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

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.