New York J. Math.17(2011) 51–74.
Hopf Galois structures on Kummer
extensions of prime power degree
Lindsay N. Childs
Abstract. Let K be a field of characteristic not p (an odd prime), containing a primitive pn-th root of unity ζ, and let L = K[z] with xpn
−a the minimal polynomial ofz overK: thus L|K is a Kummer extension, with cyclic Galois groupG=hσiacting onLviaσ(z) =ζz. T. Kohl, 1998, showed thatL|K has pn−1 Hopf Galois structures. In
this paper we describe these Hopf Galois structures.
Contents
1. Introduction 51
2. Greither–Pareigis theory and Byott’s translation 53
3. The action ofLNG onL 54
4. Regular embeddings forG cyclic 55
5. Regular subgroups of Perm(G) forG cyclic 57
6. Determining the K-Hopf algebra 59
References 73
1. Introduction
Chase and Sweedler [CS69] introduced the concept of a Hopf Galois ex-tension, generalizing the notion of a classical Galois extension of fieldsL|K
with Galois group G. Given a field extension L ⊃ K and a cocommuta-tive K-Hopf algebraH that acts on Lmaking Linto an H-module algebra (i.e., h(ab) =P
(h)h(1)(a)h(2)(b), where, following Sweedler’s notation, the
comultiplication ∆ :H→H⊗KH is given on an elementh ofH by
∆(h) =X
(h)
h(1)⊗h(2),
thenL is anH-Hopf Galois extension of K if the obvious map
L⊗KH→EndK(L)
induced from the module action ofH onL is 1-1 and onto.
Received January 31, 2010.
2000Mathematics Subject Classification. 12F10.
Key words and phrases. Hopf Galois extension, Kummer extension,p-adic logarithm.
ISSN 1076-9803/2011
IfLis a Galois extension ofK with Galois groupG, thenLis aKG-Hopf Galois extension ofK.
For some time after the concept was introduced, there was little interest in applying it to classical Galois extensions of fields, until in [GP87], Greither and Pareigis showed that a classical Galois extension L|K of fields with Galois group G could have Hopf Galois structures other than the classical one by KG, and showed how to transform the problem of determining the number of Hopf Galois structures onL|K into one depending purely on the Galois group G.
Subsequently, Byott [By96], extending [Ch89], found a translation of the group-theoretic problem that made the problem of counting Hopf Galois structures on L|K far more tractable. Thus most of the results on Hopf Galois structures on field extensions with given Galois groupGhave utilized Byott’s translation, in particular, [By96], [Ko98], [CaC99], [By02], [Ch03], [By04a], [By04b], [Ch05], [By07], [Ch07], [CCo07], [FCC11]. Other than the original paper of Greither and Pareigis [GP87] very few papers explicitly use the direct approach of [GP87] to determine Hopf Galois structures and even fewer explicitly describe the K-Hopf algebra and the action of the Hopf algebra on the field extension L. The most notable exceptions are papers that utilize the Kummer theory of formal groups to yield Hopf Galois structures (see [Ch00], Chapter 12), results where the Galois group has order
p2, p an odd prime (see [Ch96] and [By02]), and work of Kohl [Ko07], for
groups Gof order 4p,p an odd prime.
For Galois extensions L|K of local number fields with Galois group G, a classical problem in local Galois module theory is to understand the val-uation ring S of L as a module over the group ring RG, where R is the valuation ring ofK. E. Noether showed that S is a free RG-module if and only if L|K is tamely ramified. In the wild (= non-tame) case Leopoldt showed that sometimesS is a freeAmodule whereAis the associated order
of S in KG. More generally, if L|K is an H-Hopf Galois extension and A,
the associated order of S in H, is an R-Hopf order in H, then S is A-free
[CM94]. Byott ([By97a], [By97b], [By99], [By00], [By02]) has constructed examples of wild Galois extensions L|K of local fields with Galois group
G where S is not a free module over the associated order in KG, but the associated order in some other H-Hopf Galois structure on L|K is an R -Hopf order inH, and hence S is free over that associated order . Thus the existence of Hopf Galois structures other than the classical one for Galois extensions of local fields adds an array of new possibilities for the study of local Galois module structure.
The purpose of this paper is to study the Hopf Galois structures on a cyclic extension L|K of order pn, where p is an odd prime. It has been
known since [Ko98] that there are pn−1 Hopf Galois structures onL|K, but
odd prime, we describe the K-Hopf algebras H as K-algebras for each of thepn−1 Hopf Galois structures, and describe how the elements ofH act on
L.
Acknowledgements. My thanks to the Mathematics Department at Vir-ginia Commonwealth University for its hospitality while this research was conducted, and to a referee for numerous helpful comments on a previous version of this paper.
2. Greither–Pareigis theory and Byott’s translation
In order to pass from results obtained via Byott’s translation to a more explicit description of the Hopf Galois structures, we need to examine the Greither–Pareigis and Byott results in some detail.
LetL|K be a Galois extension with Galois group G. Then the map
γ :L⊗KL→HomL(LG, L) :=GL
by γ(a⊗b)(σ) =aσ(b) is an isomorphism. If{xσ :σ∈G}is the dual basis of{σ ∈G}, thenL∼=K⊗KLmaps underγ intoGLand the imageγ(1⊗b)
satisfies
γ(1⊗b)(σ) =σ(b) for all σ inG. Thus
γ(1⊗b) =X
σ∈G
σ(b)xσ,
and for τ inG,
γ(1⊗τ(b)) = X
σ∈G
σ(τ(b))xσ
= X
σ∈G
σ(b)xστ−1
= X
σ∈G
σ(b)xρ(τ)(σ)
whereρ:G→Perm(G) is the right regular representation. Thus the action of G on L ∼= K ⊗L corresponds under base change to an action of G on
GL = P
τLxτ making GL a Galois extension of L with Galois group G
acting as permutations of the dual basis {xσ}via ρ.
A subgroup N of Perm(G) is regular if N has the same order as G and the orbit inG of each element ofG under action byN is all of G.
Greither and Pareigis showed that given any Hopf Galois structure onL|K
by a K-Hopf algebra H, thenL⊗KH =LN for some regular subgroupN
of Perm(G) acting on GL by permuting the (subscripts of the) dual basis
{xσ}. Also, N is normalized by λ(G), the image in Perm(G) of G under the left regular representation λ of G, λ(σ)(τ) = στ. Conversely, each regular subgroup N of Perm(G) defines an LN-Hopf Galois structure on
that Hopf Galois structure descends to an (LN)G-Hopf Galois structure on
(GL)G∼=L, where Gacts onGL by
τ(axσ) =τ(a)xλ(τ)(σ)
forτ, σ inG,ainL, and Gacts on LN by
τ(aη) =τ(a)λ(τ)ηλ(τ)−1
forτ inG,ηinN andainL. Thus there is a bijection between Hopf Galois structures onL|K and regular subgroups of Perm(G) normalized by λ(G).
Byott’s translation works as follows. Given a Galois extension L|K with Galois group G, let N be a group with the same cardinality as G. Inside Perm(N) is Hol(N), the normalizer ofλ(N). Then Hol(N) =ρ(N)·Aut(N). Each homomorphism β : G → Hol(N) so that β(G) is a regular subgroup of Perm(N) yields a Hopf Galois structure on L|K, as follows: Given β, define the function b : G → N by b(σ) = β(σ)(e), where e is the identity element ofN. Sinceβ(G) is a regular subgroup of Perm(N),bis a bijection. Thenbdefines an isomorphism between Perm(N) and Perm(G) by the map that sends π in Perm(N) to C(b−1)(π) =b−1πb in Perm(G). So define an
embedding α:N →Perm(G) by
α(η)(σ) =b−1(λ(η)(b(σ)))
forσ inG,η inN. As Byott [By96] showed, two regular embeddingsβ, β′
:
G → Perm(N) define the same regular subgroup α(N) of Perm(G), hence the same Galois structure onL|K, iff there exists an automorphismγ ofN
so that β′(σ) =C(γ)(β(σ)) =γβ(σ)γ−1 for allσ inG.
This translation of the problem of finding regular subgroups of Perm(G) normalized byλ(G) has made the problem of determining Hopf Galois struc-tures onL|K somewhat easier by first, splitting the problem into a number of separate problems, one for each isomorphism class of groups N of the same cardinality asG, and second, replacing the problem of finding regular subgroups of Perm(G) normalized by λ(G) by finding regular embeddings of Ginto Hol(N), typically a much smaller group. As a result, most of the results on Hopf Galois structures on field extensions with given Galois group
Ghave utilized Byott’s translation, as noted above.
Moving from results using the Byott translation to explicitly describing the Hopf Galois structure involves two obstacles: first, using the function b
(rarely a homomorphism) to translate from a regular embedding β : G →
Hol(N) to the corresponding embedding α :N → Perm(G), and secondly, descending the action of α(N) onGL to an action ofL(α(N))λ(G) on L.
3. The action of LNG on L
Let L|K be a Galois extension of fields with Galois groupG of order n, let N be a group of order nand suppose β :G→ Hol(N)⊂Perm(N) is a regular embedding. Then N gives rise to a Hopf Galois structure on L|K
Proposition 1. Let L|K, G, N, β, H be as above. Then H acts on L as follows. For ξ=P
η∈Nsηη in H,sη in L anda in L, ξ(a) =X
η sηb−1
(η−1
)(a).
Proof. The regular embedding β :G→ Hol(N) yields the regular embed-dingα:N →Perm(G) via
α(η)(σ) =b−1(λ(η)(b(σ)).
Forain L, the image ofainGLis P
σσ(a)xσ, and
ξ X σ∈G
σ(a)xσ
!
=X
η
sηη X σ
σ(a)xσ
!
=X
η,σ
sησ(a)xα(η)(σ).
Since H maps GLG to itself, ξ(P
σσ(a)xσ) has the form
P
σσ(c)xσ, the
image inGLof an element cof L. Thus forainL,ξ(a) =c, the coefficient of x1 in the last expression (where 1 is the identity element ofG).
Nowb:G→N is a bijection that maps 1 inGto the identity elemente
of N, sinceβ is a homomorphism. So , 1 =α(η)(σ) =b−1
(λ(η)b(σ)) iff
ηb(σ) =e,
iff
σ=b−1(η−1).
So the coefficient of x1 is
ξ(a) = X
(η,σ),α(η)(σ)=1
sησ(a)
=X
η∈N
sηb−1(η−1)(a).
Once we find a set of generators of H =LNG, we may use the mapb−1
to describe the action ofH onL as in Proposition1.
4. Regular embeddings for G cyclic
LetLbe a Galois extension ofK with Galois groupGcyclic of orderpn, p an odd prime. Kohl [Ko98] showed that if L|K is H-Hopf Galois, then the K-Hopf algebraH has associated groupG: that is, L⊗KH ∼=LG. In
Now Hol(G)∼=G⋊Aut(G)∼=Z/pnZ⋊(Z/pnZ)×, where we view elements
of Hol(G) as of the form (a, g) with a, g integers modulo pn and g coprime
to p. Since G embeds in Perm(G) by the right regular representation ρ, (a, g) acts on h inG by (a, g)(h) = (gh−a), and so the multiplication on
Gis (a, g)(a′, g′) = (a+ga′, gg′). It is routine to verify that (a, g) has order
pn iff a is coprime to p and g ≡ 1 (mod p). Thus for G =hσi the regular
embeddings β : G → Hol(Z/pnZ) have the form β(σ) = (a,1 +dp) for a
coprime to p.
Proposition 2. Up to equivalence, the pn−1 equivalence classes of regular embeddings β :G→Hol(G) are represented byβ satisfying
β(σ) = (−1,1 +dp)
for d modulopn−1.
Proof. Given β, β′
: G→ Hol(G), β ∼ β′
ifβ′
=C(γ)β for γ in Aut(G). Now (Z/pnZ)× ∼
= Aut(G) via g 7→ γg, left multiplication by g. For h in G=Z/pnZandβ(σ) = (a,1+dp) in Hol(G) =ρ(G)·Aut(G)∼=G⋊Aut(G),
γgβ(σ)γ−1
g (h) =γg(a,1 +dp)γ
−1
g (h)
=γg(a,1 +dp)(g−1h)
=γg((1 +dp)g−1h−a)
=g((1 +dp)g−1h−a)
= (1 +dp)h−ga
= (ga,1 +dp)(h).
For eachacoprime top, there is somegso thatga≡ −1 (modpn). Sincedis
unaffected byγg, each choice of dmodulopn−1 yields a different equivalence
class.
For later use we note:
Lemma 3. Forg in (Z/pnZ)×
, (−1, g)λ(a)(−1, g)−1 =λ(ga).
Proof.
(−1, g)(g−1, g−1) = (−1 +g(g−1), gg−1) = (0,1)
and so for all h inG,
(−1, g)λ(a)(−1, g)−1
(h) = (−1, g)λ(a)(g−1
, g−1
)(h) = (−1, g)λ(a)(g−1h−g−1)
= (−1, g)(a+g−1h−g−1)
=g(a+g−1h−g−1) + 1
=ga+h
5. Regular subgroups of Perm(G) for G cyclic
As noted earlier, given a regular embedding β :G→ Hol(N), we obtain the corresponding regular subgroupα(N) of Perm(G) by using the bijective function b:G→ N defined by b(σ) = β(σ)(0), where 0 is the identity ele-ment ofN. Thenbdefines an isomorphism between Perm(N) and Perm(G) by the map that sendsπin Perm(N) toC(b−1)(π) =b−1πbin Perm(G). For
Ga cyclic group (written additively), this isomorphism yields an embedding
α:N →Perm(G) by
α(θ)(σ) =b−1
(λ(θ)(b(σ))) =b−1
(θ+b(σ))
for σ in G, θ in N. The subgroup of Perm(G) corresponding to β is then
α(N), and the action ofLNG onL is described byb−1, as in Proposition1.
ForG cyclic of orderpn,N ∼=Gand the groupM =α(G)⊂Perm(G) is
generated byη =α(1), the permutation that sendsq inGtob−1(b(q) + 1).
In particular, for q=b−1(k), we have
η(b−1
(k)) =b−1
(bb−1
(k) + 1) =b−1
(k+ 1).
Thus the cycle description of the generator η ofα(G) in Perm(G) is
(b−1(1), b−1(2), b−1(3), . . .).
To find b−1:N →G, we have
Proposition 4. Let G = Z/pnZ, p odd, and let β : G → Hol(G) with β(1) = (−1, g) for g= 1 +dpas in Proposition 2. Then for t,s in G,
β(t) =
−g
t−1 g−1, g
t
,
b(t) = (1 +dp)
t−1 dp
b−1
(s) = logp(1 +sdp) logp(1 +sp)
where logp(y) is thep-adic logarithm function.
Proof. If β:G→Hol(G) withβ(1) = (−1, g) for g= 1 +dp, then
β(t) = (−1, g)t= (−(1 +g+g2+. . .+gt−1), gt) =
−
gt−1 g−1
, gt
.
so
Forg= 1 +dp,
Solving for t is the same as solving the discrete logarithm problem in the cyclic group (1 +pZ)/(1 +pnZ).
Forx a multiple ofp, thep-adic logarithm function is
hence
b−1(3r) = 3r
b−1(1 + 3r) = 1 + 3r
b−1(2 + 3r) = (2 + 3r)−3.
As an element of Perm(G), the generator η ofα(G) has cycle description
(0,1,8,3,4,2,6,7,5).
Since λ(1) = (0,1,2,3,4,5,6,7,8), one may verify easily that
λ(1)ηλ(1)−1
=η4.
6. Determining the K-Hopf algebra
For the remainder of the paper we assume that K contains a primitive
pn-th root of unity ζ, p odd, and L is a Kummer extension of K of order pn, so thatL=K[z] and the minimal polynomial ofz overK isxpn
−afor someainK.
The regular subgroupα(G) =M of Perm(G) yields aK-Hopf algebra ac-tion onL by theK-Hopf algebraH=LMG, whose structure is determined
by howG acts on L and on howλ(G) acts on M. The action of Gon L is given. In our case L= K[z] is a Kummer extension of K, so the action is transparent. The action of λ(G) on M is also straightforward.
Proposition 6. Let G = Z/pnZ, p odd. Suppose β : G → Hol(G) is a
regular embedding withβ(1) = (−1, g)where g= 1 +dpas in Proposition 2. Then the group α(G) = M = hηi where η = α(1) = C(b−1)(λ(1)). The
group M is normalized byλ(G) in Perm(G). In fact, the action of λ(G) on
M is induced byλ(1)ηλ(1)−1 =ηg.
Proof. We know that η=C(b−1)(λ(1)), and we showed in Lemma3 that
(−1, g)λ(1)(−1, g)−1 =λ(g·1).
When we translate to Perm(G) via C(b−1), we obtain
C(b−1)(−1, g)C(b−1)(λ(1))C(b−1)(−1, g)−1 =C(b−1)(λ(1))g,
that is,
C(b−1)(−1, g)ηC(b−1)(−1, g)−1 =ηg,
using multiplicative notation for composition of permutations in Perm(G). Now fort inG=Z/pnZ, we have
b(t) = g
and
C(b−1
)(−1, g)(t) =b−1
(−1, g)b(t)
=b−1(−1, g)
gt−1 g−1
=b−1
g
gt−1 g−1
+ 1
=b−1
gt+1−1
g−1
=b−1(b(t+ 1))
=t+ 1.
So C(b−1)(−1, g) =λ(1). Thus
λ(1)ηλ(1)−1
=ηg.
Translating to multiplicative notation for the Galois group G = hσi ∼=
Z/pnZof the Kummer extensionL|K, letα(G) =M =hηias in Proposition
6. The action of G on LM is induced by the Galois action of G on L and the action on M by σ(η) =λ(σ)ηλ(σ)−1 =ηg. Thus Proposition6 enables
us to determine the K-Hopf algebraH =LMG acting on L.
As a K-module we can determine a set of generators of H by simply taking the sums of elements in the distinct orbits under the action of G of
zkηl for all k, l with 0 ≤ k, l ≤ pn−1. But we are interested in H as a K-algebra, so our objective in the remainder of the paper is to obtain an economical set of generators of H as a K-algebra.
As an initial model, we first do the known case whereG=hσiis cyclic of order p2.
6.1. G of order p2
. We suppose L is a cyclic Kummer extension ofK of order p2 with Galois group G= hσi. Thus K contains a primitive p2-root of unity ζ, andL=K[z] with σ(z) =ζz, where the minimal polynomial of
zoverK isxp2−a. LetM =hηibe cyclic of orderp2 whereσ(η) =ηg with g= 1 +dp. Thenσ(ηp) =ηp, so ηp is inLMG=H. Therefore the minimal
idempotents
e1s = 1
p p−1
X
i=0
ζ−spiηpi
of K[hηpi] for s = 0, . . . , p−1 are fixed by G. These idempotents satisfy ηpe1
s =ζspe1s. It follows that σ(z−sdpe1
sη) =ζ
−sdpz−sdpe1
sη1+dp
=ζ−sdpz−sdpe1
sηdpη
=z−sdpe1
So following the construction of Greither [Gr92], we let
av = p−1
X
s=0
vse1
s
wherev=z−dp. Then
σ(avη) =avη.
Proposition 7. H=LMG =K[ηp, avη] where v=z−dp .
Proof. Since H has dimensionp2 overK and K[ηp] =Pp−1
s=0Ke1s is a
sub-algebra of H, it suffices to show that fors= 0, . . . , p−1, K[ηp, avη]e1
s has
dimensionp overKe1s. Now
K[ηp, avη]e1s=K[avη]e1s =K[z−sdpη]e1
s,
the elements {(z−sdpe1
sη)r : 0 ≤r < p} are linearly independent over Ke1s,
and
(z−sdpe1
sη)p =z
−p2sd
ζspe1s
is inKe1
s. Thus for each s,K[ηp, avη]e1s has dimensionp over Ke1p. Hence K[ηp, avη] has dimensionp2 overK and is a subalgebra ofH, hence is equal
toH.
This result is in [Ch96, Section 1].
6.2. G of order p3
. To preview the main result, Theorem12 below, we write down the result for L|K a Kummer extension with Galois group G, cyclic of orderp3.
InKG we have the idempotents
e1t = 1
p2
p2−1 X
j=0
ζ−pjtηpj
for 0≤t < p2, and
e2t = 1
p p−1 X
j=0
ζ−p2jt
ηp2j
for 0 ≤ t < p. Then e2t is G-invariant for all t, and e1t is G-invariant if p
dividest.
Forσ(η) =η1+dpwith (p, d) = 1, then, analogous to the Greither element
for thep2-case, we let
g1,sp=z−sdp
2
e1spη, for 0≤s≤p−1, and
g1,s= p−1 X
i=0
σi(z−sdpe1
the sum of the conjugates of z−sdpe1
sη, and let
h=
p−1 X
s=1
g1,s+ p−1 X
s=0
g1,sp.
Then Theorem 12 shows that
LMG=K[h, ηp2, e20ηp, e10η].
Forσ(η) =η1+dp2, Theorem12 specializes to show that
LMG =K[h, ηp, e20η] where
h=
p2−1 X
s=0
g1,s= p2−1
X
s=0
z−sdp2
e1sη.
6.3. Gof order pn.
LetL=K[z], a cyclic Kummer extension of fields of order pn with Galois group G = hσi as at the beginning of Section 6. Let M =hηibe the regular subgroup of Perm(G) normalized byGcorresponding to the regular embedding β : G → Hol(G) where β(G) = h(−1,1 +dpν)i
with (d, p) = 1 and ν ≥1. Then the corresponding K-Hopf algebra acting on L is LMG, where G acts on L via the Galois action and acts on M by σ(η) =η1+dpν
.
Idempotents. To obtain a set of algebra generators forLMG, we first look
at the idempotents of KM.
For r= 0, . . . , n−1 and t modulo pn−r we have the pairwise orthogonal
idempotents ofK[ηpr
],
ert = 1
pn−r pn−r−1
X
i=0
ζ−tpri
ηpri.
Then
ηprert =ζprtert
so for all k≥r,
ηspkert =ζstpkert,
and
1 =
pn−r−1 X
t=0
ert.
More generally, the idempotents of K[ηpr+1
] decompose in K[ηpr
] as:
Lemma 8. Forr = 0, . . . , n−2,
p−1
X
k=0
Proof.
p−1
X
k=0
ert+kpn−r−1 =
1
pn−r p−1
X
k=0
pn−r−1 X
i=0
ζ−pri(t+kpn−r−1)
ηpri
= 1
pn−r pn−r−1
X
i=0
ηpriζ−prit
p−1 X
k=0
ζ−ikpn−1
.
and the last sum = 0 if i6≡0 modulop, and =pifi=pj. So
p−1 X
k=0
ert+kpn−r−1 =
1
pn−r−1
pn−r−1−1
X
j=0
ζ−pr+1jt
ηpr+1j =ert+1.
Corollary 9. For all k >0 and all s, t, r,
er
sert+k= 0 if t6≡s (modpn
−r−k)
=er
s if t≡s (modpn
−r−k).
Proof. Since er+1
s =ers+1ers+1, Lemma8 gives p−1
X
k=0
ers+kpn−r−1 =
p−1 X
k=0
ers+1ers+kpn−r−1.
Multiplying both sides by er
s, we get ers=ers+1ers
by pairwise orthogonality of the idempotents {er
s : s = 0, . . . , pn
−r−1}.
Since the subscript t of ert+1 is defined modulo pn−r−1, the result is then
clear fork= 1. For generalk >0, we can use the casek= 1 to write
esr=ersesr+1· · ·ers+k.
Then ert+ker
s=ers iff
ers+k=ert+k,
iff
s≡t (modpn−r−k),
and equals 0 otherwise.
The groupG=hσi acts on the idempotents by
where the subscript t(1 +dpν)−1 is modulo pn−r. To verify this G-action,
the change of variables j=i(1 +dpν) gives
σ(ert) = 1
pn−r pn−r−1
X
i=0
ζ−tpri
ηpri(1+dpν)
= 1
pn−r pn−r−1
X
j=0
ζ−tprj(1+dpν)−1
ηprj
=ert(1+dpν)−1.
Fort=spǫ with scoprime to pand for k≥0, we have
σpk(erspǫ) =er
spǫ(1+dpν)−pk.
The subscript ton er
t is modulopn
−r, and fork=n−r−ǫ−ν,
spǫ(1 +dpν)−pk
≡spǫ (modpn−r).
Hence
σp(n−r−ǫ−ν)(erspǫ) =erspǫ.
In particular, for r=n−ǫ−ν and ǫ= 0, . . . , n−1−ν, we have
σ(en−ν−ǫ
spǫ ) =ensp−ǫν−ǫ.
We may now write 1 as a sum of pairwise orthogonal G-invariant idem-potents:
Proposition 10. In LMG, we have
1 =
n−ν−2 X
ǫ=0
pν−1 X
s=1,(s,p)=1
en−ν−ǫ spǫ +
pν−1 X
s=1
e1spn−ν−1.
Proof. We just observed that all of the idempotents in the sum are fixed by G.
We have
1 =
pν−1 X
s=0
en−ν s =
pν−1 X
s=1,(s,p)=1
en−ν s +
pν−1−1
X
r=0
en−ν rp .
Now by Lemma8,
en−ν rp =
p−1
X
k=0
en−ν−1
also a sum of pairwise orthogonal idempotents, so
Repeating this decompositionn−ν−2 more times yields the desired formula.
Generators. We want to find generators of LMGen−ν−ǫ
spǫ over Kensp−ǫν−ǫ.
By analogy with the p2 case, which involves the sum of Greither generators
z−sdpe1 n > 2 what are fixed are sums of conjugates under the action of G. We therefore need to know what power ofσ fixes these elements.
Proposition 11. We have
Now we can define the generators of the K-Hopf algebra LMG.
The main result. We show that the algebra generators of H generate all of LMG:
Theorem 12. H =LMG.
Proof. The idea of the proof is to take the idempotents in Proposition 10, break up K[h] into a direct product corresponding to those idempotents, and count the dimensions over K of the direct factors.
We first show that the idempotents en−ν−ǫ
spǫ appearing in Proposition 10
are inH. For ǫ= 1, . . . , n−ν−1, we have
en−ν−ǫ
spǫ =en0−ǫensp−ǫν−ǫ
by Corollary9. Since
we may multiply both sides by en−ǫ
s are K-linear combinations of powers of ηp
n−ν
, hence are inH.
Now by Proposition 10, 1 decomposes into a sum of pairwise orthogonal
G-invariant idempotents:
We have four cases to show.
Case 4. Fors,tcoprime to p and ǫ, f ≤n−ν−2,
g1,spǫen−ν−f
tpf = 0
iff 6=ǫor iff =ǫbutt6=s, while
g1,spǫen−ν−ǫ
spǫf =g1,spǫ.
We use Corollary 9: For all k >0 and alls, t, r,
ersert+k= 0 if t6≡s (modpn−r−k)
=ers ift≡s (mod pn−r−k).
Case1: We have n−ν−f ≥2 andtis coprime to p. Now
g1,spn−ν−1en−ν−f
tpf =z
−sdpn−1
ηe1spn−ν−1en −ν−f tpf ,
and by Proposition 8, this 6= 0 if spn−ν−1 ≡ tpf (modpn−(n−ν−f)). But
since
ordp(tpf) =f ≤n−ν−2<ordp(spn−ν−1)
the congruence never holds, so Case1 is true.
Case2: Sinceg1,spn−ν−1 is a multiple ofe1spn−ν−1, Case2follows from the
orthogonality of the idempotents {e1
tpn−ν−1}.
For Cases 3 and 4 we assume s is coprime to p and n−ν −ǫ≥ 2. We write
g1,spǫ=
pn−ǫ−ν−1−1
X
i=0
σi(z−sdpǫ+ν
ηe1spǫ)
=
pn−ǫ−ν−1−1
X
i=0
z−sdpǫ+ν
ζ−sdpǫ+ν
η(1+dpν)ie1spǫ(1+dpν)−i
=
pn−ǫ−ν−1−1
X
i=0
γie1spǫ(1+dpν)−i
whereγi is the coefficient of e1spǫ(1+dpν)−i in theith summand.
Case3 is the case
g1,spǫe1
tpn−ν−1 =
pn−ǫ−ν−1−1
X
i=0
γie1spǫ(1+dpν)−ie1tpn−ν−1,
which = 0 from Corollary 9by essentially the same argument as in Case 1. Case4: Fort coprime to pand n−ν−f ≥2, we have
g1,spǫen−ν−f
tpf =
pn−ǫ−ν−1−1
X
i=0
γie1spǫ(1+dpν)−ie
n−ν−f tpf .
The term
γie1spǫ(1+dpν)−ie
n−ν−f
depending on whether or not
spǫ(1 +dpν)−i≡tpf (mod pn−(n−ν−f))
that is,
spǫ≡tpf(1 +dpν)i (modpν+f).
Sincesandtare coprime top, this congruence can hold exactly whenǫ=f
and s≡t (modpν), independent of i. Thus g
NowHdecomposes into a direct sum of subrings corresponding to the idem-potents arising in Proposition10:
H=
s, ǫ. The sum of those dimensions is less than or equal to the dimension of
H as a K-module. When we show that the sum of the dimensions is pn,
then, sinceLMGis known by descent to have dimensionpnandH ⊆LMG,
we will obtain equality.
To compute the desired dimensions, we have
Proposition 14. For r= 1, . . . , n−ǫ−ν−1 and(s, p) = 1,
gr,spp ǫ =gr+1,spǫ.
Proof. Since gr,spǫ is a sum of terms involving pairwise orthogonal
In the summation formula forgr,spp ǫwe write the index of summation in base
pn−ǫ−r−ν−1: i=j+kpn−ǫ−r−ν−1. Then
gr,spp ǫ =
pn−ǫ−r−ν−1−1
X
j=0
σj p−1 X
k=0
σkpn−ǫ−r−ν−1(z−sdpǫ+r+ν
erspǫζsp r+ǫ
).
Focusing on the part of the expression forgr,spp ǫ involving the indexk, we
have
σkpn−ǫ−r−ν−1(z−sdpǫ+r+ν
) =ζ−sdkpn−1
z−sdpǫ+r+ν
,
and
σkpn−ǫ−r−ν−1(espr ǫ) =erspǫ−sdkpn−r−1.
To verify this last formula, we see that the subscript ofer on the left side is
spǫ(1 +dpν)−kpn−ǫ−r−ν−1
,
and since the subscriptt ofer
t is defined modulo pn−r, that subscript is spǫ(1 +dpν)−kpn−ǫ−r−ν−1
≡spǫ−spǫkpn−ǫ−r−ν−1dpν
≡spǫ−skdpn−r−1 (modpn−r).
Thus
σkpn−ǫ−r−ν−1
(z−sdpǫ+r+νer spǫζsp
r+ǫ
)
=z−sdpǫ+r+ν
erspǫ−skdpn−r−1ζsp
r+ǫ−skdpn−1
.
Now we observe that
ηprerspǫ−skdpn−r−1 =ζsp
r+ǫ−skdpn−1
erspǫ−skdpn−r−1.
So
z−sdpǫ+r+ν
erspǫ−skdpn−r−1ζsp
r+ǫ−skdpn−1
=z−sdpǫ+r+ν
ηprerspǫ−skdpn−r−1,
and the sum involving kbecomes
p−1
X
k=0
z−sdpǫ+r+ν
ηprerspǫ−skdpn−r−1 =z
−sdpǫ+r+ν
ηpr p−1
X
k=0
erspǫ−skdpn−r−1
=z−sdpǫ+r+ν
ηprersp+1ǫ
by Lemma8, sincesd is coprime top. Thus
gpr,spǫ =
pn−ǫ−r−ν−1−1
X
j=0
σj(z−sdpǫ+r+ν
ηprersp+1ǫ )
Now we compute the dimension
[K[g1,spǫ] :Kensp−ǫν−ǫ]
for (s, p) = 1 andǫ < n−ν−1 and forǫ=n−ν−1 and alls. First assumes is coprime top. We have from Proposition 14that
gp1,spr ǫ =gr+1,spǫ
for all r= 1, . . . , n−ǫ−ν−1 and (s, p) = 1, and so
g1p,spn−ǫǫ−ν−1 =gn−ǫ−ν,spǫ =zsdp n−1
en−ǫ−ν spǫ ηp
n−ǫ−ν−1
.
Then
g1p,spn−ǫǫ−ν = (gp n−ǫ−ν−1
1,spǫ )p=z−sdp n
en−ǫ−ν spǫ ηp
n−ǫ−ν
is in Ken−ǫ−ν
spǫ . Thus the dimension ofK[g1,spǫ] overKespn−ǫǫ−ν is ≤pn−ν−ǫ.
Nowg1,spǫ has the form
g1,spǫ =z−sdp ǫ+ν
η
pn−ǫ−ν−r−1 X
i=0
ζdie1
spǫ(1+dpν)−i
for some exponent di. This is of the form θz−sdpǫ+ν where θ is in the
group ring L[η]. Since L[η] is a free module over K[η] with basis {zj : j= 0, . . . , pn−1} andg1pn,sp−ǫǫ−ν 6= 0 inK[η], the set
{gj1,spǫ :j= 0. . . , pn−ǫ−ν−1}
is linearly independent overK[η], hence overKe1
spǫ.
It follows that for ǫ < n−ν −1 and s coprime to p, the dimension of
K[g1,spǫ overKesp1 ǫ is =pn−ν−ǫ.
Forǫ=n−ν−1 and alls, we have
K[g1,spn−ν−1] =K[z−sdp
n−1
e1spn−ν−1η]
which clearly has dimension pas a module over Ke1spn−ν−1.
Summing these dimensions, the dimension ofH overK is
≥
n−ν−2
X
ǫ=0
pν−1 X
s=1,(s,p)=1
pn−ǫ−ν+ pν−1
X
s=0
p
= (pν−pν−1
)(pn−ν +pn−ν−1
+. . .+p2) +p(pν) =pn.
Since H ⊆LMG, we must have H =LMG, completing the proof of
Theo-rem 12.
Example 15. Let Gbe cyclic of order 27. LetM be the regular subgroup of Perm(G) corresponding to the cyclic subgroup h(−1,4)i. Then β(t) = (−1,4)t, so
b(t)≡
4t−1
3
One may verify that
Using Theorem 12 and Proposition 1, the actions of these generators of H
are as follows:
As for the components ofh, we have
fors= 1,2 (note that g1,0=e10η was done above), and since
g1,s =
2 X
i=0
σi(z−3se1
sη)
=z−3se1
sη+ζ
−3sz−3se1 7sη4+ζ
−6sz−3se1 22sη16
=z−3sη(e1
s+ζ18se17s+e122s)
= 1 9z
−3s
8 X
k=0
(ζ−3sk+ζ18s−21sk+ζ−66k)η3k+1,
we have
g1,s(a) = z−3s
9
8 X
k=0
(ζ−3sk+ζ18s−21sk+ζ−66k)σ14+6k(a)
fors= 1,2.
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Department of Mathematics and Statistics, University at Albany, Albany, NY 12222
