1. Introduction
Moduli problems in algebraic geometry can be roughly 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 objects of a more geometric nature, like certain classes of commutative or noncommutative varieties. In [Reference Artin and ZhangAZ01], Artin and Zhang provide a framework for studying the moduli of objects in a certain class of abelian categories, and they use this to construct Hilbert schemes of quotients of a noetherian object. We view this 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 in the context of finite length abelian categories.
We construct moduli spaces of objects in an abelian category
$\mathcal{C}$
satisfying the following conditions (see § 2 for definitions).
Definition 1.1. Let
$k$
be a field. An essentially small
$k$
-linear abelian category
$\mathcal{C}$
is called nearly finite if it satisfies the following:
-
(A0) is Hom-finite, that is, its Hom-spaces are finite-dimensional;
-
(A1) is finite length, that is, every object has finite length; and
-
(A2) has enough projective objects.
Our main results can be summarized as follows.
Main Theorem 1 (Theorem 4.2, Theorem 4.15). Let
$\mathcal{C}$
be a nearly finite category (Definition 1.1) and let
$\mathcal{M}_{\mathcal{C}}$
be the stack of objects in
$\mathcal{C}$
as in Definition 3.1. Then:
-
(i)
$\mathcal{M}_{\mathcal{C}}$
is a disjoint union of algebraic stacks of finite type over
$k$
; -
(ii)
$\mathcal{M}_{\mathcal{C}}$
admits a good moduli space
$M_{\mathcal{C}}$
which is a disjoint union of local artinian
$k$
-schemes; and -
(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, we point out that it refines King’s construction of quiver moduli spaces when the quiver is acyclic or when the path algebra is finite-dimensional, which we now recall [Reference KingKin94]. 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
$\mathrm{G}^{\mathrm{ eff}}({\mathcal{C}})$
, in § 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 [Reference AlperAlp13, Reference Alper, Halpern-Leistner and HeinlothAHLH23]. Moreover, we follow the treatment of the moduli of abelian categories in [Reference Alper, Halpern-Leistner and HeinlothAHLH23] using the functor of Artin and Zhang. The main contribution of this paper is the observation that the finiteness 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.
In § 5, we provide several examples of abelian categories that satisfy Definition 1.1. In particular, 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 and Zhang (before imposing any stability conditions). Indeed, Hom-finiteness (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
$k$
[Reference Alper, Halpern-Leistner and HeinlothAHLH23, Lemma 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
$\mathrm{Ind}({\mathcal{C}})$
is adically complete, as in [Reference Artin and ZhangAZ01], which is a necessary condition for
$\mathcal{M}_{{\mathcal{C}}}$
to be represented by an algebraic stack.
1.1 Related works
King [Reference KingKin94] 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 are nearly finite in the sense of Definition 1.1 and have a finite number of simple objects up to isomorphism. Further, various stability conditions have been described on a subclass of them, in [Reference Futorny, Jardim and MouraFJM08].
Artin and Zhang [Reference Artin and ZhangAZ01] 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 [Reference Alper, Halpern-Leistner and HeinlothAHLH23, § 7], the authors apply the theory of good moduli spaces to stacks of objects as defined in [Reference Artin and ZhangAZ01]. With a few technical assumptions, they prove the 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 [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Theorem 7.27] assume as an input that the moduli stack is algebraic and locally of finite type. Some technical work in § 3 is dedicated to verifying these assumptions for an abstract abelian category.
There have been constructions of moduli of objects in triangulated and differential categories (commonly referred to as dg categories). For example, Toën and Vaquié [Reference Toën and VaquiéTV07] construct derived moduli of perfect objects in dg categories, and Lieblich [Reference LieblichLie06] 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 (for example, the moduli of modules over a finite
$k$
-algebra of infinite global dimension, such as
$k[\epsilon ]/(\epsilon^2)$
).
1.2 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 [Reference KandaKan19, Theorem 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 (one can see this, for example, when we take
$\mathcal{C}$
to be the category of finite length modules over the polynomial algebra
$k[t]$
). 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.
1.3 Vistas
We view this project as the first step in our attempt to understand strange duality [Reference Donagi and TuDT94, Reference BeauvilleBea95, Reference Derksen and WeymanDW00, Reference Marian and OpreaMO07] in the context of abelian categories. Exploring the strange duality phenomena for nearly finite categories 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 § 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.
2. Abelian categories and base-change
2.1 Nearly finite categories
A
$k$
-linear abelian category
$\mathcal{C}$
is Hom-finite if for every
$V, W \in {\mathcal{C}}$
the dimension of
${\mathrm{Hom}}_{\mathcal{C}}(V,W)$
over the base field
$k$
is finite. 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 supremum of the numbers
$n$
for which there exists a (possibly infinite) filtration
\begin{align*} 0=V_0 \subsetneq V_1 \subsetneq \dots \subsetneq V_n = V. \end{align*}
Such a filtration is called a Jordan–Hölder filtration if each factor
$V_i / V_{i-1}$
is simple for
$1\leqslant i \leqslant n$
, and in this case we set
$\mathrm{gr} V = \bigoplus V_i / V_{i-1}$
.
Remark 2.1. An object has finite length if and only if it is both noetherian and artinian [Sta23, 0FCJ], and in this case
$\mathrm{gr} V$
is independent, up to isomorphism, of the choice of the Jordan–Hölder filtration [Sta23, 0FCK].
Lemma 2.2. Let
$\mathcal{C}$
be a nearly finite abelian category (Definition 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
${\mathrm{Hom}}(P_S, S^{\prime}) =0$
.
Proof.
Let
$S$
be a simple object. By Definition 1.1, 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 the finite length assumption in Definition 1.1,
$P$
admits a finite direct sum decomposition
$P = \bigoplus _j P_j$
, where each
$P_j$
is indecomposable and projective. Since
$P_j \to S$
is nonzero for some
$j$
, we can set
$P_S = P_j$
. Uniqueness follows from sequentially applying [Reference KrauseKra15, Theorem 5.5, Lemma 3.6, Corollary 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 [Reference KrauseKra15, Lemma 3.6]. This implies
$S \cong S^{\prime}$
.
Notation 2.3. 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 2.2, the latter set is independent of any choice, up to isomorphism. We call
$\{P_i\}_{i \in I}$
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 the Grothendieck groups of the given abelian category from those of the subcategory of perfect objects. We mimic the construction of Quillen’s algebraic K-theory spectrum [Reference QuillenQui73] as a group completion, with respect to direct sum as the group operation, of the groupoid of finitely generated projective modules, omitting the higher homotopy data.
Definition 2.4. Let
$\mathcal{C}$
be an essentially small 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
$\mathrm{Perf} ({\mathcal{C}})$
the full additive subcategory of
$\mathcal{C}$
consisting of perfect objects. -
− The (zeroth) K-theory of
$\mathcal{C}$
is denoted by
$\mathrm{K}_0 ({\mathcal{C}})$
and is the abelian group generated by isomorphism classes of objects in
$\mathrm{Perf} ({\mathcal{C}})$
, subject to the additivity relations in short exact sequences. -
− The (zeroth) G-theory of
$\mathcal{C}$
is denoted by
$\mathrm{G}_0 ({\mathcal{C}})$
and is the abelian group generated by isomorphism classes of objects in
$\mathcal{C}$
, subject to the additivity relations in short exact sequences. -
− We say that a G-theory class
$\alpha \in \mathrm{G}_0 ({\mathcal{C}})$
is effective if there is
$V \in {\mathcal{C}}$
such that
$\alpha = {[V]}$
. We denote by
$G^{\mathrm{ eff}} ({\mathcal{C}})$
the submonoid of effective G-theory classes. We similarly define
$\mathrm{K}^{\mathrm{ eff}} ({\mathcal{C}})$
as the set of
$\alpha \in \mathrm{K}_0({\mathcal{C}})$
such that
$\alpha = [V]$
for some perfect object
$V$
.
We will prove in Lemma 2.6 that we have a natural embedding
$\iota \colon \mathrm{K}_0({\mathcal{C}}) \to \mathrm{G}_0({\mathcal{C}})$
. We note, however, that
$\mathrm{K}^{\mathrm{ eff}}({\mathcal{C}}) \subseteq \mathrm{K}_0({\mathcal{C}}) \cap G^{\mathrm{ eff}}({\mathcal{C}})$
is not an equality in general. For example, if
$\mathcal{C}$
is the category of finite-dimensional representations of a quiver
$A_1 = \bullet \longrightarrow \bullet$
, then there are two projective objects
$P_1 = (k \to k)$
and
$P_2 = (0\to k)$
. The object
$S_1 = (k\to 0)$
is perfect, because it admits a two-step projective resolution
$0 \to P_2 \to P_1 \to S_1 \to 0$
; however, its K-theory class is not effective. The following result is an straightforward consequence of the definitions, we omit the proof.
Lemma 2.5. Suppose that
$\mathcal{C}$
is nearly finite. Let
$V$
be an object of
$\mathrm{Perf} ({\mathcal{C}})$
and
$W$
be an object of
$\mathcal{C}$
. Then we have the following:
-
(i)
$\mathrm{Ext}^i(V,W) = 0$
for all
$i \gg 0$
; -
(ii) for all
$i$
, we have
$\dim _k \mathrm{Ext}^i(V,W) \lt \infty$
; and -
(iii) there is a natural pairing
$\mathrm{K}_0({\mathcal{C}}) \otimes _{\mathbb{Z}} \mathrm{G}_0({\mathcal{C}}) \to \mathbb{Z}$
defined on effective classes asWe write
\begin{align*} ([V], [W]) \longmapsto \langle [V], [W] \rangle = \chi \left ( \mathrm{RHom}_{\mathcal{C}} (V, W) \right ). \end{align*}
$\langle V, W \rangle = \langle [V], [W] \rangle$
for brevity.
Lemma 2.6. Suppose that
$\mathcal{C}$
is nearly finite. Let
$\{P_i\}_{i \in I}$
be the canonical set of projective generators as in Notation 2.3.
-
(i) We have
$\mathrm{K}_0 ({\mathcal{C}}) \cong \mathbb{Z}^{\oplus I}$
with basis
$\{ [P_i] \}_{i \in I}$
, and
$\mathrm{G}_0 ({\mathcal{C}}) \cong \mathbb{Z}^{\oplus I}$
with basis
$\{ [S_i] \}_{i \in I}$
. These bases are dual up to positive scalar multiples, i.e.,
$\langle P_i, S_i \rangle = \tau _i \geq 1$
and
$\langle P_i, S_j \rangle = \delta _{ij} \tau _i$
. -
(ii) The pairing
$\mathrm{K}_0 ({\mathcal{C}}) \times \mathrm{G}_0 ({\mathcal{C}}) \to \mathbb{Z}$
from Lemma 2.5 is nondegenerate. -
(iii) There is a natural injection
$\iota \colon \mathrm{K}_0 ({\mathcal{C}}) \to \mathrm{G}_0 ({\mathcal{C}})$
given by
$\iota ([V]) = [V]$
. Furthermore, if
${\mathcal{C}} = \mathrm{Perf} ({\mathcal{C}})$
, then
$\iota$
is an isomorphism.
Proof.
For part (i), the isomorphism
$\mathrm{K}_0({\mathcal{C}}) \cong \mathbb{Z}^{\oplus I}$
follows from the fact that every short exact sequence of projective objects splits and from the Krull–Schmidt property [Reference KrauseKra15, Lemma 5.1, Theorems 5.5 and 4.2]. On the other hand, the isomorphism for
$\mathrm{G}_0({\mathcal{C}})$
follows from the existence of Jordan–Hölder filtrations and the uniqueness of the corresponding associated graded objects. The fact that they are dual bases up to scaling follows from the vanishing of
${\mathrm{Hom}}(P_S, S^{\prime})$
stated in Lemma 2.2. Part (ii) follows at once from the existence of dual bases.
The injectivity in part (iii) follows from the Jordan–Hölder property and from the fact that every short exact sequence of projective objects splits. Part (iii) is a direct consequence of Definition 2.4.
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
${\mathrm{Hom}}_{\mathcal{A}}(G,\underline {\ \ })$
detects zero morphisms.
Definition 2.7 [Reference PopescuPop73, pp. 91, 92]. An object
$V \in {\mathcal{A}}$
is called
-
− finitely generated (finite type) if, whenever
$V \cong \mathrm{colim}_i V_i$
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
${\mathrm{Hom}}_{\mathcal{A}}(V,\underline {\ \ })$
commutes with filtered colimits; and -
− noetherian if every subobject of
$V$
is finitely generated; equivalently, if subobjects of
$V$
satisfy the ascending chain condition.
When
${\mathcal{A}} \cong R\mathrm{ -}\textrm {Mod}$
, 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
${\mathrm{Hom}}_{{\mathcal{A}}}(V,\underline {\ \ })$
commutes with filtered colimits are called compact. These have a special importance for the moduli theory we consider.
Notation 2.8. We denote by
${\mathcal{A}}^{\mathrm{ fp}}$
the full subcategory of finitely presented objects in
$\mathcal{A}$
.
Definition 2.9. A Grothendieck category is called locally noetherian if it admits a set of noetherian projective generators.
We recall the following useful fact.
Theorem 2.10 [Reference PopescuPop73, Chapter 5, Theorem 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 [Reference Artin and ZhangAZ01].
Definition 2.11 [Reference Artin and ZhangAZ01, Section 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
$\psi \colon R \to {\mathrm{End}}_{{\mathcal{A}}}(V)$
. Objects of
${\mathcal{A}}_R$
are called
$R$
-module objects in
$\mathcal{A}$
, and the homomorphism
$\psi$
is called an
$R$
-module structure on
$V$
. The set of morphisms between two pairs
$(V, \psi ) \to (W, \xi )$
consists of those morphisms in
${\mathrm{Hom}}_{{\mathcal{A}}}(V, W)$
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 [Reference Artin and ZhangAZ01, Section B3], Artin and Zhang define a right exact functor of abelian categories
$V\otimes _R \underline {\ \ } \colon R\mathrm{ -}\textrm {Mod} \to {\mathcal{A}}_R$
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 2.12 [Reference Artin and ZhangAZ01, Proposition 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 2.13 [Reference Artin and ZhangAZ01, Section C1]. An object
$V \in {\mathcal{A}}_R$
is called
$R$
-flat if the functor
$V \otimes _R \underline {\ \ } \colon$
$R\mathrm{ -}\textrm {Mod} \to {\mathcal{A}}_R$
is exact.
For the current setup of moduli theory to work, we need the following condition.
Definition 2.14 (see [Reference Artin, Small and ZhangASZ99]). 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 Definition 2.14 in general [Reference Resco and SmallRS93]. 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 length (A0). Denote by
$\mathcal{A}$
its ind-completion
${\mathcal{A}} = \mathrm{Ind} ({\mathcal{C}})$
[Reference Kashiwara and SchapiraKS06, § 8.6]. By [Reference Kashiwara and SchapiraKS06, Theorem 8.6.5(vi)],
$\mathcal{A}$
is Grothendieck. Every finitely presented object
$V$
in
$\mathcal{A}$
can be shown to be a direct summand of an object from
$\mathcal{C}$
, because
$V$
is a filtered colimit of some objects
$C_j \in {\mathcal{C}}$
, and then the identity
$\mathrm{id}_V$
in
${\mathrm{Hom}}(V, \mathrm{colim}_j\ C_j)$
factors through some
$C_\ell$
. By [Reference Kashiwara and SchapiraKS06, Proposition 8.6.11], the subcategory
$\mathcal{C}$
is thick in
$\mathcal{A}$
, and hence we have
${\mathcal{A}}^{\mathrm{ fp}} \simeq {\mathcal{C}}$
.
Proposition 2.15. If
$\mathcal{C}$
is Hom-finite (A0) and finite length (A1), then
$\mathcal{C}$
is equivalent to the category
$H \hbox{-} \mathrm{ fdcomod}$
of finite-dimensional comodules over some
$k$
-coalgebra
$H$
.
Proof.
Our assumptions imply that
${\mathcal{A}} = \mathrm{Ind} ({\mathcal{C}})$
is of finite type in the sense of [Reference TakeuchiTak77, Definition 4.4], because
$\mathcal{A}$
is Grothendieck as noted above, and the other conditions are satisfied by (A0) and (A1). By [Reference TakeuchiTak77, Theorem 5.1],
$\mathcal{A}$
is equivalent to the category
$H \hbox{-} \mathrm{ Comod}$
of comodules over some coalgebra
$H$
. This identifies
$\mathcal{C}$
with a subcategory of
$H \hbox{-} \mathrm{ Comod}$
of objects of finite length. The fundamental theorem of coalgebra states that every
$H$
-comodule is a union of finite-dimensional subcomodules [Reference Dăscălescu, Năstăsescu and RaianuDNR01, Theorem 2.1.7], so a comodule of finite length is necessarily finite-dimensional.
Theorem 2.16. If
$\mathcal{C}$
is an essentially small
$k$
-linear abelian category which is Hom-finite (A0) and finite length (A1), then
${\mathcal{A}} = \mathrm{Ind} ({\mathcal{C}})$
is strongly locally noetherian.
Since
${\mathcal{A}} \simeq \mathrm{Ind}({\mathcal{A}}^{\mathrm{ fp}})$
, 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 length (A1).
Proof.
Since
$\mathcal{C}$
is a finite length category, it is noetherian, and hence
$\mathcal{A}$
is locally noetherian.
Let
$R$
be a noetherian
$k$
-algebra. By [Reference Artin and ZhangAZ01, Corollary 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 Proposition 2.15, where the vertical arrows are the forgetful functors.
By Proposition 2.15,
$ F(V) \otimes R$
is a finite free module over a noetherian ring
$R$
, and hence is itself noetherian. But
$F_R (V \otimes R) = F(V) \otimes R$
, which forces
$V \otimes R$
to be noetherian.
Corollary 2.17. If
$\mathcal{C}$
is an essentially small
$k$
-linear abelian category which is Hom-finite (A0) and finite length (A1), then for any field extension
$k \subset K$
, finitely presented objects in the base-change category
$((\mathrm{Ind}\, {\mathcal{C}})_K)^{\mathrm{ fp}}$
form a Hom-finite and finite length category as well.
Proof.
This follows by specializing the proof of Theorem 2.16 to the case of
$R=K$
. Indeed, we get
$(\mathrm{Ind}\, {\mathcal{C}})_K \cong (H \hbox{-} \mathrm{ Comod})_K \cong (H_K) \hbox{-} \mathrm{ Comod}$
, so the subcategory of finitely presented objects consists of comodules that are finite-dimensional over
$K$
.
Example 2.18 (Necessity of Hom-finiteness). Theorem 2.16 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 many 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
${\mathcal{A}} = \mathrm{Ind}({\mathcal{C}})$
.
Lemma 2.19 (Nakayama’s lemma [Reference Artin and ZhangAZ01, Theorem 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
$U \subset \mathrm{Spec}(R)$
such that for any
$R$
-algebra
$T$
we have
$W \otimes _R T =0$
if and only if
$\mathrm{Spec}(T) \to \mathrm{Spec}(R)$
factors through
$U$
.
Lemma 2.20. Suppose that
$\mathcal{C}$
satisfies Definition 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
${\mathcal{A}}_{R} = (\mathrm{Ind}\, {\mathcal{C}})_R$
.
Proof.
The set
$\{P_i \otimes R\}_{i \in I}$
generates
${\mathcal{A}}_{R}$
by [Reference Artin and ZhangAZ01, Corollary B3.17]. Each
$\{P_i \otimes R\}_{i \in I}$
is projective by tensor-Hom adjunction (Proposition 2.12), or by [Reference Artin and ZhangAZ01, Lemma D3.2]. Each
$P_i \otimes R$
is noetherian by Theorem 2.16.
2.5 Behavior of
$\mathrm{Hom}$
in families
Lemma 2.21. 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.
-
(i) For all
$R$
-modules
$M$
, the natural morphismis an isomorphism.
\begin{align*}{\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V) \otimes _{R} M \to {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V \otimes _{R} M) \end{align*}
-
(ii) If
$T$
is an
$R$
-algebra, then the natural morphismis an isomorphism.
\begin{align*} {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V)\otimes _{R} T \to {\mathrm{Hom}}_{{\mathcal{A}}_{T}}(P\otimes _{R} T, V \otimes _{R} T)\end{align*}
-
(iii) If
$V \in {\mathcal{A}}_R$
is
$R$
-flat, then
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P,V)$
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 the commutative diagram below.
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
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, A) \otimes _{R} M \to {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, A \otimes _{R} M)$
is an isomorphism.
This now implies part (ii), because we have natural morphisms
\begin{align*} {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V)\otimes _{R} T \to {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V\otimes _{R} T) \to {\mathrm{Hom}}_{{\mathcal{A}}_{T}}(P\otimes _{R}{T}, V \otimes _{R}{T}) , \end{align*}
and the first one is an isomorphism by part (i), while the second one is an isomorphism by tensor-Hom adjunction (Proposition 2.12).
To prove part (iii), we choose an injective morphism
$N \hookrightarrow M$
of
$R$
-modules and check that it remains injective after tensoring with
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P,V)$
. We have the following commutative diagram.
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
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(\underline {\ \ },\underline {\ \ })$
, we conclude that the bottom morphism is injective. Therefore,
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V)\otimes _{R} N \to {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P, V)\otimes _{R}M$
is injective.
Lemma 2.22. Assume that
${\mathcal{A}} = \mathrm{Ind} ({\mathcal{C}})$
, where
$\mathcal{C}$
satisfies Definition 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
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P \otimes R,V)$
is a finitely presented
$R$
-module.
Proof.
By Lemma 2.20,
$\{P_i\otimes R\}_{i \in I}$
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
$P_\ell \in {\mathcal{C}}$
being 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 the fact that
$P \otimes R$
is projective, we can apply the exact functor
${\mathrm{Hom}}_{{\mathcal{A}}_R}(P \otimes R, \underline {\ \ } )$
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
\begin{align*}\bigoplus _j {\mathrm{Hom}}_{{\mathcal{A}}_R}(P \otimes R, P_j^{\prime} \otimes R) \to \bigoplus _{\ell } {\mathrm{Hom}}_{{\mathcal{A}}_R}(P \otimes R, P_{\ell } \otimes R) \to {\mathrm{Hom}}_{{\mathcal{A}}_R}(P \otimes R, V) \to 0. \end{align*}
By Lemma 2.21(ii), we have that
\begin{align*}{\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P\otimes R, P_j^{\prime}\otimes R) \cong {\mathrm{Hom}}_{{\mathcal{A}}}(P, P_j^{\prime}) \otimes R \; \; \rm \ and\ {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P\otimes R, P_{\ell } \otimes R) \cong {\mathrm{Hom}}_{{\mathcal{A}}}(P, P_{\ell }) \otimes R\end{align*}
are finite free
$R$
-modules (recall that
${\mathrm{Hom}}_{{\mathcal{A}}}(P, P_{\ell })$
and
${\mathrm{Hom}}_{{\mathcal{A}}}(P, P_j^{\prime})$
are finite
$k$
-vector spaces). Therefore, the exact sequence above shows that
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P \otimes R,V)$
is a finitely presented
$R$
-module.
3. The stack of objects in
$\mathcal{C}$
In this section, we use the notation
${\mathcal{A}} = \mathrm{Ind} ({\mathcal{C}})$
, where
$\mathcal{C}$
is nearly finite (Definition 1.1). In particular,
$\mathcal{A}$
is strongly locally noetherian by Theorem 2.16. We fix once and for all the canonical set of projective generators
$\{P_i\}_{i \in I}$
as in Notation 2.3.
Definition 3.1. We denote by
$\mathcal{M}_{{\mathcal{C}}}$
the pseudofunctor from
$k\mathrm{ -}\mathrm{Alg}$
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
$\mathrm{Quot}(F)$
the functor from
$k$
-
$\mathrm{Alg}$
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
$\mathrm{Quot}(F)$
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 [Reference Artin and ZhangAZ01, Theorem E3.1]. The category
$\mathcal{A}$
is strongly locally noetherian by Theorem 2.16, and adically complete by [Reference Artin and ZhangAZ01, Corollary D3.3]. It is Ext-finite by the existence of projective resolutions by objects in
${\mathcal{A}}^{\mathrm{ fp}} = {\mathcal{C}}$
and Lemma 2.22. The other assumptions needed are implied directly by Definition 1.1. The valuative criterion for properness follows from [Reference Artin and ZhangAZ01, Lemma E3.3].
Definition 3.4. For all
$F \in {\mathcal{C}}$
, we let
$u_{F}: \mathrm{Quot}(F) \to \mathcal{M}_{{\mathcal{C}}}$
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
$u_{F}([q]) := F\otimes R / {\mathrm{ker}}(q)$
.
Recall that
$I$
was the indexing set for the 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
$\bigsqcup _{\vec {n}}u_{P_{\vec {n}}} \colon \bigsqcup _{\vec {n}} \mathrm{Quot}(P_{\vec {n}}) \to \mathcal{M}_{{\mathcal{C}}}$
, 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
$\bigsqcup _{\vec {n}}u_{P_{\vec {n}}} \colon \bigsqcup _{\vec {n}} \mathrm{Quot}(P_{\vec {n}}) \to \mathcal{M}_{{\mathcal{C}}}$
is schematic, smooth and surjective. Moreover, for each fixed
$\vec {n}$
the morphism
$u_{P_{\vec {n}}}\colon \mathrm{Quot}(P_{\vec {n}}) \to \mathcal{M}_{{\mathcal{C}}}$
is quasi-affine.
Before we prove this lemma, it might be helpful to have a toy example in mind. In the case when
$\mathcal{C}$
is the category of finite-dimensional vector spaces over
$k$
, we have
$\mathrm{K}_0 ({\mathcal{C}}) = \mathrm{G}_0 ({\mathcal{C}}) \cong \mathbb{Z}$
, and then the tuple
$\vec n$
is just one natural number
$n \in \mathbb{N}$
, which corresponds to the projective object
$P_n = k^{\oplus n}$
. The quot scheme
$\mathrm{Quot}(k^{\oplus n})$
is a disjoint union
$\mathrm{Quot}(k^{\oplus n}) = \sqcup _{\ell \geq 0} \mathrm{Gr}(n,\ell )$
of grassmannians, and we have
$\mathcal{M}_{{\mathcal{C}}} = \sqcup _{\ell \geq 0} B\mathrm{GL} _{\ell }$
. Then the statement of the lemma reduces to the claim that for each
$\ell$
, the morphism
$\mathrm{Gr}(n,\ell ) \to B\mathrm{GL} _\ell$
is quasi-affine.
Proof.
Let
$R$
be a
$k$
-algebra, and choose a morphism
$\mathrm{Spec}(R) \to \mathcal{M}_{{\mathcal{C}}}$
corresponding to an element
$V \in ({\mathcal{A}}_{R})^{\mathrm{ fp}}$
. We want to show that
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} (\bigsqcup _{\vec {n}} \mathrm{Quot}(P_{\vec {n}})) \to \mathrm{Spec}(R)$
is represented by a smooth surjective morphism of schemes, with each
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}})$
quasi-affine. Since both functors
$\bigsqcup _{\vec {n}} \mathrm{Quot}(P_{\vec {n}})$
and
$\mathcal{M}_{{\mathcal{C}}}$
commute with filtered colimits (cf. the proof of [Reference Artin and ZhangAZ01, Lemma E.3.4]), it suffices to check this when
$R$
is a finite type
$k$
-algebra.
We start by showing that
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}})$
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
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}})$
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
${\mathrm{Hom}}_{{\mathcal{A}}_{S}}(P_{\vec {n}} \otimes S, V \otimes _{R} S)$
. We conclude the proof of quasi-affineness and smoothness by showing the following:
-
(1)
$X$
is represented by an affine smooth
$R$
-scheme; and -
(2) the natural monomorphism of functors
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}}) \hookrightarrow X$
is represented by an open immersion.
For (1), notice that
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P \otimes R, V)$
is a vector bundle on
$\mathrm{Spec}(R)$
by Lemma 2.21(iii) and Lemma 2.22. Moreover, by Lemma 2.21(ii), for any
$R$
-algebra
$S$
we have
$X(S) = {\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P \otimes R, V) \otimes _{R} S$
. Therefore, the functor
$X$
is represented by the total space of the vector bundle
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P \otimes R, V)$
, 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
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}})$
consists of the
$S$
-points of
$X$
such that the pullback of
$\psi$
is surjective. In other words,
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} \mathrm{Quot}(P_{\vec {n}})$
is represented by the
$S$
-points of
$X$
such that the pullback of
$\mathcal{Q}$
is
$0$
. By Nakayama’s lemma (Lemma 2.19), this is represented by an open subscheme of
$X$
.
We are left to prove surjectivity. Since
$\mathrm{Spec}(R) \times _{\mathcal{M}_{{\mathcal{C}}}} (\bigsqcup _{\vec {n}} \mathrm{Quot}(P_{\vec {n}})) \to \mathrm{Spec}(R)$
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 [Reference Artin and ZhangAZ01, Theorem 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, Tag06DB].
Definition 3.8. For any object in
$V \in {\mathcal{C}}$
, we say that
$V$
has G-theory class
$\alpha$
if
$[V] = \alpha$
in
$\mathrm{G}_0({\mathcal{C}})$
.
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
$\mathrm{Spec}(R) \to \mathcal{M}_{{\mathcal{C}}}$
. Define
$R(k)_{\alpha } \subset \mathrm{Spec}(R)(k)$
by
\begin{align*} R(k)_{\alpha } := \left \{ f \colon \mathrm{Spec}(k) \to \mathrm{Spec}(R) \lvert f^*(V)\ {\rm has\ G\hbox{-}theory\ class} \alpha \right \} .\end{align*}
We need to show that there exists a (unique) open and closed subscheme
$U \subset \mathrm{Spec}(R)$
with
$U(k) = R(k)_{\alpha }$
. For each
$i \in I$
, we can similarly define
\begin{align*} R(k)^i_{\alpha } := \left \{ f \colon \mathrm{Spec}(k) \to \mathrm{Spec}(R) \lvert \langle P_i,\ f^*(V)\rangle = \langle P_i, \alpha \rangle \right \}. \end{align*}
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 2.21(iii) and Lemma 2.22, the module
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P_i, V)$
is a vector bundle on
$\mathrm{Spec}(R)$
, and by Lemma 2.21(ii) the set
$R(k)^i_{\alpha }$
consists of those closed points in
$\mathrm{Spec}(R)(k)$
such that the fiber of the vector bundle
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P_i, V)$
has dimension
$\langle P_i, \alpha \rangle$
. This is the set of closed points of a closed and open subscheme of
$\mathrm{Spec}(R)$
, concluding the proof of the claim. Now, we have
$R(k)_{\alpha } = \bigcap _{i \in I} R(k)^i_{\alpha }$
by Lemma 2.6(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
$\mathrm{Spec}(R)$
. 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.
-
(i) It has affine diagonal.
-
(ii) It is
$\Theta$
-reductive and
$S$
-complete. -
(iii) It satisfies the existence part of the valuative criterion for properness.
-
(iv) A point
$V \in |\mathcal{M}_{{\mathcal{C}}}|$
is closed if and only if
$V$
is semisimple.
Proof.
Part (i) follows from [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Lemma 7.20]. Parts (ii) and (iii) follow from [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Lemmas 7.16, 7.17, 7.18] (note that in our case we don’t need to restrict to discrete valuating rings that are essentially of finite type over
$k$
, because
$\mathcal{A}$
is strongly locally noetherian). Part (iv) follows from [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Lemma 7.19].
Proposition 3.11. For every effective G-theory class
$\alpha$
, the open and closed substack
$(\mathcal{M}_{\mathcal{C}})_\alpha \subset \mathcal{M}_{\mathcal{C}}$
is connected and quasi-compact.
Proof.
Let
$A$
be the unique semisimple object of class
$\alpha$
. We will show that every object
$V$
from
$(\mathcal{M}_{\mathcal{C}})_\alpha$
is in the same connected component as
$A$
. Indeed, if
$0 = V_0 \subsetneq V_1 \subsetneq \ldots \subsetneq V$
denotes the Jordan–Hölder filtration, then a choice of weights induces a morphism
$\Theta _k := \mathbb{A}^1_k/\mathbb{G}_{m,k} \to (\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
which sends
$1$
to
$[V]$
and sends
$0$
to the associated graded
$[\bigoplus _{i=0}^{n-1} V_{i+1}/V_i]$
(see [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Corollary 7.13]). Since
$\bigoplus _{i=0}^{n-1} V_{i+1}/V_i$
is isomorphic to the semisimple object
$A$
(Remark 2.1), it follows that
$V$
and
$A$
lie on the same connected component of
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
.
To show that
$(\mathcal{M}_{\mathcal{C}})_\alpha$
is quasi-compact, we pick any quasi-compact open substack
${\mathcal{U}} \subset (\mathcal{M}_{\mathcal{C}})_\alpha$
that contains the semisimple object
$A$
. We will show that
${\mathcal{U}} = (\mathcal{M}_{\mathcal{C}})_\alpha$
by proving that
$\mathcal{U}$
contains all
$k$
-points. Indeed, take any
$k$
-point of
$(\mathcal{M})_{\alpha }(k)$
corresponding to an element
$V$
. Let
$f:\Theta _k \to (\mathcal{M})_{\alpha }$
be a Hölder degeneration
$f:\Theta _k \to (\mathcal{M})_{\alpha }$
as in the previous paragraph. Then the preimage
$f^{-1}({\mathcal{U}})$
is an open substack of
$\Theta _k$
containing
$0$
(since
$A \in {\mathcal{U}}(k)$
). This forces
$f^{-1}({\mathcal{U}}) = \Theta _k$
, and hence we conclude that
$[V] = f(1)$
belongs to
$\mathcal{U}$
, as desired.
4. Moduli spaces
For this section, we keep the same notation and assumptions as in § 3. We assume in addition that the characteristic of
$k$
is
$0$
.
4.1 Good moduli space for
$\mathcal{M}_{{\mathcal{C}}}$
Lemma 4.1. Let
$\alpha$
be an effective G-theory class. Then
$(\mathcal{M}_{\mathcal{C}})_\alpha$
admits a good moduli space
$(M_{\mathcal{C}})_\alpha$
which is proper over
$k$
.
Proof.
We use [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Theorem A]. The base field
$k$
has characteristic
$0$
, and
$(\mathcal{M}_{\mathcal{C}})_\alpha$
is of finite type (Proposition 3.11),
$\Theta$
-reductive, S-complete (Lemma 3.10(ii)), and satisfies the existence part of the valuative criterion for properness (Lemma 3.10(iii)).
We denote by
$M_{{\mathcal{C}}}:= \bigsqcup _{\alpha \in G^{\mathrm{ eff}}({\mathcal{C}})} (M_{\mathcal{C}})_\alpha$
the algebraic space given by the disjoint union of
$(M_{\mathcal{C}})_\alpha$
.
Theorem 4.2. With notation as above:
-
(i)
$\mathcal{M}_{{\mathcal{C}}} \to M_{{\mathcal{C}}}$
is a good moduli space; -
(ii) there is a bijection between the set of effective G-theory classes in
$G^{\mathrm{ eff}}({\mathcal{C}})$
and the set
$M_{{\mathcal{C}}}(k)$
; and -
(iii) for any
$\alpha \in G^{\mathrm{ eff}} ({\mathcal{C}})$
, the moduli space of the open and closed substack
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
is a local finite artinian scheme over
$\mathrm{Spec}(k)$
.
Proof.
Since the statement of part (i) can be checked Zariski locally on the target [Reference AlperAlp13, Proposition 4.7(ii)], it is enough to recall that
$\mathcal{M}_{\mathcal{C}}$
is the disjoint union of its connected components
$(\mathcal{M}_{\mathcal{C}})_\alpha$
(Proposition 3.11), each of which admits a good moduli space by Lemma 4.1.
By [Reference AlperAlp13, Proposition 9.1] the
$k$
-points of
$M_{{\mathcal{C}}}$
are in bijection with the 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 the isomorphism classes of semisimple objects in
$\mathcal{C}$
. Note that every effective G-theory class
$[A] \in \mathrm{G}^{\mathrm{ eff}} ({\mathcal{C}})$
has a unique semisimple representative. Therefore, we have a bijection between the effective G-theory classes in
$\mathrm{G}^{\mathrm{ eff}} ({\mathcal{C}})$
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
$\mathrm{Spec}(k)$
.
Remark 4.3. Since the formation of moduli spaces commutes with base-change [Reference AlperAlp13, Proposition 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
$\mathrm{G}_0 ({\mathcal{C}}) = \mathrm{G}_0 (({\mathcal{A}}_K)^{\mathrm{ fp}})$
.
Example 4.4. When
$\mathcal{A}$
is the category of representations of an acyclic quiver, then each
$(M_{\mathcal{A}})_\alpha \cong \mathrm{Spec} k$
, see e.g. [Reference Belmans, Damiolini, Franzen, Hoskins, Makarova and TajakkaBDF+22]. However, this will not be true in general. For example, take
${\mathcal{A}} = k[\varepsilon ]\mathrm{ -}\textrm {Mod}$
, where
$\varepsilon ^2 = 0$
. Then
$\mathrm{G}_0({\mathcal{A}}^{\mathrm{ fp}}) \cong \mathbb{Z}$
, where
$[M]$
is identified with
$\dim M \in \mathbb{Z}$
. Taking
$\alpha = 1 \in \mathbb{Z}$
, we get
$(\mathcal{M}_{\mathcal{A}})_\alpha \cong \mathrm{Spec} k[\varepsilon ] \times B\mathbb{G}_{\mathrm{ m}}$
, and so
$(M_{\mathcal{A}})_\alpha \cong \mathrm{Spec} k[\varepsilon ]$
.
Remark 4.5. In view of Theorem 4.2, one can apply [Reference Ibáñez NúñezInNn24, Theorem 7.4.10] to conclude that for all
$\alpha \in G^{\mathrm{ eff}}$
the moduli stack
$\mathcal{M}_{\alpha }$
embeds as a closed substack of a stack of representations of a quiver. We note that the conclusion that the stack is linearly lit from [Reference Ibáñez NúñezInNn24, Theorem 7.4.10] does not technically guarantee the embedding if the good moduli space is not reduced, but the proof of the necessary result [Reference Ibáñez NúñezInNn24, Proposition 7.4.1] for obtaining the embedding also applies in the case when the good moduli space is the spectrum of a local artinian algebra.
In particular,
$\mathcal{M}_{\alpha }$
is isomorphic to the quotient stack of an affine scheme by an action of a product of general linear groups. We would like to stress that this GIT presentation of the stack
$\mathcal{M}_{\alpha }$
only becomes apparent a posteriori, after one proves that
$\mathcal{M}_{\alpha }$
admits a good moduli space which is a point (possibly nonreduced). Even with this quotient presentation in mind, we believe that it is conceptually clearer to approach the construction of stability conditions on
$\mathcal{M}_{\alpha }$
from the intrinsic point of view, as we do in the next subsection.
4.2 Stability of objects in
$\mathcal{C}$
Fix an effective G-theory class
$\alpha \in G^{\mathrm{ eff}} ({\mathcal{C}})$
. In this subsection we define a family of stability conditions coming from numerical invariants on
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
in the sense of [Reference Halpern-LeistnerHL21]. We first need to define some line bundles on
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
.
Definition 4.6. Let
$\beta \in \mathrm{K}_0 ({\mathcal{C}}) \cong \mathbb{Z}^{\oplus I}$
be 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
$\mathrm{ Pic}(\mathcal{M}_{{\mathcal{C}}})$
such that for every noetherian
$k$
-algebra
$R$
and every
$f \colon \mathrm{Spec}(R) \to \mathcal{M}_{{\mathcal{C}}}$
determined by an
$R$
-flat element
$A \in ({\mathcal{A}}_{R})^{\mathrm{ fp}}$
, we have
\begin{align*} f^*(\mathcal{L}_{\beta }) := \bigotimes _{i \in I}\mathrm{ det}({\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P_i \otimes R, A))^{\otimes n_i}. \end{align*}
The definition above makes sense, because each
${\mathrm{Hom}}_{{\mathcal{A}}_{R}}(P_i \otimes R, A)$
is a vector bundle on
$\mathrm{Spec}(R)$
(Lemmas 2.21 and 2.22), and
$n_i =0$
for all but finitely many
$i \in I$
.
Remark 4.7. Let
$E = \bigoplus _{w \in \mathbb{Z}} E^w$
be a graded object in
$\mathcal{C}$
, which we may equivalently think of as a morphism
$g: B(\mathbb{G}_{m})_k \to \mathcal{M}_{{\mathcal{C}}}$
. Then, the
$\mathbb{G}_m$
-weight of the pullback is given by
$\mathrm{wt}(g^*(\mathcal{L}_{\beta })) = \sum _{w\in \mathbb{Z}}w\langle \beta ,E^w\rangle$
.
Definition 4.8. Let
$\gamma = \sum _{i \in I} m_i [P_i] \in \mathrm{K}_0 ({\mathcal{C}}) \otimes _{\mathbb{Z}}\mathbb{Q}$
be a rational K-theory class such that
$m_i \geq 0$
for all
$i \in I$
. For any field
$K$
and any
$V \in ({\mathcal{A}}_{K})^{\mathrm{ fp}}$
, we define the
$\gamma$
-length of
$V$
by
\begin{align*}\ell _{\gamma }(V) := \left \langle \gamma _K, V \right \rangle \geq 0,\end{align*}
where
$\gamma _K$
denotes the base-change
$\sum _{i \in I} m_i [P_i \otimes K] \in \mathrm{K}_0(({\mathcal{A}}_K)^{\mathrm{ fp}}) \otimes _{\mathbb{Z}}\mathbb{Q}$
. Similarly for a G-theory class
$\alpha$
, we set
$\ell _{\gamma }(\alpha ) := \langle \gamma , \alpha \rangle$
.
Definition 4.9. Let
$\alpha$
be a G-theory class. 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
$a_i \gt 0$
). We say that a class
$\gamma = \sum _{i \in I} m_i [P_i] \in \mathrm{K}_0 ({\mathcal{C}})$
is
$\alpha$
-nondegenerate if
$m_i \gt 0$
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
$\ell _{\gamma }(E) \gt 0$
.
Example 4.10. 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 4.11. Let
$\beta \in \mathrm{K}_0({\mathcal{C}})$
, and let
$\gamma \in \mathrm{K}_0 ({\mathcal{C}}) \otimes _{\mathbb{Z}} \mathbb{Q}$
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
\begin{align*} \mathcal{L}_{\beta , \gamma } := \mathcal{L}_{\beta }^{\otimes \ell _{\gamma }(\alpha )} \otimes \mathcal{L}_{\gamma }^{\otimes -\langle \beta , \alpha \rangle } .\end{align*}
For the rest of this section, we fix
$\beta$
and
$\gamma$
as above.
Definition 4.12. 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
$\sigma _{\beta }(E) \leqslant \sigma _{\beta }(A)$
.
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
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }^{\beta \mathrm{ -} ss} \subset (\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
which admits a projective good moduli space over
$k$
. We would also like to develop a Harder–Narasimhan theory for this stability condition. We will accomplish this via the use of a numerical invariant in the sense of [Reference Halpern-LeistnerHL21, Definition 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
$({\mathcal{A}}_{K})^{\mathrm{ fp}}$
(cf. [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Proposition 7.12]). Using this description, we define a rational quadratic norm on graded points as in [Reference Halpern-LeistnerHL21, Definition 4.1.12].
Definition 4.13. We denote by
$b_{\gamma }$
the rational quadratic norm on graded points of
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
defined as follows. For any
$h\gt 0$
, any 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 any vector
$\vec {r} \in \mathbb{R}^h$
, we set
\begin{align*} 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 ), \end{align*}
where
$\vec {w} \cdot \vec {r} = \sum _{j=1}^h w_j r_j$
denotes the standard inner product for the two vectors
$\vec {w} = (w_j)_{j=1}^h$
and
$\vec {r} = (r_j)_{j=1}^h$
.
Definition 4.14. We define
$\mu _{\beta }$
to be the
$\mathbb{R}$
-valued numerical invariant on the stack given by
$\mu _{\beta } = \mathrm{ wt}(\mathcal{L}_{\beta , \gamma })/\sqrt {b_{\gamma }}$
.
We recall that under certain conditions, such a numerical invariant can be used to define a
$\Theta$
-stratification on the stack (cf. [Reference Halpern-LeistnerHL21, § 4.1] or [Reference Halpern-Leistner, Herrero and JonesHLFHJ24, § 2.5]).
Theorem 4.15. Fix
$\beta \in \mathrm{K}_0({\mathcal{C}})$
and
$\gamma \in \mathrm{K}_0({\mathcal{C}})\otimes _{\mathbb{Z}}\mathbb{Q}$
that is
$\alpha$
-nondegenerate.
-
(i) The numerical invariant
$\mu _{\beta }$
defines a
$\Theta$
-stratification on the stack
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }$
. -
(ii) The open
$\mu _{\beta }$
-semistable locus
$(\mathcal{M}_{{\mathcal{C}}})^{\mu _{\beta }\mathrm{ -} ss}_{\alpha }$
admits a good moduli space
$(M_{{\mathcal{C}}})_{\alpha }^{\beta \mathrm{ -} ss}$
that is projective over
$\mathrm{Spec}(k)$
. A power of the dual line bundle
$\mathcal{L}_{\beta , \gamma }^{-1}$
descends to an ample line bundle on
$(M_{{\mathcal{C}}})_{\alpha }^{\beta \mathrm{ -} ss}$
. -
(iii) The
$k$
-points of the
$\mu _{\beta }$
-semistable open locus
$(\mathcal{M}_{{\mathcal{C}}})_{\alpha }^{\mu _{\beta }\mathrm{ -} ss}$
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 Theorem 4.2. Recall that the good moduli space
$(M_{{\mathcal{C}}})_{\alpha }$
is finite over
$\mathrm{Spec}(k)$
. To conclude parts (i) and (ii), we use [Reference Halpern-LeistnerHL21, Theorem 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 part (iii). Let
$p \in (\mathcal{M}_{{\mathcal{C}}})_{\alpha }(k)$
correspond to an object
$A \in {\mathcal{C}}$
with G-theory class
$\alpha$
. By [Reference Alper, Halpern-Leistner and HeinlothAHLH23, Corollary 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
\begin{align*} \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})}}.\end{align*}
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
\begin{align*} \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})) \leqslant 0. \end{align*}
Using the additivity relations
$\langle \beta , A_{w}/A_{w+1}\rangle =\langle \beta ,A_{\omega }\rangle -\langle \beta ,A_{w+1}\rangle$
and
$\ell _{\gamma }(A_w/A_{w+1}) = \ell _{\gamma }(A_w) - \ell _{\gamma }(A_{w+1})$
as well as the identities
$\langle \beta , A \rangle = \langle \beta , \alpha \rangle$
and
$\ell _{\gamma }(A) = \ell _{\gamma }(\alpha )$
, we rewrite this as follows:
\begin{align*} \sum _{w \in \mathbb{Z}} (\ell _{\gamma }(A) \langle \beta , A_w \rangle - \ell _{\gamma }(A_w) \langle \beta , A \rangle ) \leq 0. \end{align*}
A standard argument (cf. [Reference Halpern-Leistner, Herrero and JonesHLFHJ24, Proposition 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 Definition 4.12.
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
${\mathcal{C}} = A\hbox{-} \mathrm{ fdmod}$
of finite-dimensional left modules over
$A$
is nearly finite in the sense of Definition 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.
In view of Proposition 5.1, our main results in Theorem 4.15 apply to the moduli of representations of a finite-dimensional associative
$k$
-algebra. This moduli problem and stability conditions have been considered by King [Reference KingKin94, § 4]. For comparison, we note that King is not completely explicit about the nonreduced structure of the moduli problem, opting to work with varieties instead, whereas the Artin–Zhang functor equips the moduli spaces with a natural (possibly nonreduced) scheme structure. This can, of course, also be made explicit from the point of view of [Reference KingKin94].
Example 5.2 (Representations of acyclic quivers). For the following example, we refer the reader to [Reference KingKin94] and [Reference Belmans, Damiolini, Franzen, Hoskins, Makarova and TajakkaBDF+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 Definition 1.1 by Proposition 5.1.
In this case we have
$\mathrm{G}_0 ({\mathcal{C}}) = \mathbb{Z}^{\oplus I}$
, 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
$\beta = \sum _{i \in I} n_i[P_i] \in \mathrm{K}_0({\mathcal{C}})$
and
$\gamma$
as in Example 4.10, our definition of
$\beta$
-semistability (Definition 4.12) agrees with that of King [Reference KingKin94]. Hence, we recover the moduli spaces of representations of
$Q$
constructed using GIT in [Reference KingKin94] and intrinsically in [Reference Belmans, Damiolini, Franzen, Hoskins, Makarova and TajakkaBDF+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
${\mathcal{C}} = \textrm{rep} Q$
to be the category of representations of
$Q$
whose total dimension is finite. Then
$\mathcal{C}$
is nearly finite. In this case, the connected components of
$\mathcal{M}_{{\mathcal{C}}}$
correspond to the moduli stacks of representation of finite acyclic subquivers
$W \subset Q$
with fixed dimension vector.
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
$\mathfrak{n}=\oplus _{\alpha \gt 0}\mathfrak{g}_{\alpha }$
.
Definition 5.4. The Bernstein-Gelfand-Gelfand category
$\mathcal{O}$
is the full subcategory of
$\textrm {Mod} U(\mathfrak{g})$
such that:
-
(i)
$M$
is a finitely generated
$U(\mathfrak{g})$
-module; -
(ii)
$M$
is
$\mathfrak{h}$
-semisimple, i.e.
$M=\oplus _{\lambda \in \mathfrak{h}^{\vee }}M_{\lambda }$
; and -
(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 a nearly finite abelian category.
Proof.
Since
$\mathcal{O}$
is noetherian, artinian and Hom-finite by [Reference HumphreysHum08, Theorems 1.1 and 1.11], (A0) and (A1) are satisfied. It is essentially small because simple objects are parametrized by linear functionals in
$\mathfrak{h}^{\vee }$
, by [Reference HumphreysHum08, Theorem 1.3]. Furthermore, (A2) holds by [Reference HumphreysHum08, Theorem 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
$\lambda$
[Reference HumphreysHum08, Theorem 1.3]. Each
$L(\lambda )$
has a unique indecomposable projective cover
$P(\lambda )$
[Reference HumphreysHum08, Theorem 3.9]. In this case, every object is perfect and hence the
$G$
-theory and the
$K$
-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
\begin{align*}L(\lambda ), {\quad}L(-2-\lambda )=M(-2-\lambda ),{\quad} M(\lambda )=P(\lambda ), {\quad}M(\lambda )^{\vee }, {\quad}P(-2-\lambda ).\end{align*}
For a fixed class
$\alpha$
, 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 4.10. Then we find that the
$\beta$
-slope
$\sigma _{\beta }$
is
\begin{align*}\sigma _{\beta }(L(0))=\beta _1, {\quad}\sigma _{\beta }(L(-2))=\beta _2\end{align*}
\begin{align*}\sigma _{\beta }(M(0))=\sigma _{\beta }(M(0)^{\vee })=\frac {\beta _1+\beta _2}{2}\end{align*}
\begin{align*}\sigma _{\beta }(P(-2))=\frac {\beta _1+2\beta _2}{3}.\end{align*}
It follows that
$M(0)$
is
$\beta$
-semistable if and only if
$\beta _1\geq \beta _2$
. On the other hand,
$M(0)^{\vee }$
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 [Reference Futorny, Jardim and MouraFJM08, Example 3.4].
5.3 Comodules over co-Frobenius Hopf algebras
Let
$C$
be a counital coalgebra. Set
${\mathcal{C}} = H\mathrm{ -}\mathrm{ fdcomod}$
, so
${\mathcal{A}} = H\mathrm{ -}\mathrm{ Comod} = \mathrm{Ind} ({\mathcal{C}})$
by the fundamental theorem of coalgebra. The category
$\mathcal{C}$
automatically satisfies (A0) and (A1). For the fact that
$\mathcal{C}$
is essentially small, we use that simple comodules are in bijection with simple subcoalgebras of
$H$
[Reference LarsonLar71, p. 354].
Theorem 5.7 [Reference LinLin77, Theorems 3 and 10]. The following are equivalent for a Hopf algebra
$H$
:
-
(i)
$H$
is co-Frobenius; and
-
(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
${\mathcal{C}} = H\mathrm{ -}\mathrm{ fdcomod}$
is nearly finite.
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 Theorem 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 is 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 [Reference KingKin94]; see § 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 [Reference Andruskiewitsch and DăscălescuAD03, 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, and hence there will necessarily be infinitely many simple comodules.
Acknowledgements
We would like to thank Pieter Belmans, Pavel Etingof, Dan Halpern-Leistner, Andrés Ibáñez Núñez, Kimoi Kemboi, Andrew Kwon, Tony Pantev, Maximilien Péroux, Alekos Robotis and David Rydh for their interest in the project and helpful discussions. The authors also thank the anonymous referees for their thoughtful comments and attention to detail. In particular, Proposition 3.11 (which greatly simplifies our original argument) and Remark 4.5 were suggested by a referee.
Conflicts of Interest
None.
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