DSpace at VNU: The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

DSpace at VNU: The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra tài liệu, giáo án...

DOI 10.1007/s00208-011-0698-4 Mathematische Annalen

The homomorphisms between the Dickson–Mùi

algebras as modules over the Steenrod algebra

allA-module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be

isomorphic to a quotient of the polynomial algebra on one indeterminate We provethat the reduced Dickson–Mùi algebra is atomic in the meaning that if anA-module

endomorphism of the algebra is non-zero on the least positive degree generator, then

it is an automorphism This particularly shows that the reduced Dickson–Mùi algebra

is an indecomposableA-module The similar results also hold for the odd

character-istic Dickson algebras In particular, the odd charactercharacter-istic reduced Dickson algebra

is atomic and therefore indecomposable as a module over the Steenrod algebra

Mathematics Subject Classification (2010) Primary 55S10· 55S05 ·

20G10· 20G05

1 Introduction and statement of results

LetV = Vs be an elementary abelian p-group of rank s, where p is a prime ThenV

can also be regarded as an s-dimensional vector space overFp , the prime field of p

The paper is dedicated to the memory of my late friend, Pha.m Anh Minh.

The work was supported in part by a grant of the NAFOSTED.

N H V Hu’ng (B)

Department of Mathematics, Vietnam National University, Hanoi,

334 Nguy˜ên Trãi Street, Hanoi, Vietnam

e-mail: nhvhung@vnu.edu.vn

828 N H V Hu’ng

elements Let H∗(V) denote the mod p cohomology of (a classifying space BV of)

the groupV As it is well-known

H∗(V) ∼=



F2[x1, , x s ], p = 2,

E (e1, , e s ) ⊗ F p [x1, , x s ], p > 2.

Here(x1, , x s ) is a basis of H1(V) = Hom(V, F p ) when p = 2, or a basis of

H2(V) and x i = β(e i ) for 1 ≤ i ≤ s with β the Bockstein homomorphism when

p > 2.

The general linear group G L(V) ∼ = GL(s, F p ) acts regularly on V and therefore

on H∗(V) The Dickson algebra, which was first studied and explicitly computed

by Dickson [3], is the algebra of all invariants ofFp [x1, , x s] under the action of

G L(V) The invariant algebra H∗(V) G L ( V)was explicitly computed by Mùi [10] for

p > 2 We call H∗(V) G L ( V) the Dickson–Mùi algebra and denote it by D(V), or simply by D s , in the both cases p = 2 and p an odd prime.

Being the cohomology of the classifying space BV, the group H∗(V) is equipped with a structure of module over the mod p Steenrod algebra, A = A p Each element

γ ∈ GL(V) induces a homeomorphism Bγ : BV → BV, whose induced

homomor-phism in cohomology is anA-isomorphism γ∗: H∗(V) → H∗(V) The map γ → γ∗

gives rise to the regular action of G L (V) on H∗(V) So the actions of GL(V) and A upon H∗(V) commute with each other Therefore, the Dickson–Mùi algebra inherits

a structure of module over the Steenrod algebra from H∗(V).

Let D(V) or D s be the augmentation ideal of all positive degree elements in the

Dickson–Mùi algebra D s = D(V) We call it the reduced Dickson–Mùi algebra In

Sects.3and4we will respectively define the restriction r es s ,n : D s → D nand the

transfer tr n ,r : D n → D r on the Dickson–Mùi algebras LetU and W be respectively

Fp -vector spaces of dimensions r and n The above two homomorphisms are also denoted by r es V,W : D(V) → D(W) and tr W,U : D(W) → D(U) respectively The

following theorem is one of the main result of the paper

Theorem 1.1 The A-module homomorphisms



tr n ,r r es s ,n1≤ n ≤ min{r, s}

form a basis of the vector space H om A (D s , D r ) of all A-module homomorphisms from D s to D r In particular, dimFp H om A (D s , D r ) = min{r, s}.

The main ingredients of our proof are as follows LetU and V be Fp-vector spaces

of dimensions r and s respectively.

First, according to a theorem by Carlsson [2] for p = 2 and by Miller [9] for p

odd prime, ˜H∗(U) is injective in the category of unstable reduced A-modules Hence,

eachA-module homomorphism f : D(V) → D(U) ⊂ ˜ H∗(U) can be extended to an A-module homomorphism ˆf : ˜ H∗(V) → ˜ H∗(U).

Secondly, by a theorem of Adams–Gunawardena–Miller [1], ˆf can be expressed

as

ˆf = λ1ϕ1∗+ · · · + λ k ϕ k∗,

The extension ˆf is reduced so that K er ϕ i’s are pairwise distinct for 1 ≤ i ≤ k.

Finally, the restrictions and the transfers are taken into account when we decompose

ϕ i∗ = π i∗ϕ∗i withπ i an epimorphism andϕ i a monomorphism, and then recognizethe relation between the termsλ i ϕ∗

i ’s in order to get f factoring through the G

Note that tr0,r r es s ,0simply maps 1∈ D s to 1∈ D r and vanishes on D s

The commutativity relation of the transfer and the restriction is given as follows

Theorem 1.2 (i) r es n ,r tr s ,n = tr s −n+r,r r es s ,s−n+r , for n ≥ max{r, s}.

(ii) tr n ,r r es s ,n = res s −n+r,r tr s ,s−n+r , for n ≤ min{r, s}.

Here, by convention, tr m ,r r es s ,m sends 1 to 1 and vanishes on D s for m < 0 The algebras End A (D s ) and End A (D s ) are described as follows.

Theorem 1.3 LetFp [t] be the polynomial algebra on an indeterminate t There are isomorphisms of algebras

(i) End A (D s ) ∼= Fp [t]/(t s ),

(ii) End A (D s ) ∼= Fp [t]/(t s+1− t s ),

which send tr s −1,s r es s ,s−1 to t Particularly, these algebras are commutative.

Part (ii) of the theorem adjusts the corresponding result announced in [5]

We will show that (tr s −1,s r es s ,s−1 ) i = tr s −i,s r es s ,s−i for any i by using

Theorem1.2and the two equalities that tr r ,s tr n ,r = tr n ,s (see Lemma4.4) and that

r es r ,n r es s ,r = res s ,n(see Lemma3.3) for n ≤ r ≤ s Then the theorem is proved by

combining this formula with Theorem1.1or with its consequence on H om A (D s , D r ) The vector space H om A (D s , D r ) is equipped with a bimodule structure: It is a right module over End A (D s ) and a left module over End A (D r ) By passing to the quotient, H om A (D s , D r ) is also a bimodule: a right module over End A (D s ) and a left module over End A (D r ).

Set u i = trmin(r,s)−i,r r es s ,min(r,s)−i for i ≥ 0 Denote t = tr s −1,s r es s ,s−1 in

End A (D s ) or in End A (D s ) Since s is not part of the notation, t also means

tr r −1,r r es r ,r−1 in End A (D r ) or in End A (D r ) Theorem 1.2 leads us to thefollowing

Proposition 1.4 The structures of the bimodules H om A (D s , D r ) ∼= ⊕min(r,s)

i=0 Fp u i

and H om A (D s , D r ) ∼= ⊕min(r,s)−1

i=0 Fp u i are given by

θ : Hom(U, V) → Hom A (D(V), D(U)),

ϕ → tr ϕ( U),U r es V,ϕ(U)

It is evident that, forϕ, ψ ∈ Hom(U, V), θ(ϕ) = θ(ψ) if and only if I mϕ ∼ = I mψ,

or equivalently K er ϕ ∼ = K erψ We write ϕ ∼ ψ to say that this condition is valid It

is easy to see that(Hom(U, V)/ ∼) ∼ = GL(V)\Hom(U, V)/GL(U).

Theorem1.1can be re-expressed in the following formulation: The mapθ induces

two isomorphisms of vector spaces

Fp [G L (V)\Hom(U, V)/GL(U)] −→ Hom∼ A (D(V), D(U)),

Fp [G L(V)\Hom(U, V)/GL(U)]Fp0 ∼

In order to get the second isomorphism from the first one, we observe thatθ(F p0) is exactly the subspace of homomorphisms that vanish on D(V).

The following is probably something of interest

Conjecture 1.5 For any subgroups G of G L(U) and H of GL(V), there is an

isomor-phism ofFp-vector spaces

Fp [H \Hom(U, V)/G] ∼ = Hom A (H∗(V) H , H∗(U) G ).

Note that this isomorphism happens for G = {1}, H = {1} by the theorem of Adams–Gunawardena–Miller, and for G = GL(U), H = GL(V) by Theorem1.1.Suppose that the conjecture is true Then we are interested in the following prob-lem: Describe the product inFp [H \End(V)/H] that gives rise to an isomorphism of

Let(ε1, , ε s ) be a basis of V Each equivalence class in GL(V)\End(V)/GL(V)

contains exactly one endomorphism of the formξ k : V → V for 0 ≤ k ≤ s that sends

ε i toε i −k for i > k and to 0 for i ≤ k The composition in End(V) satisfies

The following theorem is an another main result of the paper

Theorem 1.6 An A-module endomorphism f : D s → D s is an automorphism if and only if

where λ is a non-zero scalar In particular, there are exactly (p − 1)p s−1

automor-phisms of the A-module D s

Theorem 1.7 If an A-module endomorphism f : D s → D s is non-zero on the least positive degree generator of the Dickson–Mùi algebra, then it is an automorphism.

An immediate consequence of this theorem is the following corollary, which showsthe rigidity of the Dickson–Mùi algebra

Corollary 1.8 The reduced Dickson–Mùi algebra is an indecomposable module over

the Steenrod algebra.

An application of Theorem1.1is the following theorem

Theorem 1.9 Let f : D s → D r be a homomorphism of A-algebras Then

f =



λres s ,r , r ≤ s,

0, r > s, where λ is either 0 or 1.

For p an odd prime, let S∗(V) be the symmetric algebra on βV∗, a copy of the dual

spaceV∗ofV This algebra is graded by assigning degree 2 to each element in βV∗.

The Dickson algebra, denotedD(V) or D s with s = dim V, is the algebra S∗(V) G L ( V)

of all invariants of S∗(V) under the action of GL(V) The reduced Dickson algebra D(V) or D sis, by definition, the augmentation ideal of all positive degree elements inthe Dickson algebraD(V) = D s Theorem1.1is also valid for the odd characteristicDickson algebras in the same way as it is for the Dickson–Mùi algebras The following

is a consequence of this theorem

832 N H V Hu’ng

Corollary 1.10 In an odd characteristic, any A-module homomorphism between the reduced Dickson algebras ϕ : D s → D r can uniquely be extended to an A-module homomorphism between the reduced Dickson–Mùi algebras ˆϕ : D s → D r The map

ϕ → ˆϕ gives rise to an isomorphism Hom A (D s , D r ) ∼ = Hom A (D s , D r ).

The odd characteristic reduced Dickson algebras are also atomic in the meaning

of Theorem10.3, which is similar to Theorem1.7, and therefore indecomposable asmodules over the Steenrod algebra

The paper is divided into 10 sections and organized as follows The tion in Sect 1 is followed by the preliminary in Sect 2, where we collect some

introduc-needed results on the cohomology of the elementary abelian p-groups and on the

Dickson–Mùi algebra Sections3 8 deal with the characteristic p = 2 In Sects.3

and4we define respectively the restriction and the transfer on the Dickson algebrasand give a beginning study of their basic behaviors Section 5deals with the rela-tionship between the transfer and the restriction on the Dickson algebras We studytheA-module homomorphisms between the (reduced) Dickson algebras in Sect.6.Section7is devoted to the study of theA-module endomorphisms and the A-module

automorphisms of the (reduced) Dickson algebras This section also investigates the

bimodule structures of H om A (D s , D r ) and Hom A (D s , D r ) The A-algebra

homo-morphisms between the (reduced) Dickson algebras are investigated in Sect.8 Weexpress in Sect.9the changes needed for the Dickson–Mùi algebra, the case of oddprime characteristic In Sect.10we show that the results that are similar to the ones for

the Dickson–Mùi algebras also hold for the Dickson algebras in characteristic p > 2.

It should be mentioned that there is some overlap between the present paper andKechagias’ manuscript [8], where the dimension of the endomorphism ring and theindecomposability of the reduced Dickson–Mùi algebras are probably shown.Most of the paper’s contents, except Sect 10, have been announced in [5]

2 Preliminary

To make the paper self contained, we collect in this section some needed results on the

cohomology of the elementary abelian p-groups and on the Dickson–Mùi algebra.

First, following [3] and [10], we define the Dickson and the Mùi invariants.For any non-negative integers(r1, , r s ) we set [r1, , r s ] = det(x p r j

i ) In

par-ticular, we define

L s ,i = [0, , ˆi, , s], (for 0 ≤ i ≤ s),

L s = L s ,s = [0, 1, , s − 1].

Here and in what follows, by writing ˆi we mean i is deleted The Dickson invariants

are defined as follows

Q s ,i = L s ,i /L s ,

for 0≤ i < s (Particularly, we observe that Q s ,0 = L p−1

s ) They can inductively beexpressed by the formula

Q s ,i = Q p

s −1,i−1 + Q s −1,i V p−1

s , where Q s ,s = 1, Q s ,i = 0 for i < 0, and

c1, ,c s−1 ∈Fp

(c1x1+ · · · + c s−1x s−1+ x s ).

For 0≤k ≤s, the following element is first defined in EZ(e1, ., e s )⊗Z[x1, ., x s]

and then projected to E(e1, , e s ) ⊗ F p [x1, , x s]:

in which there are exactly k rows of (e1· · · e s ) (See the accurate meaning of the

deter-minants in a commutative graded algebra in Mùi [10].) The Mùi invariants are definedby

M s ,i1, ,i k = [k; 0, , ˆi1, , ˆi k , , s − 1],

R s ,i1, ,i k = M s ,i1, ,i k L p−2

s ,

for 0≤ i1< · · · < i k ≤ s − 1.

It should be noted that Q s ,i and R s ,i1, ,i kare invariant under the general linear group

G L(V), while M s ,i1, ,i k is invariant under the Sylow subgroup of G L (V) consisting

of all upper triangular matrices with 1 on the main diagonal

The Dickson algebra, which originally appeared in Invariant Theory in the early20th century, plays a key role in Algebraic Topology nowadays

Theorem 2.1 (Dickson [3])

Fp [x1, , x s]G L ( V)= Fp [Q s ,0 , , Q s ,s−1 ],

for any prime number p.

In the case when p is an odd prime, the exterior algebra is taken into account Then

the Dickson–Mùi algebra plays the role of the Dickson algebra

In the preceding two theorems, deg (Q s ,i ) = 2 s− 2i for p = 2, and deg(Q s ,i ) =

2(ps − p i ), deg(R s ,i1, ,i k ) = k + 2(p s − 1) − 2(p i1+ · · · + p i k ) for p an odd prime.

The action of the Steenrod algebra on the Dickson algebra is described as follows

Theorem 2.3 (Hu’ng [4]) For p = 2,

The analogue of this theorem for p an odd prime is given by Hu’ng–Minh [6]

The amazing theorem below was originally due to G Carlsson for p= 2, then was

proved in a dual statement by H Miller for any prime p.

Theorem 2.4 (Carlsson [2], Miller [9])

H∗(V) is injective in the category of unstable A-modules for any prime number p.

The following theorem is a surprising corollary of Theorem 1.6 in the paper byAdams et al [1]

Theorem 2.5 (Adams et al [1])

Let U, V be elementary abelian p-groups Then the obvious map

Fp [HomFp (U, V)] → Hom A (H∗(V), H∗(U))

is an isomorphism for any prime number p.

3 Restriction on the Dickson algebras

In Sects.3 8, we always work with the characteristic p= 2

LetU and V be F2-vector spaces of respectively dimensions r and s with r ≤ s.

Lemma 3.1 Suppose i U,V : U → V is an inclusion of vector spaces.

(i) The induced homomorphism in cohomology i∗

U,V : H∗(V) → H∗(U) maps

H∗(V) G L ( V) to H∗(U) G L ( U) .

(ii) The homomorphism i∗

U,V : H∗(V) G L ( V) → H∗(U) G L ( U) does not depend on

the choice of the inclusion i U,V

Proof (i) Denote i = i U,Vfor abbreviation LetW be a vector subspace of V such that

V = iU⊕W For any linear isomorphism α : U → U, we define a linear isomorphism

γ : V → V by setting

γ | iU= iαi−1,

γ |W= id|W.

It is easy to check that i α = γ i, or equivalently α∗i∗= i∗γ∗.

For Q ∈ H∗(V) G L ( V), we haveγ∗(Q) = Q by definition of invariants So we get

α∗i∗(Q) = i∗γ∗(Q) = i∗(Q).

That is, i∗(Q) is invariant under the action of any isomorphism α∗, or equivalently

i∗(Q) ∈ H∗(U) G L ( U).

(ii) Suppose i and j are two inclusions ofU into V Then, there exists an

automor-phism h of V such that j = hi Hence j∗= i∗h∗ By definition, the homomorphism

h∗, which is induced by an isomorphism h ∈ GL(V), acts identically on H∗(V) G L ( V).

That is, h∗is the identity on H∗(V) G L ( V) , therefore j∗= i∗on H∗(V) G L ( V).

The following definition is based on the preceding lemma

Definition 3.2 The restriction from D (V) to D(U), denoted by res V,U : D(V) → D(U), is the A-algebra homomorphism i∗

U,V : H∗(V) G L ( V) → H∗(U) G L ( U)induced

by an inclusion i U,V : U → V It is also denoted by res s ,r : D s → D r for r =dimU ≤ dim V = s.

Evidently, r es V,U maps D(V) to D(U) The resulting map is also denoted by

r es V,U : D(V) → D(U), or res s ,r : D s → D r

IfU ∼= V, then res V,U : D(V) → D(U) is the canonical A-algebra isomorphism induced by any isomorphism of spaces i U,V: U → V

Lemma 3.3

r es U,T r es V,U = res V,T : H∗(V) G L ( V) → H∗(T) G L ( T)

for any spaces T, U, V with dim T ≤ dim U ≤ dim V In other words,

r es r ,n r es s ,r = res s ,n : D s → D n , for any non-negative integers n ≤ r ≤ s.

The proof is straightforward

Proof Let i r ,s : U → V be an inclusion We choose a basis (x1, , x s ) of V∗ =

H1(V) and a basis (y1, , y r ) of U∗= H1(U) such that

r ,s Recall that, from [3] and [10], the Dickson invariants

can inductively be defined by the formula

Q s ,i = Q2

s −1,i−1 + Q s −1,i V s , where Q s ,s = 1, Q s ,i = 0 for i < 0, and V s denotes the Mùi invariant

The lemma is proved by backward induction on r using Lemma3.3

4 Transfer on the Dickson algebras

In the section, we define the notion of transfer on the Dickson algebras and give somebeginning study on it

LetU and V be F2-vector spaces of respectively dimensions r and s with r ≤ s Any

two epimorphismsπ1, π2: V → U with the same kernel, K, differ by tion with an automorphismα of U: more precisely, π2= απ1 So the induced maps

post-composi-π∗

1, π∗

2 : H∗(U) G L ( U) → H∗(V) are the same, as α∗acts identically on H∗(U) G L ( U).

Denote this induced map byπ∗

K The group G L (V) permutes the subspaces of sion s − r So the sum of the π∗

dimen-K’s lands in the G L(V)-invariants.

Definition 4.1 The transfer on the Dickson algebras is defined by

Obviously, the transfer is a homomorphism of A-modules.

Since the transfer tr U,V only depends on the dimensions ofU and V, it is also

denoted by tr r ,s : D r → D s , where r = dim U ≤ dim V = s Clearly, tr U,V maps

D (U) to D(V) The resulting map is also denoted by tr U,V : D(U) → D(V), or

an epimorphism, whose kernel isK There is an 1–1 correspondence between the set

of all 1-dimensional vector spaces of the formπ∗

KH1(F2) and the set of all non-zero linear forms c1x1+ · · · + c n x n with c i ∈ F2 So, by definition of the transfer, we have

In particular, each term on the right-hand side is divisible by x1x2· · · x n

It is easy to see that

838 N H V Hu’ng

On the other hand, Q n ,0 is the unique G L n-invariant of degree 2n − 1 in the n terminates x1, , x n Therefore

inde-tr1,n (x2n−1) = Q n ,0

Part (i) of the lemma is proved

(ii) Note that 2n − 1 is the degree of the top Dickson generator Q n ,0 Thus, the only

degrees j with 0 < j < 2 n − 1 in which D n is non-zero are j = deg Q n ,i = 2n− 2i

for 0< i < n So, it suffices to show that tr1,n (x2n−2i

) = 0, for 0 < i < n.

As i > 0, tr1,n (x2n −i−1) = 0 by degree information Indeed, this is a consequence

of the facts that

deg(x2n −i−1) = 2 n −i − 1 < 2 n−1= degQ n ,n−1 , and that Q n ,n−1 is the least positive degree generator in D n Then, since the transfercommutes with the Frobenius map, we get

for any spaces T, U, V with dim T ≤ dim U ≤ dim V In other words,

tr r ,s tr n ,r = tr n ,s : D n → D s , for any non-negative integers n ≤ r ≤ s.

Proof Let T, U and V be F2-vector spaces of respectively dimensions n, r and s with

n ≤ r ≤ s In order to work with the transfers, we choose some sets of epimorphisms

Kif the maps in the 2 sides differ by post-composition with an

automorphism ofT, or equivalently if ˆK = K erπˆK= K er(π1

given(s − n)-dimensional vector space ˆK.

There is an 1–1 correspondence between the set of such couples(K, H) and the

set of all (s − r)-dimensional vector subspaces K of ˆK Indeed, if πˆK  π1

K( ˆK), for which

H ∼= ˆK/K is of dimension (s − n) − (s − r) = r − n with ˆK = (π2

K)−1(H) Indeed, (π2

The number of all(s − r)-dimensional vector subspaces K of the (s −

n)-dimen-sional space ˆK is classically given by:

The lemma is proved.

5 Restriction and transfer on the Dickson algebras

The commutativity relation of the transfer and the restriction is investigated by thefollowing theorem, which is also numbered as Theorem1.2in the introduction

Theorem 5.1 (i) r es n ,r tr s ,n = tr s −n+r,r r es s ,s−n+r , for n ≥ max{r, s}.

(ii) tr n ,r r es s ,n = res s −n+r,r tr s ,s−n+r , for n ≤ min{r, s}.

Here, by convention, tr m ,r r es s ,m sends 1 to 1 and vanishes on D s for m < 0.

We prepare to prove the theorem by a couple of lemmas

Lemma 5.2 r esr +1,r tr r ,r+1 = tr r −1,r r es r ,r−1 , for any positive integer r

Proof Let us denoteF = F2for abbreviation

Any two epimorphismsπ1, π2 : Fr+1 → Fr with the same kernel,K, differ bypost-composition with an automorphismα of F r:π2 = απ1 Let C and C be the

(r + 1) × r matrices of respectively π∗

1 andπ∗

2 with respect to the bases(x1, , x r )

of H1(F r ) and (x1, , x r+1) of H1(F r+1) Also, let A be the r ×r matrix of α∗with

respect to the basis(x1, , x r ) of H1(F r ) Then π1, π2are epimorphisms if and only

if C , Care of the maximal rank r , while α is isomorphic if and only if A is invertible.

The equalityπ2 = απ1is equivalent toπ∗

2 = π∗

1α∗ : H∗(F r ) → H∗(F r+1), or to

C= C A.

Two(r + 1) × r matrices C and Care said to be in the same orbit if there is an

r × r invertible matrix A such that C= C A.

whereK runs over all vector subspaces of dimension 1 in Fr+1, while C runs over a

set of all orbit representatives of(r + 1) × r matrices with rank r.

We consider the distinct two types of matrices C’s.

Type 1: The first r rows of C are linearly independent Let A be the inverse of the r ×r matrix, whose rows are the first r rows of C Then

Type 2: The first r rows of C are linearly dependent Then its last row together with

some preceding(r − 1) rows are linearly independent There is an invertible r × r matrix A such that

where C is an r ×(r −1) matrix of the maximal rank, (r −1) Indeed, A can be chosen

as follows The first(r − 1) columns of A are any (r − 1) independent solutions of

⎞

⎟

⎟.

(The equation system has a unique solution as r ank(C) = r and as the last row of C

is non-zero.) In addition, the equality

(c r+11 c r +1r )A = (0 0 1)

842 N H V Hu’ng

shows that the last column of A can not be written as a linear combination of its

first(r − 1) columns Combining all the above properties of A we conclude that it is

invertible

It is evident that matrices of different types are not in the same orbit

Assume that C , Care two r × (r − 1) matrices of the maximal rank r − 1, and A

is an invertible r × r matrix Then

Each of the coefficients c1, , c r is independently able to be either 0 or 1 There are

2r combinations of such values of c1, , c r Hence

If C is of type 2, then we have

The lemma is completely proved

Lemma 5.3 r esn ,r tr n −1,n = tr r −1,r r es n −1,r−1 , for r ≤ n Here, by convention,

tr r −1,r r es n −1,r−1 sends 1 to 1 and vanishes on D n−1for r − 1 < 0.

Proof

r es n ,r tr n −1,n = (res n −1,r r es n ,n−1 )tr n −1,n (by Lemma3.3)

= res n −1,r (tr n −2,n−1 r es n −1,n−2 ) (by Lemma5.2)

= (res n −2,r r es n −1,n−2 )tr n −2,n−1 r es n −1,n−2 (by Lemma3.3)

= res n −2,r (tr n −3,n−2 r es n −2,n−3 )res n −1,n−2 (by Lemma5.2)

= res n −2,r tr n −3,n−2 r es n −1,n−3 (by Lemma3.3)

=

= res r ,r tr r −1,r r es n −1,r−1

= tr r −1,r r es n −1,r−1 (as res r ,r = id D r ).

Proof of Theorem5.1 (i)

r es n ,r tr s ,n = res n ,r (tr n −1,n tr s ,n−1 ) (by Lemma4.4)

844 N H V Hu’ng

= tr r −(n−s),r r es s ,r−(n−s) tr s ,s

= tr s −n+r,r r es s ,s−n+r (as tr s ,s = id D s ).

(ii) Now we show that (i) implies (ii)

Set m = s − n + r From n ≤ min{r, s} it implies that m ≥ max{r, s} By (i) we

have

r es s −n+r,r tr s ,s−n+r = res m ,r tr s ,m = tr s −m+r,r r es s ,s−m+r

= tr n ,r r es s ,n ,

as s − m + r = s − (s − n + r) + r = n Then, part (ii) holds.

The theorem is completely proved

Actually, part (i) and part (ii) of the proposition are equivalent One can show (ii)implying (i) similarly as we showed (i) implying (ii)

The following corollaries of Theorem 5.1intend to partially answer the tion: How can one relate the behavior of the transfer to the augmentation-ideal-adicfiltration?

Corollary 5.6 If n ≤ min{r, s}, then

The corollary is proved

6 TheA-module homomorphisms between the Dickson algebras

The goal of this section is to prove the following theorem, which is the p= 2 case ofTheorem1.1in the introduction

Theorem 6.1 The A-module homomorphisms



tr n ,r r es s ,n1≤ n ≤ min{r, s}

form a basis of the vector space H om A (D s , D r ) of all A-module homomorphisms from D s to D r In particular, dimF2H om A (D s , D r ) = min{r, s}.

We fix a set T of isomorphism class representatives of vector spaces T’s with

0< dim T ≤ min{dim U, dim V} For each vector subspace K ⊂ U with codim K =

dimT, let πK : U → T be an epimorphism with K erπK = K To prepare for theproof of the theorem, we need the following lemma

846 N H V Hu’ng

Lemma 6.2 The set of homomorphisms



π∗

Kr es V,T  K ⊂ U,codimK = dim T ≤ dim V

is linearly independent in H om A (D(V), ˜ H∗(U)).

Proof The proof is divided into two steps.

Step 1 Set s= dim V First we show that the elements in the set



πK∗r es V,T (Q s ,s−1 )  K ⊂ U,codimK = dim T ≤ dim V

are non-zero and pairwise distinct

Denoting n= dim T, by Lemma3.4, we have

Case 1 Let πK: U → T and πK : U → Tbe epimorphisms, whereT and Tare

distinct in the setT of isomorphism class representatives of spaces Without losing generality, assume that n= dim T< n = dim T Suppose to the contrary that

Since the Frobenius map ξ → ξ2 is a monomorphism on the polynomial algebra

H∗(U), the above equality is equivalent to the following