# Maps between schematic semi-graded rings

Andrés Chacón  Universidad Nacional de Colombia - Sede Bogotá Campus
Universitario<EMAIL_ADDRESS>,  María Camila Ramírez  Universidad
Nacional de Colombia - Sede Bogotá Campus Universitario
<EMAIL_ADDRESS>and  Armando Reyes  Universidad Nacional de Colombia
- Sede Bogotá Campus Universitario<EMAIL_ADDRESS>Dedicated to Professor
Oswaldo Lezama

###### Abstract.

Motivated by Smith’s work [47, 48] on maps between non-commutative projective
spaces of the form ${\rm Proj}_{nc}A$ in the setting of non-commutative
projective geometry developed by Rosenberg and Van den Bergh, and the notion
of schematicness introduced by Van Oystaeyen and Willaert [56] to
$\mathbb{N}$-graded rings with the aim of formulating a non-commutative scheme
theory à la Grothendieck [17], in this paper we consider a first approach to
maps in the Smith’s sense in the more general setting of non-commutative
projective spaces over semi-graded rings defined by Lezama and Latorre [28].
We extend Smith’s key result [47, Theorem 3.2], [48, Theorem 1.2] from the
category of schematic $\mathbb{N}$-graded rings to the category of schematic
semi-graded rings.

###### Key words and phrases:

Non-commutative projective space, closed immersion, semi-graded ring,
schematic ring, Ore set

###### 2020 Mathematics Subject Classification:

14A22; 16S38; 16S80; 16U20; 16W60

The authors were supported by the research fund of Faculty of Science, Code
HERMES 53880, Universidad Nacional de Colombia - Sede Bogotá, Colombia.

##  1\. Introduction

It is well-known that Serre [44] proved a theorem that describes the coherent
sheaves on a projective scheme in terms of graded modules. Following Artin and
Zhang [4, Section 1], a commutative graded $\Bbbk$-algebra is associated to a
projective scheme ${\rm Proj}A$, and the geometry of this scheme can be
described in terms of the quotient category
$\mathsf{qgr}A=\mathsf{gr}A/\mathsf{tors}$, where $\mathsf{gr}A$ denotes the
category of graded modules and $\mathsf{tors}$ denotes its subcategory of
torsion modules. For $A$ a finitely generated commutative graded
$\Bbbk$-algebra and $X$ its associated projective scheme, if $\mathsf{coh}X$
denotes the category of coherent sheaves on $X$ and $\mathcal{O}_{X}(n)$ is
the $n$th power of the twisting sheaf on $X$ [19, p. 117], then we have a
functor $\Gamma_{*}:\mathsf{coh}X\to\mathsf{qgr}\ A$ given by

$\Gamma_{*}(\mathcal{F})=\bigoplus_{d=-\infty}^{\infty}{\rm
H}^{0}(X,\mathcal{F}\otimes\mathcal{O}_{X}(d)),$

and Serre’s theorem asserts that if $A$ is generated over $\Bbbk$ by elements
of degree one, then $\Gamma_{*}$ defines an equivalence of categories
$\mathsf{coh}X\to\mathsf{qgr}\ A$ [44, Section 59, Proposition. 7.8, p. 252],
[17, 3.3.5] and [19, Proposition. II. 5.15].

Artin and Zhang [4] extended Serre’s theorem. For $A$ an $\mathbb{N}$-graded
algebra over a commutative Noetherian ring, they defined the associated
projective scheme to be the pair ${\rm Proj}\ A=({\rm qgr}\ A,\mathcal{A})$,
where ${\rm qgr}\ A$ is the quotient category above and $\mathcal{A}$ is the
object determined by the right module $A_{A}$. They showed the autoequivalence
$s$ of ${\rm qgr}\ A$ defined by the shift of degrees in ${\rm gr}\ A$. The
object $\mathcal{A}$ plays the role of the structure sheaf of ${\rm Proj}\ A$
and $s$ the role of the polarization defined by the projective embedding (this
definition is the same as is given by Verevkin [60, 61]). Since Serre’s
theorem does not hold for all commutative graded algebras, i.e., the functor
defined by $\Gamma_{*}$ need not be an equivalence, Artin and Zhang’s
definition of ${\rm Proj}\ A$ is compatible with the classical definition for
commutative graded rings only under some additional hypotheses, such as that
$A$ is generated in degree one. In the literature, the non-commutative version
of Serre’s theorem is known as Serre-Artin-Zhang-Verevkin theorem [4, 60, 61].

Van Oystaeyen and Willaert [56] studied above ${\rm Proj}\ A$ by developing a
kind of scheme theory similar to the commutative theory (note that Manin [33]
commented the failure of attempts to obtain a non-commutative scheme theory à
la Grothendieck [17] for quantized algebras). They noticed that this theory is
possible only if the connected and $\mathbb{N}$-graded algebra considered
contains “enough” Ore sets. Algebras satisfying this condition are called
schematic, and some examples are homogenizations of almost commutative
algebras, Rees rings of universal enveloping algebras of Lie algebras, and
three-dimensional Sklyanin algebras. They constructed a generalized
Grothendieck topology for the free monoid on all Ore sets of a schematic
algebra $R$, and defined a non-commutative site (see [57] for more details) as
a category with coverings on which sheaves can be defined, and formulated the
Serre’s theorem. As a consequence of their treatment, an equivalence between
the category of all coherent sheaves and the category ${\rm Proj}\ A$ was
obtained in the sense of non-commutative algebraic geometry introduced by
Artin [3]. In a series of papers [57, 58, 59, 62], and the book [55], Van
Oystaeyen and Willaert investigated several properties of schematic
$\mathbb{N}$-graded rings.

On the other hand, following Rosenberg [41, 42] and Van den Bergh’s [53] ideas
of the non-commutative projective geometry, Smith [47, 48] takes a
Grothendieck category as the basic non-commutative geometric object. More
exactly, we think of a Grothendieck category $\mathsf{Mod}X$ (the idea for the
notation $X=\mathsf{Mod}X$ is Van den Bergh’s) as “the quasi-coherent sheaves
on an imaginary non-commutative space $X$”. The standard commutative example
is the category $\mathsf{Qcoh}X$ of quasi-coherent sheaves on a quasi-
separated, quasi-compact scheme $X$. The two non-commutative models are
$\mathsf{Mod}R$, the category of right modules over a ring, and
$\mathsf{Proj}A$, the non-commutative projective spaces having a (not
necessarily commutative) graded ring $A$ as homogeneous coordinate ring, which
were defined by Verevkin [61, 60] and Artin and Zhang [4] (for more details,
see [46, Definition 2.2]). A $\mathsf{map}$ $g:Y\to X$ between two spaces is
an adjoint pair of functors $(g^{*},g_{*})$ with
$g_{*}:\mathsf{Mod}Y\to\mathsf{Mod}X$ and $g^{*}$ left adjoint to $g_{*}$. If
$g_{*}$ is faithful and has a right adjoint, then $g$ is called
$\mathsf{affine}$. An immediate example is a ring homomorphism $\varphi:R\to
S$ that induces an affine map $g:Y\to X$ between the affine spaces defined by
$\mathsf{Mod}Y:=\mathsf{Mod}S$ and $\mathsf{Mod}X:=\mathsf{Mod}R$. In his
papers, Smith considered the question on maps between non-commutative
projective spaces. He called a map $g:Y\to X$ a $\mathsf{closed\ immersion}$
if it is affine and the essential image of $\mathsf{Mod}Y$ in $\mathsf{Mod}X$
under $g_{*}$ is closed under submodules and quotients. For $J$ a graded ideal
in a not necessarily commutative $\mathbb{N}$-graded $\Bbbk$-algebra
$A=A_{0}\oplus A_{1}\oplus\dotsb$ in which ${\rm dim}_{\Bbbk}A_{i}<\infty$ for
all $i$, Smith [47, Theorem 3.2], [48, Theorem 1.2] showed that a surjective
homomorphism $A\to A/J$ of graded rings induces a closed immersion $i:{\rm
Proj}_{nc}A/J\to{\rm Proj}_{nc}A$ between the non-commutative projective
spaces with homogeneous coordinate rings $A$ and $A/J$.

With the aim of generalizing $\mathbb{N}$-graded rings, finitely
$\mathbb{N}$-graded algebras and other families of algebras appearing in ring
theory and non-commutative geometry that are not $\mathbb{N}$-graded algebras
(of course in a non-trivial sense), Lezama and Latorre [28] introduced the
semi-graded rings generated in degree one. In their paper, they investigated
geometric and algebraic properties of these objects such as generalized
Hilbert series, Hilbert polynomial and Gelfand-Kirillov dimension. In
particular, they extended the notion of non-commutative projective scheme in
the sense of Artin and Zhang above for $\mathbb{N}$-graded rings to the
setting of semi-graded rings, and generalized the Serre-Artin-Zhang-Verevkin
theorem [28, Section 6]. Since then, algebraic and homological properties of
semi-graded rings have been studied by different researchers [1, 2, 11, 22,
26, 30, 34, 35].

Recently, the first and third authors [8, 10] investigated the schematicness
of semi-graded rings and extended Van Oystaeyen and Willaert’s ideas [56] on a
scheme theory for non-commutative $\mathbb{N}$-graded rings to the class of
semi-graded rings (as a matter of fact, the condition of connectedness of the
algebra appearing in [56] is not assumed), and presented another approach to
the Serre-Artin-Zhang-Verevkin theorem for semi-graded rings [10, Section 5].

With all above facts in mind, in this paper we consider maps in the Smith’s
sense but now in the setting of non-commutative projective spaces over semi-
graded rings. We extend Smith’s key result [47, Theorem 3.2], [48, Theorem
1.2] from the category of schematic $\mathbb{N}$-graded rings to the category
of schematic semi-graded rings. This paper contributes to the research on
noncommutative geometry for semi-graded rings developed in the literature
(e.g. [11, 21, 25, 26, 28, 27, 39, 51], and references therein).

The article is organized as follows. In Section 2, we recall some definitions
and preliminaries on schematic semi-graded rings that are necessary for the
rest of the paper. In Section 3 we present the original results of the paper.
Our key results are Propositions 3.5 and 3.7, and Theorem 3.14. Finally,
Section 5 contains some ideas for future work.

Throughout the paper, the term ring means an associative ring with identity
not necessarily commutative. The symbols $\mathbb{N}$ and $\mathbb{Z}$ denote
the set of natural numbers including zero and the ring of integer numbers,
respectively. The term module will always mean left module unless stated
otherwise. The category of left $R$-modules is written as $\mathsf{Mod}R$.

##  2\. Definitions and preliminaries

Lezama and Latorre [28] presented an introduction to the non-commutative
algebraic geometry for non-$\mathbb{N}$-graded algebras and finitely non-
graded algebras by defining a new class of rings, the semi-graded rings. These
rings extend several kinds of non-commutative rings of polynomial type such as
Ore extensions [36, 37], families of differential operators generalizing Weyl
algebras and universal enveloping algebras of finite dimensional Lie algebras
such as PBW extensions [6], algebras appearing in mathematical physics [23,
39], ambiskew polynomial rings [24], 3-dimensional skew polynomial rings [7,
39, 41], bi-quadratic algebras on 3 generators with PBW bases [5], skew PBW
extensions [13, 29], and other algebras having PBW bases [15, 16, 35]. A
detailed list of examples of semi-graded rings and its relationships with
other algebras can be found in Fajardo et al. [11].

######  Definition 2.1 ([28, Definition 2.1]).

Let $R$ be a ring. $R$ is said to be semi-graded ($\mathsf{SG}$) if there
exists a collection $\\{R_{n}\\}_{n\in\mathbb{Z}}$ of subgroups $R_{n}$ of the
additive group $R^{+}$ such that the following conditions hold:

  1. (i)

$R=\bigoplus\limits_{n\in\mathbb{Z}}R_{n}$.

  2. (ii)

For every $m,n\in\mathbb{Z}$, $R_{m}R_{n}\subseteq\bigoplus\limits_{k\leq
m+n}R_{k}$.

  3. (iii)

$1\in R_{0}$.

The collection $\\{R_{n}\\}_{n\in\mathbb{Z}}$ is called a semi-graduation of
$R$, and we say that the elements of $R_{n}$ are homogeneous of degree $n$.
$R$ is positively semi-graded if $R_{n}=0$ for every $n<0$. If $R$ and $S$ are
semi-graded rings and $f:R\rightarrow S$ is a ring homomorphism, then $f$ is
called homogeneous if $f(R_{n})\subseteq S_{n}$ for every $n\in\mathbb{Z}$.

Note that $\mathbb{N}$-graded rings are positively $\mathsf{SG}$. Ring-
theoretical, algebraic and geometric properties of semi-graded rings have been
investigated by some mathematicians (e.g., [1, 2, 5, 18, 20, 22, 31, 40, 43,
50, 51, 52], and references therein).

######  Definition 2.2 ([28, Definition 2.2]).

Let $R$ be an $\mathsf{SG}$ ring and let $M$ be an $R$-module. We say that $M$
is semi-graded if there exists a collection $\\{M_{n}\\}_{n\in\mathbb{Z}}$ of
subgroups $M_{n}$ of the additive group $M^{+}$ such that the following
conditions hold:

  1. (i)

$M=\bigoplus\limits_{n\in\mathbb{Z}}M_{n}$.

  2. (ii)

For every $m\geq 0$ and $n\in\mathbb{Z}$,
$R_{m}M_{n}\subseteq\bigoplus\limits_{k\leq m+n}M_{k}$.

The collection $\\{M_{n}\\}_{n\in\mathbb{Z}}$ is called a semi-graduation of
$M$, and we say that the elements of $M_{n}$ are homogeneous of degree $n$.
$M$ is said to be positively semi-graded if $M_{n}=0$ for every $n<0$. Let
$f:M\rightarrow N$ be a homomorphism of $R$-modules, where $M$ and $N$ are
semi-graded $R$-modules, if $f(M_{n})\subseteq N_{n}$ for every
$n\in\mathbb{Z}$, then $f$ is called homogeneous. Let $R$ be an $\mathsf{SG}$
ring, $M$ an $\mathsf{SG}$ $R$-module, and $N$ a submodule of $M$, we say that
$N$ is a semi-graded ($\mathsf{SG}$) submodule of $M$ if
$N=\bigoplus\limits_{n\in\mathbb{Z}}N_{n}$ where $N_{n}=M_{n}\cap N$. In this
case, $N$ is an SG $R$-module [28, Definition 2.3].

Different properties of modules over families of semi-graded rings have been
investigated in [11, 25, 32, 34, 35, 38].

From [28, Proposition 2.6], we know that if $R$ is an $\mathsf{SG}$ ring, $M$
is an $\mathsf{SG}$ $R$-module and $N$ is a submodule of $M$, then the
following conditions are equivalent:

  1. (1)

$N$ is a semi-graded submodule of $M$.

  2. (2)

For every $z\in N$, the homogeneous components of $z$ are in $N$.

  3. (3)

$M/N$ is an $\mathsf{SG}$ $R$-module with semi-graduation given by

$(M/N)_{n}=(M_{n}+N)/N,\ n\in\mathbb{Z}.$

If $M$ is an $\mathsf{SG}$ $R$-module and $\\{N_{i}\\}_{i\in I}$ is a family
of $\mathsf{SG}$ submodules of $M$, then it is clear that
$\bigcap\limits_{i\in I}N_{i}$ is an $\mathsf{SG}$ submodule of $M$.

Let $X$ be a subset of $M$. We define the $\mathsf{SG}$ submodule generated by
$X$ as the intersection of all $\mathsf{SG}$ submodules containing $X$, and we
will denote it as $\langle X\rangle^{\mathsf{SG}}$.

In a similar way, if $R$ is a positively $\mathsf{SG}$ ring, for
$t\in\mathbb{N}$, we define $R_{\geq t}$ as the intersection of all two-sided
ideals that are $\mathsf{SG}$ submodules containing $\bigoplus\limits_{k\geq
t}R_{k}$.

Consider an element $n\in\mathbb{Z}$ and the following sets [10, Section 3.1]:

$\displaystyle R^{\prime}_{n}:=$ $\displaystyle\ \\{r\in R_{n}\mid\ {\rm for\
all}\ m\in\mathbb{Z},\ {\rm and}\ {\rm for\ all}\ h\in R_{m},rh\in
R_{n+m}\\},$ $\displaystyle R^{\prime\prime}_{n}:=$ $\displaystyle\ \\{r\in
R^{\prime}_{n}\mid\ {\rm for\ all}\ m\in\mathbb{Z},\ {\rm and}\ {\rm for\
all}\ h\in R_{m},hr\in R_{n+m}\\},$ $\displaystyle R^{\prime}:=$
$\displaystyle\ \bigcup_{n\in\mathbb{Z}}R^{\prime}_{n},$ $\displaystyle
R^{\prime\prime}:=$ $\displaystyle\
\bigcup_{n\in\mathbb{Z}}R^{\prime\prime}_{n}.$

For $R$ an $\mathsf{SG}$ ring and a left Ore set $S$ of $R$, the first and
third authors [10, Definition 3.8] called $S$ good if the following conditions
hold:

  1. (i)

$S\subseteq R^{\prime\prime}$, and

  2. (ii)

if $s\in S$ and $r\in R^{\prime}$, then there exist elements $u\in R^{\prime}$
and $v\in S$ such that $us=vr$.

If $R$ is an $\mathsf{SG}$ ring and $M$ is an $\mathsf{SG}$ $R$-module, then
$M$ is called localizable semi-graded ($\mathsf{LSG}$) if for every element
$(n,m)\in\mathbb{Z}^{2}$, the inclusion $R^{\prime}_{n}M_{m}\subseteq M_{n+m}$
holds [10, Definition 3.9]. The importance of good left Ore sets can be
appreciated in the following remark:

######  Remark 2.3 ([10, Propositions 3.10, 3.11 and 3.12]).

Let $R$ be an $\mathsf{SG}$ ring and $S$ a good left Ore set of $R$. Then:

  1. (i)

If $M$ is an $\mathsf{LSG}$ $R$-module, then $S^{-1}M$ is an $\mathsf{LSG}$
$R$-module with semi-graduation given by

$(S^{-1}M)_{n}=\left\\{\frac{f}{s}\mid f\in\bigcup_{k\in\mathbb{Z}}M_{k},\
{\rm deg}(f)-{\rm deg}(s)=n\right\\}.$

  2. (ii)

$S^{-1}R$ is an $\mathsf{SG}$ ring with semigraduation given by

$(S^{-1}R)_{n}=\left\\{\frac{f}{s}\mid f\in\bigcup_{k\in\mathbb{Z}}R_{k},\
{\rm deg}(f)-{\rm deg}(s)=n\right\\}.$

  3. (iii)

If $M$ is an $\mathsf{LSG}$ $R$-module, then $S^{-1}M$ is an $\mathsf{SG}$
$S^{-1}R$-module.

Having in mind the algebraic properties mentioned above, let $\mathsf{SGR}$ be
the category of semi-graded rings whose objects are the semi-graded rings and
whose morphisms are the homogeneous ring homomorphisms. If we fix an
$\mathsf{SG}$ ring $R$, $\mathsf{SGR}-R$ denotes the category of semi-graded
modules over $R$ whose morphisms are the homogeneous $R-$homomorphisms. It is
straightforward to see that $\mathsf{SGR}-R$ is preadditive, and that its zero
object is the trivial module. Let $f:M\rightarrow N$ be a morphism in
$\mathsf{SGR}-R$. Since ${\rm Ker}(f)$ and ${\rm Im}(f)$ are semi-graded
submodules, it follows that $N/{\rm Im}(f)$ is a semi-graded module. This fact
guarantees that the category $\mathsf{SGR}-R$ has kernels and cokernels. If
$f$ is a monomorphism of $\mathsf{SGR}-R$, then $f$ is the kernel of the
canonical homomorphism $j:N\rightarrow N/{\rm Im}(f)$. If $f$ is an
epimorphism, then $f$ is the cokernel of the inclusion $i:{\rm
Ker}(f)\rightarrow M$. In this way, the category $\mathsf{SGR}-R$ is normal
and conormal.

If $\\{M_{i}\\}_{i\in I}$ is a family of objects of $\mathsf{SGR}-R$, then
their direct sum $\bigoplus\limits_{i\in I}M_{i}$ is a semi-graded ring with
semi-graduation given by

$\biggl{(}\bigoplus_{i\in I}M_{i}\biggr{)}_{p}:=\bigoplus_{i\in
I}(M_{i})_{p},\ p\in\mathbb{Z}.$

It is easy to see that this object with the natural inclusions coincides with
the coproduct of the familiy of objects $\\{M_{i}\\}_{i\in I}$ in
$\mathsf{SGR}-R$. Therefore, $\mathsf{SGR}-R$ is an Abelian category. Finally,
let $\mathsf{LSG}-R$ be the full subcategory of $\mathsf{SGR}-R$ whose objects
are the $\mathsf{LSG}$ $R$-modules. This subcategory is closed for subobjects,
quotients and coproducts, so it is Abelian (see [10, Section 3.2] for more
details).

Let $R$ be an $\mathsf{SG}$-ring and $M$ an $R$-module. An element $m\in M$ is
a torsion element if there exist $n,t\geq 0$ such that $R_{\geq t}^{n}m=0$.
The set of torsion elements of $M$ is denoted by $T(M)$. $M$ is called a
torsion module if $T(M)=M$, and it is said to be torsion-free if $T(M)=0$.
From [28, Remark 5.4] we know that $T(M)$ is an $R$-submodule of $M$.

The following notion of schematicness introduced by Chacón and Reyes [8, 10]
to the class of semi-graded rings extends the corresponding defined by Van
Oystaeyen and Willaert [56] in the $\mathbb{N}$-graded case. Before, for a
positively $\mathsf{SG}$ ring $R$, if $R_{+}:=\bigoplus\limits_{k\geq
1}R_{k}$, then we say that a left Ore set $S$ is non-trivial if $S\cap
R_{+}\neq\emptyset$.

######  Definition 2.4 ([10, Definition 4.1]).

Let $R$ be a positively $\mathsf{SG}$ left Noetherian ring. $R$ is called
(left) schematic if there is a finite set $I$ of non-trivial good left Ore
sets of $R$ such that for each $(x_{S})_{S\in I}\in\prod\limits_{S\in I}S$,
there exist $t,m\in\mathbb{N}$ such that $(R_{\geq
t})^{m}\subseteq\sum\limits_{S\in I}Rx_{s}$.

If $R$ is schematic by considering the good left Ore sets $S_{i}$, and
defining for each $i$,

$\kappa_{S_{i}}(M)=\\{m\in M\mid\exists s\in S_{i},sm=0\\}$

then $\bigcap\limits_{i=1}^{n}\kappa_{S_{i}}(M)=T(M)$ for every $\mathsf{SG}$
$R$-module $M$. If $M$ is an $\mathsf{LSG}$ $R$-module, then for each
$i=1,\dotsc,n$, $\kappa_{S_{i}}(M)$ is an $\mathsf{SG}$ submodule of $M$, and
so $T(M)$ is also an $\mathsf{SG}$ submodule of $M$. These facts imply that
$M/T(M)$ is an $\mathsf{SG}$ $R$-module.

##  3\. Maps between schematic semi-graded rings

This section contains the original results of the paper. We start with the
following fact on morphisms in the category $\mathsf{SGR}-R$.

######  Proposition 3.1.

Let $f:N\rightarrow M$ be a morphism of $\mathsf{SGR}-R$ and $X\subseteq N$.
Then $f(\langle X\rangle^{\mathsf{SG}})=\langle f(X)\rangle^{\mathsf{SG}}$.

###### Proof.

Let $y\in\langle f(X)\rangle^{\mathsf{SG}}$. Then $y\in M^{\prime}$ for every
$\mathsf{SG}$-submodule $M^{\prime}$ of $M$ with $f(X)\subseteq M^{\prime}$.
Since $\langle X\rangle^{\mathsf{SG}}$ is an $\mathsf{SG}$-submodule of $N$
and $X\subseteq\langle X\rangle^{\mathsf{SG}}$, then $f(\langle
X\rangle^{\mathsf{SG}})$ is an $\mathsf{SG}$-submodule of $M$ such that
$f(X)\subseteq f(\langle X\rangle^{\mathsf{SG}})$, whence $y\in f(\langle
X\rangle^{\mathsf{SG}})$. This shows that $\langle
f(X)\rangle^{\mathsf{SG}}\subseteq f(\langle X\rangle^{\mathsf{SG}})$.

On the other hand, if $y\in f(\langle X\rangle^{\mathsf{SG}})$ then there
exists $x\in\langle X\rangle^{\mathsf{SG}}$ with $f(x)=y$. Let $M^{\prime}$ be
an $\mathsf{SG}$-submodule of $M$ such that $M^{\prime}\supseteq f(X)$. Then
$f^{-1}(M^{\prime})$ is an $\mathsf{SG}$-submodule of $N$ with $X\subseteq
f^{-1}(M^{\prime})$, which implies that $x\in f^{-1}(M^{\prime})$, and hence
$y=f(x)\in M^{\prime}$ and $y\in\langle f(X)\rangle^{\mathsf{SG}}$. Thus,
$f(\langle X\rangle^{\mathsf{SG}})\subseteq\langle f(X)\rangle^{\mathsf{SG}}$.
∎

Let $\mathsf{TOR}-R$ be the full subcategory of $\mathsf{SGR}-R$ consisting of
the torsion modules of $R$. Following the ideas presented by Lezama and
Latorre [28, Theorem 5.5], it is easy to see that $\mathsf{TOR}-R$ is a Serre
subcategory of $\mathsf{SGR}-R$.

For $R$ and $S$ $\mathsf{SG}$ rings, if $f:R\rightarrow S$ is a homomorphism
of $\mathsf{SG}$ rings and $M$ is an $\mathsf{SG}$ $S$-module, by defining
$rm:=f(r)m$, for elements $r\in R_{a}$ and $m\in M_{b}$, we get $f(r)\in
S_{a}$, and so $rm=f(r)m\in\bigoplus\limits_{c\leq a+b}M_{c}$. This guarantees
that $M$ is also an $\mathsf{SG}$ $R$-module with the semi-graduation given as
$S$-module. Under these conditions, it is clear that if $g:M\rightarrow N$ is
a homomorphism of $\mathsf{SG}$ $S$-modules, then $g$ is also a homomorphism
of $\mathsf{SG}$ $R$-modules. In this way, we obtain the functor

(3.1) $\displaystyle f_{*}:\mathsf{SGR-S}$ $\displaystyle\
\rightarrow\mathsf{SGR-R},$ (3.2) $\displaystyle M$ $\displaystyle\ \mapsto
f_{*}(M)=M$ (3.3) $\displaystyle g$ $\displaystyle\ \mapsto f_{*}(g)=g$

where $f_{*}(M)$ is the same module $M$ considered as an $R$-module. In
particular, if $J$ is an $\mathsf{SG}$ ideal of $R$ (that is, a two-sided
ideal that is an $\mathsf{SG}$ submodule of $R$), then $R/J$ is an
$\mathsf{SG}$ ring and the canonical morphism $f:R\rightarrow R/J$ is
homogeneous. This fact guarantees the existence of the functor

(3.4) $f_{*}:\mathsf{SGR}-R/J\rightarrow\mathsf{SGR}-R.$

As in the case of Smith [47, Theorem 3.2], [48, Theorem 1.2], we are
interested in the existence of induced $\mathsf{closed\ immersions}$ of
projective spaces, so our aim in this section is to define a pair of adjoint
functors to $f_{*}$.

######  Definition 3.2.

Consider $M$ an $\mathsf{SG}$ $R$-module. We define $f^{!}(M)$ as the largest
$\mathsf{SG}$ $R$-submodule of $M$ that is annihilated by $J$.

We have the following immediate result on $f^{!}(M)$.

######  Proposition 3.3.

$f^{!}(M)$ is an $\mathsf{SG}$ $R/J$-module by defining $\overline{r}m=rm$,
for every $\overline{r}\in R/J$ and each $m\in f^{!}(M)$.

###### Proof.

Let $r,s\in R$ with $\overline{r}=\overline{s}$. Then $r-s\in J$, and since
$m\in f^{!}(M)$, $m$ is annihilated by $J$, which implies that $0=(r-s)m=rm-
sm$, i.e., $rm=sm$. It is clear that the product defined satisfies the
definition of module. Finally, for elements
$\overline{r}\in\left(R/J\right)_{a}$ and $m\in M_{b}$, without loss of
generality we can take $r\in R_{a}$, and get
$\overline{r}m=rm\in\bigoplus_{c\leq a+b}M_{c}$, whence the assertion follows.
∎

######  Proposition 3.4.

Let $g:M\rightarrow N$ be a morphism of $\mathsf{SG}$ $R$-modules. Then
$g(f^{!}(M))\subseteq f^{!}(N)$.

###### Proof.

Consider $m\in f^{!}(M)$ and $j\in J$. Since $m$ is annihilated by $J$,
$jm=0$, and so $jg(m)=g(jm)=g(0)=0$. This shows that $g(m)$ is annihilated by
$J$, and so $g(m)\in f^{!}(N)$. ∎

Propositions 3.3 and 3.4 allow us to define the functor

(3.5) $\displaystyle f^{!}:\mathsf{SGR}-R$ $\displaystyle\
\rightarrow\mathsf{SGR}-R/J$ (3.6) $\displaystyle M$ $\displaystyle\ \mapsto
f^{!}(M)$ (3.7) $\displaystyle g$ $\displaystyle\ \mapsto
f^{!}(g)=g|_{f^{!}(M)},$

that sends every morphism to its restriction. If no confusion arises, we write
$f^{!}(g)=g$.

######  Proposition 3.5.

$(f_{*},f^{!})$ is an adjoint pair.

###### Proof.

Consider $M$ and $N$ be objects belonging to $\mathsf{SGR}-R/J$ and
$\mathsf{SGR}-R$, respectively. If $g:f_{*}(M)\rightarrow N$ is a morphism of
$\mathsf{SGR}-R$, then it is clear that ${\rm Im}(g)\subseteq f^{!}(N)$, and
hence we get the isomorphism $\mu_{MN}:{\rm Hom}(M,f^{!}(N))\rightarrow{\rm
Hom}(f_{*}(M),N)$ given by $\mu_{MN}(g)=g$. This guarantees the required
natural isomorphism. ∎

With the aim of defining the other adjoint pair, recall that in the graded
setting, to every $R$-module $M$ is assigned the $R/J$-module $M/JM$. However,
since in the case of semi-graded rings $JM$ is not necessarily an
$\mathsf{SG}$ submodule of $M$ (for instance, take $R=\Bbbk[x]$, $J=Rx$, and
$M=A_{1}(\Bbbk)$, the first Weyl algebra), then we cannot assert that $M/JM$
is an object in $\mathsf{SGR}-R/J$. This fact motivates us to consider
$f^{*}(M):=M/\langle JM\rangle^{\mathsf{SG}}$. Precisely, since $\langle
JM\rangle^{\mathsf{SG}}$ is an $\mathsf{SG}$ submodule of $M$, then $f^{*}(M)$
is also an $\mathsf{SG}$ $R$-module, and it is easy to see that $f^{*}(M)$ is
an $\mathsf{SG}$ $R/J$-module by defining
$\overline{a}\cdot\overline{m}=\overline{am}$, and considering the semi-
graduation of $f^{*}(M)$ as an $\mathsf{SG}$ $R$-module. In this way, if
$\alpha:N\rightarrow M$ is a morphism of $\mathsf{SGR}-R$ modules, then it is
possible to define

$\displaystyle\alpha^{*}:f^{*}(N)$ $\displaystyle\ \rightarrow f^{*}(M)$
$\displaystyle\overline{m}$ $\displaystyle\ \mapsto\overline{\alpha(m)}.$

More exactly,

######  Proposition 3.6.

$\alpha^{*}$ is a morphism of the category $\mathsf{SGR}-R/J$.

###### Proof.

Since $\alpha(JN)\subseteq JM$, then $\alpha(\langle
JN\rangle^{\mathsf{SG}})=\langle\alpha(JN)\rangle^{\mathsf{SG}}\subseteq\langle
JM\rangle^{\mathsf{SG}}$, which shows that $\alpha^{*}$ is well-defined. It is
immediate to see that $\alpha^{*}$ is a homogeneous $R/J$-homomorphism. ∎

From the discussion above, we assert the existence of the functor

(3.8) $f^{*}:\mathsf{SGR}-R\rightarrow\mathsf{SGR}-R/J,$

such that the following proposition holds:

######  Proposition 3.7.

$(f^{*},f_{*})$ is an adjoint pair.

###### Proof.

Consider $M$ an object in $\mathsf{SGR}-R$ and $N$ an object in
$\mathsf{SGR}-R/J$. Let $h:M\rightarrow f_{*}(N)$ a morphism in
$\mathsf{SGR}-R$. If $m\in M$ and $j\in J$, then
$h(jm)=jh(m)=\overline{j}h(m)=0$, which implies $JM\subseteq{\rm Ker}(h)$, and
hence $\langle JM\rangle^{\mathsf{SG}}\subseteq{\rm Ker}(h)$. Then the map
$\lambda_{MN}(h):f^{*}(M)\rightarrow N$ given by
$\lambda_{MN}(h)(\overline{m})=h(m)$ is well-defined, and in fact, it is clear
that it is a morphism in $\mathsf{SGR}-R/J$.

On the other hand, if $g:f^{*}(M)\rightarrow N$ is a morphism in
$\mathsf{SGR}-R/J$, it is straightforward to see that the map
$\lambda^{\prime}_{MN}(g):M\rightarrow f_{*}(N)$ defined by
$\lambda^{\prime}_{MN}(g)(m)=g(\overline{m})$ is a morphism in
$\mathsf{SGR}-R$.

Finally, note that the maps

$\displaystyle\lambda_{MN}:$ $\displaystyle\ {\rm
Hom}(M,f_{*}(N))\rightarrow{\rm Hom}(f^{*}(M),N),\ {\rm and}$
$\displaystyle\lambda^{\prime}_{MN}:$ $\displaystyle\ {\rm
Hom}(f^{*}(M),N)\rightarrow{\rm Hom}(M,f_{*}(N))$

are inverses of each other, and correspond to the natural isomorphisms that
guarantee that $(f^{*},f_{*})$ is an adjoint pair. ∎

######  Definition 3.8.

Let $\mathcal{A}$ be a subcategory of $\mathsf{SGR}-R$ (or $\mathsf{Mod}-R$).
We say that an object $E$ of $\mathcal{A}$ is $T$-injective if the following
condition holds: for an object $M$ and $N$ a subobject of $M$ (both in
$\mathcal{A}$) with $M/N$ being of $T$-torsion, then every morphism from $N$
to $E$ can be extended to $M$. If this extension is unique, then $E$ is said
to be $T$-closed.

As it occurs in the graded case, $E$ is $T$-closed if and only if $E$ is
$T$-torsionfree and $T$-injective. Note that this notion of being $T$-closed
is equivalent to the corresponding introduced by Gabriel [12].

If $M$ is an $R$-module, there exists a $T$-closed module $Q(M)$ in
$\mathsf{Mod}-R$ which is called the module of quotients of $M$ satisfying the
following property: there exists an injective homomorphism
$\Phi_{M}:M/T(M)\rightarrow Q(M)$ such that ${\rm Coker}(\Phi_{M})$ is
torsion. For more details on the construction of $Q(M)$, see Goldman [14,
Section 3], Stenstrom [49, Chapter IX] or Van Oystaeyen [54].

If $M$ is an object in $\mathsf{LSG}-R$, then $Q(M)$ is also an object in
$\mathsf{LSG}-R$ [10, Theorem 4.11]. In this case, the semi-graduation of
$Q(M)$ is defined as follows: since $M^{\prime}={\rm Im}(\Phi_{M})$ is
isomorphic to $M/T(M)$, then $M^{\prime}$ has a natural semi-graduation. Now,
for $\xi\in Q(M)$, we say that $\xi$ is homogeneous of degree $k$ if and only
if there exist elements $n,t\in\mathbb{N}$ such that $R_{\geq
n}^{t}\xi\subseteq M^{\prime}$, and for all $s\in R_{\geq t}^{n}\cap
R^{\prime}$, the element $s\xi$ is homogeneous of degree $k+{\deg}(s)$. By
considering this semi-graduation, it is straightforward to see that $\Phi_{M}$
is a morphism in $\mathsf{LSG}-R$.

######  Proposition 3.9.

If $M$ is an $\mathsf{LSG}-R$ module, then $Q(M)$ is $T$-closed in
$\mathsf{LSG}-R$.

###### Proof.

We know that $Q(M)$ is $T$-torsionfree and $T$-injective in $\mathsf{Mod}-R$.
Let $N\in\mathsf{LSG}-R$, $N^{\prime}$ an $\mathsf{SG}$ submodule of $N$ such
that $N/N^{\prime}$ is $T$-torsion, and $f:N^{\prime}\rightarrow Q(M)$ an
homogeneous $R$-homomorphism. Since $Q(M)$ is $T$-injective in
$\mathsf{Mod}-R$, there exists an $R$-homomorphism $g:N\rightarrow Q(M)$ that
extends $f$, we only have to show that $g$ is also homogeneous.

Consider $m\in N_{k}$. By using that $N/N^{\prime}$ is $T$-torsion, there
exist elements $n_{1},t_{1}\in\mathbb{N}$ with $R_{\geq
n_{1}}^{t_{1}}m\subseteq N^{\prime}$. On the other hand, let $M^{\prime}$ be
the isomorphic image of $M/T(M)$ in $Q(M)$. Since $Q(M)/M^{\prime}$ is
$T$-torsion, there exist $n_{2},t_{2}\in\mathbb{N}$ with $R_{\geq
n_{2}}^{t_{2}}g(m)\subseteq M^{\prime}$. If $n:={\rm max}\\{n_{1},n_{2}\\}$
and $t:={\rm max}\\{t_{1},t_{2}\\}$, then $R_{\geq n}^{t}g(m)\subseteq
M^{\prime}$, and for every $s\in R_{\geq n}^{t}\cap R^{\prime}$, we get that
$sm$ is homogeneous and $sm\in N^{\prime}$ (recall that $N$ is
$\mathsf{LSG}$). Hence, $sg(m)=g(sm)=f(sm)\in(M^{\prime})_{{\rm deg}(s)+k}$
due to that $f$ is homogeneous. By considering the semi-graduation of $Q(M)$,
it follows that $g(m)\in(Q(M))_{k}$, whence $g$ is homogeneous. ∎

Since in general we do not know if the canonical functor
$\pi:\mathsf{SGR}-R\rightarrow\mathsf{SGR}-R/\mathsf{TOR}-R$ has a right
adjoint, from now on we consider a schematic ring $R$ and the full subcategory
$\mathsf{LSG}-R$.

Since $\mathsf{LSG}-R$ is a subcategory of $\mathsf{SGR}-R$ closed for
subobjects and quotients, then $\tau:=\mathsf{TOR}-R\cap\mathsf{LSG}-R$ is a
Serre subcategory of $\mathsf{LSG}-R$. As a matter of fact, due to [12,
Proposition 4, Section 17], the canonical functor
$\pi:\mathsf{LSG}-R\rightarrow\mathsf{LSG}-R/\tau$ has a right adjoint, say
$\omega$. On the other hand, by [12, Corollary 4, Section 17; Proposition
13(a), Section 17] the category $\mathsf{LSG}-R/\tau$ is equivalent to the
full subcategory $\mathcal{C}-R$ of $\mathsf{LSG}-R$ consisting of the
$T$-closed modules. Note that we can define the functor

$\displaystyle Q:\mathsf{LSG}-R$ $\displaystyle\ \rightarrow\mathcal{C}-R$
$\displaystyle M$ $\displaystyle\ \mapsto Q(M),$

and in fact, $Q=\omega\pi$. Without loss of generality, from now on we assume
that $Q(M)$ is an extension of $M/T(M)$ and $\Phi_{M}$ is the inclusion map.

Having in mind that our objects of interest are schematic rings, we need to
guarantee that if $J$ is a $\mathsf{SG}$ ideal of a schematic ring $R$ then
$R/J$ is also schematic. We say that $J$ is a compatible ideal of $R$ if
$R^{\prime}/J=\left(R/J\right)^{\prime}$. The following result shows the
importance of this condition.

######  Proposition 3.10.

If $R$ is a schematic ring and $J$ is a compatible $\mathsf{SG}$ ideal of $R$,
then $R/J$ is also schematic.

###### Proof.

It is easy to see that if $S$ is a good Ore set of $R$, then $S/J$ is also a
multiplicative set that satisfies Ore’s condition. Now, by using that $R$ is
left Noetherian, then $R/J$ also is, and it is well-known that $S/J$ is an Ore
set of $R/J$. Since $J$ is compatible, we get $S/J$ is a good Ore set.

On the other hand, for every $n\in\mathbb{N}$, it can be seen that $R_{\geq
n}/J=\left(R/J\right)_{\geq n}$, which guarantees that for
$S_{1},\dotsc,S_{k}$ good Ore sets of $R$ satisfying the schematic condition,
then $S_{1}/J,\dots,S_{k}/J$ are good Ore sets of $R/J$ satisfying this
condition. ∎

Notice that if $M$ is an $\mathsf{LSG}-R/J$ module, then $f_{*}(M)$ is an
$\mathsf{LSG}-R$ module. If $J$ is a compatible ideal of $R$, then for each
$\mathsf{LSG}-R$ module $M$, we get that $f^{*}(M)$ and $f^{!}(M)$ are
$\mathsf{LSG}-R/J$ modules.

######  Proposition 3.11.

If $E$ is an object in $\mathcal{C}-R$, then $f^{!}(E)$ is an object in
$\mathcal{C}-R/J$.

###### Proof.

Consider $M$ an object in $\mathsf{LSG}-R/J$, $N$ a subobject of $M$ with
$M/N$ of torsion, and $g:N\rightarrow f^{!}(E)$ a morphism in
$\mathsf{LSG}-R/J$. Let $j:f^{!}(E)\rightarrow E$ be the inclusion. By
considering the morphism $j\circ g:f_{*}(N)\rightarrow E$ and using that $E$
is closed, there exists a homogeneous $R$-homomorphism $h:f_{*}(M)\rightarrow
E$ that extends to $j\circ g$. Besides, it is clear that
$f^{!}(h):M\rightarrow f^{!}(E)$ extends to $g$.

Finally, due to $f^{!}(E)$ is a submodule of $E$, and $E$ is torsion-free,
then $f^{!}(E)$ is torsion-free in $\mathsf{LSG}-R$, and so it is torsion-free
in $\mathsf{LSG}-R/J$. ∎

We define the morphisms

$i_{*}:\mathcal{C}-R/J\rightarrow\mathcal{C}-R,\quad
i^{!}:\mathcal{C}-R\rightarrow\mathcal{C}-R/J,\quad{\rm and}\quad
i^{*}:\mathcal{C}-R\rightarrow\mathcal{C}-R/J,$

as the restriction maps of $Qf_{*}$, $f^{!}$, and $Q^{\prime}f^{*}$,
respectively, where $Q$ is the functor assigning the module of quotients in
the category $\mathsf{LSG}-R$ and $Q^{\prime}$ is the same functor in
$\mathsf{LSG}-R/J$.

######  Lemma 3.12.

$(i_{*},i^{!})$ is an adjoint pair and $i_{*}$ is fully faithful.

###### Proof.

  1. (1)

$(i_{*},i^{!})$ is an adjoint pair: Let $M$ be an object in $\mathcal{C}-R/J$
and $N$ an object in $\mathcal{C}-R$. If $g:M\rightarrow f^{!}(N)$ is a
homogenous $R/J$-homomorphism and $j:f^{!}(N)\rightarrow N$ is the inclusion
map, then $j\circ g:M\rightarrow N$ is a homogeneous $R$-homomorphism. Since
$N$ is $T$-closed, there exists a unique map
$\overline{g}:i_{*}(M)=Q(M)\rightarrow N$ extending $j\circ g$. This fact
allows to obtain the map $\gamma_{MN}:{\rm Hom}(M,f^{!}(N))\rightarrow{\rm
Hom}(i_{*}(M),N)$ defined by $\gamma_{MN}(g)=\overline{g}$. It is clear that
$\gamma_{MN}$ is a monomorphism.

On the other hand, note that $M\subseteq f^{!}(Q(M))$ and $f^{!}(Q(M))/M$ is a
torsion module, whence $f^{!}(Q(M))$ is the module of quotients of $M$ (in
$\mathsf{LSG}-R/J$), and by using that $M$ is $T$-closed, we get
$f^{!}(Q(M))=M$. In this way, if $h:i_{*}(M)\rightarrow N$ is a homogeneous
$R$-homomorphism, then $h=\gamma_{MN}(f^{!}(h))$, and so $\gamma_{MN}$ is a
bijective map. It is straightforward to see that these maps induce the natural
isomorphism that show that $(i_{*},i^{!})$ is an adjoint pair.

  2. (2)

$i_{*}$ is fully faithful: From the definition of $f_{*}$ we know that it is
fully faithful. It is easy to show that when restricted to free torsion
modules, the functor $Q$ is also fully faithful. From these facts we get that
$i_{*}$ is fully faithful

∎

With the aim of proving the other adjunction, we require that $J$ satisfies
the following additional ($\ast$): for any elements $n,t\in\mathbb{N}$, when
$x\in J\cap R_{\geq t}^{n}$ there exist $n_{x},t_{x}\in\mathbb{N}$ such that
$R_{\geq t_{x}}^{n_{x}}x\subseteq JR_{\geq t}^{n}$. Let us just mention that
this condition holds trivially in commutative cases. The importance of
condition ($\ast$) is shown in the following lemma which is key in the proof
of Theorem 3.14.

######  Lemma 3.13.

If $J$ satisfies condition ($\ast$) and $E$ is an object in $\mathcal{C}-R/J$,
then $f_{*}(E)$ is an object in $\mathcal{C}-R$.

###### Proof.

Recall that the construction of the module of quotients in $\mathsf{LSG}-R$ is
the same as in the category $\mathsf{Mod}-R$, so it follows that a semi-graded
$R$-module $M$ is closed in $\mathsf{LSG}-R$ if and only if is closed in
$\mathsf{Mod}-R$. In this way, we only have to show that $E$ is closed in
$\mathsf{Mod}-R$.

Consider elements $n,t\in\mathbb{N},\ I=R_{\geq t}^{n}$ and $g:I\rightarrow E$
an $R$-homomorphism. We define the maps

$\displaystyle g^{*}$ $\displaystyle\ :I/JI\rightarrow
E,\quad\overline{x}\mapsto g(x)$ $\displaystyle j$ $\displaystyle\
:I/JI\rightarrow R/J,\quad\overline{x}\mapsto x.$

Since $E$ is an $R/J$-module, $g^{*}$ is well-defined. It is clear that $g$
and $j$ are $R/J$-homomorphisms. Note that ${\rm Coker}(j)$ is torsion, and by
using that $J$ satisfies ($\ast$), we get that ${\rm Ker}(j)$ is torsion. As
$E$ is closed in $\mathsf{Mod}-R/J$ there exist an $R/J$-homomorphism
$h:R/J\rightarrow E$ with $h\circ j=g^{*}$. If we take the canonical map
$\theta:R\rightarrow R/J$, then the $R$-homomorphism
$h\circ\theta:R\rightarrow E$ extends $g$, and by [14, Proposition 3.2] $E$ is
$T$-injective in $\mathsf{Mod}-R$. Besides, since $E$ is $T$-torsionfree in
$\mathsf{Mod}-R$ (due to that $E$ is closed in $\mathsf{Mod}-R/J$), it follows
that $E$ is $T$-closed in $\mathsf{Mod}-R$. ∎

Finally, we get the most important result of the paper that extends partially
Smith’s result [47, Theorem 3.2], [48, Theorem 1.2].

######  Theorem 3.14.

Let $R$ be a schematic $\mathsf{SG}$ ring and $J$ a compatible $\mathsf{SG}$
ideal of $R$ satisfying ($\ast$). Then the map
$i:\mathcal{C}-R/J\rightarrow\mathcal{C}-R$ given by the functors
$(i^{*},i_{*},i^{!})$ is a closed immersion.

###### Proof.

By Lemma 3.12, we only have to show that $(i^{*},i_{*})$ is an adjoint pair
and that the essential image of $i_{*}$ is closed for subobjects and
quotients.

Let $M$ be an object in $\mathcal{C}-R$ and $N$ an object in
$\mathcal{C}-R/J$. Due to Lemma 3.13, we get $(i_{*}=f_{*})$, and hence the
morphism $v_{MN}:{\rm Hom}(M,i_{*}(N))\rightarrow{\rm Hom}(i^{*}(M),N)$
defined by $v_{MN}=Q^{\prime}\circ\lambda_{MN}$, where $\lambda_{MN}$ is the
map defined in 3.7, induces the required natural isomorphism.

By last, note that in the category $\mathcal{C}-R$ the subobjects of an object
$M$ are the closed $\mathsf{SG}$-submodules of $M$, and if $N$ is a subobject
of $M$ then the quotient is given by $Q(M/N)$. Besides, since ${\rm
Im}(i_{*})$ is the subcategory consisting of the $\mathsf{SG}$-$R$-modules $M$
such that $f^{!}(M)=M$, and in this case we have $Q(M/N)=Q^{\prime}(M/N)$, it
follows that ${\rm Im}(i_{*})$ is closed under quotients and submodules. ∎

We get immediately Smith’s result in the schematic graded case formulated by
Van Oystaeyen and Willaert [56].

######  Corollary 3.15 ([47, Theorem 3.2]; [48, Theorem 1.2]).

Let $J$ be a graded ideal in an $\mathbb{N}$-graded schematic $\Bbbk$-algebra
$A$ satisfying ($\ast$). Then the homomorphism $A\to A/J$ induces a closed
immersion $i:{\rm Proj}_{nc}A/J\to{\rm Proj}_{nc}A$.

###### Proof.

Since $A$ is $\mathbb{N}$-graded, then the category $\mathsf{GrMod}-A$
coincides with the category $\mathsf{LSG}-A$. Finally, it is clear that in the
graded case, the graded ideals are $\mathsf{SG}$ compatible. ∎

##  4\. Acknowledgments

We would like to express our sincere thanks to Professor Jason Gaddis for
introducing us to this project.

##  5\. Conclusions and future work

For $R$ a schematic $\mathsf{SG}$ ring and $J$ a compatible $\mathsf{SG}$
ideal of $R$ that satisfies ($\ast$), we have showed that the canonical map
$R\rightarrow R/J$ induces a categorical map
$\mathcal{C}-R/J\rightarrow\mathcal{C}-R$. Since in the $\mathbb{N}$-graded
setting the categories $\mathsf{LSG}-R$ and $\mathsf{gr}R$ coincide, it
follows that the category $\mathcal{C}-R$ is equivalent to the category ${\rm
Proj}_{nc}(R)$ defined by Smith, Theorem 3.14 presents a partial
generalization of Smith’s result [47, Theorem 3.2]; [48, Theorem 1.2] since he
did not assume the condition of schematicness on the $\mathbb{N}$-graded ring.
As we saw above, in the semi-graded case we need this condition to guarantee
the existence of the module of quotients in (at least) the category
$\mathsf{LSG}-R$. An immediate task is to investigate if the category
$\mathsf{SGR}-R$, or an appropriate subcategory of this, has modules of
quotients. Thinking about it, a possible way to solve this problem is due to
Gabriel [12], which consists in to prove that $\mathsf{SGR}-R$ has enough
injectives (in fact, this would allow to formulate of notion of cohomology in
the semi-graded setting). Precisely, it is important to note that our
methodology here is very different from that considered by Smith, since he
used the well-known fact that the category of modules over an
$\mathbb{N}$-graded ring is a Grothendieck category, while in the semi-graded
context we do not know if $\mathsf{SGR}-R$ is also Grothendieck; we only know
that $\mathsf{SGR}-R$ is $\texttt{Ab}5$ [8, Section 1.5]. Of course, if we
found a positive answer then automatically we guarantee the existence of
enough injectives in $\mathsf{SGR}-R$. This will be our line of research in
the near future.

Last but not least, we are also interested in Smith’s papers [45, 46]. In the
first, he called a non-commutative space $X$ integral if there is an
indecomposable injective $X$-module $\mathscr{E}_{X}$ such that its
endomorphism ring is a division ring and every $X$-module is a subquotient of
a direct sum of copies of $\mathscr{E}_{X}$. A Noetherian scheme is integral
in this sense if and only if it is integral in the usual sense. Smith proved
that several classes of non-commutative spaces over $\mathbb{N}$-graded rings
are integral. On the other hand, in [46] he investigated the concepts of
closed points, closed subspaces, open subspaces, weakly closed and weakly open
subspaces, and effective divisors, on a non-commutative space over an
$\mathbb{N}$-graded ring. As expected, the characterization of all these
notions in the setting of (schematic) semi-graded rings is a very interesting
problem.

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$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{3}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  |  $\frac{1}{16}$ |  |  |  |  $\frac{15}{16}$
$|\mathcal{C}_{\overline{\mathrm{\textbf{6}}}\mathrm{\textbf{3}},\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  |  $\frac{5}{8}$ |  |  |  |  $\frac{3}{8}$
$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{3}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}\,\mathcal{C}_{\overline{\mathrm{\textbf{6}}}\mathrm{\textbf{3}},\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|$ |  |  $\frac{1}{4}$ |  |  |  |  $\frac{3}{4}$
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\mathrm{\textbf{8}}\mathrm{\textbf{3}}}|^{2}$ |  $\frac{1}{64}$ |  |  |  $\frac{9}{32}$ |  $\frac{45}{64}$ |
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{3}},\mathrm{\textbf{8}},\mathrm{\textbf{8}}\mathrm{\textbf{3}}}|^{2}$ |  $\frac{9}{16}$ |  |  |  $\frac{1}{8}$ |  $\frac{5}{16}$ |
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\mathrm{\textbf{8}}\mathrm{\textbf{3}}}\,\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{3}},\mathrm{\textbf{8}},\mathrm{\textbf{8}}\mathrm{\textbf{3}}}|$ |  $\frac{1}{8}$ |  |  |  $\frac{1}{4}$ |  $\frac{5}{8}$ |
$|\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{8}},\mathrm{\textbf{6}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  |  $\frac{5}{32}$ |  $\frac{9}{16}$ |  |  |  $\frac{9}{32}$
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{6}},\mathrm{\textbf{8}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  |  $\frac{1}{16}$ |  $\frac{5}{8}$ |  |  |  $\frac{5}{16}$
$|\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{8}},\mathrm{\textbf{6}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}\,\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{6}},\mathrm{\textbf{8}},\mathrm{\textbf{8}}\overline{\mathrm{\textbf{3}}}}|$ |  |  $-\frac{1}{8}$ |  $\frac{3}{4}$ |  |  |  $\frac{3}{8}$
(49)

For the $t$-channel mediated processes, represented by the last diagram in
figure 3 and the last two diagrams in figure 4, there are two combinations
that decompose onto a single channel. This is due to the fact the channels,
the color representations of the initial and final states must be the same,
i.e.
$\mathrm{\textbf{Q}}\in\mathrm{\textbf{R}}_{1}\otimes\mathrm{\textbf{R}}_{2}\cap\mathrm{\textbf{S}}_{1}\otimes\mathrm{\textbf{S}}_{2}$
. Using equation (43) we thus find that
$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{8}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}|^{2}$
and
$|\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{6}},\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}}\overline{\mathrm{\textbf{3}}}}|^{2}$
are non-zero only for initial pairs of particles in the
$\overline{\mathrm{\textbf{3}}}$ and 6 representation respectively. The
results for the remaining combinations are

Q |  1 |  3 |  $\overline{\mathrm{\textbf{3}}}$ |  6 |  $\overline{\mathrm{\textbf{6}}}$ |  8
---|---|---|---|---|---|---
$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}},\overline{\mathrm{\textbf{3}}},\overline{\mathrm{\textbf{3}}}\mathrm{\textbf{3}}}|^{2}$ |  $\frac{1}{3}$ |  |  |  |  |  $\frac{2}{3}$
$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}},\mathrm{\textbf{6}},\overline{\mathrm{\textbf{3}}}\mathrm{\textbf{3}}}|^{2}$ |  $\frac{2}{3}$ |  |  |  |  |  $\frac{1}{3}$
$|\mathcal{C}_{\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}},\mathrm{\textbf{8}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  $\frac{1}{18}$ |  |  |  |  |  $\frac{17}{18}$
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{3}},\mathrm{\textbf{8}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}|^{2}$ |  |  |  $\frac{2}{3}$ |  $\frac{1}{3}$ |  |
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{3}},\mathrm{\textbf{8}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}^{t}\,\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{3}},\mathrm{\textbf{8}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}^{u}|$ |  |  |  $2$ |  $-1$ |  |
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}},\overline{\mathrm{\textbf{3}}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  |  $\frac{1}{4}$ |  |  |  $\frac{3}{4}$ |
$|\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}},\overline{\mathrm{\textbf{3}}}\overline{\mathrm{\textbf{3}}}}^{t}\,\mathcal{C}_{\mathrm{\textbf{3}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}},\overline{\mathrm{\textbf{3}}}\overline{\mathrm{\textbf{3}}}}^{u}|$ |  |  $-\frac{1}{2}$ |  |  |  $\frac{3}{2}$ |
$|\mathcal{C}_{\mathrm{\textbf{6}}\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{3}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  $\frac{1}{6}$ |  |  |  |  |  $\frac{5}{6}$
$|\mathcal{C}_{\mathrm{\textbf{6}}\overline{\mathrm{\textbf{6}}},\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}}\mathrm{\textbf{3}}}|^{2}$ |  $\frac{4}{9}$ |  |  |  |  |  $\frac{5}{9}$
$|\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}|^{2}$ |  |  $\frac{3}{8}$ |  $\frac{5}{8}$ |  |  |
$|\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}^{t}\,\mathcal{C}_{\mathrm{\textbf{6}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\mathrm{\textbf{3}}\mathrm{\textbf{3}}}^{u}|$ |  |  $-\frac{3}{2}$ |  $-\frac{5}{2}$ |  |  |
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  $\frac{1}{8}$ |  |  |  |  |  $\frac{7}{8}$
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{3}}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}^{t}\,\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\mathrm{\textbf{3}},\overline{\mathrm{\textbf{3}}}\mathrm{\textbf{3}}}^{u}|$ |  $-1$ |  |  |  |  |  $2$
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\mathrm{\textbf{6}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}|^{2}$ |  $\frac{1}{4}$ |  |  |  |  |  $\frac{3}{4}$
$|\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\mathrm{\textbf{6}},\mathrm{\textbf{3}}\overline{\mathrm{\textbf{3}}}}^{t}\,\mathcal{C}_{\mathrm{\textbf{8}}\mathrm{\textbf{8}},\overline{\mathrm{\textbf{6}}},\overline{\mathrm{\textbf{3}}}\mathrm{\textbf{3}}}^{u}|$ |  $\frac{1}{3}$ |  |  |  |  |  $\frac{2}{3}$
(50)

In these tables some of the representations Q appearing in the decomposition
(43) are not present since they do not appear in the color decompositions of
the SM final states and therefore cannot contribute to the annihilation
processes. Since the diagram in figure 12 also has $u$-channel variants we use
the superscripts $t$ and $u$ to indicate the respective channel whenever the
color flow of the interference term is ambiguous. Now, using equation (48) and
by reading of the values for $\alpha_{\mathrm{\textbf{Q}}}$ in table (45) and
for $\beta_{\mathrm{\textbf{Q}}}$ in tables (49) and (50), we can calculate
the Sommerfeld corrections for all processes considered in this work.

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