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].

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.

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

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.

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.

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.

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.

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

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 $x^4 = U_xx^2 = U_xx^{\sharp } + \operatorname {\mathrm {Tr}}_J(x)U_xx - S_J(x)U_x1_J = \operatorname {\mathrm {Tr}}_J(x)x^3 - S_J(x)x^2 + N_J(x)x$ .

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 $\gamma _1 \gamma _2 \gamma _3 = 1$ .

7 Albert algebras are Freudenthal algebras are Jordan algebras

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

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

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.

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 $\mathfrak {a}$ .

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.

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

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:

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].

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.

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].

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 11.2 (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.

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.

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.

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, $x^{\sharp } \in J'$ and $x \times y \in J'$ for all $x, y \in J'$ . Since $w^{\sharp } = \mu (0,0,1_A)$ and $x_0 \times (0,x_1,x_2) = (0,-x_0x_1,-x_2x_0)$ for all $x_i \in A$ by (12.2), it follows that $J'$ must be all of $J(A, \mu )$ .

Proof of Proposition 12.1

Theorem 1.2 in [Reference First and ReichsteinFiR] reduces the proof to the case where R is a field F, in which case $\dim \operatorname {\mathrm {Max}} R = 0$ , so the task is to prove that three elements suffice to generate an Albert algebra J. If F is infinite, then Proposition 4.1 in [Reference First and ReichsteinFiR] reduces us to considering the case where J is split. If F is finite, then J is split by Proposition 11.3.

If $F \ne \mathbb {F}_2$ , then the split cubic étale F-algebra $E := F \times F \times F$ can be generated by a single element x as an associative algebra. On the other hand, if $F = \mathbb {F}_2$ , let $E := \mathbb {F}_8$ be the cyclic cubic extension of F, which is again generated by one element, call it x. In either case, x also generates the Jordan algebra $E^+$ , because the powers of x in $E^+$ and E are the same.

Hence, Example 12.2 (a) and Lemma 12.3 show that $\operatorname {\mathrm {Mat}}_3(F)^+$ is generated by two elements. Lemma 12.3 combined with Example 12.2 (b) shows that the split Albert algebra $J(\operatorname {\mathrm {Mat}}_3(F),1)$ is generated by three elements.

Remark 12.4. That the Jordan algebra $\operatorname {\mathrm {Mat}}_3(F)^+$ , for any field F, is generated by two elements doesn’t seem too surprising, but one should keep in mind that the analogous result for 2-by-2 matrices is false: the minimal number of generators for the Jordan algebra $\operatorname {\mathrm {Mat}}_2(F)^+$ is three.

Remark 12.5 (dichotomy of fields and the Tits construction)

The classification of Albert algebras over a field F of characteristic $\ne 3$ has a fundamentally different flavor depending on whether or not $H^3(F, \mathbb {Z}/3)$ is zero, as indicated by [Reference RostRost], [Reference Petersson and RacinePeR96], or [Reference GaribaldiGar09, Section 8]. If $H^3(F, \mathbb {Z}/3) = 0$ — as is the case for global fields, p-adic fields, and the real numbers — every Albert F-algebra is reduced, that is, of the form $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for some C and $\Gamma $ , and is not a division algebra. (It is natural to speculate that this is the reason it took many years after Albert algebras were defined — all the way until 1958 — for the first Albert division algebra to be exhibited in [Reference AlbertAlb58].) In the other case, when $H^3(F, \mathbb {Z}/3) \ne 0$ , as happens when $F = \mathbb {Q}(t)$ , for example, one can construct an Albert division algebra via the first Tits construction described above as $J(A \otimes \mathbb {Q}(t), t)$ for A an associative division algebra of dimension 9 over $\mathbb {Q}$ .

It is known that every Albert algebra over a field is obtained by the first Tits construction or second Tits construction (which we have not described here) (see [Reference McCrimmonMcC70, Theorem 10] or [Reference Petersson and RacinePeR86b, Theorem 3.1(i)]). Both constructions have been extended from the case of algebras over a field to an arbitrary base ring [Reference Petersson and RacinePeR86a]. However, in this more general setting, the Tits constructions do not produce all Albert algebras [Reference Parimala, Sridharan and ThakurPaST].

13 Isotopy

The aim of this section is to discuss the notion of isotopy of Jordan algebras, which will pay off later in the paper when we discuss groups of type $\mathsf {E}_6$ in Section 15 and $\mathsf {E}_7$ in Section 17. We include this material at this point in the paper because Corollary 13.6 is needed in the following section.

We have presented the notion of isotopy for Jordan algebras. However, there are analogous notions for other classes of algebras, which go back at least to [Reference AlbertAlb42]. For associative algebras, isotopy is the same as isomorphism. For octonion algebras, isotopy amounts to norm equivalence [Reference Alsaody and GilleAlsG, Corollary 6.7], which is a weaker condition than isomorphism (see [Reference GilleGil14] and [Reference Asok, Hoyois and WendtAsHW]).

Isotopes of cubic Jordan algebras

If J is a cubic Jordan R-algebra and $u \in J$ is invertible, then [Reference McCrimmonMcC69, Theorem 2] and its proof show that the isotope $J^{(u)}$ is a cubic Jordan algebra as well whose identity element, adjoint and norm are given by

(13.1) $$ \begin{align} 1_{J^{(u)}} = u^{-1}, \quad x^{\sharp(u)} = N_J(u)U_u^{-1}x^{\sharp}, \quad N_{J^{(u)}}(x) = N_J(u)N_J(x). \end{align} $$

Moreover, the (bi-)linear and quadratic trace of $J^{(u)}$ have the form

(13.2) $$ \begin{align} T_{J^{(u)}}(x,y) = T_J(U_ux,y), \quad \operatorname{\mathrm{Tr}}_{J^{(u)}}(x) = T_J(u,x), \quad S_{J^{(u)}}(x) = T_J(u^{\sharp},x^{\sharp}). \end{align} $$

The first equation of (13.2) is in [Reference McCrimmonMcC69, p. 500], while the second one follows from (13.1), the first, and Lemma 6.5 (1) via $\operatorname {\mathrm {Tr}}_{J^{(u)}}(x) = T_{J^{(u)}}(u^{-1},x) = T_J(U_uu^{-1},x) = T_J(u,x)$ . Similarly,

$$\begin{align*}S_{J^{(u)}}(x) = \operatorname{\mathrm{Tr}}_{J^{(u)}}(x^{\sharp(u)}) = T_J(u,N_J(u)U_u^{-1}x^{\sharp}) = T_J(N_J(u)U_u^{-1}u,x^{\sharp}) = T_J(u^{\sharp},x^{\sharp}). \end{align*}$$

Example 13.2. $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is isotopic to $\operatorname {\mathrm {Her}}_3(C)$ for every $\Gamma $ . Indeed, for

$$\begin{align*}u := \left( \begin{smallmatrix} \gamma_1 & 0 & 0 \\ 0 & \gamma_2 & 0 \\ 0 & 0 & \gamma_3 \end{smallmatrix} \right) \quad \in \operatorname{\mathrm{Her}}_3(C, \Gamma), \end{align*}$$

the map $\phi \!: \operatorname {\mathrm {Her}}_3(C, \Gamma )^{(u)} \to \operatorname {\mathrm {Her}}_3(C)$ defined by

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

is an isomorphism of Jordan algebras. One can also turn this around:

$$\begin{align*}\operatorname{\mathrm{Her}}_3(C,\Gamma) = (\operatorname{\mathrm{Her}}_3(C,\Gamma)^{(u)})^{(u^{-2})} \cong \operatorname{\mathrm{Her}}_3(C)^{(\phi(u^{-2}))} = \operatorname{\mathrm{Her}}_3(C)^{(u^{-1})}. \end{align*}$$

Jordan algebras isotopic to a reduced Freudenthal algebra

In the special case where R is a field, a Jordan algebra that is isotopic to the split Albert algebra $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ is necessarily isomorphic to it (see, for example [Reference JacobsonJ71, p. 53, Theorem 9]). This need not hold for general R: Alsaody has shown in [Reference AlsaodyAls21, Theorem 2.7] that there exists a ring R finitely generated over $\mathbb {C}$ and an Albert R-algebra that is isotopic to the split Albert R-algebra but is not isomorphic to it. Here we show that it is sufficient to assume that R is a semilocal ring (Corollary 13.4), as a consequence of a more general result (Theorem 13.3) from which we also obtain the key Corollary 13.6.

We work in a slightly more general context than semilocal rings. For the following statements, see [Reference Estes and GuralnickEsG] and [Reference McDonald and WaterhouseMcDW]. We say that R is an LG ring if whenever a polynomial $f \in R[x_1, \ldots , x_n]$ represents a unit over $R_{\mathfrak {m}}$ for every maximal ideal $\mathfrak {m}$ of R, then f represents a unit over R. Every semilocal ring is an LG ring. It is easy to see that rings $R_1$ , $R_2$ are both LG if and only if their product $R_1 \times R_2$ is LG. The ring of all algebraic integers and the ring of all real algebraic integers are LG rings. Every integral extension of an LG ring is LG [Reference Estes and GuralnickEsG, Corollary 2.3].

Theorem 13.3. Suppose J is a Jordan R-algebra that is isotopic to $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for some composition R-algebra C and some $\Gamma $ . If R is an LG ring, then J is isomorphic to $\operatorname {\mathrm {Her}}_3(C, \Gamma ')$ for some $\Gamma '$ .

Proof. As in previous proofs, we reduce to the case where C has constant rank.

In view of Example 13.2, J is isotopic to $\operatorname {\mathrm {Her}}_3(C)$ , that is, $J \cong \operatorname {\mathrm {Her}}_3(C)^{(u^{-1})}$ for some invertible $u \in \operatorname {\mathrm {Her}}_3(C)$ . The same example shows we are done if u is diagonal.

Write N for the cubic form on $\operatorname {\mathrm {Her}}_3(C)$ . In case u is not diagonal, we will apply successive elements $\eta \in \operatorname {\mathrm {GL}}(\operatorname {\mathrm {Her}}_3(C))$ such that $N \eta = N$ as polynomial laws. (In the notation of Section 15 below, $\eta \in \mathbf {Isom}(\operatorname {\mathrm {Her}}_3(C))(R)$ .) Note that each such $\eta $ defines an isomorphism of R-modules

(13.3) $$ \begin{align} \eta \colon \operatorname{\mathrm{Her}}_3(C)^{(u^{-1})} \to \operatorname{\mathrm{Her}}_3(C)^{(\eta(u)^{-1})}. \end{align} $$

We have

$$\begin{align*}N(\eta(u)^{-1}) = N(\eta(u))^{-1} = N(u)^{-1} = N(u^{-1}), \end{align*}$$

so we have by (13.1) that

$$\begin{align*}N_{\operatorname{\mathrm{Her}}_3(C)^{(u^{-1})}} = N(u)^{-1} N = N(\eta(u)^{-1}) N \eta = N_{\operatorname{\mathrm{Her}}_3(C)^{(\eta(u)^{-1})}}\eta. \end{align*}$$

Since $\eta $ is a norm isometry that maps the identity element $u^{-1}$ in the domain of (13.3) to the identity element in the codomain, it is an isomorphism of algebras by Lemma 10.2. Thus, if successive elements $\eta $ transform u into a diagonal element, the proof will be complete.

We employ the transformation $\tau _{st}(q)$ for $1 \le s \ne t \le 3$ and $q \in C$ defined by

$$\begin{align*}\tau_{st}(q) \colon A \mapsto (I_3 + q E_{st}) A (I_3 + \bar{q} E_{ts}), \end{align*}$$

where $I_3$ is the identity matrix, $E_{st}$ is the 3-by-3 matrix with a 1 in the $(s, t)$ -entry and 0 elsewhere, and juxtaposition defines naive multiplication of 3-by-3 matrices with entries in C. For example,

$$\begin{align*}\tau_{12}(q) \left( \begin{smallmatrix} \alpha_1 & c_3 & \cdot \\ \cdot & \alpha_2 & c_1 \\ c_2 & \cdot & \alpha_3 \end{smallmatrix} \right) = \left( \begin{smallmatrix} \alpha_1+n_C(q, c_3) + \alpha_2 n_C(q) & c_3 + \alpha_2 q & \cdot \\ \cdot & \alpha_2 & c_1 \\ c_2 + \bar{c}_1\bar{q} & \cdot & \alpha_3 \end{smallmatrix} \right). \end{align*}$$

These transformations appear in [Reference JacobsonJ61, Section 5] and [Reference KrutelevichKr02, Section 2]; the argument in either reference shows that $\tau _{st}(q)$ preserves N for all choices of s, t, and q. For $e = 2, 3$ , define polynomial functions $\nu _e$ from $C^e$ to the group scheme $\mathbf {G}$ of linear transformations stabilizing the norm N via

$$\begin{align*}\nu_3(q_1, q_2, q_3) = \tau_{31}(q_3) \tau_{21}(q_2) \tau_{12}(q_1) \quad \text{and} \quad \nu_2(q_1, q_2) = \tau_{32}(q_2) \tau_{23}(q_1). \end{align*}$$

Additionally, for every permutation $\pi $ of $\{ 1, 2, 3 \}$ , there is a linear transformation of $\operatorname {\mathrm {Her}}_3(C)$ that we denote also by $\pi $ , for example, the transposition $(1\,2)$ acts via

(13.4) $$ \begin{align} \left( \begin{smallmatrix} \alpha_1 & c_3 & \cdot \\ \cdot & \alpha_2 & c_1 \\ c_2 & \cdot & \alpha_3 \end{smallmatrix} \right) \mapsto \left( \begin{smallmatrix} \alpha_2 & \bar{c}_3 & \cdot \\ \cdot & \alpha_1 & \bar{c}_2 \\ \bar{c}_1 & \cdot & \alpha_3 \end{smallmatrix} \right) \end{align} $$

(see, for example, [Reference KrutelevichKr02, p. 282]). The other transpositions are constructed analogously and each evidently preserves the norm. In this way, we obtain a representation of the permutation group on $\operatorname {\mathrm {Her}}_3(C)$ .

Case: R is local: We collect some observations in the case where R is a local ring. Write

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

By hypothesis, $N(u)$ is invertible, that is, does not lie in the maximal ideal $\mathfrak {m}$ of R.

If $\alpha _1 \in R^{\times }$ or $\alpha _2, \alpha _3 \notin R^{\times }$ , then after modifying u by a transformation in the image of $\nu _3$ , we may assume that $\alpha _1 \in R^{\times }$ and $c_2 = c_3 = 0$ . Indeed, if $\alpha _1 \notin R^{\times }$ , then by (6.11) we have

$$\begin{align*}N(u) \equiv \operatorname{\mathrm{Tr}}_C(c_1 c_2 c_3) \bmod \mathfrak{m}, \end{align*}$$

whence $c_3 \notin \mathfrak {m} C$ . Since $n_C$ continues to be regular when changing scalars to $R/\mathfrak {m}$ , some $q \in C$ has $n_C(q,c_3) \notin \mathfrak {m}$ . Applying $\tau _{12}(q)$ , we may arrange $\alpha _1 \in R^{\times }$ . Then note that $\tau _{21}(q) \left ( \begin {smallmatrix} \alpha _1 & c_3 & \cdot \\ \cdot & \alpha _2 & c_1 \\ c_2 & \cdot & \alpha _3 \end {smallmatrix} \right )$ has top row entries $\alpha _1$ , $c_3 + \alpha _1 \bar {q}$ , $\bar {c_2}$ . Taking $q = -\bar {c}_3 \alpha _1^{-1}$ shows that we may assume $c_3 = 0$ . The argument that we may assume $c_2 = 0$ is similar, with the role of $\tau _{21}$ replaced by $\tau _{31}$ .

Now suppose that $\alpha _1 \in R^{\times }$ and $c_2 = c_3 = 0$ . If $\alpha _2 \in R^{\times }$ or $\alpha _3 \notin R^{\times }$ , then after modifying u by a transformation in the image of $\nu _2$ , we may assume that u is diagonal. Indeed, since u has norm $\alpha _1 (\alpha _2 \alpha _3 - n_C(c_1)) \not \in \mathfrak {m}$ , at least one of $\alpha _2$ , $\alpha _3$ , or $n_C(c_1)$ is not in $\mathfrak {m}$ . The same argument as in the preceding paragraph, with $\tau _{12}$ replaced by $\tau _{23}$ , shows that we may assume that $\alpha _2 \not \in \mathfrak {m}$ . The same argument as in the preceding paragraph, with $\tau _{21}$ replaced by $\tau _{32}$ , shows that we may assume that $c_1 = 0$ . Thus, we have transformed u into a diagonal element, as required.

General case: Return to the setting of R as in the statement of the theorem. We combine the transformations $\nu _3$ , $\nu _2$ , and permutations together into a polynomial function $C^{21} \to \mathbf {G}$ , namely

(13.5) $$ \begin{align} \left( \nu_2 \, (2\,3)\, \nu_2 \nu_3 \,(1\,3) \right)\left( \nu_2 \, (2\,3)\, \nu_2 \nu_3 \,(1\,3) \right) \left( \nu_2 \, (2\,3)\, \nu_2 \nu_3 \right), \end{align} $$

where the arguments to the various $\nu _2$ , $\nu _3$ are assigned independently. Combining this with the polynomial function on $\mathbf {G}$ that sends $g \in \mathbf {G}(R)$ to the product of the diagonal entries of $gu$ , we obtain a polynomial law in $\mathscr {P}(C^{21}, R)$ . But more is true. Because R is LG and C is projective of constant rank, C is a free module (see [Reference Estes and GuralnickEsG, Theorem 2.10] or [Reference McDonald and WaterhouseMcDW, p. 457]). Choosing a basis for C expresses this polynomial law as a polynomial with coefficients in R.

We claim that this polynomial represents a unit over $R_{\mathfrak {m}}$ for every maximal ideal $\mathfrak {m}$ of R. For a given $\mathfrak {m}$ , here is how to pick the element of $C^{21}$ that produces a unit. If $\alpha _1 \in R_{\mathfrak {m}}^{\times }$ or $\alpha _2, \alpha _3 \notin R_{\mathfrak {m}}^{\times }$ , applying $\nu _3$ to u, with arguments chosen as in the second paragraph of the local case, we obtain an element with $\alpha _1 \in R_{\mathfrak {m}}^{\times }$ and $c_3 = c_2 = 0$ . We take this to be the rightmost term in (13.5). If that element has $\alpha _2 \in R_{\mathfrak {m}}^{\times }$ or $\alpha _3 \notin R_{\mathfrak {m}}^{\times }$ , the next $\nu _2$ term can be chosen to produce a diagonal u; one takes the remaining $\nu $ terms in (13.5) to have argument 0. Otherwise, $\alpha _3$ is invertible in $R_{\mathfrak {m}}$ , and we plug 0 into the rightmost $\nu _2$ , pick the argument for the next $\nu _2$ as in the proof of the local case, and plug 0 into the remaining $\nu $ terms to the left in (13.5). The claim is verified if $\alpha _1 \in R_{\mathfrak {m}}^{\times }$ or $\alpha _2, \alpha _3 \notin R_{\mathfrak {m}}^{\times }$ .

The next case of the claim is where $\alpha _2 \in R_{\mathfrak {m}}^{\times }$ . In that case, we plug 0 into the rightmost three $\nu $ terms in (13.5). After applying the permutation $(2\,3)$ and then $(1\,3)$ , we obtain an element of $\operatorname {\mathrm {Her}}_3(C)$ with $\alpha _1$ invertible, and a well-chosen argument for the next $\nu _3$ term will assure that $c_3 = c_2 = 0$ . As in the preceding paragraph, choosing the arguments for the leftmost two $\nu _2$ terms in the middle product in (13.5) suffices to transform u into a diagonal element, verifying the claim in this case.

The last case of the claim is when $\alpha _3 \in R_{\mathfrak {m}}^{\times }$ . Plug 0 in the $\nu $ terms in the middle and right parenthetical expressions in (13.5). After applying all permutations in (13.5) besides the leftmost transposition to u, we obtain an element of $\operatorname {\mathrm {Her}}_3(C)$ with $\alpha _1 \in R_{\mathfrak {m}}^{\times }$ and the argument in the preceding paragraph, again, transforms u into a diagonal element, completing the proof of the claim.

Since R is an LG ring, the claim provides an element $g \in \mathbf {G}(R)$ such that $gu$ has $(1,1)$ -entry a unit. That is, we may assume that in the element u, $\alpha _1$ is invertible. Applying now $\tau _{21}(q)$ and $\tau _{31}(q)$ to u for values of q chosen as in the local case, we may assume that $c_3 = c_2 = 0$ .

Applying now an argument as in the preceding five paragraphs, with the function

$$\begin{align*}\nu_2 \, (2\,3) \, \nu_2 \colon C^4 \to \mathbf{G}, \end{align*}$$

we conclude that we may transform u to further assume that $\alpha _2$ is invertible, and therefore, apply a transformation $\tau _{32}(q)$ to transform it into a diagonal element, as required.

Corollary 13.4. Suppose J is a Jordan R-algebra over an LG ring R. If J is isotopic to a split Freudenthal algebra whose rank does not take the value 6, then J is itself a split Freudenthal algebra.

Proof. As in previous proofs, one is reduced to the case where J has constant rank, which is not 6. The theorem and Proposition 7.5 give the claim.

The hypothesis that the rank is not 6 is necessary, because $\operatorname {\mathrm {Her}}_3(\mathbb {R}, {\left \langle {1, -1, -1}\right \rangle })$ is isotopic to the split Freudenthal algebra $\operatorname {\mathrm {Her}}_3(\mathbb {R})$ (Example 13.2) but is not isomorphic to it (Example 6.8).

Example 13.5 (isotopy over global fields)

For $F = \mathbb {R}$ or a global field, there is a bijection between the isomorphism classes of octonion algebras and isotopy classes of Albert algebras given by $C \leftrightarrow \operatorname {\mathrm {Her}}_3(C)$ . Indeed, every Albert F-algebra is reduced (Example 11.2), so $C \mapsto \operatorname {\mathrm {Her}}_3(C)$ touches every isotopy class. For injectivity, if $C, C'$ are distinct octonion algebras, there is a real embedding $F \hookrightarrow \mathbb {R}$ such that $C \otimes \mathbb {R} \not \cong C' \otimes \mathbb {R}$ , and Corollary 13.4 shows that $\operatorname {\mathrm {Her}}_3(C) \otimes \mathbb {R}$ and $\operatorname {\mathrm {Her}}_3(C') \otimes \mathbb {R}$ are not isotopic.

Corollary 13.6. Every isotope of a Freudenthal algebra is itself a Freudenthal algebra.

Proof. Suppose J is an isotope of a Freudenthal algebra. After base change to a faithfully flat extension, J is an isotope of a split Freudenthal algebra.

The R-algebra $S := \prod _{\mathfrak {m}} R_{\mathfrak {m}}$ , where $\mathfrak {m}$ ranges over maximal ideals of R, is faithfully flat. For each $\mathfrak {m}$ , $J \otimes R_{\mathfrak {m}}$ is $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for C a split composition $R_{\mathfrak {m}}$ -algebra and some $\Gamma $ by Theorem 13.3. By Proposition 7.3, there is a faithfully flat $R_{\mathfrak {m}}$ -algebra T such that $J \otimes T$ is a split Freudenthal algebra. The product of these T’s is a faithfully flat R-algebra over which J is the split Freudenthal algebra.

We close this section by making explicit the relationship between isotopy and norm similarity between Freudenthal algebras, extending Lemma 10.2.

Proposition 13.7. Let J and $J'$ be Freudenthal R-algebras. For an R-linear map $\phi \!: J \to J'$ , the following are equivalent:

1. 1. $\phi $ is an isomorphism $J \to (J')^{(u)}$ for some invertible $u \in J'$ (“ $\phi $ is an isotopy”).

2. 2. $N_{J'} \phi = \alpha N_J$ as polynomial laws for some $\alpha \in R^{\times }$ , and $\phi $ is surjective (“ $\phi $ is a norm similarity”).

Proof. Since $(J^{\prime })^{(u)}$ is a Freudenthal algebra by Corollary 13.6, condition (2) follows from (1) by Lemma 10.2 and (13.1). Conversely, we assume (2) and prove (1). Because $N_{J'}(\phi (1_J)) = \alpha $ , the element $\phi (1_J)$ is invertible in $J'$ . We set $u := \phi (1_J)^{-1}$ and $J'' := (J')^{(u)}$ . We have

$$\begin{align*}\phi(1_J) = u^{-1} = 1_{J''}. \end{align*}$$

Also, $N_{J'}(u) = N_{J'}(\phi (1_J))^{-1} = \alpha ^{-1}$ . Then

$$\begin{align*}N_{J''} \phi = N_{J'}(u) N_{J'} \phi = N_J \end{align*}$$

as polynomial laws. Lemma 10.2 implies that $\phi $ is an isomorphism $J \xrightarrow {\sim } J''$ , as desired.