A Hopf algebra having a separable Galois extension is finite dimensional

0.13 MB·English

by  Juan Cuadra

A Hopf algebra having a separable
Galois extension is (cid:28)nite dimensional
7
0
∗
0
2 J. Cuadra

n Universidad de Almería
a
J Depto. Álgebra y Análisis Matemáti
o

1
E-04120 Almería, Spain
1

email: j
diazual.es
]
G
S
.
h 2000 Mathemati
s Subje
t Classi(cid:28)
ation: Primary 16W30.
t
a
m
Abstra
t
[
It is shown that a Hopf algebra over a (cid:28)eld admitting a Galois
2
extension separable over its subalgebra of
oinvariants is of (cid:28)nite di-
v
3 mension. This answers in the a(cid:30)rmative a question posed by Beattie
1 et al. in [Pro
. Amer. Math. So
. 128 No. 11 (2000), 3201-3203℄.
6
2
1
6 Introdu
tion
0
/
h
The notion of Hopf-Galois extension, as known nowadays, is due to Kreimer
t
a and Takeu
hi [5℄ and it is a mainstay of Hopf algebra theory. It emanated
m
from the work of Chase and Sweedler about a
tions of Hopf algebras on
:
v rings [2℄. When the Hopf algebra is the
oordinate algebra of an a(cid:30)ne group
i
X s
heme that a
ts on an a(cid:30)ne algebra, the above notion may be interpreted
r in geometri
terms and it is linked with the
on
ept of torsor or prin
ipal
a
homogeneous spa
e [7, page 168℄. Faithfully (cid:29)at Hopf-Galois extensions are

urrently widelya

epted asa non
ommutative
ounterpart of thisgeometri

on
ept.
H = k[G] G k
For , the group algebra of a group over a (cid:28)eld , [6, Theo-
H
rem 8.1.7℄ shows that an -Galois extension is pre
isely a strongly graded
k A A = ⊕ A
σ∈G σ
algebra. That is, a -algebra admitting a de
omposition as
k A A = A σ,τ ∈ G
σ τ στ
-ve
tor spa
e and satisfying for all . The subalgebra
∗
This resear
h was supported by proje
t MTM2005-03227from MCYT and FEDER.
1
A A e G
e
of
oinvariants of is ( the identity element of ). N st ses
u et al.
A ⊂ A
e

hara
terized in [4, Proposition 2.1℄ when the extension is separable.
A A G
e
In parti
ular, they found that if is separable over , then is (cid:28)nite.
It was investigated in [1℄ if an analogous result
ould hold for general Hopf
H k
algebras. To be morepre
ise, suppose that is a Hopf algebraover having
H
a Hopf-Galois extension separable over its subalgebra of
oinvariants. Is
ne
essarily of (cid:28)nite dimensional? A positive response was given under the
H
additional assumption that to be
o-Frobenius.
In this short note we answer this question in the a(cid:30)rmative. Our proof
relies on a
ombination of the properties of the separability idempotent, the
Galois maps and an old result of Sweedler. As often happens in Hopf alge-
bra theory, the new proof seems more natural and simpler than the original
H = k[G]
proof for . Combining our result with one of Cohen and Fis
hman
we provide a
hara
terization of separable Hopf-Galois extensions that gen-
eralizes to Hopf algebras the above-mentioned one of N st ses
u et al. for
strongly graded rings.
We (cid:28)x some notation and re
all the de(cid:28)nition of Hopf-Galois extension.
We expe
t that the reader is familiar with the rudiments of Hopf algebra
H
theory. Our
onventions and notations are those of [6℄. Throughout
k ε
stands for a Hopf algebra over a (cid:28)eld . Its
ounit is denoted as usual by .
k
All ve
tor spa
es
onsidered in the sequel are over , map means linear map,
⊗ k
and denotes the tensor produ
t over .
H A ρ : A → A⊗H
For a right -
omodule algebra with stru
ture map ,
Aco(H) = {a ∈ A : ρ(a) = a⊗1 }
H
its subalgebra of
oinvariants is denoted
B
by . The Galois maps are given by
′ ′ ′
can : A⊗B A → A⊗H, a⊗B a 7→ Xaa(0) ⊗a(1),
(a′)
′ ′ ′
can : A⊗B A → A⊗H, a⊗B a 7→ Xa(0)a ⊗a(1).
(a)
B ⊂ A H
Re
all from [6, De(cid:28)nition 8.1.1℄ that is said to be an -Galois ex-
can H
tension if is an isomorphism. It is known that for having bije
tive
antipode, any of the two maps to bceanisomorphism may be requirceadn′in the
de(cid:28)nition of Galois extension sin
e is bije
tive if and only if is bi-
je
tive, [6, page 124℄. However, we will not employ this fa
t. Furthermore,
can
we will only use that is surje
tive.
2
1 The main theorem
H A H
Theorem. Let be a Hopf algebra and let be a right -Galois extension
B H
separable over its subalgebra of
oinvariants . Then is (cid:28)nite dimensional.
H
Proof: We will prove that has a non-zero (cid:28)nite dimensional left ideal. In
H
virtue of [8, Corollary 2.7℄ this will imply that is (cid:28)nite dimensional.
e = n e ⊗ e′ ∈ A ⊗ A
Let Pi=1 i Bni e e′ = B1 be athee=seepaarabilityai∈deAm.potent given
cbaynh′(yep)othesis. Then Pi=1 i i A and for all Noti
e that
is non-zero sin
e
n n
′ ′ ′
1A = Xeiei = XXei(0)ε(ei(1))ei = (idA ⊗ε)can(e).
i=1 i=1 (ei)
a ∈ A,h ∈ H j = 1,...,m
j j
We pi
k non-zero elements for su
h that
m n
′ ′
Xaj ⊗hj = can(e) = XXei(0)ei ⊗ei(1)
(1)
j=1 i=1 (ei)
a h ∈ H can
j
and the 's are linearly independent. Take arbitrary. Sin
e is
c ,d ∈ A l = 1,...,r
l l
surje
tive we may (cid:28)nd for satisfying
r r
1⊗h = can(Xcl ⊗B dl) = XXcldl(0) ⊗dl(1).
(2)
l=1 l=1 (dl)
l = 1,...,r d e = ed can′
l l
For we have . Applying to this equality we get
n n
′ ′
XXXdl(0)ei(0)ei ⊗dl(1)ei(1) = XXei(0)eidl ⊗ei(1).
(3)
i=1 (ei) (dl) i=1 (ei)
Then,
m r n
( )( ) ′
Xaj ⊗hhj 1=2 XXXXcldl(0)ei(0)ei ⊗dl(1)ei(1)
j=1 l=1 (dl) i=1 (ei)
r n
( ) ′
=3 XXXclei(0)eidl ⊗ei(1)
l=1 i=1 (ei)
r m
( )
=1 XXclajdl ⊗hj.
l=1 j=1
3
ϕ ∈ A∗ ϕ (a ) = δ t = 1,...,m.
t t j tj
Let be su
h that , the Krone
ker symbol, for
ϕ ⊗id
t H
Evaluating on the pre
eding set of equalities we obtain
r m
hht = XXϕt(clajdj)hj.
l=1 j=1
h
j
This yields that the subspa
e spanned by the 's is a (cid:28)nite dimensional
H
non-zero left ideal of , as required. q.e.d.
Cohen and Fis
hman provided in [3, Theorem 1.8℄ several
hara
teriza-
tions of separable Hopf-Galois extensions for a (cid:28)nite dimensional Hopf alge-
bra. These
hara
terizations together with our result allow to
hara
terize
su
h extensions for an arbitrary Hopf algebra.
H A H
Corollary. Let be a Hopf algebra and let be a right -Galois extension.
Aco(H) ⊂ A H
Then, is separable if and only if is (cid:28)nite dimensional and one
of the equivalent
onditions (2)-(6) in [3, Theorem 1.8℄ holds.
This
orollary may be viewed as a generalization to Hopf algebras of [4,
Proposition 2.1℄
hara
terizing strongly graded rings that are separable over
its
omponent of degree one.
Referen
es
[1℄ M. Beattie, S. D s
les
u and “. Raianu, A Co-Frobenius Hopf Algebra
with a Separable Galois Extension is Finite. Pro
. Amer. Math. So
. 128
No. 11 (2000), 3201-3203.
[2℄ S.U.Chase andM.E. Sweedler, Hopf Algebras and Galois Theory.Le
ture
Notes in Mathemati
s 97. Springer-Verlag, Berlin, 1969.
[3℄ M. Cohen, D. Fis
hman, Semisimple Extensions and Elmenents of Tra
e
1. J. Algebra 149 (1992), 419-437.
[4℄ C. N st ses
u, M.Van den Bergh and F.Van Oystaeyen, Separable Fun
-
tors Applied to Graded Rings. J. Algebra 123 (1989), 397-413.
[5℄ H.F. Kreimer, M. Takeu
hi, Hopf Algebras and Galois Extensions of an
Algebra. Indiana Univ. Math. J. 30 (1981), 675-692.
[6℄ S. Montgomery, Hopf Algebras and Their A
tions on Rings. CMBS No.
82, AMS, 1993.
4
[7℄ H.-J. S
hneider, Prin
ipal Homogeneous Spa
es for Arbitrary Hopf alge-
bras. Israel J. Math. 72 Nos. 1-2 (1980), 167-195.
[8℄ M.E. Sweedler, Integrals for Hopf Algebras. Ann. of Math. (2) 89 (1969),
323-335.
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