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WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY
GENERATED MODULES
MIODRAG CRISTIAN IOVANOV
Dedicated to Fred Van Oystaeyen for his sixtieth birthday
Abstract. It is well known that the torsion part of any finitely generated module over
the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra
dual to the divided power coalgebra over Kand the torsion part of any K[[X]]-module
actually identifies with the rational part of that module. More generally, for a certain
general enough class of coalgebras - those having only finite dimensional subcomodules
- we see that the above phenomenon is preserved: the set of torsion elements of any
C∗-module is exactly the rational submodule. With this starting point in mind, given
a coalgebra Cwe investigate when the rational submodule of any finitely generated left
C∗-module is a direct summand. We prove various properties of coalgebras Chaving
this splitting property. Just like in the K[[X]] case, we see that standard examples of
coalgebras with this property are the chain coalgebras which are coalgebras whose lattice
of left (or equivalently, right, two-sided) coideals form a chain. We give some representa-
tion theoretic characterizations of chain coalgebras, which turn out to make a left-right
symmetric concept. In fact, in the main result of this paper we characterize the colo-
cal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional
coalgebras) as being exactly the chain coalgebras. This characterizes the cocommutative
coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in
particular cases and construct various and general classes of examples of coalgebras with
this splitting property.
Introduction
Let Rbe a ring and Tbe a torsion preradical on the category of left R-modules RM. Then
Ris said to have splitting property provided that T(M), the torsion submodule of M, is a
direct summand of Mfor any M∈RM. More generally, if Cis a Grothendieck category
and Ais a subcategory of C, then Ais called closed if it is closed under subobjects, quotient
objects and direct sums. To every such subcategory we can associate a preradical t(also
called torsion functor) if for every M∈ C we denote by t(M) the sum of all subobjects of
Mthat belong to A. We say that Chas the splitting property with respect to Aif t(M) is a
direct summand of Mfor all M∈ C. In the case of the category of R-modules, the splitting
property with respect to some closed subcategory is a classical problem which has been
considered by many authors. In particular, when Ris a commutative domain, the question
of when the (classical) torsion part of an Rmodule splits off is a well known problem. J.
Rotman has shown in [Rot] that for a commutative domain the torsion submodule splits
off in every R-module if and only if Ris a field. I. Kaplansky proved in [K1], [K2] that for
Key words and phrases. Torsion Theory, Splitting, Coalgebra, Rational Module.
2000 Mathematics Subject Classification. Primary 16W30; Secondary 16S90, 16Lxx, 16Nxx, 18E40.
∗This paper was partially supported by the contract nr. 24/28.09.07 with UEFISCU “Groups, quantum
groups, corings and representation theory” of CNCIS, PN II (ID 1002) and was also partially written within
the frame of the bilateral Flemish-Romanian project ”New Techniques in Hopf Algebra Theory and Graded
Rings”.
1
2 MIODRAG CRISTIAN IOVANOV
a commutative integral domain Rthe torsion part of every finitely generated R-module
Msplits in Mif and only if Ris a Pr¨ufer domain. While complete or partial results have
been obtained for different cases of subcategories of RM- such as the Dickson subcategory
- or for commutative rings (see also [T1], [T2], [T3]), the general problem remains open
for the non-commutative case and the general categorical setting.
In this paper we investigate a special and important case of rings (algebras) Rarising as
the dual algebra of a K-coalgebra C,R=C∗. We are thus situated in the realm of the
theory of coalgebras and their dual algebras, a theory intensely studied over the last two
decades. Then the category of the left R-modules naturally contains the category MCof
all right C-comodules as a full subcategory. In fact, MCidentifies with the subcategory
Rat(C∗M) of all rational left C∗-modules, which is generally a closed subcategory of C∗M.
Then it is natural to study splitting properties with respect to this subcategory, and two
questions regarding this splitting property with respect to Rat(C∗M) naturally arise: first,
when is the rational part of every left C∗-module Ma direct summand of Mand second,
when does the rational part of every finitely generated C∗-module Msplit in M. The
first problem, the splitting of C∗Mwith respect to the closed subcategory Rat(C∗M) has
been treated by C. N˘ast˘asescu and B. Torrecillas in [NT] where it is proved that if all
C∗-modules split with respect to Rat then the coalgebra Cmust be finite dimensional.
The techniques used involve some amount of category theory (localization in categories)
and strongly rely on some general results of M.L.Teply from [T1], [T2], [T3]; another proof
of this fact also based on the general results of Teply is found in [C]; see also [I1] for a
direct approach.
We consider the more general problem of when Chas the splitting property only for finitely
generated modules, that is, the problem of when is the rational part Rat(M) of Ma direct
summand in Mfor all finitely generated left C∗-modules M. We say that such a coalgebra
has the left f.g. Rat-splitting property (or we say that it has the Rat-splitting property for
finitely generated left modules). If the coalgebra Cis finite dimensional, then every left
C∗-module is rational so MCis equivalent to C∗Mand Rat(M) = Mfor all C∗-modules
Mand in this case Rat(M) trivially splits in any C∗-module. Therefore we will deal with
infinite dimensional coalgebras, as generally the infinite dimensional coalgebras produce
examples essentially different from the ones in algebra theory.
The starting and motivating point of our research is the fact that over the ring of formal
power series over a field R=K[[X]] (or a division algebra), any finitely generated module
splits into its torsion part and a complementary module. In this case, Ris the dual of the
so called divided power coalgebra, and the torsion part of any module identifies with the
rational submodule. Here the analogue with classical torsion splitting problems becomes
obvious. In fact, what turns out to be essential in this example is the structure of ideals
of K[[X]], and that is, they are linearly ordered. This suggests the consideration of more
general coalgebras, those whose left subcomodules form a chain. This turns out to be a
left-right symmetric concept, and the most basic example of infinite dimensional coalge-
bra having the f.g. Rat-splitting property (Proposition 2.3 and Theorem 2.5). One key
observation in this study is that if Chas the f.g. Rat-splitting property, then the indecom-
posable left injectives have only finite dimensional proper subcomodules, and this motivates
the introduction of comodules and coalgebras Chaving only finite dimensional proper left
subcomodules, which we call almost finite (or almost finite dimensional) comodules. This
proves to be the proper generalization of the phenomenon found in the case of K[[X]],
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 3
i.e. the set of torsion elements of a left C∗-module Mforms a submodule which coincides
exactly with the rational submodule of M(Proposition 1.5). Before turning to the study
of chain comodules and coalgebras, we give several general results for coalgebras Cwith
the f.g. Rat-splitting property: they are artinian as right C∗-module and injective as left
C∗-module, have at most countable dimension and C∗is a left Noetherian ring. Moreover,
such coalgebras have finite dimensional coradical and the f.g. Rat-splitting property is
preserved by subcoalgebras.
The f.g. Rat-splitting property has been studied before in [C] where the last two of the
above statements were proven, but with the use of very strong results of M.L. Teply;
we also include alternate direct proofs. Chain coalgebras were also studied recently in
[LS] and also briefly in [C] and [CGT]. However, our interest in chain coalgebras is of
a different nature; it is a representation theoretic one and is directed towards our main
result of this paper, that generalizes a result previously obtained [C] in the commutative
case: we characterize the coalgebras having the f.g. Rat-splitting property and that are
colocal, and show that they are exactly the chain coalgebras (Section 3, Theorem 3.4), a
result that will involve quite technical arguments. In fact, our characterizations of chain
coalgebras are done as a consequence of more general discussions such as the study chain
of comodules and more generally almost finite comodules and coalgebras. For example,
we show that almost finite coalgebras are reflexive, and that chain coalgebras are almost
finite, and thus obtain the fact that chain coalgebras are reflexive (a result also found in
the recent [LS]) from our more general framework.
We provide several nontrivial examples. One will be the construction of a noncocommu-
tative chain coalgebra with coradical isomorphic to the dual of the Hamilton algebra of
quaternions. However, we see that when the base field Kis algebraically closed or the
coalgebra is pointed, then a chain coalgebra is isomorphic to the divided power coalgebra
if it is infinite dimensional or to one of its subcoalgebras otherwise. This also characterizes
the divided power coalgebra over an algebraically closed field as the only local coalgebra
having the above mentioned splitting property. As an application of the main result, we
obtain the structure of cocommutative coalgebras having the f.g. Rat-splitting property
from [C] in a more precise form: they are finite coproducts of finite dimensional coalge-
bras and infinite dimensional chain coalgebras. Moreover, following this model, our results
allow us to generalize to the noncommutative case and show that a coalgebra that is a
finite direct sum of infinite dimensional left chain comodules (serial coalgebra) has the
left f.g. Rat-splitting property; moreover, this is again a left-right symmetric concept.
More generally, a coproduct of such a coalgebra and a finite dimensional one again has
the f.g. Rat-splitting property. We conclude by constructing a class of explicit examples
of noncocommutative coalgebras of this type over an arbitrary field, which will depend on
a positive integer qand a permutation σof qelements.
1. General Considerations
Let Cbe a coalgebra with counit εand comultiplication ∆. We use the Sweedler convention
∆(c) = c1⊗c2where we omit the summation symbol. For general facts about coalgebras
and comodules we refer to [A], [DNR] or [Sw]. For a vector space Vand a subspace W
of Vdenote by W⊥={f∈V∗|f(x) = 0,∀x∈W}and for a subspace X∈V∗denote
by X⊥={x∈V|f(x)=0,∀f∈X}(it will be understood from the context what is
the space Vwith respect to which the orthogonal is considered). Various properties of
this correspondence between subspaces of Vand V∗are well known and studied in more
4 MIODRAG CRISTIAN IOVANOV
general settings in [DNR] (Chapter 1), [AF], [AN], [I0]. Related to that, we recall the
finite topology on the dual V∗of a vector space V: a basis of 0 for this linear topology
is given by the sets W⊥with Wa finite dimensional subspace of V. Any topological
consideration will refer to this topology. We often use the following: a subspace Xof V∗
is closed (in the finite topology) if and only if (X⊥)⊥=X; also, if Wis a subspace of V,
then (W⊥)⊥=W. (see [DNR], Chapter 1)
For a coalgebra Cdenote by C0⊆C1⊆C2⊆. . . the coradical filtration of C, that is, C0
is the coradical of C, and Cn+1 ⊆Csuch that Cn+1/Cnis the socle of the right (or left)
C-comodule C/Cnfor all n∈N. Then Cnis a subcoalgebra of Cfor all n, and the same
Cnis obtained whether we take the socle of the left C-comodule C/Cnor of the right C-
comodule C/Cn. Put C−1= 0 and R=C∗. Denote J=J(C∗) the Jacobson radical of C∗.
By [DNR] we have S
n∈N
Cn=C,J=C⊥
0and (Jn+1)⊥=Cn. Then Jn⊆((Jn)⊥)⊥=C⊥
n−1
and since S
n∈N
Cn=C, we see that T
n∈N
Jn= 0.
For a left (right) C-comodule Mwith comultiplication ρ:M→C⊗M(ρ:M→M⊗C),
the Sweedler notation writes ρ(m) = m−1⊗m0(respectively ρ(m) = m0⊗m1). Moreover,
the dual M∗of Mbecomes a left (right) C∗-module by the action induced by the right
(left) C∗-action on Mby duality: for m∗∈M∗,m∈Mand c∗∈C∗, (c∗·m∗)(m) =
m∗(m·c∗) = c∗(m−1)m∗(m0) (respectively (m∗·c∗)(m) = m∗(m0)c∗(m1)).
Lemma 1.1. Let Cbe a coalgebra over a field Kand Mbe a left C-comodule. Then for
any finitely generated left submodule Xof M∗,(X⊥)⊥=X, that is, Xis closed in the
finite topology on M∗.
Proof. It is enough to prove this for cyclic submodules: if (C∗f⊥)⊥=C∗ffor all f∈M∗
and X=C∗·f1+. . . C∗·fnthen (X⊥)⊥= (
n
T
i=1
(C∗fi)⊥)⊥=
n
P
i=1
(C∗f⊥
i)⊥=
n
P
i=1
C∗fi=X
(since (
n
T
i=1
Mi)⊥=
n
P
i=1
M⊥
ifor Mi⊆M; see, for example [I0], Proposition 3 or [DNR],
Chapter 1; also [AN] and [AF]).
Let X=C∗fand u:M→C, u(m) = m−1f(m0), where for m∈M,m−1⊗m0∈C⊗M
denotes the comultiplication of m∈M; then L= (C∗f)⊥={m∈M|(hf)(m) = 0,∀h∈
C∗}={m∈M|f(h(m−1)m0) = h(m−1f(m0)) = 0,∀h∈C∗}={m∈M|m−1f(m0) =
0}, so L= ker(u) (the left C∗-module structure on M∗is induced from the right C∗-
module structure on Mby duality). If g∈L⊥⊆M∗, then ker(u)⊆ker(g) we can factor
gas g=p◦uwith p: Im(u)→K, and then defining h∈C∗as h=pon Im(u)⊆C
and 0 on some complement of Im(u) we get (hf)(m) = f(m·h) = f(h(m−1)m0) =
h(m−1)f(m0) = h(m−1f(m0)) = h(u(m)) = p◦u(m) = g(m), i.e. g∈C∗f. This shows
that (C∗f⊥)⊥=C∗f.
1.1. ”Almost finite” coalgebras and comodules.
Definition 1.2. AC-comodule Mwill be called almost finite (or almost finite dimen-
sional) if it has only finite dimensional proper subcomodules. Call a coalgebra Cleft
almost finite if CCis almost finite.
Proposition 1.3. Let Mbe a left almost finite (dimensional) C-comodule. Then:
(i) Mis artinian as left C-comodule (equivalently, as right C∗-module).
(ii) Any nonzero submodule of M∗has finite codimension; consequently M∗is (left) Noe-
therian. Moreover, all submodules of M∗are closed in the finite topology of M∗.
(iii) Mhas at most countable dimension.
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 5
Proof. (i) Obvious.
(ii) Let 0 6=I < M∗be a submodule, 0 6=f∈M∗. Then X= (C∗·f)⊥is a subcomodule
of Mwhich is finite dimensional and C∗·f=X⊥from Lemma 1.1, so C∗·fhas finite
codimension, and so does I⊇C∗·f. Thus M∗is Noetherian and the last assertion of (ii)
follows now from Lemma 1.1.
(iii) Assume Mis infinite dimensional and define inductively a sequence (mk)k≥0such
that mk+1 /∈Mk=m1·C∗+... +mk·C∗. This can be done since the Mk’s are finite
dimensional, and then S
k≥0
Mk⊆Mis infinite-countable dimensional, and thus cannot be
a proper submodule of M. Thus S
k≥0
Mk=M, and the proof is finished.
The above Proposition shows that a left almost finite coalgebra Cis coreflexive by [DNR],
Exercise 1.5.14, since every ideal of finite codimension in Cis closed (Also, by a result of
Radford, Cis coreflexive if and only any finite dimensional C∗-module is rational). Thus
we have:
Corollary 1.4. Let Cbe a left almost finite coalgebra. Then any nonzero left ideal of C∗
is closed in the finite topology on C∗and has finite codimension, C∗is Noetherian and
Jn=C⊥
n−1. Moreover, Cis coreflexive.
For a left C∗-module Mdenote by T(M) the set of all torsion elements of M, that
is, T(M) = {x∈M|annC∗x6= 0}. If Cis a finite dimensional coalgebra, it is
well known that the categories of right C-comodules and left C∗-modules are equiva-
lent. Thus, we are interested in the infinite dimensional case. As mentioned above, for
coreflexive coalgebras, Rat(M) = {x∈M|C∗·xis finite dimensional}={x∈M|
annC∗(x) has finite codimension}. For (infinite dimensional) almost finite coalgebras, we
see that the rational submodule of a C∗-module has an even more special form:
Proposition 1.5. Let Cbe an infinite dimensional left almost finite coalgebra and let
R=C∗. Then for any left R-module Mwe have Rat(M) = T(M); moreover, x∈Rat(M)
if and only if R·xis finite dimensional.
Proof. If x∈Rat(M) then R·xis finite dimensional and then annR(x) must be of finite
codimension, thus nonzero as Ris infinite dimensional. Conversely, if x∈T(M) and
x6= 0 then I= annR(x) is a nonzero left ideal of Rso it is closed by Corollary 1.4; thus
I=X⊥with X6=Ca finite dimensional subcomodule of C. Then R·x'R/annR(x) =
C∗/X⊥'X∗which is a rational left C∗-module, being the dual of a finite dimensional
subcomodule of C.
1.2. The Splitting Property.
Definition 1.6. We shall say that a coalgebra Chas the left (right) f.g. Rat-splitting
property, or that it has the left (right) Rat-splitting property for all finitely generated
modules if the rational part of any finitely generated left (right) C∗-module splits off.
The following key observation, together with the succeding study of chain coalgebras,
motivates our previous introduction of almost finite comodules and coalgebras.
Proposition 1.7. Let Cbe a coalgebra such that Rat(M)splits off in any finitely generated
left C∗-module M. Then any indecomposable injective left C-comodule Eis an almost
finite comodule.
Proof. Let Tbe the socle of E; then Tis simple and E=E(T) is the injective envelope
of T. We show that if K⊆E(T) is an infinite dimensional subcomodule then K=E(T).
6 MIODRAG CRISTIAN IOVANOV
Suppose K(E(T). Then there is a left C-subcomodule (right C∗-submodule) K(L⊂
E(T) such that L/K is finite dimensional. We have an exact sequence of left C∗-modules:
0→(L/K)∗→L∗→K∗→0
As L/K is a finite dimensional left C-comodule, we have that (L/K)∗is a rational left
C∗-module; thus Rat(L∗)6= 0. Also L∗is finitely generated as it is a quotient of E(T)∗
which is a direct summand of C∗. We have L∗=Rat(L∗)⊕Xfor some left C∗-submodule
Xof L∗. Then Rat(L∗) is finitely generated because L∗is, so it is finite dimensional.
As Lis infinite dimensional by our assumption, we have X6= 0. This shows that L∗is
decomposable and finitely generated, thus it has at least two maximal submodules, say
M, N . We have an epimorphism E(T)∗f
→L∗→0 and then f−1(M) and f−1(N) are
distinct maximal C∗-submodules of E(T)∗. But by [I], Lemma 1.4, E(T)∗has only one
maximal C∗-submodule which is T⊥, so we have obtained a contradiction.
Let C0be the coradical of C, the sum of all simple subcomodules of C. By [DNR],
Section 3.1, C0is a cosemisimple coalgebra that is a direct sum of simple subcoalgebras
C0=L
i∈I
Ciand each simple subcoalgebra Cicontains only one type of simple left (or
right) C-comodule; moreover, any simple left (or right) C-comodule is isomorphic to one
contained in some Ci. A coalgebra Cwith C0finite dimensional is called almost connected
coalgebra.
The following two Propositions have also been observed in [C] (Lemma 3.2 and Lemma
3.3), but general powerful techniques from [T3] are used there. We provide here direct
simple arguments.
Proposition 1.8. Let Cbe a coalgebra with the left f.g. Rat-splitting property. Then there
is only a finite number of isomorphism types of simple left C-comodules, equivalently, C0
is finite dimensional.
Proof. By the above considerations, if Siis a simple left C-subcomodule of Ci, we
have that (Si)i∈Iforms a set of representatives for the isomorphism types of simple left
C-comodules. Let Sbe a set of representatives for the simple right C-comodules. Let
E(Ci) be an injective envelope of the left C-comodule Ciincluded in C; then as C0is
essential in Cwe have C=L
i∈I
E(Ci) as left C-comodules or right C∗-modules. Then
C∗=Q
i∈I
E(Ci)∗as left C∗-modules. As Si⊆E(Ci), we have epimorphisms of left C∗-
modules E(Ci)∗→S∗
i→0 and therefore we have an epimorphism of left C∗-modules
C∗→Q
i∈I
S∗
i→0. But there is a one-to-one correspondence between left and right simple
C-comodules given by {Si|i∈I} 3 S7→ S∗∈ S. Hence there is an epimorphism
C∗→Q
S∈S
S→0, which shows that the left C∗-module P=Q
S∈S
Sis finitely generated
(actually generated by a single element). But then as Rat(C∗P) is a direct summand in P,
we must have that Rat(C∗P) is finitely generated, so it is finite dimensional. Therefore,
as Σ = L
S∈S
Sis a rational left C∗-module which is naturally included in P, we have
Σ⊆Rat(P). This shows that L
S∈S
Sis finite dimensional, so S(and also I) must be finite.
This is equivalent to the fact that C0is finite dimensional, because each Ciis a simple
coalgebra, thus a finite dimensional one.
Proposition 1.9. If Cis has the left f.g. Rat-splitting property then so does any subcoal-
gebra Dof C.
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 7
Proof. Let Mbe a finitely generated left D∗-module. Since C∗/D⊥'D∗,Mhas an
induced left C∗-module structure and is annihilated by D⊥(that is, D⊥·x= 0 for all
x∈M). Then a subspace of Mis a C∗-submodule if and only if it is a D∗-submodule.
There is M=T⊕Xa direct sum of C∗-modules (equivalently D∗-submodules, since D⊥
annihilates the elements in both Tand X) with Tthe rational C∗-submodule of M. It
will suffice to show that a submodule of Mis rational as C∗-module if and only if it is
rational as D∗-module. Indeed, let m∈T=RatC∗(M); then there is Pimi⊗ci∈T⊗C
such that c∗˙m=Pic∗(ci)mi; we may assume that the mi’s are linearly independent.
Then for c∗∈D⊥⊆C∗we get 0 = c∗·m=Pic∗(ci)miand so c∗(ci) = 0 since the
mi’s are independent, showing that ci∈(D⊥)⊥=D. Therefore ρ(m) = P
i
mi⊗ci∈
T⊗D, where ρis the comultiplication of T, and thus m∈RatD∗(M). The converse
inclusion RatD∗(M)⊆RatC∗(M) is obvious, since the D-comultiplication RatD∗(M)→
RatD∗(M)⊗D⊆RatD∗(M)⊗Cinduces a C-comultiplication through the canonical
inclusion D⊆C, compatible with the C∗-multiplication of M.
Proposition 1.10. Let Cbe a coalgebra that has the left f.g. Rat-splitting property. Then
the following assertions hold:
(i) Cis artinian as left C-comodule (equivalently, as right C∗-module).
(ii) C∗is left Noetherian.
(iii) Chas at most countable dimension.
(iv) Cis injective as left C∗-module.
Proof. (i) We have a direct sum decomposition C=L
i∈F
E(Si) where C0=L
i∈F
Siis the
decomposition of C0into simple left C-comodules and E(Si) are injective envelopes of
Sicontained in C. Since Fis finite as C0is finite dimensional, the result follows from
Propositions 1.7 and 1.3
(ii) Since C∗=L
i∈F
E(Si)∗, this also follows from 1.3.
(iii) Similar to (i).
(iv) By [I0] Lemma 2, it is enough to prove that E=CCsplits off in any left C∗-module
Min which it embeds (E⊆M) and such that M/E is cyclic generated by an element
ˆx∈M/E. Let H=Rat(C∗·x)⊆M; then there is X < C∗·xsuch that H⊕X=C∗·x.
Then E+His a rational C∗-module so (E+H)∩X= 0; also M=C∗·x+Eso
(E+H) + X=M, showing that E+His a direct summand in M. But as Eis an
injective comodule, we have that Esplits off in E+H, thus Emust split in Mand the
proof is finished.
2. Chain Coalgebras
Definition 2.1. We say that a left (right) C-comodule Mis a chain (or uniserial) co-
module if and only if the lattice of the left (right) subcomodules of Cis a chain, that is,
for any two subcomodules X, Y of Meither X⊆Yor Y⊆X. We say a coalgebra Cis a
left (right) chain coalgebra (or uniserial coalgebra) if Cis a left (right) chain C-comodule.
In other words, a left C-comodule Mis a chain comodule if Mis uniserial as a right
C∗-module. Part of the following proposition is a somewhat different form of Lemma 2.1
from [CGT]. However, we will need to use some of the other equivalent statements bellow.
Proposition 2.2. Let Mbe a left (right) C-comodule. The following assertions are equiv-
alent:
(i) Mis a chain comodule.
8 MIODRAG CRISTIAN IOVANOV
(ii) M∗is a chain (uniserial) left (right) C∗-module.
(iii) Mand Mn=”the n’th Loewy term in the Loewy series of Mfor n≥ −1”, are the
only subcomodules of M(M−1= 0).
(iv) M⊥
n={u∈M∗|u(x)=0,∀x∈Mn}for n≥ −1and 0 are the only submodules of
M∗.
(v) Mn/Mn−1is either simple or 0for all n≥ −1. (If Mn/Mn−1is 0for some nthen
Mk/Mk−1is 0for all k≥n.)
Proof. (iv)⇒(ii) is obvious.
(ii)⇒(i) If M∗is uniserial, then for any two subcomodules X, Y of Cwe have X⊥⊆Y⊥,
say. Thus we get X= (X⊥)⊥⊇(Y⊥)⊥=Y.
(i)⇔(iii) is obvious (note that (iii) this does not exclude the possibility that M=Mnfrom
some nonward)
(i)⇒(iv) If Mis a chain comodule, it is enough to assume that Mis infinite dimensional,
because of the duality of categories between finite dimensional left comodules and finite
dimensional right comodules. We note that each M⊥
nis generated by any un∈M⊥
n\M⊥
n+1.
Let f∈M⊥
nand denote un, f :M→C,un(m) = m−1un(m0) (f(m) = m−1un(m0)).
Then un∈M⊥
n\M⊥
n+1 shows that unis a morphism of left C-comodules that factors to a
morphism M/Mn→Cwhich does not cancel on Mn+1/Mn- the only simple subcomodule
of M/Mn. Therefore Ker (un:M/Mn→C) = 0 and we have a diagram
0//M
Mn
un//
f
C
g
C
that is completed commutatively by g(as CCis injective), so that we get g◦un=fand
then if g=ε◦gwe have, for m∈M,g(m−1)un(m0) = g(m−1un(m1)) = ε(g(un(m))) =
ε(f(m)) = ε(m−1)f(m0) = f(m). Thus g·un=f. This shows that any cyclic submodule
of M∗coincides to one of the M⊥
n, because for any 0 6=f∈M∗there is some nsuch that
f∈M⊥
n\M⊥
n+1, since M=S
n
Mn. It therefore follows that for any nonzero submodule I
of M∗there is M⊥
n⊆I; since the Mn’s are (obviously) finite dimensional, M⊥
nand Ihave
finite codimension and it now easily follow from the above considerations that I=M⊥
k,
where kis the smallest number such that M⊥
k⊆I.
(v)⇒(iii) Let Xbe a right subcomodule of Mand suppose X6=Mand X6= 0. Then there
is n≥0 such that Mn*Xand let nbe minimal with this property. Then we must have
Mn−1⊆Xby the minimality of nand we show that Mn−1=X. Indeed, if Mn−1(Xwe
can find a simple subcomodule of X/Mn−1. But then Mn−16=M, so Mn−16=Mnand as
Mn/Mn−1is the only simple subcomodule of M/Mn, we find Mn/Mn−1⊆X/Mn−1, that
is Mn⊆X, a contradiction.
(i)⇒(v) If Mn+1/Mnis nonzero and it is not simple then we can find S1=X1/Mnand
S2=X2/Mn(X1, X2⊆M) two distinct simple modules contained in M/Mn. Then
X1∩X2=Mn,X16=Mn,X26=Mn. But this shows that neither X1(X2nor X2(X1
which is a contradiction.
The following result shows that chain coalgebra is a left-right symmetric notion and also
characterizes chain coalgebras.
Proposition 2.3. The following assertions are equivalent for a coalgebra C:
(i) Cis a right chain coalgebra.
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 9
(ii) Cn+1/Cnis either 0or a simple (right) comodule for all n≥ −1.
(iii) Cn,n≥ −1and Care the only right subcomodules of C.
(iv) Jn,n≥0and 0 are the only right ideals of C∗.
(v) C∗is a right (or left) uniserial ring (chain algebra).
(vi) The left hand side version of (i)-(iv).
(vii) C1has length less or equal to 2.
Proof. The equivalence of (i)-(vi) follows from Proposition 2.2 and also by Corollary 1.4
(i)⇒(vii) is obvious and (vii)⇒(i) is a result from [C]. We note a direct argument for
this case: it is enough to deal with the case when C1has length 2; by induction, assume
Ck/Ck−1is simple or 0 for k≤n. Assume Cn6=Cn−1and note that since Cn/Cn−1is
the socle of C/Cn−1, then C/Cn−1embeds in Cand therefore if Cn+1 /Cn−1has length at
most 2, since it embeds in C1. Thus Cn+1/Cnis simple or 0.
Remark 2.4.The above Proposition includes many of the results in [LS] sections 5.1-5.3.
By Proposition 2.2 a chain module is almost finite and by 2.3 a chain coalgebra is left and
right almost finite, so the results of the first section apply here. Therefore we also obtain
that a chain coalgebra is coreflexive.
Next we show that a chain coalgebra is both a left and a right f.g. Rat-splitting property
coalgebra. Although this follows in a more general setting as in Section 4, we also provide a
direct proof that does not involve the tools used in there, but makes use of the interesting
fact that for a left almost finite coalgebra Cand any left C∗-comodule M,T(M) is a
submodule of Mand is exactly the rational submodule of M.
Theorem 2.5. If Cis a chain coalgebra, then Chas the left and right f.g. Rat-splitting
property.
Proof. Of course, we only need to consider the case when Cis infinite dimensional. First
notice that every torsion-free R-finitely generated module Mis free: indeed if x1, . . . , xnis
a minimal system of generators, then if λ1x1+· · · +λnxn= 0 with λinot all zero, we may
assume that λ16= 0. Without loss of generality we may also assume that λ1R⊇λiR, ∀i
as any two ideals of Rare comparable by Proposition 2.3. Therefore we have λi=λ1sifor
some si∈R. Then λ1x1+λ1s2x2+· · · +λ1snxn= 0 implies x1+s2x2+· · · +snxn= 0 as
Mis torsionfree and λ16= 0. Hence x1∈R< x2, . . . , xn>, contradicting the minimality
of n.
Now if Mis any left R-module and T=T(M) = Rat(M) (by Proposition 1.5) then
T(M/T (M)) = 0. Indeed take ˆx∈T(M/T (M)) and put I= annC∗ˆx6= 0 so Ihas
finite codimension and Iis a two-sided ideal by Proposition 2.3. By Corollary 1.4 and
Remark 2.4, Iis finitely generated and therefore Ix is also finitely generated. Also, since
I= annC∗ˆx, we get Ix ⊆T=Rat(M). Thus Ix is finitely generated rational, so I x
has finite dimension. We obviously have an epimorphism R
I→Rx
Ix which shows that
Rx/Ix is finite dimensional because Ihas finite codimension in R. Therefore we get that
dim(Rx) = dim(Rx/Ix) + dim(Ix)<∞, so then by Proposition 1.5 we have that Rx is
rational, thus x∈Tso ˆx= 0.
Now as M/T is torsion-free, there are x1, . . . , xn∈Mwhose images ˆx1,..., ˆxnin M/T
form a basis. Then it is easy to see that x1, . . . , xnare linearly independent in M. Then if
X=Rx1+· · · +Rxnwe have X+T=Mand X∩T= 0, because if a1x1+· · · +anxn∈T
we get a1ˆx1+· · · +anˆxn=ˆ
0 so ai= 0,∀ibecause ˆx1,..., ˆxnare independent in M/T .
Thus T(M) splits off in Mand the theorem is proved, as T(M) = RatR(M) by 1.5.
10 MIODRAG CRISTIAN IOVANOV
We will denote by Knthe coalgebra with a basis c0, c1, . . . , cn−1and comultiplication
ck7→ P
i+j=k
ci⊗cjand counit ε(ci) = δ0,i. The coalgebra S
n∈N
Knhaving a basis cn, n ∈N
and comultiplication and counit given by these equations is called the divided power
coalgebra (see [DNR]). Part of the following Lemma is discussed in [CGT] Theorem 3.2;
also part of it in the cocommutative case is observed in [C], 3.5 and 3.6. The same result
appears in [LS], but with a different proof. Also Theorem 2.7 below can be obtained as a
consequence of the general theory of serial coalgebras developed in [CGT] (Theorem 2.10
(iii) and Remark 2.12); in this respect, Lemma 2.6 could then be obtained as a consequence
of Theorem 2.7. We provide here a direct argument.
Lemma 2.6. Let Cbe a finite dimensional chain coalgebra over a field Kand suppose
that either Kis algebraically closed or Cis pointed. Then Cis isomorphic to Knfor some
n∈N.
Proof. Let A=C∗; we have dimC0= 1 because Kis algebraically closed (thus EndAC0
is a skewfield containing K). Thus dim Ck=kfor all kfor which Ck6=C. As C∗is finite
dimensional Jn= 0 for some nand let nbe minimal with this property. By Corollary
1.4 Jk=C⊥
k−1. Then Jk/J k+1 has dimension equal to the dimension of Ck/Ck−1which
is 1 for k < n, because Ck+1/Ckit is a simple comodule isomorphic to C0. We then
have that Jk/Jk+1 is generated by any of its nonzero elements. Choose x∈J\J2.
We prove that xn−16= 0. Suppose the contrary holds and take y1, . . . , yn−1∈J. As x
generates J/J 2, there is λ∈Ksuch that y1−λx ∈J2and then y1xn−2−λxn−1∈Jn,
so y1xn−2∈Jn= 0 because xn−1= 0. Again, there is µ∈Ksuch that y2−µx ∈J2
and then y1y2−µy1x∈J3so y1y2xn−3∈Jn(y1xn−2= 0). By continuing this procedure,
one gets that y1y2. . . yn−2x= 0 and then we again find α∈Kwith yn−1−αx ∈J2, thus
y1. . . yn−1−αy1. . . yn−2x∈Jn= 0. This shows that y1. . . yn−1= 0 for all y1, . . . , yn−1∈
J. Thus Jn−1= 0, a contradiction.
As xn−16= 0 we see that xk∈Jk\Jk+1 for all k= 0, . . . , n −1, so Jk/J k+1 is generated
by the class of xk. Now if y∈A, there is λ0∈Ksuch that y−λ0·1A∈J(either y∈J
or ygenerates A/J). As J/J2is 1 dimensional and generated by the image of x, there
is λ1∈Ksuch that y−λ0−λ1x∈J2. Again, as J2/J 3is 1 dimensional generated by
the image of x2, there is λ2∈Ksuch that y−λ0−λ1x−λ2x2∈J3. By continuing this
procedure we find λ0, . . . , λn−1∈Ksuch that y−λ0−λ1x− · · · − λn−1xn−1∈Jn= 0,
so y=λ0+λ1x+· · · +λn−1xn−1. This obviously gives an isomorphism between A
and K[X]/(Xn). Therefore Cis isomorphic to Kn, because there is an isomorphism of
K-algebras K∗
n'K[X]/(Xn).
Theorem 2.7. If Kis an algebraically closed field and Cis an infinite dimensional chain
coalgebra, then Cis isomorphic to the divided power coalgebra. The same conclusion holds
provided the infinite dimensional chain coalgebra Cis pointed.
Proof. By the previous Lemma we have that Cn'Knfor all n. If e∈C0, ∆(e) =
λe ⊗e, λ ∈K, then for c0=λe we get ∆(c0) = c0⊗c0. Suppose we constructed
a basis c0, c1, . . . , cn−1for Cn−1with ∆(ck) = P
i+j=k
ci⊗cj,ε(ci) = δ0,i. Denote by
An=C∗
nthe dual of Cn; for the rest of this proof, if V⊆Cnis a subspace of Cnwe
write V⊥for the set of the functions of Anwhich are 0 on V. Choose E1∈C⊥
0\C⊥
1;
then En
16= 0 and En+1
1= 0 as in the proof of Lemma 2.6 (E1∈An). This shows
that Ek
1∈C⊥
k−1\C⊥
k, that ε|Cn, E1, . . . , En
1exhibits a basis for Anand that there is an
isomorphism of algebras An'K[X]/(Xn+1) taking E1to ˆ
X. We can easily see that
Ei
1(cj) = δij,∀k= 0,1, . . . , n −1 and then by a standard linear algebra result we can find
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 11
cn∈Cnsuch that En
1(cn) = 1 and En
1(ci) = 0 for i<n. Then by dualization, the relations
Ei
1(cj) = δij,∀i, j = 0,1, . . . , n become ∆(ck) = P
i+j=k
ci⊗cj,∀k= 0,1, . . . , n. Therefore
we may inductively build the basis (cn)n∈Nwith ε(ck) = δ0kand ∆(cn) = P
i+j=n
ci⊗cj,∀n.
A non-trivial example. In the following we construct an example of a chain coalgebra
that is not cocommutative and thus different of the divided power coalgebra over K. Recall
that if Ais a kalgebra, ϕ:A→Ais a morphism and δ:A→Ais a ϕ-derivation (that
is a linear map such that δ(ab) = δ(a)b+ϕ(a)δ(b) for all a, b ∈A), we may consider the
Ore extension A[X, ϕ, δ] which is A[X] as a vector space and with multiplication induced
by Xa =ϕ(a)X+δ(a). Let Kbe a subfield of R, the field of real numbers. Let Dbe
the subalgebra of Hamilton’s quaternion algebra having the set B={1, i, j, k}as a vector
space basis over K. Recall that multiplication is given by the rules i·j=−j·i=k;
j·k=−k·j=i;k·i=−i·k=j;i2=j2=k2=−1. Denote by σ:D→Dthe linear
map defined on the basis of Dby
σ=1i j k
1j k i
It is not difficult to see then that σis an algebra automorphism, and that Dis a division
algebra (skewfield). Our example will be constructed with the aid of such an Ore extension
constructed with a trivial derivation: denote by Dσ[X] = D[X, σ, 0] the Ore extension
of Dconstructed by σwith the derivation ϕequal to 0 everywhere. Then a basis for
Dσ[X] over Kconsists of the elements uXk, with u∈Band k∈N. Also denote by
An=Dσ[X]/<Xn>the algebra obtained by factoring out the two-sided ideal generated
by Xnfrom Dσ[X].
Proposition 2.8. The two sided ideal < Xn>of Dσ[X]consists of elements of the form
f=
n+m
P
l=n
alXl. Moreover, the only (left, right, two-sided) ideals containing < Xn>are
the ideals < Xl>,l= 0, . . . , n and consequently Anis a chain K-algebra.
Proof. It is clear by the multiplication rule Xa =σ(a)Xfor a∈Bthat elements of
Dσ[X] are of the type
N
P
l=0
alXland that every element of Anis a ”polynomial” of the
form f=a0+a1x+· · · +an−1xn−1, with al∈Dand where xrepresents the class of X.
Such an element fis invertible if and only if a06= 0. To see this, first note that if a0= 0
then fis nilpotent, as xis nilpotent and one has fl∈< xl>by successively using the
relation xa =σ(a)x. Conversely write f=a0·(1 + a−1
0a1x+· · · +a−1
0an−1xn−1) and note
that the element g=a−1
0a1x+· · · +a−1
0an−1xn−1is nilpotent as before, so 1 + gmust
be invertible in Anand therefore fmust be invertible. Thus we may write every element
f=alxl+...an−1xn−1of Anas the product f= (al+al+1x+· · · +an−1xn−1−l)·xl=g·xl
with invertible g. Then if Iis a left ideal of Anand f∈I, we have f=g·xlfor an
invertible element gand some l≤n. Hence it follows that xl∈I. Taking the smallest
number lwith the property xl∈I, we obviously have that I=< xl>.
Let Cndenote the coalgebra dual to An. Note that Anhas a Kbasis B={axl|a∈B, l ∈
0,1, . . . , n −1}and we have the relations (axi)(bxj) = aσi(b)xi+j. Let (Ea
i)a∈B,i∈0,n−1be
the basis of Cnwhich is dual to B, that is, Ea
i(bxj) = δijδab for all a, b ∈Band i, j ∈N.
Also, for i∈Nand a∈Bdenote by i·a=σi(a) the action of Non Binduced by σ.
12 MIODRAG CRISTIAN IOVANOV
Proposition 2.9. With the above notations, denoting by ∆nand εnthe comultiplication
and respectively, the counit of Cnwe have
∆n(Ec
p) = X
i+j=p;a(i·b)=±c
c−1a(i·b)Ea
i⊗Eb
j
and
εn(Ec
p) = δp,0δc,1.
Proof. For u, v ∈Band k, l ∈Nwe have Ec
p(uxk·vxl) = Ec
p(u(k·v)xk+l) and as k·v∈B
by the formulas defining Dwe have that if d=u(k·v) then either d∈Bor −d∈B.
Then Ec
p(uxk·vxl) = Ec
p(dxk+l) = δk+l,pδu(k·v),±cc−1u(k·v) as the sign of this expression
must be 1 if d∈Band −1 if d /∈B, and this is exactly c−1u(k·v) when u(k·v) = ±c.
We also have
X
i+j=p;a(i·b)=±c
c−1a(i·b)Ea
i(uxk)Eb
j(vxl) = X
i+j=p;a(i·b)=±c
δk,iδu,a δl,j δv,bc−1a(i·b)
=δk+l,pδu(k·v),±cc−1u(k·v)
and therefore we get
X
i+j=p;a(i·b)=±c
c−1a(i·b)Ea
i(uxk)Eb
j(vxl) = Ec
p(uxk·vxl)
As this is true for all uxk, vxl∈ B, by the definition of the comultiplication of the coalgebra
dual to an algebra, we get the first equality in the statement of the proposition. The second
one is obvious, as εn(Ec
p) = Ec
p(1 ·X0) = δp,0δc,1.
Now notice that there is an injective map Cn⊂Cn+1 taking Ec
ifrom Cnto Ec
ifrom Cn+1.
Therefore we can regard Cnas subcoalgebra of Cn+1. Denote by C=S
n∈N
Cn; it has a
basis formed by the elements Ec
n, n ∈N, c ∈Band comultiplication ∆ and counit εgiven
by
∆(Ec
n) = X
i+j=n;a(i·b)=±c
c−1a(i·b)Ea
i⊗Eb
j
and
ε(Ec
n) = δn,0δc,1.
By Proposition 2.8 we have that Anis a chain algebra and therefore Cn=A∗
nis a chain
coalgebra. Therefore, we get that the coradical filtration of Cis C0⊆C1⊆C2⊆. . . and
that this is a chain coalgebra which is obviously non-cocommutative.
3. The co-local case
Throughout this section we will assume (unless otherwise specified) that Chas the left
f.g. Rat-splitting property and that it is a colocal coalgebra, that is, C0is a simple left
(and consequently simple right) C∗-module. Then as J=C⊥
0,C∗is a local algebra.
We will also assume that Cis not finite dimensional, thus by Proposition 1.10 Chas a
countable basis. We have that Cis the injective envelope of C0as left comodules, thus by
Proposition 1.7 we have that every left subcomodule of Cis finite dimensional (all Cnare
finite dimensional). Then if Iis a left nonzero ideal of C∗different from C∗, by Corollary
1.4 Iis finitely generated and of finite codimension. Denote again R=C∗. Also for a left
R-module Mdenote by J(M) the Jacobson radical of M.
Proposition 3.1. With the above notations, Ris a domain.
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 13
Proof. Let S= End(CC, CC). Note that Sis a ring with multiplication equal to the
composition of morphisms and that Sis isomorphic to Rby an isomorphism that takes
every morphism of left C-comodules f∈Sto the element ε◦f∈R. Then it is enough
to show that Sis a domain. If f:C→Cis a nonzero morphism of left Ccomodules,
then Ker(f)(Cis a proper left subcomodule of Cso it must be finite dimensional. Then
as Cis not finite dimensional we see that Im(f)'C/Ker(f) is an infinite dimensional
subcomodule of C. Thus Im(f) = C, and therefore every nonzero morphism of left
comodules from Cto Cmust be surjective. Now if f, g ∈Sare nonzero then they are
surjective so f◦gis surjective and thus f◦g6= 0.
Proposition 3.2. Rsatisfies ACCP on right ideals and also on left ideals.
Proof. Suppose there is an ascending chain of right ideals x0·R(x1·R(x2·R(. . .
that is not stationary. Then there are (λn)n∈Nin Rsuch that xn=xn+1 ·λn+1. Note
that λn+1 ∈J, because otherwise λn+1 would be invertible in Ras Ris local and then we
would have xn+1 =xn·λ−1
n. This would yield xn·R=xn+1 ·R, a contradiction. Then
x1=xn+1 ·λn+1λn. . . λ2, so x1∈Jnfor all n∈N, showing that x1∈T
n∈N
Jn= 0. Thus
we obtain a contradiction: x0·R(x1·R= 0. The statement is obvious for left ideals as
RRis Noetherian.
The next proposition together with the following theorem contain the main ideas of the
result.
Proposition 3.3. Suppose αR and βR are two right ideals that are not comparable, i.e.
neither one is a subset of the other. Then any two principal right ideals of Rcontained in
αR ∩βR are comparable.
Proof. Take aR, bR ⊆αR ∩βR, so a=αx =βy and b=αu =βv; we may obviously
assume that a, b 6= 0 as otherwise the assertion is obvious. Then α, β, x, y, u, v are nonzero.
Denote by Lthe left submodule of R×Rgenerated by (x, u) and by Mthe quotient module
R×R
L. We write (s, t) for the image of the element (s, t) through the canonical projection
π:R×R→M. We have (y, v)6= (0,0) as otherwise (y , v) = λ(x, u) for some λ∈R;
then we would have y=λx,v=λu so βy =βλx =αx and then βλ =α(because Ris a
domain), a contradiction to αR (βR. Also β·(y, v ) = α·(x, u) = (0,0) with β6= 0. This
shows that (0,0) 6= (y, v)∈T=T(M), so T(M)6= 0. Take X < M such that M=T⊕X.
We must have X6= 0, as otherwise (1,0) ∈Tso there would be a nonzero λ∈Rand a
µ∈Rsuch that λ·(1,0) = µ·(x, u)∈L. But then λ=µx, 0 = µu, so µ= 0 (u6= 0)
showing that λ= 0, a contradiction.
Now note that xand uare not invertible, as otherwise, for xinvertible, αx =βy implies
α∈βR so αR ⊆βR; the same can be inferred if uis invertible. Therefore x, u ∈Jas
Ris local so L⊆J×J. Hence J(M) = J×J/L so M/J (M) = R×R/L
J×J/L 'R×R/J ×J
which has dimension 2 as a module over the skewfield R/J. Since M=T⊕Xand Mis
finitely generated, then so are Tand Xand therefore J(X)6=Xand J(T)6=T. Then as
M
J(M)=T
J(T)⊕X
J(X)has dimension 2 over R/J, it follows that both T /J(T) and X/J (X)
are simple. Hence Tand Xare local, and as they are finitely generated, it follows that they
are generated by any element not belonging to their Jacobson radical. Let T0(respectively
X0) be the inverse images of T(and Xrespectively) in R×Rand t∈T0and s∈X0be
such that Rt +L=T0and Rs +L=X0. We have R×R=T0+X0=Rt +L+Rs +L=
(Rt +Rs) + L⊆(Rt +Rs) + J×J⊆R×Rso (Rt +Rs) + J×J=R×R. Therefore
we obtain Rt +Rs =R×Rbecause J×Jis small in R×R.
14 MIODRAG CRISTIAN IOVANOV
Write t= (p, q)∈T0. Then t=t+L∈Timplies that there is λ6= 0 in Rsuch that
λt = 0 ∈Mand therefore there is µ∈Rwith λ(p, q) = µ(x, u). We show that either
p /∈Jor q /∈J. Indeed assume otherwise: t= (p, q)∈J×J. Then we get Rt ⊆J×J.
Because Rt +Rs =R×Rwe see that R×R/J ×Jmust be generated over Rby the
image of s. This shows that the R/J module R×R/J ×J= (R/J)2has dimension 1 and
this is obviously a contradiction.
Finally, suppose p /∈Jso pis invertible; then the equations λp =µx,λq =µu imply
λ=µxp−1and µxp−1q=µu. But µ6= 0 because pis invertible and λ6= 0. Therefore we
obtain u=xp−1q; thus b=αu =αxp−1q=ap−1qshowing that b∈aR i.e. bR ⊆aR.
Similarly if qis invertible, we get aR ⊆bR.
Theorem 3.4. If Cis an (infinite dimensional) local coalgebra with the left f.g. Rat-
splitting property, then Cis a chain coalgebra.
Proof. We first show that every two principal left ideals of Rare comparable. Suppose
there are two left ideals of R,R·x0and R·y0that are not comparable. Then as they
have finite codimension and C∗is infinite dimensional, we have Rx0∩Ry06= 0 and take
06=αx0=βy0∈Rx0∩Ry0. Then the right ideals αR and βR are not comparable, as
otherwise, if for example αR ⊆βR, we would have a relation α=βλ so αx0=βλx0=βy0.
As β6= 0 we get λx0=y0because Ris a domain, and then Ry0⊆Rx0, a contradiction.
By Proposition 3.2 the set {λR |λR ⊆αR ∩βR}is Noetherian (relative to inclusion)
and let λR be a maximal element. If x∈αR ∩βR then by Proposition 3.3 we have that
xR and λR are comparable and by the maximality of λR it follows that xR ⊆λR, so
x∈λR. Therefore αR ∩βR =λR. Note that λ6= 0, because αR and βR are nonzero
ideals of finite codimension. Then we see that λR 'Ras right Rmodules, because Ris
a domain, and again by Proposition 3.3 any two principal right ideals of λR =αR ∩βR
are comparable, so the same must hold in RR. But this is in contradiction with the fact
that αR and βR are not comparable, and therefore the initial assertion is proved.
Now we prove that Jn/Jn+1 is a simple right module for all n. As R/J is semisimple
(it is a skewfield) and Jn/Jn+1 has an R/J module structure, it follows that Jn/Jn+1
is a semisimple left R/J-module and then Jn/J n+1 is semisimple also as R-module. If
we assume that it is not simple, then there are f, g ∈Jn\Jn+1 such that Rˆ
f= (Rf +
Jn+1)/J n+1 and Rˆg= (Rg+Jn+1)/J n+1 are different simple R-modules, so Rˆ
f∩Rˆg=ˆ
0 in
Jn/Jn+1. Then (Rf +Jn+1)∩(Rg +Jn+1 ) = Jn+1 which shows that Rf and Rg cannot
be comparable, a contradiction. As Jn=C⊥
n−1, we see that dim(Cn−1) = codim(Jn).
Then for n≥1, dim(Cn/Cn−1) = dim(Cn)−dim(Cn−1) = codimR(Jn)−codimR(Jn+1 ) =
dim(Jn/Jn+1) = dim(C0). Because C0is the only type of simple right C-comodule, this
last relation shows that the right C-comodule Cn/Cn−1must be simple. Therefore Cmust
be a chain coalgebra.
We may now combine the results of Sections 2 and 3 and obtain
Corollary 3.5. Let Cbe a co-local (infinite dimensional) coalgebra. Then Cis a left
(right) finite splitting coalgebra if and only if Cis a chain coalgebra. Moreover, if the base
field Kis algebraically closed or the coalgebra Cis pointed, then this is further equivalent
to the fact that Cis isomorphic to the divided power coalgebra.
Proof. This follows from Theorems 2.5, 2.7 and 3.4.
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 15
4. Serial coalgebras and General Examples
In this section we provide some nontrivial general examples of non-colocal coalgebras for
which this splitting property holds.
Lemma 4.1. Let C=D⊕Ebe coproduct of two coalgebras Dand E. Then Chas the
left f.g. Rat-splitting property if and only if Dand Ehave the Rat-splitting property.
Proof. Assume Chas the left f.g. Rat-splitting property. It is well known that the
category of modules over C∗'D∗×E∗is isomorphic to the product of the category of D∗-
modules with that of E∗-modules; in this respect, if Mis a left C∗-module, then M=N⊕P
where N=E⊥·M,P=D⊥·Mare C∗submodules that have an induced D∗=C∗/D⊥-
and respectively E∗=C∗/E⊥-module structure (since D⊥·N=0=E⊥·P). Also, one
can check that a D∗-module Xis rational if and only if it is rational as C∗-module with
its induced C∗-module structure: if ρ:X→X⊗Cis a C-comultiplication then we must
have ρ(X)⊆X⊗Dsince D⊥cancels X, and ρbecomes a D-comultiplication. Indeed, if
ρ(x) = P
i
xi⊗yi+P
j
x0
j⊗y0
jwith xi, x0
j∈Xassumed linearly independent, yi∈Dand
y0
j∈E, then for any e∗∈C∗such that e∗|D= 0, we have 0 = e∗·x=P
j
e∗(y0
j)x0
j, so
e∗(y0
j) = 0 by linearly independence. This shows that x0
j∈(D⊥)⊥=Dso x0
j= 0 for all
j. Thus, we obtain that Rat(D∗N) = Rat(C∗N) and Rat(E∗P) = Rat(C∗P), and we have
direct sums N=Rat(N)⊕N0and P=Rat(P)⊕P0in D∗Mand E∗M; but N0and P0also
have an induced C∗-module structure with E∗=D⊥acting as 0, and we finally observe
that this yields a direct sum of C∗modules M=Rat(C∗N)⊕N0⊕Rat(C∗P)⊕P0=
Rat(C∗M)⊕(N0⊕P0).
The other implication follows from Proposition 1.9.
We note now the following proposition which was also proved in [C], but with techniques
involving general results of M. Teply from [T1] and [T3].
Proposition 4.2. Assume Cis a cocommutative coalgebra. Then Cis is a f.g. Rat-
splitting coalgebra if and only if it is a finite coproduct of finite dimensional coalgebras and
infinite dimensional chain coalgebras. Moreover, these chain coalgebras are isomorphic to
the divided power coalgebra in any of the cases:
(i) the base field is algebraically closed;
(ii) Cis pointed.
Proof. Since Cis cocommutative, C=
n
L
i=1
Ci, where Ciare colocal subcoalgebras of C.
Now each of the Cimust have the splitting property for finitely generated modules by
Proposition 1.9, and therefore they must be either finite dimensional or be chain coalge-
bras. The converse follows from the previous Lemma and the results of Section 2. The
final assertion comes from Theorem 2.7.
Recall, for example from [F], 25.1.12 that a module Mis called serial if it is a direct
sum of uniserial (chain) modules; a ring Ris said to be left (right) serial if Ris serial
when regarded as left (right) R-module, and serial when Ris both left and right serial.
In analogy to these definitions, for a C-comodule Mwe say that Mis serial if it is
serial when regarded as C∗-module (so it is a direct sum of serial -or chain- comodules).
A coalgebra will be called left (right) serial if and only if it is serial as a right (left) C∗-
module, i.e. as a left (right) C-comodule, and serial if it is both left and right serial. These
definitions coincide with those in [CGT]. We note at this point that in our definitions, a
16 MIODRAG CRISTIAN IOVANOV
uniserial coalgebra is the same as a chain coalgebra, while a uniserial coalgebra in [CGT]
is understood as a homogeneous uniserial coalgebra, that is, a coalgebra Cthat is serial
and the composition factors of each indecomposable injective comodule are isomorphic
(see Definition 1.3 [CGT]). The following is a generalization of Proposition 1.6, [CGT].
Proposition 4.3. Let Cbe a coalgebra. Then the following are equivalent:
(i) Cis a right serial coalgebra and C0is finite dimensional.
(ii) C∗is a right serial algebra.
Consequently C∗is serial if an only if Cis serial and C0is finite dimensional, equivalently,
Cis serial and C∗is semilocal.
Proof. (i)⇒(ii) Let C0=
k
L
i=1
Sibe a decomposition of C0into simple right comodules,
E(Si) be an injective envelope of Sicontained in C; then C=
k
L
i=1
E(Si) in MCand
C∗M. Since any other decomposition of Cin MCis equivalent to this one, we have that
E(Si) are chain comodules and then E(Si)∗are chain modules by Proposition 2.2. As
C∗=
n
L
i=1
E(Si)∗in MC∗we get that C∗is right serial.
(ii)⇒(i) If C∗is right serial, it is a direct sum of uniserial modules C∗=L
i
Mi, each of
which has to be cyclic; then we easily see that these modules have to be local (for example
by [F], 25.4.1B) and indecomposable (a finitely generated local module is indecomposable).
Since there can be only a finite number of Mi’s in a decomposition of C∗, and each of the
Mi’s are local we get that C∗is semilocal, and then C∗/J is semisimple (J=C⊥
0). But
C∗/J =C∗/C⊥
0=C∗
0and thus C0is cosemisimple finite dimensional. Then C∗=
k
L
i=1
Mi
with Milocal uniserial. Let Ei= (L
j6=i
Mj)⊥; since L
j6=i
Mjis finitely generated, it is closed in
the finite topology of C∗and therefore E⊥
i=L
j6=i
Mj, so E∗
i'C∗/E⊥
i=C∗/(L
j6=i
Mj)'Mi.
Then by Proposition 2.2 we get that Eiis a right chain C-comodule; also because of the
anti-isomorphism of latices between the right subcomodules of Cand closed right C∗-
modules of C∗(see [DNR] or [I0], Theorem 1), we get that C=
k
L
i=1
Ei, with Eiright chain
comodules. Thus Cis a left serial coalgebra.
We say that a coalgebra Cis purely infinite dimensional serial if it is serial and the
uniserial left (and also the uniserial right) comodules into which it decomposes are infinite
dimensional. Equivalently, one can say that injective envelopes of any left (and also
every right) simple C-comodule is infinite dimensional. It is not difficult to see that for an
almost connected coalgebra it is enough to ask that only left injective envelopes are infinite
dimensional: let C=
k
L
i=1
E(Si) be a decomposition of Cwith Sisimple left comodules and
E(Si) an injective envelope for each Si. Assume Cis serial; then each E(Si) is uniserial.
Then if LnE(Si) is the n-th term in the Loewy series of E(Si) then Cn=
k
L
i=1
LnE(Si)
and E(Si) is infinite dimensional for all iif and only if Ln−1E(Si)6=LnE(Si) for all i
and all n≥0 (L−1= 0), equivalently, Cn/Cn−1'
k
L
i=1
LnE(Si)/Ln−1E(Si) has length k
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 17
(as a module) for all n. Since this last condition is a left-right symmetric condition, the
assertion follows. The next proposition provides the general example of this section:
Proposition 4.4. Let Cbe a purely infinite dimensional serial coalgebra which is almost
connected. Then Chas the left (and also the right) f.g. Rat-splitting property.
Proof. By the previous proposition, C∗is serial. Let Mbe a left finitely generated
C∗-module. Let C=
k
L
i=1
E(Si) be a decomposition as above, in CM, with E(Si) chain
comodules; then C∗=L
i∈I
E(Si)∗in C∗M. By Remark 2.4 and Proposition 1.3 the E(Si)∗’s
are noetherian. Hence C∗is Noetherian (both left and right, since Cis left and right serial).
This shows that every finitely generated C∗-module is also finitely presented. Then, by
[F], Corollary 25.3.4, M=
n
L
j=1
Mjwith Mjcyclic uniserial left C∗-modules. For each j
there are two possibilities:
•Mjis finite dimensional. Let mjbe a generator of the left C∗-module Mj, and then let
I=annC∗(mj). Then Iis a left ideal of C∗and it is finitely generated (C∗is Noetherian),
so I=X⊥,X⊆C(Lemma 1.1). Moreover, C∗/I 'C∗·mj=Mjand so Ihas finite
codimension since Mjis finite dimensional. Hence Xis finite dimensional and is a left
subcomodule of C. Then Mj'C∗/X ⊥'X∗, following that Mjis rational as a dual of
the rational right C∗-module X. So Rat(Mj) = Mj.
•Mjis infinite dimensional. Let mjbe a generator of Mjas before, and S=Mj/J(Mj)
which is a simple module because Mjis local since it is cyclic and uniserial. Let Pi=
E(Si)∗; since C∗/J =
k
L
i=1
Pi/JPiand all Piare local, there is some isuch that Pi/J Pi'S.
Then we have a diagram
Pi
p
u
~~}
}
}
}
}
}
}
}
Mjπ
//S//0
completed commutatively by usince Piis projective, and p, π are the canonical maps.
Note that uis surjective, since otherwise Im(u)⊆Ker (π) because Ker (π) is the only
maximal submodule of the finitely generated module Mi. This cannot happen since πu =
p6= 0. By Remark 2.4 and Proposition 1.3, we see that any nonzero submodule of
Pi=E(Si)∗has finite codimension. Then if Ker (u)6= 0, Mj= Im(u)'Pi/Ker (u) would
be finite dimensional, which is excluded by the hypothesis on Mj. This shows that uis an
isomorphism so Mj'E(Si)∗and we now get that Mjhas no finite dimensional submodules
besides 0 (again by Remark 2.4 and Proposition 1.3). This shows that Rat(Mj)=0
Finally, if we set F={j|Mjfinite dimensional}, we see that Rat(M) =
n
L
j=1
Rat(Mj) =
L
j∈F
Mj, and this shows that Rat(M) is a direct summand in M=
n
L
j=1
Mj.
Example 4.5. Let Kbe a field, q≥1and σ∈Sqbe a permutation of {1,2, . . . , q}.
Denote by Kq
σ[X]the vector space with basis xp,n with p∈ {1,2...,q}and n≥0. Define
18 MIODRAG CRISTIAN IOVANOV
a comultiplication ∆and a counit εon Kq
σ[X]as follows:
∆(xp,n) = X
i+j=n
xp,i ⊗xσi(p),j
ε(xp,n) = δn,0,∀p∈ {1,2, . . . , q}, n ≥0
It is easy to see that ∆is coassociative and εbecomes a counit, so Kq
σ[X]becomes a
coalgebra:
(∆ ⊗I)∆(xp,n) = (∆ ⊗I)( X
i+j=n
xp,i ⊗xσi(p),j)
=X
i+j=nX
s+t=i
xp,s ⊗xσs(p),t ⊗xσi(p),j
=X
s+t+j=n
xp,s ⊗xσs(p),t ⊗xσs+t(p),j
=X
s+u=n
xp,s ⊗X
t+j=u
xσs(p),t ⊗xσt(σs(p)),j
= (I⊗∆)( X
s+u=n
xp,s ⊗xσs(p),u)
= (I⊗∆)∆(xp,n)
Also, we have P
i+j=n
ε(xp,i)xσi(p),j =
n
P
i=0
δi,0xσi(p),n−i=xp,n and P
i+j=n
xp,iε(xσi(p),j ) =
P
i+j=n
xp,iδj,0=xp,n, showing that Kq
σ[X]together with these morphisms is a coalgebra.
Let Epbe the vector subspace of Kq
σ[X]with basis xp,n,n≥0. Note that the Ep’s are right
subcomodules of Kq
σ[X](obviously by the definition of ∆and ε). We show Epare chain
comodules in several steps:
(i) Let Ep,n =< xp,0, xp,1, . . . , xp,n >be the space with basis {xp,0, xp,1, .. . , xp,n}; it is
actually a right subcomodule of Ep. We note that Ep/Ep,n 'Eσn+1(p). Indeed, if xdenotes
the image of x∈Epin Ep/Ep,n, we have the following formulas for the comultiplication
of Ep/Ep,n
xp,m 7−→ X
i+j=m,i≥n+1
xp,i ⊗xσi(p),j =X
i+j=m−n−1
xp,i+n+1 ⊗xσi(σn+1(p)),j
for m≥n+ 1. The comultiplication of Eσn+1(p)is given by the formulas:
xσn+1(p),s 7−→ X
i+j=s
xσn+1(p),i ⊗xσi(σn+1 (p)),j
These relations show that the correspondence xp,i+n+1 7−→ xσn+1(p),i is an isomorphism of
Kq
σ[X]-comodules.
(ii) Let x=λ0xp,0+λ1xp,1+· · · +λp,nxp,n ∈Epand assume λn6= 0. Let f∈Kq
σ[X]∗
be equal to 1on xp,n and 0on the rest of the elements of the basis xt,i. Then one easily
sees that f·x=P
i+j≤n
λi+jxp,if(xσi(p),j ) = λnxp,0(the only terms remaining are the one
having j=n,i= 0, and such a term occurs only once in this sum). Since λn6= 0, we get
that xp,0belongs to the subcomodule generated by x. This shows that Ep,0is contained in
any submodule of Ep. This shows that that Epis colocal and Ep,0is its socle (which is a
simple comodule).
(iii) An inductive argument now sows that Ep,n are chain comodules for all n. Indeed, by
WHEN DOES THE RATIONAL TORSION SPLIT OFF FOR FINITELY GENERATED MODULES 19
the isomorphism in (i) and by (ii), we have that Ep,n+1/Ep,n 'Eσn+1 (p),0. This shows
that Epis a chain comodule by Proposition 2.2.
Since Kq
σ[X] =
q
L
p=1
Epas right Kq
σ[X]-comodules, we see that Kq
σ[X]is right serial, so it
is serial by Proposition 4.3 and even purely infinite dimensional, and thus constitutes an
example of a left and right f.g. Rat-splitting coalgebra by Proposition 4.4.
More examples can be obtainted by
Corollary 4.6. If C=D⊕Ewhere Dis a finite dimensional coalgebra and Eis a
purely infinite serial dimensional coalgebra, then Chas the both the left and the right f.g.
Rat-splitting property.
Remark 4.7.The fact that Kq
σ[X] is also left serial (and then purely infinite dimensional)
can also follow by noting that Kq
σ[X]op 'Kq
σ−1[X] as coalgebras. It is also interesting to
note that if σ=σ1. . . σris a decomposition of σinto disjoint cycles of respective lengths
q1, . . . , qr(or, more generally, into mutually commuting permutations), then there is an
isomorphism of coalgebras
Kq
σ[X]'
r
M
i=1
Kqi
σi[X]
We omit the proofs here. As a final comment, we note that by the above results, some
natural questions arise: is the concept of f.g. Rat-splitting left-right symmetric? That
is, does the left f.g. Rat-splitting property of a coalgebra also imply the right f.g. Rat-
splitting property? One should note that all the above examples have both the left and
the right Rat-splitting property. Also, it would be interesting to know whether a general-
ization of the results in the local case hold in the general non-cocommutative case as the
cocommutative case of this section and the above non-cocommutative examples seem to
suggest: if Chas the left f.g. Rat-splitting property, can it be written as a direct sum
of finite dimensional injectives and infinite dimensional chain injectives (likely in CM),
or maybe a decomposition of coalgebras as in Corollary 4.6, and to what extent such a
decomposition would characterize this property?
Acknowledgment
The author wishes to thank C. N˘ast˘asescu for useful remarks on the subject as well as for
his support throughout the past years.
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Miodrag Cristian Iovanov
University of Bucharest, Faculty of Mathematics, Str. Academiei 14
RO-010014, Bucharest, Romania
and
State University of New York @ Buffalo
Department of Mathematics, 244 Mathematics Building
Buffalo, NY 14260-2900, USA
E–mail address:yovanov@gmail.com; e-mail@yovanov.net
