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arXiv:1502.03130v2 [math.CT] 1 Mar 2015
A TORSION THEORY IN THE CATEGORY OF COCOMMUTATIVE HOPF ALGEBRAS
MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
Abstract. The purpose of this article is to prove that the category of cocommutative Hopf K-
algebras, over a field Kof characteristic zero, is a semi-abelian category. Moreover, we show that this
category contains a torsion theory whose torsion-free and torsion parts are given by the category of
groups and by the category of Lie K-algebras, respectively.
1. Introduction
The starting point of this article on Hopf algebras is a well-known result due to A. Grothendieck,
as outlined in [Swe69], saying that the category of finite-dimensional, commutative and cocommutative
Hopf K-algebras over a field Kis abelian. This result was extended by M. Takeuchi to the category of
all commutative and cocommutative Hopf K-algebras, not necessarily finite-dimensional [Tak72]. The
category HopfK,coc of cocommutative Hopf K-algebras is not additive, thus it can not be abelian.
In the present paper we investigate some of its fundamental exactness properties, showing that it is a
homological category (Section 3), and that it is Barr-exact (Section 5), leading to the conclusion that
the category HopfK,coc is semi-abelian [JMT02] when the field Kis of characteristic zero (Theorem
5.1) This result establishes a new link between the theory of Hopf algebras and the more recent one of
semi-abelian categories, both of which can be viewed as wide generalizations of group theory. Since a
category Cis abelian if and only both Cand its dual Cop are semi-abelian, this observation can be seen
as a “non-commutative” version of Takeuchi’s theorem mentioned above. This result was independently
obtained by Clemens Berger and Stephen Lack.
We furthermore prove the existence of a non-abelian torsion theory (T,F)in Hopf K,coc , where the
torsion subcategory Tis the category of primitive Hopf K-algebras, which is equivalent to the category
of Lie K-algebras, and the torsion-free subcategory Fis the category of group Hopf K-algebras, which
is equivalent to the category of groups.
The categories of groups and of Lie K-algebras are two typical examples of semi-abelian categories:
this shows again that the theories of cocommutative Hopf algebras and of semi-abelian categories are
strongly intertwined. The category HopfK,coc inherits some fundamental exactness properties from
groups and Lie algebras thanks to the well-known canonical decomposition of a cocommutative Hopf
algebra into a semi-direct product of a group Hopf algebra and a primitive Hopf algebra (a result
associated with the names Cartier-Gabriel-Kostant-Milnor-Moore). The present work opens the way to
some new applications of categorical Galois theory [Jan91] in the category of cocommutative Hopf K-
algebras, since the reflection from this category to the torsion-free subcategory of group Hopf algebras
enjoys all the properties needed for this kind of investigations, as we briefly explain in Section 4.
Date: March 3, 2015.
2010 Mathematics Subject Classification. 18E40, 18E10, 20J99, 16T05, 16S40.
Key words and phrases. semi-abelian category, torsion theory, cocommutative Hopf algebra.
1
2 MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
2. Preliminaries
2.1. Semi-abelian categories.
Semi-abelian categories [JMT02] are finitely complete, pointed, exact in the sense of M. Barr [Bar71],
protomodular in the sense of D. Bourn [Bou91], with finite coproducts. These categories have been
introduced to capture some typical algebraic properties valid for non-abelian algebraic structures such
as groups, Lie algebras, rings, crossed modules, varieties of Ω-groups in the sense of P. Higgins [Hig56]
and compact groups. As already mentioned in the introduction, every abelian category is in particular
semi-abelian.
Although protomodularity is a property that can be expressed in any category with finite limits, in
the pointed context, i.e. when there is a zero object 0in C, protomodularity amounts to the fact that
the following formulation of the Split Short Five Lemma holds: given a commutative diagram
0//Kk//
κ
A
f//
α
B
β
s
oo
0//K′
k′//A′f′
//B′
s′
oo
where k=ker(f),k′=ker(f′),f◦s= 1B, and f′◦s′= 1B′(i.e. fare f′are split epimorphisms
with sections sand s′), if both κand βare isomorphisms, then so is α.
A useful lemma which holds in protomodular categories is the following ([Bou91], Proposition 11):
Lemma 2.1. Given a split short exact sequence in a pointed protomodular category
0//Kk//A
f//B
s
oo//0
the pair of morphisms (k, s)is jointly epimorphic.
Any protomodular category Cis a Mal’tsev category [CLP91]: this means that every (internal)
reflexive relation Cis an (internal) equivalence relation. Recall that a reflexive relation on an object X
is a diagram of the form
(2.1) R
p1//
p2//X,
δ
oo
where p1and p2are jointly monic, and p1◦δ= 1X=p2◦δ; such a reflexive relation Ris an equivalence
relation when, moreover, there exist σ:R−→ Rand τ:R×XR−→ Ras in the diagram
R×XRτ//R
σ
p1//
p2//X
δ
oo
such that p1◦σ=p2and p2◦σ=p1(symmetry), and p1◦τ=p1◦π1and p2◦τ=p2◦π2(transitivity).
In the present article, by a regular category is meant a finitely complete category where every morphism
can be factorized as a regular epimorphism followed by a monomorphism, and where regular epimorphisms
are pullback stable. A regular category Cis said to be Barr-exact if, moreover, every equivalence relation
is effective, i.e. every equivalence relation is the kernel pair of a morphism in C. A category which is
pointed, protomodular and regular is said to be homological [BB04]. In this context several basic
diagram lemmas of homological algebra hold true (such as the snake lemma, the 3-by-3-Lemma, etc.).
A TORSION THEORY I N THE CATEGORY OF COCOMMUTATIVE HOPF ALGE BRAS 3
We end these preliminaries with the following diagram indicating some implications between the
different contexts recalled above:
semi-abelian
tt✐✐✐✐✐✐✐✐✐
**
❚
❚
❚
❚
❚
❚
❚
❚
❚
homological
,,
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩
❩Barr-exact
protomodular
regular
Mal’tsev
2.2. The category HopfK,coc of cocommutative Hopf K-algebras.
The category we study in this article is the category of Hopf K-algebras over a field K, denoted by
Hopf K. The objects in Hopf Kare Hopf K-algebras, i.e. sextuples (H, M , u, ∆, ǫ, S)where (H, M , u)
is a K-algebra and (H, ∆, ǫ)is a K-coalgebra, such that these two structures are compatible, i.e. maps
Mand uare K-coalgebras morphisms, making (H, M , u, ∆, ǫ)aK-bialgebra. The linear map Sis
called the antipode, and makes the following diagram commute:
H
ǫ
))
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
❙
∆//H⊗HS⊗id //
id⊗S//H⊗HM//H
K
u
55
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
❦
Morphisms in Hopf Kare exactly morphisms of K-bialgebras (i.e. morphisms that are both mor-
phisms of K-algebras and K-coalgebras), as morphisms of K-bialgebras always preserve antipodes.
To denote the comultiplication map of a Hopf algebra H, we will use the Sweedler notation: ∀h∈H,
∆(h) = h1⊗h2by omitting the summation sign. A Hopf algebra His said to be cocommutative if its
comultiplication map ∆makes the following diagram commute, where ∀x, y ∈H,σ(x⊗y) := y⊗x
H⊗H
σ
%%
❑
❑
❑
❑
❑
❑
❑
❑
❑
❑
H∆//
∆;;
✇
✇
✇
✇
✇
✇
✇
✇
✇H⊗H
The category of cocommutative Hopf K-algebras will be denoted by Hopf K,coc . In the category
Hopf K,coc there are two full subcategories which will be of importance for our work: the category
GrpHopfKof group Hopf algebras, and the category PrimHopf Kof primitive Hopf algebras, whose
definitions we are now going to recall.
(1) The group Hopf algebra on a group G, denoted by K[G], is the free vector space on Gover the
field K, i.e. K[G] = {Pg∈Gαgg, where (αg)g∈Gis a family of scalars with only a finite number
being non zero}and {g|g∈G}is a basis of K[G]. The group Hopf algebra K[G]can be
equipped with a structure of cocommutative Hopf algebra, by taking the multiplication induced
by the group law, and comultiplication ∆ : K[G]−→ K[G]⊗K[G], counit ǫ:K[G]→K
and antipode S:K[G]→K[G]the linear maps defined on the base elements respectively by
∆(g) = g⊗g,ǫ(g) = 1 and S(g) = g−1,∀g∈G.
4 MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
This assignment defines a functor K[−] : Grp →Hopf Kfrom the category of groups to the
category of Hopf algebras, which has a right adjoint G:Hopf K→Grp, that associates to any
Hopf algebra Hits group of grouplike elements G(H) = {x∈H|∆(x) = x⊗x, ǫ(x) = 1}. If
we restrict ourselves from the category of Hopf algebras to the full subcategory GrpHopfKof
group Hopf algebras, then the functor K[−]: Grp →GrpHopf Kis by construction surjective
on objects and moreover even an isomorphism of categories. Indeed, let us recall why the
functor G ◦ K[−]is the identity on the category of groups. If Gis a group and K[G]the group
Hopf K-algebra on G, one clearly has that G⊆ GK[G](in fact, this inclusion is the unit
of the adjunction described above). Conversely, let x=Pg∈Gαgg∈K[G]be a group-like
element. We have ∆(x) = Pg∈Gαgg⊗gbut also ∆(x) = x⊗x=Pg,h∈Gαgαhg⊗h. Since
{g⊗h|g, h ∈G}forms a basis of K[G]⊗K[G], we have ∀g, h ∈G:α2
g=αg, and αgαh= 0
if g6=h. It follows that all αg’s should be zero except one that should be 1K, thus xis in G.
(2) The universal enveloping algebra of a Lie algebra L, denoted by U(L), is defined by the quotient
U(L) = T(L)/I, where T(L)is the tensor algebra on the vector space underlying L, and Iis the
two-sided ideal of T(L)generated by the elements of the form x⊗x′−x′⊗x−[x, x′],∀x, x′∈L.
Remark that the elements of Lgenerate U(L)as an algebra. The universal enveloping algebra
U(L)can be equipped with a structure of cocommutative (and non commutative) Hopf algebra,
by taking the concatenation as multiplication, and comultiplication ∆ : U(L)−→ U(L)⊗U(L),
counit ǫ:U(L)→Kand antipode S:U(L)→U(L)the algebra maps defined on the
generators by ∆(x) = x⊗1 + 1 ⊗x,ǫ(x) = 0 and S(x) = −x,∀x∈L.
Recall that for any Hopf algebra Han element x∈His called a primitive element if
∆(x) = x⊗1 + 1 ⊗x(and consequently, ǫ(x) = 0). The above constructions lead to a
pair of adjoint functors, where the functor U:LieAlgK→Hopf Kis a left adjoint to
P:Hopf K→LieAlgK. We can now consider the category PrimHopf K, which is the
full subcategory of Hopf Kwhose objects are primitive Hopf algebras, that is Hopf algebras
generated as algebra by primitive elements. In the case where Kis of characteristic 0, the
category PrimHopfKis known to be isomorphic to the category LieAlgKof Lie K-algebras
[MM65, Theorem 5.18].
Remark 2.2.As can be seen from the formula for comultiplication, both group Hopf algebras and
primitive Hopf algebras are cocommutative. Therefore the categories GrpHopfKand PrimHopfKare
also full subcategories of Hopf K,coc . The functors U, P, K [−],Gand their adjunctions are represented
in the following diagram:
Grp
K[−]
⊥//Hopf K,coc
P
⊤//
G
ooLieAlgK
U
oo
3. The category of cocommutative Hopf algebras over a field of characteristic
zero is homological
The category HopfK,coc is certainly pointed, with the zero object K, that will be denoted by 0, from
now on. HopfK,coc is complete and cocomplete, since it is locally presentable [Por11]. We will now
establish its protomodularity and regularity.
3.1. Protomodularity of the category HopfK.Let us consider the following commutative diagram
of short exact sequences in the category Hopf K,coc:
A TORSION THEORY I N THE CATEGORY OF COCOMMUTATIVE HOPF ALGE BRAS 5
(3.1) 0//A//
idA
C1//
θ
B
idB
//0
0//A//C2//B//0
Thanks to the explicit descriptions of equalizers and coequalizers given in [AD95] one can easily prove
that the kernel and the cokernel of θare the zero object. This proves that θis a monomorphism and an
epimorphism of Hopf K-algebras. Since monomorphisms are injections and epimorphisms are surjections
in the category Hopf K,coc [Chi10, NT94], this shows that θis an isomorphism of Hopf K-algebras and
so the category Hopf K,coc is protomodular.
The category Hopf K,coc is actually even strongly protomodular, [Bou00] since it has finite limits
and it can be viewed as the category of internal groups in the category of cocommutative K-coalgebras
(see [Str07], for instance. The category of cocommutatitve K-coalgebras is studied in detail in [GP87]).
This argument is general and can be applied to cocommutatitve Hopf algebras in any braided monoidal
category.
Nevertheless, this result holds more generally for the category Hopf Kof arbitrary Hopf K-algebras,
that is also protomodular. This follows from the following result (Lemma 3.2.19 in [AD95]) by taking into
account the fact that any split extension induces a cleft exact sequence in the sense of by Andruskiewitsch
and Devoto [AD95]:
Theorem 3.1. Consider a diagram of the form (3.1) in the category of Hopf K-algebras, where the
above exact sequence is cleft. Then the bottom exact sequence is also cleft and θis an isomorphism.
3.2. Semi-direct products of cocommutative Hopf algebras. Let Bbe a cocommutative Hopf
algebra. A B-module Hopf algebra is an Hopf algebra Athat is at the same time a left B-module with
action ρ:B⊗A→A, ρ(b⊗a) = b·asuch that ρis a morphism of bialgebras. The semi-direct product
(also known as smash product) of Band A, denoted by A⋊B, is the Hopf algebra whose underlying
vector space is the tensor product A⊗Band with the following structure. The unit is uA⋊B=uA⊗uB
and multiplication given by
(a⊗b)(a′⊗b′) = a(b1·a′)⊗b2b′,
for all a, a′∈Aand b, b′∈B. The coalgebra structure is given by the tensor product coalgebra, i.e.
∆A⋊B= (idA⊗σ⊗idB)(∆A⊗∆B)and ǫA⋊B=ǫA⊗ǫB. The antipode is given by SA⋊B(a⊗b) =
SB(b1)·SA(a)⊗SB(b2).
The following Lemma is a reformulation of Theorem 4.1in [Mol77]:
Lemma 3.2. Every split short exact sequence in Hopf K,coc
0//Ak//H
p//B
s
oo//0
is canonically isomorphic to the semi-direct product exact sequence
0//Ai1//A⋊B
p2//B
i2
oo//0
where i1=idA⊗uB,i2=uA⊗idBand p2=ǫA⊗idB.
6 MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
Proof. The arrow h:A⋊B→Hin the diagram below is given by h(a⊗b) = k(a)s(b)for all
a⊗b∈A⋊B.
0//Ai1//A⋊B
p2//
h
B
i2
oo//0
0//Ak//H
p//B
s
oo//0
This is a morphism of split short exact sequences, and therefore his an isomorphism by protomodularity
of HopfK,coc.
We use this lemma to reformulate the well-known structure theorem for cocommutative Hopf algebras
over an algebraically closed field of characteristic zero (see for instance [Swe69], page 279 in combination
with Lemma 8.0.1(c)) in terms of split exact sequences.
Theorem 3.3 (Cartier-Gabriel-Moore-Milnor-Kostant).Every cocommutative Hopf K-algebra H, over
an algebraically closed field Kof characteristic 0, is isomorphic to the semi-direct product
H∼
=U(LH)⋊K[GH]
of the universal enveloping algebra of a Lie algebra U(LH)with the group Hopf algebra K[GH], where
LHand GHare given respectively by the space of primitive elements and set of group-like elements
of H. Consequently, for each H, there exists a canonical split exact sequence of cocommutative Hopf
algebras of following form
0//U(LH)iH//H
pH//K[GH]
sH
oo//0
3.3. Regularity of the category HopfK,coc.
3.3.1. The regular epimorphism/monomorphism factorization in Hopf K,coc .Let f:A−→ Bbe a
morphism of cocommutative Hopf K-algebras. By the protomodularity of HopfK,coc, it is well known
that regular epimorphisms are the same as cokernels, i.e. normal epimorphisms. Thus, to construct
the regular epimorphism/monomorphism factorization of the morphism f, we consider the kernel i:
Hker(f)→Aof fand the cokernel p:A→HC oker(i)of i, both computed in the category of
Hopf K,coc :
Hker(f)i//Af//
p
B
HCoker(i)22m
22
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
❢
The existence of this factorization msuch that m◦p=ffollows from the universal property of
the cokernel pof i. It remains to prove that mis a monomorphism, which is equivalent in Hopf K,coc
to showing that mis an injection. In the category Hopf K,coc the above factorization is obtained as
in the category of vector spaces since HCoker(i) = A
AHker(f)+A(by [AD95]), and ker(f) =
AHker(f)+A, (by [Shu88, New75]).
Note that any epimorphism of cocommutative Hopf K-algebras is then a normal epimorphism, and
the following classes of epimorphisms coincide in HopfK,coc:
normal epis = regular epis = epis = surjective morphisms.
A TORSION THEORY I N THE CATEGORY OF COCOMMUTATIVE HOPF ALGE BRAS 7
Remark 3.4.The two full subcategories GrpHopf Kand PrimHopf K, of HopfK,coc, are closed
under quotients since morphisms of Hopf K-algebras preserve group-like and primitive elements.
3.3.2. Pullback stability of regular epimorphisms in the category Hopf K,coc .To prove the pullback
stability of regular epimorphisms in the category Hopf K,coc , the approach we follow is to apply the
pullback stability of regular epimorphisms in the two full subcategories GrpHopf Kand PrimHopf K
of Hopf K,coc, which are both semi-abelian, and closed under pullbacks and quotients in HopfK,coc.
From the regularity of these two categories and the decomposition Theorem 3.3 we deduce the regularity
of HopfK,coc
Remark 3.5.In the following we shall assume that Kis an algebraically closed field. It can be checked
that this is not a restriction: indeed, given a field Kand φ:K→Kan embedding of Kin an algebraic
closure K, one has the adjunction
Hopf K,coc
Rφ
⊥11HopfK,coc
Lφ
qq
where Rφis the “restriction of scalars functor” and Lφ=− ⊗KKits left adjoint, the “extension of
scalars” functor. Being a left adjoint, Lφpreserves regular epimorphisms and moreover Lφreflects regular
epimorphisms and preserves finite limits. Accordingly, knowing that Hopf K ,coc is regular (respectively,
exact), one can deduce from this that Hopf K,coc is regular (resp. exact) as well.
The following result concerning split short exact sequences in Hopf K,coc will be useful in the proof
of the regularity of this category:
Lemma 3.6. Given the following commutative diagram of split short exact sequences in Hopf K,coc:
0//A1iH1
//
hA
H1pH1//
h
B1
hB
//
sH1
ss0
0//A2iH2
//H2pH2//B2//
sH2
ss0
We have that his surjective if and only if both hAand hBare surjective.
Proof. We apply Lemma 3.2 to the exact sequences in the statement of the Lemma, we obtain the
following commutative diagram which is canonically isomorphic to the previous one:
0//A1i1//
hA
A1⋊B1p1//
hA⊗hB
ξ1
rrB1
hB
//
s1
qq0
0//A2i2//A2⋊B2p2//
ξ2
rrB2//
s2
qq0
Hence, the morphism his surjective if and only if hA⊗hB:A1⋊B1→A2⋊B2is surjective.
If hAand hBare surjective, then hA⊗hBis surjective by considering this morphism on its underlying
vector space. For the converse implication, if hA⊗hBis surjective, let us note that for each semi-direct
product Ai⋊Bi, the underlying coalgebra is exactly the categorical product of the coalgebras Aiand
Bi; we denoted ξi=idAi⊗ǫBifor the coalgebra-projection of Ai⋊Bionto Ai(which is not a Hopf
algebra morphism).
8 MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
It is clear that hA◦ξ1=ξ2◦h(as coalgebra morphisms). Since ξ2is a split epimorphism and his
surjective, we conclude that hAis surjective. It is clear that hBis surjective whenever his.
Theorem 3.7. Consider the following pullback (P, πA, πB)in the category HopfK,coc:
(3.2) PπB//
πA
B
g
Af//C
if fis a regular epimorphism then πBis a regular epimorphism.
Proof. The fact that the subcategory GrpHopfK(resp., PrimHopf K) is semi-abelian and closed in
Hopf K,coc under pullbacks and regular epimorphisms implies that regular epimorphisms are pullback
stable whenever the Hopf algebras A,Band Cin diagram of the form (3.2) belong to GrpHopfK(or
to PrimHopf K, respectively).
Let us now consider A,Band Ccocommutative Hopf K-algebras over a field Kof characteristic
zero. By Theorem 3.3, we have: A∼
=U(LA)⋊K[GA],B∼
=U(LB)⋊K[GB]and C∼
=U(LC)⋊K[GC].
We will consider the following commutative diagram where (P1, πA1, πB1)is the pullback of f1and
g1,(P2, πA2, πB2)is the pullback of f2and g2.
P1
πB1//
πA1
iP++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱U(LB)iB
++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
g1
PπB//
πA
pP++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱B
pB++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
g
P2
πB2//
sP
nn
πA2
K[GB]
sB
nn
g2
U(LA)f1//
iA++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱U(LC)iC
++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
Af//
pA++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱C
pC++
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
❱
K[GA]
sA
nn
f2//K[GC]
sC
nn
When fis surjective, the surjectivity of f1and f2follow both from Lemma 3.6 applied to the lower
part of the diagram. The front and back faces of the diagram are in GrpHopfKand in PrimHopf K,
respectively, thus πB1and πB2are surjective. Applying Lemma 3.6 again (in converse direction), we
obtain that πBis also surjective.
4. A torsion theory in the category HopfK,coc
In the non-abelian context of homological categories it is natural to define and study a general notion
of torsion theory, that extends the one introduced by S.E. Dickson in the frame of abelian categories
[Dic98]. This study was first initiated in [BG06], and further developed in [DEG12] [EG14], also in
relationship with semi-abelian homology theory.
Let us recall the definition of a torsion theory in the homological context:
A TORSION THEORY I N THE CATEGORY OF COCOMMUTATIVE HOPF ALGE BRAS 9
Definition 4.1. In a homological category C, a torsion theory is given by a pair (T,F) of full and
replete (i.e. isomorphism closed) subcategories of Csuch that:
i. For any object Xin C, there exists a short exact sequence:
0//TtX//XηX//F//0
where 0is the zero object in C,T∈Tand F∈F.
ii. The only morphism f:T−→ Ffrom T∈Tto F∈Fis the zero morphism.
When (T,F) is a torsion theory, Tis called the torsion subcategory of C, and Fits torsion-free
subcategory. Among the many examples in the homological context, let us just mention the following
ones:
Example 4.2.
(1) Every torsion theory in an abelian category C. For instance: the pair (Abt,Abtf ) in the
category of abelian groups Ab. Where Abtand Abtf denote the full and replete subcategories
of the category of abelian groups whose objects are torsion and torsion-free abelian groups,
respectively.
(2) The pair (NilCRng,RedCRng) in the category of commutative rings CRng, where NilCRng
and RedCRng denote the full subcategories of nilpotent commutative rings, and of reduced
commutative rings (i.e. commutative rings without non trivial nilpotent elements), respectively.
(3) The pair (Grp(Ind),Grp(Haus)) in the category of topological groups Grp(Top), where
Grp(Ind)and Grp(Haus)denote the full subcategories of indiscrete groups and of Hausdorff
groups, respectively.
From now on, in the homological category of cocommutative Hopf K-algebras, Twill always de-
note the (full) subcategory PrimHopfKof primitive Hopf algebras, and Fthe (full) subcategory
GrpHopfKof group Hopf algebras.
Thanks to Theorem 3.3, we know that we can associate the following short exact sequence with any
cocommutative Hopf K-algebra H:
0//U(LH)iH//HpH//K[GH]//0
Any morphism from T∈Tto F∈Fis the zero morphism Hopf K,coc since any primitive Hopf
algebra U(L)is generated by its primitive elements, which are preserved by a morphism of Hopf algebras
and a group Hopf algebra K[G]does not contain any primitive element. It follows that (T,F) is a
torsion theory which is actually hereditary, namely the torsion subcategory Tis closed in Hopf K,coc
under subobjects:
Proposition 4.3. The pair (T,F) is a hereditary torsion theory in Hopf K,coc .
Proof. Given any monomorphism m:AU(L)in HopfK,coc with codomain a primitive Hopf algebra
in T. The morphism mpreserves group-like elements as a Hopf algebra morphism, and the group
of group-like elements is trivial in a primitive Hopf algebra [Sch94]: since mis injective, by applying
Theorem 3.3 to Awe see that Acan not contain any group-like element (different from 1A) and therefore
has to be primitive.
10 MARINO GRAN, GABRIEL KADJO, AND JOOST VERCRUYSSE
As it follows from the results in [BG06] the reflector Iin the adjunction
F
H
⊥11Hopf K,coc
I
rr
is semi-left-exact in the sense of Cassidy-Hebert-Kelly [CHK85], i.e. it preserves all pullbacks of the form
Pp2//
p1
Y
ηY
H(X)H(f)//HI(Y)
where ηYis the Y-component of the unit of the adjunction and flies in the subcategory F. The
adjunction is then admissible in the sense of categorical Galois theory [Jan91]: this opens the way to
further investigations in the direction of semi-abelian homology [DEG12]. The fact that the torsion theory
is hereditary and HopfK,coc a homological category implies that the corresponding Galois coverings are
precisely those regular epimorphisms f:A→Bin HopfK,coc with the property that the kernel Hker(f)
is in F(by applying Theorem 4.5in [GR07]). This fact is crucial to describe generalized Hopf formulae
for homology, as explained in [EG14].
5. The category of cocommutative Hopf algebras over a field of characteristic
zero is semi-abelian
In order to prove that HopfK,coc is semi-abelian, it remains to show that equivalence relations are
effective. For this, we shall show that any equivalence relation Ras in diagram (2.1) in Hopf K,coc is
the kernel pair of its coequalizer q:X→X. We first apply Theorem 3.3 to the equivalence relation R,
obtaining the following commutative diagram, where the morphisms q1,q2and qare the coequalizers
of p11 and p21,p12 and p22 ,p1and p2, respectively, and (Eq(q),π1,π2) is the kernel pair of q.
0//U(LR)
i′
R
--
iR//
p21
p11
R
p2
p1
pR//
θ
''
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆K[GR]
p22
p12
//0
Eq(q)
π2
ww♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
π1
ww♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
♣
p′
R
::
0//U(LX)iX//
q1
∆X1
OO
XpX//
∆X
OO
q
K[GX]
q2
∆X2
OO
//0
U(LX)iX
//XpX//K[GX]
The subcategory PrimHopf Kof primitive Hopf algebras is semi-abelian, is closed under products
[MM65] and subobjects (Proposition 4.3), thus under pullbacks in Hopf K,coc ; since it is also closed
under quotients, the left column is exact, i.e. U(LR) = Eq(q1). On the other hand, let us then explain
why the right column is also exact. First observe that K[GR]is a reflexive relation on K[GX]: for this,
one can use Lemma 5.2in [BG06] and the fact that the reflector I:Hopf K,coc →GrpHopf Kpreseves
A TORSION THEORY I N THE CATEGORY OF COCOMMUTATIVE HOPF ALGE BRAS 11
binary products. The category GrpHopf Kof group Hopf algebras is exact Mal’tsev and closed under
pullbacks and quotients in Hopf K,coc: it follows that K[GR]is the kernel pair of q2in Hopf K,coc.
The universal property of (Eq(q),π1,π2) gives the unique arrow θ:R→Eq(q)with π1◦θ=p1and
π2◦θ=p2. By applying the Split Short Five Lemma to the commutative diagram
0//U(LR)iR//
idU(LR)
RpR//
θ
K[GR]
idK[GR]
//0
0//U(LR)i′
R
//Eq(q)p′
R
//K[GR]//0
it follows that the morphism θis an isomorphism, and the equivalence relation Ris effective. One
accordingly has the following:
Theorem 5.1. For any field Kof characteristic 0, the category Hopf K,coc of cocommutative Hopf
K-algebras is semi-abelian.
Acknowledgement
The authors are very grateful to Clemens Berger and Stephen Lack for many useful suggestions on
the subject of the article.
Funding : The second author is funded by a grant from the National Fund for Scientific Research of Belgium
F.R.S. - FNRS.
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(Marino Gran, Gabriel Kadjo) Institut de recherche en Mathématique et Physique, Université catholique
de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
E-mail address:marino.gran@uclouvain.be
E-mail address:gabriel.kadjo@uclouvain.be
(Joost Vercruysse) Département de Mathématique, Université Libre de Bruxelles, Campus de la
Plaine, Boulevard du Triomphe, 1050 Bruxelles, Belgium.
E-mail address:jvercruy@ulb.ac.be
