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Induced maps in Galois cohomology Nguyen, Bich-Ngoc Cao 2011

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Induced Maps in Galois Cohomology by Bich-Ngoc Cao Nguyen B.Sc., University of Manitoba, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011 c© Bich-Ngoc Cao Nguyen 2011 Abstract Galois cohomology is an important tool in algebra that can be used to clas- sify isomorphism classes of algebraic objects over a field. In this thesis, we show that many objects of interest in algebra can be described in cohomo- logical terms. Some objects that we discuss include quadratic forms, Pfister forms, G-crossed product algebras, and tuples of central simple algebras. We also provide cohomological interpretation to some induced maps that naturally occur in short exact sequences. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . v Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A long exact sequence in group cohomology . . . . . . . . . . 3 2.3 An action of CG on H1(G,A) . . . . . . . . . . . . . . . . . . 4 2.4 An action of H1(G,A) on H1(G,B) . . . . . . . . . . . . . . 4 2.5 H1(k,G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.6 The Brauer group H2(k,Gm) . . . . . . . . . . . . . . . . . . 5 3 Quadratic forms and Pfister forms . . . . . . . . . . . . . . . 6 3.1 Pfister forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Scaled Pfister forms . . . . . . . . . . . . . . . . . . . . . . . 9 4 Galois cohomology of special orthogonal groups . . . . . . 10 5 Central simple algebras . . . . . . . . . . . . . . . . . . . . . . 13 5.1 Central simple algebras . . . . . . . . . . . . . . . . . . . . . 13 5.2 G-crossed product algebras . . . . . . . . . . . . . . . . . . . 15 iii 6 Trace forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.1 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 iv Acknowledgements I would like to thank my supervisor Zinovy Reichstein for introducing me to the topic, and for his invaluable suggestions to improve this thesis. I am very grateful for his time and patience in working with me throughout my program. I also thank the Faculty of Graduate Studies at the University of British Columbia for their financial support. Lastly, I thank Alexander Duncan, Jerome Lefebvre, and Shane Cernele for their useful comments and discussions. v Dedication I would like to dedicate this thesis to my beloved sister, who loves mathe- matics just as much as I do. vi Chapter 1 Introduction In order to facilitate the study of certain mathematical objects over a field, it is common to try and establish a systematic way of organizing them into families. An example of such is the complete classification of connected com- pact Lie groups over C by Cartan and Weil. However, over non-algebraically closed fields, complications arise, and a complete classification is more com- plicated to establish. Over C, a real circle and a real line with its origin removed are isomorphic, while over R, they are not. Thus, to classify ob- jects over such fields, one needs to rely on a theory called Galois cohomology. Galois cohomology is often used to classify isomorphism classes of alge- braic objects over a field. More precisely, given an algebraic structure A over a field k and G = Autk(A), then H 1(k,G) is in bijective correspondence with k-isomorphism classes of twisted forms of A, that is, classes of algebraic ob- jects A′ that become isomorphic to A over a separable closure of k (cf. [4]). For example, in [6], T.A. Springer had shown that H1(k,On(ksep)) classifies isomorphism classes of quadratic forms of dimension n over k. For further examples of algebraic objects that can be classified via Galois cohomology, refer to [3]. In this thesis, we show that many objects of interest in algebra can be naturally described by Galois cohomology sets or characterized as images of certain induced maps in Galois cohomology. An example of the classification is to show that elements of H1(k,GLn /µd) where d | n can be identified with central simple algebras over k of degree n and exponent dividing d (cf. Lemma 2.6 [1]). The structure of this thesis is as follows. In Chapter 2, we introduce the notations and some background material in group and Galois cohomology. In particular, we list some known examples of the first Galois cohomology 1 sets that can be described via isomorphism classes of algebraic objects. In chapters 3 and4, we will discuss some applications of Galois cohomology in quadratic form theory. In particular, we describe quadratic forms of dis- criminant 1, Pfister forms and scaled Pfister forms in cohomological terms. Chapter 5 is used to discuss some applications of Galois cohomology in the theory of central simple algebras. We give a cohomological description of l-tuples (A1, . . . , Al) of central simple k-algebras whose Brauer classes satisfy a given system of linear equations of the form n1[A1] + . . . + nr[Ar] = 0 in the Brauer group Br(k). We also describe G-crossed product algebras as the image of certain induced maps in Galois cohomology. Finally, in chapter 6, we give a cohomological interpretation of the trace form in alternative and Jordan algebras. 2 Chapter 2 Preliminaries 2.1 Notation We will use the following notation throughout the thesis: k - a base field of characteristic 0 containing a primitive 4-th root of unity ksep - a separable closure of k Γ - the absolute Galois group of k K - a finite Galois extension of k ΓK - the relative Galois group of K over k µp - the group of p-th roots of unity 〈a1, . . . , an〉 - the diagonal quadratic form (x1, . . . , xn) 7→ n∑ i=1 aix 2 i . 2.2 A long exact sequence in group cohomology Let Γ be a group. Suppose that A,B and C are Γ-groups where A is a central subgroup of B and they satisfy the following short exact sequence of Γ-groups: 0→ A→ B → C → 0 Then, we obtain the following long exact sequence of pointed sets: 0→ AΓ → BΓ → CΓ → H1(Γ, A)→ H1(Γ, B)→ H1(Γ, C)→ H2(Γ, A) 3 2.3 An action of CG on H1(G,A) CG has a right action on H1(G,A) given as follow. Let c ∈ C, lift c to b ∈ B, then the image of as ∈ H1(G,A) under c is the cocycle b−1assb. We have the following proposition: Proposition 2.3.1. (Proposition 39 p.52 [5]) Two elements of H1(G,A) have the same image in H1(G,B) if and only if they are in the same CG- orbit. 2.4 An action of H1(G,A) on H1(G,B) H1(G,A) has a left action on H1(G,B) given as follow. Let α ∈ H1(G,A) and β ∈ H1(G,B) , then α · β = ι?(α)β. Similar to above, in this context, we have the following proposition: Proposition 2.4.1. (Proposition 42 p.54 [5]) Two elements of H1(G,B) have the same image in H1(G,C) if and only if they are in the same orbit under the action of H1(G,A). 2.5 H1(k,G) Given a field k and an algebraic groupG defined over ksep, we define H 1(k,G) to be the group cohomology set H1(Γ, G). Here are some basic known examples of Galois cohomology: 1. H1(k,GLn(ksep)) = {1} (Hilbert’s Theorem 90). 2. H1(k, µd) ∼= k×/(k×)d, where given a ∈ k×/(k×)d, the corresponding cocycle γ : Γ→ µd is defined by γ(s) = b−1s(b) for all s ∈ Γ, and some b ∈ k×sep such that bd = a. 3. H1(k, Sn) ∼= {isomorphism classes of étale algebra of dimension n over k}, where an étale k-algebra L of dimension n is a direct sum of n finite separable field extensions of k. 4 4. H1(k,G) ∼= {isomorphism classes of G-Galois algebras of dimension n over k} for a finite group G, where a G-Galois k-algebra L is an étale algebra over k endowed with an action by G such that LG = k (cf. p.287-288 [3]). 5. H1(k,On(ksep)) ∼= {isomorphism classes of quadratic forms of dimension n over k}, where a quadratic form of dimension n is a homogeneous polynomial of degree 2 in n variables. 6. H1(k,PGLn(ksep)) ∼= {isomorphism classes of central simple algebras of degree n over k}, where an associative algebra A over k is a central simple algebra if Z(A) = k, and A contains no nontrivial proper two sided ideals. 7. Let G1, . . . , Gn be n algebraic groups over ksep. Then, H1(k, n∏ i=1 Gi) ∼= n∏ i=1 H1(k,Gi). 2.6 The Brauer group H2(k,Gm) One can identify elements of H2(k,Gm) with Brauer equivalence classes of central simple algebras over k. Recall that from the short exact sequence: 1→ Gm(ksep)→ GLn(ksep)→ PGLn(ksep)→ 0, the connecting map δ : H1(k,PGLn(ksep))→ H2(k,Gm(ksep)) maps a central simple algebra A to its equivalence class in the Brauer group. 5 Chapter 3 Quadratic forms and Pfister forms Consider the following embedding of algebraic groups: ι :µn2 −→ On(ksep) (ε1, . . . , εn) 7→ diag(ε1, . . . , εn) Recall that we may identify H1(k, µn2 ) with (k ×/(k×)2)n. We observe that: Lemma 3.0.1. The induced map ι? : H 1(k, µn2 ) → H1(k,On(ksep)) is pre- cisely the map which sends (a1, . . . , an) to 〈a1, . . . , an〉, where ai ∈ k×/(k×)2 for 1 ≤ i ≤ n. Proof. Let α : Γ→ µn2 be a cocycle in Z1(k, µn2 ). Since (ai)ni=1 ∈ (k×/(k×)2)n is an n-tuple representing α, one has that for all s ∈ Γ, α(s) = (b−11 s(b1), . . . , b −1 n s(bn)), for some b 2 i = ai and bi ∈ k×sep. Fix one such n-tuple (b1, . . . , bn) ∈ (k×sep)n where b2i = ai. Denote by K the compositum of Galois extensions k(bi) in ksep for 1 ≤ i ≤ n. Then, K is also a finite Galois extension over k. Let {ei}ni=1 be a basis of kn, and {e?i }ni=1 the corresponding dual basis. Fix a tensor x = n∑ i=1 e?i ⊗ e?i ∈ ((kn)?)⊗2, which represents the split quadratic form over kn. Since ι?(α) ∈ Z1(k,On(ksep)) ⊆ Z1(k,GLn(ksep)), it follows from H 1(k,GLn(ksep)) being trivial that ι?(α) is cohomologous to 0 in Z1(ΓK ,GLn(K)). Thus, there exists f ∈ GL(Kn) such that for all s ∈ ΓK , ι?(α)(s) = f−1s(f). Suppose that there exists h ∈ GL(Kn) such that f−1s(f) = h−1s(h). Then, hf−1 = s(hf−1) for 6 all s ∈ ΓK , and hence, hf−1 ∈ GL(kn). Therefore, hf−1(x) defines an isometric quadratic form to x on kn, which further implies that h(x) defines an isometric quadratic form to f(x) on Kn. So, it suffices to consider f being the matrix diag(b1, . . . , bn) ∈ GL(Kn). Then, f satisfies the equation ι?(α)(s) = f −1s(f) trivially. Thus, f(x) = n∑ i=1 bie ? i ⊗ bie?i = n∑ i=1 aie ? i ⊗ e?i . So, ι?(α) corresponds to the quadratic space (K n, f(x)), where f(x) is pre- cisely the quadratic form represented by 〈a1, . . . , an〉. Corollary 3.0.2. The map ι? : H 1(k, µn2 ) → H1(k,On(ksep)) defined in Lemma 3.0.1 is surjective. Proof. Let q be a quadratic form corresponding to a cohomology class α in H1(k,On(ksep)). Since every quadratic form is diagonalizable, there exists (a1, . . . , an) ∈ kn such that q is equivalent to a diagonal quadratic form q′ = diag(a1, . . . , an). Then, q and q′ are isometric, which means that q′ also represents α. Thus, if pi : k× → k×/(k×)2 is the projection map, then (pi(ai)) n i=1 ∈ (k×/(k×)2)n ∼= (H1(k, µ2))n is such that ι?((pi(ai))ni=1) = α. Lemma 3.0.3. Let φ : µm2 → µn2 be the map defined by φ((εj)mj=1) = ( m∏ j=1 ε ei,j j ) n i=1. Then, the induced map φ? : H 1(k, µm2 )→ H1(k, µn2 ) is precisely the map φ?((aj)mj=1) = ( m∏ j=1 a ei,j j ) n i=1, where ai ∈ k×/(k×)2 for 1 ≤ i ≤ m. Proof. Let α : Γ→ µm2 be a cocycle in Z1(k, µm2 ). Since (aj)mj=1 ∈ (k×/(k×)2)m corresponds to the cocycle α, one has that for all s ∈ Γ, α(s) = (b−1j s(bj)) m j=1, for some b 2 j = aj and bj ∈ k×sep. Hence, the resulting cocycle φ?(α) : Γ→ µn2 is the map φ?(α)(s) = ( m∏ j=1 (b−1j s(bj)) ei,j )ni=1 = ( m∏ j=1 ((b ei,j j ) −1s(bei,jj )) n i=1 for all s ∈ Γ. 7 Thus, an n-tuple of elements in k×/(k×)2 which corresponds to φ?(α) is ( m∏ j=1 (b ei,j j ) 2)ni=1 = ( m∏ j=1 a ei,j j ) n i=1, as desired. Corollary 3.0.4. Let {χi}2ni=1 be the set of all distinct maps of the form χi : µ n 2 → µ2 such that χi((εj)nj=1) = n∏ j=1 ε di,j j for di,j ∈ {0, 1}. Consider an embedding map λ : µn2 → O2n(ksep) via λ((εj)nj=1) = diag(χ1((εj)nj=1), . . . , χ2n((εj)nj=1). Then, the induced map λ? : H 1(k, µn2 )→ H1(k,O2n(ksep)) is defined by: λ?((aj) n j=1) = 〈χ1((aj)nj=1), . . . , χ2n((aj)nj=1〉, where ai ∈ k×/(k×)2 for 1 ≤ i ≤ n. Proof. Observe that λ = ι ◦ φ, where ι : µ2n2 → On(ksep) is the embedding defined in Lemma 3.0.1, and φ : µn2 → µ2 n 2 defined in Lemma 3.0.3 with ei,j = di,j . Hence, it follows from the aforementioned Lemmas that λ? = ι? ◦ φ? is the desired map. 3.1 Pfister forms A 2n-dimensional quadratic form η is an n-fold Pfister form if η = n⊗ i=1 〈1, ai〉. Using above results, one arrives at the following characterization: Corollary 3.1.1. A 2n-dimensional quadratic form η is an n-fold Pfister form if and only if η ∈ λ?(H1(k, µn2 )), where λ? is the map defined in Corol- lary 3.0.4. Proof. The statement follows from the observation that an n-fold Pfister form  a1, . . . , an  can be expressed as:  a1, . . . , an = n⊗ i=1 〈1, ai〉 = 〈χ1((aj)nj=1), . . . , χ2n((aj)nj=1〉, 8 where {χi}2ni=1 are precisely the maps defined in Corollary 3.0.4. 3.2 Scaled Pfister forms A 2n-dimensional quadratic form γ is an n-fold scaled Pfister form if γ = a0η for some n-fold Pfister form η. Similar to above, one obtains the following: Corollary 3.2.1. Let ψ : µn+12 → On(ksep) be the map defined by: ψ((εj) n j=0) = ε0 diag(χ1((εj) n j=1), . . . , χ2n((εj) n j=1)), where {χi}2ni=1 are the maps defined in Corollary 3.0.4. Then, a 2n-dimensional quadratic form η is an n-fold scaled Pfister form if and only if η ∈ ψ?(H1(k, µn+12 )). Proof. Observe that ψ = ι ◦ φ, where ι : µ2n2 → On(ksep) is the embed- ding defined in Lemma 3.0.1, and φ : µn+12 → µ2 n 2 defined in Lemma 3.0.3 with ei,j = di,j for 1 ≥ j, and ei,0 = 1 for all i. Thus, by Lemma 3.0.1 and Lemma 3.0.3, ψ? = ι? ◦ φ? is the map such that given (aj)nj=0 ∈ (k×/(k×)2)n+1, one has that ψ?((aj) n j=0) = 〈a0χ1((aj)nj=1), . . . , a0χ2n((aj)nj=1〉, which is precisely the scaled Pfister form 〈a0〉 ⊗ n⊗ i=1 〈1, ai〉. 9 Chapter 4 Galois cohomology of special orthogonal groups In order to study H1(k,SOn(ksep)), we will rely on the long exact sequence in cohomology to relate H1(k,SOn(ksep)) to H 1(k,On(ksep)), since the latter is better understood. Then, by giving an interpretation to the induced maps between them, we will give a cohomological intepretation to this Galois cohomology set. Recall the following discriminant map: disc : H1(k,On(ksep))→ ksep 〈a1, . . . , an〉 7→ n∏ j=1 aj We observe that: Lemma 4.0.2. Let det : On(ksep) → µ2 be the determinant map. Then, det? : H 1(k,On(ksep))→ H1(k, µ2) is the discriminant map. Proof. Denote by ξ the inclusion map of SOn(ksep) into On(ksep), and ι : µ n 2 → On(ksep) the map defined in Lemma 3.0.1. Furthermore, let φ : µn−12 → µn2 be the em- bedding such that φ((εi) n−1 i=1 ) = (εi) n i=1, where εn = n−1∏ i=1 εi, and λ : µ n 2 → µ2 the map which maps (εi) n i=1 to n∏ i=1 εi. Also, since Im(ιφ) ⊆ SOn(ksep), de- note by τ : µn−12 → SOn(ksep) the composition of φ and ι. Then, observe that one has the following commutative diagram where the two horizontal 10 sequences are exact: 0 // µn−12 φ // τ  µn2 λ // ι  µ2 // ∼=  0 0 // SOn(ksep) ξ // On(ksep) det // µ2 // 0 So, one obtains the following commutative diagram of long exact se- quences in cohomology: Z/2Z δ ′ // ∼=  (k×/(k×)2)n−1 φ? // τ?  (k×/(k×)2)n λ? // ι?  k×/(k×)2 ∼=  Z/2Z δ // H1(k,SOn) ξ? // H1(k,On) det? // k×/(k×)2 (4.1) Given a cohomology class α ∈ H1(k, SOn(ksep)), one may represent ζ?(α) by a diagonal quadratic form 〈a1, . . . , an〉 for some aj ∈ k×/(k×)2. It follows from the definition of ι? that (aj) n j=1 is an n-tuple in (k ×/(k×)2)n such that ι?((aj) n j=1) = α. By Lemma 3.0.3, λ?((aj) n j=1) = n∏ j=1 aj . Thus, det ?(α) = det ?ι?((aj) n j=1) = λ?((aj) n j=1) = n∏ j=1 aj , which is precisely the discriminant map. Lemma 4.0.3. The map ξ? : H 1(k, SOn(ksep)) → H1(k,On(ksep)) in Dia- gram 4.1 is injective. Proof. Suppose that ξ? is not injective. So, there exist two distinct cohomol- ogy classes α and β in H1(k, SOn(ksep)) such that ξ?(α) = ξ?(β). By Propo- sition 2.3.1, α and β belong to the same Z/2Z-orbit, that is, s(α) = β, where s denotes the non-trivial element of Z/2Z. Since ι? is surjective, there exists an element x ∈ (k×/(k×)2)n such that ι?(x) = ξ?(α). From diagram 4.1, 11 one has that λ?(x) = det ?(ι?(x)) = det ?(ξ?(α)) = 0. So, it follows from the sequence being exact that there exists an element x0 ∈ (k×/(k×)2)n−1 such that φ?(x0) = x. Observe that ξ?(τ?(x0)) = ι?(φ?(x0)) = ι?(x) = ξ?(α) = ξ?(β). Thus, by Proposition 2.3.1, τ?(x0) belongs to the same Z/2Z-orbit as α and β. Since a Z/2Z-orbit can only have at most two distinct elements, without loss of generality, one may assume that τ?(x0) = α. By Lemma 3.0.3, φ? is a group homomorphism such that φ((ai) n−1 i=1 ) = (ai) n i=1, where an = n−1∏ i=1 ai. It is clear from its definition that φ? is injective. Hence, by Proposition 2.3.1, the action of Z/2Z on (k×/(k×)2)n−1 is trivial. So, since τ? respects the actions of Z/2Z on (k×/(k×)2)n−1 and H1(k, SOn(ksep)) , one has that α = τ?(x0) = τ?(s(x0)) = s(τ?(x0)) = s(α) = β, which contradicts the assumption that α and β are distinct. Thus, one concludes that ξ? is an injective map, as desired. By applying the above two lemmas, one obtains the following result: Theorem 4.0.4. H1(k,SOn(ksep)) is naturally in bijection with the set of classes of quadratic k-forms with discriminant 1. Proof. By Lemma 4.0.3, since ξ? is injective, one may identify H 1(k,SOn(ksep)) with the kernel of map det?, which is the discriminant map by Lemma 4.0.2. Thus, the statement of the theorem follows. 12 Chapter 5 Central simple algebras 5.1 Central simple algebras Consider a subgroup C ⊆ Glm. Via the embedding of each element of Glm as an l-tuple of scalar matrices, one obtains an embedding of C in l∏ i=1 GLni . Let G = ( l∏ i=1 GLni)/C. Lemma 5.1.1. The induced map pi? : H 1(k,G) → H1(k, l∏ i=1 PGLni) is in- jective. Proof. Consider the short exact sequence: 1→ Glm/C → G→ l∏ i=1 PGLni → 1, which induces the following exact sequence in cohomology: . . .→ H1(k,Glm/C)→ H1(k,G)→ H1(k, l∏ i=1 PGLni) Since Glm/C ∼= Glm, we have that H1(k,Glm/C) ∼= H1(k,Glm) = {1}. So, the action of H1(k,Glm/C) is trivial on H1(k,G). Hence, it follows from Proposition 2.4.1 that pi? : H 1(k,G)→ H1(k, l∏ i=1 PGLni) is injective. Let (χ1, . . . , χl) be a set of generators for the group of characters of Glm which vanish on C. Consider the map piC : Glm → Glm/C where piC(g) = (χ1(g), . . . , χl(g)). 13 Lemma 5.1.2. (γ1, . . . , γl) is in the kernel of pi C ? : H 2(k,Glm)→ H2(k,Glm/C) if and only if for every character χ = (a1, . . . , al) ∈ Zl of Glm that vanishes on C, γ⊗a11 ⊗ . . .⊗ γ⊗all = 0 in H2(k,Gm). Proof. Let γ = (γ1, . . . , γl) ∈ ker(piC? ). We have the following short exact sequence of abelian groups: 1→ C → Glm → Glm/C → 1, which induces the following exact sequence in cohomology: . . .→ H2(k,C)→ H2(k,Glm)→ H2(k,Glm/C)→ . . . So, ker(piC? ) = Im(ι?). Then, since ι is the inclusion map, as a cocycle, γ ∈ Z2(k,C). So, χi◦γ = 0 for all i. Let χ be a character of Glm that vanishes on C, then χ = l∑ i=1 di(χi). Clearly, χ ◦ γ = 0. Hence, χ? : H2(k,Glm) → H2(k,Gm) maps γ to the trivial cocycle in H2(k,Gm). On the other hand, χ(γ) = l⊗ i=1 γ⊗aii . Therefore, γ ⊗a1 1 ⊗ . . . ⊗ γ⊗all = 0 in H2(k,Gm) for every χ = (a1, . . . , al) that vanishes on C. Conversely, let γ ∈ H2(k,Glm) such that γ⊗a11 ⊗ . . . ⊗ γ⊗all = 0 in H2(k,Gm) for every χ = (a1, . . . , al) that vanishes on C. Then, Im(γ) ⊆ C, since otherwise, there exists a character χ such that χ ◦ γ 6= 0. Hence, γ ∈ Im(ι?) = ker(piC? ), as desired. Using the above lemmas, we arrive at the following theorem: Theorem 5.1.3. H1(k,G) are in one-to-one correspondence with k-isomorphism classes of l-tuples (A1, . . . , Al) of central simple algebras over k, such that each Ai is of degree ni, and for every character χ = (a1, . . . , al) ∈ Zl of Glm that vanishes on C, A⊗a11 ⊗ . . .⊗A⊗all = 0 in H2(k,Gm). Proof. Consider the following commutative diagram where the two rows are 14 exact: 1 // Glm ι // piC  l∏ i=1 GLni pi // pi  l∏ i=1 PGLni // ∼=  1 1 // Glm/C ι // G pi // l∏ i=1 PGLni // 1 which induces the following commutative diagram in Galois cohomology: 1 = H1(k, l∏ i=1 GLni) //  H1(k, l∏ i=1 PGLni) δ // ∼=  H2(k,Glm) piC?  H1(k,G) pi? // H1(k, l∏ i=1 PGLni) δC // H2(k,Glm/C) By Lemma 5.1.1, the map pi? is injective. Thus, we can identify each element in H1(k,G) with its image in H1(k, l∏ i=1 PGLni) ∼= l∏ i=1 H1(k,PGLni), which is a k-isomorphism class of l-tuples (A1, . . . , Al) where each Ai is a central simple algebra of degree ni. Since the bottom row is exact and the diagram is commutative, Im(pi?) = ker(δC) = ker(pi C ? ◦ δ). Moreover, it follows from the top row being exact that ker δ = {1}, and hence, δ(Im(pi?)) = ker(piC? ). Observe that δ maps an l-tuple of central simple algebras (A1, . . . , Al) to the l-tuple ([A1], . . . , [Al]) of its class in the Brauer group. By Lemma 5.1.2, ([A1], . . . , [Al]) ∈ ker(piC? ) if and only if for every character χ = (a1, . . . , al) ∈ Zl of Glm that vanishes on C, [A ⊗a1 1 ⊗ . . .⊗A⊗all ] = 0. Thus, the statement of the proposition follows. 5.2 G-crossed product algebras A G-crossed product algebra A over k is a central simple k-algebra that contains a maximal G-Galois sub-algebra L. Let G be a finite group of order n. Denote by T ∼= Gm/∆ the split max- 15 imal torus consisting of diagonal matrices up to scalar multiples in PGLn. Since Sn acting on T by permutation, the embedding of G into Sn induces an embedding of T o G into T o Sn, which is a subgroup of PGLn. Thus, we have an embedding ι of T oG into PGLn. Proposition 5.2.1. H1(k, ToG) classifies isomorphism classes of G-crossed product algebras over k. Proof. Consider pairs of the form (A,L), where A is a central simple al- gebra of degree n over k, L is a faithful left G-module (where G acts by permutation), and L is also an n-dimensional commutative semisimple sub- algebra of A. A morphism f of two such pairs (A1, L1) and (A2, L2) is a k-algebra homomorphism f : A1 → A2, such that f(L1) ⊆ L2 and f |L1 is G-equivariant. Let the split object of these pairs be (A0, L0) where A0 = Mn(k) and L0 = ⊕ g∈G kg, embedded into A0 as the diagonal matrices. We first show that Aut((A0 ⊗ ksep, L0 ⊗ ksep)) ∼= T o G. Clearly, T o G ⊆ Aut((A0⊗ksep, L0⊗ksep)) since every elements of T oG is invertible, and G acts on T by permutation. Conversely, let f ∈ Aut((A0 ⊗ ksep, L0 ⊗ ksep)). Then, f |L0⊗ksep is a G-equivariant k-algebra automorphism. Observe that if f(1e⊗ 1) = ∑ g∈G 1g ⊗ ag, then f being G-equivariant implies that for any g′ ∈ G, f(1g′ ⊗ 1) = g′ · f(1e⊗ 1) = g′ · ∑ g∈G 1g ⊗ ag  = ∑ g∈G 1(g′g)⊗ ag. So, f is determined uniquely by f(1e⊗1). Thus, since L0⊗ksep is embedded in A) ⊗ ksep ∼= Mn(ksep) as the diagonal matrices, f is represented by an element of T oG. Since elements of T oG are invertible, f can be extended uniquely to an automorphism of A0 ⊗ ksep by the Skolem-Noether theorem. Thus, Aut((A0 ⊗ ksep, L0 ⊗ ksep) ∼= T oG. By Galois descent, H1(k, T oG) classifies isomorphism classes of twisted forms of (A0, L0) over k. Observe that A0 is also the split object for the central simple algebras over k, and as a subalgebra of A0, elements of L0 are 16 invertible. So, L0 ∼= kn, which is the split G-Galois algebra. Hence, twisted forms of (A0, L0) are precisely the G-crossed product algebras over k. Corollary 5.2.2. Let A be a G-cross product algebra over k, and α ∈ Z1(k,PGLn) that corresponds to A. Then, α ∈ ι?(H1(k, T oG)). Proof. The proposition follows from the previous lemma and the observation that since ι? is induced by the inclusion map, it maps the class of a pair (A,L) to the class of A in H1(k,PGLn(ksep)). 17 Chapter 6 Trace forms 6.1 Trace Recall that a k-algebra A is strictly power-associative if for all x ∈ A and a, b ∈ Z, xaxb = xa+b. Let A be a strictly power-associative unital k-algebra of dimension n, and (ei) n i=1 a basis of A over k. Consider a generic element x = ∑ xiei ∈ A⊗ k(x1, . . . , xn). There exists a monic polynomial PA,x(t) = t m − s1(x)tm−1 + . . .+ (−1)msm(x), of least degree such that PA,x(x) = 0, and each si is a homogeneous poly- nomial in terms of x1, . . . , xn with coefficients in k. Then, s1(x) = TA is the generic trace of the algebra A over k. Define the following bilinear trace map: TrA/k : A×A→ k (x, y) 7→ TA(xy). If A is an alternative algebra or a Jordan algebra, then TrA/k is an associa- tive, symmetric bilinear form (cf. Corollary 4 p.227 [2]). In particular, TrA/k is non-singular if and only if A is separable (cf. Lemma 32.4 [3]). Moreover, if σ ∈ Aut(A), then σ(TA(x)) = TA(σ(x)), that is, the trace form TrA/k is invariant under the k-automorphism group of A (cf. Theorem 1 p.224 [2]). Hence, there exists an embedding ι : Autk(A)→ On(ksep,TrA/k). Theorem 6.1.1. Let A be an alternative or Jordan n-dimensional k-algebra. Then, the induced map ι? : H 1(k,Autk(A)) → H1(k,On(ksep,TrA/k)) maps 18 a twisted form A′ of A to its trace form TrA′/k. Proof. Let A′ be a twisted form of A in H1(k,Autk(A)), and K a finite Galois extension of k such that A′ ∈ H1(K/k,Autk(A)). Let V = kn, and x ∈ V ?⊗V ?⊗V the structure constant of A. Then, (V, x) corresponds to the trivial cocycle in H1(K/k,Autk(A)). Suppose that (V, x ′) is the associated element to A′ in E(K/k, (V, x)). So, there exists a K-isomorphism f : VK → VK such that f(xK) = x′K , and the cocycle responding to A′ is γ : ΓK → Autk(A) such that γ(s) = f−1s(f) for s ∈ ΓK . Consequently, ι?(γ) = ι◦γ : ΓK → On(ksep,TrA/k). With V defined as above, and y ∈ V ?⊗ V ? representing the trace form TrA/k, one obtains that the quadratic space corresponding to ι?(γ) is (V, f(yK)). Denote by y ′ the tensor corresponding to TrA′/k. It remains to be shown that (VK , yK) ∼=K (VK , y′K) via f and (V, f(y)) ∼=k (V, y′) via the identity map. Clearly, f is a K-isomorphism of VK , and for any a, b ∈ (VK ,TrA′,k⊗K), one has that y′K(a, b) = TA′(x ′ K(a, b))) = TA′(f(xK)(a, b)) = TA(xK(f −1(a), f−1(b))) = yK(f−1(a), f−1(b)). Hence, (VK , yK) ∼=K (VK , y′K) via f . Since f(xK) = x′K , f is a k-algebra homomorphism. So, f(yK) = y ′ K implies that f(y) = y ′. Thus, (V, f(y)) ∼=k (V, y′) via the identity map. Thus, the associated quadratic form to ι?(γ) is TrA′,k, as desired. Remark 6.1.2. We can apply the above theorem to the following two im- portant classes of algebras: étale algebras and G-Galois algebras. Recall that étale algebras of dimension n and G-Galois algebras over k are clas- sified by H1(k, Sn(ksep)) and H 1(k,G) for a finite group G, respectively. So, let ι : Sn(ksep) → On(ksep) be the embedding of permutation matri- ces into the orthogonal group, and ρ : G → Sn the Cayley representation map. Then, ι? : H 1(k, Sn(ksep)) → H1(k,On(ksep)) is the map which sends an étale k-algebra L to its trace form TrL/k. Similarly, the induced map (ι ◦ ρ)? : H1(k,G)→ H1(k,On(ksep)) is the map sending a G-Galois algebra L of dimension n to its trace form TrL/k. 19 Chapter 7 Conclusion In this thesis, we show that many objects of interest in algebra can be nat- urally described by Galois cohomology sets or characterized as images of certain induced maps. This is an attempt to establish a connection between Galois cohomology and abstract algebra, allowing us to view common al- gebraic objects from a different perspective. In many cases, this alternate view leads to new approaches to existing problems in the area. A limitation of this work is that the treatment for each case is specific to the objects being considered. There is no reason to expect that an induced map has a natural interpretation because the same cohomological set may have various reasonable algebraic meanings. For example, H1(k, Sn) can be thought of as isomorphism classes of étale algebras or of Sn-Galois algebras. Thus, as in this thesis, working with individual cases is necessary in order to understand the connection between these two areas of mathematics. 20 Bibliography [1] G. Berhuy and Z. Reichstein. On the notion of canonical dimension for algebraic groups. Adv. Math., 198(1):128–171, 2005. [2] Nathan Jacobson. Structure and representations of Jordan algebras. American Mathematical Society Colloquium Publications, Vol. XXXIX. American Mathematical Society, Providence, R.I., 1968. [3] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol. The book of involutions, volume 44 of American Mathemati- cal Society Colloquium Publications. American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. [4] Jean-Pierre Serre. Local fields, volume 67 of Graduate Texts in Mathe- matics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. [5] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathe- matics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author. [6] T. A. Springer. On the equivalence of quadratic forms. Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math., 21:241–253, 1959. 21

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