Tame comodule type, roiter bocses, and a geometry context for coalgebras
We study the class of coalgebras $C$ of $fc$-tame comodule type introduced by the author. With any basic computable $K$-coalgebra $C$ and a bipartite vector $v = (v′|v″) ∈ K_0(C) × K_0(C)$, we associate a bimodule matrix problem $\textbf{Mat}^v_C(ℍ)$, an additive Roiter bocs $\textbf{B}^C_v$, an aff...
Saved in:
| Date: | 2009 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3060 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509086070603776 |
|---|---|
| author | Simson, D. Сімсон, Д. |
| author_facet | Simson, D. Сімсон, Д. |
| author_sort | Simson, D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:24Z |
| description | We study the class of coalgebras $C$ of $fc$-tame comodule type introduced by the author. With any basic computable $K$-coalgebra $C$ and a bipartite vector $v = (v′|v″) ∈ K_0(C) × K_0(C)$, we associate a bimodule matrix problem $\textbf{Mat}^v_C(ℍ)$, an additive Roiter bocs $\textbf{B}^C_v$, an affine algebraic $K$-variety $\textbf{Comod}^C_v$, and an algebraic group action $\textbf{G}^C_v × \textbf{Comod}^C_v → \textbf{Comod}^C_v$.
We study the $fc$-tame comodule type and the fc-wild comodule type of $C$ by means of $\textbf{Mat}^v_C(ℍ)$, the category $\textbf{rep}_K (\textbf{B}^C_v)$ of $K$-linear representations of $\textbf{B}^C_v$, and geometry of $\textbf{G}^C_v$ -orbits of $\textbf{Comod}_v$.
For computable coalgebras $C$ over an algebraically closed field $K$, we give an alternative proof of the $fc$-tame-wild dichotomy theorem. A characterization of $fc$-tameness of $C$ is given in terms of geometry of $\textbf{G}^C_v$-orbits of $\textbf{Comod}^C_v$. In particular, we show that $C$ is $fc$-tame of discrete comodule type if and only if the number of $\textbf{G}^C_v$-orbits in $\textbf{Comod}^C_v$ is finite for every $v = (v′|v″) ∈ K_0(C) × K_0(C)$. |
| first_indexed | 2026-03-24T02:35:30Z |
| format | Article |
| fulltext |
UDC 512.5
D. Simson (Nicolaus Copernicus Univ., Toruń, Poland)
TAME COMODULE TYPE, ROITER BOCSES,
AND A GEOMETRY CONTEXT FOR COALGEBRAS*
РУЧНИЙ КОМОДУЛЬНИЙ ТИП, БОКСИ РОЙТЕРА
I ГЕОМЕТРИЧНИЙ КОНТЕКСТ ДЛЯ КОАЛГЕБР
Dedicated to the memory of Andrey Vladimirovich Roiter
We study the class of coalgebras C of fc-tame comodule type introduced by the author. To any basic
computable K-coalgebra C and a bipartite vector v = (v′|v′′) ∈ K0(C)×K0(C), we associate a bimodule
matrix problem Matv
C(H), an additive Roiter bocs BC
v , an affine algebraic K-variety ComodC
v , and an
algebraic group action GC
v × ComodC
v −→ ComodC
v . We study the fc-tame comodule type and the
fc-wild comodule type of C by means of Matv
C(H), the category repK(BC
v ) of K-linear representations
of BC
v , and geometry of GC
v -orbits of ComodC
v . For computable coalgebras C over an algebraically
closed field K, we give an alternative proof of the fc-tame-wild dichotomy theorem. A characterisation of
fc-tameness of C is given in terms of geometry of GC
v -orbits of Comodv . In particular, we show that C is
fc-tame of discrete comodule type if and only if the number of GC
v -orbits in ComodC
v is finite, for every
v = (v′|v′′) ∈ K0(C)×K0(C).
Вивчено клас коалгебр C fc-ручного комодульного типу, що введений автором. Кожну базову злiченну
K-коалгебру C та дводольний вектор v = (v′|v′′) ∈ K0(C) ×K0(C) пов’язано з бiмодульною мат-
ричною задачею Matv
C(H), адитивними боксами Ройтера BC
v , афiнним алгебраїчним K-рiзновидом
ComodC
v та алгебраїчним груповим оператором GC
v ×ComodC
v −→ ComodC
v . Дослiдження fc-
ручного та fc-дикого комодульних типiв C проведено з використанням Matv
C(H), категорiї repK(BC
v )
K-лiнiйних зображень BC
v та геометрiї GC
v -орбiт ComodC
v . Для злiченних коалгебр C над алгебра-
їчно замкненим полем K наведено альтернативне доведення теореми про fc-ручну дику дихотомiю.
Характеризацiю fc-ручної властивостi для C подано через геометрiю GC
v -орбiт Comodv . Показано,
зокрема, що C належить до fc-ручного дискретного комодульного типу тодi i тiльки тодi, коли кiлькiсть
GC
v -орбiт в ComodC
v скiнченна для кожного v = (v′|v′′) ∈ K0(C)×K0(C).
1. Introduction. Throughout this paper, we use the terminology and notation introduced
in [21, 22, 28]. We fix a field K. Given a K-coalgebra C, we denote by C-Comod
and C-comod the categories of left C-comodules and left C-comodules of finite K-
dimension. We recall that C is said to be basic if the left C-comodule CC has a
decomposition
CC =
⊕
j∈IC
E(j) (1.1)
into a direct sum of pairwise non-isomorphic indecomposable injective left comodules
E(j). Throughout this paper, given j ∈ IC , we denote by S(j) the unique simple
subcomodule of E(j). Hence, socC =
⊕
j∈IC S(j). Following [26], the coalgebra C is
called Hom-computable (or computable, in short) if dimK HomC(E(i), E(j)) is finite,
for all i, j ∈ IC .A leftC-comoduleM is said to be computable if dimK HomC(M,E(j))
is finite, for all j ∈ IC .
Given a computable comodule M, we denote by lgthM = (`j(M))j∈IC ∈ ZIC
the composition length vector of M, where `j(M) < ∞ is the number of simple
composition factors of M isomorphic to the simple comodule S(j). It is clear that
lgthM ∈ Z(IC), if M is of finite K-dimension. We recall from [21] that the map
*Supported by Polish Research Grant 1 P03A № N201/2692/ 35/2008-2011.
c© D. SIMSON, 2009
810 ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 811
M 7→ lgthM defines a group isomorphism lgth : K0(C) '−−→ Z(IC), where K0(C) =
= K0(C-comod) is the Grothendieck group of the category C-comod and Z(IC) is the
direct sum of IC copies of Z.
We recall from [21] and [25] that an arbitrary K-coalgebra C is defined to be
of K-wild comodule type (or K-wild, in short), if the category C-comod of finite
dimensional C-comodules is of K-wild representation type [18, 21, 23] in the sense that
there exists an exact K-linear representation embedding T : modΓ3(K) −→ C-comod,
where Γ3(K) =
(
K K3
0 K
)
. A K-coalgebra C is defined to be of K-tame comodule
type [25] (or K-tame, in short), if the category C-comod of finite dimensional left
C-comodules is of K-tame representation type ([18], Section 14.4, [22]), that is, for
every vector v ∈ K0(C) ∼= Z(IC), there exist C-K[t]-bicomodules L(1), . . . , L(rv), that
are finitely generated free K[t]-modules, such that all but finitely many indecomposable
left C-comodules M with lgthM = v are of the form M ∼= L(s) ⊗K1
λ, where s ≤ rv
and
K1
λ = K[t]/(t− λ), λ ∈ K. (1.2)
Equivalently, there exist a non-zero polynomial h(t) ∈ K[t] and C-K[t]h-bicomodules
L(1), . . . , L(rv), that are finitely generated free K[t]h-modules, such that all but finitely
many indecomposable left C-comodules M with lgthM = v are of the form M ∼=
∼= L(s)⊗K1
λ, where s ≤ rv and K[t]h = K[t, h(t)−1] is a rational K-algebra, see [7] or
[18] (Section 14.4). In this case, we say that L(1), . . . , L(rv) form an almost parametri-
sing family for the family indv(C-comod) of all indecomposable C-comodules M with
lgthM = v.
Here, by a C-K[t]h-bicomodule CLK[t]h we mean a K-vector space L equipped
with a left C-comodule structure and a right K[t]h-module structure satisfying the
obvious associativity conditions. In [28], a K-tame-wild dichotomy theorem is proved
for left (or right) semiperfect coalgebras and for acyclic hereditary coalgebras over an
algebraically closed field K by reducing the problem to the fc-tame-wild dichotomy
theorem [28] (Theorem 2.11) and, consequently, to the tame-wild dichotomy theorem
for finite dimensional K-algebras proved in [7] and [3].
The aim of the paper is to study the classes of coalgebras C of fc-tame comodule
type and of fc-wild comodule type introduced in [28]. We recall that C is of of fc-tame
comodule type if, for every coordinate vector v = (v′|v′′) ∈ K0(C) × K0(C), the
indecomposable finitely copresented C-comodules N such that cdn(N) = (v′|v′′) form
at most finitely many one-parameter families, see Section 2 for a precise definition.
We study mainly computable fc-tame and fc-wild basic coalgebras C by means of
a bimodule matrix problem MatvC(H), the additive category repK(BC
v ) of K-linear
representations an additive Roiter bocs BC
v , an affine algebraic K-variety MapCv , an
algebraic (parabolic) group action GC
v ×MapCv −→ MapCv , and a Zariski open GC
v -
invariant subset ComodCv ⊆ MapCv , associated to C and to any bipartite vector
v = (v′|v′′) ∈ K0(C)×K0(C). It is shown in Section 4 that there is a bijection between
the GC
v -orbits of ComodCv and the isomorphism classes of comodules in C-Comodfc.
On this way, we get in Theorem 4.1 a characterisation of fc-tameness and fc-wildness
of computable colagebras by means of MatvC(H), the K-linear representations of the
Roiter bocs BC
v , and in terms of geometry of the GC
v -orbits of ComodCv .
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
812 D. SIMSON
We show in Section 4 that a computable colagebra C is fc-tame of discrete
comodule type if and only if the number of GC
v -orbits in ComodCv is finite, for
every bipartite vector v = (v′|v′′) ∈ K0(C) × K0(C). Moreover, we prove that a
computable colagebra C is fc-tame if and only if, for every bipartite vector v =
= (v′|v′′) ∈ K0(C)×K0(C), there exists a constructible subset C(v) of the constructi-
ble set indComodCv ⊆ ComodCv (defined by the indecomposable C-comodules) such
that GC
v ∗ C(v) = indComodCv and dim C(v) ≤ 1, see Theorem 4.1.
We also give an alternative proof of the following fc-tame-wild dichotomy theorem
proved in [28]: If C is a basic computable coalgebra over an algebraically closed field
K then C is either fc-tame or fc-wild, and these two types are mutually exclusive.
We prove it in Section 3 by a reduction to the tame-wild dichotomy theorem of Drozd
[7] for representations of additive Roiter bocses, by applying the bimodule problems
technique introduced in [5] and developed in [3, 4, 9, 17, 19, 20].
Throughout this paper we freely use the coalgebra representation theory notation and
terminology introduced in [2, 16, 21, 22, 28]. The reader is referred to [1, 8, 10, 18] for
representation theory terminology and notation, and to [3, 4, 7, 9, 13] for a background
on the representation theory of bocses.
In particular, given a ring R with an identity element, we denote by Mod(R) the
category of all unitary right R-modules, and by mod(R) ⊇ fin(R) the full subcategories
of Mod(R) formed by the finitely generated R-modules and the finite dimensional R-
modules, respectively. Given a K-coalgebra C and a left C-comodule M, we denote by
socM the socle of M, that is, the sum of all simple C-subcomodules of M.
A comoduleN inC-Comod is said to be socle-finite ifN is a subcomodule of a finite
direct sum of indecomposable injective comodules, or equivalently, dimK socN is finite.
We say that N is finitely copresented if N admits a socle-finite injective copresentation,
that is, an exact sequence 0 −→ N −→ E0
ψ−→ E1 in C-Comod, where each of the
comodules E0 and E1 is a finite direct sum of indecomposable injective comodules.
If E0, E1 ∈ add(E), for some socle-finite injective C-comodule E, the comodule N
is called finitely E-copresented. We denote by C-Comodfc ⊇ C-ComodEfc the full
subcategories of C-Comod whose objects are the finitely copresented comodules and
finitely E-copresented comodules, respectively. Here by add(E) we mean the full addi-
tive subcategory of C-Comod whose objects are finite direct sum of indecomposable
injective comodules isomorphic to direct summands of E.
2. Preliminaries on fc-comodule types for coalgebras. Throughout we assume
that K is an algebraically closed field and C is a basic K-coalgebra with a fixed
decomposition (1.1). Following [28], given a finitely copresented C-comodule N in
C-Comodfc, with a minimal injective copresentation 0 −→ N −→ EN0
g−→ EN1 , we
define the coordinate vector of N to be the bipartite vector
cdn(N) = (cdnN0 | cdnN1 ) ∈ K0(C)×K0(C) = Z(IC) × Z(IC), (2.1)
where cdnN0 = lgth(socEN0 ) and cdnN1 = lgth(socEN1 ). We call a bipartite vector
v = (v′|v′′) ∈ Z(IC) × Z(IC) proper if v′ 6= 0 and v′′ has non-negative coordinates.
Note that an indecomposable comodule N in C-Comodfc is injective if and only if the
vector cdn(N) is proper and has the form v = (ej |v′′), where v′′ = 0 and ej is the jth
standard basis vector of Z(IC), for some j ∈ IC .
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 813
The support of a bipartite vector v = (v′|v′′) ∈ Z(IC) × Z(IC) is the finite subset
supp(v) = {j ∈ IC ; v′j 6= 0 or v′′j 6= 0} of IC .
We recall from [28] that K-coalgebra C is defined to be of fc-wild comodule
type (or fc-wild, in short), if the category C-Comodfc of finitely copresented C-
comodules is of K-wild representation type [18, 23, 25] in the sense that there exists
an exact K-linear representation embedding T : modΓ3(K) −→ C-Comodfc, where
Γ3(K) =
[
K K3
0 K
]
.
A C-K[t]h-bicomodule CLK[t]h is defined to be finitely copresented if there is a
C-K[t]h-bicomodule exact sequence 0→ CLK[t]h → E′⊗K[t]h
ψ−→ E′′⊗K[t]h, such
that E′, E′′ are socle-finite injective C-comodules. If E′, E′′ are finitely E-copresented,
we call CLK[t]h finitely E-copresented.
A K-coalgebra C is defined to be of fc-tame comodule type (or fc-tame, in short),
if the category C-Comodfc is of fc-tame representation type [18] (Section 14.4), that is,
for every bipartite vector v = (v′|v′′) ∈ K0(C) ×K0(C) ∼= Z(IC) × Z(IC), there exist
C-K[t]h-bicomodules L(1), . . . , L(rv), that are finitely copresented, such that all but
finitely many indecomposable left C-comodules N in C-Comodfc, with cdn(N) = v,
are of the form N ∼= L(s) ⊗K1
λ, where s ≤ rv,
K1
λ = K[t]/(t− λ),
and λ ∈ K. In this case, we say that L(1), . . . , L(rv) is a finitely copresented almost
parametrising family for the family indv(C-Comodfc) of all indecomposable C-
comodules N with cdn(N) = v. Obviously, one can restrict the definition to proper
bipartite vectors v = (v′|v′′).
We recall from [28] that the growth function µ̂1
C : K0(C)×K0(C) −−−→ N of C
associates to any bipartite vector v = (v′|v′′) ∈ K0(C) ×K0(C), the minimal number
µ̂1
C(v) = rv ≥ 1 of non-zero finitely copresented C-K[t]h-bicomodules L(1), . . . , L(rv)
forming an almost parametrising family for indv(C-Comodfc).We set µ̂1
C(v) = rv = 0,
if there is no such a family of bicomodules, that is, there is only a finite number of
comodules N in indv(C-Comodfc), up to isomorphism.
An fc-tame coalgebra C is defined to be of fc-discrete comodule type if µ̂1
C = 0,
that is, the number of the isomorphism classes of the indecomposable C-comodules N
in C-Comodfc with cdn(N) = v is finite, for every bipartite vector v = (v′|v′′) ∈
∈ K0(C)×K0(C).
By the main result in [28], the definition is left-right symmetric, for any computable
coalgebra C. Note also that the K-tameness and K-wildness of a coalgebra are defined
by means of finite dimensional comodules, but the fc-tame comodule type and fc-wild
comodule type are defined by means of the category C-Comodfc of finitely copresented
comodules that usually contains a lot of infinite dimensional comodules.
In the proof of our main results, we need the following construction that associates to
any v = (v′|v′′) ∈ K0(C) ×K0(C) and any finitely copresented C-K[t]h-bicomodule
CLK[t]h a new one CL̃K[t]h , called fc-localising v-corrected C-K[t]h-bicomodule.
Construction 2.1. Let C be a basic K-coalgebra with a decomposition (1.1), and
let v = (v′|v′′) ∈ K0(C)×K0(C) = Z(IC) × Z(IC) be a proper bipartite vector.
Let Uv = supp(v) ⊆ IC be the support of v = (v′|v′′). We call the socle-finite
injective C-comodules
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
814 D. SIMSON
E(v′) =
⊕
i∈IC
E(i)v
′
i and E(v′′) =
⊕
j∈IC
E(j)v
′′
j (2.2)
the standard injective C-comodules with cdnE(v′) = (v′|0) and cdnE(v′′) = (v′′|0).
We fix a rational K-algebra S = K[t]h and note that
Ev = EUv =
⊕
a∈Uv
E(a) (2.3)
is a socle-finite injective direct summand of CC.
Assume that CLS is a finitely copresented C-S-bicomodule with a fixed injective
C-S-bicomodule copresentation
0 −→ CLS −→ E0 ⊗ S
ψ−→ E1 ⊗ S (2.4)
where E0, E1 are socle-finite injective comodules such that E(v′) ⊆ E0 and E(v′′) ⊆
⊆ E1.
We construct in three steps a finitely Ev-copresented C-S-bicomodule CL̃S , called
a localising fc-correction of CLS as follows.
Step 1◦. Fix a decomposition E0 = E′0 ⊕ E′′0 , where E′0 is the injective envelope
of the semisimple subcomodule S(v) generated by the simple subcomodules of E0 that
are isomorphic to S(j), with j ∈ Uv. Obviously, every simple subcomodule S of E′′0
has the form S ∼= S(a), where a 6∈ Uv.
Step 2◦. Define a C-S-subbicomodule CL
′
S of CLS to be the kernel of the
composite C-S-bicomodule homomorhism E′0 ⊗ S
u′0⊗S−→ E0 ⊗ S
ψ−→ E1 ⊗ S, where
u′0 : E′0 ↪→ E0 is the canonical embedding.
Step 3◦. Let ev : C → K be the idempotent of the algebra C∗ = HomK(C,K)
defined by the direct summand Ev of CC. An fc-localising correction of CLS is the
C-S-bicomodule
CL̃S = evC�evCev [resEv (CL
′)S ], (2.5)
where resEv : C-Comodfc −→ evCev-Comodfc is the exact restriction functor and
evC�evCev (−) : evCev-Comodfc −→ C-Comodfc is the left exact cotensor product
functor defined in [11] and [25] ((2.9), see also [29]).
The following fc-localising correction lemma is of importance.
Lemma 2.1. Let K be an algebraically closed field, C a basic K-coalgebra with
the decomposition (1.1), v = (v′|v′′) ∈ K0(C) × K0(C) = Z(IC) × Z(IC) a proper
bipartite vector, S = K[t]h, and CLS a finitely copresented C-S-bicomodule with a
fixed injective C-S-bicomodule copresentation (2.4) as in Construction 2.1.
(a) The C-S-bicomodule CL̃S (2.5) has an injective C-S-bicomodule copresentation
0 −→ CL̃S −→ Ẽ0 ⊗ S
ψ̃−→ Ẽ1 ⊗ S (2.6)
and the comodules Ẽ0 = E′0, Ẽ1 lie in add(EUv ).
(b) If N is an indecomposable comodule in C-Comodfc such that cdn(N) = v and
N ∼= CLS ⊗ K1
λ, with λ ∈ K, then the restriction û′0 : CL′S ↪→ CLS of the splitting
monomorphism u′0⊗S : E′0⊗S ↪→ E0⊗S to CL
′
S is an embedding of C-S-bicomodules
and induces isomorphisms CL̃S ⊗K1
λ
∼= CL
′
S ⊗K1
λ
∼= N of C-comodules.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 815
Proof. (a) By the construction, there are a decomposition E0 = E′0 ⊕E′′0 and exact
sequence
0 −→ CL
′
S −→ E′0 ⊗ S
ψ′
−→ E1 ⊗ S
of C-S-bicomodules, where ψ′ = ψ◦(u′0⊗S) and u′0 = (idE′
0
, 0) : E′0 ↪→ E0 = E′0⊕E′′0
is the canonical embedding into the direct summand E′0 of E0. We recall from [11] and
[25] (Section 2) that the restriction functor resEv : C-Comodfc −→ evCev-Comodfc
is exact and the cotensor product functor evC�evCev (−) : evCev-Comodfc −→
−→ C-Comodfc is left exact. Then we derive an exact sequence
0 −→ CL̃S −→ Ẽ0 ⊗ S
ψ′
−→ E∨1 ⊗ S
of C-S-bicomodules, where Ẽ0 = evC�evCev resEv (E
′
0) and
E∨1 = evC�evCev resEv (E1).
Since E0 is a direct summand of EUv , then by [11] and [25] (Proposition 2.7
and Theorem 2.10), there is an isomorphism Ẽ0
∼= E0, the socle of resEv (E1) is
a finite dimensional subcomodule of the coalgebra evCev and the socle of E∨1 =
= evC�evCev resEv (E1) is a finite direct sum of comodules S(a), with a ∈ Uv. It
follows that the injective envelope Ẽ1 = EC(E∨1 ) of the C-comodule E∨1 lies in
add(EUv ). Hence we get the exact sequence (2.6) and (a) follows.
(b) The canonical embedding u′0 = (idE′
0
, 0) : E′0 ↪→ E0 = E′0 ⊕ E′′0 into the direct
summand E′0 of E0 induces the commutative diagram of C-S-bicomodules
0 −→ CLS −→ (E′0 ⊕ E′′0 )⊗ S ψ−→ E1 ⊗ S
û′0
x u′0⊗S
x idE1⊗S
x
0 −→ CL
′
S −→ E′0 ⊗ S
ψ′
−→ E1 ⊗ S
with exact rows, where û′0 is the restriction of the monomorphism u′0 ⊗ S : E′0 ⊗ S ↪→
↪→ E0 ⊗ S to CL
′
S . Obviously, û′0 is an embedding of C-S-bicomodules.
Let N be an indecomposable comodule in C-Comodfc such that cdn(N) = v =
= (v′|v′′) and N ∼= CLS ⊗ K1
λ, with λ ∈ K. Then N has a minimal injective
copresentation 0 −→ N −→ E(v′)
g−→ E(v′′). Recall that cdnE(v′) = (v′|0) and
cdnE(v′′) = (v′′|0). Then we get a commutative diagram of C-comodules in
C-Comodfc
0 −→ N −→ E(v′)
g−→ E(v′′)y∼= f0
y f1
y
0 −→ CL⊗S K1
λ −→ (E′0 ⊕ E′′0 )⊗K1
λ
ψ⊗id−→ E1 ⊗K1
λ
û′0
x u′0⊗id
x id
x
0 −→ CL
′ ⊗S K1
λ −→ E′0 ⊗K1
λ
ψ′
−→ E1 ⊗K1
λ
with exact rows. Since the upper row is a minimal injective copresentation of N, then f0
and f1 are monomorphisms, and f0 has a factorisation E(v′)
f ′0−→ E′0⊗K1
λ
u′0⊗S−→ E′0⊗K1
λ
through the subcomodule E′0⊗K1
λ of E0⊗K1
λ, because the socle of E′′0 ⊗K1
λ contains
no simple comodules S(a), with a ∈ Uv. It follows that f ′0 restricts to a monomorphism
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
816 D. SIMSON
f̂ ′0 : N → CL
′⊗SK1
λ such that the composite mapN
f̂0−→ CL
′⊗SK1
λ
û′0−→ CL⊗SK1
λ is
an isomorphism. Consequently, f̂ ′0 : N → CL
′⊗SK1
λ is an isomorphism ofC-comodules.
Hence, in the notation of Construction 2.1, we get the isomorphisms
CL⊗S K1
λ = [evC�evCev resEv (CL
′)]⊗S K1
λ
∼=
∼= evC�evCev [resEv (CL
′ ⊗S K1
λ)] ∼= evC�evCev [resEv (N)] ∼= N
of C-comodules, because N is finitely EUv -copresented and [25] (Theorem 2.10 (d))
applies to N.
The lemma is proved.
3. fc-Tameness, fc-wildness and Roiter bocses for coalgebras. We show in this
section how the study of fc-tame and fc-wild coalgebras can be reduced to the study
of bimodule matrix problems in the sense of Drozd [5], to representations of additive
Roiter bocses [3 – 7], and to the study of propartite modules over a class of bipartite
algebras [19, 20].
To formulate our main results on fc-tame and fc-wild computable coalgebras, we
recall some notation, see [25] and [26]. Given a socle-finite injective direct summand
E = EU =
⊕
u∈U
E(u) (3.1)
of CC =
⊕
j∈IC E(j), with a finite subset U of IC , we define the category C-ComodEUfc
to be fc-tame if for every bipartite vector v = (v′|v′′) ∈ ZU × ZU , there is a finitely
E-copresented almost parametrising family for indv(C-ComodEUfc ).
We start with the following fc-parametrisation correction lemma.
Lemma 3.1. Let K be an algebraically closed field, C a basic K-coalgebra with
the decomposition (1.1), and E = EU a socle-finite injective direct summand (3.1)
of CC.
(a) If C is fc-tame then the category C-ComodEUfc is fc-tame.
(b) If v = (v′|v′′) ∈ K0(C)×K0(C) = Z(IC) × Z(IC) is a proper bipartite vector,
S = K[t]h, and L(1), . . . , L(rv) is a finitely copresented almost parametrising fami-
ly of C-S-bicomodules for indv(C-ComodEUfc ) then the fc-localising v-corrected C-
S-bicomodules L̃(1), . . . , L̃(rv) in the sense of Construction 2.1 form a finitely EU -
copresented almost parametrising family for indv(C-ComodEUfc ).
Proof. It is sufficient to prove (b), because (a) is a direct consequence of (b).
Assume that v = (v′|v′′) ∈ K0(C) × K0(C) = Z(IC) × Z(IC) is a proper biparti-
te vector and L(1), . . . , L(rv) is a finitely copresented almost parametrising family for
indv(C-ComodEUfc ). Assume that rv ≥ 0 is a minimal number of such non-zero bi-
modules. If rv = 0 then there is nothing to prove, because the number of the isomorphism
classes of indecomposable comodules in indv(C-ComodEUfc ) is finite.
Assume that rv ≥ 1. Then, for each 1 ≤ j ≤ rv, there is an indecomposable
comodule N such that cdn(N) = v and N ∼= L(j)⊗S K1
λ(j), for some λ(j) ∈ K. Then
N has a minimal injective copresentation 0 −→ N −→ E(v′)
g−→ E(v′).
Since CL
(j)
S is a finitely copresented C-S-bicomodule then it has an injective C-S-
bicomodule copresentation
0 −→ CL
(j)
S −→ E
(j)
0 ⊗ S
ψ(j)
−→ E
(j)
1 ⊗ S
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 817
where E(j)
0 , E
(j)
1 are socle-finite injective C-comodules. Since N ∼= L(j)⊗S K1
λ(j) then
there are C-comodule monomorphisms E(v′) ⊆ E0 and E(v′′) ⊆ E1, because the
sequence
0 −→ CL
(j) ⊗S K1
λ(j) −→ E
(j)
0 ⊗K1
λ(j)
ψ̂(j)
−→ E
(j)
1 ⊗K1
λ(j)
induced by the previous one is exact and is a socle-finite injective copresentation of
N ∼= L(j) ⊗S K1
λ(j). Then the Construction 2.1 applies to CL
(j)
S , for j = 1, . . . , rv.
By applying Lemma 2.1 to the finitely copresented C-S-bicomodule L(j) we get a
finitely EU -copresented C-S-bicomodule L̃(j) such that the fc-localising v-corrected
C-S-bicomodules L̃(1), . . . , L̃(rv) form a finitely EU -copresented almost parametrising
family for indv(C-ComodEUfc ).
The lemma is proved.
Following [25, 26, 28] given a socle-finite injective direct summand E = EU (3.1),
we consider the K-algebra
RE = EndCE =
⊕
u∈U
euRE , (3.2)
where euRE = HomC(E,E(u)) is viewed as an indecomposable projective right ideal
of RE and eu is the primitive idempotent of RE defined by the summand E(u) of E.
Since the set U is finite then
∑
u∈U
eu is the identity of RE . It is easy to see that the
Jacobson radical J(RE) of RE has the form J(RE) = {h ∈ EndCE;h(socE) = 0}. It
follows that the algebra RE is semiperfect and pseudocompact with respect to the K-
linear topology defined by the left ideals aβ = HomC(E/Vβ , E) ⊆ RE , where {Vβ}β
is the directed set of all finite dimensional subcomodules of E. Since E =
⋃
β Vβ , then
there are isomorphisms
RE = EndCE ∼= lim
←−β
HomC(Vβ , E) ∼= lim
←−β
RE/aβ . (3.3)
Following [3, 7, 28], we consider the homomorphism category Map1(E) whose
objects are the triples (E0, E1, ψ) with E0, E1 comodules in add(E) and ψ : E0 −→ E1
a homomorphism of C-comodules such that ψ(socE0) = 0; and whose morphisms are
the pairs (f0, f1), where f0 : E0 −→ E′0, f1 : E1 −→ E′1 and ψ′◦f0 = f1◦ψ. Denote by
Map2(E) the full subcategory of Map1(E) whose objects are the triples (E0, E1, ψ)
such that soc Imψ = socE1. or equivalently, ψ : E0 −→ E1 has no non-zero direct
summand of the form 0 −→ E′′. We define the coordinate vector of (E0, E1, ψ) to be
the bipartite vector
cdn(E0, E1, ψ) = (lgth(socE0)|lgth(socE1)) ∈ ZU × ZU = K0(RE)×K0(RE).
(3.4)
Following [7], [3] (Section 6) and [28], we denote by P1(R
op
E ) the category whose
objects are the triples (P1, P0, φ) with P0, P1 finitely generated projective left RE-
modules and φ : P1 −→ rad(P0) = P0J(RE) a homomorphism of left RE-modules;
and whose morphisms are the pairs (g1, g0), where g0 : P0 −→ P ′0, g1 : P1 −→ P ′1 and
φ′ ◦ g1 = g0 ◦ φ. Denote by P2(R
op
E ) the full subcategory of P1(R
op
E ) whose objects
are the triples (P1, P0, φ) with Kerφ ⊆ rad(P1). or equivalently, φ : P1 −→ P0 has no
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
818 D. SIMSON
non-zero direct summand of the form P −→ 0. We define the coordinate vector of
(P1, P0, φ) to be the bipartite vector
cdn(P1, P0, φ) = (lgth(topP1)|lgth(topP0)) ∈ ZU × ZU = K0(R
op
E )×K0(R
op
E ).
We call cdn(Cokerφ) = cdn(P1, P0, φ) the coordinate vector of the RE-module
Cokerφ.
We start with the following important result. Here we freely use the terminology and
notation introduced in [3] (Section 6), [7], and [28].
Theorem 3.1. Let K be an algebraically closed field, C a basic K-coalgebra
with the decomposition (1.1), E a socle-finite injective direct summand (3.1) of CC,
and assume that the K-algebra RE = EndCE (3.2) is finite-dimensional. Let BE =
= (A,AVA) be the additive Roiter bocs associated to theK-algebra RopE in [3] (Proposi-
tion 6.1). Then there is a commutative diagram
Map1(E) HE−→
'
P1(R
op
E ) G←−
'
repK(BE)
kerE
y cokE
y
C-ComodfcE
h•E−→
'
mod(RopE ),
(3.5)
where HE and h•E = HomC(•, E) are K-linear contravariant equivalences of categori-
es, G is a covariant K-linear equivalence of categories, h•E is an exact functor,
kerE(E0, E1, ψ) = Kerψ, cokE(P1, P0, φ) = Cokerφ, and the following conditions
are satisfied.
(a) The functors cokE and kerE are full dense and restrict to the representation
equivalences kerE :Map2(E) −→ C-ComodEfc and cokE : P2(R
op
E ) −→ mod(RopE ).
The right-hand part in the diagram is defined as in [7] (Section 5) and [3, p. 476, 478],
with RopE , G, cokE and Λ, Ξ, cok interchanged.
(b) If N is an indecomposable comodule in C-ComodEfc then there exists a uni-
que, up to isomorphism, indecomposable object (E0, E1, ψ) in Map1(E) such that
kerE(E0, E1, ψ) ∼= N. In this case (E0, E1, ψ) lies inMap2(E) and
cdn(N) = cdn(E0, E1, ψ) = σ(cdnHE(E0, E1, ψ)) = dimG−1HE(E0, E1, ψ)),
where we set σ(v′|v′′) = (v′′|v′).
(c) If the category C-ComodEfc is not of K-wild representation type (shortly, K-
wild) then the additive category repK(BE) of the K-linear representations of BE is not
wild and, given a non-negative vector
v = (v′|v′′) ∈ ZU × ZU ⊆ Z(IC) × Z(IC) ∼= K0(C)×K0(C),
there exist minimal bocses B1, . . . ,Bn, with Bi = (Bi,Wi), finitely E-copresented
C-Bi-bicomodules Ti and full functors Fi : repK(Bi) −→ C-Comodfc which reflect
isomorphisms such that
(c1) Fi(X) = Ti ⊗Bi X, for all representations X in repK(Bi),
(c2) every indecomposable comodule N in C-ComodEfc, with cdn(N) = v, is
isomorphic to Fi(X), for some i and some representation X in repK(Bi),
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 819
(c3) the functors Fi induce group homomorphisms K0(Bi) −→ ZU ⊆ Z(IC) ∼=
∼= K0(C) taking the dimension vector dim(X) of X to cdnFi(X).
Proof. By our assumption, the injective comodule E = EU is socle-finite and
the K-algebra RE = EndCE is finite dimensional. Let D : modRopE −→ modRE be
the standard duality given by L 7→ D(L) = HomK(L,K), for any L in modRopE .
We define the contravariant functor h•E by setting h
(−)
E = HomC(−, E). Since E is
injective, the functor h•E is exact and, by [26] (Proposition 2.13), h•E is an equivalence
of categories such that (lgthN)u = (dimhNE )u = dimK(hNE )eu, for any comoduleN in
C-ComodfcEU and all u ∈ U, where dimN ′ is the dimension vector of a left RU -module
N ′. This means that resU (lgthN) = dimhNE , for any comodule N in C-ComodfcEU ,
where resU : Z(IE) −→ ZU is the restriction homomorphism.
We define the functor HE on objects by setting HE(E0, E1, ψ) = (hE1
E , hE0
E , hψE),
and on morphisms by setting HE(f0, f1) = (hf1E , h
f0
E ). A direct calculation shows that
(hE1
E , hE0
E , hψE) belongs to P1(R
op
E ), if (E0, E1, ψ) ∈ Map1(E) and that HE is well
defined.
For a purpose of next steps of the proof (and in order to see a nature of Map1(E)
as the bimodule problem in the sense of Drozd [5], see also [4, 17]), we give a different
detailed proof of the above fact.
Let K = add(E) be the full additive subcategory of C-Comod formed by finite
direct sums of the injective C-comodules E(u), with u ∈ U, and let H = HE be the
K-K-bimodule H(−, ·) = HE(−, ·) : Kop ×K −→ modK defined by the formula
H(E′, E′′) = {g ∈ HomC(E′, E′′);ψ(socE′) = 0} ⊆ HomC(E′, E′′),
with E′, E′′ ∈ K. Note that H(E,E) = {ψ ∈ EndCE; ψ(socE) = 0} = J(RE) is the
Jacobson radical of the algebra RE .
We construct HE as the composite functor
Map1(E) H′
−→
'
Mat(KHE
K ) H′
−→
'
P1(R
op
E ), (3.6)
where Mat(KHE
K ) is the additiveK-category of KHE
K -matrices in the sense of Drozd [5],
see also [4], [10], [18] (Chapter 17), [20] (Section 2) for details. Recall that the objects
of Mat(KHE
K ) are the triples (E′, E′′, ψ), where E′, E′′ ∈ obK and ψ ∈ H(E′, E′′),
and morphisms are defined in a natural way.
The functor H ′ is defined by attaching to any object (E0, E1, ψ) of Map1(E),
with ψ ∈ HomC(E0, E1) = H(E0, E1) and E0, E1 ∈ K, the triple H ′(E0, E1, ψ) =
= (E0, E1, ψ), viewed as an object of Mat(KHE
K ). Given a morphism (f0, f1):
(E0, E1, ψ) −→ (E′0, E
′
1, ψ
′), we set H ′(f0, f1) = (f0, f1). It is easy to see that H ′ is
a K-linear equivalence of categories.
Now we construct the functor H ′′. In the notation of [20] (Section 2), we denote
by RE-pr the category of finitely generated projective left RE-modules and we define
the Nakayama equivalence ω : K '−→ (RE-pr)op that associates, to any object x of K,
the finitely generated projective left RE-module ω(x) = hxE = HomC(x,E). Hence, by
applying the formula (2.9) in [20] to K = L = K = add(E) and the bimodule M = H,
we conclude that, for any pair x = E′, y = E′′ of objects in K, the (contravariant!)
functor ω induces the natural isomorphisms
H(E′, E′′) = H(x, y) ∼= HomRE (hyE ,H(x,E)) ∼=
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
820 D. SIMSON
∼= HomRE (hyE ,H(E,E)⊗RE hxE) ∼=
∼= HomRE (hyE , J(RE)⊗RE hxE) ∼=
∼= HomRE (hyE , radhxE) = HomRE (hE
′′
E , radhE
′
E ) ∼=
∼= HomRE (J(RE)+ ⊗RE h
y
E , h
x
E) ∼=
∼= HomRE (J(RE)+ ⊗RE hE
′′
E , hE
′
E ), (3.7)
where J(RE)+ = HomRE (J(RE), RE) is viewed as an RE-RE-bimodule.
Hence, if (E0, E1, ψ) is an object of Map1(E) (or of Mat(KHK)) then ψ ∈
∈ H(E′, E′′) and its image ψ̂ : hE
′′
E −→ radhE
′
E under the composite isomorphism (3.7)
is such that hψE = u · ψ̂, where u : radhE
′
E ↪→ hE
′
E is the embedding. It follows that
(hE
′′
E , hE
′
E , h
ψ
E) lies in P1(RE) if and only if (E0, E1, ψ) lies in Map1(E). We define
H ′′ (and HE) on objects (E0, E1, ψ) by setting
H ′′(E0, E1, ψ) = HE(E0, E1, ψ) = (hE
′′
E , hE
′
E , hψE),
and on morphisms (f0, f1) by H ′′(f0, f1) = HE(f0, f1) =
(
hf1E , h
f0
E
)
. Obviously,
H = H ′′ ◦ H ′. Since, up to isomorphism, all objects of P1(RE) are of the form
(hE
′′
E , hE
′
E , h
ψ
E), with (E0, E1, ψ) ∈ Map1(E), then the functors H ′′ and HE are
equivalences of categories making the square in (3.5) commutative.
(a) The fact that the functors ker and cok are full and dense follows immedi-
ately form the definitions. It is easy to see that (P1, P0, φ) is an object of P1(RE)
if and only if P1
φ−→ P0 → Cokerφ → 0 is a minimal projective presentation of
Cokerψ in mod(RopE ). Analogously, (E0, E1, ψ) is an object of Map1(E) if and only
if 0 → Kerψ → E0
ψ−→ E1 is a minimal injective E-copresentation of Kerψ. Hence
easily follows that the functors cokE and kerE restrict to the representation equi-
valences kerE :Map2(E) −→ C-ComodEfc and cokE : P2(R
op
E ) −→ mod(RopE ). The
remaining statements in (a) follow from the definitions and [3] (Section 6).
(b) LetN be an indecomposable comodule inC-ComodEfc. ThenN admits a minimal
injective E-copresentation 0→ N → E0
ψ−→ E1 in C-Comod, with E0, E1 ∈ add(E)
and, therefore, (E0, E1, ψ) is an object ofMap1(E). It follows that
HE(E0, E1, ψ) = (hE1
E , hE0
E , hψE) ∈ P2(RE)
and, hence, hE1
E
hψE−→ hE0
E −→ hNE → 0 is a minimal projective presentation of hNE
in modRopE . Hence the equalities cdn(N) = cdn(E0, E1, ψ) = σ(cdnHE(E0, E1, ψ))
easily follow. The equality σ(cdnHE(E0, E1, ψ)) = dimG−1HE(E0, E1, ψ)) is proved
in [7] (Section 5) and [3] (Section 6).
(c) First we show that the functor G in (3.5) is the composite functor
P1(R
op
E ) G′
←−
'
R̂E-modprpr
G′′
←−
'
repK(BE), (3.8)
where R̂E-modprpr is the additiveK-category of finite dimensional propartite left modules
over the finite dimensional bipartite K-algebra
R̂E =
[
RE J(RE)+
0 RE
]
(3.9)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 821
in the sense of [20], with J(RE)+ = HomRE (J(RE), RE). First we note that if X =
= (X ′, X ′′, ξ : J(RE)+⊗RE X ′ −→ X ′′), is a propartite left R̂E-module then, up to
isomorphism, the projective left RE-modules X ′, X ′′ have the forms X ′ = hE
′′
E ,
X ′′ = hE
′
E , where E′, E′′ ∈ add(E). Then, in view of the isomorphisms
HomRE (J(RE)+ ⊗RE hE
′′
E , hE
′
E ) ∼=
∼= HomRE (hE
′′
E , J(RE)⊗RE hE
′
E ) ∼= HomRE (hE
′′
E , radhE
′
E )
given in (3.7), we can view X as the triple X = (X ′, X ′′, ξ̃), where ξ̃ = u ◦ ξ is the
composition hE
′′
E
ξ−→ radhE
′
E
u−→ hE
′
E of the image ξ of ξ ∈ HomRE (J(RE)+ ⊗RE
hE
′′
E , hE
′
E ), under the composite isomorphism, with the canonical embedding u. In other
words, the triple G′(X) = (X ′, X ′′, φ) = (hE
′′
E , hE
′
E , φ) is an object of P1(R
op
E ).
This defines the equivalence G′, and we set G′′ = G ◦ (G′)−1. It is clear that the
functor TK = (G′′)−1 is the equivalence TK : R̂E-modprpr
'−→ repK(BE) defined in
[20] ((4.11)).
Following an observation of Drozd [7] (see also [3] and [20, p. 44, 45]), given a fini-
tely generatedK-algebra S, the category rep(BE , S) of right S-module representations
of the bocs BE = (A,AVA) has as objects the A-S-bimodules AXS in modfp(A⊗Sop)
(the category of finitely presented left (A⊗ Sop)-modules), which are finitely generated
projective, when viewed as right S-modules, see [7], [3] and [20, p. 44, 45] for details.
We set repK(BE) = rep(BE ,K).
By [20] (Proposition 4.9), there is an equivalence of categories
TS : (R̂E ⊗ Sop)-modprpr
'−−→ rep(BE , S), (3.10)
for any finitely generated K-algebra S, where
(R̂E ⊗ Sop) =
[
RE ⊗ Sop J(RE)+ ⊗ Sop
0 RE ⊗ Sop
]
.
The objects of (R̂E⊗Sop)-modprpr are R̂E-S-bimodules that are (RE⊗Sop)-(RE⊗Sop)-
propartite and finitely generated projective as left S-modules.
Following the above construction of the functor G′, we can construct equivalences
of categories
P1((RE ⊗ Sop)op)
G′
E,S←−−− (R̂E ⊗ Sop)-modprpr
G′′
E,S←−−− rep(BE , S), (3.11)
and we extend the diagram (3.5) to the following commutative diagram
Map1(E ⊗ Sop)
HE,S−→
'
P1((RE ⊗ Sop)op)
GE,S←−
'
rep(BE , S)
ker
y cok
y
(C ⊗ Sop)-ComodfcE⊗S
op h•S−→
'
mod((RE ⊗ Sop)op),
(3.12)
where GE,S = G′E,S ◦ G′′E,S and T−1
S = G′′E,S . We set Ĉ = C ⊗ Sop and view it as
an Sop-coalgebra with the comultiplication ∆̂ = ∆ ⊗ Sop and the counit ε̂ = ε ⊗ Sop.
Then Ê = E⊗Sop is an injective object in the category Ĉ-Comod of left Ĉ-comodules,
which is projective, when viewed as a right S-module.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
822 D. SIMSON
We define Ĉ-ComodfcÊ = (C ⊗ Sop)-ComodfcE⊗S
op
to be the full subcategory
Ĉ-Comod whose objects are the finitely Ê-copresented Ĉ-comodules, that is, finitely
E ⊗Sop-copresented Ĉ-bicomodules. The categoriesMap1(E ⊗Sop), P1(RE ⊗Sop),
and the functors ker = kerE⊗Sop , cok = cokRE⊗Sop are defined in an obvious way.
We only prove that the functor h•S : (C ⊗ Sop)-ComodfcE⊗S
op
−→ mod((RE ⊗
⊗Sop)op) in (3.12) defined by Z 7→ hZS = HomĈ(Z, Ê), is an equivalence of categories.
The fact that HE,S is an equivalence of categories can be proved by applying the
properties of h•S and the isomorphism
χE′,E′′ : HomC(E′, E′′)⊗ Sop −→ HomĈ(E′ ⊗ Sop, E′′ ⊗ Sop), (3.13)
with E′, E′′ ∈ add(E), given by g⊗s 7→ [(g⊗ id) ·s : E′⊗Sop −→ E′′⊗Sop], because
the bimodule problem arguments used above extend almost verbatim to our situation.
The homomorphism χE′,E′′ is an isomorphism of S-modules, for each pair E′, E′′ of
comodules in add(E), because it is functorial with respect to homomorphisms E′ → E′1
and E′′ → E′′1 of C-comodules and it is proved in [28] ((2.10)) that χE′,E′′ is bijective,
for E′ = E′′ = E, if the algebra RE is finite dimensional.
Hence easily follows that a left Ĉ-comodule Z lies in (C ⊗ Sop)-ComodfcE⊗S
op
if and only if there is an exact sequence 0 −→ Z −→ E0 ⊗ Sop −→ E1 ⊗ Sop, with
E0, E1 ∈ add(E). By applying HomĈ(−, E ⊗ Sop) and the isomorphism χE′,E′′ , we
get the exact sequence
h
E′
1
E ⊗ S
op −→ hE0
E ⊗ S
op −→ hZS −→ 0
of left (RE⊗Sop)-modules, that is a projective presentation of hZS = HomĈ(Z,E⊗Sop).
Hence, we conclude that the functor h•S in (3.12) is an equivalence of categories. It
follows that the functor HE,S in (3.12) is an equivalence of categories making the
diagram (3.12) commutative.
Note that, by [20] (Proposition 4.9(b)) and the definition of the functors GE,S , HE,S
in (3.12) and the functors GE and HE in (3.5), for every module L in the category
fin(Sop) of finite dimensional left S-modules and every R̂E-S-bimodule R̂E
XS in the
category R1((RE ⊗ Sop)op) there exist isomorphisms
G−1
E (R̂EX ⊗S L) ∼= G−1
E,S(R̂EXS)⊗S L,
and
H−1
E (R̂EX ⊗S L) ∼= H−1
E,S(R̂EXS)⊗S L
that are functorial with respect to the S-module homomorphisms L → L′ and R̂E-S–
bimodule homomorphisms R̂EXS → R̂E
X ′S .
By applying the diagram (3.12), we reduce the proof of (c) to [7] (Propositi-
ons 11 and 13), and to [3] (Theorem B). Here we follow closely the notation and
the proof of [3] (Theorem B). We recall that our functor GE in (3.5) is just the functor
Ξ: repK(BE) −−−−→ P1(RE) in [3, p. 476], where repK(BE) = rep(BE ,K).
Assume that the category C-ComodEfc is notK-wild. Then the category C-ComodEfc
is not K-wild and, by [28] (Proposition 2.8 (a)), the finite dimensional K-algebras RE
and RopE are not wild. Hence, according to [3] (Theorem B) and its proof, the category
repK(BE) is not wild and there exist minimal bocses B1, . . . ,Bn, with Bi = (Bi,Wi),
finitely generated RE-Bopi -bimodules T ′i and full functors F ′i : repK(Bi) −→ RE-mod
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 823
which reflect isomorphisms such that the conditions (c1), (c2) and (c3) stated in (c) are
satisfied with C-ComodEfc, Fi : repK(Bi) −→ C-ComodEfc and RE-mod, F ′i :
repK(Bi) −→ RE-mod interchanged. Moreover, it is shown in the proof of [3] (Theo-
rem B) that, for each i = 1, . . . , n, the RE-Bopi -bimodules T ′i are of the form T ′i =
= cokBi(T̂
′
i ), where T̂ ′i ∈ P1(RE⊗Biop), and F̂ ′i (X) = T̂ ′i ⊗BiX, for all representati-
ons X of the bocs Bi.
Let T̂i = H−1
E,Bi
(T ′i ) ∈ Map1(E ⊗ Biop) be the preimage of T ′i under the functor
HE,S in (3.12), with S = Bi. Finally, let Ti = ker(T̂i) ∈ Ĉ-ComodfcÊ be the image of
T̂i under the functor ker in (3.12), applied to S = Bi. Then Ti is a finitely E-copresented
C-Bi-bicomodule and we set Fi(−) = Ti ⊗Bi (−).
In view of (a), (b) and the properties of the functors F ′i : repK(Bi) −→ RE-mod
listed above, the conditions (c1) – (c3) are satisfied, because the arguments given in the
proof of [3] (Theorem B) extends almost verbatim. The details are left to the reader.
Corollary 3.1. Under the assumption made in Theorem 3.1, for a given socle-
finite injective direct summand E of CC such that dimK EndCE < ∞, the following
conditions are equivalent.
(a) The category C-ComodEfc is K-wild.
(b) C-ComodEfc is properly fc-wild (or smooth) [20] (Section 6), that is, for every
finitely generated K-algebra Λ (equivalently, for Λ = K〈t1, t2〉, or Λ = Γ3(K)) there
exists a finitely E-copresented C-Λ-bicomodule CNΛ that induces a representation
embedding CN ⊗Λ (−) : fin(Λop) −→ C-ComodEfc.
(c) The finite dimensional K-algebras RopE and RE are wild.
(d) The additive K-category repK(BE) is wild, where BE is the Roiter bocs of
RopE , see (3.5).
(e) The additive K-category R̂E-modprpr is wild, where R̂E is the bipartite
algebra (3.9).
Proof. Since the functor h•E : C-ComodEfc −→ RE-mod in (3.5) is an exact equi-
valence of categories then the condition (a) implies (c). The inverse implication (c)⇒ (a)
and the equivalence of (a) and (b) follows from [28] (Corollary 2.12). The implication
(d) ⇒ (a) follows from Theorem 3.1 (c). The equivalence (d) ⇔ (e) follows from [20]
(Proposition 4.9). Since (c) ⇔ (d) follows from [7] (Section 5) and [3], then the proof
is complete.
In the proof of the fc-tame-wild dichotomy we use the following lemma.
Lemma 3.2. Under the assumption made in Theorem 3.1, for a given socle-finite
injective direct summand E = EU of CC such that RE = EndCE is of finite dimension,
(a) the fc-tameness of the category C-ComodEfc implies the tameness of the additive
K-categories Map1(E) ∼= repK(BE) ∼= R̂E-modprpr and the tameness of the algebras
RE and Rop, where R̂E is the bipartite algebra (3.9) and BE is the Roiter bocs of RopE ,
see (3.5),
(b) given a proper bipartite vector v = (v′|v′′) ∈ ZU × ZU ⊆ K0(C)×K0(C) we
have
µ̂1
C(v) = µ̂1
R̂E
(σ(v)) = µ̂1
RopE
(σ(v)),
where µ̂1
R̂E
(σ(v)) and µ̂1
RopE
(σ(v)) is the minimal cardinality of an almost parametrising
family for indσ(v)(R̂E-modprpr) and indσ(v)(mod(RopE )), respectively.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
824 D. SIMSON
Proof. Assume that the category C-ComodEfc is fc-tame, that is, for any proper
non-negative bipartite vector v = (v′|v′′) ∈ ZU × ZU ⊆ K0(C) × K0(C), there
exist a non-zero polynomial h ∈ K[t], C-K[t]h-bicomodules L(1), . . . , L(rv), that are
finitely E-K[t]h-copresented and form an almost parametrising family for the family
indv(C-ComodEfc) of all indecomposable C-comodules M with cdnM = v. It follows
that all L(j) lie in C-ComodfcE⊗K[t]h . Then, for each j ∈ {1, . . . , rv}, there is an exact
sequence
0 −→ CL
(j)
K[t]h
−→ E
(j)
0 ⊗K[t]h
ψ(j)
−→ E
(j)
1 ⊗K[t]h
in C-ComodfcE⊗K[t]h , with E(j)
0 , E
(j)
1 in add(E), such that
L̂(j) = (E(j)
0 ⊗K[t]h, E
(j)
1 ⊗K[t]h, ψ(j))
is an object ofMap1(E⊗K[t]h), see (3.12). By applying Theorem 3.1, one can show that
the objects L̂(1), . . . , L̂(rv) form a finitely E-copresented almost parametrising family
for indv(Map1(E)), that is, all but finitely many indecomposable objects (E′, E′′, g)
inMap1(E), with cdn(E′, E′′, g) = v, are of the form
(E′, E′′, g) ∼= L̂(s) ⊗K[t]h := (E(s)
0 ⊗K[t]h, E
(s)
1 ⊗K[t]h, ψ(j) ⊗K1
λ),
where s ≤ rv, K1
λ = K[t]/(t− λ) and λ ∈ K. This shows that the categoryMap1(E)
is tame. The functor G−1
S ◦HE,S in the diagram (3.12), with S = K[t]h, carries each
of the objects L̂(s) to some object U (s) ∈ rep(BE ,K[t]h)) such that all but finitely
many indecomposable objects X in repK(BE), with dim(X) = σ(v), are of the form
X ∼= U (j)⊗K1
λ, where s ≤ rv. This shows that the category repK(BE) is tame and, by
[3] (Section 6) and [7], the algebra RopE and RE are tame. Since, by Proposition 4.9 (b)
and Theorem 6.5 in [20], the category repK(BE) is tame if and only if R̂E-modprpr is
tame then the proof of (a) is complete.
Moreover, it follows that, given a proper vector v = (v′|v′′) ∈ ZU × ZU , any
almost parametrising family for indv(C-ComodEfc) consisting of finitely E-copresented
bicomodules L(1), . . . , L(rv) leads to an almost parametrising family L̂(1), . . . , L̂(rv) ∈
∈Map1(E⊗S), with S = K[t]h, for indv(Map1(E)) . By applying the functor HE,S
in (3.12) and then the functor (G′E,S)−1 in (3.11), to L̂(1), . . . , L̂(rv), we get an almost
parametrising family ̂̂
L(1), . . . ,
̂̂
L(rv) ∈ (R̂E ⊗ S)-modprpr, for indv(R̂E-modprpr). Since
the vector v = (v′|v′′) is proper then, up to a localisation of S = K[t]h, by applying the
functor cok in (3.12) we get an almost parametrising family cok(̂̂
L(1)), . . . , cok(̂̂
L(rv))
for indσ(v)(mod(RopE )).
By Lemma 3.1, any finitely copresented family for indv(C-Comodfc) can be
corrected to a finitely E-copresented almost parametrising family for
indv(C-Comodfc) = indv(C-ComodEfc), for any v = (v′|v′′) ∈ ZU × ZU . Hence
(b) follows and the proof is complete.
Now we are able to give an alternative proof of the fc-tame-wild dichotomy for
computable coalgebras established in [28].
Theorem 3.2. Assume that C is a basic coalgebra over an algebraically closed
fieldK such that dimK HomK(E′, E′′) is finite, for each pair E′, E′′ of indecomposable
direct summands of CC. Then C is either of tame fc-comodule type or of wild fc-
comodule type, and these two types are mutually exclusive.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 825
Proof. Since C is basic, CC has a decomposition (1.1). Assume that C is not
of fc-wild comodule type. To show that C is of fc-tame comodule type, fix a non-
negative bipartite vector v = (v′|v′′) ∈ Z(IC) × Z(IC) ∼= K0(C) × K0(C). Since the
support Uv = supp(v) of v is a finite subset of IC then the injective C-comodule E =
= EUv =
⊕
j∈Uv E(j) is socle-finite and, according to our assumption the algebraRE =
= EndCE is finite dimensional. Moreover, every left C-comodule N, with cdn(N) =
= v lies in the subcategory C-ComodEfc of C-Comodfc. Then indv(C-Comodfc) =
= indv(C-ComodEfc) and, by our assumption, the category C-ComodEfc is not of K-
wild comodule type. Then, by Theorem 3.1, there exist minimal bocses B1, . . . ,Bn,
with Bi = (Bi,Wi), finitely E⊗Ri-copresented C-Bi-bicomodules Ti and full functors
Fi(−) = Ti⊗Bi (−) : repK(Bi) −→ C-ComodEfc which reflect isomorphisms such that
the conditions (c1) – (c3) in Theorem 3.1 are satisfied. In particular, every indecomposable
comodule N in C-ComodEfc with cdn(N) = v is isomorphic to Fi(X), for some i
and some representation X in repK(Bi). Hence we conclude, as in the proof of [3]
(Corollary C), that there is a finite set of pairs (Ri, L(i)), where each Ri = K[t]h is a
localisation of K[t] and L(i) is a finitely E-copresented C-Ri-bicomodule such that
L(i) ∈ (C ⊗Ropi )-ComodfcE⊗R
op
i (3.14)
and all but finitely many indecomposable left C-comodules N in C-Comodfc, with
cdn(N) = v, are of the form N ∼= L(s) ⊗ Y, for some i and some indecomposable
Ri-module Y. Hence we conclude, as in the proof of Theorem 14.18 in [18, p. 297], that
there exist finitely E-copresented C-K[t]h-bicomodules L̂(1), . . . , L̂(rv) such that all but
finitely many indecomposable left C-comodules N in C-Comodfc, with cdn(N) = v,
are of the form N ∼= L̂(s) ⊗ K1
λ, where s ≤ rv, K
1
λ = K[t]/(t − λ) and λ ∈ K.
Consequently, the coalgebra is of fc-tame comodule type.
It remains to prove that the coalgebra C can not be both of fc-tame and of fc-
wild comodule type. Assume to the contrary, that C is of fc-tame and of fc-wild
comodule type. Let T : modΓ3(K) −→ C-Comodfc be an exactK-linear representation
embedding, where Γ3(K) =
[
K K3
0 K
]
. Let S1 be the unique simple injective right
Γ3(K)-module, and let S2 be the unique simple projective right Γ3(K)-module, up to
isomorphism. Since T (S1) and T (S2) lie in C-Comodfc, then there are exact sequences
0 → T (S1) → E
(1)
0 −→ E
(1)
1 and 0 → T (S2) → E
(2)
0 −→ E
(2)
1 , where E(1)
0 , E
(1)
1 ,
E
(2)
0 , E
(2)
1 are socle-finite injective C-modules.
Let E be a socle-finite direct summand of C such that the comodules E(1)
0 , E
(1)
1 ,
E
(2)
0 , E
(2)
1 lies in add(E). We show that ImT ⊆ C-ComodEfc. Indeed, if N = T (X)
lies in ImF, where X is a module in modΓ3(K), then there is an exact sequence
0→ Sn2 → X → Sm1 → 0, with n,m ≥ 0. Since T is exact, we get the exact sequence
0→ T (S2)n → N → T (S1)m → 0 in C-Comod. The comodules T (S1)m and T (S2)n
obviously lie in C-ComodEfc and, hence, also N lies in C-ComodEfc. This shows that
ImT ⊆ C-ComodEfc and, hence, the category C-ComodEfc is fc-wild and, according to
Corollary 3.1, the finite dimensional algebra RE is wild.
On the other hand, in view of the fc-parametrisation correction lemma (Lemma 3.1),
the assumption that C is of fc-tame comodule type implies that C-ComodEfc is fc-
tame. Hence, by Lemma 3.2, the finite dimensional algebra RE is tame and we get a
contradiction with the tame-wild dichotomy [7] for finite dimensional K-algebras.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
826 D. SIMSON
Now we can complete [28] (Proposition 2.8 (a)) as follows.
Corollary 3.2. Under the assumption made in Theorem 3.1, for a given socle-finite
injective direct summand E = EU of CC such that the algebra RE = EndCE is of
finite dimension, the following conditions are equivalent.
(a) The category C-ComodEfc is fc-tame.
(b) The finite dimensional K-algebra RE is tame.
(c) The additive K-categories Map1(E) ∼= repK(BE) are tame, where BE is the
additive Roiter bocs of RopE , see (3.5).
(d) The additive K-category R̂E-modprpr is tame, where R̂E is the bipartite
algebra (3.9).
Moreover, if C-ComodEfc is fc-tame then, given a proper bipartite vector v =
= (v′|v′′) ∈ ZU×ZU ⊆ K0(C)×K0(C), we have µ̂1
C(v) = µ̂1
R̂E
(σ(v)) = µ̂1
RopE
(σ(v)).
In particular, C-ComodEfc is of polynomial growth if and only if R̂E-modprpr is of
polynomial growth.
Proof. The equivalence (b) ⇔ (c) follows from the theorem of Drozd [7] (see also
[3], [28] (Proposition 2.8) and from the proof of Theorem 3.1. The equivalence (c)⇔ (d)
follows from [20] (Theorem 6.5) (or from the proof of Theorem 3.1). To prove (c)⇒ (a),
note that, according to [7], if repK(BE) is tame, it is not wild. Then, by Theorem 3.2
and its proof, the category C-ComodEfc is fc-tame. Since (a) ⇒ (c) follows from
Lemma 3.2 (a), the conditions (a) – (d) are equivalent. The remaining statement follows
from Lemma 3.2 (b).
Corollary 3.3. Let C be a basic coalgebra over an algebraically closed field K
such that dimK HomK(E′, E′′) is finite, for each pair E′, E′′ of indecomposable direct
summands of CC. The following conditions are equivalent.
(a) The coalgebra C is of tame fc-comodule type.
(b) For any proper bipartite vector v = (v′|v′′) ∈ K0(C) × K0(C), there is a
finitely EUv -copresented almost parametrising family for indv(C-Comodfc) =
= indv(C-ComodfcEUv ), where Uv = supp(v) ⊆ Z(IC) is the support of v and
EUv =
⊕
j∈Uv E(j).
(c) For any socle-finite direct summand E of CC, C-ComodEfc is fc-tame.
(d) For any socle-finite direct summand E of CC, C-ComodEfc is not fc-wild.
(e) For any socle-finite direct summand E of CC, the finite dimensional K-algebra
RE = EndCE is tame.
(f) For any socle-finite direct summand E of CC, the category R̂E-modprpr is tame,
where R̂E is the bipartite algebra (3.9).
The coalgebra C is of fc-discrete comodule type if and only if, for any proper
bipartite vector v = (v′|v′′) ∈ K0(C) × K0(C), the family indv(C-ComodfcEUv ), wi-
th Uv = supp(v) ⊆ Z(IC), is finite up to isomorphism, or equivalently, the family
indv(R̂EUv−modprpr) is finite up to isomorphism.
Proof. The implication (b) ⇒ (a) is obvious. The implication (c) ⇒ (b) and the
equivalence of the statements (c) – (f) is an immediate consequence of previous results.
To prove (a) ⇔ (b), we fix a proper bipartite vector v = (v′|v′′) ∈ Z(IC) × Z(IC)
and set Uv = supp(v), EUv =
⊕
j∈Uv E(j). It is clear that indv(C-Comodfc) =
= indv(C-ComodfcEUv ). Since C is fc-tame then there are finitely copresented
C-K[t]h-bimodules L(1), . . . , L(rv) forming an almost parametrising family of for
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 827
indv(C-Comodfc). By Lemma 3.1, the family corrects to an almost parametrising fami-
ly L̃(1), . . . , L̃(rv) for indv(C-Comodfc) = indv(C-ComodfcEUv ) consisting of finitely
EUv -copresented bicomodules. Hence (b) follows and the conditions (a) – (f) are equi-
valent. Since the remaining statement of corollary is a consequence of Lemma 3.2 (b),
the proof is complete.
4. A geometry context for computable coalgebras. Througouht we assume that
K is an algebraically closed field and C a basic computable K-coalgebra with a fixed
decomposition CC =
⊕
j∈IC E(j) (1.1). Following [7, 17, 19, 20], we introduce in
Definitions 4.1 and 4.2 a geometry context for a coalgebra C, compare with [15]. We
use it in the study of comodules over a K-coalgebra C by applying the geometry of
orbits. In particular, we give a geometric characterisation of fc-tame coalgebras.
Definition 4.1. Given a computable K-coalgebra C (1.1) and a bipartite non-
negative vector v = (v′|v′′) ∈ Z(IC) × Z(IC), we define an action
∗ : GC
v ×MapCv −→ MapCv (4.1)
of an algebraic (parabolic) group GC
v on an affine K-variety MapCv as follows.
(a) GC
v = AutCE(v′) × AutCE(v′′) viewed as an algebraic group with respect
to Zariski topology, where E(v′) =
⊕
i∈IC E(j)v
′
i and E(v′′) =
⊕
j∈IC E(j)v
′′
j are
the standard injective C-comodules (2.2) with lgthE(v′) = (v′|0) and lgthE(v′′) =
= (v′′|0).
(b) MapCv = {ψ ∈ HomC(E(v′),E(v′′));ψ(socE(v′)) = 0} ⊆ HomC(E(v′),
E(v′′)) is viewed as an affine K-variety (Zariski closed subset of the affine space
HomC(E(v′),E(v′′)) of finite K-dimension).
(c) The algebraic group (left) action (4.1) of GC
v on MapCv is defined by the
conjugation (f ′, f ′′) ∗ ψ = f ′′ ◦ g ◦ (f ′)−1, where ψ ∈ MapCv , f
′ ∈ AutCE(v′) and
f ′′ ∈ AutCE(v′′).
Definition 4.2. Given a computable K-coalgebra C and a bipartite non-negative
vector v = (v′|v′′) ∈ Z(IC) × Z(IC) = K0(C)×K0(C), the open subset
ComodCv = {ψ ∈MapCv ; socE(v′′) ⊆ Imψ} (4.2)
of the variety MapCv is called a variety of C-comodules N with cdn(N) = v.
We start with the following useful facts.
Lemma 4.1. Let C be a computable K-coalgebra and v = (v′|v′′) ∈ Z(IC) ×
× Z(IC) = K0(C)×K0(C) a non-negative bipartite vector.
(a) ComodCv is a GC
v -invariant and Zariski open subset of the affine variety MapCv .
(b) The map ψ 7→ Kerψ defines a bijection between the GC
v -orbits of ComodCv and
the isomorphism classes of comodules N in C-Comodfc such that cdn(N) = v.
Proof. (a) To see that ComodCv is a Zariski open subset of MapCv , note that,
given a ∈ supp(v′′) ⊆ IC , the subset Da of MapCv consisting of all ψ ∈ MapCv
such that ψ : E(v′) −→ E(v′′) has a factorisation through the subcomodule E(v′′)a =
=
⊕
j 6=aE(j)v
′′
j of E(v′′) is Zariski closed. Since the set supp(v′′) is finite then
D =
⋃
a∈supp(v′′)
Da is closed and therefore ComodCv = MapCv \D is open. The fact
that ComodCv is a GC
v -invariant subset of MapCv follows by applying the definitions.
(b) Note that a C-comodule homomorphism ψ : E(v′) −→ E(v′′) is an element
of ComodCv if and only if 0 −→ Kerψ −→ E(v′)
ψ−→ E(v′′) is a minimal injective
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
828 D. SIMSON
copresentation of Kerψ in C-Comodfc. Hence every comodule N in C-Comodfc, with
cdn(N) = v, is isomorphic to Kerψ, for some ψ : E(v′) −→ E(v′′) in ComodCv .
Obviously, two elements ψ : E(v′) −→ E(v′′) and ψ′ : E(v′) −→ E(v′′) of ComodCv
lie in the same GC
v -orbits if and only if the comodules Kerψ and Kerψ′ are isomorphic.
Hence (b) follows.
The lemma is proved.
Now we characterise computable K-colagebras of fc-discrete comodule type in
terms of the GC
v -orbits of ComodCv as follows.
Proposition 4.1. Let K be an algebraically closed field and C a computable
K-coalgebra. The following four conditions are equivalent.
(a) The coalgebra C is fc-tame of discrete comodule type.
(b) For every bipartite vector v = (v′|v′′) ∈ K0(C)×K0(C), there is only a finite
number of indecomposable objects (E0, E1, ψ) inMap1(EUv ) with cdn(E0, E1, ψ) =
= v, up to isomorphism, where Uv = supp(v).
(c) The number of GC
v -orbits in ComodCv is finite, for every bipartite vector v =
(v′|v′′) ∈ K0(C)×K0(C).
(d) The number of GC
v -orbits in MapCv is finite, for every bipartite vector v =
= (v′|v′′) ∈ K0(C)×K0(C).
Proof. (a) ⇒ (b) Assume that C is fc-tame of discrete comodule type. Let
v = (v′|v′′) be a bipartite vector in K0(C) × K0(C) and let (E0, E1, ψ) be an
indecomposable object of Map1(EU )) such that cdn(E0, E1, ψ) = (v′|v′′), where
we set U = Uv = supp(v).
If v′ = 0 then E0 = 0, E1
∼= E(a), with a ∈ U, and therefore the number of
the indecomposable objects (E0, E1, ψ) ofMap1(EU )) with cdn(E0, E1, ψ) = (0|v′′)
equals the cardinality of the finite subset U = supp(v) of IC .
Assume that v′ 6= 0, that is, the vector v is proper. Since (E0, E1, ψ) is indecompos-
able, it lies in Map2(EU ), because it has no non-zero direct summand of the form
(0, Z, 0), By Proposition 4.1 (a), with E and EU interchanged, the functor kerEU in
the diagram (3.5) restrict to the representation equivalence kerEU : Map2(EU ) −→
−→ C-ComodEUfc . Then Kerψ = kerEU (E0, E1, ψ) is an indecomposable comodule
in C-ComodEUfc such that cdn(Kerψ) = cdn(E0, E1, ψ) = v, see Proposition 4.1 (b).
Since C is fc-tame of discrete comodule type then the number of the isomorphism
classes of such comodules is finite and, hence, the number of the isomorphism classes
of indecomposable objects (E0, E1, ψ) inMap1(EU ) with cdn(E0, E1, ψ) = v is also
finite.
(b) ⇒ (d) Let v = (v′|v′′) ∈ K0(C) × K0(C) be a vector with non-negative
coordinates and let (E0, E1, ψ) be an object in Map1(EU ). Since the coalgebra C is
assumed to be computable then the endomorphism ring End(ψ) of (E0, E1, ψ) is a finite
dimension K-algebra, and End(ψ) is a local algebra if (E0, E1, ψ) is indecomposable. It
follows thatMap1(EU ), with U = supp(v) ⊆ IC , is a Krull – Schmidt category such
that each of its objects is a finite direct sum of indecomposable objects, and every such a
decomposition is unique up to isomorphism and a permutation of the indecomposables.
By our assumption, there is only a finite number of indecomposable objects (E′0, E
′
1,
ψ′) inMap1(EUv ) with cdn(E′0, E
′
1, ψ
′) ≤ v, up to isomorphism. Let E1, . . . ,Esv be
a complete set of such indecomposable objects. Then, up to isomorphism, any object
(E0, E1, ψ) inMap1(EUv ), with cdn(E0, E1, ψ) = v, has the form
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 829
(E(v′),E(v′′), ψ) ∼= E`11 ⊕ . . .⊕ E`svsv
where `(E(v′),E(v′′), ψ) = (`1, . . . , `sv ) ∈ Nsv is a vector with non-negative coordi-
nates such that
`1 · cdn(E1) + . . .+ `sv · cdn(Esv ) = v.
Obviously, the number of such vectors (`1, . . . , `sv ) is finite. The unique decomposition
property inMap1(EUv ) yields
`(E(v′),E(v′′), ψ) = `(E(v′),E(v′′), ψ′)
if and only if (E(v′),E(v′′), ψ) ∼= (E(v′),E(v′′), ψ′),
or equivalently, if and only if the elements ψ and ψ′ of MapCv lie in the same GC
v -orbit.
Hence the number of GC
v -orbits in MapCv is finite and (d) follows.
Since the implication (d) ⇒ (c) is obvious and the implication (c) ⇒ (a) follows
from Lemma 4.1 (b), the proof is complete.
Now we present a characterisation of computable fc-tame colagebras in terms of
geometry of the GC
v -orbits of ComodCv .
Theorem 4.1. Let K be an algebraically closed field and C a computable K-
coalgebra.
(a) C is fc-tame.
(b) For every bipartite vector v = (v′|v′′) ∈ K0(C)×K0(C), the categoryMap1(EUv ),
with Uv = supp(v), is tame.
(c) For every bipartite vector v = (v′|v′′) ∈ K0(C) × K0(C), the subset
indComodCv of ComodCv defined by the indecomposable C-comodules is constructi-
ble and there exists a constructible subset C(v) of indComodCv such that
GC
v ∗ C(v) = indComodCv and dim C(v) ≤ 1.
(d) For every bipartite vector v = (v′|v′′) ∈ K0(C)×K0(C), the subset indMapCv
of MapCv defined by the indecomposable C-comodules is constructible and there exists
a constructible subset Ĉ(v) of indMapCv such that
GC
v ∗ Ĉ(v) = indMapCv and dim Ĉ(v) ≤ 1.
Proof. (a) ⇒ (b) Apply Lemma 3.2 (a) to E = EU =
⊕
j∈U E(j), where U =
= supp(v) ⊆ IC .
(b)⇒ (a) Apply Corollary 3.3.
We prove the equivalence of (b), (c) and (d) by applying the arguments used by
Drozd [7], see also [3], [18] (Section 15.2) and [20] (Theorem 6.5).
(b) ⇒ (d) Fix a bipartite vector v = (v′|v′′) ∈ K0(C) × K0(C) and assume that
the category Map1(EUv ), with Uv = supp(v), is tame. Then there is a parametrising
family of functors
L̂(1), . . . , L̂(r) : ind1(K[t]h) −→ Map1(EUv )
for the family indv(Map1(EUv ),where h ∈ K[t] and Uv = supp(v).Here ind1(K[t]h)
is the category of one-dimensional K[t]h-modules. Hence we conclude, as in [18]
(Lemma 14.30, Remark 14.27) that the functors L̂(1), . . . , L̂(r) induce regular maps
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
830 D. SIMSON
`1, . . . , `r : modK[t]h(1) −→ MapCv
such that every point of indMapCv belongs to an GC
v –orbit of the set
Ĉ(v) = Im `1 ∪ . . . ∪ Im `r.
Here modK[t]h(1) is the variety of one-dimensional K[t]h-modules. Since we have
dimmodK[t]h(1) = 1 then, according to the Chevalley Theorem, the subsets Im `1, . . .
. . . , Im `r of indMapCv are constructible and therefore Ĉ(v) is a constructible subset of
indMapCv . Moreover, it follows that dim(Im `j) ≤ 1, for j = 1, . . . , r, and therefore
dim Ĉ(v) ≤ 1, compare with [15] and [18, p. 317].
The equivalence (d)⇔(c) easily follows from the fact that indMapCv \indComodCv
is a finite set and ComodCv is an open subset of MapCv , by Lemma 4.1.
(d) ⇒ (b) Assume to the contrary that there is a bipartite vector v = (v′|v′′) ∈
∈ K0(C) × K0(C) such that the category Map1(EUv ), with Uv = supp(v), is not
tame. By Corollary 3.2, the finite dimensional algebra RUv is not tame. Then RUv is
wild [7] and therefore the categoryMap1(EUv ) is wild, by [3] (Section 6) and the proof
of Theorem 3.1.
Let W = K〈t1, t2〉 be the free polynomial K-algebra in two non-commuting
indeterminates t1 and t2. Since the category Map1(EUv ) is wild then there exists an
object CNW = (E′ ⊗W, E′′ ⊗W, ψ) inMap1(EUv ⊗W), with E′, E′′ in add(EUv ),
such that the functor
N̂ = CN ⊗W (−) : fin(W) −→ Map1(EUv )
preserves the indecomposability and respects the isomorphism classes.
Let w = (w′|w′′), where w′ = lgth(socE′) and w′′ = lgth(socE′′). It is
well known that indMapCw is a constructible subset of MapCw , compare with [18]
(Lemma 14.32).
Note that Uw = supp(w) ⊆ Uv, cdn(N̂(X)) = w, and N̂(X)) ∼= (E(w′),E(w′′),
ψ), for some ψ ∈ indMapCw ⊆ MapCw , if X ∈ fin(W) and dimK X = 1. It follows
that the restriction N̂ : ind1(W) −→ Map1(EUv ) of N̂ to ind1(W) induces a regular
map (see [18], Lemma 14.30)
`N : modW(1) −→ indMapCw ⊆MapCw .
Since modW(1) ∼= K2, the map `N is injective, and according to the Chevalley
Theorem the set Im `N is constructible then the variety dimension dim(Im `N ) of Im `N
equals two. Hence, in view of (d) with v and w interchanged, we get the contradiction
2 = dim(Im `N ) ≤ dim C(v) ≤ 1 (apply [12] (Lemma 3.16) or [18] (Lemma 15.15)).
This completes the proof.
5. On fc-tameness for arbitrary coalgebras. The fc-tame-wild dichotomy for
an arbitrary basic coalgebra C over an algebraically closed field K remains an open
problem. Some suggestions for the proof in case C is not computable is given in the
following proposition that collects important consequences of the technique described
in Section 3. In particular, it shows that the coalgebra C is fc-tame if and only if every
socle-finite colocalisation CE ∼= R◦E of C (in the sense of [11, 25]) is fc-tame.
Proposition 5.1. Assume that K is an algebraically closed field and C is an
arbitrary basic coalgebra with a decomposition CC =
⊕
j∈IC E(j) (1.1).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 831
(a) Given a socle-finite injective direct summand E = EU =
⊕
u∈U E(u) (3.1)
of CC, with a finite subset U of IC , the K-algebra RE = EndCE is semi-perfect
and pseudocompact with respect to the topology defined by (5.2) below. There is a
commutative diagram
Map1(E) HE−→
'
P1(R
op
E ) G′
←−
'
R̂E-modprpr
kerE
y cokE
y
CE-Comodfc ∼= C-ComodfcE
h•E−→
'
modfp(R
op
E ),
(5.1)
where CE ∼= R◦E is the colocalisation of C at E in the sense of [11, 25], modfp(R
op
E )
is the category of finitely presented left RE-modules, R̂E-modprpr is the category of
finitely generated propartite left modules over the bipartite K-algebra R̂E (3.9), HE
and h•E = HomC(•, E) are K-linear contravariant equivalences of categories defined
as in (3.5), G′ is the covariant K-linear equivalence of categories defined in (3.9), h•E
is an exact functor, kerE(E0, E1, ψ) = Kerψ, cokE(P1, P0, φ) = Cokerφ.
(b) For any socle-finite comoduleE = EU as in (a), the fc-tameness of the coalgebra
C implies that the category C-ComodEUfc is fc-tame, that is, the coalgebra CEU is fc-
tame.
(c) Conversely, if the category CEU -Comodfc ∼= C-ComodEUfc is fc-tame, for all
socle-finite injective direct summands E = EU , then the coalgebra C is fc-tame.
Proof. (a) Let E = EU be a socle-finite direct summand of C as in (a). The
K-algebra RE = EndCE has the decomposition RE =
⊕
u∈U euRE , where euRE =
= HomC(E,E(u)) is an indecomposable projective right ideal of RE and eu is the
primitive idempotent of RE defined by the summand E(u) of E. Since the set U is
finite then
∑
u∈U eu is the identity of RE , see [25, 26, 28]. It is easy to see that the
Jacobson radical J(RE) of RE has the form J(RE) = {h ∈ EndCE;h(socE) = 0}. It
follows that the algebra RE is semiperfect and pseudocompact with respect to the K-
linear topology defined by the left ideals aβ = HomC(E/Vβ , E) ⊆ RE , where {Vβ}β
is the directed set of all finite dimensional subcomodules of E. Since E =
⋃
β Vβ , then
there are isomorphisms
RE = EndCE ∼= lim
←−β
HomC(Vβ , E) ∼= lim
←−β
RE/aβ . (5.2)
The remaining statements in (a) follow from the proof of Theorem 3.1.
For the proof of (b) and (c), apply Lemma 3.1 and the arguments used in the proof
of Theorem 3.1.
It follows from [28] (Corollaries 2.12 and 2.13) and the results of Section 3 that the
fc-tameness and fc-wildness of a computable coalgebra C is equivalent, respectively,
to the K-tameness and the K-wildness of the finite dimensional algebra RE , for every
socle-finite direct summand of C. Proposition 5.1 shows that the fc-tameness and fc-
wildness of a basic coalgebra C (that is not necessarily computable) can be studied by
means of the tameness and wildness of the categories R̂E-modprpr and modfp(R
op
E ) over
the semiperfect algebras R̂E and RE that are not finite dimensional, in general.
We recall from [26] (Corollary 2.10) that a socle-finite coalgebra C is computable if
and only if dimK C is finite. Hence, if C is a cocommutative noncomputable coalgebra
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
832 D. SIMSON
with simple socle then C is infinite dimensional and, in view of Proposition 5.1, we have
the following consequence of Drozd [6].
Corollary 5.1. Assume that K is an algebraically closed field and C is a basic
infinite dimensional cocommutative K-coalgebra with a unique simple subcoalgebra S.
If S is finitely copresented and C is not fc-wild then
(i) C is a subcoalgebra of the pathK-coalgebraK2(L2,Ω) (see [21] (Example 6.18),
[22], [24]), where L2 is the two loop quiver
tameness, bocses and gemetry for coalgebras 21
is an indecomposable projective right ideal of RE and eu is the primitive idempotent of
RE defined by the summand E(u) of E. Since the set U is finite then
∑
u∈U eu is the
identity of RE, see [27], [28] and [30]. It is easy to see that the Jacobson radical J(RE)
of RE has the form J(RE) = {h ∈ EndCE; h(socE) = 0}. It follows that the algebra
RE is semiperfect and pseudocompact with respect to the K-linear topology defined by
the left ideals aβ = HomC(E/Vβ, E) ⊆ RE, where {Vβ}β is the directed set of all finite
dimensional subcomodules of E. Since E =
⋃
β Vβ, then there are isomorphisms
(5.3) RE = EndCE ∼= lim
←−β
HomC(Vβ, E) ∼= lim
←−β
RE/aβ.
The remaining statements in (a) follow from the proof of Theorem 3.6.
For the proof of (b) and (c), apply Lemma 3.2 and the arguments used in the proof
of Theorem 3.6. �
It follows from [30, Corollaries 2.12 and 2.13] and the results of Section 3 that the
fc-tameness and fc-wildness of a computable coalgebra C is equivalent, respectively, to
the K-tameness and the K-wildness of the finite dimensional algebra RE, for every socle-
finite direct summand of C. Proposition 5.1 shows that the fc-tameness and fc-wildness
of a basic coalgebra C (that is not necessarily computable) can be studied by means of the
tameness and wildness of the categories R̂E-modprpr and modfp(R
op
E ) over the semiperfect
algebras R̂E and RE that are not finite dimensional, in general.
We recall from [28, Corollary 2.10] that a socle-finite coalgebra C is computable if and
only if dimK C is finite. Hence, if C is a cocommutative noncomputable coalgebra with
simple socle then C is infinite dimensional and, in view of Proposition 5.1, we have the
following consequence of Drozd [6].
Corollary 5.4. Assume that K is an algebraically closed field and C is a basic
infinite dimensional cocommutative K-coalgebra with a unique simple subcoalgebra S. If
S is finitely copresented and C is not fc-wild then
(i) C is a subcoalgebra of the path K-coalgebra K2(L2,Ω) (see [23, Example 6.18], [24],
[26]), where L2 is the two loop quiver
L2 : β1�•� β2
and Ω ⊆ KL2 is the ideal of the path algebra KL2 generated by the two zero-relations
β1β2 and β2β1, and
(ii) K2(L2,Ω) is a string coalgebra in the sense of [24, Section 6],
(iii) the colagebras K2(L2,Ω) and C are of tame comodule type, and K2(L2,Ω) is of
non-polynomial growth.
Proof. By our assumption, C has a simple socle S and C = E(S) is the injec-
tive envelope of S, that is, the set IC in the decomposition (1.1) has one element and
Theorem 5.1 applies to E = E(S) = C. It follows that the K-algebra RE is pseudo-
compact, infinite dimensional, commutative, local, and complete. Since C is not fc-wild,
the category modfp(RE) is not K-wild, by Theorem 5.1. Since S is finitely copresented
then C–comod ⊆ C–Comodfc and therefore fin(RE) ⊆ modfp(RE). It follows that the
category fin(RE) is not not K-wild. Hence, by [6], the unique maximal ideal J(RE) of
RE is generated by at most two elements and RE is isomorphic to a quotient of the K-
algebra K[[t1, t2]]/(t1t2), where K[[t1, t2]] is the power series K-algebra in two commuting
indeterminates t1, t2 and (t1t2) is the ideal of K[[t1, t2]] generated by t1t2.
It is easy to see that the path coalgebra K2(L2,Ω) = Ω⊥ ⊆ K2L2 is isomorphic with
the coalgebra
and Ω ⊆ KL2 is the ideal of the path algebra KL2 generated by the two zero-relations
β1β2 and β2β1, and
(ii) K2(L2,Ω) is a string coalgebra in the sense of [22] (Section 6),
(iii) the colagebras K2(L2,Ω) and C are of tame comodule type, and K2(L2,Ω) is
of non-polynomial growth.
Proof. By our assumption, C has a simple socle S and C = E(S) is the injective
envelope of S, that is, the set IC in the decomposition (1.1) has one element and Proposi-
tion 5.1 applies to E = E(S) = C. It follows that the K-algebra RE is pseudocompact,
infinite dimensional, commutative, local, and complete. Since C is not fc-wild, the
category modfp(RE) is not K-wild, by Proposition 5.1. Since S is finitely copresented
then C-comod ⊆ C-Comodfc and therefore fin(RE) ⊆ modfp(RE). It follows that
the category fin(RE) is not K-wild. Hence, by [6], the unique maximal ideal J(RE)
of RE is generated by at most two elements and RE is isomorphic to a quotient of
the K-algebra K[[t1, t2]]/(t1t2), where K[[t1, t2]] is the power series K-algebra in two
commuting indeterminates t1, t2 and (t1t2) is the ideal of K[[t1, t2]] generated by t1t2.
It is easy to see that the path coalgebra K2(L2,Ω) = Ω⊥ ⊆ K2L2 is isomorphic with
the coalgebra
K[t1, t2]� = K ⊕
∞⊕
n=1
Kt
n
1 ⊕
∞⊕
m=1
Kt
m
2 ,
where the comultiplication ∆: K[t1, t2]� −−−−−→ K[t1, t2]�⊗K[t1, t2]� and the couni-
ty ε : K[t1, t2]� −→ K are defined by the formulae ∆(tmj ) =
∑
r+s=m
t
r
j⊗t
s
j for j = 1, 2,
ε(1) = 1 and ε(tsj) = 0 for s ≥ 1 and j = 1, 2, see [21] (Example 6.18).
Moreover, it follows from [24] that C is isomorphic to a subcoalgebra of K2(L2,Ω).
Since K2(L2,Ω) is a string coalgebra then, according to [21] (Example 6.18) and
[22] (Theorem 6.2) K2(L2,Ω) ∼= K[t1, t2]� is of tame comodule type and, hence,
the coalgebra C is of tame comodule type, too. It is shown in [21] (Example 6.18) that
K2(L2,Ω) ∼= K[t1, t2]� is tame of non-polynomial growth.
1. Assem I., Simson D., Skowroński A. Elements of the representation theory of associative algebras. Vol. 1.
Techniques of representation theory // London Math. Soc. Student Texts 65. – Cambridge; New York:
Cambridge Univ. Press, 2006.
2. Chin W. A brief introduction to coalgebra representation theory // Lect. Notes Pure and Appl. Math. –
2004. – 237. – P. 109 – 131.
3. Crawley-Boevey W. W. On tame algebras and bocses // Proc. London Math. Soc. – 1988. – 56. –
P. 451 – 483.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
TAME COMODULE TYPE, ROITER BOCSES, AND A GEOMETRY CONTEXT FOR COALGEBRAS 833
4. Crawley-Boevey W. W. Matrix problems and Drozd’s theorem // Topics Algebra, Pt I: Rings and
Represent. Algebras / Eds S. Balcerzyk, T. Józefiak, J. Krempa, D. Simson, W. Vogel. – Warszawa:
PWN, 1990. – P. 199 – 222.
5. Drozd Ju. A. Matrix problems and categories of matrices // Zap. Nauchn. Sem. Leningrad. Otdel. Mat.
Inst. Steklov (LOMI). – 1972. – 28. – P. 144 – 153 (in Russian).
6. Drozd Ju. A. Representations of commutative algebras // Funkc. Analiz i Pril. – 1972. – 6. – P. 41 – 43
(in Russian).
7. Drozd Ju. A. Tame and wild matrix problems // Represent. and Quadr. Forms. – Kiev: Inst. Mat. Akad.
Nauk USSR, 1979. – P. 39 – 74 (in Russian).
8. Drozd Ju. A., Kirichenko V. V. Finite dimensional algebras. – Berlin etc.: Springer, 1994.
9. Drozd Ju. A., Ovsienko S. A., Furchin B. Ju. Categorical constructions in representation theory // Algebr.
Structures and Appl. – Kiev: UMK VO, 1988. – P. 43 – 73 (in Russian).
10. Gabriel P., Roiter A. V. Representations of finite dimensional algebras // Algebra VIII: Encyclopedia
Math. Sci. – 1992. – 73.
11. Jara P., Merino L., Navarro G. Localization in tame and wild coalgebras // J. Pure and Appl. Algebra.
– 2007. – 211. – P. 342 – 359.
12. Kasjan S., Simson D. Varieties of poset representations and minimal posets of wild prinjective type //
Represent. Algebras: Can. Math. Soc. Conf. Proc. – 1993. – 14. – P. 245 – 284.
13. Kleiner M., Roiter A. V. Representations of differential graded categories // Matrix Problems. – Kiev:
Inst. Math. Akad. Nauk USSR, 1977. – P. 5 – 70 (in Russian).
14. Kleiner M., Reiten I. Abelian categories, almost split sequences and comodules // Trans. Amer. Math.
Soc. – 2005. – 357. – P. 3201 – 3214.
15. Kraft H., Riedtmann K. Geometry of representations of quivers // Represent. Algebras: London Math.
Soc. Lect. Notes 116. – 1986. – P. 109 – 145.
16. Montgomery S. Hopf algebras and their actions on rings // CMBS. – 1993. – № 82.
17. de la Peña J. A., Simson D. Prinjective modules, reflection functors, quadratic forms and Auslander –
Reiten sequences // Trans. Amer. Math. Soc. – 1992. – 329. – P. 733 – 753.
18. Simson D. Linear representations of partially ordered sets and vector space categories // Algebra, Logic
and Appl. – 1992. – 4.
19. Simson D. Representation embedding problems, categories of extensions and prinjective modules //
Represent. Theory Algebras / Eds R. Bautista, R. Martinez-Villa, J. A. de la Peña: Can. Math. Soc. Conf.
Proc. – 1996. – 18. – P. 601 – 639.
20. Simson D. Prinjective modules, propartite modules, representations of bocses and lattices over orders //
J. Math. Soc. Jap. – 1997. – 49. – P. 1 – 68.
21. Simson D. Coalgebras, comodules, pseudocompact algebras and tame comodule type // Colloq. math. –
2001. – 90. – P. 101 – 150.
22. Simson D. Path coalgebras of quivers with relations and a tame-wild dichotomy problem for coalgebras
// Lect. Notes Pure and Appl. Math. – 2004. – 236. – P. 465 – 492.
23. Simson D. On Corner type Endo-Wild algebras // J. Pure and Appl. Algebra. – 2005. – 202. – P. 118 – 132.
24. Simson D. Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally
nilpotent representations // Colloq. math. – 2007. – 109. – P. 307 – 343.
25. Simson D. Localising embeddings of comodule categories with applications to tame and Euler coalgebras
// J. Algebra. – 2007. – 312. – P. 455 – 494.
26. Simson D. Hom-computable coalgebras, a composition factors matrix and the Euler bilinear form of an
Euler coalgebra // Ibid. – 2007. – 315. – P. 42 – 75.
27. Simson D. Representation-directed incidence coalgebras of intervally finite posets and the tame-wild
dichotomy // Communs Algebra. – 2008. – 36. – P. 2764 – 2784.
28. Simson D. Tame-wild dichotomy for coalgebras // J. London Math. Soc. – 2008. – 78. – P. 783 – 797.
29. Woodcock D. Some categorical remarks on the representation theory of coalgebras // Communs Algebra.
– 1997. – 25. – P. 775 – 2794.
Received 10.02.09
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
|
| id | umjimathkievua-article-3060 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:30Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/d484e4d2837814ffe05c6d837a4f540d.pdf |
| spelling | umjimathkievua-article-30602020-03-18T19:44:24Z Tame comodule type, roiter bocses, and a geometry context for coalgebras Ручний комодульний тип, бокси ройтера i геометричний контекст для коалгебр Simson, D. Сімсон, Д. We study the class of coalgebras $C$ of $fc$-tame comodule type introduced by the author. With any basic computable $K$-coalgebra $C$ and a bipartite vector $v = (v′|v″) ∈ K_0(C) × K_0(C)$, we associate a bimodule matrix problem $\textbf{Mat}^v_C(ℍ)$, an additive Roiter bocs $\textbf{B}^C_v$, an affine algebraic $K$-variety $\textbf{Comod}^C_v$, and an algebraic group action $\textbf{G}^C_v × \textbf{Comod}^C_v → \textbf{Comod}^C_v$. We study the $fc$-tame comodule type and the fc-wild comodule type of $C$ by means of $\textbf{Mat}^v_C(ℍ)$, the category $\textbf{rep}_K (\textbf{B}^C_v)$ of $K$-linear representations of $\textbf{B}^C_v$, and geometry of $\textbf{G}^C_v$ -orbits of $\textbf{Comod}_v$. For computable coalgebras $C$ over an algebraically closed field $K$, we give an alternative proof of the $fc$-tame-wild dichotomy theorem. A characterization of $fc$-tameness of $C$ is given in terms of geometry of $\textbf{G}^C_v$-orbits of $\textbf{Comod}^C_v$. In particular, we show that $C$ is $fc$-tame of discrete comodule type if and only if the number of $\textbf{G}^C_v$-orbits in $\textbf{Comod}^C_v$ is finite for every $v = (v′|v″) ∈ K_0(C) × K_0(C)$. Вивчено клас коалгебр $C$ $fc$-ручного комодульного типу, що введений автором. Кожну базову злічєнну $K$-коалгебру $C$ та дводольний вектор $v = (v′|v″) ∈ K_0(C) × K_0(C)$ пов'язано з бімодульною матричною задачею $\textbf{Mat}^v_C(ℍ)$, адитивними боксами Ройтера $\textbf{B}^C_v$, афінним алгебраїчним $K$-різновидом $\textbf{Comod}^C_v$ та алгебраїчним груповим оператором $\textbf{G}^C_v × \textbf{Comod}^C_v → \textbf{Comod}^C_v$. Дослідження $fc$-ручного та $fc$-дикого комодульних типів $C$ проведено з використанням $\textbf{Mat}^v_C(ℍ)$, категорії $\textbf{rep}_K (\textbf{B}^C_v)$ $K$-лінійних зображень $\textbf{B}^C_v$ та геометрії $\textbf{G}^C_v$-орбіт Comod^. Для зліченних коалгебр $C$ над алгебраїчно замкненим полем $K$ наведено альтернативне доведення теореми про $fc$-ручну дику дихотомію. Характеризацію $fc$-ручної властивості для $C$ подано через геометрію $\textbf{G}^C_v$-орбіт $\textbf{Comod}_v$. Показано, зокрема, що $C$ належить до $fc$-ручного дискретного комодульного типу тоді i тільки тоді, коли кількість $\textbf{G}^C_v$-орбіт в $\textbf{Comod}^C_v$ скінченна для кожного $v = (v′|v″) ∈ K_0(C) × K_0(C)$. Institute of Mathematics, NAS of Ukraine 2009-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3060 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 6 (2009); 810-833 Український математичний журнал; Том 61 № 6 (2009); 810-833 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3060/2869 https://umj.imath.kiev.ua/index.php/umj/article/view/3060/2870 Copyright (c) 2009 Simson D. |
| spellingShingle | Simson, D. Сімсон, Д. Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title | Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title_alt | Ручний комодульний тип, бокси ройтера i геометричний контекст для коалгебр |
| title_full | Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title_fullStr | Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title_full_unstemmed | Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title_short | Tame comodule type, roiter bocses, and a geometry context for coalgebras |
| title_sort | tame comodule type, roiter bocses, and a geometry context for coalgebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3060 |
| work_keys_str_mv | AT simsond tamecomoduletyperoiterbocsesandageometrycontextforcoalgebras AT símsond tamecomoduletyperoiterbocsesandageometrycontextforcoalgebras AT simsond ručnijkomodulʹnijtipboksirojteraigeometričnijkontekstdlâkoalgebr AT símsond ručnijkomodulʹnijtipboksirojteraigeometričnijkontekstdlâkoalgebr |