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# Moduli of objects in finite length abelian categories

Andres Fernandez Herrero  Emma Lennen  and Svetlana Makarova

###### Abstract

We construct moduli spaces of objects in an abelian categories satisfying some
finiteness hypotheses. Our approach is based on the work of Artin-Zhang [AZ01]
and the intrinsic construction of moduli spaces for stacks developed by Alper-
Halpern-Leistner-Heinloth [AHLH19].

###### Contents

  1. 1 Introduction
  2. 2 Abelian categories and base-change
    1. 2.1 Finite categories
    2. 2.2 K-theory and G-theory
    3. 2.3 Finiteness conditions in Grothendieck categories
    4. 2.4 Base-change of abelian categories
    5. 2.5 Behavior of $\operatorname{Hom}$ in families
  3. 3 The stack of objects in $\mathcal{C}$
  4. 4 Moduli spaces
    1. 4.1 Good moduli space for $\mathcal{M}_{\mathcal{C}}$
    2. 4.2 Stability of objects in $\mathcal{C}$
  5. 5 Examples
    1. 5.1 Modules over a finite dimensional associative algebra
    2. 5.2 Category $\mathcal{O}$
    3. 5.3 Comodules over co-Frobenius Hopf algebras

##  1 Introduction

Moduli problems in algebraic geometry can roughly be divided into two classes:
“linear” and “nonlinear”. Linear moduli problems classify objects that can be
broken down to linear algebraic data, like representations, sheaves and
complexes. Nonlinear moduli problems often classify more geometric objects,
like certain classes of varieties, commutative or noncommutative. In [AZ01],
Artin and Zhang provide a framework to study the moduli of objects in a
certain class of abelian categories, and they use it to construct Hilbert
schemes of quotients of a noetherian object. We view it as a general
construction in the realm of linear moduli problems. The goal of this paper is
to study the range of applicability of their theory.

We construct moduli spaces of objects in an abelian category $\mathcal{C}$
satisfying the following conditions, see Section 2 for definitions.

######  Standing Assumptions 1.1.

$\mathcal{C}$ is an essentially small $k$-linear abelian category that

  1. (A0)

is Hom-finite,

  2. (A1)

is finite,

  3. (A2)

has enough projective objects.

Our main results can be summarized as follows.

######  Main Theorem (4.5, 4.17).

Let $\mathcal{C}$ be a category satisfying the 1.1 and let
$\mathcal{M}_{\mathcal{C}}$ be the stack of objects in $\mathcal{C}$ as in
3.1. Then,

  1. (i)

$\mathcal{M}_{\mathcal{C}}$ is a disjoint union of algebraic stacks of finite
type over $k$.

  2. (ii)

$\mathcal{M}_{\mathcal{C}}$ admits a good moduli space $M_{\mathcal{C}}$ which
is a disjoint union of local artinian $k$-schemes.

  3. (iii)

K-theory classes naturally give rise to a notion of slope stability, which
induces a $\Theta$-stratification on each connected component of
$\mathcal{M}_{\mathcal{C}}$. The corresponding semistable locus admits a
projective good moduli space.

To illustrate the main theorem above, let us recall the analogous results by
King [Kin94], which apply to the special case when $\mathcal{C}$ is the
category of finite dimensional representations of an acyclic quiver. In that
case the connected components of $\mathcal{M}_{\mathcal{C}}$ are in bijection
with dimension vectors for the representations (more generally, in our setting
we show that the connected components of $\mathcal{M}_{\mathcal{C}}$ are in
natural bijection with effective G-theory classes in
$\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$ as in Section 2.2). The
K-theory classes are in natural correspondence with tuples of integers indexed
by the vertices of the quiver. For each such tuple, King defined a notion of
semistability for representations of quivers, and showed that the semistable
locus admits a projective good moduli space over $k$.

Our proof of the main theorem uses the intrinsic criteria for the existence of
moduli spaces developed in [Alp13, AHLH19]. Moreover, we follow the treatment
of the moduli of abelian categories in [AHLH19] using the functor by Artin-
Zhang. The main contribution of this paper is the observation that the mild
conditions that we impose on our abelian categories guarantee that the moduli
functor $\mathcal{M}_{\mathcal{C}}$ is represented by an algebraic stack which
admits a good moduli space. A novelty in our approach is the use of ind-
algebraic spaces to prove the existence of the good moduli space before
knowing quasicompactness of the connected components of the moduli stack, and
then concluding a posteriori that each connected component is quasicompact. We
hope that this type of technique could apply to other stacks where
quasicompactness is known only for the irreducible components (Remark 4.6).

In Section 5, we provide several examples of abelian categories satisfying
1.1. We observe that for the examples in Section 5.3 it is not a priori clear
if the classical GIT methods can be applied at all, because the Hilbert scheme
constructed by [AZ01] is only known to be an algebraic space rather than a
projective scheme.

Hence, our results provide new moduli spaces – moduli of comodules over Hopf
algebras of functions on quantum groups, which, to our knowledge, have not
been constructed before, yet may be of interest to the representation theory
community.

Even though our assumptions on the category $\mathcal{C}$ seem quite
restrictive, we believe that they are not very far from being sharp for the
existence of a moduli space for the whole moduli functor
$\mathcal{M}_{\mathcal{C}}$ in the sense of Artin-Zhang (before imposing any
stability conditions). Indeed, (A0) is necessary in order for the functor to
be represented by an algebraic stack. The finite length condition in (A1) is
necessary for the functor $\mathcal{M}_{\mathcal{C}}$ to admit a good moduli
space of finite type over $k$ by [AHLH19, Lem. 7.19]. Finally, the assumption
on the existence of enough projective objects (A2) is to our knowledge the
only general categorical condition that guarantees that
$\operatorname{Ind}(\mathcal{C})$ is adically complete as in [AZ01], which is
a necessary condition for $\mathcal{M}_{\mathcal{C}}$ to be represented by an
algebraic stack.

### Related works

King [Kin94] applies the methods of GIT to construct moduli of finite-
dimensional representations of finite-dimensional associative algebras. These
categories are known to have a categorical description, namely, they satisfy
1.1 and have a finite number of simple objects up to isomorphism. Further,
various stability conditions have been described on a subclass of those in
[FJM08].

Artin and Zhang [AZ01] develop methods for defining families of objects in
more general abelian categories, and prove that the Hilbert functor of
quotients is representable by an algebraic space. This is a striking general
result that they apply to constructing moduli spaces for noncommutative graded
algebras. However, it relies on the assumption that the category is adically
complete. This is guaranteed by the assumption that the category has enough
projective objects, and it can also be checked directly for categories of
coherent sheaves on projective schemes (commutative or noncommutative).

In [AHLH19, Section 7], the authors apply the theory of good moduli spaces to
stacks of objects as defined in [AZ01]. With a few technical assumptions, they
prove existence of a proper good moduli space of Bridgeland semistable objects
on the derived category of a proper smooth variety. Some of their more general
results [AHLH19, Thm. 7.27] assume as an input that the moduli stack is
algebraic and locally of finite type. Some technical work in Section 3 is
dedicated to verifying these assumptions for an abstract abelian category.

There have been constructions of moduli of objects in triangulated and dg
categories. For example, Toën and Vaquié [TV07] construct derived moduli of
perfect objects in dg categories, and Lieblich [Lie06] constructs classical
moduli of perfect universally gluable complexes on a proper morphism. While
their constructions work in vast generality, their moduli stacks only see
perfect objects. On the other hand, in our work, we also construct moduli of
objects that are not necessarily perfect.

### Limitations

Our framework leaves out interesting examples such as the moduli of coherent
sheaves on a proper scheme or the moduli of objects in the heart of a
Bridgeland stability condition. In these cases the whole moduli functor does
not necessarily admit a good moduli space, but instead there is a
semistability condition such that the semistable locus admits a good moduli
space.

The requirement that the category $\mathcal{C}$ has enough projectives seems
too restrictive to address some of these classical moduli problems (see
[Kan19, Thm. 1.1] for a criterion of having enough projectives).
Unfortunately, applying the theory of Artin and Zhang directly to arbitrary
abelian categories without enough projectives does not usually yield functors
represented by algebraic stacks. It seems natural to attempt to modify the
moduli functor, or to consider a natural embedding of the abelian category
into one where the Artin-Zhang formalism is better behaved. This is part of
ongoing investigations by the authors.

### Vistas

We view this project as the first step in our attempt to understand strange
duality [DT94, Bea95, DW00, MO07] in the context of abelian categories.
Exploring the strange duality phenomena for categories satisfying 1.1 will be
part of our future research.

A possible avenue of exploration for the interested reader would be to
understand concretely the wall crossing behavior of the stability conditions
that we provide. It should be possible to use the $\Theta$-stratification and
the linearity of the moduli problem to keep track of the variation of the
moduli spaces in some of the examples in Section 5. Furthermore, we have not
attempted to investigate further the geometry of the moduli spaces that can be
constructed using our theory, so this is another natural direction to
investigate.

### Notation

Unless otherwise stated, all of our rings are assumed to be associative and
unital. We shall work over a fixed algebraically closed ground field $k$. All
of the rings, schemes and stacks are implicitly equipped with a morphism to
$\operatorname{Spec}k$. For any two schemes (or stacks) $X$, $Y$ we write
$X\times Y$ to denote the fiber product over $\operatorname{Spec}k$, any
unadorned tensor product means tensoring over $k$, i.e.,
$\otimes=\otimes_{k}$. We denote by $k\text{-}\operatorname{Alg}$ the category
of commutative $k$-algebras.

We will work with a fixed category $\mathcal{C}$ that satisfies 1.1. Denote by
$\mathcal{A}=\operatorname{Ind}(\mathcal{C})$ the ind-completion of
$\mathcal{C}$. We will write $V$, $W$ for objects in abelian categories; $P$
and $S$ will usually denote projective and simple objects, respectively. We
reserve letters $R$, $T$ to talk about $k$-algebras and $M$, $N$ for modules
over them.

### Acknowledgements

We would like to thank Pieter Belmans, Pavel Etingof, Dan Halpern-Leistner,
Kimoi Kemboi, Tony Pantev, Maximilien Péroux, David Rydh and Alekos Robotis
for their interest in the project and helpful discussions.

##  2 Abelian categories and base-change

###  2.1 Finite categories

######  Definition 2.1.

A $k$-linear abelian category $\mathcal{C}$ is _Hom-finite_ if for every
$V,W\in\mathcal{C}$, the dimension of $\operatorname{Hom}_{\mathcal{C}}(V,W)$
over the base field $k$ is finite.

######  Definition 2.2.

Let $\mathcal{C}$ be a $k$-linear abelian category.

  * •

An object $V\in\mathcal{C}$ is _simple_ if it has exactly two subobjects: $0$
and $V$.

  * •

The _length_ of an object $V\in\mathcal{C}$ is the largest natural number $n$
for which there exists a filtration

$0=V_{0}\subset V_{1}\subset\dots\subset V_{n}=V$

such that each factor $V_{i}/V_{i-1}$ is simple for $1\leq i\leq n$. Such a
filtration is called a _Jordan-Hölder filtration_ , and we set
$\operatorname{gr}V=\bigoplus V_{i}/V_{i-1}$.

  * •

$\mathcal{C}$ is _finite_ if every object has finite length [Pop73, p. 370].

######  Remark 2.3.

An object has finite length if and only if it is both noetherian and artinian
[Sta23, 0FCJ], and $\operatorname{gr}V$ is independent of the choice of the
Jordan-Hölder filtration [Sta23, 0FCK].

######  Lemma 2.4.

Let $\mathcal{C}$ satisfy 1.1. For every simple object $S\in\mathcal{C}$,
there is a unique (up to isomorphism) indecomposable projective object
$P_{S}\in\mathcal{C}$ that admits an epimorphism $P_{S}\twoheadrightarrow S$.
If $S^{\prime}$ is another simple object that is not isomorphic to $S$, then
$\operatorname{Hom}(P_{S},S^{\prime})=0$.

###### Proof.

Let $S$ be a simple object. By (A2), there is a projective object $P$ in
$\mathcal{C}$ and a nonzero morphism $P\to S$, which is necessarily an
epimorphism since $S$ is simple. By (A1), $P$ admits a finite direct sum
decomposition $P=\bigoplus_{j}P_{j}$, where each $P_{j}$ is indecomposable
projective. Since $P_{j}\to S$ is nonzero for some $j$, we can set
$P_{S}=P_{j}$. Uniqueness follows from sequentially applying [Kra15, Thm. 5.5,
Lem. 3.6, Cor. 3.5].

Let $S^{\prime}$ be any other simple object in $\mathcal{C}$, and suppose that
there is a nonzero morphism $P_{S}\to S^{\prime}$, which necessarily is an
epimorphism. Since both $S^{\prime}$ and $S$ are simple, the kernels of
$P_{S}\to S$ and $P_{S}\to S^{\prime}$ are maximal proper subobjects of
$P_{S}$, and so they must agree by [Kra15, Lem. 3.6]. This implies $S\cong
S^{\prime}$. ∎

######  Notation 2.5.

Let $I$ denote the set of isomorphism classes of simple objects in
$\mathcal{C}$. We denote by $\\{S_{i}\\}_{i\in I}$ a set of simple objects
from each of the isomorphism classes, and choose their respective
indecomposable projective covers $\\{P_{i}\\}_{i\in I}$. By Lemma 2.4, the
latter set is independent of any choices, up to isomorphism.

######  Definition 2.6.

We call $\\{P_{i}\\}_{i\in I}$ as above the _canonical set of projective
generators_.

###  2.2 K-theory and G-theory

In this subsection, we recall the standard Grothendieck group construction. In
algebraic geometry, this construction is often applied to the category of
coherent sheaves on a smooth variety, which happens to coincide with the
category of perfect objects. In general however, because abelian categories
may not have finite global dimension, we need to distinguish Grothendieck
groups of the given abelian category and of the subcategory of perfect
objects. We mimic the construction of Quillen’s algebraic K-theory spectrum
[Qui10] as a group completion of the groupoid of finitely generated projective
modules, omitting the higher homotopy data.

######  Definition 2.7.

Let $\mathcal{C}$ be an abelian category.

  * •

An object $V$ of $\mathcal{C}$ is _perfect_ if it admits a finite resolution
by projective objects in $\mathcal{C}$. We denote by
$\operatorname{Perf}(\mathcal{C})$ the full additive subcategory of
$\mathcal{C}$ consisting of perfect objects.

  * •

The _(zeroth) K-theory_ of $\mathcal{C}$ is denoted by
$\operatorname{K_{0}}(\mathcal{C})$ and is an abelian group generated by
isomorphism classes of objects in $\operatorname{Perf}(\mathcal{C})$, subject
to the relations of additivity in short exact sequences.

######  Definition 2.8.

Let $\mathcal{C}$ be an abelian category. The _(zeroth) G-theory_ of
$\mathcal{C}$ is denoted by $\operatorname{G_{0}}(\mathcal{C})$ and is an
abelian group generated by isomorphism classes of objects in $\mathcal{C}$,
subject to the relations of additivity in short exact sequences.

######  Definition 2.9.

We say that a G-theory class $\alpha\in\operatorname{G_{0}}(\mathcal{C})$ is
_effective_ if there is $V\in\mathcal{C}$ such that $\alpha={[V]}$. We denote
by $\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$ the submonoid of effective
G-theory classes. We similarly define
$\operatorname{K^{\mathrm{eff}}}(\mathcal{C})$.

######  Lemma 2.10.

Suppose that $\mathcal{C}$ satisfies 1.1. Let $V$ be an object of
$\operatorname{Perf}(\mathcal{C})$ and $W$ be an object of $\mathcal{C}$, then
we have the following.

  1. (i)

$\operatorname{Ext}^{i}(V,W)=0$ for all $i\gg 0$.

  2. (ii)

For all $i$, we have $\dim_{k}\operatorname{Ext}^{i}(V,W)<\infty$.

  3. (iii)

There is a natural pairing
$\operatorname{K_{0}}(\mathcal{C})\otimes_{\mathbb{Z}}\operatorname{G_{0}}(\mathcal{C})\to\mathbb{Z}$
defined on effective classes as

$([V],[W])\longmapsto\langle[V],[W]\rangle=\chi\left(\operatorname{RHom}_{\mathcal{C}}(V,W)\right).$

We write $\langle V,W\rangle=\langle[V],[W]\rangle$ for brevity. ∎

######  Lemma 2.11.

Suppose that $\mathcal{C}$ satisfies 1.1. Let $\\{P_{i}\\}_{i\in I}$ be the
canonical set of projective generators as in 2.6.

  1. (i)

We have $\operatorname{K_{0}}(\mathcal{C})\cong\mathbb{Z}^{\oplus I}$ with
basis $\\{[P_{i}]\\}_{i\in I}$, and
$\operatorname{G_{0}}(\mathcal{C})\cong\mathbb{Z}^{\oplus I}$ with basis
$\\{[S_{i}]\\}_{i\in I}$. These bases are dual up to a scalar multiple, i.e.,
$\langle P_{i},S_{i}\rangle=\tau_{i}\geq 1$ and $\langle
P_{i},S_{j}\rangle=\delta_{ij}\tau_{i}$.

  2. (ii)

The pairing
$\operatorname{K_{0}}(\mathcal{C})\times\operatorname{G_{0}}(\mathcal{C})\to\mathbb{Z}$
from Lemma 2.10 is nondegenerate.

  3. (iii)

If $\mathcal{C}=\operatorname{Perf}(\mathcal{C})$, then $\iota$ is an
isomorphism.

###### Proof.

Part (i) follows from Lemma 2.4 and 2.7. Part (ii) follows at once from the
existence of dual bases. Part (iii) follows from 2.7. ∎

###  2.3 Finiteness conditions in Grothendieck categories

In this subsection, we fix a Grothendieck category $\mathcal{A}$. Recall that
an abelian category is called _Grothendieck_ if it has arbitrary direct sums,
filtered colimits are exact, and it has a generator. A _generator_ is an
object $G\in\mathcal{A}$ such that
$\operatorname{Hom}_{\mathcal{A}}(G,\underline{\ \ })$ detects zero morphisms.

######  Definition 2.12 ([Pop73, pp. 91, 92]).

An object $V\in\mathcal{A}$ is called

  * •

_finitely generated (finite type)_ if, whenever
$V\cong\operatorname{colim}_{i}V_{i}$ for a filtered diagram of subobjects
$V_{i}\subset V$, there is an index $i$ for which $V=V_{i}$;

  * •

_finitely presented_ if it is finitely generated, and the kernel of any
epimorphism $W\twoheadrightarrow V$ from a finitely generated object $W$ is
finitely generated itself; equivalently, if
$\operatorname{Hom}_{\mathcal{A}}(V,\underline{\ \ })$ commutes with filtered
colimits;

  * •

_noetherian_ if every subobject of $V$ is finitely generated; equivalently, if
subobjects of $V$ satisfy the ascending chain condition.

When $\mathcal{A}\cong R\text{-}\operatorname{Mod}$, the three notions are
equivalent to the algebraic notions of finite generation, finite presentation
and being noetherian, respectively. In a more general category admitting
filtered colimits, objects $V$ for which
$\operatorname{Hom}_{\mathcal{A}}(V,\underline{\ \ })$ commutes with filtered
colimits are called _compact_. These have a special importance for the moduli
theory we consider.

######  Notation 2.13.

We denote by $\mathcal{A}^{\mathrm{fp}}$ the full subcategory of finitely
presented objects in $\mathcal{A}$.

######  Definition 2.14.

A Grothendieck category is called _locally noetherian_ if it admits a set of
noetherian projective generators.

We recall the following useful fact.

######  Theorem 2.15 ([Pop73, Ch. 5, Thm. 8.7]).

Let $\mathcal{A}$ be a locally noetherian category, then the three notions of
finiteness for objects coincide. Hence, $\mathcal{A}^{\mathrm{fp}}$ is an
abelian category whose kernels and cokernels coincide with those in
$\mathcal{A}$.

###  2.4 Base-change of abelian categories

Let $\mathcal{A}$ be a $k$-linear Grothendieck abelian category and $R$ a
commutative $k$-algebra. We recall some notions from [AZ01].

######  Definition 2.16 ([AZ01, B2]).

We denote by $\mathcal{A}_{R}$ the category of pairs $(V,\psi)$ where
$V\in\mathcal{A}$ and $\psi$ is a homomorphism of $k$-algebras $\psi\colon
R\to\operatorname{End}_{\mathcal{A}}(V)$. The set of morphisms between two
pairs $(V,\psi)\to(W,\xi)$ consists of those morphisms in
$\operatorname{Hom}_{\mathcal{A}}(V,W)$ that are compatible with the
corresponding $R$-module structures on $V$ and $W$.

For any two commutative $k$-algebras $R$, $T$ with a homomorphism $R\to T$,
there is an associated forgetful functor
$|_{R}\colon\mathcal{A}_{T}\to\mathcal{A}_{R}$.

In [AZ01, B3], Artin and Zhang define a right exact functor of abelian
categories $V\otimes_{R}\underline{\ \ }\colon
R\text{-}\operatorname{Mod}\to\mathcal{A}_{R}$ for an object $V$ of
$\mathcal{A}_{R}$. For any fixed $R$-module $M$, they similarly define a right
exact functor $\underline{\ \
}\otimes_{R}M\colon\mathcal{A}_{R}\to\mathcal{A}_{R}$. If $M=T$ is equipped
with the structure of a $k$-algebra, then $\underline{\ \ }\otimes_{R}T$
factors through the forgetful functor $\mathcal{A}_{T}\to\mathcal{A}_{R}$, and
so we get a base-change functor $\underline{\ \
}\otimes_{R}T\colon\mathcal{A}_{R}\to\mathcal{A}_{T}$.

######  Proposition 2.17 ([AZ01, B3.16]).

Let $R\to T$ be a map of $k$-algebras. Then the functor $\underline{\ \
}\otimes_{R}T\colon\mathcal{A}_{R}\to\mathcal{A}_{T}$ is the left adjoint of
$|_{R}\colon\mathcal{A}_{T}\to\mathcal{A}_{R}$. ∎

######  Definition 2.18 ([AZ01, C1]).

An object $V\in\mathcal{A}_{R}$ is called _$R$ -flat_ if the functor
$V\otimes_{R}\underline{\ \ }\colon
R\text{-}\operatorname{Mod}\to\mathcal{A}_{R}$ is exact.

For the current setup of moduli theory to work, we need the following
condition.

######  Definition 2.19 (see [ASZ99]).

The category $\mathcal{A}$ is called _strongly locally noetherian_ if for all
noetherian $k$-algebras $R$, the base-change $\mathcal{A}_{R}$ is locally
noetherian.

It is nontrivial to check it in general [RS93]. We prove that under our
assumptions, the strong noetherian property is automatic.

Let $\mathcal{C}$ be a $k$-linear abelian category which is Hom-finite (A0)
and finite (A1). Denote by $\mathcal{A}$ its _ind-completion_
$\mathcal{A}=\operatorname{Ind}(\mathcal{C})$ [KS06, §8.6]. By [KS06, Thm.
8.6.5(vi)], $\mathcal{A}$ is Grothedieck, and by [KS06, Prop. 8.6.11], we have
$\mathcal{A}^{\mathrm{fp}}\simeq\mathcal{C}$.

######  Proposition 2.20.

If $\mathcal{C}$ is Hom-finite (A0) and finite (A1), then it is equivalent to
the category $H\text{-}\mathrm{fdcomod}$ of finite dimensional comodules over
some $k$-coalgebra $H$.

###### Proof.

Our assumptions imply that $\mathcal{A}=\operatorname{Ind}(\mathcal{C})$ is of
finite type in the sense of [Tak77, Def. 4.4], because $\mathcal{A}$ is
Grothendieck as noted above, and the other conditions are satisfied by (A0)
and (A1). By [Tak77, Thm. 5.1], $\mathcal{A}$ is equivalent to the category
$H\text{-}\mathrm{Comod}$ of comodules over some coalgebra $H$. This
identifies $\mathcal{C}$ with a subcategory of $H\text{-}\mathrm{Comod}$ of
objects of finite length. The fundamental theorem of coalgebra states that
every $H$-comodule is a union of finite dimensional subcomodules [DNR01, Thm.
2.1.7], so a comodule of finite length is necessarily finite dimensional. ∎

######  Theorem 2.21.

If $\mathcal{C}$ is an essentially small $k$-linear abelian category which is
Hom-finite (A0) and finite (A1), then
$\mathcal{A}=\operatorname{Ind}(\mathcal{C})$ is strongly locally noetherian.

Since $\mathcal{A}\simeq\operatorname{Ind}(\mathcal{A}^{\mathrm{fp}})$, this
automatically implies that $\mathcal{A}$ is strongly locally noetherian
whenever it is a locally noetherian Grothendieck category for which
$\mathcal{A}^{\mathrm{fp}}$ is Hom-finite (A0) and finite (A1).

###### Proof.

Since $\mathcal{C}$ is a finite category, it is noetherian, hence
$\mathcal{A}$ is locally noetherian.

Let $R$ be a noetherian $k$-algebra. By [AZ01, B3.17], $\mathcal{C}\otimes
R=\\{V\otimes R\mid V\in\mathcal{C}\\}$ generates $\mathcal{A}_{R}$. If we
show that for every $V\in\mathcal{C}$, its base change $V\otimes R$ is
noetherian, then the claim will be proved. So pick $V\in\mathcal{C}$ and
consider the following commutative diagram of functors, which exists by 2.20,
where the vertical arrows are the forgetful functors.

${{(H\text{-}\mathrm{Comod})_{R}}}$${{H\text{-}\mathrm{Comod}}}$${{R\text{-}\operatorname{Mod}}}$${{\mathrm{Vect}_{k}}}$$\scriptstyle{R\otimes\underline{\
\ }}$$\scriptstyle{F}$$\scriptstyle{F_{R}}$$\scriptstyle{R\otimes\underline{\
\ }}$

By 2.20, $F(V)\otimes R$ is a finite free module over a noetherian ring $R$,
hence itself noetherian. But $F_{R}(V\otimes R)=F(V)\otimes R$, so it forces
$V\otimes R$ to be noetherian. ∎

######  Corollary 2.22.

If $\mathcal{C}$ is an essentially small $k$-linear abelian category which is
Hom-finite (A0) and finite (A1), then for any field extension $k\subset K$,
finitely presented objects in the base change category
$((\operatorname{Ind}\mathcal{C})_{K})^{\mathrm{fp}}$ form a Hom-finite and
finite category as well.

###### Proof.

This follows by specializing the proof of 2.21 to the case of $R=K$. Indeed,
we get
$(\operatorname{Ind}\mathcal{C})_{K}\cong(H\text{-}\mathrm{Comod})_{K}\cong(H_{K})\text{-}\mathrm{Comod}$,
so the subcategory of finitely presented objects consists of comodules that
are finite dimensional over $K$. ∎

######  Example 2.23 (Necessity of Hom-finiteness).

2.21 does not hold if we remove the assumption that $\mathcal{C}$ is Hom-
finite. As an example, let $\mathcal{C}$ denote the category of finite
dimensional vector spaces over the algebraic closure $\overline{k(t)}$ of the
field of rational functions $k(t)$. We can view $\mathcal{C}$ as a $k$-linear
category which is not Hom-finite. Every element in $\mathcal{C}$ has finite
length. The ind-completion $\mathcal{A}$ is the category of all vector spaces
over $\overline{k(t)}$. We set $R=\overline{k(t)}$. Then the base-change
$\mathcal{A}_{\overline{k(t)}}$ is the abelian category of
$\overline{k(t)}\otimes_{k}\overline{k(t)}$-modules. The trivial rank 1 module
$\overline{k(t)}\otimes_{k}\overline{k(t)}$ is finitely presented, but it is
not noetherian. Indeed, we have a surjection
$\overline{k(t)}\otimes_{k}\overline{k(t)}\to\overline{k(t)}\otimes_{k(t)}\overline{k(t)}$
and the algebra $\overline{k(t)}\otimes_{k(t)}\overline{k(t)}$ has infinitely
minimal primes corresponding to elements of the Galois group
$Gal(\overline{k(t)}/k(t))$. This means that $\mathcal{A}$ is not strongly
noetherian.

We end this subsection by stating two useful lemmas about the base-change to a
noetherian algebra $R$. As usual, we use the notation
$\mathcal{A}=\operatorname{Ind}(\mathcal{C})$.

######  Lemma 2.24 (Nakayama’s lemma [AZ01, Thm. C4.3]).

Suppose that $\mathcal{C}$ satisfies (A0) and (A1). Let $R$ be a noetherian
$k$-algebra, and let $W\in(\mathcal{A}_{R})^{\mathrm{fp}}$ be a noetherian
object. Then there is an open subscheme $U\subset\operatorname{Spec}(R)$ such
that for any $R$-algebra $T$ we have $W\otimes_{R}T=0$ if and only if
$\operatorname{Spec}(T)\to\operatorname{Spec}(R)$ factors through $U$.

######  Lemma 2.25.

Suppose that $\mathcal{C}$ satisfies 1.1, and let $\\{P_{i}\\}_{i\in I}$ be a
set of projective generators in $\mathcal{C}$. For any noetherian $k$-algebra
$R$, we have that $\\{P_{i}\otimes R\\}_{i\in I}$ is a set of noetherian
projective generators in
$\mathcal{A}_{R}=(\operatorname{Ind}\mathcal{C})_{R}$.

###### Proof.

The set $\\{P_{i}\otimes R\\}_{i\in I}$ generates $\mathcal{A}_{R}$ by [AZ01,
Cor. B3.17]. Each $\\{P_{i}\otimes R\\}_{i\in I}$ is projective by tensor-Hom
adjunction (2.17), or by [AZ01, Lem. D3.2]. Each $P_{i}\otimes R$ is
noetherian by 2.21. ∎

###  2.5 Behavior of $\operatorname{Hom}$ in families

######  Lemma 2.26.

Let $\mathcal{A}$ be a Grothendieck $k$-linear category. Let $R$ be a
$k$-algebra. Let $P\in(\mathcal{A}_{R})^{\mathrm{fp}}$ be a finitely generated
projective object and let $V\in\mathcal{A}_{R}$ be any object. Then we have
the following.

  1. (i)

For all $R$-modules $M$, the natural morphism

$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}M\to\operatorname{Hom}_{\mathcal{A}_{R}}(P,V\otimes_{R}M)$

is an isomorphism.

  2. (ii)

If $T$ is an $R$-algebra, then the natural morphism

$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}T\to\operatorname{Hom}_{\mathcal{A}_{T}}(P\otimes_{R}T,V\otimes_{R}T)$

is an isomorphism.

  3. (iii)

If $V\in\mathcal{A}_{R}$ is $R$-flat, then
$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)$ is an $R$-flat module.

###### Proof.

For part (i), choose a presentation $R^{\oplus J}\to R^{\oplus L}\to M\to 0$
of the module $M$. This induces a commutative diagram below.

${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)^{\oplus
J}}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)^{\oplus
L}}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}M}$${0}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V^{\oplus
J})}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V^{\oplus
L})}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V\otimes_{R}M)}$${0.}$

Both rows are exact, because tensor products are right exact and $P$ is
projective. Since $P$ is finitely presented, the two leftmost vertical
morphisms are isomorphisms, and we conclude by the five lemma that
$\operatorname{Hom}_{\mathcal{A}_{R}}(P,A)\otimes_{R}M\to\operatorname{Hom}_{\mathcal{A}_{R}}(P,A\otimes_{R}M)$
is an isomorphism.

This now implies part (ii), because we have natural morphisms

$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}T\to\operatorname{Hom}_{\mathcal{A}_{R}}(P,V\otimes_{R}T)\to\operatorname{Hom}_{\mathcal{A}_{T}}(P\otimes_{R}{T},V\otimes_{R}{T}),$

and the first one is an isomorphism by part (i), while the second one is an
isomorphism by tensor-Hom adjunction (2.17).

To prove part (iii), we choose an injective morphism $N\hookrightarrow M$ of
$R$-modules and check that it remains injective after tensoring with
$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)$. We have a commutative diagram.

${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}N}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}M}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V\otimes_{R}N)}$${\operatorname{Hom}_{\mathcal{A}_{R}}(P,V\otimes_{R}M).}$

By part (i), both vertical morphisms are isomorphisms. Since $V$ is $R$-flat,
the morphism $V\otimes_{R}N\to V\otimes_{R}M$ is a monomorphism. By left
exactness of $\operatorname{Hom}_{\mathcal{A}_{R}}(\underline{\ \
},\underline{\ \ })$, we conclude that the bottom morphism is injective.
Therefore,
$\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}N\to\operatorname{Hom}_{\mathcal{A}_{R}}(P,V)\otimes_{R}M$
is injective. ∎

######  Lemma 2.27.

Assume $\mathcal{A}=\operatorname{Ind}(\mathcal{C})$, where $\mathcal{C}$
satisfies 1.1. Let $R$ be a $k$-algebra, and let
$P\in\mathcal{A}^{\mathrm{fp}}$ be a finitely presented projective object. If
$V\in\mathcal{A}_{R}$ is finitely presented, then
$\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)$ is a finitely presented
$R$-module.

###### Proof.

By Lemma 2.25 $\\{P_{i}\otimes R\\}_{i\in I}$ is a set of noetherian
projective generators in $\mathcal{A}_{R}$. Since $V$ is finitely generated,
there exists an epimorphism $\bigoplus_{\ell}P_{\ell}\otimes
R\twoheadrightarrow V$ for a finite set of indices $\ell$, with each
$P_{\ell}\in\mathcal{C}$ projective. Since $V$ is finitely presented, the
kernel $K$ of $\bigoplus_{\ell}P_{\ell}\otimes R\twoheadrightarrow V$ is
finitely generated, and so it also admits a surjection
$\bigoplus_{j}P_{j}^{\prime}\otimes R$ for another finite tuple of projective
objects $P^{\prime}_{j}\in\mathcal{C}$. Using that $P\otimes R$ is projective,
we can apply the exact functor $\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes
R,\underline{\ \ })$ to the presentation $\bigoplus_{j}P_{j}^{\prime}\otimes
R\to\bigoplus_{\ell}P_{\ell}\otimes R\to V$ in order to get an exact sequence

$\bigoplus_{j}\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes
R,P_{j}^{\prime}\otimes
R)\to\bigoplus_{\ell}\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes
R,P_{\ell}\otimes R)\to\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)\to
0.$

By Lemma 2.26 (ii), we have that

$\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,P_{j}^{\prime}\otimes
R)\cong\operatorname{Hom}_{\mathcal{A}}(P,P_{j}^{\prime})\otimes R\;\;\text{
and}\;\;\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,P_{\ell}\otimes
R)\cong\operatorname{Hom}_{\mathcal{A}}(P,P_{\ell})\otimes R$

are finite free $R$-modules (recall that
$\operatorname{Hom}_{\mathcal{A}}(P,P_{\ell})$ and
$\operatorname{Hom}_{\mathcal{A}}(P,P_{j}^{\prime})$ are finite $k$-vector
spaces). Therefore, the exact sequence above shows that
$\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)$ is a finitely presented
$R$-module. ∎

##  3 The stack of objects in $\mathcal{C}$

In this section, we use the notation
$\mathcal{A}=\operatorname{Ind}(\mathcal{C})$, where $\mathcal{C}$ satisfies
1.1. In particular, $\mathcal{A}$ is strongly locally noetherian by 2.21. We
fix once and for all the canonical set of projective generators
$\\{P_{i}\\}_{i\in I}$ as in 2.6.

######  Definition 3.1.

We denote by $\mathcal{M}_{\mathcal{C}}$ the pseudofunctor from
$k\text{-}\operatorname{Alg}$ into groupoids that sends a $k$-algebra $R$ into
the groupoid of $R$-flat objects of finite presentation in the abelian
category $\mathcal{A}_{R}$.

######  Definition 3.2.

Let $F$ denote an object in $\mathcal{A}^{\mathrm{fp}}\simeq\mathcal{C}$. We
denote by $\operatorname{Quot}(F)$ the functor from $k$-$\operatorname{Alg}$
into sets that sends a $k$-algebra $R$ into the set of equivalence classes of
quotients $F\otimes R\twoheadrightarrow V$ in $\mathcal{A}_{R}$, where $V$ is
$R$-flat and of finite presentation. Two such quotients are considered
equivalent if they have the same kernel.

######  Proposition 3.3.

For any $F\in\mathcal{C}$, the functor $\operatorname{Quot}(F)$ is represented
by a separated algebraic space locally of finite type over $k$. Moreover, it
satisfies the valuative criterion of properness.

###### Proof.

It suffices to show that the $k$-linear abelian category $\mathcal{A}$
satisfies the hypotheses of [AZ01, Thm. E3.1, Lem. E3.3]. The category
$\mathcal{A}$ is strongly locally noetherian by 2.21, and adically complete by
[AZ01, Cor. D3.3]; the other assumptions needed are implied directly by 1.1. ∎

######  Definition 3.4.

For all $F\in\mathcal{C}$, we let
$u_{F}:\operatorname{Quot}(F)\to\mathcal{M}_{\mathcal{C}}$ denote the morphism
of functors that for every $k$-algebra $R$ sends an equivalence class of
quotients $[q\colon F\otimes R\twoheadrightarrow V]$ to the object
$u_{F}([q]):=F\otimes R/\operatorname{ker}(q)$.

Recall that $I$ was the indexing set for canonical projective generators
$\\{P_{i}\\}_{i\in I}$.

######  Notation 3.5.

For every tuple of nonnegative integers $\vec{n}=(n_{i})_{i\in J}$ indexed by
a finite subset $J\subset I$, we set $P_{\vec{n}}=\bigoplus_{i\in
J}P_{i}^{\oplus n_{i}}$

Consider the morphism
$\bigsqcup_{\vec{n}}u_{P_{\vec{n}}}\colon\bigsqcup_{\vec{n}}\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$,
where the disjoint union runs over all tuples $\vec{n}$ of nonnegative
integers indexed by finite subsets $J\subset I$ as above.

######  Lemma 3.6.

The morphism of functors
$\bigsqcup_{\vec{n}}u_{P_{\vec{n}}}\colon\bigsqcup_{\vec{n}}\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$
is schematic, smooth and surjective. Moreover, for each fixed $\vec{n}$ the
morphism
$u_{P_{\vec{n}}}\colon\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$
is quasi-affine.

###### Proof.

Let $R$ be a $k$-algebra, and choose a morphism
$\operatorname{Spec}(R)\to\mathcal{M}_{\mathcal{C}}$ corresponding to an
element $V\in(\mathcal{A}_{R})^{\text{fp}}$. We want to show that
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}(\bigsqcup_{\vec{n}}\operatorname{Quot}(P_{\vec{n}}))\to\operatorname{Spec}(R)$
is represented by a smooth surjective morphism of schemes, with each
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})$
quasi-affine. Since both functors
$\bigsqcup_{\vec{n}}\operatorname{Quot}(P_{\vec{n}})$ and
$\mathcal{M}_{\mathcal{C}}$ commute with filtered colimits (cf. the proof of
[AZ01, E.3.4]), it suffices to check this when $R$ is a finite type
$k$-algebra.

We start by showing that
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})$
is represented by a quasi-affine smooth $R$-scheme for any fixed $\vec{n}$.
For every $R$-algebra $S$, the set of $S$-points of
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})$
is the set of epimorphisms $P_{\vec{n}}\otimes S\twoheadrightarrow
V\otimes_{R}S$. Consider the functor $X$ that sends an $R$-algebra $S$ to the
set $\operatorname{Hom}_{\mathcal{A}_{S}}(P_{\vec{n}}\otimes
S,V\otimes_{R}S)$. We conclude the proof of quasi-affineness and smoothness by
showing the following.

  1. (1)

$X$ is represented by an affine smooth $R$-scheme.

  2. (2)

The natural monomorphism of functors
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})\hookrightarrow
X$ is represented by an open immersion.

For (1), notice that $\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)$ is a
vector bundle on $\operatorname{Spec}(R)$ by Lemma 2.26 (iii) and Lemma 2.27.
Moreover, by Lemma 2.26(ii), for any $R$-algebra $S$ we have
$X(S)=\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)\otimes_{R}S$.
Therefore, the functor $X$ is represented by the total space of the vector
bundle $\operatorname{Hom}_{\mathcal{A}_{R}}(P\otimes R,V)$, which is affine
and smooth over $R$. For (2), consider the universal homomorphism
$\psi:P\otimes\mathcal{O}_{X}\to V\otimes_{R}\mathcal{O}_{X}$, and denote by
$\mathcal{Q}$ the cokernel of $\psi$. The subfunctor
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})$
consists of the $S$-points of $X$ such that the pullback of $\psi$ is
surjective. In other words,
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}\operatorname{Quot}(P_{\vec{n}})$
is represented by the $S$-points of $X$ such that the pullback of
$\mathcal{Q}$ is $0$. By Nakayama’s lemma (Lemma 2.24), this is represented by
an open subscheme of $X$.

We are left to prove surjectivity. Since
$\operatorname{Spec}(R)\times_{\mathcal{M}_{\mathcal{C}}}(\bigsqcup_{\vec{n}}\operatorname{Quot}(P_{\vec{n}}))\to\operatorname{Spec}(R)$
is a smooth morphism of schemes locally of finite type over $k$, it suffices
to check surjectivity on $k$-points. This amounts to showing that for every
$V\in\mathcal{C}$ there exists some $\vec{n}$ and a surjection
$P_{\vec{n}}\twoheadrightarrow V$, which holds because $\\{P_{i}\\}_{i\in I}$
is a set of generators. ∎

######  Proposition 3.7.

The pseudofunctor $\mathcal{M}_{\mathcal{C}}$ is represented by an algebraic
stack locally of finite type over $k$.

###### Proof.

By [AZ01, C8.6], the pseudofunctor $\mathcal{M}_{\mathcal{C}}$ is a stack in
the fppf topology. Since we have exhibited an algebraic space atlas locally of
finite type over $k$ (Lemma 3.6), it follows that $\mathcal{M}_{\mathcal{C}}$
is an algebraic stack locally of finite type over $k$ [Sta23, Tag 06DB]. ∎

######  Definition 3.8.

For any object in $V\in\mathcal{C}$, we say that $V$ has G-theory class
$\alpha$ if $[V]=\alpha$ in $\operatorname{G_{0}}(\mathcal{C})$.

######  Proposition 3.9.

For every effective G-theory class $\alpha$, the subset of $k$-points of
$\mathcal{M}_{\mathcal{C}}$ with G-theory class $\alpha$ are exactly the
$k$-points of an open and closed substack
$(\mathcal{M}_{\mathcal{C}})_{\alpha}\subset\mathcal{M}_{\mathcal{C}}$.

###### Proof.

Let $R$ be a finite type $k$-algebra and let $V$ be a $R$-flat noetherian
object in $\mathcal{A}_{R}$, corresponding to a point
$\operatorname{Spec}(R)\to\mathcal{M}_{\mathcal{C}}$. Define
$R(k)_{\alpha}\subset\operatorname{Spec}(R)(k)$ by

$R(k)_{\alpha}:=\left\\{f\colon\operatorname{Spec}(k)\to\operatorname{Spec}(R)\;\lvert\;\text{$f^{*}(V)$
has G-theory class $\alpha$}\right\\}.$

We need to show that there exists a (unique) open and closed subscheme
$U\subset\operatorname{Spec}(R)$ with $U(k)=R(k)_{\alpha}$. For each $i\in I$,
we can similarly define

$R(k)^{i}_{\alpha}:=\left\\{f\colon\operatorname{Spec}(k)\to\operatorname{Spec}(R)\;\lvert\;\text{$\langle
P_{i},f^{*}(V)\rangle=\langle P_{i},\alpha\rangle$}\right\\}.$

We claim that for all $i\in I$ there is a (unique) open and closed subscheme
$U^{i}$ with $U^{i}(k)=R(k)^{i}_{\alpha}$. By Lemma 2.26 (iii) and Lemma 2.27,
the module $\operatorname{Hom}_{\mathcal{A}_{R}}(P_{i},V)$ is a vector bundle
on $\operatorname{Spec}(R)$, and by Lemma 2.26 (ii) the set
$R(k)^{i}_{\alpha}$ consists of those closed points in
$\operatorname{Spec}(R)(k)$ such that the fiber of the vector bundle
$\operatorname{Hom}_{\mathcal{A}_{R}}(P_{i},V)$ has dimension $\langle
P_{i},\alpha\rangle$. This is the set of closed points of a closed and open
subscheme of $\operatorname{Spec}(R)$, concluding the proof of the claim. Now,
we have $R(k)_{\alpha}=\bigcap_{i\in I}R(k)^{i}_{\alpha}$ by Lemma 2.11(ii).
We have shown that each $R(k)^{i}_{\alpha}$ is the set of $k$-points of the
union $U_{i}$ of some connected components of $\operatorname{Spec}(R)$. Since
$R$ is of finite type over $k$, it has finitely many open and closed connected
components. In particular, the intersection $R(k)_{\alpha}=\bigcap_{i\in
I}R(k)^{i}_{\alpha}$ eventually stabilizes, thus showing that $R(k)_{\alpha}$
is the set of $k$-points of the closed and open subscheme $U_{i_{1}}\cap
U_{i_{2}}\cap\ldots\cap U_{i_{l}}$ for some finite set of indexes
$i_{1},i_{2},\ldots,i_{l}\in I$. ∎

######  Lemma 3.10.

The algebraic stack $\mathcal{M}_{\mathcal{C}}$ satisfies the following
properties.

  1. (i)

It has affine diagonal.

  2. (ii)

It is $\Theta$-reductive and $S$-complete.

  3. (iii)

It satisfies the existence part of the valuative criterion for properness.

  4. (iv)

A point $V\in|\mathcal{M}_{\mathcal{C}}|$ is closed if and only if $V$ is
semisimple.

###### Proof.

Part (i) follows from [AHLH19, Lem. 7.20]. Parts (ii)—(iii) follow from
[AHLH19, Lem. 7.16, 7.17, 7.18] (note that in our case we don’t need to
restrict to DVRs that are essentially of finite type over $k$, because
$\mathcal{A}$ is strongly locally noetherian). Part (iv) follows from [AHLH19,
Lem. 7.19]. ∎

######  Proposition 3.11.

For every effective G-theory class $\alpha$, there exists $\vec{n}$ such that
the (open) image of
$\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$ contains
$(\mathcal{M}_{\mathcal{C}})_{\alpha}$.

###### Proof.

Write $\alpha=\sum_{j\in J}n_{j}[S_{j}]$ for some finite subset $J\subset I$.
Set $\vec{n}=(n_{i})_{i\in J}$. We conclude by showing that the open image of
$\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$ contains
$(\mathcal{M}_{\mathcal{C}})_{\alpha}$. Since everything is locally of finite
type over $k$, it suffices to check this at the level of $k$-points. Choose
$A\in(\mathcal{M}_{\mathcal{C}})_{\alpha}(k)$, which amounts to an object
$A\in\mathcal{C}$ with G-theory class $\alpha$. We can express $A$ as an
iterated extension of simple objects $A_{1},A_{2},\ldots A_{m}$. For each $l$,
set $\beta_{l}:=[A_{l}]$ to be the G-theory class of $A_{l}$, so we have
$\alpha=\sum_{l=1}^{m}\beta_{l}$. By the equality
$\alpha=\sum_{l=1}^{m}\beta_{l}$, we have $b_{l}=S_{i_{l}}$ for some $i_{l}\in
J$. There exists a nonzero homomorphism $f_{l}:P_{i_{l}}\to A_{l}$, which is
necessarily epic since $A_{l}$ is simple. Since all of the $P_{i_{l}}$ are
projective, we can lift these homomorphisms to an epimorphism
$\bigoplus_{l=1}^{m}P_{i_{l}}\twoheadrightarrow A$ onto the iterated extension
$A$. The equality $\alpha=\sum_{l=1}^{m}\beta_{l}$ implies that
$\bigoplus_{l=1}^{m}P_{i_{l}}\cong P_{\vec{n}}$, thus concluding the proof. ∎

######  Proposition 3.12.

Every irreducible component of $\mathcal{M}_{\mathcal{C}}$ is quasicompact.

###### Proof.

Let $\mathcal{Z}\subset|\mathcal{M}_{\mathcal{C}}|$ be an irreducible
component of $\mathcal{M}_{\mathcal{C}}$. There exists some
$\alpha\in\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$ such that
$\mathcal{Z}$ is contained in the open and closed substack
$(\mathcal{M}_{\mathcal{C}})_{\alpha}$. By 3.11, there exists some $\vec{n}$
such that $\mathcal{Z}$ is contained in the image of
$\operatorname{Quot}(P_{\vec{n}})\to\mathcal{M}_{\mathcal{C}}$. Let
$Z\subset\operatorname{Quot}(P_{\vec{n}})$ denote the closed inverse image of
$\mathcal{Z}$. The proof of Lemma 3.6 shows that $Z$ is represented by an open
subset of the total space of a vector bundle on $\mathcal{Z}$. In particular,
$Z$ is irreducible, and so it is contained in an irreducible component of
$\operatorname{Quot}(P_{\vec{n}})$. To conclude, we just need to show that $Z$
is quasicompact, which will follow if we can show that every irreducible
component of $\operatorname{Quot}(P_{\vec{n}})$ is quasicompact. Since
$\operatorname{Quot}(P_{\vec{n}})$ satisfies the valuative criterion for
properness, this follows from [HL21, Lem. 2.2.7]. ∎

##  4 Moduli spaces

For this section, we keep the same notation and assumptions as in Section 3.
We assume in addition that the characteristic of $k$ is $0$.

###  4.1 Good moduli space for $\mathcal{M}_{\mathcal{C}}$

######  Definition 4.1.

We denote by $J_{\text{irr}}$ the set consisting of all finite unions of
irreducible components of $\mathcal{M}_{\mathcal{C}}$. We equip
$J_{\text{irr}}$ with the partial order $\leq$ determined by inclusion in
$|\mathcal{M}_{\mathcal{C}}|$.

For each element $\mathcal{Z}$ in $J_{\text{irr}}$, we can equip $\mathcal{Z}$
with a canonical structure of a closed substack of $\mathcal{M}_{\mathcal{C}}$
by viewing it as the scheme theoretic closure of the largest open
$\mathcal{U}\subset\mathcal{Z}$ that does not intersect any other irreducible
component of $\mathcal{M}_{\mathcal{C}}$. The existence of such $\mathcal{U}$
follows from the fact that $\mathcal{Z}$ is quasicompact and
$\mathcal{M}_{\mathcal{C}}$ quasiseparacted and locally of finite type.

######  Lemma 4.2.

Let $\mathcal{Z}\subset\mathcal{M}_{\mathcal{C}}$ be an element of
$J_{\text{irr}}$. Then $\mathcal{Z}$ admits a good moduli space $Z$ which is
proper over $k$.

###### Proof.

We use [AHLH19, Thm. A]. The base field $k$ has characteristic $0$, and
$\mathcal{Z}$ is of finite type (3.12), $\Theta$-reductive, S-complete (Lemma
3.10 (ii)) and satisfies the existence part of the valuative criterion for
properness (Lemma 3.10 (iii)). ∎

Note that each inclusion $\mathcal{Z}_{1}\subset\mathcal{Z}_{2}$ in
$J_{\text{irr}}$ induces a closed immersion $Z_{1}\subset Z_{2}$ at the level
of good moduli spaces.

######  Definition 4.3.

We denote by $M_{\mathcal{C}}:=\operatorname{colim}_{\mathcal{Z}\in
J_{\text{irr}}}Z$ the ind-proper strict ind-algebraic space given by the
colimit of the good moduli spaces for elements of $J_{\text{irr}}$.

######  Proposition 4.4.

The strict ind-algebraic space $M_{\mathcal{C}}$ is represented by a separated
algebraic space locally of finite type over $k$. Moreover, $M_{\mathcal{C}}$
satisfies the valuative criterion for properness over $k$.

###### Proof.

The strict ind-algebraic space $M_{\mathcal{C}}$ is a functor from
$k$-algebras into sets that satisfies descent for the fpqc topology (cf.
[Ric19, §1.2]). In order to show that it is represented by an algebraic space
locally of finite type over $k$, it suffices to find a Zariski cover by
subfunctors that are represented by algebraic spaces of finite type over $k$.
For each $k$-point $z\in M_{\mathcal{C}}(k)$, there exists some
$\mathcal{Z}\in J_{\text{irr}}$ such that $z$ belongs to the good moduli space
$Z$. Since $\mathcal{Z}\to Z$ is a good moduli space morphism, there exists a
closed point $x\in|\mathcal{Z}|$ that maps to $z$. Note that $x$ is also
closed in the larger stack $\mathcal{M}_{\mathcal{C}}$, because
$\mathcal{Z}\subset\mathcal{M}_{\mathcal{C}}$ is a closed immersion. Let
$\mathcal{Q}_{x}$ be a quasicompact open neighborhood of $x$ in
$\mathcal{M}_{\mathcal{C}}$. We equip the closed complement
$\mathcal{Y}_{x}=\mathcal{M}_{\mathcal{C}}\setminus\mathcal{Q}_{x}$ with the
reduced substack structure.

Choose any $\mathcal{W}\in J_{\text{irr}}$ containing $x$, with associated
good moduli space $\varphi\colon\mathcal{W}\to W$ (Lemma 4.2). The image of
$\varphi(\mathcal{Y}_{x}\cap\mathcal{W})$ is closed and does not contain the
point $z\in W\subset\mathcal{M}_{\mathcal{C}}$ [Alp13, Thm. 4.16 (ii)-(iii)].
We denote by $U_{z}^{W}$ the open complement of
$\varphi(Z_{x}\cap\mathcal{W})$ in $W$. The open algebraic spaces
$U_{z}^{W}\subset W$ as $\mathcal{W}$ runs through $J_{\text{irr}}$ are
compatible, and therefore yields an open subfunctor
$U_{z}:=\operatorname{colim}_{\mathcal{W}\in J_{\text{irr}}}U_{z}^{W}$ of
$M_{\mathcal{C}}$. The open substack
$\mathcal{Q}_{x}\subset\mathcal{M}_{\mathcal{C}}$ is contained in only
finitely many irreducible components of $\mathcal{M}_{\mathcal{C}}$, and so it
is contained in some fixed $\mathcal{W}_{0}\in J_{\text{irr}}$ such that
$\mathcal{Q}_{x}$ is disjoint from any other component not in
$\mathcal{W}_{0}$. For any $\mathcal{W}\supset\mathcal{W}_{0}$, we have
$U_{z}^{W}=U_{z}^{W_{0}}$. Therefore, the colimit
$\operatorname{colim}_{\mathcal{W}^{\prime}\in J_{\text{irr}}}U_{z}^{W}$
eventually stabilizes for a cofinal set of indexes, and we have that
$U_{z}=U_{z}^{W_{0}}$ is an algebraic space of finite type over $k$. Since
each closed stratum of $M_{\mathcal{C}}$ is an algebraic space locally of
finite type over $k$, the ind-algebraic space $M_{\mathcal{C}}$ is Jacobson.
This means that the set of open sub-algebraic spaces $\\{U_{z}\\}_{z\in
M_{\mathcal{C}}(k)}$ as $z$ runs over all $k$-points is a Zariski cover of
$M_{\mathcal{C}}$, as desired.

The fact that $M_{\mathcal{C}}$ satisfies the valuative criterion for
properness comes from the fact that every quasicompact closed stratum is
proper. From the colimit description, it follows that every quasicompact open
sub-algebraic space is separated. Therefore, $M_{\mathcal{C}}$ is a decent
algebraic space [Sta23, Tag 03I8], and hence quasi-separated [Sta23, Tag
0BB6]. Since $M_{\mathcal{C}}$ satisfies the valuative criterion for
properness, this implies that $M_{\mathcal{C}}$ is separated [Sta23, Tag
03KV]. ∎

######  Theorem 4.5.

With notation as above:

  1. (i)

$\mathcal{M}_{\mathcal{C}}\to M_{\mathcal{C}}$ is a good moduli space.

  2. (ii)

There is a bijection between the set of effective G-theory classes in
$\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$ and the set
$M_{\mathcal{C}}(k)$.

  3. (iii)

For any $\alpha\in\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$, the moduli
space of the open and closed substack $(\mathcal{M}_{\mathcal{C}})_{\alpha}$
is a local finite artinian scheme over $\operatorname{Spec}(k)$.

###### Proof.

We use the same notation as in the proof of 4.4. Since the statement can be
checked Zariski locally on the target [Alp13, Prop. 4.7 (ii)], we can restrict
to one of the quasicompact opens $U_{z}$. Note that $U_{z}$ is contained in
the good moduli space $W_{0}$ for $\mathcal{W}_{0}$. We have a cartesian
diagram

${U_{z}\times_{M_{\mathcal{C}}}\mathcal{M}_{\mathcal{C}}}$${\mathcal{W}_{0}}$${U_{z}}$${W_{0}.}$

Indeed, if a point $w$ is not contained in $\mathcal{W}_{0}$, then it is not
contained in $\mathcal{Q}_{x}$, which means that it must be contained in
$\mathcal{Y}_{x}\cap\mathcal{W}$ for some $\mathcal{W}$. It follows that the
image $\varphi(w)$ is not in $U_{z}$.

Since $\mathcal{W}_{0}\to W_{0}$ is a good moduli space, the base-change
$U_{z}\times_{M_{\mathcal{C}}}\mathcal{M}_{\mathcal{C}}\to U_{z}$ is a good
moduli space [Alp13, Prop. 4.7 (i)].

By [Alp13, Prop. 9.1] the $k$-points of $M_{\mathcal{C}}$ are in bijection
with closed $k$-points of the stack $\mathcal{M}_{\mathcal{C}}$. By Lemma 3.10
(iv), the closed points of $\mathcal{M}_{\mathcal{C}}$ are in bijection with
isomorphism classes of semisimple objects in $\mathcal{C}$. Note that every
effective G-theory class $[A]\in\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$
has a unique semisimple representative. Therefore, we have a bijection between
the effective G-theory classes in
$\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$ and the set
$M_{\mathcal{C}}(k)$. This also implies that the moduli space of each open and
closed substack $(\mathcal{M}_{\mathcal{C}})_{\alpha}$ has a single $k$-point,
and so must be a local finite artinian scheme over $\operatorname{Spec}(k)$. ∎

######  Remark 4.6.

The arguments above can be applied to show a mild strengthening of some of the
results in [AHLH19]. Let $\mathcal{M}$ be a stack locally of finite type with
affine automorphism groups and separated diagonal over a field $k$ of
characteristic $0$. Suppose that $\mathcal{M}$ is $\Theta$-reductive and
$S$-complete, and that every irreducible component of $\mathcal{M}$ is
quasicompact. Then, $\mathcal{M}$ admits a separated good moduli space. We
have seen that this small improvement is quite useful in context.

######  Remark 4.7.

Since the formation of moduli spaces commutes with base-change [Alp13, Prop.
4.7(i)], it follows from our description of $M_{\mathcal{C}}$ that for any
over-field $K\supset k$ the base-change functor $(-)\otimes K$ induces a
canonical identification of G-theory groups
$\operatorname{G_{0}}(\mathcal{C})=\operatorname{G_{0}}((\mathcal{A}_{K})^{fp})$

######  Example 4.8.

When $\mathcal{A}$ is the category of representations of an acyclic quiver,
then each $(M_{\mathcal{A}})_{\alpha}\cong\operatorname{Spec}k$, see e.g.
[BDF+22]. However, this will not be true in general. For example, take
$\mathcal{A}=k[\varepsilon]\text{-}\operatorname{Mod}$, where
$\varepsilon^{2}=0$. Then
$\operatorname{G_{0}}(\mathcal{A}^{\mathrm{fp}})\cong\mathbb{Z}$, where $[M]$
is identified with $\dim M\in\mathbb{Z}$. Taking $\alpha=1\in\mathbb{Z}$, we
get
$(\mathcal{M}_{\mathcal{A}})_{\alpha}\cong\operatorname{Spec}k[\varepsilon]\times
B\mathbb{G}_{\mathrm{m}}$, and so
$(M_{\mathcal{A}})_{\alpha}\cong\operatorname{Spec}k[\varepsilon]$.

###  4.2 Stability of objects in $\mathcal{C}$

Fix an effective G-theory class
$\alpha\in\operatorname{G^{\mathrm{eff}}}(\mathcal{C})$. In this subsection we
define a family of stability conditions coming from numerical invariants on
$(\mathcal{M}_{\mathcal{C}})_{\alpha}$ in the sense of [HL21]. We first need
to define some line bundles on $(\mathcal{M}_{\mathcal{C}})_{\alpha}$.

######  Definition 4.9.

Let $\beta\in\operatorname{K_{0}}(\mathcal{C})\cong\mathbb{Z}^{\oplus I}$ of
the form $\sum_{i\in I}n_{i}[P_{i}]$, for some tuple of integers $n_{i}$. We
define $\mathcal{L}_{\beta}$ to be the line bundle in
$\text{Pic}(\mathcal{M})$ such that for every noetherian $k$-algebra $R$ and
every $f\colon\operatorname{Spec}(R)\to\mathcal{M}_{\mathcal{C}}$ determined
by an $R$-flat element $A\in(\mathcal{A}_{R})^{\mathrm{fp}}$, we have

$f^{*}(\mathcal{L}_{\beta}):=\bigotimes_{i\in
J}\text{det}(\operatorname{Hom}_{\mathcal{A}_{R}}(P_{i}\otimes R,A))^{\otimes
n_{i}}.$

The definition above makes sense, because each
$\operatorname{Hom}_{\mathcal{A}_{R}}(P_{i}\otimes R,A)$ is a vector bundle on
$\operatorname{Spec}(R)$ (Lemmas 2.26 and 2.27), and $n_{i}=0$ for all but
finitely many $i\in I$.

######  Definition 4.10.

Let $\gamma=\sum_{i\in
I}m_{i}[P_{i}]\in\operatorname{K_{0}}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{Q}$
be a rational K-theory class such that $m_{i}\geq 0$ for all $i\in I$. For any
field $K$ and any $V\in(\mathcal{A}_{K})^{\text{fp}}$, we define the
$\gamma$-length of $V$ by

$\ell_{\gamma}(V):=\left\langle\gamma_{K},V\right\rangle\geq 0,$

where $\gamma_{K}$ denotes the base-change $\sum_{i\in I}m_{i}[P_{i}\otimes
K]\in\operatorname{K_{0}}((\mathcal{A}_{K})^{\mathrm{fp}})\otimes_{\mathbb{Z}}\mathbb{Q}$.
Similarly for a G-theory class $\alpha$, we set
$\ell_{\gamma}(\alpha):=\langle\gamma,\alpha\rangle$.

######  Definition 4.11.

Let $J_{\alpha}\subset I$ be the set of all $j\in I$ such that $\langle
P_{j},\alpha\rangle\neq 0$ (equivalently, we have $\alpha=\sum_{i\in
J_{\alpha}}a_{i}[S_{i}]$ for some $a_{i}>0$). We say that a class
$\gamma=\sum_{i\in I}m_{i}[P_{i}]\in\operatorname{K_{0}}(\mathcal{C})$ is
$\alpha$-nondegenerate if $m_{i}>0$ for all $i\in J_{\alpha}$.

If $\gamma$ is an $\alpha$-nondegenerate class, then for any $B\in\mathcal{C}$
with G-theory class $\alpha$ and any $0\neq E\subset B$, we have
$\ell_{\gamma}(E)>0$.

######  Example 4.12.

Let $\gamma=\sum_{i\in J_{\alpha}}\frac{1}{\langle
P_{i},S_{i}\rangle}[P_{i}]$. Then, for all $B\in\mathcal{C}$ with G-theory
class $\alpha$ and all $E\subset B$, the $\gamma$-length $\ell_{\gamma}(E)$ is
just the length of $E$ in the abelian category $\mathcal{C}$ (i.e. the size of
its Jordan-Hölder filtration).

######  Definition 4.13.

Let $\beta\in K_{0}(\mathcal{C})$, and let
$\gamma\in\operatorname{K_{0}}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{Q}$ be
an $\alpha$-nondegenerate class. Then $\mathcal{L}_{\beta,\gamma}$ is defined
to be the line bundle on $(\mathcal{M}_{\mathcal{C}})_{\alpha}$ given by

$\mathcal{L}_{\beta,\gamma}:=\mathcal{L}_{\beta}^{\otimes\ell_{\gamma}(\alpha)}\otimes\mathcal{L}_{\gamma}^{\otimes-\langle\beta,\alpha\rangle}.$

For the rest of this section, we fix $\beta$ and $\gamma$ as above.

######  Definition 4.14.

For any object $E\in\mathcal{C}$ with $\ell_{\gamma}(E)\neq 0$, we define the
$\beta$-slope of $E$ by
$\sigma_{\beta}(E):=\frac{\langle\beta,E\rangle}{\ell_{\gamma}(E)}$. An object
$A\in\mathcal{C}$ with numerical class $\alpha$ is called _$\beta$
-semistable_ if for all nonzero subobjects $0\neq E\subset A$, we have the
inequality of slopes $\sigma_{\beta}(E)\leq\sigma_{\beta}(A)$.

Our goal is to show that the set of $\beta$-semistable objects with G-theory
class $\alpha$ are the $k$-points of an open substack
$(\mathcal{M}_{\mathcal{C}})_{\alpha}^{\beta\text{-}ss}\subset(\mathcal{M}_{\mathcal{C}})_{\alpha}$
which admits a projective good moduli space over $k$. We would also like to
develop Harder-Narasimhan theory for this stability condition. We will
accomplish this via the use of a numerical invariant in the sense of [HL21,
Def. 0.0.3].

For any field $K\supset k$ and any positive integer $h$, morphisms
$B(\mathbb{G}_{m})^{h}_{K}\to\mathcal{M}_{\mathcal{C}}$ amount to
$\mathbb{Z}^{h}$-graded objects
$\bigoplus_{\vec{w}\in\mathbb{Z}^{h}}A_{\vec{w}}$ in
$(\mathcal{A}_{K})^{\text{fp}}$ (cf. [AHLH19, Prop. 7.12]). Using this
description, we define a rational quadratic norm on graded points as in [HL21,
Def. 4.1.12].

######  Definition 4.15.

We denote by $b_{\gamma}$ the rational quadratic norm on graded points of
$(\mathcal{M}_{\mathcal{C}})_{\alpha}$ defined as follows. For any $h>0$,
morphism $g\colon
B(\mathbb{G}_{m})^{h}_{K}\to(\mathcal{M}_{\mathcal{C}})_{\alpha}$
corresponding to a graded object
$\bigoplus_{\vec{w}\in\mathbb{Z}^{h}}A_{\vec{w}}$, and vector
$\vec{r}\in\mathbb{R}^{h}$, we set

$b_{\gamma}(g)(\vec{r})=\sum_{\vec{w}\in\mathbb{Z}^{h}}\left(\vec{w}\cdot\vec{r}\right)^{2}\ell_{\gamma}\left(A_{\vec{w}}\right).$

where $\vec{w}\cdot\vec{r}=\sum_{j=1}^{h}w_{j}r_{j}$ denotes the standard
inner product for two vectors $\vec{w}=(w_{j})_{j=1}^{h}$ and
$\vec{r}=(r_{j})_{j=1}^{h}$.

######  Definition 4.16.

We define $\mu_{\beta}$ to be the $\mathbb{R}$-valued numerical invariant on
the stack given by
$\mu_{\beta}=\text{wt}(\mathcal{L}_{\beta,\gamma})/\sqrt{b_{\gamma}}$

We recall that under certain conditions, such numerical invariant can be used
to define a $\Theta$-stratification on the stack (cf. [HL21, §4.1] or [HLHJ21,
§2.5]).

######  Theorem 4.17.

Fix $\beta\in\operatorname{K_{0}}(\mathcal{C})$ and
$\gamma\in\operatorname{K_{0}}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{Q}$
that is $\alpha$-nondegenerate.

  1. (i)

The numerical invariant $\mu_{\beta}$ defines a $\Theta$-stratification on the
stack $(\mathcal{M}_{\mathcal{C}})_{\alpha}$.

  2. (ii)

The open $\mu_{\beta}$-semistable locus
$(\mathcal{M}_{\mathcal{C}})^{\mu_{\beta}\text{-}ss}_{\alpha}$ admits a good
moduli space $(M_{\mathcal{C}})_{\alpha}^{\beta\text{-}ss}$ that is projective
over $\operatorname{Spec}(k)$. A power of the dual line bundle
$\mathcal{L}_{\beta,\gamma}^{-1}$ descends to an ample line bundle on
$(M_{\mathcal{C}})_{\alpha}^{\beta\text{-}ss}$.

  3. (iii)

The $k$-points of the $\mu_{\beta}$-semistable open locus
$(\mathcal{M}_{\mathcal{C}})_{\alpha}^{\mu_{\beta}\text{-}ss}$ are exactly the
$\beta$-semistable objects.

###### Proof.

Let
$\varphi:(\mathcal{M}_{\mathcal{C}})_{\alpha}\to(M_{\mathcal{C}})_{\alpha}$ be
the separated good moduli space of the open and closed substack
$(\mathcal{M}_{\mathcal{C}})_{\alpha}\subset\mathcal{M}_{\mathcal{C}}$, which
exists by 4.5. Recall that the good moduli space $(M_{\mathcal{C}})_{\alpha}$
is finite over $\operatorname{Spec}(k)$. To conclude (i) and (ii), we use
[HL21, Thm. 5.6.1 (1)+(2)] with
$\mathfrak{X}=\mathfrak{Y}=(\mathcal{M}_{\mathcal{C}})_{\alpha}$ and
$\mathcal{L}=\mathcal{L}^{-1}_{\beta,\gamma}$ and $\|\bullet\|=b_{\gamma}$. In
this case, the weak $\Theta$-stratification will automatically be a
$\Theta$-stratification because the characteristic of $k$ is $0$.

We are left to prove (iii). Let $p\in(\mathcal{M}_{\mathcal{C}})_{\alpha}(k)$
corresponding to an object $A\in\mathcal{C}$ with G-theory class $\alpha$. By
[AHLH19, Cor. 7.13], $\Theta$-filtrations of $p$ correspond to
$\mathbb{Z}$-weighted filtrations $(A_{w})_{w\in\mathbb{Z}}$, where

  * •

$A_{w}\subset A_{w-1}\subset A$ for all $w\in\mathbb{Z}$.

  * •

$A_{w}=0$ for $w\gg 0$, and $A_{w}=A$ for $w\ll 0$.

For any such filtration $f$ corresponding to $(A_{w})_{w\in\mathbb{Z}}$, a
direct computation shows that the value of the numerical invariant is

$\mu_{\beta}(f)=\frac{\sum_{w\in\mathbb{Z}}w\left(\ell_{\gamma}(\alpha)\cdot\langle\beta,A_{w}/A_{w+1}\rangle-\langle\beta,\alpha\rangle\cdot\ell_{\gamma}(A_{w}/A_{w+1})\right)}{\sqrt{\sum_{w\in\mathbb{Z}}w^{2}\ell_{\gamma}(A_{w}/A_{w+1})}}.$

Therefore the point $p=A$ is $\mu_{\beta}$-semistable if and only if for all
weighted filtrations $(A_{w})_{w\in\mathbb{Z}}$ we have

$\sum_{w\in\mathbb{Z}}w(\ell_{\gamma}(\alpha)\cdot\langle\beta,A_{w}/A_{w+1}\rangle-\langle\beta,\alpha\rangle\cdot\ell_{\gamma}(A_{w}/A_{w+1}))\leq
0.$

Using summation by parts and the identities
$\langle\beta,A\rangle=\langle\beta,\alpha\rangle$ and
$\ell_{\gamma}(A)=\ell_{\gamma}(\alpha)$, we rewrite this as

$\sum_{w\in\mathbb{Z}}(\ell_{\gamma}(A)\langle\beta,A_{w}\rangle-\ell_{\gamma}(A_{w})\langle\beta,A_{w}\rangle)\leq
0.$

A standard argument (cf. [HLHJ21, Prop. 5.16]) shows that the inequality above
is satisfied for all filtrations $(A_{w})_{w\in\mathbb{Z}}$ if and only if $A$
is $\beta$-semistable in the sense of 4.14. ∎

##  5 Examples

###  5.1 Modules over a finite dimensional associative algebra

######  Proposition 5.1.

Let $A$ be an associative $k$-algebra that is finite-dimensional as a
$k$-vector space. Then the category $\mathcal{C}=A\text{-}\mathrm{fdmod}$ of
finite dimensional left modules over $A$ satisfies 1.1.

###### Proof.

All objects of $\mathcal{C}$ have finite dimension as $k$-vector spaces, and
so they are of finite length. It also follows that $\mathcal{C}$ is Hom-
finite. Since $A$ regarded as an $A$-module is a projective generator,
$\mathcal{C}$ satisfies (A2). Moreover, every simple element of $\mathcal{C}$
is a quotient module of $A$, and so the isomorphism classes of simple objects
in $\mathcal{C}$ form a set.

∎

######  Example 5.2 (Representations of acyclic quivers).

For the following example, we refer the reader to [Kin94] and [BDF+22] and
references therein. Let $Q$ be an acyclic quiver. The category $\mathcal{C}$
of finite dimensional representations of $Q$ is isomorphic to finite
dimensional representations of its path algebra, which is an associative
algebra of finite dimension over $k$. In particular, $\mathcal{C}$ satisfies
1.1 by 5.1.

In this case we have $\operatorname{G_{0}}(\mathcal{C})=\mathbb{Z}^{\oplus
I}$, where $I$ is the set of vertices of the quiver $Q$. For each vertex $i\in
I$, there is an associated projective object $P_{i}$ in $\mathcal{C}$. For any
representation $V$ and $i\in I$, the pairing $\langle P_{i},V\rangle$ is the
dimension of the underlying $k$-vector space $V_{i}$ at the vertex $i$. Given
a class $\beta=\sum_{i\in I}n_{i}[P_{i}]\in\operatorname{K_{0}}(\mathcal{C})$
and $\gamma$ as in Example 4.12, our definition of $\beta$-semistability
(4.14) agrees with the one defined by King [Kin94]. Hence, we recover the
moduli spaces of representations of $Q$ constructed using GIT in [Kin94] and
intrinsically in [BDF+22].

We offer another quiver example that is related to representations of finite-
dimensional algebras, even though it does not strictly fall into that
framework.

######  Example 5.3 (Representations of semi-infinite quivers).

Let $Q$ be a quiver with possibly infinitely many vertices and edges. We say
that $Q$ is _projectively semi-infinite_ if for any vertex $i$, the number of
paths starting in $i$ is finite. Equivalently, we require that any
indecomposable projective representation $P$ has finite total dimension
$\sum_{i\in Q_{0}}\dim P_{i}$.

Set $\mathcal{C}=\operatorname{rep}Q$ to be the category of representations of
$Q$ whose total dimension is finite. Then $\mathcal{C}$ satisfies 1.1.

###  5.2 Category $\mathcal{O}$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Fix a Cartan
subalgebra $\mathfrak{h}$ and let $\mathfrak{h}^{\vee}$ be its dual. Fix a
system of simple roots $\Phi$ and let $\Phi^{+}$ be the positive roots. Let
$\mathfrak{n}=\oplus_{\alpha>0}\mathfrak{g}_{\alpha}$.

######  Definition 5.4.

The _BGG category_ $\mathcal{O}$ is the full subcategory of
$\operatorname{Mod}U(\mathfrak{g})$ such that

  1. (i)

$M$ is a finitely generated $U(\mathfrak{g})$-module.

  2. (ii)

$M$ is $\mathfrak{h}$-semisimple, i.e.
$M=\oplus_{\lambda\in\mathfrak{h}^{\vee}}M_{\lambda}$.

  3. (iii)

$M$ is locally $\mathfrak{n}$-finite, i.e. $\forall v\in
M,U(\mathfrak{n})\cdot v$ is finite dimensional.

######  Theorem 5.5.

Category $\mathcal{O}$ is an abelian category satisfying conditions 1.1.

###### Proof.

Since $\mathcal{O}$ is noetherian, artinian and Hom-finite by [Hum08, Thm.
1.1, 1.11], (A0) and (A1) are satisfied. It is essentially small because
simple objects are parametrized by linear functionals in $\mathfrak{h}^{\vee}$
by [Hum08, Thm. 1.3]. Furthermore, (A2) holds by [Hum08, Thm. 3.8]. ∎

Let $M(\lambda)$ denote the Verma module associated to
$\lambda\in\mathfrak{h}^{\vee}$. Its unique simple quotient $L(\lambda)$ is
the unique simple object of highest weight $\lambda$ [Hum08, Thm. 1.3]. Each
$L(\lambda)$ has a unique indecomposable projective cover $P(\lambda)$ [Hum08,
Thm. 3.9]. In this case, every object is perfect and hence the $G$-theory and
$K$-theory are the same. For any $M\in\mathcal{O}$, its $K$-theory class is
given by $\sum_{\lambda}[M:L(\lambda)][L(\lambda)]$ where $[M:L(\lambda)]$ is
composition factor multiplicity.

######  Example 5.6.

Consider $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$. Then for any block
$\mathcal{O}_{\chi_{\lambda}}$, there are five non-isomorphic indecomposable
modules

$L(\lambda),L(-2-\lambda)=M(-2-\lambda),M(\lambda)=P(\lambda),M(\lambda)^{\vee},P(-2-\lambda).$

For a fixed class $\alpha$, then there are only finitely many objects with
that class. Hence, our moduli spaces would be a finite number of points.

Consider the principal block $\mathcal{O}_{\chi_{0}}$, that is $\lambda=0$.
Fix a class $\beta=\beta_{1}[L(0)]+\beta_{2}[L(-2)]$ and take $\gamma$ as in
Example 4.12. Then we find that the $\beta$-slope $\sigma_{\beta}$ is

$\sigma_{\beta}(L(0))=\beta_{1},\sigma_{\beta}(L(-2))=\beta_{2}$
$\sigma_{\beta}(M(0))=\sigma_{\beta}(M(0)^{\vee})=\frac{\beta_{1}+\beta_{2}}{2}$
$\sigma_{\beta}(P(-2))=\frac{\beta_{1}+2\beta_{2}}{3}.$

It follows that $M(0)$ is $\beta$-semistable if and only if
$\beta_{1}\geq\beta_{2}$. Also, $M(0)^{\vee}$ is $\beta$-semistable if and
only if $\beta_{2}\geq\beta_{2}$. Meanwhile, $P(-2)$ is $\beta$-semistable
only when $\beta_{1}=\beta_{2}$, which gives the trivial stability condition.
We have recovered the stability conditions given in [FJM08, Ex. 3.4].

###  5.3 Comodules over co-Frobenius Hopf algebras

Let $C$ be a counital coalgebra. Set $\mathcal{C}=H\text{-}\mathrm{fdcomod}$,
so $\mathcal{A}=H\text{-}\mathrm{Comod}=\operatorname{Ind}(\mathcal{C})$ by
the fundamental theorem of coalgebra. The category $\mathcal{C}$ automatically
satisfies (A0) and (A1). For the fact that $\mathcal{A}$ is essentially small,
we use that simple comodules are in bijection with simple subcoalgebras of $H$
[Lar71, pg. 354].

######  Theorem 5.7 ([Lin77, Thms 3, 10]).

The following are equivalent for a Hopf algebra $H$.

  1. (i)

$H$ is co-Frobenius.

  2. (ii)

The injective hull of a simple $H$-comodule is finite-dimensional.

For the purpose of the present paper, we use this theorem as a definition of a
co-Frobenius Hopf algebra.

######  Corollary 5.8.

Let $H$ be a co-Frobenius Hopf algebra. Then
$\mathcal{C}=H\text{-}\mathrm{fdcomod}$ satisfies 1.1.

###### Proof.

It remains to check that $\mathcal{C}$ has enough projective objects (A2). For
this, we show that every simple object $S\in\mathcal{C}$ admits a surjection
from a projective object $P\to S$. Since $H$ is a Hopf algebra, using its
antipode ensures that taking the $k$-linear dual $(\underline{\ \
})^{\vee}\colon\mathcal{C}^{\mathrm{op}}\to\mathcal{C}$ is a self-
antiequivalence of $\mathcal{C}$. The comodule $S^{\vee}$ is still simple, so
by 5.7, $S^{\vee}\in\mathcal{C}$ admits a finite-dimensional injective hull
$S^{\vee}\to E$, so $E\in\mathcal{C}$. The comodule $P=E^{\vee}$ is
projective, since an antiequivalence sends injective objects to projective.
Hence, we get the desired surjection $P\to S$. ∎

We next present an example of a co-Frobenius Hopf algebra which has infinitely
many simple comodules. This relevant because, if there were only finitely
many, the category of comodules would be equivalent to the category of
representations of a finite-dimensional associative algebra, but moduli spaces
have been known for them since 1994 [Kin94], see Section 5.1 for more details.

######  Example 5.9 (Finite dimensional comodules over quantum groups).

An example known to representation theorists includes the algebra
$\mathcal{O}_{q}(\mathfrak{g})\subset U_{q}(\mathfrak{g})^{\vee}$ of functions
on Lusztig’s quantum group at a root of unity $q$ attached to a simple Lie
algebra $\mathfrak{g}$. The Hopf algebra $\mathcal{O}_{q}(\mathfrak{g})$ was
proved to be co-Frobenius in [AD03, Example 4.1]. Let $\mathcal{C}$ denote the
category of finite-dimensional comodules over $\mathcal{O}_{q}(\mathfrak{g})$.
We claim that the set of simple objects in $\mathcal{C}$ is infinite. Indeed,
every simple comodule is a subcomodule of $\mathcal{O}_{q}(\mathfrak{g})$. By
the fundamental theorem of coalgebra, every element of a comodule is contained
in a finite-dimensional subcomodule; in particular, every simple comodule is
finite-dimensional, and every element of $\mathcal{O}_{q}(\mathfrak{g})$ is
contained in a simple subcomodule. But $\mathcal{O}_{q}(\mathfrak{g})$ is
infinite-dimensional, hence there will necessarily be infinitely many simple
comodules.

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