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...

Math Ann (2012) 353:827–866 DOI 10.1007/s00208-011-0698-4 Mathematische Annalen The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra ˜ H V Hu’ng Nguyên Received: 19 December 2010 / Revised: 15 June 2011 / Published online: August 2011 © Springer-Verlag 2011 Abstract The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group It is a module over the Steenrod algebra, A We determine explicitly all the A-module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A-module automorphisms of the (reduced) Dickson–Mùi algebras The algebra of all A-module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A-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 indecomposable A-module The similar results also hold for the odd characteristic Dickson algebras In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra Mathematics Subject Classification (2010) 20G10 · 20G05 Primary 55S10 · 55S05 · Introduction and statement of results Let V = Vs be an elementary abelian p-group of rank s, where p is a prime Then V can also be regarded as an s-dimensional vector space over F p , 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, ˜ Trãi Street, Hanoi, Vietnam 334 Nguyên e-mail: nhvhung@vnu.edu.vn 123 828 N H V Hu’ng elements Let H ∗ (V) denote the mod p cohomology of (a classifying space BV of) the group V As it is well-known H ∗ (V) ∼ = F2 [x1 , , xs ], E(e1 , , es ) ⊗ F p [x1 , , xs ], p = 2, p > Here (x1 , , xs ) is a basis of H (V) = H om(V, F p ) when p = 2, or a basis of H (V) and xi = β(ei ) for ≤ i ≤ s with β the Bockstein homomorphism when p > The general linear group G L(V) ∼ = G L(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 of F p [x1 , , xs ] under the action of G L(V) The invariant algebra H ∗ (V)G L(V) was explicitly computed by Mùi [10] for p > We call H ∗ (V)G L(V) the Dickson–Mùi algebra and denote it by D(V), or simply by Ds , in the both cases p = 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 ∈ G L(V) induces a homeomorphism Bγ : BV → BV, whose induced homomorphism in cohomology is an A-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 G L(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 Ds = D(V) We call it the reduced Dickson–Mùi algebra In Sects and we will respectively define the restriction r ess,n : D s → D n and the transfer trn,r : D n → Dr on the Dickson–Mùi algebras Let U and W be respectively F p -vector spaces of dimensions r and n The above two homomorphisms are also denoted by r esV,W : D(V) → D(W) and trW,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 trn,r r ess,n ≤ n ≤ min{r, s} form a basis of the vector space H om A (D s , Dr ) of all A-module homomorphisms from D s to Dr In particular, dimF p H om A (D s , Dr ) = min{r, s} The main ingredients of our proof are as follows Let U and V be F p -vector spaces of dimensions r and s respectively First, according to a theorem by Carlsson [2] for p = and by Miller [9] for p odd prime, H˜ ∗ (U) is injective in the category of unstable reduced A-modules Hence, each A-module homomorphism f : D(V) → D(U) ⊂ H˜ ∗ (U) can be extended to an A-module homomorphism fˆ : H˜ ∗ (V) → H˜ ∗ (U) Secondly, by a theorem of Adams–Gunawardena–Miller [1], fˆ can be expressed as fˆ = λ1 ϕ1∗ + · · · + λk ϕk∗ , 123 The homomorphisms between the Dickson–Mùi algebras 829 where λi ∈ F p and ϕi∗ is the homomorphism induced in cohomology by some linear map ϕi : U → V for any i Then ϕi∗ is a homomorphism of A-algebras (and not only a homomorphism of A-modules) The extension fˆ is reduced so that K er ϕi ’s are pairwise distinct for ≤ i ≤ k Finally, the restrictions and the transfers are taken into account when we decompose ϕi∗ = πi∗ ϕ i∗ with πi an epimorphism and ϕ i a monomorphism, and then recognize the relation between the terms λi ϕi∗ ’s in order to get f factoring through the G L(U)invariants By means of the A-module decomposition Ds = F p · ⊕ D s , we get the following consequence of the preceding theorem: The A-module homomorphisms trn,r r ess,n ≤ n ≤ min{r, s} form a basis of the vector space H om A (Ds , Dr ) of all A-module homomorphisms from Ds to Dr In particular, dimF p H om A (Ds , Dr ) = min{r, s} + Note that tr0,r r ess,0 simply maps ∈ Ds to ∈ Dr and vanishes on D s The commutativity relation of the transfer and the restriction is given as follows Theorem 1.2 (i) r esn,r trs,n = trs−n+r,r r ess,s−n+r , for n ≥ max{r, s} (ii) trn,r r ess,n = r ess−n+r,r trs,s−n+r , for n ≤ min{r, s} Here, by convention, trm,r r ess,m sends to and vanishes on D s for m < The algebras EndA (D s ) and EndA (Ds ) are described as follows Theorem 1.3 Let F p [t] be the polynomial algebra on an indeterminate t There are isomorphisms of algebras (i) EndA (D s ) ∼ = F p [t]/(t s ), ∼ (ii) EndA (Ds ) = F p [t]/(t s+1 − t s ), which send trs−1,s r ess,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 (trs−1,s r ess,s−1 )i = trs−i,s r ess,s−i for any i by using Theorem 1.2 and the two equalities that trr,s trn,r = trn,s (see Lemma 4.4) and that r esr,n r ess,r = r ess,n (see Lemma 3.3) for n ≤ r ≤ s Then the theorem is proved by combining this formula with Theorem 1.1 or with its consequence on H om A (Ds , Dr ) The vector space H om A (Ds , Dr ) is equipped with a bimodule structure: It is a right module over EndA (Ds ) and a left module over EndA (Dr ) By passing to the quotient, H om A (D s , Dr ) is also a bimodule: a right module over EndA (D s ) and a left module over EndA (Dr ) Set u i = trmin(r,s)−i,r r ess,min(r,s)−i for i ≥ Denote t = trs−1,s r ess,s−1 in EndA (Ds ) or in EndA (D s ) Since s is not part of the notation, t also means trr −1,r r esr,r −1 in EndA (Dr ) or in EndA (Dr ) Theorem 1.2 leads us to the following min(r,s) Proposition 1.4 The structures of the bimodules H om A (Ds , Dr ) ∼ = ⊕i=0 F p u i min(r,s)−1 and H om A (D s , Dr ) ∼ F p u i are given by = ⊕i=0 123 830 N H V Hu’ng (i) u i t = u i+1 , (ii) tu i = u i+1 , where u min(r,s)+1 = u min(r,s) in H om A (Ds , Dr ) and u min(r,s) = in H om A (D s , Dr ) The result on H om A (Ds , Dr ) adjusts the corresponding one announced in [5] Let us study the map θ : H om(U, V) → H om A (D(V), D(U)), ϕ → trϕ(U),Ur esV,ϕ(U) It is evident that, for ϕ, ψ ∈ H om(U, V), θ (ϕ) = θ (ψ) if and only if I mϕ ∼ = I mψ, or equivalently K er ϕ ∼ = K er ψ We write ϕ ∼ ψ to say that this condition is valid It is easy to see that (H om(U, V)/ ∼) ∼ = G L(V)\H om(U, V)/G L(U) Theorem 1.1 can be re-expressed in the following formulation: The map θ induces two isomorphisms of vector spaces ∼ = F p [G L(V)\H om(U, V)/G L(U)] −→ H om A (D(V), D(U)), ∼ = F p [G L(V)\H om(U, V)/G L(U)] F p −→ H om A (D(V), D(U)) In order to get the second isomorphism from the first one, we observe that θ (F p 0) 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 G L(V), there is an isomorphism of F p -vector spaces F p [H \H om(U, V)/G] ∼ = H om 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 = G L(U), H = G L(V) by Theorem 1.1 Suppose that the conjecture is true Then we are interested in the following problem: Describe the product in F p [H \End(V)/H ] that gives rise to an isomorphism of algebras F p [H \End(V)/H ] ∼ = EndA (H ∗ (V) H ) Even in the case of H = G L(V), the product on the left-hand side is not derived from the composition in End(V) Indeed, the algebra EndA (D(V)) is commutative by Theorem 1.3, while rank(ϕψ) = rank(ψϕ) in general, for ϕ, ψ ∈ End(V) (Actually, the composition in End(V) does not induce canonically a product in G L(V)\End(V)/G L(V).) Let (ε1 , , εs ) be a basis of V Each equivalence class in G L(V)\End(V)/G L(V) contains exactly one endomorphism of the form ξk : V → V for ≤ k ≤ s that sends 123 The homomorphisms between the Dickson–Mùi algebras 831 εi to εi−k for i > k and to for i ≤ k The composition in End(V) satisfies ξk ξ = ξk+ , k + ≤ s, 0, k + > s Based on Theorem 1.3 we get an isomorphism of algebras F p [G L(V)\End(V)/G L(V)] ∼ = EndA (D(V)), where the multiplication on the left-hand side is defined by the above product of the representatives ξk ’s (0 ≤ k ≤ s) 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 s−1 f = λid D s + λn trn,s r ess,n (λn ∈ F p ), n=1 where λ is a non-zero scalar In particular, there are exactly ( p − 1) p s−1 automorphisms 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 shows the 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 Theorem 1.1 is the following theorem Theorem 1.9 Let f : D s → Dr be a homomorphism of A-algebras Then f = λr ess,r , 0, r ≤ s, r > s, where λ is either or For p an odd prime, let S ∗ (V) be the symmetric algebra on βV∗ , a copy of the dual space V∗ of V This algebra is graded by assigning degree to each element in βV∗ The Dickson algebra, denoted D(V) or Ds with s = dim V, is the algebra S ∗ (V)G L(V) of all invariants of S ∗ (V) under the action of G L(V) The reduced Dickson algebra D(V) or Ds is, by definition, the augmentation ideal of all positive degree elements in the Dickson algebra D(V) = Ds Theorem 1.1 is also valid for the odd characteristic Dickson algebras in the same way as it is for the Dickson–Mùi algebras The following is a consequence of this theorem 123 832 N H V Hu’ng Corollary 1.10 In an odd characteristic, any A-module homomorphism between the reduced Dickson algebras ϕ : Ds → Dr can uniquely be extended to an A-module homomorphism between the reduced Dickson–Mùi algebras ϕˆ : D s → Dr The map ϕ → ϕˆ gives rise to an isomorphism H om A (Ds , Dr ) ∼ = H om A (D s , Dr ) The odd characteristic reduced Dickson algebras are also atomic in the meaning of Theorem 10.3, which is similar to Theorem 1.7, and therefore indecomposable as modules over the Steenrod algebra The paper is divided into 10 sections and organized as follows The introduction in Sect is followed by the preliminary in Sect 2, where we collect some needed results on the cohomology of the elementary abelian p-groups and on the Dickson–Mùi algebra Sections 3–8 deal with the characteristic p = In Sects and we define respectively the restriction and the transfer on the Dickson algebras and give a beginning study of their basic behaviors Section deals with the relationship between the transfer and the restriction on the Dickson algebras We study the A-module homomorphisms between the (reduced) Dickson algebras in Sect Section is devoted to the study of the A-module endomorphisms and the A-module automorphisms of the (reduced) Dickson algebras This section also investigates the bimodule structures of H om A (Ds , Dr ) and H om A (D s , Dr ) The A-algebra homomorphisms between the (reduced) Dickson algebras are investigated in Sect We express in Sect the changes needed for the Dickson–Mùi algebra, the case of odd prime characteristic In Sect 10 we show that the results that are similar to the ones for the Dickson–Mùi algebras also hold for the Dickson algebras in characteristic p > It should be mentioned that there is some overlap between the present paper and Kechagias’ manuscript [8], where the dimension of the endomorphism ring and the indecomposability of the reduced Dickson–Mùi algebras are probably shown Most of the paper’s contents, except Sect 10, have been announced in [5] 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 p For any non-negative integers (r1 , , rs ) we set [r1 , , rs ] = det(xi ticular, we define rj ) In par- ˆ , s], (for ≤ i ≤ s), L s,i = [0, , i, L s = L s,s = [0, 1, , s − 1] Here and in what follows, by writing iˆ we mean i is deleted The Dickson invariants are defined as follows Q s,i = L s,i /L s , 123 The homomorphisms between the Dickson–Mùi algebras 833 p−1 for ≤ i < s (Particularly, we observe that Q s,0 = L s expressed by the formula p p−1 Q s,i = Q s−1,i−1 + Q s−1,i Vs ) They can inductively be , where Q s,s = 1, Q s,i = for i < 0, and Vs = (c1 x1 + · · · + cs−1 xs−1 + xs ) c1 , ,cs−1 ∈F p For ≤ k ≤ s, the following element is first defined in E Z (e1 , ., es )⊗Z[x1 , ., xs ] and then projected to E(e1 , , es ) ⊗ F p [x1 , , xs ]: [k; rk+1 , , rs ] = e1 · e1 prk+1 k! x1 · prs x1 · · · es ··· · · · · es prk+1 , · · · xs ··· · prs · · · xs in which there are exactly k rows of (e1 · · · es ) (See the accurate meaning of the determinants in a commutative graded algebra in Mùi [10].) The Mùi invariants are defined by Ms,i1 , ,ik = [k; 0, , iˆ1 , , iˆk , , s − 1], p−2 Rs,i1 , ,ik = Ms,i1 , ,ik L s , for ≤ i < · · · < i k ≤ s − It should be noted that Q s,i and Rs,i1 , ,ik are invariant under the general linear group G L(V), while Ms,i1 , ,ik is invariant under the Sylow subgroup of G L(V) consisting of all upper triangular matrices with on the main diagonal The Dickson algebra, which originally appeared in Invariant Theory in the early 20th century, plays a key role in Algebraic Topology nowadays Theorem 2.1 (Dickson [3]) F p [x1 , , xs ]G L(V) = F p [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 123 834 N H V Hu’ng Theorem 2.2 (Mùi [10]) H ∗ (V)G L(V) = F p [Q s,0 , , Q s,s−1 ] ⊕ s k=1 ⊕ 0≤i s, then for any n ≤ min{r, s} = s we get either n < s < r or n = s < r In the both cases, as shown above, we have s f (Q s,i ) = λn trn,r r ess,n (Q s,i ) = n=1 for ≤ i < s, and s f (Rs,i1 , ,ik ) = trn,r r ess,n (Rs,i1 , ,ik ) = 0, n=1 for ≤ i < · · · < i k ≤ s − Since f is a homomorphism of A-algebras, and the Dickson–Mùi algebra Ds is algebraically generated by Q s,0 , , Q s,s−1 , Rs,i1 , ,ik with ≤ i < · · · < i k ≤ s − 1, this implies f = Case If r ≤ s, then the above equality shows that r −1 f (Q s,i ) = λr r ess,r (Q s,i ) + λn trn,r r ess,n (Q s,i ) n=1 = λr r ess,r (Q s,i ), for ≤ i < s Additionally, by Lemma 9.1, we get r −1 f (Rs,i1 , ,ik ) = λr r ess,r (Rs,i1 , ,ik ) + λn trn,r r ess,n (Rs,i1 , ,ik ) n=1 = λr r ess,r (Rs,i1 , ,ik ), for ≤ i < · · · < i k ≤ s − Hence f = λr ess,r with λ = λr Apply λr ess,r to Q 2s,s−1 in order to see that it is an A-algebra homomorphism if and only if λ2 = λ, or equivalently λ is equal to either or The theorem is proved 123 864 N H V Hu’ng 10 The A-homomorphisms between the odd characteristic Dickson algebras The purpose of this section is to prove that the results that are similar to the ones for the Dickson–Mùi algebras also hold for the odd characteristic Dickson algebras We always assume that p is an odd prime in this section For an F p -vector space V, let βV∗ be a copy of the dual space V∗ of V The spaces ∗ V and βV∗ are graded by assigning their elements to degrees and respectively Let S ∗ (V) be the graded symmetric algebra on βV∗ For a linear map ϕ : U → V between the two vector spaces, let S ∗ (ϕ) : S ∗ (V) → S ∗ (U) be the homomorphism induced by ϕ We also denote S ∗ (ϕ) by ϕ ∗ for abbreviation It is classically known that S ∗ is a contravariant functor from the category of F p -vector spaces to the category of F p -algebras The general linear group G L(V) acts on V and therefore on S ∗ (V) The Dickson algebra, denoted D(V) or Ds with s = dim V, is the algebra S ∗ (V)G L(V) of all invariants of S ∗ (V) under the action of G L(V) The space S ∗ (V) can be regarded as an A-subalgebra and a G L(V)-subalgebra of ∗ H (V) ∼ = E ∗ (V)⊗S ∗ (V), where E ∗ (V) denotes the graded exterior algebra on V∗ and ∗ β : V → βV∗ the Bockstein isomorphism The two actions of A and G L(V) upon S ∗ (V) commute with each other So the Dickson algebra inherits a structure of module over the Steenrod algebra from S ∗ (V) Obviously, the Dickson algebra D(V) = S ∗ (V)G L(V) is an A-subalgebra of the Dickson–Mùi algebra D(V) = H ∗ (V)G L(V) The reduced Dickson algebra D(V) or Ds is, by definition, the augmentation ideal of all positive degree elements in the Dickson algebra D(V) = Ds Restriction and transfer on the Dickson algebras in characteristic p > can respectively be defined by Definition 3.2 and Definition 4.1, in which the cohomology functor H ∗ is replaced by the symmetric algebra functor S ∗ They commute with the restriction and the transfer on the Dickson–Mùi algebras via the canonical inclusions of the Dickson algebras into the Dickson–Mùi algebras The following is similar to Theorem 1.1 Theorem 10.1 The A-module homomorphisms trn,r r ess,n ≤ n ≤ min{r, s} form a basis of the vector space H om A (Ds , Dr ) of all A-module homomorphisms from Ds to Dr In particular, dimF p H om A (Ds , Dr ) = min{r, s} The proof of this theorem is similar to that of Theorem 1.1 given in Sect for p an odd prime The main change is that the cohomology functor is replaced by the symmetric algebra functor An another change is the way of using the injectivity of the cohomology of a vector space From a theorem by Miller [9], H˜ ∗ (U) is injective in the category of unstable reduced A-modules Hence, each A-module homomorphism f : D(V) → D(U) ⊂ H˜ ∗ (U) can be extended to an A-module homomorphism fˆ : H˜ ∗ (V) → H˜ ∗ (U) We leave the detailed changes in the proof of the theorem to the reader The following is also numbered as Corollary 1.10 in the introduction 123 The homomorphisms between the Dickson–Mùi algebras 865 Corollary 10.2 Any A-module homomorphism between the reduced Dickson algebras ϕ : Ds → Dr can uniquely be extended to an A-module homomorphism between the reduced Dickson–Mùi algebras ϕˆ : D s → Dr The map ϕ → ϕˆ gives rise to an isomorphism H om A (Ds , Dr ) ∼ = H om A (D s , Dr ) Proof Let ϕ = n λn trn,r r ess,n be the linear expression of ϕ : Ds → Dr in terms of the basis given in Theorem 10.1 Then an extension of it to an A-module homomorphism between the reduced Dickson–Mùi algebras is the homomorphism of the same formulation ϕˆ = n λn trn,r r ess,n Suppose ϕ = n λn trn,r r ess,n : D s → Dr is also an extension of ϕ to an A-module homomorphism between the reduced Dickson–Mùi algebras Then, the restrictions of ϕˆ and ϕ on Ds are the same, namely ϕ So we have ϕ= λn trn,r r ess,n = n λn trn,r r ess,n n The linear independence of the basis given by Theorem 10.1 shows that λn = λn for ≤ n ≤ min{r, s}, and therefore ϕˆ = ϕ The corollary is proved Based on this corollary, we easily state the results for the odd characteristic Dickson algebras that are similar to most of the ones for the Dickson–Mùi algebras given in the introduction However, note that the least positive degree generator Q s,s−1 of the Dickson algebra Ds is not the least positive degree generator Rs,0,1, ,s−1 of the corresponding Dickson–Mùi algebra Ds So, Theorem 1.7 needs to be restated and reproved for the odd characteristic Dickson algebras as follows Theorem 10.3 If an A-module endomorphism f : Ds → Ds is non-zero on the least positive degree generator Q s,s−1 of the Dickson algebra, then it is an automorphism Proof The proof is similar to that of Theorem 7.2 for the p=2 Dickson algebra Since Q s,s−1 is the unique non-zero generator in the least positive degree p s−1 of Ds , and f is a degree-preserving endomorphism, we observe that f (Q s,s−1 ) = λQ s,s−1 for some λ ∈ F p By the hypothesis, f (Q s,s−1 ) = 0, it implies that λ = On the other hand, from Theorem B in [6], we have Pp s−2 (Q s,s−1 ) = Q s,s−2 , P p s−3 (Q s,s−2 ) = Q s,s−3 , , P (Q s,1 ) = Q s,0 This link between Q s,s−1 and Q s,0 by Steenrod operations shows that the equality f (Q s,s−1 ) = λQ s,s−1 implies f (Q s,i ) = λQ s,i for ≤ i < s By Theorem 10.1, the endomorphism f can be written in the form s s−1 λn trn,s r ess,n = λs idDs + f = n=1 λn trn,s r ess,n , n=1 123 866 N H V Hu’ng where λ1 , , λs ∈ F p Applying the both side of this equality to Q s,0 we obtain s−1 ( f − λs idDs )(Q s,0 ) = (λ − λs )Q s,0 = λn trn,s r ess,n (Q s,0 ) = n=1 Here the last equality comes from the fact that r ess,n (Q s,0 ) = for any n < s, by Lemma 9.1 It implies that λ − λs = 0, or equivalently λs = λ = By the theorem for the odd characteristic Dickson algebras that is similar to Theorem 1.6, f = λidDs + s−1 n=1 λn trn,s r ess,n with λ = is an automorphism of Ds The theorem is proved An alternative proof of the theorem can be obtained by a simple combination of the results for the odd characteristic Dickson algebras that are similar to Corollary 9.5(i) for r = s and to Theorem 1.6 An immediate consequence of the above theorem is that the odd characteristic reduced Dickson algebra is indecomposable as a module over the Steenrod algebra The analogue of Theorem 1.9 also holds for the odd characteristic Dickson algebras Restricting the arguments of Theorem 1.9’s proof on the odd characteristic Dickson algebra as a part of the Dickson-Mùi algebra, we get a proof for the analogue Acknowledgments I would like to thank the referee for many helpful suggestions and comments particularly on Theorem 1.3(ii), Definition 4.1, Lemma 6.2 and on the proof of Theorem 7.1, which have led to improvement of the paper’s exposition I also express my warm thanks to Lê M Hà and Võ T N Qu`ynh for several fruitful discussions References Adams, J.F., Gunawardena, J.H., Miller, H.R.: The Segal conjecture for elementary abelian p-groups Topology 24, 435–460 (1985) Carlsson, G.: G B Segal’s burnside ring conjecture for (Z /2)k Topology 22, 83–103 (1983) Dickson, L.E.: A fundamental system of invariants of the general modular linear group with a solution of the form problem Trans Am Math Soc 12, 75–98 (1911) Hu’ng, N.H.V.: The action of the Steenrod squares on the modular invariants of linear groups Proc Am Math Soc 113, 1097–1104 (1991) Hu’ng, N.H.V.: The homomorphisms between the Dickson-Mùi algebras as modules over the Steenrod algebra C R Acad Sci Paris Serie I 348, 1001–1004 (2010) Hu’ng, N.H.V., Minh, P.A.: The action of the mod p Steenrod operations on the modular invariants of linear groups Vietnam J Math 23, 39–56 (1995) Hu’ng, N.H.V., Peterson, F.P.: Spherical classes and the Dickson algebra Math Proc Camb Phil Soc 124, 253–264 (1998) Kechagias, N.E.: A Steenrod-Milnor action ordering on Dickson invariants http://www.math.uoi.gr/ ~nondas_k Miller, H.R.: The Sullivan conjecture on maps from classifying spaces Ann Math (2) 120, 39– 87 (1984) 10 Mùi, H.: Modular invariant theory and cohomology algebras of symmetric groups J Fac Sci Univ Tokyo Sect IA Math 22, 319–369 (1975) 123 ... that the results that are similar to the ones for the Dickson–Mùi algebras also hold for the Dickson algebras in characteristic p > It should be mentioned that there is some overlap between the. .. The homomorphisms between the Dickson–Mùi algebras 847 The last equality comes from the fact that 2n−n −1 is not divisible by 2n−n So, applying the Steenrod operation to the both side of the. .. contradicts the preceding theorem The corollary is proved The A -algebra homomorphisms between the Dickson algebras The following theorem is also numbered as Theorem 1.9 in the introduction Theorem