Cohomological invariants of odd degree Jordan 8 algebras 0 0 2 Mark L. MacDonald n a February 2, 2008 J 0 1 Abstract ] In this paper we determine all possible cohomological invariants of A Aut(J)-torsors in Galois cohomology with mod 2 coefficients (character- R istic of the base field not 2), for J a split central simple Jordan algebra . of odd degree n≥3. This hasalready been donefor J of orthogonal and h t exceptional type, and we extend these results to unitary and symplectic a type. Wewilluseourresultstocomputetheessentialdimensionsofsome m groups, for example we show that ed(PSp )=n+1 for n odd. 2n [ 1 1 Introduction v 4 TheStiefel-Whitneyclassesofquadraticformsoverk defineinvariantsinGalois 3 6 cohomologyH∗(k,Z/2Z)uptoisometry[De62],[Mi70]. Itisshownin[GMS03, 1 ch. VI] that the even Stiefel-Whitney classes form a basis of all cohomological 1. invariants of SOn-torsors for n ≥ 3 odd. This is done by identifying the SOn- 0 torsorswithisomorphismclassesofdeterminant1quadraticformsofdimension 8 n. These torsorsmay be further identified with isomorphismclassesofalgebras 0 with orthogonal involution of degree n, by sending q to its adjoint involution : v adq (see [KMRT, Thm. 4.2, p.42]). These classes may further be identified i with isomorphism classes of central simple Jordan algebras of degree n whose X associated composition algebra C is one dimensional (see [Ja68, p.210]). We r a wish to extend these results to dim(C) = 2 or 4, which is to say, odd degree algebras with unitary and symplectic involutions. In fact, in the octonion case dim(C) = 8, we only have a Jordan algebra when the degree is 3, and then it is called an Albert algebra. The mod 2 cohomological invariants of Albert algebrashavebeendeterminedin[GMS03,ch.VI].Nevertheless,weincludethis case here for completeness. For any n ≥ 3 we will define Jr := (M (C),−) over k, as the split Jor- n n + dan algebra of hermitian elements, where C is the split composition algebra of dim(C) = 2r. If r = 3 then we insist that n = 3. Here − denotes the conju- gate transposeinvolution. The followingtable summarizes,for n=2m+1, the (split) automorphism groups Aut(Jr), together with their mod 2 cohomologi- n 1 cal invariants (see Thm. 4.7). We list the degrees of invariants which form an ∗ H (k )-basis of all invariants. 0 r Aut(Jr ) Degrees of H∗-basis of Inv(Aut(Jr)) 2m+1 n 0 SO 0, 2, 4, ··· , 2m 2m+1 1 Z/2Z⋉PGL 0, 1, 3, ··· , 2m+1 2m+1 2 PSp 0, 2, 4, ··· , 2m+2 2(2m+1) 3 F 0, 3, 5 4 Here m ≥ 1, and F denotes the split simple group of type F . To specify 4 4 the semi-direct product in r = 1, we just need to describe how the non-trivial element of Z/2Z acts on PGL . This action is defined by sending any 2m+1 [a]∈PGL to its inverse transpose [(at)−1] (see [KMRT, 29.20]). 2m+1 We will show (Thm. 4.7) that for r = 1,2 and 3 and n ≥ 3 odd, the nor- malizedinvariantsofAut(Jr)indegreegreaterthanzeroareinone-to-onecor- n respondence with the even Stiefel-Whitney classes of n-dimensional quadratic forms (which are the invariants of Aut(J0)). Under this bijection the degree n zero Stiefel-Whitney class correspondsto the degree r invariantwhich classifies the associated composition algebra. The Aut(J1)-invariants are discussed in detail in [KMRT, §19.B, §30.C] or 3 [HKRT], including a mod 3 invariant of degree 2. These invariants determine an Aut(J1)-torsor up to isomorphism (considered there as a degree 3 algebra 3 with unitary involution). Recently in [GPT07] Garibaldi, Parimala and Tignol have classified mod 2 invariants of degree ≤3 for Aut(J2)-torsors for n even. n In the finalsectionwe determine the essentialdimensionat 2for ourgroups Aut(Jr) for n ≥ 3 odd. In each case it is equal to the lower bound given by n [CS06, Theorem 1]. In particular, we find that ed(PSp ) = n+1 for n ≥ 3 2n odd,wherepreviouslythebestupperboundwasgivenby2n2−3n−1in[Le04]. 2 Preliminaries Throughout we will let k be a field extension of a fixed base field k of charac- 0 teristic not 2, and k will be a separable closure of k. s For any k-algebra A (by which we will mean finite dimensional, not neces- sarilyassociative,withidentity),wewillusetheusualnotionsfromGaloiscoho- mology [KMRT, §29] to identify Aut (A)-torsors over k with k-isomorphism alg classes of k-algebras B such that Aks ∼= Bks. Similarly for algebras with invo- lution. Inthe Introductionwedefinedthe splitJordanalgebrasJr :=(M (C),−) n n + forr =0,1,2whenn≥3,andalsoforr =3whenn=3. Here,andthroughout this paper we will write dim(C)=2r. These Jordanalgebras are pairwise non- isomorphic, and over k they represent nearly all simple Jordan algebras by a s 2 theoremofAlbert(see[Ja68,Ch.V.6,p.204]). WewillsaythatasimpleJordan algebra is of degree n (for some n≥3), if it becomes isomorphic to Jr over k . n s These are the only kind of Jordan algebras that we will consider. Furthermore,forr=0,1,2wehavethat(M (C),−)isacentralsimplealge- n brawithorthogonal(resp.unitary,symplectic)involution,whereC isagainthe split composition algebra of dimension 2r. They are pairwise non-isomorphic, andoverk theyformallk -isomorphismclassesofcentralsimplealgebraswith s s involution. We say that a central simple algebra with involution has degree n if it becomes isomorphic to (M (C),−) over k . n s One must be careful with the potentially confusing terminology here. A centralsimple algebrawithinvolutionoverk isdefinedtobe centralandsimple as analgebra-with-involution,andmight not be centralor simple as an algebra over k (see [Ja68, p.208] or [KMRT, §2] for precise definitions). Forr =0,1,2wehavea1-to-1correspondencebetweenisomorphismclasses of these two types of objects given by (A,σ) ↔ (A,σ) . So we can view + Aut(Jr)-torsors as either isomorphism classes of algebras with involution or n as isomorphism classes of Jordan algebras (see [Ja68, Ch. V.7, p.210]). An ad- vantage of the Jordan algebra point of view is that it includes the exceptional r =3 case. 2.1 Invariants of quadratic forms ∗ ∗ We will follow the notation of [GMS03] and write H (k) or even H for the GaloiscohomologyringH∗(Gal(k /k),Z/2Z). Fora∈k∗/(k∗)2, wewilldenote s thecorrespondingelement(a)∈H1(k)sothatwehave(a·b)=(a)+(b). Welet Quad (k) be the pointed set of k-isometry classes of n-dimensional quadratic n,1 forms of determinant 1. And we let Pfr(k) be the pointed set of k-isometry classes of r-Pfister forms. We will write Inv(G) = Inv (H1(−,G)) for the k0 ∗ group of cohomologicalinvariants in H of G-torsors. To define the Stiefel-Whitney classes of a quadratic form q over k, take a diagonalizationq ∼=ha1,··· ,ani. Thendefine wi(q) tobe the ithdegreepartof the product n ∗ w(q)= Y(1+(aj))∈H (k). j=1 ThisproductiscalledthetotalStiefel-Whitneyclass. Itisindependentofthedi- agonalization,whichcanbeshownbyachainequivalenceargument(see[Mi70]). For an r-Pfister form, q =hha ,··· ,a ii:=h1,−a i⊗···⊗h1,−a i, 1 r 1 r define aninvariantbye (q)=(a )···(a )∈Hr(k). Itis shownin[GMS03, VI] r 1 r that the H∗-module of invariants of r-Pfister forms, Inv(Pf ), has an H∗(k )- r 0 basis consisting of {1,e }. r 3 3 An upper bound for the invariants In this section we show that Inv(Aut(Jr)) injectively embeds into the tensor n product of two groups of invariants that we understand well (see Cor. 3.6). For any associative composition algebra C over k, and hermitian form h on a free n-dimensionalC-module V, the trace form, q is the quadraticform over h k on V such that qh(x) = h(x,x). If φ is the norm of C, and V ∼= C ⊗V0 as k-vector spaces, then qh ∼=φ⊗q0 for some n-dimensional quadratic form q0 on V . 0 Lemma 3.1. Let C be an associative composition algebra over k with conju- gation involution, and let h, h′ be hermitian forms over C. Then h ∼= h′ iff qh ∼=qh′. Proof. This is shown in [Sch85, 10.1.1,10.1.7]. Proposition 3.2. Let C be an associative composition algebra over k of di- mension 1 (resp. 2, 4). Then isomorphism classes of involutions on M (C) of n orthogonal (resp. unitary, symplectic) type correspond to similarity classes of n-dimensional hermitian forms over C, under ad ↔h. h Proof. This is proved in [KMRT, Prop. 4.2, p.43]. We will calla Jordanalgebraover k reduced if it is isomorphicto one of the form(M ⊗C,ad ⊗−) forsomecompositionk-algebraC, andn-dimensional n q + quadratic form q over k. NoticethisimpliesJacobson’s[Ja68]definitionofreducedintermsoforthog- onal idempotents, and his definition implies this one, by his Coordinatization theorem for n≥3. Lemma 3.3. Let J be a central simple Jordan algebra of odd degree n and of type r =0 or 2. Then J is reduced. Proof. So J ∼=(A,σ) , where (A,σ) is a central simple algebra with an involu- + tion of the first kind. So its index is a power of 2 (see [KMRT, p.18, 2.8]). Case r = 0: The degree of A as a central simple algebra is n, so the index divides n, and hence A is split. So J is reduced by Prop. 3.2 (or [KMRT, p.1]). Case r =2: The degree of A as a central simple algebra is 2n, so the index divides2,andhenceJ isreducedbyProp.3.2andtheremarksatthebeginning of this section. Lemma 3.4. Let J be a central simple Jordan algebra of odd degree. Then J becomes reduced after extending scalars by a field extension of odd degree. Proof. The only non-reduced algebras are when r =3 or r = 1 by Lemma 3.3. Forther=3case,asstatedintheintroduction,wemusthaven=3. By[SV00, 6.1] any non-reduced Albert algebra becomes reduced after a cubic extension. For r =1 there are two cases, as follows. 4 Case J ∼= (B ×Bop,σ) : Here σ is the exchange involution, and B is a + central simple algebra over k of degree n odd. Then any maximal subfield L is a splitting field for B of degree dividing n. Then J is reduced by Prop. 3.2. L Case J ∼= (A,σ) : Here A is a central simple algebra over K of degree n + odd,whereK isaquadraticextensionofk,andσ isaninvolutionofthe second kind (i.e. unitary) over k. Since the Brauer group of a finite field is trivial, we may assume that k is infinite. LetL/kbeanodddegreeextensionsuchthatind(A )=disminimal,where L A isasimple associativealgebrawithcentreK =K⊗ L. disoddsinced|n. L L k Let D be the division K -algebra that is Brauer equivalent to A , and hence L L ofdegreed asa centralsimple algebra. ThenD has aninvolutionofthe second kind τ that fixes L by [KMRT, 3.1, p.31]. We want to show that d = 1, so assume that d > 1. Then we can take a non-scalarelementa inthe degreedJordanalgebra(D,τ) . Since k is infinite, + we may choose a to be of maximal degree in the sense of [Ja68, p.224]. In other words, a is such that deg(m ) = d, where m (λ) ∈ L[λ] is the minimal a a polynomial of a in (D,τ) , for some indeterminate λ. Here we are using the + fact from [Ja68, p.233] that the degree of the generic minimal polynomial of a generic element is equal to the degree of the Jordan algebra as defined in the Preliminaries. Also, the coefficients of m are in L by [KMRT, 32.1.2, p.452]. a If we let α be a root of m in k , then the minimal polynomial of α is m . a s a ThisisbecauseDisadivisionalgebraandhencem isirreducible. SoE =L(α) a is a field extension of degree d over L, and in particular, of odd degree over k. ′ Then by considering the generic norm of a := α1 − a ∈ D , we see that E E ′ ′ n(a) = m (α) = 0 [Ja68, p.224]. So a 6= 0 and is non-invertible, and hence a D isnotadivisionalgebra. Soind(A )=ind(D )<d,whichcontradictsthe E E E minimality of d. Therefore d=1, and hence AL ∼=Mn(KL). So by Prop. 3.2 we see that JL is reduced. Proposition 3.5. For n ≥ 3 odd, let J be an Aut(Jr)-torsor over k. Then n there is an odd-degree extension L/k such that J is in the image of L H:Pf (L)×Quad (L)→H1(L,Aut(Jr)) r n,1 n (φ,q) 7→(M ⊗C ,ad ⊗−) . n φ q + Moreover, if r =2 the map is a surjection, and for r =0 it is a bijection. Proof. FromLemma3.3and3.4wegetL/ksuchthatJ isreduced,andsincen L is odd, we canscaleq sothat det(q)=1. Lemma 3.3givesthe r =2surjection, and the r =0 bijection is well-known. Corollary 3.6. We have an injective map of invariants Inv(Aut(Jr))֒→Inv(Pf )⊗Inv(Quad ). n r n,1 5 Proof. By [Ga07, Lemma 5.3] we can use the surjectivities from Prop. 3.5 to induce an injective map on invariants. Then from [GMS03, Ex. 16.5], we can expressthe invariantsofthe directproductPf ×Quad asthe tensorproduct r n,1 of the invariants of each factor. 4 Construction of the invariants Now it is a matter of deciding which of these invariants occur. In other words, we wish to determine the image of the injective map in Cor. 3.6. It turns out to be the constant invariants together with all multiples of e , the degree r r invariant of Pf (Thm. 4.7). r Theorem 4.1. For n odd, r =0,1,2 or 3, the invariants e ⊗w ∈Inv(Pf )⊗ r 2i r Inv(Quad ) extend uniquely to Aut(Jr)-invariants of degree r+2i, which we n,1 n will call v . i If the invariants extend, then by Cor. 3.6 they are unique. First we will show how to constructthe invariants onreduced Jordanalgebras,and then use [Ga07, Prop. 7.2] to extend them to all Jordan algebras. For a reduced Jordan algebra J = (M (C),ad ⊗−) , we call C the com- n q + position algebra associated to J. It is determined up to isomorphism by the isomorphism class of J (see [Ja68] or [Mc04, 16]). We will usually denote its norm form φ, which is a Pfister form. Lemma 4.2. The quadratic form φ⊗q is determined up to similarity by the isomorphism class of the reduced Jordan algebra J =(M (C),ad ⊗−) . n q + Proof. First consider the case when C is associative, which is to say, r 6= 3. By [Ja68, p.210] the Jordan algebra J determines the isomorphism class of the algebra with involution (M (C),σ). Then Prop. 3.2 lets us associate up to a n scalar,the n-dimensionalhermitian formh on C. Finally, Lemma 3.1 allowsus to determine its trace form φ⊗q up to similarity. In the case r = 3, we use the following argument. For any reduced Jordan algebra of degree n, the quadratic (reduced) trace form, T (x) = Trd(x2) = J trace(x2) is determined up to isometry by the isomorphismclass ofJ, and is of the form TJ ∼=nh1i⊥h2iφ⊗∧2q. But since q is similar to ∧2(q) for 3-dimensional forms, we see that φ⊗q is determined up to similarity for n=3, and in particular when r =3. Remark 4.3. In the r = 2 case, this observation was noted in [GQMT01, Lemma 4.2]. Lemma 4.4. Let φ be an r-fold Pfister form, and q, q′ quadratic forms over k. Then φ⊗q ∼=φ⊗q′ implies er(φ)w(q)=er(φ)w(q′)∈H∗(k). 6 Proof. This is an extension of [De62] or [Mi70] where it is shown for r =0. We need the fact from [WS77] that says if φ⊗q ∼=φ⊗q′ then q and q′ are φ-chain equivalent. Twoquadraticformsaresimplyφ-equivalent iftheycanbothbediagonalized in such a way that q ∼= ha1,··· ,ani and q′′ ∼= hλa1,a2,··· ,ani, where λ is represented by φ. Then two forms are φ-chain equivalent if there is a finite chain of simple φ-equivalences from one to the other. This immediately reduces the problem to showing that equality holds at eachstage of the chain equivalence. This is the same as showing e (φ)w(hai)= r e (φ)w(λhai) for λ represented by φ. For such a λ, we have φ ⊗ h1,−λi is r isotropic,and hence e (φ)·(λ)=0∈Hr+1(k). Expanding w(λhai)=1+(λ)+ r (a), the result clearly follows. SothefollowingLemmashowsthatthequadraticformφ⊗q,wheredet(q)= 1, is in fact determined up to isometry by the isomorphism class of a reduced Jordan algebra. Since q is odd dimensional, there is always a determinant 1 quadratic form similar to it. We will write d(q) = det(q) for the element of k∗/(k∗)2 corresponding to w (q)∈H1(k). 1 Lemma 4.5. Let φ be an r-Pfister form, λ∈ k∗, and q,q′ quadratic forms. If φ⊗q ∼=φ⊗λq′ then φ⊗d(q)q ∼=φ⊗d(q′)q′. Proof. e (φ)·w (q)=e (φ)·w (λq′)∈Hr+1(k)⇔e (φ)·(λd(q)d(q′))=0 r 1 r 1 r ′ ⇔φ⊗hhλd(q)d(q )ii is hyperbolic ′ ∗ ⇔d(q)d(q )=λ mod D(φ) ⇒φ⊗d(q)q ∼=φ⊗d(q′)q′. Proof of Thm. 4.1. First we will show that the invariants e ⊗w extend to r 2i invariants on k-isomorphism classes of reduced Jordan algebras. Consider the reduced Jordan algebra J = (M (C),ad ⊗ −) with n = n q + 2m+1. Then we can assume det(q) = 1. Lemma 4.2 together with Lemma 4.5 show that φ ⊗ q is determined up to isometry by the isomorphism class of J. Then by Lemma 4.4 we can define v (J) = e (φ)w (q) ∈ Hr+2i(k) on i r 2i k-isomorphism classes of reduced Jordan algebras, for 1 ≤ i ≤ m. This clearly extends e ⊗w . r 2i Finally, by Lemma 3.4, any odd degree Jordan algebra becomes reduced after an odd degree extension. So by [Ga07, Prop. 7.2] these invariants may be extended to non-reduced Jordan algebras as well, and in other words, to all Aut(Jr)-torsors. ByCor.3.6,v istheuniqueinvariantextendinge ⊗w . n i r 2i Remark 4.6. For r = 1, there is an invariant closely related to v defined on 1 conjugacy classes of algebras with unitary involution of odd degree in [KMRT, p.438, 31.44]. They related it to the Rost invariant. 7 Now we can state and prove our main theorem. Theorem 4.7. Inv (Aut(Jr)) is a free H∗(k )-module with a basis consisting k0 n 0 of the invariants {1,v ,v ,v ,··· ,v }. 0 1 2 m Proof. For r = 0 this is shown in [GMS03, ch. VI], noting that in this case v =1, causing a redundancy in the set of basis elements. So take r >0. From 0 Cor. 3.6 we know that every Aut(Jr)-invariant restricts to some n 1⊗a+e ⊗b∈Inv(Pf )⊗Inv(Quad ), r r n,1 for some uniquely defined a,b∈Inv(Quad ). We know from [GMS03, ch. VI] n,1 that any b ∈ Inv(Quad ) is in the H∗(k)-span of the even Stiefel-Whitney n,1 classes, so by Thm. 4.1, e ⊗b is the restriction of some Aut(Jr)-invariant in r n ∗ the H (k)-span of {v ,v ,··· ,v }. So all that remains to show is that if 1⊗a 0 1 m is the restriction of an Aut(Jr)-invariant, then a is constant. n Let a′ be an Aut(Jr)-invariant that restricts to 1⊗a for some Quad - n n,1 invariant a. If we let C be the split composition algebra of dimension 2r, then s J =(M (C ),ad ⊗−) n s q + is isomorphic to the split algebra Jr (by Prop. 3.2 for r 6= 3 and by [SV00, n ′ Cor. 5.8.2] for r = 3). So for any such J, we must have that a(J) = a(q) is a constant,independentofq. Sincewecantakeqtobeanarbitraryn-dimensional formofdeterminant1,this implies a is constant. This completes the proof. Remark 4.8. Onemayaskto whatextentdothese v determine the Aut(Jr)- i n torsors? Thereareexamplesofnon-isometricquadraticformswithdeterminant 1 in each dimension ≥ 4 that have equal total Stiefel-Whitney classes [Sch67, Beispiel 3.4.1]. So one can use these examples to write down two different reduced Aut(Jr)-torsors for n≥4 odd, whose invariants agree. n In the case of n = 3, on the other hand, for r = 0 or 2, the torsors are determined completely by v and v . This is because they are determined by 0 1 their quadratic trace form [SV00, §5]. But for r = 1 and r = 3 this is not the case, because (for n = 3) the trace form of any non-reduced algebra is isometrictothetraceformofsomereducedalgebra. Nevertheless,forr =1one may define a degree 2, mod 3 invariant, which together with v and v , classify 0 1 Aut(J1)-torsors [KMRT, §19.B, §30.C]. For r = 3, one may define a degree 3, 3 mod3 invariant,butitisanopenproblemwhether thisinvarianttogetherwith v and v , classify Aut(J3)-torsors [Se95, 9.4]. 0 1 3 5 Essential dimension Given an algebraic group G over k , and a G-torsor E over k, the essential 0 dimension of E is defined to be the minimum transcendence degree over k 0 of all fields of definition of E. The essential dimension of an algebraic group is defined to be the supremum of the essential dimensions of all of its torsors 8 ([RY00], [BF03], [CS06]). The essential dimension of most simple algebraic groupsis unknown. We will determine the value ofthe essentialdimension at2 forsomegroupsG, whichwewilldenoteby ed(G;2)(see[CS06]fora definition of the essential dimension at a prime). In all cases that we consider, ed(G;2) is equal to the lower bound given in [CS06, Theorem 1] (or [RY00, Thm. 7.8] for characteristic 0). Proposition 5.1. For n≥3 odd, we have ed(Aut(Jr);2)=r+n−1. n Proof. BythesurjectivityinLemma3.5wehavethatforanyAut(Jr)-torsorJ n overk,thereisanodddegreeextensionL/ksuchthatJ isreduced. Sobyusing L [BF03,Lemma1.11]wehavethated(J;2)≤edL(JL)≤ed(Pfr)+ed(Quadn,1)= r+n−1. This gives us the upper bound ed(Aut(Jr);2)≤r+n−1. n The lower bound follows from [CS06, Theorem 1]. Alternatively, we could deduce the lower bound by using the non-triviality of the degree r + n − 1 cohomological invariant v . This follows from a slight modification of [BF03, m Cor.3.6],thatifthereisadegreedinvariantmod2,thentheessentialdimension at 2 is at least d. Let us consider what Prop. 5.1 says for different r and n≥3 odd. For r =0 we get the well known fact that ed(SO )=ed(SO ;2)=n−1. n n For r =1 we geted(Z/2Z⋉PGL )≥ed(Z/2Z⋉PGL ;2)=n. The exact n n value of ed(Z/2Z⋉PGL ) is unknown to the author for any n≥3. n For r = 2 we get ed(PSp ) = ed(PSp ;2) = n+ 1, since all PSp - 2n 2n 2n torsors are reduced. Previously, the best known upper bound for ed(PSp ) 2n was 2n2−3n−1, which holds for n even as well [Le04]. For r = 3 we get ed(F ) ≥ ed(F ;2) = 5, which is the best known lower 4 4 bound for ed(F ). The best published upper bound for the essential dimension 4 is ed(F ) ≤ 19 in [Le04]. [Ko00] claimed to show that ed(F ) = 5, but there 4 4 was a mistake in the proof. 6 Acknowledgements I would like to thank Jean-Pierre Tignol for pointing out the r = 2 case of Lemma 3.3, and Skip Garibaldi for suggesting the application to essential di- mension, and catching several mistakes in a previous version of this paper. I would also like to thank my Ph.D. supervisor Burt Totaro, as well as Carl Mc- Tague and Arthur Prendergast-Smithfor many useful discussions. I also thank the referee for their comments and suggestions. References [BF03] G. Berhuy and G. Favi. Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279-330. 9 [CS06] V. Chernousov and J.-P. Serre. Lower bounds for essential dimen- sions via orthogonal representations.J. Algebra 305 (2006), 1055-1070. [De62] A. Delzant. D´efinition des classes de Stiefel-Whitney d’un module quadratique sur un corps de caractristique diff´erente de 2. (French) C. R. Acad. Sci. 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