline{\alpha})={\begin{pmatrix}R(\alpha_{1}-\alpha_{1})&R(\alpha_{2}-\alpha_{1})&\cdots&R(\alpha_{n}-\alpha_{1})\\\
R(\alpha_{1}-\alpha_{2})&R(\alpha_{2}-\alpha_{2})&\cdots&R(\alpha_{n}-\alpha_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
R(\alpha_{1}-\alpha_{n})&R(\alpha_{2}-\alpha_{n})&\cdots&R(\alpha_{n}-\alpha_{n})\end{pmatrix}}$
(2.9)

For each $\gamma\in\Gamma$, $\mathbb{M}_{n}(R)(\overline{\alpha})_{\gamma}$,
the $\gamma$-homogeneous elements, are the $n\times n$-matrices over $R$ with
the degree shifted (suspended) as follows

$\mathbb{M}_{n}(R)(\overline{\alpha})_{\gamma}={\begin{pmatrix}R_{\gamma+\alpha_{1}-\alpha_{1}}&R_{\gamma+\alpha_{2}-\alpha_{1}}&\cdots&R_{\gamma+\alpha_{n}-\alpha_{1}}\\\
R_{\gamma+\alpha_{1}-\alpha_{2}}&R_{\gamma+\alpha_{2}-\alpha_{2}}&\cdots&R_{\gamma+\alpha_{n}-\alpha_{2}}\\\
\vdots&\vdots&\ddots&\vdots\\\
R_{\gamma+\alpha_{1}-\alpha_{n}}&R_{\gamma+\alpha_{2}-\alpha_{n}}&\cdots&R_{\gamma+\alpha_{n}-\alpha_{n}}\end{pmatrix}}.$
(2.10)

We identify graded finitely generated projective modules with homogeneous
idempotent matrices of degree zero. This is taken from [17, Lemma 3.2.3].

######  Lemma 2.5.

We have the following statements.

  1. (1)

Any graded finitely generated projective module gives rise to a homogeneous
idempotent matrix of degree zero.

  2. (2)

Any homogeneous idempotent matrix of degree zero gives rise to a graded
finitely generated projective module.

###### Proof.

For (1), let $P$ be a graded finitely generated projective $R$-module. Then
there is a graded module $Q$ such that $P\oplus Q\cong
R^{n}(\overline{\alpha})$ for some $n\in\mathbb{N}$ and
$\overline{\alpha}=(\alpha_{1},\cdots,\alpha_{n})\in\Gamma^{n}$. Define the
homomorphism $e\in\operatorname{End}_{R}(R^{n}(\overline{\alpha}))$ which
sends $Q$ to zero and acts as identity on $P$. Clearly, $e$ is an idempotent
and graded homomorphism of degree zero. Thus
$e\in\operatorname{End}_{R}(R^{n}(\overline{\alpha}))_{0}\cong\mathbb{M}_{n}(R)(\overline{\alpha})_{0}$.

For (2), let $e\in\mathbb{M}_{n}(R)(\overline{\alpha})_{0}$ be a homogeneous
idempotent matrix of degree zero, where
$\overline{\alpha}=(\alpha_{1},\cdots,\alpha_{n})$ with $\alpha_{i}\in\Gamma$.
Then $1-e\in\mathbb{M}_{n}(R)(\overline{\alpha})_{0}$ and

$R^{n}(\overline{\alpha})=R^{n}e(\overline{\alpha})\oplus
R^{n}(1-e)(\overline{\alpha}).$

This shows that $R^{n}e(\overline{\alpha})$ is a graded finitely generated
projective $R$-module. ∎

In the first part of the paper we assume $\mathsf{k}$ to be a commutative
graded ring. In the second part, for the realisation of conical
$\Gamma$-monoids, we assume $\mathsf{k}$ to be a (graded) field concentrated
in degree zero.

###  2.4. Non-stable Graded $K$-theory

For a $\Gamma$-graded ring $R$ with identity and a graded finitely generated
projective (left) $R$-module $P$, let $[P]$ denote the class of graded
$R$-modules graded isomorphic to $P$. Then the set

$\mathcal{V}^{\operatorname{gr}}(R)=\big{\\{}\,[P]\mid P\text{ is a graded
finitely generated projective R-module}\,\big{\\}}$ (2.11)

with direct sum as addition constitutes a $\Gamma$-monoid structure defined as
follows: $[P]+[Q]=[P\oplus Q]$ and for $\gamma\in\Gamma$ and
$[P]\in\mathcal{V}^{\operatorname{gr}}(R)$, ${}^{\gamma}[P]=[P(\gamma)]$. The
group completion of $\mathcal{V}^{\operatorname{gr}}(R)$ is called the _graded
Grothendieck group_ and is denoted by $K^{\operatorname{gr}}_{0}(R)$, which as
the above discussion shows is a $\mathbb{Z}[\Gamma]$-module.

Following the notion of order unit for a $\Gamma$-monoid in §2.1, an element
$[P]\in\mathcal{V}^{\operatorname{gr}}(R)$ is an order unit for
$\mathcal{V}^{\operatorname{gr}}(R)$ if for any
$[Q]\in\mathcal{V}^{\operatorname{gr}}(R)$ we have $[Q]\leq\sum_{1\leq i\leq
n}[P(\gamma_{i})]$ for some $n\in\mathbb{N}$ and $\gamma_{i}\in\Gamma$. Since
for any graded finitely generated projective module $Q$, there is a graded
module $P$ such that $Q\oplus P\cong R(\gamma_{1})\oplus
R(\gamma_{2})\oplus\cdots\oplus R(\gamma_{t})$, it follows that that $[R]$ is
an order unit (see [17, §3]).

The following easy proposition shows that the order unit $[R]$ in
$\mathcal{V}^{\operatorname{gr}}(R)$ can determine when $R$ is a strongly
graded or a crossed product ring (see also [21, Theorems 2.10.1 and 2.10.2]).
This will be used in Theorem 8.1.

######  Proposition 2.6.

Let $R$ be a $\Gamma$-graded ring with identity and
$\mathcal{V}^{\operatorname{gr}}(R)$ the monoid of graded finitely generated
projective modules. Then

  1. (1)

The ring $R$ is strongly graded if and only if $[R]$ is an strong order unit.

  2. (2)

The ring $R$ is crossed-product if and only if $[R]$ is an invariant order
unit.

###### Proof.

(1) The ring $R$ is strongly graded if and only if, for any $\alpha\in\Gamma$,
$1=\sum_{i=1}^{n_{\alpha}}r_{i}s_{i}$, where $r_{i}\in R_{\alpha}$ and
$s_{i}\in R_{-\alpha}$. This is equivalent to having matrices
${\mathbf{r}}=(r_{1},\dots,r_{n_{\alpha}})\in\mathbb{M}_{1\times
n_{\alpha}}(R)[\overline{\alpha}][0]$ and
${\mathbf{s}}=(s_{1},\dots,s_{n_{\alpha}})^{t}\in\mathbb{M}_{n_{\alpha}\times
1}(R)[0][\overline{\alpha}]$ with ${\mathbf{r}}{\mathbf{s}}=1$. In turn this
is equivalent to having an epimorphism $\bigoplus_{n_{\alpha}}R\rightarrow
R(\alpha)\rightarrow 0$, which is equivalent to $[R(\alpha)]\leq
n_{\alpha}[R]$ in $\mathcal{V}^{\operatorname{gr}}(R)$. This is now equivalent
to $[R]$ being a strong order unit.

(2) This follows from the observation that there is a graded $R$-module
isomorphism $R\cong R(\alpha)$ if and only if there is an invertible element
in $R_{\alpha}$. ∎

###  2.5. Graded hereditary rings

Recall that a unital ring $R$ is called a left hereditary ring if any left
ideal of $R$ is a projective $R$-module [28, Chapter 4]. Analogously, we say a
$\Gamma$-graded ring $R$ is called a _left graded hereditary ring_ if any
graded left ideal of $R$ is a graded projective module.

The following proposition is the graded version of well-known equivalent
properties of hereditary rings whose proof follows mutatis mutandis, with an
attention given to the grading.

######  Proposition 2.7.

Let $R$ be a $\Gamma$-graded ring. The following statements are equivalent.

  1. (1)

$R$ is left graded hereditary;

  2. (2)

Any graded submodule of a graded projective left $R$-module is graded
projective;

  3. (3)

Any graded submodule of a graded projective left $R$-module is projective;

  4. (4)

The graded left global dimension is less than two;

  5. (5)

${\rm Ext}_{R\text{-}\operatorname{{\bf Gr}}}^{2}(-,-)$ vanishes.

Proposition 2.7(3) implies that a $\Gamma$-graded ring which is hereditary is
indeed graded hereditary. However, the following example shows that a graded
hereditary ring is not necessarily hereditary. This is in contrast with the
fact that graded regular rings and regular rings which are graded (i.e., their
(graded) global dimension is finite) coincide.

Consider a strongly $\Gamma$-graded ring $R$. By Dade’s theorem (see Section
2.2), the categories $R$-$\operatorname{{\bf Gr}}$ and
$R_{0}$-$\operatorname{{\bf Mod}}$ are equivalent. An application of
Proposition 2.7 shows that if $R_{0}$ is hereditary, then $R$ is a graded
hereditary ring. The $\mathbb{Z}$-graded ring $R=\mathbb{Z}[x,x^{-1}]$ with
$\deg(x)=1$ and $\deg(x^{-1})=-1$ is a strongly graded ring with the zero-
component the ring $\mathbb{Z}$ which is hereditary (PID rings are
hereditary). Thus $R$ is a graded hereditary ring. However, $R$ is not
hereditary: for example, the (non-graded) ideal $\langle 2,x-1\rangle$ is not
a projective module.

Note that the graded ring $\mathsf{k}[x,x^{-1}]$, where $\mathsf{k}$ is a
field, is indeed a PID ring and thus a hereditary ring. We will see the graded
rings coming out of the graded Bergman constructions are indeed hereditary and
thus they are graded hereditary as well.

###  2.6. Smash products of graded algebras

####  2.6.1. Smash products

Let $R$ be a $\Gamma$-graded ring. The smash product ring $R\\#\Gamma$ is
defined as the set of all formal sums
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}$, where $r^{(\gamma)}\in
R$ for any $\gamma\in\Gamma$, the $p_{\gamma}$’s are symbols, and all but
finitely many coefficients $r^{(\gamma)}$ are zero. Addition is defined
component-wise and multiplication is defined by linear extension of the rule
$(rp_{\alpha})(sp_{\beta})=rs_{\alpha-\beta}p_{\beta}$, where $r,s\in R$ and
$\alpha,\beta\in\Gamma$. Instead of $1p_{\gamma}$ we may write $p_{\gamma}$.
If $R$ is a $\Gamma$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is a
commutative graded ring concentrated in degree $0$, then $R\\#\Gamma$ is a
(not-necessarily unital) $\mathsf{k}$-algebra with scalar multiplication
$\lambda\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}=\sum\limits_{\gamma\in\Gamma}\lambda
r^{(\gamma)}p_{\gamma}$, for any $\lambda\in\mathsf{k}$ and
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}\in R\\#\Gamma$.

######  Lemma 2.8.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is a
commutative graded ring concentrated in degree $0$. Then the
$\mathsf{k}$-algebra $R\\#\Gamma$ has the presentation

$\displaystyle R\\#\Gamma=\Big{\langle}rp_{\gamma}~{}(r\in
R^{h},\gamma\in\Gamma)\mid{}$ $\displaystyle
rp_{\gamma}+sp_{\gamma}=(r+s)p_{\gamma}\quad(r,s\in
R_{\delta};~{}\gamma,\delta\in\Gamma),$
$\displaystyle~{}rp_{\gamma}sp_{\delta}=rs_{\gamma-\delta}p_{\delta}\quad(r,s\in
R^{h};~{}\gamma,\delta\in\Gamma)\Big{\rangle}.$

###### Proof.

Denote by $F$ the free $\mathsf{k}$-algebra generated by the set
$\\{rp_{\gamma}~{}(r\in R^{h},\gamma\in\Gamma)\\}$, and by $I$ the two-sided
ideal of $F$ generated by the set

$\big{\\{}rp_{\gamma}+sp_{\gamma}-(r+s)p_{\gamma}\mid r,s\in
R_{\delta};~{}\gamma,\delta\in\Gamma\big{\\}}\bigcup\big{\\{}rp_{\gamma}sp_{\delta}-rs_{\gamma-\delta}p_{\delta}\mid
r,s\in R^{h};~{}\gamma,\delta\in\Gamma\big{\\}}.$

Clearly the $\mathsf{k}$-algebra homomorphism $\theta:F\to R\\#\Gamma$ defined
by $\theta(rp_{\gamma})=rp_{\gamma}~{}(r\in R^{h},\gamma\in\Gamma)$ induces a
$\mathsf{k}$-algebra homomorphism $\bar{\theta}:F/I\to R\\#\Gamma$ (since
$I\subseteq\ker(\theta)$). Since the elements $rp_{\gamma}~{}(r\in
R^{h},\gamma\in\Gamma)$ generate the $\mathsf{k}$-algebra $R\\#\Gamma$, the
homomorphism $\bar{\theta}$ is surjective. It remains to show that
$\bar{\theta}$ is injective. Suppose that $\bar{\theta}(x)=0$, where $x\in
F/I$ is non-zero. Clearly we can write
$x=r_{1}p_{\gamma_{1}}+\dots+r_{n}p_{\gamma_{n}}$, where $r_{1},\dots,r_{n}\in
R^{h}\setminus\\{0\\}$ and $\gamma_{1},\dots,\gamma_{n}\in\Gamma$. For
$\gamma\in\Gamma$ set $J(\gamma):=\\{1\leq i\leq n\mid\gamma_{i}=\gamma\\}$.
We may assume that

$\deg(r_{i})\neq\deg(r_{j}),\text{ for any }\gamma\in\Gamma\text{ and }i\neq
j\in J(\gamma).$ (2.12)

It follows from

$0=\bar{\theta}(x)=r_{1}p_{\gamma_{1}}+\dots+r_{n}p_{\gamma_{n}}=\sum_{\gamma\in\Gamma}\Big{(}\sum_{i\in
J(\gamma)}r_{i}\Big{)}p_{\gamma}$

that $\sum_{i\in J(\gamma)}r_{i}=0$, for any $\gamma\in\Gamma$. In view of
(2.12) this implies $r_{i}=0$, for any $1\leq i\leq n$, a contradiction. Thus
we have shown that $\bar{\theta}$ is injective. ∎

####  2.6.2. Smash products of quotients and extensions

We continue with the assumption that $R$ is a $\Gamma$-graded
$\mathsf{k}$-algebra where $\mathsf{k}$ is a commutative graded ring
concentrated in degree $0$. If $I$ is a graded ideal of $R$, then we denote by
$I\\#\Gamma$ the ideal of $R\\#\Gamma$ consisting of all elements
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}$ such that
$r^{(\gamma)}\in I$, for any $\gamma\in\Gamma$. Note that the quotient ring
$R/I$ is $\Gamma$-graded ring.

######  Proposition 2.9.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is a
commutative graded ring concentrated in degree $0$, and $I$ a graded ideal of
$R$. Then

$(R/I)\\#\Gamma\cong R\\#\Gamma/I\\#\Gamma.$

###### Proof.

We define the map

$\displaystyle f:(R/I)\\#\Gamma$ $\displaystyle\longrightarrow
R\\#\Gamma/I\\#\Gamma$
$\displaystyle\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}+I)p_{\gamma}$
$\displaystyle\longmapsto\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}+I\\#\Gamma.$

Suppose that
$\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}+I)p_{\gamma}=\sum\limits_{\gamma\in\Gamma}(s^{(\gamma)}+I)p_{\gamma}$
in $(R/I)\\#\Gamma$. Then
$\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}-s^{(\gamma)}+I)p_{\gamma}=0$ and
hence $r^{(\gamma)}-s^{(\gamma)}\in I$, for any $\gamma\in\Gamma$. It follows
that $\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}-s^{(\gamma)})p_{\gamma}\in
I\\#\Gamma$ and hence
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}+I\\#\Gamma=\sum\limits_{\gamma\in\Gamma}s^{(\gamma)}p_{\gamma}+I\\#\Gamma$.
Thus $f$ is well-defined. Clearly $f$ is a surjective $\mathsf{k}$-algebra
homomorphism. It remains to show $f$ is injective. Suppose that
$f(\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}+I)p_{\gamma})=0$, i.e.,
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}+I\\#\Gamma=0$. Then
$\sum\limits_{\gamma\in\Gamma}r^{(\gamma)}p_{\gamma}\in I\\#\Gamma$ and hence
$r^{(\gamma)}\in I$ for any $\gamma\in\Gamma$. It follows
$\sum\limits_{\gamma\in\Gamma}(r^{(\gamma)}+I)p_{\gamma}=0$ and thus $f$ is
injective. ∎

If $X$ is a set, we denote by $R\langle X\rangle$ the $\mathsf{k}$-algebra
obtained from $R$ by adjoining the set $X$ (i.e., $R\langle X\rangle$ is the
free product of $R$ and the free $\mathsf{k}$-algebra $k\langle X\rangle$
generated by $X$). If in addition we are given a map $\deg:X\to\Gamma$, then
$R\langle X\rangle$ becomes a $\Gamma$-graded $\mathsf{k}$-algebra with the
induced grading. We denote by $X\\#\Gamma$ the set $\\{xp_{\gamma}\mid x\in
X,\gamma\in\Gamma\\}$.

######  Proposition 2.10.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is a
commutative graded ring concentrated in degree $0$, and $X$ a set and
$\deg:X\to\Gamma$ a map. Consider $R\langle X\rangle$ as a $\Gamma$-graded
$\mathsf{k}$-algebra with the induced grading. Then

$R\langle X\rangle\\#\Gamma\cong R\\#\Gamma\langle X\\#\Gamma\rangle/J,$

where $J$ is the ideal of $R\\#\Gamma\langle X\\#\Gamma\rangle$ generated by
the set

$\big{\\{}(xp_{\gamma})(1p_{\gamma})-xp_{\gamma},~{}(1p_{\gamma+\deg(x)})(xp_{\gamma})-xp_{\gamma}\mid
x\in X,\gamma\in\Gamma\big{\\}}.$

###### Proof.

It follows from Lemma 2.8 that $R\\#\Gamma\langle X\\#\Gamma\rangle/J$ has the
presentation

$\begin{split}R\\#\Gamma\langle
X\\#\Gamma\rangle/J=\Big{\langle}rp_{\gamma},xp_{\gamma}~{}(r\in R^{h},x\in
X,\gamma\in\Gamma)\mid{}&rp_{\gamma}+sp_{\gamma}=(r+s)p_{\gamma}\quad(r,s\in
R_{\delta};~{}\gamma,\delta\in\Gamma),\\\
&~{}rp_{\gamma}sp_{\delta}=rs_{\gamma-\delta}p_{\delta}\quad(r,s\in
R^{h};~{}\gamma,\delta\in\Gamma),\\\
&~{}(xp_{\gamma})(1p_{\gamma})=xp_{\gamma}\quad(x\in X,~{}\gamma\in\Gamma),\\\
&~{}(1p_{\gamma+\deg(x)})(xp_{\gamma})=xp_{\gamma}\quad(x\in
X,~{}\gamma\in\Gamma)\Big{\rangle}.\end{split}$ (2.13)

Since the relations in presentation (2.13) above are satisfied in $R\langle
X\rangle\\#\Gamma$, there is a $k$-algebra homomorphism

$f:R\\#\Gamma\langle X\\#\Gamma\rangle/J\rightarrow R\langle
X\rangle\\#\Gamma,$

such that $f(rp_{\gamma})=rp_{\gamma}$ and $f(xp_{\gamma})=xp_{\gamma}$, for
any $r\in R^{h},x\in X,\gamma\in\Gamma$. Clearly the elements
$rp_{\gamma},xp_{\gamma}$, where $r\in R^{h},x\in X,\gamma\in\Gamma$ generate
the $k$-algebra $R(X)\\#\Gamma$ and hence $f$ is surjective. It remains to
show that $f$ is injective.

Let $w$ be a non-empty and finite word over the alphabet
$\\{rp_{\gamma},xp_{\gamma}~{}(r\in R^{h},x\in X,\gamma\in\Gamma)\\}$. Then
$w=y_{1}p_{\gamma_{1}}\dots y_{n}p_{\gamma_{n}}$ for some
$y_{1},\dots,y_{n}\in R\cup X$ and $\gamma_{1},\dots,\gamma_{n}\in\Gamma$. We
call $w$ a path (ending in $\gamma_{n}$) if
$\gamma_{i-1}-\gamma_{i}=\deg(y_{i})~{}(i=2,\dots,n)$. We denote the set of of
all paths by $W$, and for any $\gamma\in\Gamma$ the set of all paths ending in
$\gamma$ by $W_{\gamma}$. Define a map $\zeta:W\to R\langle X\rangle$ by
$\zeta(y_{1}p_{\gamma_{1}}\dots y_{n}p_{\gamma_{n}})=y_{1}\dots y_{n}$. Now
suppose that $f(a)=0$ for some $a\in R\\#\Gamma\langle X\\#\Gamma\rangle/J$.
It follows from the presentation (2.13) that $a=\sum_{w\in W}\lambda_{w}w$
where $\lambda_{w}\in k~{}(w\in W)$. Hence

$0=f(a)=f\Big{(}\sum_{w\in
W}\lambda_{w}w\Big{)}=f\Big{(}\sum_{\gamma\in\Gamma}\sum_{w\in
W_{\gamma}}\lambda_{w}w\Big{)}=\sum_{\gamma\in\Gamma}\sum_{w\in
W_{\gamma}}\lambda_{w}\zeta(w)p_{\gamma}.$

It follows that $\sum_{w\in W_{\gamma}}\lambda_{w}\zeta(w)=0$ in $R(X)$ for
any $\gamma\in\Gamma$. One checks easily that this implies that $\sum_{w\in
W_{\gamma}}\lambda_{w}w=0$ in $R\\#\Gamma\langle X\\#\Gamma\rangle/J$, for any
$\gamma\in\Gamma$. Hence $a=0$ and thus $f$ is injective. ∎

######  Corollary 2.11.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra where $\mathsf{k}$ is a
commutative graded ring concentrated in degree $0$, $X$ a set and
$\deg:X\to\Gamma$ a map. Consider $R\langle X\rangle$ as a $\Gamma$-graded
$\mathsf{k}$-algebra with the induced grading. Let $I$ be a graded ideal of
$R\langle X\rangle$. Then

$(R\langle X\rangle/I)\\#\Gamma\cong R\\#\Gamma\langle X\\#\Gamma\rangle/J,$

where $J$ is the ideal of $R\\#\Gamma\langle X\\#\Gamma\rangle$ generated by
the image of $I\\#\Gamma$ and the set

$\big{\\{}(xp_{\gamma})(1p_{\gamma})-xp_{\gamma},~{}(1p_{\gamma+\deg(x)})(xp_{\gamma})-xp_{\gamma}\mid
x\in X,\gamma\in\Gamma\big{\\}}.$ (2.14)

###### Proof.

It follows from Proposition 2.9 that $(R\langle X\rangle/I)\\#\Gamma\cong
R\langle X\rangle\\#\Gamma/I\\#\Gamma$. By Proposition 2.10, there is an
isomorphism $g:R\langle X\rangle\\#\Gamma\to R\\#\Gamma\langle
X\\#\Gamma\rangle/J^{\prime}$, where $J^{\prime}$ is the ideal of
$R\\#\Gamma\langle X\\#\Gamma\rangle$ generated by (2.14). Hence

$(R\langle X\rangle/I)\\#\Gamma\cong R\langle
X\rangle\\#\Gamma/I\\#\Gamma\cong(R\\#\Gamma\langle
X\\#\Gamma\rangle/J^{\prime})/g(I\\#\Gamma).$

The assertion of the corollary follows. ∎

####  2.6.3. An isomorphism of categories

Recall that $R$-$\operatorname{{\bf Gr}}$ denotes the category of
$\Gamma$-graded $R$-modules and $R$-$\operatorname{{\bf
Gr}}_{\operatorname{proj}}$ the full subcategory of $R$-$\operatorname{{\bf
Gr}}$ whose objects are the projective objects of $R$-$\operatorname{{\bf
Gr}}$ that are finitely generated as a $R$-module. Moreover,
$R\\#\Gamma$-$\operatorname{{\bf Mod}}$ denotes the category of unital
$R\\#\Gamma$-modules and $R\\#\Gamma$-$\operatorname{{\bf
Mod}}_{\operatorname{proj}}$ the full subcategory of
$R\\#\Gamma$-$\operatorname{{\bf Mod}}$ whose objects are the projective
objects of $R\\#\Gamma$-$\operatorname{{\bf Mod}}$ that are finitely generated
as a $R\\#\Gamma$-module. In [3] it was shown that there is an isomorphism
$\psi:R$-$\operatorname{{\bf Gr}}\to R\\#\Gamma$-$\operatorname{{\bf Mod}}$,
and in [24] it was shown that $\psi$ restricts to an isomorphism
$R$-$\operatorname{{\bf Gr}}_{\operatorname{proj}}\to
R\\#\Gamma$-$\operatorname{{\bf Mod}}_{\operatorname{proj}}$. The isomorphism
$\psi$ maps an object $M$ of $R$-$\operatorname{{\bf Gr}}$ to the object
$M\\#\Gamma$ of $R\\#\Gamma$-$\operatorname{{\bf Mod}}$ which is defined as
follows. As abelian group $M\\#\Gamma=M$. The $R\\#\Gamma$-action on
$M\\#\Gamma$ is defined by $(rp_{\alpha})m=rm_{\alpha}$ for any $r\in R$,
$\alpha\in\Gamma$ and $m\in M=M\\#\Gamma$. On morphisms, $\psi$ is just the
identity map.

##  3\. Leavitt path algebras

Let $\mathsf{k}$ be a field. Consider the symbols $X=(x_{ij})$ and
$X^{*}=(x^{*}_{ji})$, where $1\leq i\leq n$ and $1\leq j\leq n+k$:

$X=\left(\begin{matrix}x_{11\phantom{n+{},}}&x_{12\phantom{n+{},}}&\dots&x_{1,n+k}\\\
x_{21\phantom{n+{},}}&x_{22\phantom{n+{},}}&\dots&x_{2,n+k}\\\
\vdots&\vdots&\ddots&\vdots\\\
x_{n,1\phantom{n+{},}}&x_{n,2\phantom{n+{},}}&\dots&x_{n,n+k}\end{matrix}\right),\qquad
X^{*}=\left(\begin{matrix}x^{*}_{11\phantom{n+{},}}&x^{*}_{12\phantom{n+{},}}&\dots&x^{*}_{1,n\phantom{n+{},}}\\\
x^{*}_{21\phantom{n+{},}}&x^{*}_{22\phantom{n+{},}}&\dots&x^{*}_{2,n\phantom{n+{},}}\\\
\vdots&\vdots&\ddots&\vdots\\\
x^{*}_{n+k,1}&x^{*}_{n+k,2}&\dots&x^{*}_{n+k,n}\end{matrix}\right).$ (3.1)

The _Leavitt algebra_ $L_{\mathsf{k}}(n,n+k)$ is defined as

$L_{\mathsf{k}}(n,n+k)=\frac{\mathsf{k}\langle X,X^{*}\rangle}{\langle
XX^{*}=I_{n},\,X^{*}X=I_{n+k}\rangle}.$ (3.2)

Here $\mathsf{k}\langle X,X^{*}\rangle$ is the free $\mathsf{k}$-algebra
generated by the symbols $x_{ij}$ and $x_{ji}^{*}$’s. Furthermore, the
relations $XX^{*}=I_{n}$ and $X^{*}X=I_{n+k}$, where
$I_{n}\in\mathbb{M}_{n}(\mathsf{k})$ and
$I_{n+k}\in\mathbb{M}_{n+k}(\mathsf{k})$ are the identity matrices, stand for
the collection of relations after multiplying the matrices and comparing the
two sides of the equations.

These rings were first studied by Leavitt [19] and [20, p.130, footnote 6].
Leavitt showed the type of the ring $L_{\mathsf{k}}(n,n+k)$ is $(n,n+k)$.
Furthermore, $L_{\mathsf{k}}(n,n+k)$, for $n>1$ and $k\geq 1$, is a domain,
whereas $L_{\mathsf{k}}(1,k+1)$, is a simple ring.

If in (3.2) we only consider the relation $X^{*}X=I_{n+k}$, then we call the
resulting algebra _Cohn algebra_

$C_{\mathsf{k}}(n,n+k)=\frac{\mathsf{k}\langle X,X^{*}\rangle}{\langle
X^{*}X=I_{n+k}\rangle},$ (3.3)

which was considered by Cohn [11], who showed that $C_{\mathsf{k}}(n,n+k)$,
for $n>1$ and $k\geq 1$, is a domain, whereas $C_{\mathsf{k}}(1,k+1)$ has only
one non-trivial proper ideal generated by relation $XX^{*}=I_{n}$. This result
also implies the simplicity of $L_{\mathsf{k}}(1,k+1)$.

The simple rings $L_{\mathsf{k}}(1,k+1)$ received significant attentions as
they are the discrete versions of Cuntz algebras. Modeled on this, Leavitt
path algebras are introduced [1], which attach to a directed graph a certain
algebra. In the case that the graph has one vertex and $k+1$ loops, it
recovers Leavitt algebra $L_{\mathsf{k}}(1,k+1)$.

In order to recover the Leavitt algebras $L_{\mathsf{k}}(n,n+k)$, for $n>1$, a
plethora of generalisations of Leavitt path algebras were introduced, such as
Leavitt path algebra of separated graphs, weighted graphs and ultra graphs.
Here, following [24], we introduce the Leavitt path algebras of hypergraphs.
We show that Bergman algebras (see Section 4) with coefficients in a
commutative semisimple ring are precisely Leavitt path algebras of hypergraphs
(Theorem 4.10).

###  3.1. Leavitt path algebras of hypergraphs

For sets $I$ and $X$, a function $x:I\rightarrow X$, $i\mapsto x_{i}=x(i)$ is
called a family of elements in $X$ indexed by $I$. A family $x$ of elements in
$X$ indexed by $I$ is usually denoted as $(x_{i})_{i\in I}$. We call the
family $(x_{i})_{i\in I}$ non-empty if $I\neq\emptyset$.

A (directed) hypergraph is a quadruple $H=(H^{0},H^{1},s,r)$, where $H^{0}$
and $H^{1}$ are sets and $s$ and $r$ are maps associating to each $h\in H^{1}$
a non-empty family $s(h)=(s(h)_{i})_{i\in I_{h}}$, respectively,
$r(h)=(r(h)_{j})_{j\in J_{h}}$ of elements in $H^{0}$. The elements of $H^{0}$
are called vertices and the elements of $H^{1}$ hyperedges. In this paper all
hypergraphs are assumed to be ${\it regular}$, i.e. for any hyperedge $h$ the
sets $I_{h}$ and $J_{h}$ are finite.

######  Definition 3.1.

Let $H$ be a hypergraph. The not necessarily unital $\mathsf{k}$-algebra
$L_{\mathsf{k}}(H)$ presented by the generating set

$\\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},i\in I_{h},j\in J_{h}\\}$

and the relations

  1. (i)

$uv=\delta_{uv}u\quad(u,v\in H^{0})$,

  2. (ii)

$s(h)_{i}h_{ij}=h_{ij}=h_{ij}r(h)_{j},~{}r(h)_{j}h_{ij}^{*}=h_{ij}^{*}=h_{ij}^{*}s(h)_{i}\quad(h\in
H^{1},i\in I_{h},j\in J_{h})$,

  3. (iii)

$\sum\limits_{j\in
J_{h}}h_{ij}h_{i^{\prime}j}^{*}=\delta_{ii^{\prime}}s(h)_{i}\quad(h\in
H^{1},~{}i,i^{\prime}\in I_{h})$ and

  4. (iv)

$\sum\limits_{i\in
I_{h}}h_{ij}^{*}h_{ij^{\prime}}=\delta_{jj^{\prime}}r(h)_{j}\quad(h\in
H^{1},~{}j,j^{\prime}\in J_{h})$

is called the Leavitt path algebra of the hypergraph $H$. We will sometimes
call a Leavitt path algebra of a hypergraph a hyper Leavitt path algebra.

It is not difficult to check that if $H$ is a hypergraph, then
$L_{\mathsf{k}}(H)$ has a set of local units, namely the set of all finite
sums of distinct elements of $H^{0}$. Furthermore, $L_{K}(H)$ is a unital ring
if and only if $H^{0}$ is finite ([24, Proposition 8]).

######  Remark 3.2.

Let $H$ be a hypergraph. We can describe the relations (ii)-(iv) in Definition
3.1 using matrices as follows. For any hyperedge $h\in H^{1}$, let $e_{h}$ be
the $I_{h}\times I_{h}$-matrix whose entry at position $(i,i^{\prime})$ is
$\delta_{ii^{\prime}}s(h)_{i}$, and $f_{h}$ the $J_{h}\times J_{h}$-matrix
whose entry at position $(j,j^{\prime})$ is $\delta_{jj^{\prime}}r(h)_{j}$.
Moreover, we denote for any hyperedge $h\in H^{1}$ the $I_{h}\times
J_{h}$-matrix whose entry at position $(i,j)$ is $h_{ij}$ also by $h$, and the
$J_{h}\times I_{h}$-matrix whose entry at position $(j,i)$ is $h^{*}_{ij}$ by
$h^{*}$. Then $L_{\mathsf{k}}(H)$ is the $\mathsf{k}$-algebra presented by the
generating set $\\{v,h_{ij},h_{ij}^{*}\mid v\in H^{0},h\in H^{1},i\in
I_{h},j\in J_{h}\\}$ and relations

  1. (I)

the elements of $H^{0}$ are pairwise orthogonal idempotents,

  2. (II)

$h=e_{h}hf_{h},\quad h^{*}=f_{h}h^{*}e_{h}\quad(h\in H^{1})$,

  3. (III)

$hh^{*}=e_{h}\quad(h\in H^{1})$ and

  4. (IV)

$h^{*}h=f_{h}\quad(h\in H^{1})$.

######  Remark 3.3.

Let $H$ be a hypergraph and $\Gamma$ an abelian group. For any $h\in H^{1}$
choose an $i_{h}\in I_{h}$ and a $j_{h}\in J_{h}$. A map $w$ which associates
to any generator $h_{ij}~{}(h\in H^{1},i\in I_{h},j\in J_{h})$ an element from
$\Gamma$, such that

$w(h_{ij})=w(h_{ij_{h}})-w(h_{i_{h}j_{h}})+w(h_{i_{h}j})$

for any $h\in H^{1}$, $i\in I_{h}$ and $j\in J_{h}$ is called a
$\Gamma$-weight map for $H$. By [24, Lemma 70], a $\Gamma$-weight map $w$ for
$H$ induces a $\Gamma$-grading on $L_{\mathsf{k}}(H)$ such that $\deg(v)=0$,
$\deg(h_{ij})=w(h_{ij})$ and $\deg(h_{ij}^{*})=-w(h_{ij})$ for any $v\in
H^{0}$, $h\in H^{1}$, $i\in I_{h}$ and $j\in J_{h}$.

######  Example 3.4 (Leavitt path algebras).

Suppose that $E$ is a directed graph, i.e. a quadruple $E=(E^{0},E^{1},r,s)$,
where $E^{0}$ and $E^{1}$ are sets and $r,s$ are maps from $E^{1}$ to $E^{0}$.
The elements of $E^{0}$ are called vertices and the elements of $E^{1}$ edges.
We may think of an edge $e\in E^{1}$ as an arrow from $s(e)$ to $r(e)$. Assume
that $E$ is _row-finite_ , i.e. for each vertex $v\in E^{0}$ there are at most
finitely many edges in $s^{-1}(v)$. Denote by $E^{\operatorname{reg}}$ the set
of regular vertices of the graph $E$, i.e. the set of all vertices $v$ for
which $s^{-1}(v)$ is not the empty set. The $\mathsf{k}$-algebra
$L_{\mathsf{k}}(E)$ presented by the generating set $\\{v,e,e^{*}\mid v\in
E^{0},e\in E^{1}\\}$ and the relations

  1. (i)

$uv=\delta_{uv}u\quad(u,v\in E^{0})$,

  2. (ii)

$s(e)e=e=er(e),~{}r(e)e^{*}=e^{*}=e^{*}s(e)\quad(e\in E^{1})$,

  3. (iii)

$\sum_{e\in s^{-1}(v)}ee^{*}=v\quad(v\in E^{\operatorname{reg}})$ and

  4. (iv)

$e^{*}f=\delta_{ef}r(e)\quad(e,f\in E^{1})$

is called the Leavitt path algebra of $E$ (cf. for example [1]). For each
$v\in E^{\operatorname{reg}}$ write $s^{-1}(v)=\\{e^{v,j}\mid j\in J_{v}\\}$.
Define a hypergraph $H=(H^{0},H^{1},s^{\prime},r^{\prime})$ by $H^{0}=E^{0}$,
$H^{1}=\\{h^{v}\mid v\in E^{\operatorname{reg}}\\}$,
$s^{\prime}(h^{v})=(v)_{i\in\\{1\\}}$ and
$r^{\prime}(h^{v})=(r(e^{v,j}))_{j\in J_{v}}$. There is a $\mathsf{k}$-algebra
isomorphism $\phi:L_{\mathsf{k}}(E)\rightarrow L_{\mathsf{k}}(H)$ such that
$\phi(u)=u$, $\phi(e^{v,j})=h^{v}_{1j}$ and
$\phi((e^{v,j})^{*})=(h^{v}_{1j})^{*}$ for any $u\in E^{0}$, $v\in
E^{\operatorname{reg}}$ and $j\in J_{v}$.

######  Example 3.5 (Vertex-weighted Leavitt path algebras).

Suppose that $(E,w)$ is a vertex-weighted graph, i.e. $E$ is a directed graph
and $w$ is a map associating to each regular vertex a positive integer. We
again assume that $E$ is row-finite. The $\mathsf{k}$-algebra
$L_{\mathsf{k}}(E,w)$ presented by the generating set
$\\{v,e_{i},e_{i}^{*}\mid v\in E^{0},e\in E^{1},1\leq i\leq w(s(e))\\}$ and
the relations

  1. (i)

$uv=\delta_{uv}u\quad(u,v\in E^{0})$,

  2. (ii)

$s(e)e_{i}=e_{i}=e_{i}r(e),~{}r(e)e_{i}^{*}=e_{i}^{*}=e_{i}^{*}s(e)\quad(e\in
E^{1},1\leq i\leq w(s(e)))$,

  3. (iii)

$\sum_{e\in s^{-1}(v)}e_{i}e_{j}^{*}=\delta_{ij}v\quad(v\in
E^{\operatorname{reg}};~{}1\leq i,j\leq w(v))$ and

  4. (iv)

$\sum_{1\leq i\leq w(v)}e_{i}^{*}f_{i}=\delta_{ef}r(e)\quad(v\in
E^{\operatorname{reg}};~{}e,f\in s^{-1}(v))$

is called the Leavitt path algebra of $(E,w)$. For each $v\in
E^{\operatorname{reg}}$ write $s^{-1}(v)=\\{e^{v,j}\mid j\in J_{v}\\}$. Define
a hypergraph $H=(H^{0},H^{1},s^{\prime},r^{\prime})$ by $H^{0}=E^{0}$,
$H^{1}=\\{h^{v}\mid v\in E^{0}_{\operatorname{reg}}\\}$,
$s^{\prime}(h^{v})=(v)_{i\in\\{1,\dots,w(v)\\}}$ and
$r^{\prime}(h^{v})=(r(e^{v,j}))_{j\in J_{v}}$. There is a $\mathsf{k}$-algebra
isomorphism $\psi:L_{\mathsf{k}}(E,w)\rightarrow L_{\mathsf{k}}(H)$ such that
$\psi(u)=u$, $\psi(e^{v,j}_{i})=h^{v}_{ij}$ and
$\psi((e^{v,j}_{i})^{*})=(h^{v}_{ij})^{*}$ for any $u\in E^{0}$, $v\in
E^{\operatorname{reg}}$, $i\in\\{1,\dots,w(v)\\}$ and $j\in J_{v}$.

##  4\. Bergman algebras

Let $R$ be a ring and $e\in\mathbb{M}_{m}(R)$ and $f\in\mathbb{M}_{n}(R)$ be
idempotent matrices. Recall that $e$ and $f$ are called _equivalent_ , denoted
$e\sim f$, if there are matrices $h\in\mathbb{M}_{m\times n}(R)$ and $h^{*}\in
M_{n\times m}(R)$ such that $e=hh^{*}$ and $f=h^{*}h$.

######  Definition 4.1.

Let $R$ be a $\mathsf{k}$-algebra, where $\mathsf{k}$ is a commutative ring.
Let $e\in\mathbb{M}_{m}(R)$ and $f\in\mathbb{M}_{n}(R)$ be idempotent
matrices. Let $h=(h_{ij})$ and $h^{*}=(h^{*}_{ji})$, where $1\leq i\leq m$ and
$1\leq j\leq n$, be a collection of symbols. Then we define a chain of
_Bergman algebras_

$B^{1}_{R}(e,f)\longrightarrow B^{2}_{R}(e,f)\longrightarrow
B^{3}_{R}(e,f)\longrightarrow B_{R}(e,f),$

as follows:

$\displaystyle B^{1}_{R}(e,f)$
$\displaystyle:=R\big{\langle}h\big{\rangle}\big{/}\big{\langle}ehf=h\big{\rangle}$
$\displaystyle B^{2}_{R}(e,f)$
$\displaystyle:=R\big{\langle}h,h^{*}\big{\rangle}\big{/}\big{\langle}ehf=h,fh^{*}e=h^{*}\big{\rangle}$
$\displaystyle B^{3}_{R}(e,f)$
$\displaystyle:=R\big{\langle}h,h^{*}\big{\rangle}\big{/}\big{\langle}ehf=h,fh^{*}e=h^{*},hh^{*}=e\big{\rangle}$

and

$\displaystyle B_{R}(e,f)=B^{4}_{R}(e,f)$
$\displaystyle:=R\big{\langle}h,h^{*}\big{\rangle}\big{/}\big{\langle}ehf=h,fh^{*}e=h^{*},hh^{*}=e,h^{*}h=f\big{\rangle}.$
(4.1)

We further define,

$B_{R}(e)=R\big{\langle}h\big{\rangle}\big{/}\big{\langle}ehe=h,h^{2}=h\big{\rangle}.$

Here $R\langle h,h^{*}\rangle$ is the free ring generated by symbols $h_{ij}$
and $h^{*}_{ji}$’s with coefficients from the $\mathsf{k}$-algebra $R$. We
assume the symbols commute with $\mathsf{k}$. In fact $R\langle
h,h^{*}\rangle$ is the coproduct $R*_{\mathsf{k}}\mathsf{k}\langle
h_{ij},h^{*}_{ji}\mid 1\leq i\leq m,1\leq j\leq n\rangle$. Furthermore, the
relations such as $ehf=h$ stand for the collection of relations after
multiplying the matrices and comparing the two sides of the equation. Clearly
there is a canonical $\mathsf{k}$-algebra homomorphim $R\rightarrow
B_{R}^{i}(e,f)$, where the images of $e$ and $f$ in $B_{R}(e,f)$ become
equivalent.

In this note, we particularly concentrate on the Bergman algebra $B_{R}(e,f)$
defined in (4.1), as many of combinatorial algebras can be realised as such
Bergman algebras.

######  Example 4.2.

We can recover classical Cohn and Leavitt algebras (3.3) and (3.2),
respectively, as follows. For a field $\mathsf{k}$, and idempotent matrices
$1\in\mathsf{k}$ and $I_{n}\in\mathbb{M}_{n}(\mathsf{k})$, we have

$\displaystyle B^{1}_{\mathsf{k}}(1,I_{n})$
$\displaystyle\cong\mathsf{k}\langle x_{1},x_{2},\dots,x_{n}\rangle$
$\displaystyle B^{2}_{\mathsf{k}}(1,I_{n})$
$\displaystyle\cong\mathsf{k}\langle
x_{1},x_{2},\dots,x_{n},x_{1}^{*},x_{2}^{*},\dots,x_{n}^{*}\rangle$
$\displaystyle B^{3}_{\mathsf{k}}(I_{n},1)$ $\displaystyle\cong
C_{\mathsf{k}}(1,n)$ $\displaystyle
B_{\mathsf{k}}(1,I_{n})=B^{4}_{\mathsf{k}}(1,I_{n})$ $\displaystyle\cong
L_{\mathsf{k}}(1,n),$

furthermore

$\displaystyle B(I_{n},I_{n+k})=B^{4}(I_{n},I_{n+k})$ $\displaystyle\cong
L_{\mathsf{k}}(n,n+k)$ $\displaystyle B^{3}(I_{n+k},I_{n})$
$\displaystyle\cong C_{\mathsf{k}}(n,n+k)$

######  Remark 4.3.

In this note we use categories of left modules over rings. One can also use
categories of right modules. In that case, one would have
$B^{3}_{\mathsf{k}}(1,I_{n})\cong C_{\mathsf{k}}(1,n)$ returning to right
$R$-modules.

######  Example 4.4.

This example demonstrates how rich the structure of seemingly easy to
construct Bergman algebras of (4.1) could be. Consider the Bergman algebra
$B_{\mathbb{M}_{2}(\mathsf{k})}(1,1)$. Here $1$ is the identity matrix of the
$\mathsf{k}$-algebra $\mathbb{M}_{2}(\mathsf{k})$. One can immediately see
that

$B_{\mathbb{M}_{2}(\mathsf{k})}(1,1)\cong\mathbb{M}_{2}(\mathsf{k})*_{\mathsf{k}}\mathsf{k}[x,x^{-1}].$

Furthermore, the discussion in Section 4 of [7] shows that the Leavitt algebra
$L_{\mathsf{k}}(2,2)$ coincides with a corner of this Bergman algebra,
specifically,

$L_{\mathsf{k}}(2,2)\cong e_{11}B_{\mathbb{M}_{2}(\mathsf{k})}(1,1)e_{11}.$

######  Lemma 4.5.

Let $R$ be a $\mathsf{k}$-algebra, where $\mathsf{k}$ is a commutative ring.
Let $e\in\mathbb{M}_{m}(R)$ and $f\in\mathbb{M}_{n}(R)$ be idempotent
matrices. If idempotent matrices $e^{\prime}\in\mathbb{M}_{k}(R)$ and
$f^{\prime}\in\mathbb{M}_{l}(R)$ are equivalent to $e$ and $f$, respectively,
then $B_{R}^{i}(e,f)$ is $\mathsf{k}$-algebra isomorphic to
$B_{R}^{i}(e^{\prime},f^{\prime})$, where $1\leq i\leq 4$.

###### Proof.

We show that if $e\sim e^{\prime}$ then $B_{R}^{i}(e,f)\cong
B_{R}^{i}(e^{\prime},f)$. The general result follows by symmetry.

Since $e\sim e^{\prime}$, there are matrices $x\in\mathbb{M}_{m\times k}(R)$,
$y\in\mathbb{M}_{k\times m}(R)$ such that $e=xy$, $e^{\prime}=yx$. Define the
map $\phi:B_{R}^{i}(e,f)\rightarrow B_{R}^{i}(e^{\prime},f)$ by sending
$h\mapsto xh^{\prime}$ and $h^{*}$$\mapsto{h^{*}}^{\prime}y$ (if $h^{*}$ is
present in the definition of $B_{R}^{i}(e,f)$). The converse map
$\psi:B_{R}^{i}(e^{\prime},f)\rightarrow B_{R}^{i}(e,f)$ is defined by sending
$h^{\prime}\mapsto yh$ and ${h^{*}}^{\prime}\mapsto h^{*}x$. We leave it to
the reader to check these assignments respect the defining relations and give
a $\mathsf{k}$-algebra isomorphism between the two algebras. ∎

######  Example 4.6.

We note that if the idempotents $e\in\mathbb{M}_{m}(R)$ and
$f\in\mathbb{M}_{n}(R)$ are equivalent, i.e., $e=hh^{*}$ and $f=h^{*}h$, then
by considering $h_{1}=ehf$ and $h_{1}^{*}=fhe$ we have $e=h_{1}h_{1}^{*}$ and
$f=h_{1}^{*}h_{1}$, with, $eh_{1}f=h_{1}$ and $fh_{1}^{*}e=h_{1}^{*}$. However
the algebras

$B^{\prime}(e,f):=R\big{\langle}h,h^{*}\big{\rangle}\big{/}\big{\langle}hh^{*}=e,h^{*}h=f\big{\rangle}.$

and

$B_{R}(e,f)=R\big{\langle}h,h^{*}\big{\rangle}\big{/}\big{\langle}ehf=h,fh^{*}e=h^{*},hh^{*}=e,h^{*}h=f\big{\rangle}.$

are not necessarily isomorphic, as the following example shows.

Suppose that $\mathsf{k}$ is a field (or more generally an integral domain).
Let $e=f\in\mathbb{M}_{2}(\mathsf{k})$ be the idempotent matrix which has a
one at position $(1,1)$ and zeros elsewhere (note that the projective
$\mathsf{k}$-module $P$ defined by $e$ is just $\mathsf{k}$, the free
$\mathsf{k}$-module of rank 1).

Let $R$ be the free $\mathsf{k}$-algebra $\mathsf{k}\langle h,h^{*}\rangle$
subject to the relations $\\{hh^{*}=e,h^{*}h=e,ehe=h,eh^{*}e=h^{*}\\}$. It
follows from the relations $ehe=h$, $eh^{*}e=h^{*}$ that all entries of $h$
and $h^{*}$ that are not at position $(1,1)$ are zero. So $R$ is just the
Laurent polynomial ring $\mathsf{k}[x,x^{-1}]$.

Let $S$ be the free $\mathsf{k}$-algebra $\mathsf{k}\langle h,h^{*}\rangle$
subject to the relations $\\{hh^{*}=e,h^{*}h=e\\}$. Since

$e=hh^{*}=hh^{*}hh^{*}=heh^{*}=heeh^{*}$

implies $e=eheeh^{*}e,$ it follows that $h_{11}h^{*}_{11}=1$. But we also have

$h_{11}h^{*}_{11}+h_{12}h^{*}_{21}=(hh^{*})_{11}=e_{11}=1$

and hence $h_{12}h^{*}_{21}=0$. If both $h_{12}$ and $h^{*}_{21}$ are non-zero
in $S$, then the ring $S$ has zero divisors, while $R=\mathsf{k}[x,x^{-1}]$
has not. Suppose now that one of the entries $h_{12}$ or $h^{*}_{21}$ is zero
in $S$. Clearly

$h^{*}_{21}h_{12}+h^{*}_{22}h_{22}=(h^{*}h)_{22}=e_{22}=0$

and hence $h^{*}_{22}h_{22}=0$ since $h_{12}=0$ or $h^{*}_{21}=0$. We show
that $h^{*}_{22}$ and $h_{22}$ are non-zero in $S$, which implies that $S$ has
zero divisors.

Let $R^{\prime}$ be the ring obtained from $R=\mathsf{k}[x,x^{-1}]$ by
adjoining two elements $y$ and $z$ subject to $yz=0$ and $zy=0$. Then $y$ and
$z$ are non-zero in $R^{\prime}$. Since

$\operatorname{diag}(x,y)\operatorname{diag}(x^{-1},z)=\operatorname{diag}(1,0)=\operatorname{diag}(x^{-1},z)\operatorname{diag}(x,y),$

there is a ring homomorphism $S\rightarrow R^{\prime}$ mapping any $r\in R$ to
itself, and

$h_{11}\mapsto x,h^{*}_{11}\mapsto x^{-1},h_{22}\mapsto y,h^{*}_{22}\mapsto
z,$

and the other entries of $h$ and $h^{*}$ to zero. Since the image of $h_{22}$
under this homomorphism is non-zero, $h_{22}$ must be non-zero as well.
Similarly it follows that $h^{*}_{22}$ is non-zero. Thus $S$ has zero divisors
while $R$ has not.

The rings introduced in Definition 4.1, come directly from Bergman’s machinery
of universal constructions, by working with idempotent matrices instead of
projective modules. Let $P$ and $Q$ be non-zero finitely generated projective
$R$-modules. In [7], Bergman constructed a ring

$S_{1}=R\big{\langle}h:\overline{P}\rightarrow\overline{Q}\big{\rangle}$

such that there is a “universal” $S_{1}$-module homomorphism
$h:S_{1}\otimes_{R}P\rightarrow S_{1}\otimes_{R}Q$. He further constructed
universal algebras

$S_{2}=R\big{\langle}h,h^{*}:\overline{P}\rightarrow\overline{Q},\overline{Q}\rightarrow\overline{P}\big{\rangle}$

and

$S_{3}=R\big{\langle}h,h^{*}:\overline{P}\rightarrow\overline{Q},\overline{Q}\rightarrow\overline{P},h^{*}h=1\big{\rangle}$

and finally

$S=S_{4}=R\big{\langle}h,h^{*}:\overline{P}\cong\overline{Q}\big{\rangle}.$

Thus, over $S$ we have $S\otimes_{R}P\cong S\otimes_{R}Q$.

Representing $P$ and $Q$ as idempotent matrices $e$ and $f$, respectively, in
§5.3 we will establish that the rings $S_{i}$, above coincide with the Bergman
algebras $B^{i}_{R}(e,f)$, $1\leq i\leq 4$, in Definition 4.1, which explains
our terminology. In fact in §5.3 we will carry out the graded version of these
constructions. By setting the grade group trivial we obtain the above
statements. We note that the paper [7] works with right modules whereas we
work here with left modules.

We can extend the definition of Bergman algebras to families of pairs of
idempotents. Let $(e,f):=\\{(e_{i},f_{i})\\}_{i\in I}$, be a collection of
pairs of idempotents. Then we define

$B_{R}(e,f):=R\big{\langle}h_{i},h_{i}^{*}\mid i\in
I\big{\rangle}\big{/}\big{\langle}e_{i}h_{i}f_{i}=h_{i},f_{i}h_{i}^{*}e_{i}=h_{i}^{*},h_{i}h_{i}^{*}=e_{i},h_{i}^{*}h_{i}=f_{i}\mid
i\in I\big{\rangle}.$ (4.2)

Similarly we define $B^{i}_{R}(e,f)$, for $1\leq i\leq 3$ as well as
$B_{R}(e)$.

######  Example 4.7.

Let $(e,f)$ be the collection $\\{(1,1),(1,1)\\}$. Then
$B_{\mathsf{k}}(e,f)=\mathsf{k}\langle x,x^{-1},y,y^{-1}\rangle,$ the (non-
commutative) Laurent polynomial ring with two variables.

Next we define the graded version of the Bergman algebras of Definition 4.1.

######  Definition 4.8.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra, where $\mathsf{k}$ is a
$\Gamma$-graded commutative ring. Let
$e\in\mathbb{M}_{m}(R)(\overline{\beta})_{0}$ and
$f\in\mathbb{M}_{n}(R)(\overline{\gamma})_{0}$ be idempotent matrices, where
$\overline{\beta}=(\beta_{1},\dots,\beta_{m})$ and
$\overline{\gamma}=(\gamma_{1},\dots,\gamma_{n})$. Furthermore, let
$h=(h_{ij})$ and $h^{*}=(h^{*}_{ji})$, where $1\leq i\leq m$ and $1\leq j\leq
n$, be a collection of symbols. To the symbols $h_{ij}$ assign the degree
$\gamma_{j}-\beta_{i}$ and $h^{*}_{ji}$ the degree $\beta_{i}-\gamma_{j}$ for
$1\leq i\leq m$ and $1\leq j\leq n$. Then the Bergman algebra $B_{R}^{i}(e,f)$
defined in Definition 4.1 form $\Gamma$-graded $\mathsf{k}$-algebras. For
$(e,f):=\\{(e_{i},f_{i})\\}_{i\in I}$, a collection of pairs of homogeneous
idempotents, the graded Bergman algebra $B^{i}_{R}(e,f)$ is defined similarly
as in (4.2)

It has already been established in the literature that Leavitt path algebras
can be realised as Bergman algebras (using the module theoretical language).
This identification was then used to describe the non-stable $K$-theory of
Leavitt path algebras (Theorem 3.2.5 in [1]).

######  Lemma 4.9.

Suppose that $R=\Pi_{t\in T}\mathsf{k}$, where $\mathsf{k}$ is field and $T$
some finite set. For any $t\in T$, let $\epsilon_{t}$ be the element of $R$
whose $t$-component is $1$ and whose other components are $0$. Then any
idempotent matrix with entries in $R$ is equivalent to a matrix of the form
$\operatorname{diag}(\epsilon_{t_{1}},\dots,\epsilon_{t_{n}})$, where
$t_{1},\dots,t_{n}\in T$.

###### Proof.

Recall that there is a $1-1$ correspondence between equivalence classes of
idempotent matrices over $R$ and isomorphism classes of finitely generated
projective $R$-modules. For any $t\in T$, let $P_{t}=R\epsilon_{t}$. It is
well-known that any finitely generated projective $R$-module $P$ is isomorphic
to a direct sum $P=P_{t_{1}}\oplus\dots\oplus P_{t_{n}}$, where
$t_{1},\dots,t_{n}\in T$. It follows that the corresponding idempotent matrix
equals (up to equivalence)
$(\epsilon_{t_{1}})\oplus\dots\oplus(\epsilon_{t_{n}})=\operatorname{diag}(\epsilon_{t_{1}},\dots,\epsilon_{t_{n}})$.
∎

The following theorem shows that if the $\mathsf{k}$-algebra $R$ is
commutative $\mathsf{k}$-semisimple, then the Bergman algebras $B_{R}(e,f)$ of
(4.2) coincide with the class of hyper Leavitt path algebras of Section 3.1.

######  Theorem 4.10.

Let $\mathsf{k}$ be a field, $\Gamma$ an abelian group and $R$ a commutative
semisimple $\mathsf{k}$-algebra considered as $\Gamma$-graded algebra
concentrated in degree zero. Let $(e,f)$ be a family of pairs of homogeneous
idempotents. Then the Bergman algebra $B_{R}(e,f)$ as defined in (4.2) is
$\Gamma$-graded isomorphic to a unital hyper Leavitt path algebra
$L_{\mathsf{k}}(H)$ whose grading is induced by a weight map. Conversely, any
unital hyper Leavitt path algebra $L_{\mathsf{k}}(H)$ whose grading is induced
by a weight map is graded isomorphic to a Bergman algebra $B_{R}(e,f)$, where
$R$ is a commutative semisimple $\mathsf{k}$-algebra concentrated in degree
zero.

###### Proof.

First we show that if $R$ is a $\Gamma$-graded commutative semisimple
$\mathsf{k}$-algebra concentrated in degree zero, then $B_{R}(e,f)$ is graded
isomorphic to a unital hyper Leavitt path algebra $L_{\mathsf{k}}(H)$ whose
grading is induced by a weight map. Write $R=\Pi_{t\in T}\mathsf{k}$, where
$T$ is some finite set. For any $t\in T$ we let $\epsilon_{t}$ be the element
of $R$ whose $t$-component is $1$ and whose other components are $0$.
Moreover, write $(e,f)=\\{(e_{u},f_{u})\\}_{u\in U}$, where for any $u\in U$,
$e_{u}\in\operatorname{\mathbb{M}}_{m_{u}}(R)(\overline{\alpha}_{u})_{0}$ and
$f_{u}\in\operatorname{\mathbb{M}}_{n_{u}}(R)(\overline{\beta}_{u})_{0}$,
$\overline{\alpha}_{u}\in\Gamma^{m_{u}}$ and
$\overline{\beta}_{u}\in\Gamma^{n_{u}}$, are homogeneous idempotent matrices.
Since $R=\Pi_{t\in T}\mathsf{k}$, by Lemma 4.5 and 4.9, we may assume that for
any $u\in U$,
$e_{u}=\operatorname{diag}(\epsilon_{t_{u,1}},\dots,\epsilon_{t_{u,m_{u}}})$
and
$f_{u}=\operatorname{diag}(\epsilon_{t^{\prime}_{u,1}},\dots,\epsilon_{t^{\prime}_{u,n_{u}}})$,
for some
$t_{u,1},\dots,t_{u,m_{u}},t^{\prime}_{u,1},\dots,t^{\prime}_{u,n_{u}}\in T$.
Define a hypergraph $H=(H^{0},H^{1},s,r)$ by $H^{0}=\\{v_{t}\mid t\in T\\}$,
$H^{1}=\\{h_{u}\mid u\in U\\}$, $I_{h_{u}}=\\{1,\dots,m_{u}\\}~{}(h_{u}\in
H^{1})$, $J_{h_{u}}=\\{1,\dots,n_{u}\\}~{}(h_{u}\in H^{1})$,
$s(h_{u})_{i}=v_{t_{u,i}}~{}(h_{u}\in H^{1},i\in I_{h_{u}})$ and
$r(u)_{j}=v_{t^{\prime}_{u,j}}~{}(h_{u}\in H^{1},j\in J_{h_{u}})$. It follows
from Remark 3.2 that $B_{R}(e,f)\cong L_{\mathsf{k}}(H)$ as
$\mathsf{k}$-algebras. Since $H^{0}=T$ is finite, $L_{\mathsf{k}}(H)$ is
unital (see [24, Proposition 8]). Now define a $\Gamma$-weight map $w$ for $H$
by $w((h_{u})_{ij})=(\beta_{u})_{j}-(\alpha_{u})_{i}$ for any $h_{u}\in
H^{1}$, $i\in I_{h_{u}}$ and $j\in J_{h_{u}}$. One checks easily that $w$ is
indeed a $\Gamma$-weight map for $H$, and that the isomorphism
$B_{R}(e,f)\cong L_{\mathsf{k}}(H)$ is graded with respect to the
$\Gamma$-grading on $L_{\mathsf{k}}(H)$ induced by $w$.

Now let $L_{\mathsf{k}}(H)$ be a unital hyper Leavitt path algebra whose
grading is induced by a $\Gamma$-weight map $w$. Since $L_{\mathsf{k}}(H)$ is
unital, $H^{0}$ is finite. Let $R:=\Pi_{v\in H^{0}}\mathsf{k}$. For any $v\in
H^{0}$ we let $\epsilon_{v}$ be the element of $R$ whose $v$-component is $1$
and whose other components are $0$. We may assume that for any $h\in H^{1}$,
$I_{h}=\\{1,\dots,m_{h}\\}$ and $J_{h}=\\{1,\dots,n_{h}\\}$, and moreover that
$i_{h}=1=j_{h}$. Since $w$ is a $\Gamma$-weight map for $H$, we have
$w(h_{ij})=w(h_{i1})-w(h_{11})+w(h_{1j})$ for any $h\in H^{1}$. For any $h\in
H^{1}$, $1\leq i\leq m_{h}$ and $1\leq j\leq n_{h}$ set
$\alpha_{h,i}:=-w(h_{i1})+w(h_{11})$ and $\beta_{h,j}:=w(h_{1j})$. Moreover,
set $\overline{\alpha}_{h}:=(\alpha_{h,1},\dots,\alpha_{h,m_{h}})$ and
$\overline{\beta}_{h}:=(\beta_{h,1},\dots,\beta_{h,n_{h}})$. For any $h\in
H^{1}$ let
$e_{h}\in\operatorname{\mathbb{M}}_{m_{h}}(R)(\overline{\alpha}_{h})_{0}$ be
the idempotent matrix whose entry at position $(i,i^{\prime})$ is
$\delta_{ii^{\prime}}\epsilon_{s(h)_{i}}$, and
$f_{h}\in\operatorname{\mathbb{M}}_{n_{h}}(R)(\overline{\beta}_{h})_{0}$ the
idempotent matrix whose entry at position $(j,j^{\prime})$ is
$\delta_{jj^{\prime}}\epsilon_{r(h)_{j}}$. It follows from Remark 3.2 that
$L_{\mathsf{k}}(H)\cong B_{R}(e,f)$ as $\mathsf{k}$-algebras, where
$(e,f)=\\{(e_{h},f_{h})\\}_{h\in H^{1}}$. One checks easily that this
isomorphism is graded as well. ∎

##  5\. Graded Bergman universal construction

In this section we start by extending Bergman’s work to the graded setting. We
will see that if one wants to realise a conical $\Gamma$-monoid as a non-
stable $K$-theory of a ring, that ring needs to be $\Gamma$-graded and the
type of $K$-theory to be considered should be the graded $K$-theory.

Throughout this section, $\Gamma$ is an abelian group, $\mathsf{k}$ a
commutative $\Gamma$-graded ring and $R$ a $\Gamma$-graded
$\mathsf{k}$-algebra, i.e., $R$ is equipped with a graded homomorphism
$\mathsf{k}\rightarrow Z(A)$.

###  5.1. Graded universal morphisms

We start with a simple lemma which will be used throughout the paper.

######  Lemma 5.1.

Let $F,M$ and $X,Y$ be $\Gamma$-graded $R$-modules with the exact sequence of
$\Gamma$-graded $R$-modules

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and a graded idempotent endomorphism $e:F\xrightarrow{}F$ (i.e., $e\circ
e=e$). Set $P:={\rm Im}\,e$. Then for any graded module homomorphism
$g:M\xrightarrow{}P$, there is a unique graded module homomorphism
$h:X\xrightarrow{}F$ in the following diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{h}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{0}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{1-e}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{F}$

satisfying $h\circ u=0$, $(1-e)\circ h=0$ and $e\circ h=g\circ\varepsilon$. On
the other hand, for any graded module homomorphism $h:X\xrightarrow{}F$ with
$h\circ u=0=(1-e)\circ h$, there is a unique graded module homomorphism
$g:M\xrightarrow{}P$ such that $e\circ h=g\circ\varepsilon$. Furthermore, we
have that $g:M\xrightarrow[]{}P$ is the zero map if and only if the
corresponding $h:X\xrightarrow[]{}F$ is the zero map.

###### Proof.

For the idempotent map $e:F\xrightarrow{}F$, it is a fact that

$\displaystyle\begin{cases}\operatorname{Ker}e={\rm Im}(1-e),&\\\ {\rm
Im}\,e=\operatorname{Ker}(1-e),&\\\ F=\operatorname{Ker}e\oplus{\rm
Im}\,e.&\end{cases}$

Define $h(x)=(g\circ\varepsilon)(x)$ for any $x\in X$. As $e(p)=p$ for any
$p\in P$, we have $e\circ h=h=g\circ\varepsilon$. Hence $(1-e)\circ
h=(1-e)\circ e\circ h=0$ and $h\circ u=g\circ\varepsilon\circ u=0$. For the
uniqueness of $h$, suppose that there exists $h^{\prime}:X\xrightarrow{}F$
such that $e\circ h^{\prime}=g\circ\varepsilon$, $h^{\prime}\circ u=0$ and
$(1-e)\circ h^{\prime}=0$. Then $h^{\prime}(x)=h^{\prime}(x)-((1-e)\circ
h^{\prime})(x)=e\circ h^{\prime})(x)=(g\circ\varepsilon)(x)=h(x)$ for any
$x\in X$. Therefore $h^{\prime}=h$. One observes that if
$g:M\xrightarrow[]{}P$ is the zero map, one takes $h$ to be the zero map which
is the desired map. Obviously if $h:X\xrightarrow[]{}F$ is the zero map, then
$g=0$.

On the other hand, when $h\circ u=0$ by the universal property of cokernel
there exists $\tau:M\xrightarrow{}F$ such that $h=\tau\circ\varepsilon$. Set
$g=e\circ\tau$. Then $e\circ h=g\circ\varepsilon$. For the uniqueness of $g$,
suppose there exists another $g^{\prime}:M\xrightarrow[]{}{}P$ satisfying
$e\circ h=g^{\prime}\circ\varepsilon$. Then we have $g\circ\varepsilon=e\circ
h=g^{\prime}\circ\varepsilon$. Since $\varepsilon$ is surjective, we have
$g=g^{\prime}$. ∎

We are in position to extend two crucial theorems of Bergman on universal ring
constructions [7, Theorems 3.1 and 3.2] to the setting of graded rings.

######  Theorem 5.2.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra. Suppose that $M$ is a
$\Gamma$-graded $R$-module and $P$ a graded finitely generated projective
$R$-module. Then there exists a $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$,
$S$, with a universal graded module homomorphism
$f:S\otimes_{R}M\rightarrow{}S\otimes_{R}P$; that is, given any
$\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$, $T$, and any graded $T$-module
homomorphism $g:T\otimes_{R}M\rightarrow{}T\otimes_{R}P$, there exists a
unique graded homomorphism $S\rightarrow{}T$ of $R\text{-rings}_{\mathsf{k}}$
such that $g=T\otimes_{S}f$.

###### Proof.

We first write $M$ as a cokernel of the graded $R$-module homomorphism
$u:\bigoplus_{i\in I}R(\alpha_{i})\xrightarrow{}\bigoplus_{j\in
J}R(\beta_{j})$ of graded free $R$-modules; that is

$\bigoplus_{i\in I}R(\alpha_{i})\stackrel{{\scriptstyle
u}}{{\longrightarrow}}\bigoplus_{j\in
J}R(\beta_{j})\stackrel{{\scriptstyle\varepsilon}}{{\longrightarrow}}M\longrightarrow
0,$

is an exact sequence of graded $R$-modules. Using the matrix representation of
§2.3, we have

$u=(u_{ij})_{i\in I,j\in J}\in\mathbb{M}_{I\times
J}(R)[\overline{\alpha}][\overline{\beta}].$

Here $\overline{\alpha}=(\alpha_{i})_{i\in I}$ and
$\overline{\beta}=(\beta_{j})_{j\in J}$. We also write $P$ as a direct summand
of a graded free $R$-module $\bigoplus_{m\in K}R(\gamma_{m})$ (see (2.4)).
Hence, $P$ can be written as the image of a graded idempotent endomorphism
$e:\bigoplus_{m\in K}R(\gamma_{m})\xrightarrow{}\bigoplus_{m\in
K}R(\gamma_{m})$. Similarly, by §2.3 we have
$1-e=(v_{mm^{\prime}})_{m,m^{\prime}\in K}\in\mathbb{M}_{K\times
K}(R)[\overline{\gamma}][\overline{\gamma}]$ with
$\overline{\gamma}=(\gamma_{m})_{m\in K}$.

For any $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ $T$ and any graded module
homomorphism $g:T\otimes_{R}M\xrightarrow{}T\otimes_{R}P$, we have the
following diagram with the first two rows exact

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
102.64917pt\raise-69.7511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
169.21783pt\raise-40.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T\otimes_{R}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
230.04607pt\raise-40.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
230.04607pt\raise-40.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-3.0pt\raise-80.2511pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
61.57182pt\raise-80.2511pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{T\otimes_{R}(\bigoplus_{m\in
K}R(\gamma_{m}))}$}}}}}}}{\hbox{\kern
183.88629pt\raise-80.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
232.54607pt\raise-80.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ (5.1)

where by Lemma 5.1, $g$ is uniquely determined by a graded module homomorphism
$h$ with $h\circ\overline{u}=0=\overline{1-e}\circ h$. But in matrix form,

$h=(h_{jm})_{j\in J,m\in K}\in\mathbb{M}_{J\times
K}(T)[\overline{\beta}][\overline{\gamma}],$

with $h_{jm}\in T_{\gamma_{m}-\beta_{j}}$. The condition
$h\circ\overline{u}=0$ is equivalent to

$(u_{ij})_{i\in I,j\in J}\cdot(h_{jk})_{j\in J,m\in K}=0$ (5.2)

in $\mathbb{M}_{I\times K}(T)[\overline{\alpha}][\overline{\gamma}]$, that is,

$\sum_{j\in J}u_{ij}h_{jm}=0$ (5.3)

for $i\in I,m\in K$. Similarly, the condition $\overline{1-e}\circ h=0$ is
equivalent to

$(h_{jm})_{j\in J,m\in K}\cdot(v_{mm^{\prime}})_{m,m^{\prime}\in K}=0$ (5.4)

in $\mathbb{M}_{J\times K}(T)[\overline{\beta}][\overline{\gamma}]$, that is

$\sum_{m\in K}h_{jm}v_{mm^{\prime}}=0$ (5.5)

for $j\in J,m^{\prime}\in K$.

Let us define $S_{0}$ as the $R\text{-ring}_{\mathsf{k}}$ obtained by
adjoining generators $h_{jm}$, for $j\in J,m\in K$ to the ring $R$. Set the
degrees of $h_{jm}$ to $\gamma_{m}-\beta_{j}$, where $j\in J$ and $m\in K$.
Then $S_{0}$ is a $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$. Define $S$ to
be the quotient of $S_{0}$ subject to the two relations (5.3) and (5.5) which
are homogeneous of degree $\gamma_{m}-\alpha_{i}$ and
$\gamma_{m^{\prime}}-\beta_{j}$, respectively, in the graded ring $S_{0}$.
Then $S$ is the desired $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$. ∎

######  Remark 5.3.

  1. (1)

The $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ $S$ is independent of the
choices of resolutions for $M$ and $P$. One observes that $S$ has the
universal property for fixed resolutions of $M$ and $P$.

  2. (2)

We prove that applying the forgetful functor $\mathcal{F}$ to the
$\Gamma$-graded ring $S$ with a universal graded module homomorphism in
Theorem 5.2 we have the ring in Bergman’s Theorem 3.1 [7]. Actually when we
apply the forgetful functor $\mathcal{F}$ to the graded projective resolution
$\bigoplus_{i\in I}R(\alpha_{i})\xrightarrow{u}\bigoplus_{j\in
J}R(\beta_{j})\xrightarrow{\varepsilon}M\xrightarrow{}0$, we have the non-
graded projective resolution $\bigoplus_{i\in
I}R\xrightarrow{u}\bigoplus_{j\in
J}R\xrightarrow{\varepsilon}M\xrightarrow{}0$. Similarly if we apply the
forgetful functor $\mathcal{F}$ to the idempotent endomorphism
$e:\bigoplus_{m\in K}R(\gamma_{m})\xrightarrow{}\bigoplus_{m\in
K}R(\gamma_{m})$, then we have the idempotent endomorphism
$e:\bigoplus_{K}R\xrightarrow{}\bigoplus_{K}R$.

Given us any $R\text{-}\text{ring}_{\mathsf{k}}$ $T^{\prime}$ and any module
homomorphism
$g^{\prime}:T^{\prime}\otimes_{R}M\xrightarrow{}T^{\prime}\otimes_{R}P$,

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
55.6023pt\raise-82.02882pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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where $h^{\prime}$ is uniquely determined by $g^{\prime}$; see Lemma 5.1. This
$h^{\prime}$ gives us a ring homomorphism $S\xrightarrow{}T^{\prime}$ . Hence
$S$ satisfies the universal property as the ring in Bergman’s Theorem 3.1.
Therefore $S$ is the ring in Bergman’s Theorem 3.1.

  3. (3)

The desired $\Gamma$-graded $\mathsf{k}$-algebra $S$ in Theorem 5.2 has the
graded $\mathsf{k}$-homomorphism $R\xrightarrow[]{}S$ which sends $r$ to
$\overline{r}$ for $r\in R$.

######  Theorem 5.4.

Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra, $M$ a graded $R$-module,
$P$ a graded projective $R$-module, and $f:M\xrightarrow{}P$ a graded module
homomorphism. Then there exists a $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$
$S$ such that $S\otimes_{R}f=0$, and $S$ is universal for this property: Given
any $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ $T$ with $T\otimes_{R}f=0$,
there exists a unique graded homomorphism of $R\text{-rings}_{\mathsf{k}}$,
$S\xrightarrow{}T$.

###### Proof.

We write $P$ as a direct summand of a graded free module $\bigoplus_{m\in
K}R(\gamma_{m})$. Then for any graded $R\text{-ring}_{\mathsf{k}}$ $T$, the
map $\overline{i}:T\otimes_{R}P\xrightarrow{}\bigoplus_{m\in K}T(\gamma_{m})$
induced by the inclusion $i:P\xrightarrow{}\bigoplus_{m\in K}R(\gamma_{m})$ is
again an inclusion. Hence for each homogeneous elements $x\in M$, we write

$f(x)=\sum_{m\in K}f_{mx}$

where $f_{mx}$ is a homogeneous element in $R(\gamma_{m})$. Observe that for
any $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ $T$, $T\otimes_{R}f=0$ if and
only if for any homogeneous elements $x\in\bigcup_{\alpha\in\Gamma}M_{\alpha}$
all the $f_{mx}$ is zero in $T$. So the desired universal ring $S$ is the
quotient of $R$ by the two-sided ideal generated by these homogeneous elements
$f_{mx}$. ∎

The ring constructed in Theorem 5.2 will be denoted by $R\langle
f:\overline{M}\xrightarrow{}\overline{P}\rangle$ (or
$R\langle\overline{M}\xrightarrow{}\overline{P}\rangle$), and that of Theorem
5.4 by $R\langle\overline{f}=0\rangle$. Note that the graded
$\mathsf{k}$-homomorphism from $R$ to $R\langle\overline{f}=0\rangle$ in
Theorem 5.4 is a canonical projection.

#### 5.1.1.

Combining Theorems 5.2 and 5.4 we can construct graded rings with rich
structures. Let $f:P\rightarrow Q$ be a module homomorphism between graded
finitely generated projective $R$-modules $P$ and $Q$. By Theorem 5.2 we
construct a graded ring with a homomorphism
$g:\overline{Q}\rightarrow\overline{P}$ and then by two applications of
Theorem 5.4 we set $1-g\overline{f}$ and $1-\overline{f}g$ to zero. We denote
the graded ring obtained by $R\langle{\overline{f}}^{-1}\rangle$ where the
homomorphism $\overline{f}$ is now invertible.

In a similar fashion, for graded finitely generated projective $R$-modules $P$
and $Q$, we can adjoin a universal graded isomorphism between $\overline{P}$
and $\overline{Q}$ by first freely adjoining a graded map
$\overline{P}\xrightarrow[]{}\overline{Q}$, then adjoining an inverse. We
denote the resulting $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ by

$R\langle i,i^{-1}:\overline{P}\cong_{\operatorname{gr}}\overline{Q}\rangle.$

Given a single graded finitely generated projective module $P$, one may obtain
a $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ by adjoining to $R$ a universal
idempotent graded endomorphism of $\overline{P}$. This ring will be denoted by

$R\langle i:\overline{P}\xrightarrow[]{}\overline{P};i^{2}=i\rangle.$

###  5.2. Universal morphisms for strongly graded rings

Let $\mathsf{k}$ be a commutative $\Gamma$-graded ring and $R$ a
$\Gamma$-graded $\mathsf{k}$-algebra. As mentioned in the introduction, in
certain sections, such as §6 and §7.2 we need to assume that the graded ring
$\mathsf{k}$ is concentrated in degree zero. Here we also need to make this
assumption, namely, $R$ is a $\Gamma$-graded $\mathsf{k}$-algebra, where
$\mathsf{k}$ is concentrated in degree zero. Note that, by our assumption
$\mathsf{k}\subseteq R_{0}$. Let $M$ be a $\Gamma$-graded $R$-module and $P$ a
$\Gamma$-graded finitely generated projective $R$-module. It is natural to
compare the Bergman algebras $R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ and $R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle$, in case
$P_{0}$ is a finitely generated projective $R_{0}$-module. We will show that
if $R$ is strongly graded, then there is a $\Gamma$-graded isomorphism of
algebras,

$R\langle f:\overline{M}\longrightarrow\overline{P}\rangle\cong
R\ast_{R_{0}}R_{0}\langle
f_{0}:\overline{M_{0}}\longrightarrow\overline{P_{0}}\rangle.$

This ties the two representable functors (1.4) and (1.5) together.

We first consider the grading for coproducts of two $\Gamma$-graded
$\mathsf{k}$-algebras. Let $R$ be a $\Gamma$-graded $\mathsf{k}$-algebra and
$S$ an $R_{0}\text{-ring}_{\mathsf{k}}$ with $\Gamma$-grading. Then the
coproduct $R\ast_{R_{0}}S$ has a $\Gamma$-grading. The coproduct is, as a
graded vector space, the space of all words in $R$ and $S$, and the
homogeneous elements of pure degree are words where the letters are
homogeneous, and the degree is given by the sum of the degrees. One observes
that when $S$ is an $R_{0}\text{-ring}_{\mathsf{k}}$ with trivial grading, the
zeroth component of the coproduct $R\ast_{R_{0}}S$ contains
$R_{0}\ast_{R_{0}}S\cong S$. And the zeroth component of $R\ast_{R_{0}}S$ can
contain elements such as $rsr^{\prime}$, where $r,r^{\prime}\in R$ are
homogeneous and the sum of the degrees are zero.

######  Proposition 5.5.

Let $\mathsf{k}$ be a commutative $\Gamma$-graded ring concentrated in degree
zero, and $R$ a strongly $\Gamma$-graded $\mathsf{k}$-algebra. Suppose that
$M$ is a $\Gamma$-graded $R$-module and $P$ a graded finitely generated
projective $R$-module. Then we have the following isomorphism of
$\Gamma$-graded $\mathsf{k}$-algebras

$R\ast_{R_{0}}R_{0}\langle
f_{0}:\overline{M_{0}}\longrightarrow\overline{P_{0}}\rangle\cong R\langle
f:\overline{M}\longrightarrow\overline{P}\rangle.$

###### Proof.

Choose a projective resolution

$\bigoplus_{i\in I}R_{0}\stackrel{{\scriptstyle
u_{0}}}{{\longrightarrow}}\bigoplus_{j\in
J}R_{0}\stackrel{{\scriptstyle\varepsilon_{0}}}{{\longrightarrow}}M_{0}\longrightarrow
0$

for $M_{0}$ as $R_{0}$-module. Since $P$ is a graded finitely generated
projective $R$-module, and $R$ is strongly graded, its zeroth component
$P_{0}$ is a finitely generated projective $R_{0}$-module. We write $P_{0}$ as
the image of the idempotent morphism $e_{0}:\bigoplus_{m\in
K}R_{0}\xrightarrow[]{}\bigoplus_{m\in K}R_{0}$ of $R_{0}$-modules.

For any $R_{0}\text{-ring}_{\mathsf{k}}$ $T_{0}$ and any module homomorphism
$g_{0}:T_{0}\otimes_{R_{0}}M_{0}\xrightarrow{}T_{0}\otimes_{R_{0}}P_{0}$, we
have the following diagram with the first two rows exact

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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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216.56665pt\raise-40.47331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
216.56665pt\raise-40.47331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
56.0087pt\raise-80.91774pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{T_{0}\otimes_{R_{0}}(\bigoplus_{m\in
K}R_{0})}$}}}}}}}{\hbox{\kern 167.91795pt\raise-80.91774pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
219.06665pt\raise-80.91774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where by Lemma 5.1, $g_{0}$ is uniquely determined by a module homomorphism
$h_{0}$ with $h_{0}\circ\overline{u_{0}}=0=\overline{1-e_{0}}\circ h_{0}$.
Then we have

$R_{0}\langle
f_{0}:\overline{M_{0}}\longrightarrow\overline{P_{0}}\rangle\cong R_{0}\langle
h_{0}\rangle/\langle u_{0}h_{0}=0=h_{0}(1-e_{0})\rangle.$

Here $u_{0}h_{0}=0$ and $h_{0}(1-e_{0})=0$ follows from the matrix relations
(5.2) and (5.4); compare Remark 5.3 (2).

Since the graded ring $R$ is strongly graded, $M\cong R\otimes_{R_{0}}M_{0}$
and we have a projective resolution of $M$ as follows:

$\bigoplus_{i\in I}R\stackrel{{\scriptstyle
1\otimes_{R_{0}}u_{0}}}{{\longrightarrow}}\bigoplus_{j\in
J}R\stackrel{{\scriptstyle
1\otimes_{R_{0}}\varepsilon_{0}}}{{\longrightarrow}}R\otimes_{R_{0}}M_{0}\longrightarrow
0.$

For any $\Gamma$-graded $R\text{-ring}_{\mathsf{k}}$ $T$ and any graded module
homomorphism $g:T\otimes_{R}M\xrightarrow{}T\otimes_{R}P$, we have the
following diagram with the first two rows exact

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
42.70247pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{T\otimes_{R}R\otimes_{R_{0}}(\bigoplus_{i\in
I}R_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 68.60396pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.60396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 181.35057pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
112.02652pt\raise-29.97331pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
181.35057pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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where by Lemma 5.1, $g$ is uniquely determined by a graded module homomorphism
$h$ with $h\circ\overline{u_{0}}=0=\overline{1-e_{0}}\circ h$. Then we have

$\begin{split}R\langle f:\overline{M}\longrightarrow\overline{P}\rangle&\cong
R\langle h\rangle/\langle u_{0}h=0=h(1-e_{0})\rangle\end{split}$ (5.7)

By Theorem 5.2, $R\langle f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$
is a $\Gamma$-graded $\mathsf{k}$-algebra. One observes that by the proof of
Theorem 5.2 the elements of the matrix $h$ as additional generators of
$R\langle f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ are of zero
degree. By Lemma 5.1, it follows that the universal homomorphism of
$R_{0}\langle f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle$
is the restriction of the universal homomorphism of $R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ to the zeroth component.

Now we prove that $R\ast_{R_{0}}R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle\cong R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ as $\mathsf{k}$-algebras.
One notes that we have the natural inclusion $\iota:R_{0}\xrightarrow[]{}R$,
and the natural homomorphisms $\pi:R_{0}\xrightarrow[]{}R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle$ sending $r_{0}$
in $R_{0}$ to $\overline{r_{0}}$, as well as
$\pi^{\prime}:R\xrightarrow[]{}R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ sending $r$ in $R$ to
$\overline{r}$. The homomorphism $\varphi:R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle\xrightarrow[]{}R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ is induced by the
homomorphism $R_{0}\langle h_{0}\rangle\xrightarrow[]{}R\langle h\rangle$
since the matrix $h$ is induced by $h_{0}$, equivalently, we have
$\varphi(r_{0})=r_{0}$ for $r_{0}\in R_{0}$ and $\varphi(h_{ij})=h_{ij}$ for
$h_{0}=(h_{ij})$. It follows directly that in the diagram below,
$\varphi\circ\pi=\pi^{\prime}\circ\iota$. In order to prove that $R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ is the coproduct
$R*_{R_{0}}R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle$, one only needs
to check that $R\langle f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ has
the universal property. That is, taking any $R_{0}\text{-ring}_{\mathsf{k}}$
$S$ and any homomorphisms $\alpha:R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle\xrightarrow[]{}S$
and $\beta:R\xrightarrow[]{}S$ with $\alpha\circ\pi=\beta\circ\iota$, we need
to show that there exists a unique homomorphism $\theta:R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle\xrightarrow[]{}S$ in the
following diagram

$\textstyle{R_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\beta}$$\textstyle{R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\varphi}$$\textstyle{R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{S}$

such that $\alpha=\theta\circ\varphi$ and $\beta=\theta\circ\pi^{\prime}$.
First we define $\theta:R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle\xrightarrow[]{}S$, where
$\theta(r)=\beta(r)$, for $r\in R$ and $\theta(h_{ij})=\alpha(h_{ij})$ for
$h=(h_{ij})$. We observe that $\theta$ is well defined as $\alpha$ sends the
generators of the two-sided ideal $\langle u_{0}h_{0}=0=h_{0}(1-e_{0})\rangle$
to zero. And we can check directly that $\alpha=\theta\circ\varphi$ and
$\beta=\theta\circ\pi^{\prime}$. The uniqueness of $\theta$ follows directly
from the commutativity of the two triangles in the above diagram. One observes
that $\theta$ preserves the grading in the graded algebras setting. Therefore,
we obtain that $R\ast_{R_{0}}R_{0}\langle
f_{0}:\overline{M_{0}}\xrightarrow[]{}\overline{P_{0}}\rangle\cong R\langle
f:\overline{M}\xrightarrow[]{}\overline{P}\rangle$ as $\mathsf{k}$-algebras. ∎

######  Proposition 5.6.

Let $R$ be a strongly $\Gamma$-graded $\mathsf{k}$-algebra. Suppose that $M$
is a $\Gamma$-graded $R$-module, $P$ a graded finitely generated projective
$R$-module and $f:M\xrightarrow{}P$ is a graded module homomorphism. Then we
have the following isomorphism of $\Gamma$-graded $\mathsf{k}$-algebras

$R\ast_{R_{0}}R_{0}\langle\overline{f_{0}}=0\rangle\cong
R\langle\overline{f}=0\rangle.$

Here, we have $f_{0}=(-)_{0}(f)$. Furthermore, the zeroth component of the
$\Gamma$-graded $\mathsf{k}$-algebra $R\langle\overline{f}=0\rangle$ is
isomorphic to $R_{0}\langle\overline{f_{0}}=0\rangle$.

###### Proof.

As $R$ is a strongly $\Gamma$-graded $\mathsf{k}$-algebra and $P$ is a graded
finitely generated projective $R$-module, $P_{0}$ is a finitely generated
projective $R_{0}$-module. We write $P_{0}$ as the summand of the free
$R_{0}$-module $\bigoplus_{c\in C}R_{0}$. The fact that it is a direct summand
insures that for any $R_{0}\text{-}{\rm ring}_{\mathsf{k}_{0}}$ $T$, the map
$\overline{i}:T\otimes_{R_{0}}P_{0}\xrightarrow[]{}T\otimes_{R_{0}}(\bigoplus_{c\in
C}R_{0})$ induced by the inclusion $i:P_{0}\xrightarrow[]{}\bigoplus_{c\in
C}R_{0}$ is again an inclusion. Hence if we write for each $m_{0}\in M_{0}$

$f_{0}(m_{0})=\sum_{c\in C}f_{m_{0},c}$

with $f_{m_{0},c}$ in the component $R_{0}$ corresponding to $c$. We can see
that for any $R_{0}\text{-}{\rm ring}_{\mathsf{k}}$ $T$,
$T\otimes_{R_{0}}f_{0}$ will be zero if and only if all the $f_{m_{0},c}$ go
to zero in $T$. Hence

$R_{0}\langle\overline{f_{0}}=0\rangle\cong R_{0}/\langle
f_{m_{0},c}\rangle_{m_{0}\in M_{0},c\in C}.$

Here $\langle f_{m_{0},c}\rangle_{m_{0}\in M_{0},c\in C}$ is the two-sided
ideal of $R_{0}$ generated by the elements $f_{m_{0},c}$ for $m_{0}\in
M_{0},c\in C$.

One observes that $R\otimes_{R_{0}}P_{0}$ is a direct summand of the free
$R$-module $R\otimes_{R_{0}}(\bigoplus_{c\in C}R_{0})\cong\bigoplus_{c\in
C}R$. Actually we have the maps

$R\otimes_{R_{0}}M_{0}\longrightarrow{}R\otimes_{R_{0}}P_{0}\longrightarrow{}R\otimes_{R_{0}}(\bigoplus_{c\in
C}R_{0})\cong\bigoplus_{c\in C}R.$

We write for each $m_{0}\in M_{0}$