Table Of ContentEXPLICIT CONSTRUCTIONS OF RAMANUJAN
COMPLEXES
5
0
0 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE
2
n
Abstract. In this paper we present for every d 2 and every
a ≥
localfieldF ofpositivecharacteristic,explicitconstructionsofRa-
J
manujan complexes which are quotients of the Bruhat-Tits build-
5
ing d(F) associated with PGLd(F).
B
]
O
C
.
h
1. Introduction
t
a
m In [LSV] we defined and proved the existence of Ramanujan com-
[ plexes, see also [B1], [CSZ] and [Li]. The goal of this paper is to
present an explicit construction of such complexes.
2
v Our work is based on the lattice constructed by Cartwright and Ste-
7
ger [CS]. This remarkable discrete subgroup Γ of PGL (F), when F
1 d
2 is a local field of positive characteristic, acts simply transitively on the
6 vertices of the Bruhat-Tits building (F), associated with PGL (F).
d d
0 B
By choosing suitable congruence subgroups of Γ, we are able to present
4
0 the 1-skeleton of the corresponding finite quotients of (F) as Cayley
d
/ B
h graphs of explicit finite groups, with specific sets of generators. The
t simplicial complex structure is then defined by means of these genera-
a
m tors.
: Let [d] denote the number of subspaces of dimension k of Fd.
v k q q
i
X
Theorem 1.1. Let q be a prime power, d 2, e 1 (e > 1 if q = 2).
ar Then, the group G = PGLd(Fqe) has an (≥explicit≥) set S of [d1]q+[d2]q+
+[ d ] generators, such that the Cayley complex of G with respect
··· d−1 q
to S is a Ramanujan complex, covered by (F), when F = F ((y)).
d q
B
The Cayley complex of G with respect to a set of generators S is
the simplicial complex whose 1-skeleton is the Cayley graph Cay(G;S),
where asubset ofi+1vertices isani-celliff every two vertices comprise
an edge. The generators in Theorem 1.1 are explicitly given in Section
9.
Date: Received: Aug. 31, 2003, Revised: Feb. 25, 2004.
This researchwas supported by the NSF and the BSF (U.S.-Israel).
1
2 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE
In the case d = 2 there are two types of Ramanujan graphs, bi-
partite, and non-bi-partite ([LPS], [Mo], [Lu1]). Here too, given r
dividing d, we produce r-partite complexes, by proving an analog of
Theorem 1.1 for every subgroup of PGL (F ) containing PSL (F ).
d qe d qe
There is also a version of the theorem for subgroups of PGL (L) where
d
L is a finite local ring.
The paper is organized as follows: In Section 2 we describe affine
buildings of type A˜ in general, in the language of -sublattices of Fd
d
O
(where is the valuation ring of F). The Cartwright-Steger group Γ
O
is constructed in Sections 3–4. Our construction slightly differs from
theirs, but is more convenient for the computations to follow. The
simply transitive action of Γ on the building is used in Section 5 to
describe the defining relations of Γ.
In Section 6 we present and analyze the finite Ramanujan quotients.
Inparticular,weuse[LSV]toshowthatthecomplexesconstructedhere
are Ramanujan. We should note that the proof of this result in [LSV]
relies on what is called the global ‘Jacquet-Langlands correspondence’
for function fields (a correspondence between automorphic representa-
tions of a division algebra and of GL , cf. [HT, Thm VI.1.1] for the
d
characteristic zero case). This correspondence in the function field case
is also considered to be true by experts, as the main ingredients of the
proof are known; though the task of writing down a complete proof has
not been carried out yet.
Originally, Γ is a group of d2 d2 matrices over a ring R. Section
×
8 provides an explicit embedding of Γ into d d matrices over a finite
×
extension of R. In Section 9 this embedding is refined to identify finite
quotients of Γ with subgroups of PGL (L) which contain PSL (L),
d d
where L is a finite local ring; a detailed algorithm is given, with an
example in Section 10. In particular the generators of Γ are given as
d d matrices over F [x]. For the convenience of the reader, we include
q
×
a glossary in the final section.
Added in Proof: We recently learned that Alireza Sarveniazi [Sa] has
also given an explicit construction of Ramanujan complexes.
2. Buildings
To every reductive algebraicgroupover alocalfieldonecanassociate
a building, which is a certain simplicial complex, on which the group
acts (see [R]). This complex plays the role of a symmetric space for
Lie groups.
Recall that a complex is a structure composed of i-cells, where the
0-cells are called the vertices, and an i-cell is a set of i + 1 vertices.
EXPLICIT CONSTRUCTIONS OF RAMANUJAN COMPLEXES 3
A complex is simplicial if every subset of a cell is also a cell. The i-
skeleton is the set of all i-cells in the complex. Buildings are in fact
clique complexes, which means that a set of i + 1 vertices is a cell
iff every two vertices form a 1-cell. This property holds for quotient
complexes, which will be the subject of Section 6.
We will now describe the affine building associated to PGL (F),
d
where F is a local field. These are called ‘buildings of type A˜ ’
d−1
because of the Dynkin diagram of the associated Weil group (which is
isomorphic to S ⋉Zd−1). Let denote the valuation ring of F; choose
d
a uniformizer ̟ (so for F =OF ((y)), = F [[y]] and ̟ = y), and
q q
O
assume /̟ = F . Consider the -lattices of full rank in Fd. For
q
every latOticeLO, ̟Lisasublattice, anOdasF -vector spaces L/̟L = Fd.
q ∼ q
We define an equivalence relation by setting L sL for every s F×.
∼ ∈
Remark 2.1. Let ̟ denote the multiplicative subgroup of F× gen-
h i
erated by ̟, and let × be the invertible elements of . Then F× =
O O
̟ ×.
h i·O
Since L = sL for any element s ×, the equivalence classes have
∈ O
the form [L] = ̟iL . Let 0 be the graph whose vertices are the
{ }i∈Z B
equivalence classes. There is an edge from [L] to a class x 0, iff
∈ B
there is a representative L′ x such that ̟L L′ L. Notice that
∈ ⊂ ⊂
this is a symmetric relation, since then ̟L′ ̟L L′.
⊂ ⊂
The vertices of 0 form the 0-skeleton of a complex , and the edges
B B
are the 1-skeleton, 1. As i-cells of we take the complete subgraphs
B B
of size i+1 of 0, which correspond to flags
B
̟L L L L ;
0 i 1 0
⊂ ⊂ ··· ⊂ ⊂
the i-skeleton is denoted i. It immediately follows that has (d
B B −
1)-cells (corresponding to maximal flags in quotients L/̟L). It also
follows that there are no higher dimensional cells.
The groupGL (F) actstransitively onlattices byits actiononbases.
d
moreover note that the action preserves inclusion of lattices. We call
L = d Fd the standard lattice . If τ GL (F) has entries in ,
0 d
O ⊆ ∈ O
then τL L . The stabilizer of L in G is thus the maximal compact
0 0 0
⊆
subgroup GL ( ). According to the definition of the equivalence rela-
d
O
tion, the scalar matrices of GL (F) act trivially on , so the action of
d
B
GL (F)inducesawelldefinedactionofPGL (F)on(theverticesof) ,
d d
B
which is easily seen to be an action of an automorphism group. Again,
the stabilizer of [L ] is the maximal compact subgroup PGL ( ). The
0 d
O
set of vertices can thus be identified with PGL (F)/PGL ( ).
d d
O
Since the only ideals of are powers of ̟ , The Invariant Factor
O h i
Theorem for F asserts that any matrix in GL (F) can be decomposed
d
4 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE
asaga′ fora,a′ GL ( ) andg = diag(̟i1,...,̟id), wherei
d 1
∈ O ≤ ··· ≤
i are integers. If L = aga′L = agL , then ̟−i1L = a(̟−i1g)L L ,
d 0 0 0 0
⊆
and on the other hand ̟−idL = a(̟−idg)L L , so any two lattices
0 0
⊇
of maximal rank are commensurable. Moreover in this case, L /̟−i1L
0
is annihilated by ̟id−i1, and so is a module over /̟id−i1 , a local
O O
ring of order qid−i1. In particular [L :̟−i1L] is a q power.
0
This basic fact allows us to define a color function ̺: 0 Z/d by
B →
̺(L) = log [L :̟iL] for large enough i; this function is well de-
q 0
fined since [̟iL:̟i+1L] = qd. Also notice that ̺([τL ]) = ν (det(τ))
0 0
(mod d), when ν is the valuation of F. This shows that SL (F) is
0 d
color preserving, while on the other hand, τ = diag(̟,1,...,1) has
̟
determinant ̟, so ̺(τ (L)) = ̺(L) + 1 for every L. It follows that
̟
GL (F) acts transitively on colors.
d
The color is additive in the sense that if L′′ L′, then ̺(L′′) =
⊆
̺(L′)+log [L′:L′′]. Similarly if L L and τ GL (F), then ̺(τL) =
q ⊆ 0 ∈ d
̺(τL )+̺(L).
0
The colors provide us with d Hecke operators, defined on functions
of 0 by summation over the neighbors of fixed color-shift:
B
A f(x) = f(y).
k
X
y∼x,̺(y)−̺(x)≡k
These operators generate the Hecke algebra H(PGL (F),PGL ( ))
d d
O
(see [LSV, Sec. 2] for more details).
For 1 t < d, let [t] denote the graph defined on the vertices 0,
≤ B B
with the edges (x,x′) 1 for which there are L x and L′ x′ such
∈ B ∈ ∈
that L′ L and [L:L′] = qt (in particular, ̺(x) ̺(x′) = t).
⊆ −
Remark 2.2. If L′ L is a sublattice of index q, then [L],[L′] are
⊂
connected in 1.
B
Proof. We need to prove that ̟L L′, but this is obvious since L/L′
is annihilated (as an -module) by⊆multiplication by ̟. (cid:3)
O
For every L′ L there is a composition series of sublattices L′ =
⊆
L(m) L(m−1) L(0) = L, such that [L(i):L(i+1)] = q. It
⊆ ⊆ ··· ⊆
follows that [1] is a connected (directed) subgraph of . In fact, [1]
B B B
determines 1, and thus all of :
B B
Proposition 2.3. Vertices x,x′ are connected in 1 iff there is a chain
B
x ,x ,...,x 0 such that x = x = x, (x ,x ) [1] for i =
0 1 d 0 d i i+1
∈ B ∈ B
0,...,d 1, and such that x′ x ,...,x .
1 d−1
− ∈ { }
Proof. Firstassumethatsuchachainexists, andchooserepresentatives
L x with L = ̟L . By the definition of [1], we may assume that
i i d 0
∈ B
EXPLICIT CONSTRUCTIONS OF RAMANUJAN COMPLEXES 5
[L :L ] = q and then L L L L . In particular
i i+1 d 2 1 0
⊂ ··· ⊂ ⊂ ⊂
([L ],[L ]) 1 for every i.
i 0
∈ B
On the other hand, if
(1) ̟L L′ L
0 0
⊂ ⊂
are lattices, we can lift a maximal flag in L/̟L = Fd to a maximal
0 ∼ q
chain of lattices refining (1), resulting in a chain x ,...,x . (cid:3)
0 d
Corollary 2.4. If (x,x′) [t], then there is a path of length t in [1]
∈ B B
from x to x′.
Using this criterion, it is easy to see that if the greatest common
divisor (t′,d) equals t, and (x,x′) [t′], then there is a path from x
∈ B
to x′ in [t]. In particular, [t′] has the same connected components
B B
as [t]. The final result of this section is not needed in the rest of the
B
paper.
We thank the referee for some simplification in the proof of the fol-
lowing proposition.
Proposition 2.5. Let t be a divisor of d. If t divides ̺(x′) ̺(x), then
−
there is a path from x to x′ in the (directed) graph [t] defined above.
B
Proof. Wemay assume x = [L ], andx′ = [L] withL = aga′L = agL ,
0 0 0
a,a′ GL ( ) and g = diag(̟i1,...,̟id). We may assume 0 i
d 1
∈ O ≤ ≤
i , and in particular L L . If the claim is true for gL L ,
d 0 0 0
··· ≤ ⊆ ⊆
then acting with a we get a chain from aL = L to L; we can thus
0 0
assume a = 1.
Before going on by induction, we multiply L by a power of ̟ so that
t
(2) i (i i ).
1 d 1
≥ d t −
−
Now, by assumption, i + +i = rt for some r N.
1 d
··· ∈
Case 1. If i > i , then we can lower t of the entries i ,...,i ,
d−t+1 1 2 d
each by 1, keeping the increasing order. Let i′,...,i′ denote the re-
1 d
sulting values and g′ = diag(̟i′1,...,̟i′d). Then gL0 g′L0 L0
⊂ ⊆
with [g′L :gL ] = qt and ̟g′L gL , so (g′L ,gL ) [t]. Since
0 0 0 0 0 0
⊆ ∈ B
the condition i d(i i ) still holds (as i was not changed), we are
1 ≥ t d− 1 1
done by induction on r.
Case 2. Now assume i = i ; let j be maximal with i = i .
d−t+1 1 j 1
If j = d then i = = i so L L and we are done. Therefore
1 d 0
··· ≡
assume j < d. Of course j d t+ 1. If i = 0 then i = 0 too by
1 d
≥ −
the assumption (2), so L = L and again we are done. Assume i > 0
0 1
and j < d. In the first step we can lower i ,...,i , but we also have
j+1 d
to lower i ,...,i in order to change exactly t components. We
1 t−(d−j)
continue lowering the highest entries, using the remaining d t entries
−
6 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE
F ((y))
qd
ppppppp NNNNNNN
k = F (y) F = F ((y))
1 qd q
NNNNNNN ppppppp
k = F (y)
q
R = F [[y]]
q
oooooooOoo
R
T
Figure 1. Subrings of F ((y))
qd
i ,...,i whenever necessary. In each of the next d/t 1 steps,
t−(d−j)+1 j
−
i is lowered by one with i unharmed—so at the end our condition
d 1
(2) is still met. The d/t modules we constructed form a chain in L ,
0
climbing from L using [t] steps, and ending with indices which now
sum up to rt d, so agBain we are done by induction. (cid:3)
−
Vertices x and x′ for which (t,d) does not divide ̺(x′) ̺(x) can-
−
not be connected in [t], so we proved that [t] has (t,d) connected
B B
components for every 1 t < d.
≤
3. The arithmetic lattice
Let F denote the field of order q (a prime power), and F the
q qd
extension of dimension d. Let φ denote a generator of the Galois group
Gal(F /F ). Fix a basis ζ ,...,ζ for F over F , where ζ = φi(ζ ).
qd q 0 d−1 qd q i 0
Extend φ to an automorphism of the function field k = F (y) by
1 qd
setting φ(y) = y; the fixed subfield is k = F (y), of co-dimension d. Let
q
ν denote the valuation defined by ν (a ym+ +a yn) = m (a = 0,
y y m n m
m < n), and set F = F ((y)), the completion··w·ith respect to ν ,6and
q y
= F [[y]], its ring of integers.
q
O
Let
1 1
R = F y, , k,
q
(cid:20) y 1+y(cid:21) ⊆
and let R denote the subring
T
1
(3) R = F y, .
T q
(cid:20) 1+y(cid:21)
Since 1+y is invertible in , R .
T
O ⊆ O
EXPLICIT CONSTRUCTIONS OF RAMANUJAN COMPLEXES 7
For a commutative R -algebra S (namely a commutative ring with
T
unit which is an R -module, e.g. k, F, R or R/I for an ideal I(cid:1)R), we
T
denote by y the element y 1 S. For such S we define an S-algebra
· ∈
(S), by
A
d−1
(S) = Sζ zj,
i
A
iM,j=0
with the relations
(4) zζ = φ(ζ )z, zd = 1+y.
i i
The center of (S) is S. We will frequently use the fact that for an
A
R -algebra S, (S) = (R ) S. It is well known that (k) is a
T A A T ⊗RT A
central simple algebra, and so there is a norm map (k)× k×, which
A →
induces a norm map (S)× S× for every S k. The norm map is a
A → ⊆
homogeneous form of degree d (in the coefficients of the basis elements
ζ zj ), so the norm is also defined for quotients (S/I). We remark
i
{that }Fqd⊗FqS is Galois over S and 1 + y is inverAtible in S, so A(S)
is an Azumaya algebra over S (see [DI], where they are called ‘central
separable algebras’). This fact will not be used in the rest of the paper.
If (S) = M (S), we say that (S) is split. We need a criterion for
∼ d
thistAohappen. IfS1 = Fqd⊗FqS isAafield(sonecessarilyS isasubfield),
then (S)isthecyclicalgebra(S /S,φ,1+y) = S [z]withtherelations
1 1
A
in (4). This is a simple algebra of degree d over its center S. Recall
Wedderburn’s norm criterion for cyclic algebras [Jac, Cor. 1.7.5]: the
algebra (S) = (S /S,φ,1 + y) splits iff 1 + y is a norm in the field
1
A
extension S /S. More generally, the exponent of (S /S,φ,1+y) (i.e.
1 1
its order in the Brauer group Br(S)) is the minimal i > 0 such that
(1+y)i is a norm. In particular, if this exponent is d = [S :S], (S)
1
A
is a division algebra (since the exponent of a central simple algebra is
always bounded by the degree of the underlying division algebra).
The algebra (R) will later be used to construct the desired com-
A
plexes. Asmentionedabove, (R) (k),k beingtheringoffractions
A ⊆ A
of R; moreover, (k) is the ring of central fractions of this algebra,
A
(k) = (R 0 )−1 (R).
A −{ } A
We now consider completions of (k). The global field k = F (y)
q
has the minus degree valuation, defiAned for f,g F [y] by ν(f/g) =
q
∈
deg(g) deg(f). Also recall that the other nonarchimedean discrete
valuatio−ns of F (y) are in natural correspondence with the prime poly-
q
nomials of F [y]. For a prime polynomial p F [y], the valuation is de-
q q
∈
fined by ν (pif/g) = i, where f,g are polynomials prime to p. The ring
p
of p-adic integers in k is F [y] = f/g: (p,g) = 1 . The completion
q p
{ }
8 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE
det
GL (F) // F×
d
OO OO
(cid:31)? N (cid:31)?
(R)× // R×
A
Figure 2. Norm and determinant
with respect to a valuation ν = ν is k = F [y] ((p)) = ∞ α pi
p ν q p { i=−v i }
(which we will also denote by kp). The ring of integers ofPkp is p =
F [y] [[p]]. The notation ν is used for the degree valuation, sincOe the
q p 1/y
completion of k with respect to this valuation is F ((1/y)); moreover,
q
filtration by the ideal 1/y of the ring of integers F [[1/y]] determines
q
h i
the valuation.
The Albert-Brauer-Hasse-Noether Theorem describes division alge-
bras over k in terms of their local invariants, which translates to an
injection Br(k) Br(k ). More precisely, the d-torsion part of Br(k )
p p
→⊕
is cyclic of order d for every valuation ν . Taking the unramified exten-
p
sion k′/k of dimension d, these d classes can be written as the cyclic
p p
algebras (k′/k ,φ,̟i) for i = 0,...,d 1, where ̟ is a uniformizer.
p p −
If (k /k,φ,c) is a cyclic k-algebra, the local invariants are determined
1
by the values ν(c) ([P, Chaps. 17–18] is a standard reference, though
the focus is on number fields).
There are only two valuations ν of k for which ν(1+y) = 0, namely
6
ν and ν , for which the values are 1 and 1, respectively. We thus
1+y 1/y
−
have
Proposition 3.1. The completions (F ((1/y))) and (F ((1+y))) of
q q
A A
(k) are division algebras. On the otherhand, forany other completion
A
k of k, (k ) splits.
ν ν
A
In particular
(5) (R) (k) F = (F) = M (F).
k ∼ d
A ⊆ A ⊗ A
The same argument embeds
(6) (R ) ( ) = M ( ).
T ∼ d
A ⊆ A O O
We use the algebras (S) to define algebraic group schemes. For an
A
R -algebra S, let G˜′(S) = (S)×, the invertible elements of (S), and
T
A A
G′(S) = (S)×/S×. Recall that for every R -algebra S, one can define
T
A
a multiplicative function (S) S× called the reduced norm (e.g. by
A →
taking the determinant in a splitting extension of S). In particular, the
diagram in Figure 2 commutes. We can thus define G˜′(S) as the set of
1
EXPLICIT CONSTRUCTIONS OF RAMANUJAN COMPLEXES 9
elements of G˜′(S) of norm 1, and G′(S) as the image of G˜′(S) under
1 1
the map G˜′(S) G′(S) (see the square in the middle of Figure 5).
→
Remark 3.2. The sequence
1 µ (S) G˜′(S) G′(S) 1
−→ d −→ 1 −→ 1 −→
is exact, where µ (S) is the group of d-roots of unity in S.
d
˜
ThesegroupschemesareformsoftheclassicalgroupsG(S) = GL (S),
d
G(S) = PGL (S), G˜ (S) = SL (S) and G (S) = PSL (S). If (S) is
d 1 d 1 d
A
a matrix ring, we have that G˜′(S) = G˜(S) and G′(S) = G(S).
It is useful to have equivalent definitions for the groups G′(S) for
various rings S. Fix the ordered basis
ζ ,...,ζ ,ζ z,...,ζ z,...,ζ zd−1,...,ζ zd−1
0 d−1 0 d−1 0 d−1
{ }
of (S) over S. Conjugation by an invertible element a (S) is a
A ∈ A
linear transformation of the algebra. Let i :G′(S) GL (S) be the
S d2
→
induced embedding. If S S′, then the diagram
⊆
G′(S′) iS′ // GLd2(S′)
OO OO
G′(cid:31)?(S) iS // GL(cid:31)d?2(S)
commutes.
Proposition 3.3. Let R S S′ be commutative rings, such that
T
⊆ ⊆
S is a Noetherian unique factorization domain. Then
i G′(S) = i G′(S′) GL (S),
S S′ d2
∩
the intersection taken in GL (S′).
d2
Proof. The inclusion i G′(S) i G′(S′) GL (S) is trivial since
S S′ d2
⊆ ∩
(S) (S′). Let α i G′(S′) GL (S), then α is an isomor-
S′ d2
A ⊆ A ∈ ∩
phism of algebras (since it is induced by an element of (S′)) and
A
preserves S (as it belongs to GL (S)). It is thus an automorphism of
d2
(S), which must be inner [AG, Thm. 3.6]. (cid:3)
A
The proposition covers, in particular, S = R ,R,k,k , , as well as
T ν ν
R¯ , and R¯ which are defined in Section 8, with an arbitrOary extension
T
S′ (usually taken to be from the same list).
Proposition 3.4. G′(R) is a discrete subgroup of G(F).
10 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE
Proof. The ring R = F [1,y, 1 ] embeds (diagonally) as a discrete
q y 1+y
subgroup of the product
F k k .
1/y p
× ×
This can be seen by letting a = fn(y) be a sequence of non-zero
n yin(1+y)jn
elements in R (with f (λ) F [λ] and i ,j 0) such that a 0. We
n q n n n
∈ ≥ →
then have that ν (a ) , which implies i = 0 for n large enough
y n n
→∞
and ν (f (y)) . Likewise, ν (a ) , so j = 0 for n large enough.
y n p n n
→∞ →∞
This implies that ν (a ) , which is a contradiction.
1/y n
→−∞
It follows that the diagonal embedding of G′(R) into
G′(F) G′(k ) G′(k )
1/y p
× ×
is discrete. But an algebraic group over a local field is compact iff it
has rank zero [PR]. Therefore, by Proposition 3.1, G′(k ) and G′(k )
1/y p
are compact, and G′(R) is discrete in the other component G′(F) =
G(F). (cid:3)
In fact, from general results it follows that G′(R) is a cocompact
lattice in G′(F), but we will show it directly when we demonstrate
that G′(R) acts transitively on the vertices of the affine building of
G(F) = PGL (F).
d
Consider G( ) = PGL ( ), a maximal compact subgroup of G(F),
d
O O
which is equal to G′( ) by Equation (6). Viewing K = i G′( ) and
O
O O
i G′(R) as subgroups of i G′(F) GL (F), the intersection
R F d2
⊆
(7) i G′(R ) = K i G′(R)
RT T ∩ R
isfinite, being the intersection ofdiscrete andcompact subgroups (note
that R = R ).
T
∩O
Proposition 3.5. G′(R ) is a semidirect product of z = Z/dZ acting
T ∼
on F×/F×. h i
qd q
Proof. Recall that R = F [y,1/(1+y)], so that (R ) = F [y,1/(1+
T q T qd
A
y),z]withtherelations zαz−1 = φ(α)(α F ) andzd = 1+y. Setting
qd
∈
y = zd 1, we see that (R ) = F [z,z−1] is a skew polynomial ring
T qd
with on−e invertible variaAble over F . Every element of (R ) has a
qd T
A
monomial αzr (α F×) with r maximal, called the upper monomial
∈ qd
(with respect to z), and similarly every element has a lower monomial.
The upper monomial of a product fg is equal to the product of the
respective upper monomial, and likewise for the lower monomials.
Now let f,g (R ) be elements with fg = 1, then the product of
T
∈ A
the upper monomials and that of the lower monomials are both equal
to1, provingthatf andg aremonomials. Thus, theinvertible elements
