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Albert algebras over $\mathbb {Z}$ and other rings

Published online by Cambridge University Press:  14 March 2023

Skip Garibaldi
Affiliation:
IDA Center for Communications Research-La Jolla, 4320 Westerra Ct, San Diego, CA 92121, USA; E-mail: skip@garibaldibros.com
Holger P. Petersson
Affiliation:
Fakultät für Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany; E-mail: holger.petersson@fernuni-hagen.de
Michel L. Racine
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur Pvt, Ottawa, ON, K1N 6N5, Canada; E-mail: mracine@uottawa.ca

Abstract

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

Type
Algebra
Creative Commons
Creative Common License - CCCreative Common License - BY
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

1 Introduction

Albert algebras, which are a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

Why Albert algebras?

In the setting of semisimple algebraic groups over a field, a standard technique for computing with elements of a group — especially an anisotropic group — is to interpret the group in terms of automorphisms of some algebraic structure, such as viewing an adjoint group of type $\mathsf {B}_n$ as the special orthogonal group of a quadratic form of dimension $2n+1$ , or an adjoint group of inner type $\mathsf {A}_n$ as the automorphism group of an Azumaya algebra of rank $(n+1)^2$ . This approach can be seen in many references, from [Reference WeilWeil], through [Reference Knus, Merkurjev, Rost and TignolKnMRT] and [Reference ConradConrad]. In this vein, Albert algebras appear as a natural tool for computations related to $\mathsf {F}_4$ , $\mathsf {E}_6$ , and $\mathsf {E}_7$ groups, as we do below.

In the setting of nonassociative algebras, Albert algebras also arise naturally. Among commutative not-necessarily-associative algebras under additional mild hypotheses (the field has characteristic $\ne 2, 3, 5$ and the algebra is metrized), every algebra satisfying a polynomial identity of degree $\le 4$ is a Jordan algebra (see [Reference Chayet and GaribaldiChG, Proposition A.8]). Jordan algebras have an analogue of the Wedderburn-Artin theory for associative algebras [Reference JacobsonJ68, p. 201, Corollary 2], and one finds that all the simple Jordan algebras are closely related to associative algebras (more precisely, “are special”) except for one kind, the Albert algebras (see, for example [Reference JacobsonJ68, p. 210, Theorem 11] or [Reference McCrimmon and Zel’manovMcCZ]).

Our contribution

In the setting of nonassociative algebras, we prove a classification of Albert algebras over $\mathbb {Z}$ (Theorem 14.3), which was viewed as an open question in the context of nonassociative algebra; here, we see that it is equivalent to the classification of groups of type $\mathsf {F}_4$ , which was known (see [Reference ConradConrad], which leverages [Reference GrossGr] and [Reference Elkies and GrossElkiesGr]). We also prove new results about ideals in Albert algebras (Theorem 8.2), about isotopy of Albert algebras over semilocal rings (Theorem 13.3), and about the number of generators of an Albert algebra (Proposition 12.1). We have not seen Lemma 15.1 in the literature, even in the case of a base field of characteristic different from 2 and 3.

In the setting of affine group schemes, the language of Albert algebras provides a way to give concrete descriptions of the affine group schemes over $\mathbb {Z}$ (see Section 18). In that language, a clever computation in [Reference Elkies and GrossElkiesGr] appears as an example of a general mechanism known as isotopy (see Definition 14.1). To facilitate these applications, we present the definition of Albert algebras in a streamlined way (see Definition 7.1). Note that they are defined as a type of what was formerly called a “quadratic” Jordan algebra — because instead of a bilinear multiplication, one has a quadratic map, the U-operator — and that the definition makes sense whether or not 2 is invertible in the base ring. Applying the definition here allows one to replace, in some proofs, “global” computations over $\mathbb {Z}$ as one finds in [Reference ConradConrad] with “local” computations over an algebraically closed field that exist in several places in the literature (see, for example, the proof of Lemma 9.1 and Section 18).

A different definition

The definition of Albert algebra over a ring R given here (Definition 7.2) is in the context of para-quadratic algebras as recalled at the beginning of Section 5 — such an algebra is an R-module M with a distinguished element $1_M$ and a quadratic map $U \!: M \to \operatorname {\mathrm {End}}_R(M)$ such that $U_{1_M} = \operatorname {\mathrm {Id}}_M$ . There are no other axioms to check. We then define a specific para-quadratic algebra, $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ , in Definition 6.7, and define an Albert R-algebra J to be a para-quadratic R-algebra such that $J \otimes S \cong \operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R)) \otimes S$ for some faithfully flat R-algebra S.

A different approach is taken by references such as [Reference PeterssonPe19, Section 6.1] or [Reference AlsaodyAls21]. They define an Albert R-algebra to be a cubic Jordan R-algebra (Definition 6.2) J whose underlying R-module is projective of rank 27 and $J \otimes F$ is a simple algebra for every homomorphism from R to a field F. This definition involves axioms that in principle need to be verified over all R-algebras. The two definitions give the same objects. Theorem 17 in [Reference PeterssonPe19] states that an Albert algebra in the sense of that paper is an Albert algebra in the sense of this paper by proving the existence of the required faithfully flat R-algebra; a detailed proof has not been published but closely follows an argument for octonion algebras from [Reference Loos, Petersson and RacineLoPR]. For the converse, an Albert algebra in the sense of this paper has a projective underlying module (because $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ does), is a cubic Jordan algebra (Proposition 10.1), and satisfies the simplicity condition (Corollary 8.5).

2 Notation

Rings, by definition, have a 1. We put ${\mathbb {Z}\text {-}\textbf {alg}}$ for the category of commutative rings, where $\mathbb {Z}$ is an initial object. For any $R \in {\mathbb {Z}\text {-}\textbf {alg}}$ , we put ${R\text {-}\textbf {alg}}$ for the category of pairs $(S, f)$ with $S \in {\mathbb {Z}\text {-}\textbf {alg}}$ and $f \!: R \to S$ , that is, the coslice category $R \downarrow {\mathbb {Z}\text {-}\textbf {alg}}$ . Below, R will typically denote an element of ${\mathbb {Z}\text {-}\textbf {alg}}$ . (The interested reader is invited to mentally replace R by a base scheme $\mathbf {X}$ , ${R\text {-}\textbf {alg}}$ with the category of schemes over $\mathbf {X}$ , finitely generated projective R-modules with vector bundles over $\mathbf {X}$ , etc., thereby translating results below into a language closer to that in [Reference Calmès and FaselCalF].) An R-algebra S is said to be fppf if it is faithfully flat and finitely presented.

We write $\operatorname {\mathrm {Mat}}_n(R)$ for the ring of n-by-n matrices with entries from R, $\operatorname {\mathrm {Id}}$ for the identity matrix, and ${\left \langle {\alpha _1, \ldots , \alpha _n}\right \rangle } \in \operatorname {\mathrm {Mat}}_n(R)$ for the diagonal matrix whose $(i,i)$ -entry is $\alpha _i$ . The transpose of a matrix x is denoted $x^{\intercal }$ . We write $\operatorname {\mathrm {GL}}_n(R)$ for the group of invertible elements in $\operatorname {\mathrm {Mat}}_n(R)$ .

Suppose now that $\mathbf {G}$ is a finitely presented group scheme over R. For each fppf $S \in {R\text {-}\textbf {alg}}$ , we write $H^1(S/R, \mathbf {G})$ for Čech cohomology of the sheaf of groups $\mathbf {G}$ relative to $R \to S$ (see, for example [Reference GiraudGir, Section III.3.6] or [Reference WaterhouseWa, Chapter 17]). The set $H^1(S/R, \mathbf {G})$ does not depend on the choice of structure homomorphism $R \to S$ , and more is true: Every morphism $S \to T$ in ${k\text {-}\textbf {alg}}$ gives a morphism $H^1(S/R, \mathbf {G}) \to H^1(T/R, \mathbf {G})$ that is injective and does not depend on the choice of arrow $S \to T$ [Reference GiraudGir, Remark III.3.6.5]. The subcategory of fppf elements of ${R\text {-}\textbf {alg}}$ has a small skeleton, so the colimit

$$\begin{align*}H^1(R, \mathbf{G}) := \varinjlim_{\text{fppf } S \in {R\text{-}\textbf{alg}}} H^1(S/R, \mathbf{G}) \end{align*}$$

is a set. We call it the nonabelian fppf cohomology of $\mathbf {G}$ . In case $\mathbf {G}$ is smooth, it agrees with étale $H^1$ . If additionally R is a field, then it agrees with the nonabelian Galois cohomology defined in, for example, [Reference SerreSerre].

Unimodular elements

Let M be an R-module. An element $m \in M$ is said to be unimodular if $Rm$ is a free R-module of rank 1 and a direct summand of M, equivalently, if there is some $\lambda \in M^*$ (the dual of M) such that $\lambda (m) = 1$ . When M is finitely generated projective, this is equivalent to: $m \otimes 1$ is not zero in $M \otimes F$ for every field $F \in {R\text {-}\textbf {alg}}$ (see, for example [Reference LoosLo, 0.3]). If $m \in M$ is unimodular, then so is $m \otimes 1 \in M \otimes S$ for every $S \in {R\text {-}\textbf {alg}}$ . In the opposite direction, if M is finitely generated projective, S is a Zariski cover of R (i.e., $\operatorname {\mathrm {Spec}} S \to \operatorname {\mathrm {Spec}} R$ is surjective), and $m \otimes 1$ is unimodular in $M \otimes S$ , it follows that m is unimodular as an element of M.

3 Background on polynomial laws

We may identify an R-module M with a functor $\mathbf {W}(M)$ from ${R\text {-}\textbf {alg}}$ to the category of sets defined via $S \mapsto M \otimes S$ . For R-modules M, N, a polynomial law (in the sense of [Reference RobyRoby] or [Reference BourbakiBouA2, Section IV.5, Exercise 9]) $f \colon \mathbf {W}(M) \to \mathbf {W}(N)$ is a morphism of functors, that is, a collection of set maps $f_S \colon M \otimes S \to N \otimes S$ varying functorially with S. We put $\mathscr {P}_R(M, N)$ for the collection of polynomial laws $\mathbf {W}(M) \to \mathbf {W}(N)$ , omitting the subscript R when it is understood. Note that $\mathscr {P}_R(M, N)$ is an R-module.

Lemma 3.1. Let M be a finitely generated projective R-module, and suppose $f \in \mathscr {P}(M, N)$ is such that $f_R(0) = 0$ . If $m \in M$ has $f_R(m)$ unimodular in N, then m is unimodular.

In the case $N = R$ , the condition that $f_R(m)$ is unimodular means that $f_R(m) \in R^{\times }$ .

Proof. Replacing f with $\lambda f$ , where $\lambda \in N^*$ is such that $\lambda (f_R(m)) = 1$ , we may assume $N = R$ and $f_R(m) = 1$ . If m is not unimodular, then there is a field $F \in {R\text {-}\textbf {alg}}$ , such that $m \otimes 1 = 0$ in $M \otimes F$ , and $f_F(m \otimes 1) = 0$ , whence $f_R(m)$ belongs to the kernel of $R \to F$ a contradiction.

A polynomial law is homogeneous of degree $d \ge 0$ if $f_S(sx) = s^d f_S(x)$ for every $S \in {R\text {-}\textbf {alg}}$ , $s \in S$ , and $x \in M \otimes S$ (see [Reference RobyRoby, p. 226]). We put $\mathscr {P}^d_R(M, N)$ for the submodule of $\mathscr {P}_R(M, N)$ of polynomial laws that are homogeneous of degree d. The polynomial laws that are homogeneous of degree 0 are constants, and those of degree 1 are linear transformations, that is, the natural maps

$$\begin{align*}N \to \mathscr{P}_R^0(M, N) \quad \text{and} \quad \operatorname{\mathrm{Hom}}_R(M, N) \to \mathscr{P}_R^1(M, N) \end{align*}$$

are isomorphisms (see [Reference RobyRoby, pp. 230, 231]). For degree $2$ , $\mathscr {P}_R^2(M,N)$ is canonically identified with the maps $f \colon M \to N$ that are quadratic in the sense that $f(rm) = r^2f(m)$ and the map $M\times M \to N$ defined by $f(m_1,m_2) := f(m_1 + m_2) - f(m_1) - f(m_2)$ is bilinear [Reference RobyRoby, p. 236, Proposition II.1]. A form of degree d on M is a polynomial law $\mathbf {W}(M) \to \mathbf {W}(R)$ that is homogeneous of degree d. The forms of degree 2 are commonly known as quadratic forms on M.

Directional derivatives

For $f \in \mathscr {P}(M, N)$ , $v \in M$ , t an indeterminate, and $n \ge 0$ , we define a polynomial law $\nabla _v^n f$ as follows. For $S \in {R\text {-}\textbf {alg}}$ and $x \in M \otimes S$ , $f_{S[t]}(x + v \otimes t)$ is an element of $N \otimes S[t]$ , and we define $\nabla _v^n f_S(x) \in N \otimes S$ to be the coefficient of $t^n$ . This defines a polynomial law called the n-th directional derivative $\nabla _v^n f$ of f in the direction v. One finds that $\nabla ^0_v f = f$ regardless of v. We abbreviate $\nabla _v f := \nabla _v^1 f$ ; it is linear in v.

If f is homogeneous of degree d and $0 \le n \le d$ , then $\nabla _v^n f(x)$ is homogeneous of degree $d - n$ in x and degree n in v. The symmetry implicit in the definition of the directional derivative gives $\nabla _v^n f(x) = \nabla _x^{d-n} f(v)$ for $x \in M$ .

Lemma 3.2. Suppose M, N are R-modules and A is a unital associative R-algebra and $g \in \mathscr {P}(M , A)$ is a polynomial law such that there is an element $m \in M$ such that $g_R(m) \in A$ is invertible. If $f \in \mathscr {P}^d(M, N)$ satisfies

$$\begin{align*}g_S(x) \in (A \otimes S)^{\times} \Rightarrow f_S(x) = 0 \end{align*}$$

for all $S \in {R\text {-}\textbf {alg}}$ and $x \in M \otimes S$ , then $f = 0$ .

Proof. Since the hypotheses are stable under base change, it suffices to show that $f(v) = 0$ for all $v \in M$ . Replacing g by $L \circ g \in \mathscr {P}(M, A)$ , where $L \in \operatorname {\mathrm {End}}_R(A)$ is multiplication in A on the left by the inverse of $g_R(m)$ , we may assume $g_R(m) = 1_A$ . Set $S := R[\varepsilon ]/(\varepsilon ^{d+1})$ . For $v \in M$ , the element

$$\begin{align*}g_S(m + \varepsilon v) = 1_A + \sum_{n=1}^d \varepsilon^n \nabla_v^n g_R(m) \end{align*}$$

is invertible in $A_S$ , so by hypothesis,

$$\begin{align*}0 = f_S(m+\varepsilon v) = \sum_{n = 0}^d \varepsilon^n \nabla_v^n f_R(m). \end{align*}$$

Focusing on the coefficient of $\varepsilon ^d$ in that equation gives

$$\begin{align*}0 = \nabla_v^d f_R(m) = \nabla_m^0 f_R(v) = f_R(v), \end{align*}$$

as required.

While Lemma 3.2 has some similarity with the principle of extension of algebraic identities as in [Reference BourbakiBouA2, Section IV.2.3], that result imposes some hypothesis on R.

The module of polynomial laws

In the following, we write $\mathsf {S}^n M$ for the n-th symmetric power of M, that is, the R-module $\otimes ^n M$ modulo the submodule generated by elements $x - \sigma (x)$ for $x \in \otimes ^n M$ and $\sigma $ a permutation of the n factors.

Lemma 3.3. Let M and N be finitely generated projective R-modules. Then for each $d \ge 0$ :

  1. 1. $\mathscr {P}^d(M, N)$ is a finitely generated projective R-module.

  2. 2. If $T \in {R\text {-}\textbf {alg}}$ is flat, the natural map $\mathscr {P}^d_R(M, N) \otimes T \to \mathscr {P}^d_T(M \otimes T, N \otimes T)$ is an isomorphism.

  3. 3. The natural map $\mathsf {S}^d(M^*) \otimes N \to \mathscr {P}^d(M, N)$ is an isomorphism.

  4. 4. The natural map $\mathscr {P}^d(M, R) \otimes N \to \mathscr {P}^d(M, N)$ is an isomorphism.

Proof. To establish notation, we write $R = \prod _{i=0}^n R_i$ for some n such that $M = \prod _i M_i$ and $N = \prod _i N_i$ with each $M_i$ , $N_i$ an $R_i$ -module of finite constant rank.

Next write $\Gamma _d(M)$ for the module of degree d divided powers on M as defined in [Reference BourbakiBouA2, Section IV.5, Exercise 2]. We claim that it is finitely generated projective, and therefore, by [Stacks, Tag 00NX], finitely presented. If M is free, then $\Gamma _d(M)$ is free. If M is projective of constant rank, then there exists $S \in {R\text {-}\textbf {alg}}$ faithfully flat such that $M \otimes S$ is free. Because $\Gamma _d$ commutes with base change [Reference BourbakiBouA2, Section IV.5, Exercise 7], $\Gamma _d(M) \otimes S \cong \Gamma _d(M \otimes S)$ is free, and we again find that $\Gamma _d(M)$ is finitely generated projective [Stacks, Tags 03C4, 05A9]. In the general case, $\Gamma _d(M) = \prod _i \Gamma _d(M_i)$ , and the claim is verified.

To verify (2), we note that $\mathscr {P}^d_R(M, N)$ is naturally isomorphic to $\operatorname {\mathrm {Hom}}_R(\Gamma _d(M), N)$ by [Reference RobyRoby, Theorem IV.1]. Then $\mathscr {P}^d_R(M, N) \otimes T \cong \operatorname {\mathrm {Hom}}_R(\Gamma _d(M), N) \otimes T$ , which in turn is $\operatorname {\mathrm {Hom}}_T(\Gamma _d(M) \otimes T, N \otimes T)$ because T is flat and $\Gamma _d(M)$ is finitely presented [Reference BourbakiBouCA, Section I.2.10, Proposition 11]. Since $\Gamma _d$ commutes with base change, we have verified (2).

(3): If M and N are free modules, then the map is an isomorphism by [Reference RobyRoby, p. 232]. If M and N have constant rank, then there is a faithfully flat $T \in {R\text {-}\textbf {alg}}$ such that $M \otimes T$ and $N \otimes T$ are free. Since (3) holds over T by the free case, (2) and faithfully flat descent give that (3) holds. In the general case, $\mathscr {P}^d(M, N) = \prod \mathscr {P}^d(M_i, N_i)$ and $\mathsf {S}^d(M^*) \otimes N = \prod (\mathsf {S}^d(M_i^*) \otimes N_i)$ and the claim follows by the constant rank case.

(4) follows trivially from (3). For (1), note that $M^*$ is finitely generated projective, so are $\mathsf {S}^d(M^*)$ and the tensor product $\mathsf {S}^d(M^*) \otimes N$ . Applying (3) gives the claim.

One can create new polynomial laws from old by twisting by a line bundle, that is, by a rank 1 projective module.

Lemma 3.4. Let M and N be finitely generated projective R-modules. Then for every $d \ge 0$ and every line bundle L, we have:

  1. 1. There is a natural isomorphism $\mathscr {P}^d(M, N) \otimes (L^*)^{\otimes d} \to \mathscr {P}^d(M \otimes L, N)$ .

  2. 2. There is a natural isomorphism $\mathscr {P}^d(M, N) \cong \mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d})$ .

Proof. For (1), since $L^*$ is a line bundle, the natural map $(L^*)^{\otimes d} \to \mathsf {S}^d(L^*)$ is an isomorphism because it is so after faithfully flat base change. Since $\mathsf {S}^d(M^*) \otimes \mathsf {S}^d(L^*)$ is naturally identified with $\mathsf {S}^d((M \otimes L)^*)$ , combining Lemma 3.3(3),(4) then gives (1).

For (2), there are isomorphisms $\mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d}) \xrightarrow {\sim } \mathscr {P}^d(M, N) \otimes (L^*)^{\otimes d} \otimes L^{\otimes d}$ by (1) and Lemma 3.3(4). Since $L^{\otimes d} \otimes (L^*)^{\otimes d} \cong R$ , the claim follows.

Example 3.5. (References: [Stacks, Tag 03PK], [Reference Calmès and FaselCalF, Section 2.4.3], [Reference KnusKn, Section III.3]) Suppose L is a line bundle and there is an isomorphism $h \colon L^{\otimes d} \to R$ for some $d \ge 1$ . We call such a pair $[L, h]$ a d-trivialized line bundle. (In the case $d = 2$ , they are sometimes called discriminant modules.) Applying h to identify $N \otimes L^{\otimes d} \xrightarrow {\sim } N$ in Lemma 3.4(2) gives a construction that takes $f \in \mathscr {P}^d(M, N)$ and gives an element of $\mathscr {P}^d(M \otimes L, N)$ , which we denote by $[L, h] \cdot (M, f)$ .

For example, for each $\alpha \in R^{\times }$ , define ${\left \langle {\alpha }\right \rangle }$ to be $[L, h]$ as in the preceding paragraph, where $L = R$ and h is defined by $h(\ell _1 \otimes \cdots \otimes \ell _d) = \alpha \prod \ell _i$ . Clearly, ${\left \langle {\alpha \beta ^d}\right \rangle } \cong {\left \langle {\alpha }\right \rangle }$ for all $\alpha , \beta \in R^{\times }$ . Applying the construction in the previous paragraph, we find ${\left \langle {\alpha }\right \rangle } \cdot (M, f) \cong (M, \alpha f)$ .

Every $[L, h]$ with $L = R$ is necessarily isomorphic to ${\left \langle {\alpha }\right \rangle }$ for some $\alpha \in R^{\times }$ . In particular, if $\operatorname {\mathrm {Pic}}(R)$ has no d-torsion elements other than zero — for example, if R is a semilocal ring or a UFD [Stacks, Tags 0BCH, 02M9] — then each $[L, h]$ is isomorphic to ${\left \langle {\alpha }\right \rangle }$ for some $\alpha $ . The group scheme $\mu _d$ of d-th roots of unity is the automorphism group of each $[L, h]$ , where $\mu _d$ acts by multiplication on L. The group $H^1(R, \mu _d)$ classifies pairs $[L, h]$ up to isomorphism.

We say that homogeneous polynomial laws related by the isomorphism in Lemma 3.4(2) are projectively similar, imitating the language from [Reference Auel, Bernardara and BolognesiAuBB, Section 1.2] for the case of quadratic forms ( $d = 2$ ). (This relationship was called “lax-similarity” in [Reference Balmer and CalmèsBC].) We say that homogeneous degree d laws f and $[L, h] \cdot f$ for $[L, h] \in H^1(R, \mu _d)$ as in the preceding example are similar. If $\operatorname {\mathrm {Pic}}(R)$ is d-torsion, the two notions coincide.

For $f \in \mathscr {P}^d(M, N)$ , we define $\operatorname {\mathrm {Aut}}(f)$ to be the subgroup of $\operatorname {\mathrm {GL}}(M)$ consisting of elements g such that $fg = f$ as polynomial laws. In case M and N are finitely generated projective, so is $\mathscr {P}^d(M, N)$ , whence the functor $\mathbf {Aut}(f)$ from ${R\text {-}\textbf {alg}}$ to groups defined by $\mathbf {Aut}(f)(T) = \operatorname {\mathrm {Aut}}(f_T)$ is a closed sub-group-scheme of $\mathbf {GL}(M)$ .

Lemma 3.6. Let f and $f'$ be homogeneous polynomial laws on finitely generated projective modules. If f and $f'$ are projectively similar, then their automorphism groups are isomorphic.

Proof. By hypothesis, $f \in \mathscr {P}^d(M, N)$ and $f' \in \mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d})$ for some modules M and N, line bundle L, and $d \ge 0$ . The group scheme $\mathbf {Aut}(f)$ is the closed sub-group-scheme of $\mathbf {GL}(M)$ stabilizing the element f in $\mathsf {S}^d(M^*) \otimes N$ . Now, any element of $\mathbf {GL}(M)$ acts on $\mathsf {S}^d((M \otimes L)^*) \otimes (N \otimes L^{\otimes d})$ by defining it to act as the identity on L. In this way, we find a homomorphism $\mathbf {Aut}(f) \to \mathbf {Aut}(f')$ . Viewing M as $(M \otimes L) \otimes L^*$ and N as $(N \otimes L^{\otimes d}) \otimes (L^*)^{\otimes d}$ , and repeating this construction, we find an inverse mapping $\mathbf {Aut}(f') \to \mathbf {Aut}(f)$ .

4 Background on composition algebras

A not-necessarily-associative R-algebra C is an R-module with an R-linear map $C \otimes _R C \to C$ , which we view as a multiplication and write as juxtaposition. Such a C is unital if it has an element $1_C \in C$ such that $1_C c = c 1_C = c$ for all $c \in C$ (see, for example, [Reference SchaferSch]). A composition R-algebra as in [Reference PeterssonPe93] is such a C that is finitely generated projective as an R-module, is unital, and has a quadratic form $n_C \!: C \to R$ that allows composition (that is, such that $n_C(xy) = n_C(x)n_C(y)$ for all $x, y \in C$ ), satisfies $n_C(1_C) = 1$ , and whose bilinearization defined by $n_C(x, y) := n_C(x+y) - n_C(x) - n_C(y)$ gives an isomorphism $C \to C^*$ via $x \mapsto n_C(x, \cdot )$ . We say that a symmetric bilinear form with this property is regular. The quadratic form $n_C$ (which is unique by Proposition 4.3 below) is called the norm of C.

Remark 4.1. In the definition above, one can swap the condition $n_C(1_C) = 1$ with the requirement that the rank of C is nowhere zero, that is, $C \otimes F \ne 0$ for every field $F \in {R\text {-}\textbf {alg}}$ .

We put $\operatorname {\mathrm {Tr}}_C(x) := n_C(x, 1_C)$ , a linear map $C \to R$ , called the trace of C. Trivially, $\operatorname {\mathrm {Tr}}_C(1_C) = 2$ . Lemma 3.1 gives that $1_C$ is unimodular, so we may identify R with $R1_C$ , and C is a faithful R-module. The unimodularity of $1_C$ is equivalent to the existence of some $\lambda \in C^*$ such that $\lambda (1_C) = 1$ , that is, some $x \in C$ such that $\operatorname {\mathrm {Tr}}_C(x) = 1$ , whence $\operatorname {\mathrm {Tr}}_C \colon C \to R$ is surjective.

The class of composition algebras is stable under base change. That is, if C is a composition R-algebra with norm $n_C$ , then for every $S \in {R\text {-}\textbf {alg}}$ , $C \otimes S$ is a composition S-algebra with norm $n_C \otimes S$ . The following two results are essentially well known [Reference PeterssonPe93, 1.2 $-$ 1.4]. For convenience, we include their proof.

Lemma 4.2 (“Cayley-Hamilton”)

Let C be a composition algebra with norm $n_C$ , and define $\operatorname {\mathrm {Tr}}_C$ as above. Then

$$\begin{align*}x^2 - \operatorname{\mathrm{Tr}}_C(x) x + n_C(x) 1_C = 0 \end{align*}$$

for all $x \in C$ .

Proof. Linearizing the composition law $n_C(xy) = n_C(x) n_C(y)$ , we find

(4.1) $$ \begin{align} n_C(xy, x) = n_C(x) \operatorname{\mathrm{Tr}}_C(y) \quad \text{and} \end{align} $$
(4.2) $$ \begin{align} n_C(xy, wz) + n_C(wy, xz) = n_C(x,w) n_C(y,z) \end{align} $$

for all $x, y, z, w \in C$ . Setting $z = x$ and $w = 1_C$ in (4.2), we find:

$$\begin{align*}n_C(xy, x) + n_C(y, x^2) = \operatorname{\mathrm{Tr}}_C(x) n_C(x, y). \end{align*}$$

Combining these with (4.1), we find:

$$\begin{align*}n_C(x^2 - \operatorname{\mathrm{Tr}}_C(x) x + n_C(x) 1_C, y) = 0 \quad \text{for all } x, y \in C. \end{align*}$$

Since the bilinear form $n_C$ is regular, the claim follows.

A priori, a composition algebra is a unital algebra together with a quadratic form, the norm. The next result shows that these data are redundant.

Proposition 4.3. If C is a composition algebra, then the norm $n_C$ is uniquely determined by the algebra structure of C.

Proof. Let $n' \colon C \to R$ be any quadratic form making C a composition algebra, and write $\operatorname {\mathrm {Tr}}'$ for the corresponding trace $\operatorname {\mathrm {Tr}}'(x) := n'(x + 1_C) - n'(x) - n'(1_C)$ . Then $\lambda := \operatorname {\mathrm {Tr}}_C - \operatorname {\mathrm {Tr}}'$ (respectively, $q := n_C - n'$ ) is a linear (respectively, quadratic) form on C and the Cayley-Hamilton property yields

(4.3) $$ \begin{align} \lambda(x) x = q(x) 1_C \quad \text{for all } x \in C. \end{align} $$

We aim to prove that $q = 0$ . Because $1_C$ is unimodular, it suffices to prove $\lambda = 0$ . This can be checked locally, so we may assume that R is local and, in particular, $C = R1_C \oplus M$ for a free module M. Now, $\operatorname {\mathrm {Tr}}_C(1_C) = 2 = \operatorname {\mathrm {Tr}}'(1_C)$ , so $\lambda (1_C) = 0$ . For $m \in M$ a basis vector, $\lambda (m)m$ belongs to $M \cap R1_C$ by (4.3), so it is zero, whence $\lambda (m) = 0$ , proving the claim.

Corollary 4.4. Let C be a unital R-algebra. If there is a faithfully flat $S \in {R\text {-}\textbf {alg}}$ such that $C \otimes S$ is a composition S-algebra, then C is a composition algebra over R.

Proof. Because the norm $n_{C \otimes S}$ of $C \otimes S$ is uniquely determined by the algebra structure, one obtains by faithfully flat descent a quadratic form $n_C \colon C \to R$ such that $n_C \otimes S = n_{C \otimes S}$ . Because $n_{C \otimes S}$ satisfies the properties required to make $C \otimes S$ a composition algebra and S is faithfully flat over R, it follows that the same properties hold for $n_C$ .

The following facts are standard, see, for example [Reference KnusKn, Section V.7]: Composition algebras are alternative algebras. The map $\bar {\ } \!: C \to C$ defined by $\overline {x} := \operatorname {\mathrm {Tr}}_C(x)1_C - x$ is an involution, that is, an R-linear antiautomorphism of period 2.

Composition algebras of constant rank

In case R is connected — that is, $R \not \cong R_1 \times R_2$ , where neither $R_1$ nor $R_2$ are the zero ring — a composition R-algebra has rank $2^e$ for $e \in \{ 0, 1, 2, 3 \}$ [Reference KnusKn, p. 206, Theorem V.7.1.6]. Therefore, specifying a composition R-algebra C is equivalent to writing

(4.4) $$ \begin{align} R = \prod_{e=0}^3 R_e \quad \text{and} \quad C = \prod_{e=0}^3 C_e, \end{align} $$

where $C_e$ is a composition $R_e$ -algebra of constant rank $2^e$ .

If C is a composition algebra of rank 1, then since $1_C$ is unimodular, C is equal to R. The bilinear form $n_C(\cdot , \cdot )$ gives an isomorphism $C \to C^*$ and $n_C(1_C, \alpha 1_C) = 2\alpha $ , and we deduce that $2$ is invertible in R. Conversely, if 2 is invertible, then R is a composition algebra by setting $n_C(\alpha ) = \alpha ^2$ ; in this case, we say that R is a split composition algebra.

A composition algebra whose rank is 2 is not just an associative and commutative ring, it is an étale algebra [Reference KnusKn, p. 43, Theorem I.7.3.6]. Conversely, every rank 2 étale algebra is a composition algebra. Among rank 2 étale algebras, there is a distinguished one, $R \times R$ , which is said to be split.

A composition algebra whose rank is 4 is associative and is an Azumaya algebra, commonly known as a quaternion algebra. (Note that our notion of quaternion algebra is more restrictive than the one in the books [Reference KnusKn, see p. 43] and [Reference VoightVo].) Among quaternion R-algebras, there is a distinguished one, the 2-by-2 matrices $\operatorname {\mathrm {Mat}}_2(R)$ , which is said to be split.

A composition algebras whose rank is 8 is known as an octonion algebra. Among octonion R-algebras, there is a distinguished one that is said to be split, called the Zorn vector matrices and denoted $\operatorname {\mathrm {Zor}}(R)$ (see [Reference Loos, Petersson and RacineLoPR, 4.2]). As a module, we view it as $\left ( \begin {smallmatrix} R & R^3 \\ R^3 & R \end {smallmatrix} \right )$ with multiplication

$$\begin{align*}\left( \begin{smallmatrix} \alpha_1 & u \\ x & \alpha_2 \end{smallmatrix} \right) \left( \begin{smallmatrix} \beta_1 & v \\ y & \beta_2 \end{smallmatrix} \right) = \left( \begin{smallmatrix} \alpha_1 \beta_1 - u^{\intercal} y & \alpha_1 v + \beta_2 u + x \times y \\ \beta_1 x + \alpha_2 y + u \times v & {-x^{\intercal} v + \alpha_2 \beta_2}\end{smallmatrix} \right), \end{align*}$$

where $\times $ is the ordinary cross product on $R^3$ . The quadratic form is

$$\begin{align*}n_{\operatorname{\mathrm{Zor}}(R)}\left( \begin{smallmatrix} \alpha_1 & u \\ x & \alpha_2 \end{smallmatrix} \right) = \alpha_1 \alpha_2 + u^{\intercal} x. \end{align*}$$

One says that a composition R-algebra C is split if, when we write R and C as in (4.4), $C_e$ is isomorphic to the split composition $R_e$ -algebra for $e \ge 1$ .

It is well known in the case where $R = \mathbb {R}$ , the real numbers, that a composition algebra is determined up to isomorphism by its dimension and whether it is split. That is, there are only seven isomorphism classes of composition $\mathbb {R}$ -algebras, consisting of four split ones and four division algebras, namely, $\mathbb {R}$ , $\mathbb {C}$ , $\mathbb {H}$ , and $\mathbb {O}$ ; note that both collections of four contain $\mathbb {R}$ .

Example 4.5. The real octonions $\mathbb {O}$ are a composition $\mathbb {R}$ -algebra with basis $1_{\mathbb {O}}$ , $e_1$ , $e_2$ , $\ldots $ , $e_7$ which is orthonormal with respect to the quadratic form $n_{\mathbb {O}}$ with multiplication table

$$\begin{align*}e_r^2 = -1 \quad \text{and} \quad e_r e_{r+1} e_{r+3} = -1 \end{align*}$$

for all r with subscripts taken modulo 7, and the displayed triple product is associative.

The $\mathbb {Z}$ -sublattice $\mathcal {O}$ of $\mathbb {O}$ spanned by $1_{\mathbb {O}}$ , the $e_r$ , and

$$ \begin{align*} h_1 = (1 + e_1 + e_2 + e_4)/2, \quad h_2 = (1 + e_1 + e_3 + e_7)/2, \\ h_3 = (1 + e_1 + e_5 + e_6)/2 \quad \text{and} \quad h_4 = (e_1 + e_2 + e_3 + e_5)/2 \end{align*} $$

is a composition $\mathbb {Z}$ -algebra. It is a maximal order in $\mathcal {O} \otimes \mathbb {Q}$ , and all such are conjugate under the automorphism group of $\mathcal {O} \otimes \mathbb {Q}$ . (As a consequence, there is some choice in the way one presents this algebra. We have followed [Reference Elkies and GrossElkiesGr].) As a subring of $\mathbb {O}$ , it has no zero divisors. For more on this, see [Reference DicksonDi, Section 19], [Reference CoxeterCox], [Reference Conway and SmithConwS, Section 9], or [Reference ConradConrad, Section 5]. The nonuniqueness of this choice of maximal order and its relationship to other orders like $\mathbb {Z} \oplus \mathbb {Z} e_1 \oplus \cdots \oplus \mathbb {Z} e_7$ can be understood in terms of the Bruhat-Tits building of the group $\operatorname {\mathrm {Aut}}(\operatorname {\mathrm {Zor}}(\mathbb {Q}_2))$ of type $\mathsf {G}_2$ over the 2-adic numbers, compare [Reference Gan and YuGanY, Section 9].

5 Background on Jordan algebras

Para-quadratic algebras

A (unital) para-quadratic algebra over a ring R is an R-module J together with a quadratic map $U \colon J \to \operatorname {\mathrm {End}}_R(J)$ — that is, U is an element of $\mathscr {P}^2(J, \operatorname {\mathrm {End}}_R(J))$ — called the U-operator, and a distinguished element $1_J \in J$ such that $U_{1_J} = \operatorname {\mathrm {Id}}_J$ . A homomorphism $\phi \colon J \to J'$ of para-quadratic R-algebras is an R-linear map such that $\phi (1_J) = 1_{J'}$ and $U^{\prime }_{\phi (x)} \phi (y) = \phi (U_x y)$ for all $x, y \in J$ , where $U'$ denotes the U-operator in $J'$ .

Jordan algebras

As a notational convenience, we define a linear map $J \otimes J \otimes J \to J$ denoted $x \otimes y \otimes z \mapsto \{ xyz\}$ via

(5.1) $$ \begin{align} \{ x y z \} := (U_{x+z} - U_x - U_z)y. \end{align} $$

Evidently, $\{ x y z \} = \{ z y x \}$ for all $x, y, z \in J$ . A para-quadratic R-algebra J is a Jordan R-algebra if the identities

(5.2) $$ \begin{align} U_{U_x y} = U_x U_y U_x \quad \text{and} \quad U_x \{ yxz \} = \{ (U_xy) z x \} \end{align} $$

hold for all $x, y, z \in J\otimes S$ for all $S \in {R\text {-}\textbf {alg}}$ . (Alternatively, one can define a Jordan R-algebra entirely in terms of identities concerning elements of J, avoiding the “for all $S \in {R\text {-}\textbf {alg}}$ ”, at the cost of requiring a longer list of identities (see [Reference McCrimmonMcC66, Section 1]).) Note that if J is a Jordan R-algebra, then $J \otimes T$ is a Jordan T-algebra for every $T \in {R\text {-}\textbf {alg}}$ (“Jordan algebras are closed under base change”). If J is a para-quadratic algebra and $J \otimes T$ is Jordan for some faithfully flat $T \in {R\text {-}\textbf {alg}}$ , then J is Jordan.

For x in a Jordan algebra J and $n \ge 0$ , we define the n-th power $x^n$ via

(5.3) $$ \begin{align} x^0 := 1_J, \quad x^1 := x, \quad x^n = U_x x^{n-2} \ \text{for } n \ge 2. \end{align} $$

An element $x \in J$ is invertible with inverse y if $U_x y = x$ and $U_x y^2 = 1$ [Reference McCrimmonMcC66, Section 5]. It turns out that x is invertible if and only if $U_x$ is invertible if and only if $1$ is in the image of $U_x$ ; when these hold, the inverse of x is $y = U_x^{-1} x$ , which we denote by $x^{-1}$ . It follows from (5.2) that $x,y \in J$ are both invertible if and only if $U_xy$ is invertible, and in this case, $(U_xy)^{-1} = U_{x^{-1}}y^{-1}$ .

Example 5.1. Let A be an associative and unital R-algebra. Define $U_xy := xyx$ for $x, y \in A$ . Then $\{ xyz\} = xyz + zyx$ and A endowed with this U-operator is a Jordan algebra denoted by $A^+$ . Note that for $x \in A$ and $n \ge 0$ , the n-th powers of x in A and $A^+$ are the same.

Relations with other kinds of algebras

Suppose for this paragraph and the next that 2 is invertible in R. Given a para-quadratic algebra J as in the preceding paragraph, one can define a commutative (bilinear) product $\bullet $ on J via

(5.4) $$ \begin{align} x \bullet y := \frac12 \{x1_Jy \} \quad \text{for } x, y \in J. \end{align} $$

(In the case where J is constructed from an associative algebra as in Example 5.1, one finds that $x \bullet y = \frac 12 (xy + yx)$ . If, additionally, the associative algebra is commutative, $\bullet $ equals the product in that associative algebra.) If J is Jordan, then $\bullet $ satisfies

(5.5) $$ \begin{align} (x \bullet y) \bullet (x \bullet x) = x \bullet (y \bullet (x \bullet x)), \end{align} $$

which is the axiom classically called the “Jordan identity”.

In the opposite direction, given an R-module J with a commutative product $\bullet $ with identity element $1_J$ , we obtain a para-quadratic algebra by setting

(5.6) $$ \begin{align} U_x y := 2x \bullet (x \bullet y) - (x \bullet x) \bullet y \quad \text{for } x, y \in J. \end{align} $$

If the original product satisfied the Jordan identity, then the para-quadratic algebra so obtained satisfies (5.2), that is, is a Jordan algebra in our sense (see, for example [Reference JacobsonJ69, Section 1.4]).

Definition 5.2 (hermitian matrix algebras)

Let C be a composition R-algebra and $\Gamma = {\left \langle {\gamma _1, \gamma _2, \gamma _3}\right \rangle } \in \operatorname {\mathrm {GL}}_3(R)$ . We define $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ to be the R-submodule of $\operatorname {\mathrm {Mat}}_3(C)$ consisting of elements fixed by the involution $x \mapsto \Gamma ^{-1} \bar {x}^{\intercal } \Gamma $ and with diagonal entries in R. Note that, as an R-module, $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is a sum of three copies of C and three copies of R, so it is finitely generated projective.

In the special case where 2 is invertible in R, one can define a multiplication $\bullet $ on $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ via $x \bullet y := \frac 12 (xy + yx)$ , where juxtaposition denotes the usual product of matrices in $\operatorname {\mathrm {Mat}}_3(C)$ . It satisfies the Jordan identity [Reference JacobsonJ68, p. 61, Corollary], and therefore, the U-operator defined via (5.6) makes $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ into a Jordan algebra.

6 Cubic Jordan algebras

In this section, we define cubic Jordan algebras and the closely related notion of cubic norm structure. They provide a useful alternative language for computation.

Definition 6.1. Following [Reference McCrimmonMcC69] (see [Reference Petersson and RacinePeR86a, p. 212] for the terminology), we define a cubic norm R-structure as a quadruple $\mathbf {M} = (M,1_{\mathbf {M}},\sharp ,N_{\mathbf {M}})$ consisting of an R-module M; a distinguished element $1_{\mathbf {M}} \in M$ (the base point); a quadratic map $\sharp \colon M \to M$ , written $x \mapsto x^{\sharp }$ (the adjoint) with (symmetric bilinear) polarization $x \times y := (x + y)^{\sharp } - x^{\sharp } - y^{\sharp };$ and a cubic form $N_{\mathbf {M}}\colon M \to R$ (the norm) such that the following axioms are fulfilled. Define a bilinear form $T_{\mathbf {M}}\colon M \times M \to R$ by

(6.1) $$ \begin{align} T_{\mathbf{M}}(x,y) := (\nabla_xN_{\mathbf{M}})(1_{\mathbf{M}})(\nabla_yN_{\mathbf{M}})(1_{\mathbf{M}}) - (\nabla_x\nabla_yN_{\mathbf{M}})(1_{\mathbf{M}}) \end{align} $$

(the bilinear trace), which is symmetric since the directional derivatives $\nabla _x$ , $\nabla _y$ commute [Reference RobyRoby, p. 241, Proposition II.5], and a linear form $\operatorname {\mathrm {Tr}}_{\mathbf {M}}\colon M \to R$ by

(6.2) $$ \begin{align} \operatorname{\mathrm{Tr}}_{\mathbf{M}}(x) := T_{\mathbf{M}}(x,1_{\mathbf{M}}) \end{align} $$

(the linear trace). For $\mathbf {M}$ to be a cubic norm structure, we require that the identities

(6.3) $$ \begin{align} 1_{\mathbf{M}}^{\sharp} = 1_{\mathbf{M}}, \quad N_{\mathbf{M}}(1_{\mathbf{M}}) = 1, \end{align} $$
(6.4) $$ \begin{align} 1_{\mathbf{M}} \times x = \operatorname{\mathrm{Tr}}_{\mathbf{M}}(x)1_{\mathbf{M}} - x,\; (\nabla_yN_{\mathbf{M}})(x) = T_{\mathbf{M}}(x^{\sharp},y),\:x^{\sharp\sharp} = N_{\mathbf{M}}(x)x \end{align} $$

hold in all scalar extensions $M \otimes S$ , $S\in {R\text {-}\textbf {alg}}$ .

For such a cubic norm structure $\mathbf {M}$ , we then define a U-operator by

(6.5) $$ \begin{align} U_xy := T_{\mathbf{M}}(x,y)x - x^{\sharp} \times y, \end{align} $$

which together with $1_{\mathbf {M}}$ converts the R-module M into a Jordan R-algebra $J = J(\mathbf {M})$ [Reference McCrimmonMcC69, Theorem 1]. In the sequel, we rarely distinguish carefully between the cubic norm structure $\mathbf {M}$ and the Jordan algebra $J(\mathbf {M})$ . By abuse of notation, we write $1_J = 1_{\mathbf {M}}$ , $N_J = N_{\mathbf {M}}$ , $T_J = T_{\mathbf {M}}$ , and $\operatorname {\mathrm {Tr}}_J:= \operatorname {\mathrm {Tr}}_{\mathbf {M}}$ if there is no danger of confusion, even though, in general, J does not determine $\mathbf {M}$ uniquely [Reference Petersson and RacinePeR86a, p. 216].

Definition 6.2. A Jordan R-algebra J is said to be cubic if there exists a cubic norm R-structure $\mathbf {M}$ as in Definition 6.1 such that (i) $J = J(\mathbf {M})$ and (ii) $J = M$ is a finitely generated projective R-module. With the quadratic form $S_J\colon M \to R$ defined by $S_J(x) := \operatorname {\mathrm {Tr}}_J(x^{\sharp })$ for $x \in J$ (the quadratic trace), the cubic Jordan algebra J satisfies the identities

(6.6) $$ \begin{align} (U_xy)^{\sharp} = U_{x^{\sharp}}y^{\sharp}, \quad N_J(U_xy)U_xy = N_J(x)^2N_J(y)U_xy, \end{align} $$
(6.7) $$ \begin{align} U_xx^{\sharp} = N_J(x)x, \quad U_x(x^{\sharp})^2 = N_J(x)^21_J, \end{align} $$
(6.8) $$ \begin{align} x^{\sharp} = x^2 - \operatorname{\mathrm{Tr}}_J(x)x + S_J(x)1_J \quad \text{and} \end{align} $$
(6.9) $$ \begin{align} x^3 - \operatorname{\mathrm{Tr}}_J(x)x^2 + S_J(x)x - N_J(x)1_J = 0 = x^4 - \operatorname{\mathrm{Tr}}_J(x)x^3 + S_J(x)x^2 - N_J(x)x \end{align} $$

for all $x \in J$ . For (6.6)–(6.8) and the first equation of (6.9), see [Reference McCrimmonMcC69, p. 499], while the second equation of (6.9) follows from the first, (6.7), and (6.8) via $x^4 = U_xx^2 = U_xx^{\sharp } + \operatorname {\mathrm {Tr}}_J(x)U_xx - S_J(x)U_x1_J = \operatorname {\mathrm {Tr}}_J(x)x^3 - S_J(x)x^2 + N_J(x)x$ .

Remark 6.3. Note that the second equality of (6.9) derives from the first through formal multiplication by x. But, due to the para-quadratic character of Jordan algebras, this is not a legitimate operation unless $2$ is invertible in R. In fact, cubic Jordan algebras exist that contain elements x satisfying $x^2 = 0 \neq x^3$ [Reference JacobsonJ69, 1.31–1.32].

Example 6.4 (3-by-3 matrices)

We claim that $\operatorname {\mathrm {Mat}}_3(R)^+$ is a cubic Jordan algebra, in particular, it is $J(\mathbf {M})$ for $\mathbf {M} := (\operatorname {\mathrm {Mat}}_3(R), \operatorname {\mathrm {Id}}, \sharp , \det )$ , where $\sharp $ denotes the classical adjoint. We first verify that $\mathbf {M}$ is a cubic norm structure. Computing directly from the definition (6.1), we find that $T_{\mathbf {M}}(x,y) = \operatorname {\mathrm {Tr}}_{\operatorname {\mathrm {Mat}}_3(R)}(xy)$ , where the juxtaposition on the right is usual matrix multiplication. The formulas in (6.3) are obvious. For (6.4), the first two equations can be verified directly and the third equation is a standard property of the classical adjoint, completing the proof that $\mathbf {M}$ is a cubic norm structure. Similarly, one can check directly that the U-operator defined from the cubic norm structure by (6.5) equals the U-operator defined from the usual matrix product in Example 5.1, that is, $J(\mathbf {M}) = \operatorname {\mathrm {Mat}}_3(R)^+$ .

Lemma 6.5. Let J be a cubic Jordan R-algebra and $x,y \in J$ .

  1. 1. x is invertible in J if and only if $N_J(x)$ is invertible in R. In this case

    $$\begin{align*} x^{-1} = N_J(x)^{-1}x^{\sharp} \quad \text{and} \quad N_J(x^{-1}) = N_J(x)^{-1}. \end{align*}$$
  2. 2. Invertible elements of J are unimodular.

  3. 3. $N_J(U_xy) = N_J(x)^2N_J(y)$ and $N_J(x^2) = N_J(x)^2 = N_J(x^{\sharp })$ .

Proof. (1): If $N_J(x)$ is invertible in R, then (6.7) shows that so is x, with inverse $x^{-1} = N_J(x)^{-1}x^{\sharp }$ . Conversely, assume x is invertible in J. Then $y := (x^{-1})^2$ satisfies $U_xy = 1_J$ , and (6.6) yields $1_J = N_J(U_xy)U_xy = N_J(x)^2N_J(y)1_J$ , hence

$$\begin{align*}N_J(x)^2N_J(y) = 1 \end{align*}$$

since $1_J$ is unimodular by Lemma 3.1 and (6.3). Thus, $N_J(x) \in R^{\times }$ . Before proving the final formula of (1), we deal with (2), (3).

(2) follows immediately from Lemma 3.1 combined with the first part of (1).

(3): Applying Lemma 3.2 to the polynomial law $g\colon J \times J \to \operatorname {\mathrm {End}}_R(J)$ defined by $g(x,y) := U_{U_xy}$ in all scalar extensions, we may assume that $U_xy$ is invertible. By (2), therefore, $U_xy$ is unimodular, and the first equality follows from (6.6). The second equality follows from the first for $y = 1_J$ , while in the third equality, we may again assume that x is invertible, hence, unimodular. Then (6.7) combines with the first equality to imply $N_J(x)^4 = N_J(N_J(x)x) = N_J(U_xx^{\sharp }) = N_J(x)^2N_J(x^{\sharp })$ , as desired.

Now the second equality of (1) follows from the first and (3) via

$$\begin{align*}N_J(x^{-1}) = N_J(x)^{-3}N_J(x^{\sharp}) = N_J(x)^{-1}.\end{align*}$$

Without the assumption that J is finitely generated projective as an R-module, Lemma 6.5 would be false [Reference Petersson and RacinePeR85, Theorem 10].

Example 6.6. We endow the R-module $M := \operatorname {\mathrm {Her}}_3(C,\Gamma )$ from Definition 5.2 with a cubic norm R-structure $\mathbf {M}= (M,1_{\mathbf {M}},\sharp ,N_{\mathbf {M}})$ , where $1_{\mathbf {M}}$ is the 3-by-3 identity matrix. An element $x \in \operatorname {\mathrm {Her}}_3(C, \Gamma )$ may be written as

$$\begin{align*}x = \left( \begin{smallmatrix} \alpha_1 & \gamma_2 c_3 & \gamma_3 \bar{c}_2 \\ \gamma_1 \bar{c}_3 & \alpha_2 & \gamma_3 c_1 \\ \gamma_1 c_2 & \gamma_2 \bar{c}_1 & \alpha_3 \end{smallmatrix} \right) \end{align*}$$

for $\alpha _i \in R$ and $c_i \in C$ . Because three of the entries are determined by symmetry, we may denote such an element by

(6.10) $$ \begin{align} x := \sum\nolimits_{i=1}^3 \left( \alpha_i \varepsilon_i + \delta_i^{\Gamma}(c_i) \right), \end{align} $$

where $\varepsilon _i$ has a 1 in the $(i, i)$ entry and zeros elsewhere, and $\delta _i^{\Gamma }(c)$ has $\gamma _{i+2}c$ in the $(i+1, i+2)$ entry — where the symbols $i+1$ and $i+2$ are taken modulo 3 — and zeros in the other entries not determined by symmetry. In the literature on Jordan algebras, one finds the notation $c[(i+1)(i+2)]$ for what we denote $\delta _i^{\Gamma }(c)$ . We define the adjoint $\sharp $ by

$$\begin{align*}x^{\sharp} := \sum\nolimits_{i=1}^3\Big(\big(\alpha_{i+1}\alpha_{i+2} - \gamma_{i+1}\gamma_{i+2}n_C(c_i)\big)\varepsilon_i + \delta_i^{\Gamma}\big({-\alpha_ic_i} + \gamma_i\overline{c_{i+1}c_{i+2}}\big)\Big) \end{align*}$$

with indices modulo (mod) 3, and the norm $N_{\mathbf {M}}$ by

(6.11) $$ \begin{align} N_{\mathbf{M}}(x) := \alpha_1\alpha_2\alpha_3 - \sum\nolimits_{i=1}^3\gamma_{i+1}\gamma_{i+2}\alpha_in_C(c_i) + \gamma_1\gamma_2\gamma_3\mathrm{Tr}_C(c_1c_2c_3) \end{align} $$

in all scalar extensions, where the last summand on the right of (6.11) is unambiguous since $\mathrm {Tr}_C((c_1c_2)c_3) = \mathrm {Tr}_C(c_1(c_2c_3))$ [Reference McCrimmonMcC85, Theorem 3.5]. By [Reference McCrimmonMcC69, Theorem 3], $\mathbf {M}$ is indeed a cubic norm structure. The corresponding cubic Jordan algebra will again be denoted by $J := \operatorname {\mathrm {Her}}_3(C,\Gamma ) := J(\mathbf {M})$ .

(In case 2 is invertible in R, the commutative product $\bullet $ on $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ defined from the U-operator by (5.4) equals the product $x \bullet y := \frac 12 (xy + yx)$ from Definition 5.2. In order to see this, it suffices to note that the square of $x \in \operatorname {\mathrm {Her}}_3(C, \Gamma )$ as defined in (5.3) is the same as the square of x in the matrix algebra $\operatorname {\mathrm {Mat}}_3(C)$ . This in turn follows immediately from (6.8), (6.11), and the definition of the adjoint.)

For x as above and $y = \sum (\beta _i\varepsilon _i + \delta _i^{\Gamma }(d_i))$ , with $\beta _i \in R$ , $d_i \in C$ , evaluating the bilinear trace at $x,y$ yields

(6.12) $$ \begin{align} T_J(x,y) = \sum\nolimits_{i=1}^3 \big(\alpha_i\beta_i + \gamma_{i+1}\gamma_{i+2}n_C(c_i, d_i)\big). \end{align} $$

Since the bilinearization of $n_C$ is regular, so is $T_J$ .

Here is an important special case.

Definition 6.7. For the special case where $\Gamma = \operatorname {\mathrm {Id}}$ , we define $\operatorname {\mathrm {Her}}_3(C) := \operatorname {\mathrm {Her}}_3(C, \operatorname {\mathrm {Id}})$ and write $\delta _i$ for $\delta _i^{\Gamma }$ . It can be useful to write elements of $\operatorname {\mathrm {Her}}_3(C)$ as

$$\begin{align*}\left( \begin{smallmatrix} \alpha_1 & c_3 & \cdot \\ \cdot & \alpha_2 & c_1 \\ c_2 & \cdot & \alpha_3 \end{smallmatrix} \right), \end{align*}$$

where $\cdot $ denotes an entry that is omitted because it is determined by symmetry. As an example of the triple product defined from (5.1) and (6.5), we mention that for $x = \sum \alpha _i \varepsilon _i$ diagonal, we have

(6.13) $$ \begin{align} \{ \delta_i(a) \delta_{i+1}(b) x \} = \delta_{i+2}(\overline{ab}) \alpha_i \quad \text{and} \quad \{ \delta_{i+1}(b) \delta_i(a) x \} = \delta_{i+2}(\overline{ab}) \alpha_{i+1} \end{align} $$

for $i \in 1, 2, 3$ taken mod 3 and $a, b \in C$ .

Note that, for the Jordan algebra $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ , if we multiply $\Gamma $ by an element of $R^{\times }$ or any entry in $\Gamma $ by the square of an element of $R^{\times }$ , we obtain an algebra isomorphic to the original. Therefore, replacing $\Gamma $ by ${\left \langle { (\det \Gamma )^{-1} \gamma _1, (\det \Gamma )^{-1} \gamma _2, (\det \Gamma ) \gamma _3}\right \rangle }$ does not change the isomorphism class of $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ , and we may assume that $\gamma _1 \gamma _2 \gamma _3 = 1$ .

Example 6.8. When studying the Jordan R-algebras $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ in the special case $R = \mathbb {R}$ , the preceding paragraph shows that it is sufficient to consider two choices for $\Gamma $ , namely, ${\left \langle {1, s, s}\right \rangle }$ for $s = \pm 1$ . We compute $T_{\operatorname {\mathrm {Her}}_3(C, \Gamma )}$ for each choice of C and $\Gamma $ . Regular symmetric bilinear forms over $\mathbb {R}$ are classified by their dimension and signature (an integer), so it suffices to specify the signature. If $C = \mathbb {R}$ , $\mathbb {C}$ , $\mathbb {H}$ , or $\mathbb {O}$ , the signature of $n_C$ is $2^r$ for $r = 0$ , 1, 2, 3, respectively. By (6.12), $T_J$ has signature $3(1 + 2^r)$ for $J = \operatorname {\mathrm {Her}}_3(C)$ and $3 - 2^r$ for $J = \operatorname {\mathrm {Her}}_3(C, {\left \langle {1, -1, -1}\right \rangle })$ . For $C$ the split composition algebra of rank $2^r$ for $r = 1$ , 2, or 3 and any $\Gamma$ , the signature of $n_C$ is 0 and the signature of $T_{\operatorname {\mathrm {Her}}_3(C, \Gamma)}$ is 3.

Remark 6.9. Alternatively, one could define the Jordan algebra structure on $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for an arbitrary ring R without referring to cubic norm structures as follows. Writing out the formulas for the U-operator from Definition 5.2 in case $R = \mathbb {Q}$ , one finds that the formulas do not involve any denominators other than $\gamma _i$ terms and therefore make sense for any R regardless of whether 2 is invertible. This makes $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ a para-quadratic algebra. Because it is a Jordan algebra in case $R = \mathbb {Q}$ as in Definition 5.2, we conclude that $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is a Jordan algebra with no hypothesis on R by extension of identities [Reference BourbakiBouA2, Section IV.2.3, Theorem 2]. This alternative definition gives the same objects but is much harder to work with.

7 Albert algebras are Freudenthal algebras are Jordan algebras

Definition 7.1. A split Freudenthal R-algebra is a Jordan algebra $\operatorname {\mathrm {Her}}_3(C)$ as in Definition 6.7 for some split composition R-algebra C. Because split composition algebras are determined up to isomorphism by their rank function, so are split Freudenthal algebras.

A para-quadratic R-algebra J is a Freudenthal algebra if $J \otimes S$ is a split Freudenthal S-algebra for some faithfully flat $S \in {R\text {-}\textbf {alg}}$ . It is immediate that every Freudenthal algebra is a Jordan algebra. Since every split Freudenthal R-algebra is finitely generated projective as an R-module for every R, the same is true for every Freudenthal R-algebra J, and by the same reasoning, we see that the identity element $1_J$ is unimodular. Because the rank of a composition algebra takes values in $\{ 1, 2, 4, 8 \}$ , the rank of a Freudenthal algebra takes values in $\{ 6, 9, 15, 27 \}$ .

We are now prepared to define the objects named in the title of this paper.

Definition 7.2. An Albert R-algebra is a Freudenthal R-algebra of rank 27.

We continue to prove results about Freudenthal algebras, rather than merely Albert algebras. The extra generality comes at a low cost.

Proposition 7.3. For every composition R-algebra C and every $\Gamma \in \operatorname {\mathrm {GL}}_3(R)$ , $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is a Freudenthal algebra.

Proof. Replacing R with $R_e$ as in (4.4), we may assume that C has constant rank. There is a faithfully flat $S \in {R\text {-}\textbf {alg}}$ such that $C \otimes S$ is a split composition algebra.

Consider $T := S[t_1, t_2, t_3] / (t_1^2 - \gamma _1, t_2^2 - \gamma _2, t_3^2 - \gamma _3)$ . It is a free S-module, so faithfully flat. Then $\operatorname {\mathrm {Her}}_3(C, \Gamma ) \otimes T$ is isomorphic to $\operatorname {\mathrm {Her}}_3(C \otimes T)$ as Jordan algebras, and the latter is a split Freudenthal algebra.

A Freudenthal algebra is said to be reduced if it is isomorphic to $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for some C and $\Gamma $ .

Example 7.4. Let J be a Freudenthal R-algebra. If $x \in J$ has $U_x = \operatorname {\mathrm {Id}}_J$ , then $x = \zeta 1_J$ for some $\zeta \in R$ such that $\zeta ^2 = 1$ . To see this, first suppose that J is $\operatorname {\mathrm {Her}}_3(C)$ for some composition algebra C and write $x = \sum (\alpha _i \varepsilon _i + \delta _i(c_i))$ for $\alpha _i \in R$ and $c_i \in C$ . We find

$$\begin{align*}U_x \varepsilon_i = \alpha_i^2 \varepsilon_i + \delta_{i+2}(\alpha_i c_{i+2}) + \cdots \end{align*}$$

for each i, so $\alpha _i^2 = 1$ and $c_{i+2} = 0$ for all i. Then

$$\begin{align*}U_x \delta_i(1_C) = \delta_i(\alpha_{i+1} \alpha_{i+2} 1_C). \end{align*}$$

Since $1_C$ is unimodular, $\alpha _{i+1} \alpha _{i+2} = 1$ for all i, proving the claim for this J.

For general J, let $S \in {R\text {-}\textbf {alg}}$ be faithfully flat such that $J \otimes S$ is split. Then $x \in J$ maps to an element of $R1_J \otimes S \subseteq J \otimes S$ and so belongs to $R1_J \subseteq J$ . Since $U_{\zeta 1_J} = \zeta ^2 \operatorname {\mathrm {Id}}_J$ for $\zeta \in R$ , the claim follows.

The following result is well known when R is a field or perhaps a local ring (see, for example [Reference PeterssonPe19, Proposition 20]). We impose no hypothesis on R.

Proposition 7.5. Suppose C is a split composition R-algebra of constant rank at least 2, that is, C is $R \times R$ , $\operatorname {\mathrm {Mat}}_2(R)$ , or $\operatorname {\mathrm {Zor}}(R)$ . Then $\operatorname {\mathrm {Her}}_3(C, \Gamma ) \cong \operatorname {\mathrm {Her}}_3(C)$ for all $\Gamma $ .

Proof. Define $\gamma _i$ via $\Gamma = {\left \langle {\gamma _1, \gamma _2, \gamma _3}\right \rangle }$ . We may assume $\gamma _1\gamma _2\gamma _3 = 1$ . Since $n_C$ is universal, there are invertible $p, q \in C$ such that $\gamma _2 = n_C(q^{-1})$ and $\gamma _3 = n_C(p^{-1})$ , so $\gamma _1 = n_C(pq)$ . We define $C^{(p,q)}$ to be a not-necessarily-associative R-algebra with the same underlying R-module structure and with multiplication $\cdot _{(p,q)}$ defined by

$$\begin{align*}x \cdot_{(p,q)} y := (xp)(qy), \end{align*}$$

where the multiplication on the right is the multiplication in C. Certainly $(pq)^{-1}$ is an identity element in $C^{(p,q)}$ . The algebra $C^{(p,q)}$ is called an isotope of C and is studied in [Reference McCrimmonMcC71a], where it is proved to be alternative. One checks that it is a composition algebra with quadratic form $n_{C^{(p,q)}} = n_C(pq) n_C$ (see [Reference McCrimmonMcC71a, Proposition 5] for a more general statement in case R is a field).

Define $\phi \!: \operatorname {\mathrm {Her}}_3(C^{(p,q)}) \to \operatorname {\mathrm {Her}}_3(C, \Gamma )$ via $\phi (\sum x_i \varepsilon _i + \delta _i(c_i)) = \sum x_i \varepsilon _i + \delta _i^{\Gamma }(c^{\prime }_i)$ , where

$$\begin{align*}c^{\prime}_1 = (pq) c_1 (pq), \quad c^{\prime}_2 = c_2p, \quad \text{and} \quad c^{\prime}_3 = qc_3. \end{align*}$$

It is evidently an isomorphism of R-modules, and one checks that it is an isomorphism of Jordan algebras, compare [Reference McCrimmonMcC71a, Theorem 3]. Therefore, we are reduced to verifying that $C^{(p,q)}$ is split.

If C is associative, then the R-linear map

$$\begin{align*}L_{pq} \colon C^{(p,q)} \to C\text{ such that} \quad L_{pq}(x) = pqx \end{align*}$$

is an isomorphism of R-algebras. So assume $C = \operatorname {\mathrm {Zor}}(R)$ .

At the beginning, when we chose p and q, we were free to pick $\xi _i, \eta _i \in R^{\times }$ such that $p = \left ( \begin {smallmatrix} \xi _1 & 0 \\ 0 & \xi _2 \end {smallmatrix} \right )$ and $q = \left ( \begin {smallmatrix} \eta _1 & 0 \\ 0 & \eta _2 \end {smallmatrix} \right )$ . Let $A \in \operatorname {\mathrm {Mat}}_3(R)$ be any matrix such that $\det A = (\xi _1 \xi _2^2 \eta _1^2 \eta )^{-1}$ and put $B := \xi _2 \eta _1 (A^{\sharp })^{\intercal }$ , where $\sharp $ denotes the classical adjoint. With $\zeta _i := (\xi _i \eta _i)^{-1}$ , one checks, using the formula $(Sx) \times (Sy) = (S^{\sharp })^{\intercal }(x \times y)$ for $\times $ the usual cross product in $R^3$ , that the assignment

$$\begin{align*}\left( \begin{smallmatrix} \alpha_1 & u_1 \\ u_2 & \alpha_2 \end{smallmatrix} \right) \mapsto \left( \begin{smallmatrix} \zeta_1 \alpha_1 & Au_1 \\ Bu_2 & \zeta_2 \alpha_2 \end{smallmatrix} \right) \end{align*}$$

defines an isomorphism $C \xrightarrow {\sim } C^{(p,q)}$ .

8 The ideal structure of Freudenthal algebras

It is a standard exercise to show that every (two-sided) ideal in the matrix algebra $\operatorname {\mathrm {Mat}}_n(R)$ is of the form $\operatorname {\mathrm {Mat}}_n(\mathfrak {a})$ for some ideal $\mathfrak {a}$ in R. More generally, every ideal in an Azumaya R-algebra A is of the form $\mathfrak {a} A$ some ideal $\mathfrak {a}$ of R [Reference Knus and OjangurenKnO, p. 95, Corollary III.5.2].

A similar result holds for every octonion R-algebra C: Every one-sided ideal in C is a two-sided ideal that is stable under the involution on C. The maps $I \mapsto I \cap R$ and are bijections between the set of ideals of C and ideals in R. See [Reference PeterssonPe21, Section 4] for a proof in this generality and the references therein for earlier results of this type going back to [Reference MahlerMa].

We now prove a similar result for Freudenthal algebras.

Definition 8.1. An ideal in a para-quadratic R-algebra J is the kernel of a homomorphism, that is, an R-submodule I such that

$$\begin{align*}U_I J + U_J I + \{ J J I \} = I, \end{align*}$$

where we have written $U_I J$ for the R-span of $U_x y$ with $x \in I$ and $y \in J$ . (This is sometimes written with a $\subseteq $ instead of $=$ , but the two are equivalent since $U_J I \supseteq U_{1_J} I = I$ .) An R-submodule I is an outer ideal if

(8.1) $$ \begin{align} U_J I + \{ J J I \} = I. \end{align} $$

Here are some observations about outer ideals:

  1. 1. Every ideal is an outer ideal.

  2. 2. If 2 is invertible in R, then for every $x \in I$ and $y \in J$ , $U_x y = \frac 12 \{ xyx \} \in \{ J J I \}$ , so the notions of ideal and outer ideal coincide, and both agree with the notion of ideal for the commutative bilinear product $\bullet $ defined in (5.4).

  3. 3. For every ideal $\mathfrak {a}$ in R, the R-submodule $\mathfrak {a} J$ is an ideal of J.

  4. 4. If $1_J$ is unimodular, then for every outer ideal I of J, $I \cap R1_J$ is an ideal in R, for the trivial reason that I is an R-module.

  5. 5. If $\mathfrak {a}$ is an ideal in R and $1_J$ is unimodular, then $\mathfrak {a} 1_J = (\mathfrak {a} J) \cap R 1_J$ . The containment $\subseteq $ is clear. To see the opposite containment, suppose $\alpha 1_J \in \mathfrak {a} J \cap R1_J$ for some $\alpha \in R$ and write $\alpha 1_J = \sum \alpha _i y_i$ with $\alpha _i \in \mathfrak {a}$ and $y_i \in J$ . There is some R-linear $\lambda \!: J \to R$ such that $\lambda (1_J) = 1$ . Then $\alpha = \lambda (\alpha 1_J) = \sum \alpha _i \lambda (y_i)$ is in $\mathfrak {a}$ .

Theorem 8.2. Let J be a Freudenthal R-algebra. Every outer ideal of J is an ideal. The maps $I \mapsto I \cap R1_J$ and are bijections between the set of outer ideals of J and the set of ideals of R.

Proof. It suffices to show that the stated maps are bijections, because then observation (3) implies that every outer ideal is of the form $\mathfrak {a} J$ and therefore an ideal. In view of (5) (noting that $1_J$ is unimodular), it suffices to verify that $(I \cap R1_J)J = I$ for every outer ideal I. First suppose that $J = \operatorname {\mathrm {Her}}_3(C)$ for some composition R-algebra C and write $\mathfrak {a} := I \cap R1_J$ . The Peirce projections relative to the diagonal frame of J, that is, $U_{\varepsilon _i}$ and $x \mapsto \{\varepsilon _j x \varepsilon _l \}$ for $i, j, l = 1, 2, 3$ [Reference McCrimmonMcC66, p. 1074] stabilize I, and we find

$$\begin{align*}I = \sum_i (I \cap R\varepsilon_i) + (I \cap \delta_i(C)). \end{align*}$$

Set $B := \{ c \in C \mid \delta _1(c) \in I \}$ . We claim that B is an ideal in C. Note that $U_{\delta _1(1_C)} \delta _1(b) = \delta _1(\bar {b})$ , so B is stable under the involution.

We leverage (6.13). Repeatedly applying this with $a = 1_C$ and using that B is stable under the involution, we conclude that $\delta _i(B) = I \cap \delta _i(C)$ for all i. For $c \in C$ and $b \in B$ , I contains $\{ 1_J \delta _2(\bar {c}) \delta _1(\bar {b}) \} = \delta _3(c b)$ , so $cB \subseteq B$ , that is, B is an ideal in C and therefore $B = \mathfrak {a} C$ for some ideal $\mathfrak {a}$ of R.

For $c \in C$ , I contains $\{ \delta _i(1_C) \varepsilon _{i+1} \delta _i(\mathfrak {a} c) \} = \operatorname {\mathrm {Tr}}_C(\mathfrak {a} c) \varepsilon _{i+2}$ . Since $\operatorname {\mathrm {Tr}}_C$ is surjective, $\mathfrak {a} \varepsilon _j \subseteq I$ for all j.

In the other direction, if $\alpha _i \varepsilon _i \in I$ , then so is

$$\begin{align*}\{ \delta_{i+1}(1_C) 1_J (\alpha_i \varepsilon_i) \} = \delta_{i+1}(\alpha_i 1_C). \end{align*}$$

It follows that $I \cap R\varepsilon _i = \mathfrak {a} R$ for all i and, in particular, $I \cap R1_J = \mathfrak {a} R$ and $I = \mathfrak {a} J$ .

We now treat the general case. Suppose I is an outer ideal in a Freudenthal R-algebra J. There is a faithfully flat $S \in {R\text {-}\textbf {alg}}$ , such that $J \otimes S$ is a split Freudenthal algebra. We have

$$\begin{align*}((I \cap R1_J) J) \otimes S = ((I \otimes S) \cap S1_J) (J \otimes S) = I \otimes S, \end{align*}$$

where the first equality is because S is flat and the second is by the previous case, since $I \otimes S$ is an outer ideal. It follows that $I = (I \cap R1_J)J$ as desired.

Remark 8.3. In the proof above, the inclusion $(I \cap R1_J) J \subseteq I$ could instead have been argued as follows. Define $\operatorname {\mathrm {Sq}}(J)$ as the R-submodule of J generated by $x^2$ for $x \in J$ . Since $1_J$ is unimodular, one finds that $(I \cap R1_J) \operatorname {\mathrm {Sq}}(J) \subseteq I$ . Then, one argues that $\operatorname {\mathrm {Sq}}(J) = J$ for a split Freudenthal algebra, and that $\operatorname {\mathrm {Sq}}(J \otimes S) = \operatorname {\mathrm {Sq}}(J) \otimes S$ for all flat $S \in {R\text {-}\textbf {alg}}$ .

Corollary 8.4. Let $\phi \colon J \to A$ be a homomorphism of para-quadratic R algebras. If J is a Freudenthal algebra and $1_A$ is unimodular in A, then $\phi $ is injective.

In particular, the corollary applies to every homomorphism between Freudenthal R-algebras.

Proof. The kernel of $\phi $ is an ideal of J and therefore $\mathfrak {a} J$ for some ideal $\mathfrak {a}$ of R. For $\alpha \in \mathfrak {a}$ , we have $0 = \phi (\alpha 1_J) = \alpha \phi (1_J) = \alpha 1_A$ , so $\alpha = 0$ because $1_{A}$ is unimodular.

Because a Freudenthal algebra J is a projective R-module of rank $\ge 3$ , the corollary says that there is no homomorphism of para-quadratic R-algebras $J \to R^+$ . This might be stated as J is not augmented or J has no counit.

A para-quadratic algebra J is said to be simple if the underlying module is not the zero module and if every ideal in J equals 0 or J. Theorem 8.2 immediately gives:

Corollary 8.5. If J is a Freudenthal R-algebra and R is a field, then J is simple. $\Box $

Remark 8.6. There is also the notion of an inner ideal in a Jordan algebra (see [Reference McCrimmonMcC71b, Theorem 8] for a description of them for $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ ). The inner ideals are related to the projective homogeneous varieties associated with the group of isometries described in Section 15 and “outer automorphisms” relating these varieties (see [Reference RacineRac] and [Reference Carr and GaribaldiCarrG]).

9 Groups of type $\mathsf {F}_4$ and $\mathsf {C}_3$

In the following, for a Jordan R-algebra J, we write $\operatorname {\mathrm {Aut}}(J)$ for the ordinary group of R-linear automorphisms of J and $\mathbf {Aut}(J)$ for the functor from ${R\text {-}\textbf {alg}}$ to groups such that $S \mapsto \operatorname {\mathrm {Aut}}(J \otimes S)$ . Recall that for every simple root datum, there is a unique simple group scheme over $\mathbb {Z}$ called a Chevalley group [Reference Demazure and GrothendieckDemG, Corollary XXIII.5.4], and every split simple algebraic group over a field is obtained from a unique Chevalley group by base change [Reference MilneMilne, Section 23g].

Lemma 9.1. Let J be a Freudenthal algebra of rank 15 or 27 over a ring R. Then $\mathbf {Aut}(J)$ is a semisimple R-group scheme that is adjoint (i.e., its center is the trivial group scheme). Its root system has type $\mathsf {C}_3$ if J has rank 15 and type $\mathsf {F}_4$ if J has rank 27. If J is the split Freudenthal algebra, then the group scheme $\mathbf {Aut}(J)$ is obtained from the Chevalley group over $\mathbb {Z}$ by base change.

Proof. First suppose that $R = \mathbb {Z}$ and J is split. If J has rank 15, then the proof of 14.19 in [Reference SpringerSp73] shows that the automorphisms of $J \otimes F$ for every field F are exactly the automorphisms of the algebra $\operatorname {\mathrm {Mat}}_6(F)$ with the split symplectic involution, which is the split adjoint group $\operatorname {\mathrm {PGSp}}_6$ of type $\mathsf {C}_3$ . For J of rank 27, $\mathbf {Aut}(J) \times F$ is split of type $\mathsf {F}_4$ by [Reference JacobsonJ71, Section 6] (written for Lie algebras), [Reference FreudenthalFr85, Satz 4.11] (written for $\mathbb {R}$ ), [Reference Springer and VeldkampSpV, Theorem 7.2.1] (if $\operatorname {\mathrm {char}} F \ne 2, 3$ ), or [Reference SpringerSp73, 14.24] in general.

Note that $\mathbf {Aut}(J) \times F$ is connected and smooth as a group scheme over F, and $\mathbf {Aut}(J)$ is finitely presented (because $\mathbb {Z}$ is noetherian and J is a finitely generated module), so it follows by [Reference Gan and YuGanY, Proposition 6.1] or [Reference Alsaody and GilleAlsG, Lemma B.1] that $\mathbf {Aut}(J)$ is smooth as a scheme over the Dedekind domain $\mathbb {Z}$ . In summary, $\mathbf {Aut}(J)$ is semisimple and adjoint of the specified type. Moreover, because $\mathbf {Aut}(J) \times \mathbb {Q}$ is split, $\mathbf {Aut}(J)$ is a Chevalley group [Reference ConradConrad, Theorem 1.4].

In the case of general R and J, let $S \in {R\text {-}\textbf {alg}}$ be faithfully flat such that $J \otimes S$ is split. Then $\mathbf {Aut}(J) \times S$ is semisimple adjoint of the specified type. Certainly, $\mathbf {Aut}(J)$ is also smooth. Moreover, for each $\mathfrak {p} \in \operatorname {\mathrm {Spec}} R$ , there is a $\mathfrak {q} \in \operatorname {\mathrm {Spec}} S$ such that $\mathfrak {q} \cap R = \mathfrak {p}$ . Then the field of fractions $R(\mathfrak {p})$ of $R/\mathfrak {p}$ embeds in the field $S(\mathfrak {q})$ , so the algebraic closure $\overline {R(\mathfrak {p})}$ includes in the algebraic closure $\overline {S(\mathfrak {q})}$ . Because $\mathbf {Aut}(J) \times \overline {S(\mathfrak {q})}$ is adjoint semisimple of the specified type and this property is unchanged by replacing one algebraically closed field by a smaller one, the same holds over $\overline {R(\mathfrak {p})}$ . Since this holds for every $\mathfrak {p}$ , the claim is verified.

Remark 9.2. In case R is a field, the automorphism group of the split Freudenthal algebra of rank 6 or 9 can be deduced in a similar manner, referring to 14.17 and 14.16 in [Reference SpringerSp73]. The automorphism group of the split Freudenthal algebra of rank 9 is $\operatorname {\mathrm {PGL}}_3 \rtimes \mathbb {Z}/2$ . The automorphism group of the split Freudenthal algebra of rank 6 is the special orthogonal group of the quadratic form $x^2 + y^2 + z^2$ , that is, the group commonly denoted $\mathrm {SO}(3)$ . In particular, it is not smooth when R is a field of characteristic 2, and indeed, one can give examples of Freudenthal algebras of rank 6 over a field F of characteristic 2 that are not split by any separable field extension of F. (From this, it follows that such an algebra is not split by any étale F-algebra R.)

Suppose J, $J_0$ are Jordan R-algebras and there is an fppf $S \in {R\text {-}\textbf {alg}}$ and an isomorphism $f \colon J \otimes S \to J_0 \otimes S$ . The isomorphism f gives a class in $H^1(S/R, \mathbf {Aut}(J_0))$ that depends only on the isomorphism class of J as a Jordan R-algebra. Analogous statements apply when the role of Jordan algebras is replaced with affine group schemes. This is the source of the diagonal arrows in the theorem below.

Theorem 9.3. Let $J_0$ be a Freudenthal R-algebra of rank $r = 15$ or 27. In the diagram

all arrows are bijections that are functorial in R.

Proof. The top arrow has the claimed codomain by Lemma 9.1.

Every Freudenthal algebra is split by some faithfully flat R-algebra by definition, so the discussion preceding the theorem of the statement yields the lower left arrow.

As another consequence of Lemma 9.1, the conjugation map $\mathbf {Aut}(J_0) \to \mathbf {Aut}(\mathbf {Aut}(J_0))$ is an isomorphism that we use to identify $H^1(R, \mathbf {Aut}(J_0)) \xrightarrow {\sim } H^1(R, \mathbf {Aut}(\mathbf {Aut}(J_0)))$ . Every semisimple group scheme is split by some faithfully flat R-algebra (even an étale cover) [Reference Demazure and GrothendieckDemG, Corollary XXIV.4.1.6], and the remarks before the statement of the theorem now define the lower right arrow.

The facts that the diagram commutes and that the diagonal arrows are bijective are general features of the machinery of descent as in [Reference WaterhouseWa, Theorem 17.6] or see [Reference Calmès and FaselCalF, 2.2.4.5].

The machinery of descent shows a more refined statement, where each of the boxes in the theorem are replaced by groupoids and the arrows are equivalences of groupoids. In that statement, the bottom box is replaced by the groupoid of $\mathbf {Aut}(J_0)$ -torsors and the diagonal arrows are provided by [Reference GiraudGir, p. 151, Theorem III.2.5.1].

In the theorem, the set $H^1(R, \mathbf {Aut}(J_0))$ is naturally a pointed set and the bijections are actually of pointed sets, where the distinguished elements are $J_0$ in the upper left and $\mathbf {Aut}(J_0)$ in the upper right.

In case R is a field of characteristic different from 2, 3 and $r = 27$ , the theorem goes back to [Reference HijikataHij]. Or see [Reference Knus, Merkurjev, Rost and TignolKnMRT, 26.18]. The statement of the theorem is similar to various statements over a field that can be found in [Reference SerreSerre, Section III.1].

Corollary 9.4. For each Freudenthal R-algebra J of rank 15 or 27, there is an étale cover $S \in {R\text {-}\textbf {alg}}$ such that $J \otimes S$ is a split Freudenthal algebra.

Proof. Let $J_0$ be the split Freudenthal R-algebra of the same rank as J. The image $\mathbf {Iso}(J, J_0)$ of J in $H^1(R, \mathbf {Aut}(J_0))$ is an $\mathbf {Aut}(J_0)$ -torsor. Since $\mathbf {Aut}(J_0)$ is smooth (Lemma 9.1), there is an étale cover of R that trivializes $\mathbf {Iso}(J, J_0)$ .

Note that exactly the same kind of argument gives analogues of Lemma 9.1 and Theorem 9.3 for composition algebras, where $r = 4$ or 8, and the group is of type $\mathsf {A}_1$ or $\mathsf {G}_2$ , respectively.

10 Generic minimal polynomial of a Freudenthal algebra

Polynomials with polynomial-law coefficients

Let J be a Jordan R-algebra, $\mathscr {P}(J, R)$ the R-algebra of polynomial laws from J to R, and t a variable. Consider a polynomial $\mathbf {p}(t) = \sum _{i=0}^n f_it^i$ with $f_i \in \mathscr {P}(J, R)$ for $0 \leq i \leq n$ . For $S \in {R\text {-}\textbf {alg}}$ , $x \in J \otimes S$ , we have $\mathbf {p}(t,x) := \sum _{i=0}^nf_{iS}(x)t^i \in S[t]$ , and we define

$$\begin{align*}\mathbf{p}(x,x) := \sum_{i=0}^nf_{iS}(x)x^i \quad \in J \otimes S. \end{align*}$$

The algebra J is said to satisfy $\mathbf {p}$ if $\mathbf {p}(x,x) = 0 = (t\mathbf {p})(x,x)$ for all $S \in {R\text {-}\textbf {alg}}$ and $x \in J \otimes S$ . Note that the second equation follows from the first if $2$ is invertible in R but not in general (see Remark 6.3).

The generic minimal polynomial

Let $J := \operatorname {\mathrm {Her}}_3(C, \Gamma )$ as in Example 6.6. With a variable t, we recall from (6.9) that J satisfies the monic polynomial

(10.1) $$ \begin{align} \mathbf{m}_J = t^3 - \operatorname{\mathrm{Tr}}_J\cdot t^2 + S_J\cdot t - N_J \quad \in \mathscr{P}(J, R)[t]. \end{align} $$

More precisely, by [Reference LoosLo, 2.4(b)], J is generically algebraic of degree $3$ in the sense of [Reference LoosLo, 2.2] and $\mathbf {m}_J$ is the generic minimal polynomial of J, that is, the unique monic polynomial in $\mathscr {P}(J,R)[t]$ of minimal degree satisfied by J [Reference LoosLo, 2.7]. It follows that the Jordan algebra J determines the polynomial $\mathbf {m}_J$ uniquely. In particular, the generic norm $N_J$ , the generic trace $T_J$ (or $\operatorname {\mathrm {Tr}}_J$ ) and, in fact, the cubic norm structure underlying J in the sense of Definition 6.2 are uniquely determined by J as a Jordan algebra. By faithfully flat descent, every Freudenthal algebra J has a uniquely determined generic minimal polynomial of the form (10.1), and a uniquely determined underlying cubic norm structure. We conclude:

Proposition 10.1. Every Freudenthal algebra J is a cubic Jordan algebra and the bilinear trace form $T_J$ is regular. $\Box $

The preceding discussion shows that, for a Freudenthal R-algebra J, the Jordan algebra structure of J alone (ignoring that J is a cubic Jordan algebra) determines the bilinear form $T_J$ . (For example, the 11 Freudenthal $\mathbb {R}$ -algebras discussed in Example 6.8 have distinct trace forms and therefore are distinct.) When R is a field of characteristic $\ne 2, 3$ and J and $J'$ are reduced Freudenthal algebras, Springer proved that the converse also holds, that is, $J \cong J'$ if and only if $T_J \cong T_{J'}$ [Reference Springer and VeldkampSpV, Theorem 5.8.1]. We do not use Springer’s result in this paper.

The following result can also be found in [Reference PeterssonPe19, Corollary 18(b)], based on the definition of Albert algebra appearing there.

Lemma 10.2. Let J and $J'$ be Freudenthal R-algebras. An R-linear map $\phi \!: J \to J'$ is an isomorphism of para-quadratic algebras if and only if $\phi $ is surjective, $\phi (1_J) = 1_{J'}$ , and $N_{J'} = N_J \phi $ as polynomial laws.

Proof. The “only if” direction follows from the uniqueness of the generic minimal polynomial as in (10.1), so we show “if”. The equality $N_{J'} = N_J \phi $ of polynomial laws and the definition of the directional derivative in Section 3 gives formulas, such as

$$\begin{align*}\nabla_y N_J(x) = \nabla_{\phi(y)} N_{J'}(\phi(x)). \end{align*}$$

Since $\phi (1_J) = 1_{J'}$ , the definition of the bilinear forms $T_J$ and $T_{J'}$ in (6.1) give:

$$\begin{align*}T_{J'}(\phi (x), \phi (y)) = T_J(x, y) \end{align*}$$

for all $x, y$ . Therefore, on the one hand we have

$$\begin{align*}\nabla_y N_J(x) = T_J(x^{\sharp}, y) = T_{J'}(\phi(x^{\sharp}), \phi(y)). \end{align*}$$

On the other hand, we have

$$\begin{align*}\nabla_y N_J(x) = \nabla_{\phi(y)} N_{J'}(\phi(x)) = T_{J'}((\phi(x))^{\sharp}, \phi(y)). \end{align*}$$

Therefore, $\phi (x^{\sharp }) = \phi (x)^{\sharp }$ for all x. In summary, $\phi $ commutes with $\sharp $ and preserves $T_J$ . Therefore, by (6.5), $\phi $ is a homomorphism of Jordan algebras.

Suppose that x is in $\ker \phi $ . Then for all $y \in J$ , $T_J(x, y) = T_{J'}(\phi (x), \phi (y)) = 0$ , so $x = 0$ since the bilinear form $T_J$ is regular. Since $\phi $ is both surjective and injective, it is an isomorphism.

Example 10.3. We claim that $\operatorname {\mathrm {Her}}_3(R \times R)$ , the split Freudenthal algebra of rank 9, is isomorphic to $\operatorname {\mathrm {Mat}}_3(R)^+$ . To see this, define $\pi _i \!: R \times R \to R$ to be the projection on the i-th coordinate and define $\phi \colon \operatorname {\mathrm {Her}}_3(R \times R) \to \operatorname {\mathrm {Mat}}_3(R)^+$ by sending

$$\begin{align*}\left( \begin{smallmatrix} \alpha_1 & c_3 & \cdot \\ \cdot & \alpha_2 & c_1 \\ c_2 & \cdot & \alpha_3 \end{smallmatrix} \right) \mapsto \left( \begin{smallmatrix} \alpha_1 & \pi_1(c_3) & \pi_2(c_2)\\ \pi_2(c_3) & \alpha_2 & \pi_1(c_1) \\ \pi_1(c_2) & \pi_2(c_2) & \alpha_3 \end{smallmatrix} \right) \quad \text{for } \alpha_i \in R \text{ and } c_i \in R \times R. \end{align*}$$

This map is obviously R-linear and surjective and sends the identity to the identity. One checks directly that $\phi $ preserves norms, that is, that $\det (\phi (x))$ equals $N(x)$ according to (6.11). Because $\operatorname {\mathrm {Her}}_3(R \times R)$ and $\operatorname {\mathrm {Mat}}_3(R)^+$ are both cubic Jordan algebras with regular trace bilinear forms, the proof of the “if” direction of Lemma 10.2 shows that $\varphi $ is an isomorphism of Jordan algebras.

11 Basic classification results for Albert algebras

In the case where R is a field, such as the real numbers, a finite field, a local field, or a global field, one can find in many places in the literature classifications of Albert algebras proved using techniques involving algebras as in [Reference Springer and VeldkampSpV, Section 5.8]. For such an R, groups of type $\mathsf {F}_4$ can be classified using techniques from algebraic groups, such as in [Reference Platonov and RapinchukPlRap, Chapter 6] or [Reference GilleGil19]. The two approaches are equivalent by Theorem 9.3.

Example 11.1 (Albert algebras over $\mathbb {R}$ )

Up to isomorphism, there are three Albert $\mathbb {R}$ -algebras, namely, the split one $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(\mathbb {R}))$ , $\operatorname {\mathrm {Her}}_3(\mathbb {O}, {\left \langle {1, -1, -1}\right \rangle })$ , and $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ . Rather than proving this in the language of Jordan algebras as in [Reference Albert and JacobsonAlbJ, Theorem 10], one may leverage Theorem 9.3 as follows. The three algebras are pairwise nonisomorphic because their trace forms are (Example 6.8). At the same time, a computation in the Weyl group of $\mathsf {F}_4$ as in [Reference SerreSerre, Section III.4.5], [Reference Borovoi and EvenorBorE, 14.1], or [Reference Adams and TaïbiAdT, Table 3] shows that $H^1(\mathbb {R}, \mathbf {Aut}(\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(\mathbb {R}))))$ has three elements. That is, there are exactly three isomorphism classes of simple affine group schemes over $\mathbb {R}$ of type $\mathsf {F}_4$ , so we have found all of them.

Example 11.2 (Albert algebras over global fields)

Let A be an Albert F-algebra for F a global field. Put $\Omega $ for the (finite) set of inequivalent embeddings $\omega \colon F \hookrightarrow \mathbb {R}$ . Since $\mathbf {Aut}(A)$ is a simple and simply connected affine group scheme, the natural map

$$\begin{align*}H^1(F, \mathbf{Aut}(A)) \xrightarrow{\prod \omega} \prod_{\omega \in \Omega} H^1(\mathbb{R}, \mathbf{Aut}(A)) \end{align*}$$

is a bijection by [Reference HarderHa66] or [Reference Platonov and RapinchukPlRap, p. 286, Theorem 6.6]. Because $H^1(\mathbb {R}, \mathbf {Aut}(A))$ has three elements by the preceding example, there are exactly $3^{|\Omega |}$ isomorphism classes of Albert F-algebras. This result, for number fields and with a proof in the language of Albert algebras, dates back to [Reference Albert and JacobsonAlbJ, p. 417, Corollary of Theorem 12].

The same argument goes through for octonion algebras, and one finds that there are $2^{|\Omega |}$ isomorphism classes of octonion F-algebras. This result, for number fields and with a proof in the language of octonion algebras, dates back to [Reference ZornZ, p. 400].

In the special case where F has a unique real embedding (e.g., $F = \mathbb {Q}$ ), the two isomorphism classes of octonion algebras are $\operatorname {\mathrm {Zor}}(F)$ and $\mathcal {O} \otimes F$ , and the three isomorphism classes of Albert algebras are $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(F))$ and $\operatorname {\mathrm {Her}}_3(\mathcal {O} \otimes F, {\left \langle {1, s, s}\right \rangle })$ for $s = \pm 1$ .

Below, we will focus our attention on classification results in the case where R is not a field. We translate known results about cohomology of affine group schemes into the language of Albert algebras.

Proposition 11.3. If R is: (1) a complete discrete valuation ring whose residue field is finite or (2) a finite ring, then every Freudenthal R-algebra of rank 15 or 27 and every quaternion or octonion R-algebra is split.

Proof. In view of Theorem 9.3 and its analogue for composition algebras, it suffices to prove that $H^1(R, \mathbf {G}) = 0$ for $\mathbf {G}$ a simple R-group scheme of type $\mathsf {F}_4$ or $\mathsf {C}_3$ obtained by base change from a Chevalley group over $\mathbb {Z}$ . In case (1), this is [Reference ConradConrad, Proposition 3.10]. In case (2), we apply the following lemma.

Lemma 11.4. If R is a finite ring and $\mathbf {G}$ is a smooth connected R-group scheme, then $H^1(R, \mathbf {G}) = 0$ .

Proof. If R is not connected, then it is a finite product $R = \prod R_i$ , where each ring $R_i$ is finite, so $H^1(R, \mathbf {G}) = \prod H^1(R_i, \mathbf {G} \times R_i)$ . Therefore, it suffices to assume that R is connected.

Suppose $\mathbf {X}$ is a $\mathbf {G}$ -torsor. Our aim is to show that $\mathbf {X}$ is the trivial torsor, that is, $\mathbf {X}(R)$ is nonempty. Put $\mathfrak {a}$ for the nil radical $\operatorname {\mathrm {Nil}}(R)$ of R. Because R is finite, there is some minimal $m \ge 1$ such that $\mathfrak {a}^m = 0$ . We proceed by induction on m. If $m = 1$ , then R is reduced and connected, so it is a finite field and $H^1(R, \mathbf {G}) = 0$ by Lang’s theorem. For the case $m \ge 2$ , put $I := \mathfrak {a}^{m-1}$ . The ring $R/I$ has $\operatorname {\mathrm {Nil}}(R/I)^{m-1} = (\operatorname {\mathrm {Nil}}(R)/I)^{m-1} = 0$ , so by induction, $\mathbf {X}(R/I)$ is nonempty. On the other hand, $I^2 = \mathfrak {a}^{2m-2} = \mathfrak {a}^m \cdot \mathfrak {a}^{m-2} = 0$ and $\mathbf {X}$ is smooth, so the natural map $\mathbf {X}(R) \to \mathbf {X}(R/I)$ is surjective.

Example 11.5. Suppose R is a Dedekind domain, and write F for its field of fractions. For $\mathbf {G}$ a Chevalley group of type $\mathsf {G}_2$ , $\mathsf {F}_4$ , or $\mathsf {E}_8$ , the map $H^1(R, \mathbf {G}) \to H^1(F, \mathbf {G})$ has zero kernel [Reference HarderHa67, Satz 3.3]. Consequently, if A is an Albert or octonion R-algebra and $A \otimes F$ is split, then the R-algebra A is split.

In particular, if F is a global field with no real embeddings, then every Albert or octonion F-algebra is split, so every Albert or octonion R-algebra is split.

In the case where F is a number field with a real embedding, we provide the following partial result, which relies on Example 11.1.

Proposition 11.6. Suppose F is a number field and R is a localization of its ring of integers at finitely many primes. If A is an Albert (respectively, octonion) F-algebra such that $A \otimes \mathbb {R}$ is not isomorphic to $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ (respectively, $\mathbb {O}$ ) for every embedding $F \hookrightarrow \mathbb {R}$ , then there is an Albert (respectively, octonion) R-algebra B such that $B \otimes F \cong A$ and B is uniquely determined up to R-isomorphism.

Proof. Write $\mathbf {G}$ for the automorphism group of the split Albert (respectively, octonion) F-algebra. Write $H^1_{\mathrm {ind}}(R, \mathbf {G}) \subseteq H^1(R, \mathbf {G})$ for the isomorphism classes of R-algebras B such that $B \otimes F_v$ is not $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ (respectively, $\mathbb {O}$ ), that is, such that $\mathbf {Aut}(B) \times F_v$ is not compact, for all real places v of F. Since $\mathbf {G}$ is simply connected, Strong Approximation gives that the natural map $H^1_{\mathrm {ind}}(R, \mathbf {G}) \to H^1_{\mathrm {ind}}(F, \mathbf {G})$ is an isomorphism [Reference HarderHa67, Satz 4.2.4], which is what is claimed.

12 The number of generators of an Albert algebra

The goal of this section is to prove Proposition 12.1, which is inspired by the work of First-Reichstein [Reference First and ReichsteinFiR] generalizing the Forster-Swan theorem. Let J be a para-quadratic R-algebra. By a (para-quadratic) subalgebra of J, we mean a submodule $J' \subseteq J$ containing $1_J$ and closed under the U-operator, that is, such that $U_x y \in J'$ for all $x,y \in J'$ . For any subset $S \subseteq J$ , the smallest subalgebra of J containing S is called the subalgebra generated by S; if this subalgebra is all of J, we say that S generates J.

Proposition 12.1. For every noetherian ring R, every Albert R-algebra can be generated in the sense of the preceding paragraph by $3 + \dim \operatorname {\mathrm {Max}} R$ elements.

In the statement, $\operatorname {\mathrm {Max}} R$ is the topological space whose points are the maximal ideals of R, endowed with the subspace topology inherited from $\operatorname {\mathrm {Spec}} R$ . It is evident that $\dim \operatorname {\mathrm {Max}} R \le \dim \operatorname {\mathrm {Spec}} R$ , also known as the Krull dimension. Beyond this inequality, the two numbers may be quite different (e.g., for a local ring, $\dim \operatorname {\mathrm {Max}} R = 0$ and $\dim \operatorname {\mathrm {Spec}} R$ can be any number).

If 2 is invertible in R, the bound in the proposition is Corollary 4.2c in [Reference First and ReichsteinFiR]. The contribution here is to remove the hypothesis on 2. In the special case where R contains an infinite field of characteristic $\ne 2$ , a different (and possibly smaller) upper bound is given in [Reference First, Reichstein and WilliamsFiRW, Section 13].

The results in [Reference First and ReichsteinFiR] reduce the proof of the proposition to the case where R is a field. For a field of characteristic not $2$ , a proof may be read off from [Reference McCrimmonMcC04, p. 112]; we give a characteristic-free proof using the first Tits construction of cubic Jordan algebras [Reference McCrimmonMcC69, pp. 507–509]. We briefly recall its details.

The first Tits construction

Let A be a (finite-dimensional) separable associative algebra of degree $3$ over a field F, so its generic norm $N_A$ is a cubic form on A and its trace $T_A\colon A \times A \to F$ defined as in (6.1) is a nondegenerate symmetric bilinear form. Given any nonzero scalar $\mu \in F$ , we obtain a cubic norm structure

$$\begin{align*}\mathbf{M}:= (A \times A \times A,1_{\mathbf{M}},\sharp,N_{\mathbf{M}}) \end{align*}$$

by defining

(12.1) $$ \begin{align} 1_{\mathbf{M}}:= (1_A,0,0), \end{align} $$
(12.2) $$ \begin{align} x^{\sharp} := (x_0^{\sharp} - x_1x_2,\mu^{-1}x_2^{\sharp} - x_0x_1,\mu x_1^{\sharp} - x_2x_0) \quad \text{and} \end{align} $$
(12.3) $$ \begin{align} N_{\mathbf{M}}(x) := N_A(x_0) + \mu N_A(x_1) + \mu^{-1}N_A(x_2) - T_A(x_0x_1x_2) \end{align} $$

for all $x = (x_0,x_1,x_2)$ in all scalar extensions of $A \times A \times A$ . By [Reference McCrimmonMcC69, Theorem 6], $\mathbf {M}$ is a cubic norm structure, so $J(\mathbf {M})$ is a cubic Jordan F-algebra which we denote by $J(A, \mu )$ .

Example 12.2. (a): Let E be a cubic étale F-algebra. If E is either split — that is, isomorphic to $F \times F \times F$ — or a cyclic cubic field extension of F, then $J(E,1) \cong \operatorname {\mathrm {Her}}_3(F \times F, {\left \langle {1, -1, -1}\right \rangle })$ [Reference Petersson and RacinePeR84b, Theorem 3], that is, is the split Freudenthal algebra of rank 9 (Proposition 7.5), which is $\operatorname {\mathrm {Mat}}_3(F)^+$ by Example 10.3.

(b): Let A be a central simple associative F-algebra of degree $3$ . Then $J(A,1)$ is a split Albert algebra [Reference Petersson and RacinePeR84a, Corollary 4.2].

Lemma 12.3. Let A be a separable associative F-algebra of degree $3$ and $\mu \in F^{\times }$ . Then the first Tits construction $J(A,\mu )$ is generated by $A^+$ (identified in $J(A,\mu )$ through the initial summand) and $(0,1_A,0)$ .

Proof. Let $J'$ be the subalgebra of $J(A, \mu )$ generated by $A^+$ and $w := (0,1_A,0)$ . As a subalgebra, it is closed under $\sharp $ by (6.8), that is,