1 Introduction
Albert algebras, which are a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.
Why Albert algebras?
In the setting of semisimple algebraic groups over a field, a standard technique for computing with elements of a group — especially an anisotropic group — is to interpret the group in terms of automorphisms of some algebraic structure, such as viewing an adjoint group of type $\mathsf {B}_n$ as the special orthogonal group of a quadratic form of dimension $2n+1$ , or an adjoint group of inner type $\mathsf {A}_n$ as the automorphism group of an Azumaya algebra of rank $(n+1)^2$ . This approach can be seen in many references, from [Reference WeilWeil], through [Reference Knus, Merkurjev, Rost and TignolKnMRT] and [Reference ConradConrad]. In this vein, Albert algebras appear as a natural tool for computations related to $\mathsf {F}_4$ , $\mathsf {E}_6$ , and $\mathsf {E}_7$ groups, as we do below.
In the setting of nonassociative algebras, Albert algebras also arise naturally. Among commutative notnecessarilyassociative algebras under additional mild hypotheses (the field has characteristic $\ne 2, 3, 5$ and the algebra is metrized), every algebra satisfying a polynomial identity of degree $\le 4$ is a Jordan algebra (see [Reference Chayet and GaribaldiChG, Proposition A.8]). Jordan algebras have an analogue of the WedderburnArtin theory for associative algebras [Reference JacobsonJ68, p. 201, Corollary 2], and one finds that all the simple Jordan algebras are closely related to associative algebras (more precisely, “are special”) except for one kind, the Albert algebras (see, for example [Reference JacobsonJ68, p. 210, Theorem 11] or [Reference McCrimmon and Zel’manovMcCZ]).
Our contribution
In the setting of nonassociative algebras, we prove a classification of Albert algebras over $\mathbb {Z}$ (Theorem 14.3), which was viewed as an open question in the context of nonassociative algebra; here, we see that it is equivalent to the classification of groups of type $\mathsf {F}_4$ , which was known (see [Reference ConradConrad], which leverages [Reference GrossGr] and [Reference Elkies and GrossElkiesGr]). We also prove new results about ideals in Albert algebras (Theorem 8.2), about isotopy of Albert algebras over semilocal rings (Theorem 13.3), and about the number of generators of an Albert algebra (Proposition 12.1). We have not seen Lemma 15.1 in the literature, even in the case of a base field of characteristic different from 2 and 3.
In the setting of affine group schemes, the language of Albert algebras provides a way to give concrete descriptions of the affine group schemes over $\mathbb {Z}$ (see Section 18). In that language, a clever computation in [Reference Elkies and GrossElkiesGr] appears as an example of a general mechanism known as isotopy (see Definition 14.1). To facilitate these applications, we present the definition of Albert algebras in a streamlined way (see Definition 7.1). Note that they are defined as a type of what was formerly called a “quadratic” Jordan algebra — because instead of a bilinear multiplication, one has a quadratic map, the Uoperator — and that the definition makes sense whether or not 2 is invertible in the base ring. Applying the definition here allows one to replace, in some proofs, “global” computations over $\mathbb {Z}$ as one finds in [Reference ConradConrad] with “local” computations over an algebraically closed field that exist in several places in the literature (see, for example, the proof of Lemma 9.1 and Section 18).
A different definition
The definition of Albert algebra over a ring R given here (Definition 7.2) is in the context of paraquadratic algebras as recalled at the beginning of Section 5 — such an algebra is an Rmodule M with a distinguished element $1_M$ and a quadratic map $U \!: M \to \operatorname {\mathrm {End}}_R(M)$ such that $U_{1_M} = \operatorname {\mathrm {Id}}_M$ . There are no other axioms to check. We then define a specific paraquadratic algebra, $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ , in Definition 6.7, and define an Albert Ralgebra J to be a paraquadratic Ralgebra such that $J \otimes S \cong \operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R)) \otimes S$ for some faithfully flat Ralgebra S.
A different approach is taken by references such as [Reference PeterssonPe19, Section 6.1] or [Reference AlsaodyAls21]. They define an Albert Ralgebra to be a cubic Jordan Ralgebra (Definition 6.2) J whose underlying Rmodule is projective of rank 27 and $J \otimes F$ is a simple algebra for every homomorphism from R to a field F. This definition involves axioms that in principle need to be verified over all Ralgebras. The two definitions give the same objects. Theorem 17 in [Reference PeterssonPe19] states that an Albert algebra in the sense of that paper is an Albert algebra in the sense of this paper by proving the existence of the required faithfully flat Ralgebra; a detailed proof has not been published but closely follows an argument for octonion algebras from [Reference Loos, Petersson and RacineLoPR]. For the converse, an Albert algebra in the sense of this paper has a projective underlying module (because $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ does), is a cubic Jordan algebra (Proposition 10.1), and satisfies the simplicity condition (Corollary 8.5).
2 Notation
Rings, by definition, have a 1. We put ${\mathbb {Z}\text {}\textbf {alg}}$ for the category of commutative rings, where $\mathbb {Z}$ is an initial object. For any $R \in {\mathbb {Z}\text {}\textbf {alg}}$ , we put ${R\text {}\textbf {alg}}$ for the category of pairs $(S, f)$ with $S \in {\mathbb {Z}\text {}\textbf {alg}}$ and $f \!: R \to S$ , that is, the coslice category $R \downarrow {\mathbb {Z}\text {}\textbf {alg}}$ . Below, R will typically denote an element of ${\mathbb {Z}\text {}\textbf {alg}}$ . (The interested reader is invited to mentally replace R by a base scheme $\mathbf {X}$ , ${R\text {}\textbf {alg}}$ with the category of schemes over $\mathbf {X}$ , finitely generated projective Rmodules with vector bundles over $\mathbf {X}$ , etc., thereby translating results below into a language closer to that in [Reference Calmès and FaselCalF].) An Ralgebra 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 nbyn 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
is a set. We call it the nonabelian fppf cohomology of $\mathbf {G}$ . In case $\mathbf {G}$ is smooth, it agrees with étale $H^1$ . If additionally R is a field, then it agrees with the nonabelian Galois cohomology defined in, for example, [Reference SerreSerre].
Unimodular elements
Let M be an Rmodule. An element $m \in M$ is said to be unimodular if $Rm$ is a free Rmodule of rank 1 and a direct summand of M, equivalently, if there is some $\lambda \in M^*$ (the dual of M) such that $\lambda (m) = 1$ . When M is finitely generated projective, this is equivalent to: $m \otimes 1$ is not zero in $M \otimes F$ for every field $F \in {R\text {}\textbf {alg}}$ (see, for example [Reference LoosLo, 0.3]). If $m \in M$ is unimodular, then so is $m \otimes 1 \in M \otimes S$ for every $S \in {R\text {}\textbf {alg}}$ . In the opposite direction, if M is finitely generated projective, S is a Zariski cover of R (i.e., $\operatorname {\mathrm {Spec}} S \to \operatorname {\mathrm {Spec}} R$ is surjective), and $m \otimes 1$ is unimodular in $M \otimes S$ , it follows that m is unimodular as an element of M.
3 Background on polynomial laws
We may identify an Rmodule 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 Rmodules 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 Rmodule.
Lemma 3.1. Let M be a finitely generated projective Rmodule, and suppose $f \in \mathscr {P}(M, N)$ is such that $f_R(0) = 0$ . If $m \in M$ has $f_R(m)$ unimodular in N, then m is unimodular.
In the case $N = R$ , the condition that $f_R(m)$ is unimodular means that $f_R(m) \in R^{\times }$ .
Proof. Replacing f with $\lambda f$ , where $\lambda \in N^*$ is such that $\lambda (f_R(m)) = 1$ , we may assume $N = R$ and $f_R(m) = 1$ . If m is not unimodular, then there is a field $F \in {R\text {}\textbf {alg}}$ , such that $m \otimes 1 = 0$ in $M \otimes F$ , and $f_F(m \otimes 1) = 0$ , whence $f_R(m)$ belongs to the kernel of $R \to F$ a contradiction.
A polynomial law is homogeneous of degree $d \ge 0$ if $f_S(sx) = s^d f_S(x)$ for every $S \in {R\text {}\textbf {alg}}$ , $s \in S$ , and $x \in M \otimes S$ (see [Reference RobyRoby, p. 226]). We put $\mathscr {P}^d_R(M, N)$ for the submodule of $\mathscr {P}_R(M, N)$ of polynomial laws that are homogeneous of degree d. The polynomial laws that are homogeneous of degree 0 are constants, and those of degree 1 are linear transformations, that is, the natural maps
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 nth 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 $\nabla _v^n f(x) = \nabla _x^{dn} f(v)$ for $x \in M$ .
Lemma 3.2. Suppose M, N are Rmodules and A is a unital associative Ralgebra 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
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
is invertible in $A_S$ , so by hypothesis,
Focusing on the coefficient of $\varepsilon ^d$ in that equation gives
as required.
While Lemma 3.2 has some similarity with the principle of extension of algebraic identities as in [Reference BourbakiBouA2, Section IV.2.3], that result imposes some hypothesis on R.
The module of polynomial laws
In the following, we write $\mathsf {S}^n M$ for the nth symmetric power of M, that is, the Rmodule $\otimes ^n M$ modulo the submodule generated by elements $x  \sigma (x)$ for $x \in \otimes ^n M$ and $\sigma $ a permutation of the n factors.
Lemma 3.3. Let M and N be finitely generated projective Rmodules. Then for each $d \ge 0$ :

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

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

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

4. The natural map $\mathscr {P}^d(M, R) \otimes N \to \mathscr {P}^d(M, N)$ is an isomorphism.
Proof. To establish notation, we write $R = \prod _{i=0}^n R_i$ for some n such that $M = \prod _i M_i$ and $N = \prod _i N_i$ with each $M_i$ , $N_i$ an $R_i$ module of finite constant rank.
Next write $\Gamma _d(M)$ for the module of degree d divided powers on M as defined in [Reference BourbakiBouA2, Section IV.5, Exercise 2]. We claim that it is finitely generated projective, and therefore, by [Stacks, Tag 00NX], finitely presented. If M is free, then $\Gamma _d(M)$ is free. If M is projective of constant rank, then there exists $S \in {R\text {}\textbf {alg}}$ faithfully flat such that $M \otimes S$ is free. Because $\Gamma _d$ commutes with base change [Reference BourbakiBouA2, Section IV.5, Exercise 7], $\Gamma _d(M) \otimes S \cong \Gamma _d(M \otimes S)$ is free, and we again find that $\Gamma _d(M)$ is finitely generated projective [Stacks, Tags 03C4, 05A9]. In the general case, $\Gamma _d(M) = \prod _i \Gamma _d(M_i)$ , and the claim is verified.
To verify (2), we note that $\mathscr {P}^d_R(M, N)$ is naturally isomorphic to $\operatorname {\mathrm {Hom}}_R(\Gamma _d(M), N)$ by [Reference RobyRoby, Theorem IV.1]. Then $\mathscr {P}^d_R(M, N) \otimes T \cong \operatorname {\mathrm {Hom}}_R(\Gamma _d(M), N) \otimes T$ , which in turn is $\operatorname {\mathrm {Hom}}_T(\Gamma _d(M) \otimes T, N \otimes T)$ because T is flat and $\Gamma _d(M)$ is finitely presented [Reference BourbakiBouCA, Section I.2.10, Proposition 11]. Since $\Gamma _d$ commutes with base change, we have verified (2).
(3): If M and N are free modules, then the map is an isomorphism by [Reference RobyRoby, p. 232]. If M and N have constant rank, then there is a faithfully flat $T \in {R\text {}\textbf {alg}}$ such that $M \otimes T$ and $N \otimes T$ are free. Since (3) holds over T by the free case, (2) and faithfully flat descent give that (3) holds. In the general case, $\mathscr {P}^d(M, N) = \prod \mathscr {P}^d(M_i, N_i)$ and $\mathsf {S}^d(M^*) \otimes N = \prod (\mathsf {S}^d(M_i^*) \otimes N_i)$ and the claim follows by the constant rank case.
(4) follows trivially from (3). For (1), note that $M^*$ is finitely generated projective, so are $\mathsf {S}^d(M^*)$ and the tensor product $\mathsf {S}^d(M^*) \otimes N$ . Applying (3) gives the claim.
One can create new polynomial laws from old by twisting by a line bundle, that is, by a rank 1 projective module.
Lemma 3.4. Let M and N be finitely generated projective Rmodules. Then for every $d \ge 0$ and every line bundle L, we have:

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

2. There is a natural isomorphism $\mathscr {P}^d(M, N) \cong \mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d})$ .
Proof. For (1), since $L^*$ is a line bundle, the natural map $(L^*)^{\otimes d} \to \mathsf {S}^d(L^*)$ is an isomorphism because it is so after faithfully flat base change. Since $\mathsf {S}^d(M^*) \otimes \mathsf {S}^d(L^*)$ is naturally identified with $\mathsf {S}^d((M \otimes L)^*)$ , combining Lemma 3.3(3),(4) then gives (1).
For (2), there are isomorphisms $\mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d}) \xrightarrow {\sim } \mathscr {P}^d(M, N) \otimes (L^*)^{\otimes d} \otimes L^{\otimes d}$ by (1) and Lemma 3.3(4). Since $L^{\otimes d} \otimes (L^*)^{\otimes d} \cong R$ , the claim follows.
Example 3.5. (References: [Stacks, Tag 03PK], [Reference Calmès and FaselCalF, Section 2.4.3], [Reference KnusKn, Section III.3]) Suppose L is a line bundle and there is an isomorphism $h \colon L^{\otimes d} \to R$ for some $d \ge 1$ . We call such a pair $[L, h]$ a dtrivialized line bundle. (In the case $d = 2$ , they are sometimes called discriminant modules.) Applying h to identify $N \otimes L^{\otimes d} \xrightarrow {\sim } N$ in Lemma 3.4(2) gives a construction that takes $f \in \mathscr {P}^d(M, N)$ and gives an element of $\mathscr {P}^d(M \otimes L, N)$ , which we denote by $[L, h] \cdot (M, f)$ .
For example, for each $\alpha \in R^{\times }$ , define ${\left \langle {\alpha }\right \rangle }$ to be $[L, h]$ as in the preceding paragraph, where $L = R$ and h is defined by $h(\ell _1 \otimes \cdots \otimes \ell _d) = \alpha \prod \ell _i$ . Clearly, ${\left \langle {\alpha \beta ^d}\right \rangle } \cong {\left \langle {\alpha }\right \rangle }$ for all $\alpha , \beta \in R^{\times }$ . Applying the construction in the previous paragraph, we find ${\left \langle {\alpha }\right \rangle } \cdot (M, f) \cong (M, \alpha f)$ .
Every $[L, h]$ with $L = R$ is necessarily isomorphic to ${\left \langle {\alpha }\right \rangle }$ for some $\alpha \in R^{\times }$ . In particular, if $\operatorname {\mathrm {Pic}}(R)$ has no dtorsion elements other than zero — for example, if R is a semilocal ring or a UFD [Stacks, Tags 0BCH, 02M9] — then each $[L, h]$ is isomorphic to ${\left \langle {\alpha }\right \rangle }$ for some $\alpha $ . The group scheme $\mu _d$ of dth roots of unity is the automorphism group of each $[L, h]$ , where $\mu _d$ acts by multiplication on L. The group $H^1(R, \mu _d)$ classifies pairs $[L, h]$ up to isomorphism.
We say that homogeneous polynomial laws related by the isomorphism in Lemma 3.4(2) are projectively similar, imitating the language from [Reference Auel, Bernardara and BolognesiAuBB, Section 1.2] for the case of quadratic forms ( $d = 2$ ). (This relationship was called “laxsimilarity” in [Reference Balmer and CalmèsBC].) We say that homogeneous degree d laws f and $[L, h] \cdot f$ for $[L, h] \in H^1(R, \mu _d)$ as in the preceding example are similar. If $\operatorname {\mathrm {Pic}}(R)$ is dtorsion, the two notions coincide.
For $f \in \mathscr {P}^d(M, N)$ , we define $\operatorname {\mathrm {Aut}}(f)$ to be the subgroup of $\operatorname {\mathrm {GL}}(M)$ consisting of elements g such that $fg = f$ as polynomial laws. In case M and N are finitely generated projective, so is $\mathscr {P}^d(M, N)$ , whence the functor $\mathbf {Aut}(f)$ from ${R\text {}\textbf {alg}}$ to groups defined by $\mathbf {Aut}(f)(T) = \operatorname {\mathrm {Aut}}(f_T)$ is a closed subgroupscheme of $\mathbf {GL}(M)$ .
Lemma 3.6. Let f and $f'$ be homogeneous polynomial laws on finitely generated projective modules. If f and $f'$ are projectively similar, then their automorphism groups are isomorphic.
Proof. By hypothesis, $f \in \mathscr {P}^d(M, N)$ and $f' \in \mathscr {P}^d(M \otimes L, N \otimes L^{\otimes d})$ for some modules M and N, line bundle L, and $d \ge 0$ . The group scheme $\mathbf {Aut}(f)$ is the closed subgroupscheme of $\mathbf {GL}(M)$ stabilizing the element f in $\mathsf {S}^d(M^*) \otimes N$ . Now, any element of $\mathbf {GL}(M)$ acts on $\mathsf {S}^d((M \otimes L)^*) \otimes (N \otimes L^{\otimes d})$ by defining it to act as the identity on L. In this way, we find a homomorphism $\mathbf {Aut}(f) \to \mathbf {Aut}(f')$ . Viewing M as $(M \otimes L) \otimes L^*$ and N as $(N \otimes L^{\otimes d}) \otimes (L^*)^{\otimes d}$ , and repeating this construction, we find an inverse mapping $\mathbf {Aut}(f') \to \mathbf {Aut}(f)$ .
4 Background on composition algebras
A notnecessarilyassociative Ralgebra C is an Rmodule with an Rlinear 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 Ralgebra as in [Reference PeterssonPe93] is such a C that is finitely generated projective as an Rmodule, is unital, and has a quadratic form $n_C \!: C \to R$ that allows composition (that is, such that $n_C(xy) = n_C(x)n_C(y)$ for all $x, y \in C$ ), satisfies $n_C(1_C) = 1$ , and whose bilinearization defined by $n_C(x, y) := n_C(x+y)  n_C(x)  n_C(y)$ gives an isomorphism $C \to C^*$ via $x \mapsto n_C(x, \cdot )$ . We say that a symmetric bilinear form with this property is regular. The quadratic form $n_C$ (which is unique by Proposition 4.3 below) is called the norm of C.
Remark 4.1. In the definition above, one can swap the condition $n_C(1_C) = 1$ with the requirement that the rank of C is nowhere zero, that is, $C \otimes F \ne 0$ for every field $F \in {R\text {}\textbf {alg}}$ .
We put $\operatorname {\mathrm {Tr}}_C(x) := n_C(x, 1_C)$ , a linear map $C \to R$ , called the trace of C. Trivially, $\operatorname {\mathrm {Tr}}_C(1_C) = 2$ . Lemma 3.1 gives that $1_C$ is unimodular, so we may identify R with $R1_C$ , and C is a faithful Rmodule. 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 Ralgebra with norm $n_C$ , then for every $S \in {R\text {}\textbf {alg}}$ , $C \otimes S$ is a composition Salgebra 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 (“CayleyHamilton”)
Let C be a composition algebra with norm $n_C$ , and define $\operatorname {\mathrm {Tr}}_C$ as above. Then
for all $x \in C$ .
Proof. Linearizing the composition law $n_C(xy) = n_C(x) n_C(y)$ , we find
for all $x, y, z, w \in C$ . Setting $z = x$ and $w = 1_C$ in (4.2), we find:
Combining these with (4.1), we find:
Since the bilinear form $n_C$ is regular, the claim follows.
A priori, a composition algebra is a unital algebra together with a quadratic form, the norm. The next result shows that these data are redundant.
Proposition 4.3. If C is a composition algebra, then the norm $n_C$ is uniquely determined by the algebra structure of C.
Proof. Let $n' \colon C \to R$ be any quadratic form making C a composition algebra, and write $\operatorname {\mathrm {Tr}}'$ for the corresponding trace $\operatorname {\mathrm {Tr}}'(x) := n'(x + 1_C)  n'(x)  n'(1_C)$ . Then $\lambda := \operatorname {\mathrm {Tr}}_C  \operatorname {\mathrm {Tr}}'$ (respectively, $q := n_C  n'$ ) is a linear (respectively, quadratic) form on C and the CayleyHamilton property yields
We aim to prove that $q = 0$ . Because $1_C$ is unimodular, it suffices to prove $\lambda = 0$ . This can be checked locally, so we may assume that R is local and, in particular, $C = R1_C \oplus M$ for a free module M. Now, $\operatorname {\mathrm {Tr}}_C(1_C) = 2 = \operatorname {\mathrm {Tr}}'(1_C)$ , so $\lambda (1_C) = 0$ . For $m \in M$ a basis vector, $\lambda (m)m$ belongs to $M \cap R1_C$ by (4.3), so it is zero, whence $\lambda (m) = 0$ , proving the claim.
Corollary 4.4. Let C be a unital Ralgebra. If there is a faithfully flat $S \in {R\text {}\textbf {alg}}$ such that $C \otimes S$ is a composition Salgebra, then C is a composition algebra over R.
Proof. Because the norm $n_{C \otimes S}$ of $C \otimes S$ is uniquely determined by the algebra structure, one obtains by faithfully flat descent a quadratic form $n_C \colon C \to R$ such that $n_C \otimes S = n_{C \otimes S}$ . Because $n_{C \otimes S}$ satisfies the properties required to make $C \otimes S$ a composition algebra and S is faithfully flat over R, it follows that the same properties hold for $n_C$ .
The following facts are standard, see, for example [Reference KnusKn, Section V.7]: Composition algebras are alternative algebras. The map $\bar {\ } \!: C \to C$ defined by $\overline {x} := \operatorname {\mathrm {Tr}}_C(x)1_C  x$ is an involution, that is, an Rlinear 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 Ralgebra has rank $2^e$ for $e \in \{ 0, 1, 2, 3 \}$ [Reference KnusKn, p. 206, Theorem V.7.1.6]. Therefore, specifying a composition Ralgebra C is equivalent to writing
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 Ralgebras, there is a distinguished one, the 2by2 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 Ralgebras, 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
where $\times $ is the ordinary cross product on $R^3$ . The quadratic form is
One says that a composition Ralgebra 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
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
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 BruhatTits building of the group $\operatorname {\mathrm {Aut}}(\operatorname {\mathrm {Zor}}(\mathbb {Q}_2))$ of type $\mathsf {G}_2$ over the 2adic numbers, compare [Reference Gan and YuGanY, Section 9].
5 Background on Jordan algebras
Paraquadratic algebras
A (unital) paraquadratic algebra over a ring R is an Rmodule J together with a quadratic map $U \colon J \to \operatorname {\mathrm {End}}_R(J)$ — that is, U is an element of $\mathscr {P}^2(J, \operatorname {\mathrm {End}}_R(J))$ — called the Uoperator, and a distinguished element $1_J \in J$ such that $U_{1_J} = \operatorname {\mathrm {Id}}_J$ . A homomorphism $\phi \colon J \to J'$ of paraquadratic Ralgebras is an Rlinear map such that $\phi (1_J) = 1_{J'}$ and $U^{\prime }_{\phi (x)} \phi (y) = \phi (U_x y)$ for all $x, y \in J$ , where $U'$ denotes the Uoperator in $J'$ .
Jordan algebras
As a notational convenience, we define a linear map $J \otimes J \otimes J \to J$ denoted $x \otimes y \otimes z \mapsto \{ xyz\}$ via
Evidently, $\{ x y z \} = \{ z y x \}$ for all $x, y, z \in J$ . A paraquadratic Ralgebra J is a Jordan Ralgebra if the identities
hold for all $x, y, z \in J\otimes S$ for all $S \in {R\text {}\textbf {alg}}$ . (Alternatively, one can define a Jordan Ralgebra 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 Ralgebra, then $J \otimes T$ is a Jordan Talgebra for every $T \in {R\text {}\textbf {alg}}$ (“Jordan algebras are closed under base change”). If J is a paraquadratic 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 nth power $x^n$ via
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 Ralgebra. Define $U_xy := xyx$ for $x, y \in A$ . Then $\{ xyz\} = xyz + zyx$ and A endowed with this Uoperator is a Jordan algebra denoted by $A^+$ . Note that for $x \in A$ and $n \ge 0$ , the nth 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 paraquadratic algebra J as in the preceding paragraph, one can define a commutative (bilinear) product $\bullet $ on J via
(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
which is the axiom classically called the “Jordan identity”.
In the opposite direction, given an Rmodule J with a commutative product $\bullet $ with identity element $1_J$ , we obtain a paraquadratic algebra by setting
If the original product satisfied the Jordan identity, then the paraquadratic 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 Ralgebra 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 Rsubmodule 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 Rmodule, $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ 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 Uoperator 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 Rstructure as a quadruple $\mathbf {M} = (M,1_{\mathbf {M}},\sharp ,N_{\mathbf {M}})$ consisting of an Rmodule 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
(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
(the linear trace). For $\mathbf {M}$ to be a cubic norm structure, we require that the identities
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 Uoperator by
which together with $1_{\mathbf {M}}$ converts the Rmodule M into a Jordan Ralgebra $J = J(\mathbf {M})$ [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 Ralgebra J is said to be cubic if there exists a cubic norm Rstructure $\mathbf {M}$ as in Definition 6.1 such that (i) $J = J(\mathbf {M})$ and (ii) $J = M$ is a finitely generated projective Rmodule. 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
for all $x \in J$ . For (6.6)–(6.8) and the first equation of (6.9), see [Reference McCrimmonMcC69, p. 499], while the second equation of (6.9) follows from the first, (6.7), and (6.8) via $x^4 = U_xx^2 = U_xx^{\sharp } + \operatorname {\mathrm {Tr}}_J(x)U_xx  S_J(x)U_x1_J = \operatorname {\mathrm {Tr}}_J(x)x^3  S_J(x)x^2 + N_J(x)x$ .
Remark 6.3. Note that the second equality of (6.9) derives from the first through formal multiplication by x. But, due to the paraquadratic character of Jordan algebras, this is not a legitimate operation unless $2$ is invertible in R. In fact, cubic Jordan algebras exist that contain elements x satisfying $x^2 = 0 \neq x^3$ [Reference JacobsonJ69, 1.31–1.32].
Example 6.4 (3by3 matrices)
We claim that $\operatorname {\mathrm {Mat}}_3(R)^+$ is a cubic Jordan algebra, in particular, it is $J(\mathbf {M})$ for $\mathbf {M} := (\operatorname {\mathrm {Mat}}_3(R), \operatorname {\mathrm {Id}}, \sharp , \det )$ , where $\sharp $ denotes the classical adjoint. We first verify that $\mathbf {M}$ is a cubic norm structure. Computing directly from the definition (6.1), we find that $T_{\mathbf {M}}(x,y) = \operatorname {\mathrm {Tr}}_{\operatorname {\mathrm {Mat}}_3(R)}(xy)$ , where the juxtaposition on the right is usual matrix multiplication. The formulas in (6.3) are obvious. For (6.4), the first two equations can be verified directly and the third equation is a standard property of the classical adjoint, completing the proof that $\mathbf {M}$ is a cubic norm structure. Similarly, one can check directly that the Uoperator defined from the cubic norm structure by (6.5) equals the Uoperator defined from the usual matrix product in Example 5.1, that is, $J(\mathbf {M}) = \operatorname {\mathrm {Mat}}_3(R)^+$ .
Lemma 6.5. Let J be a cubic Jordan Ralgebra and $x,y \in J$ .

1. x is invertible in J if and only if $N_J(x)$ is invertible in R. In this case
$$\begin{align*} x^{1} = N_J(x)^{1}x^{\sharp} \quad \text{and} \quad N_J(x^{1}) = N_J(x)^{1}. \end{align*}$$ 
2. Invertible elements of J are unimodular.

3. $N_J(U_xy) = N_J(x)^2N_J(y)$ and $N_J(x^2) = N_J(x)^2 = N_J(x^{\sharp })$ .
Proof. (1): If $N_J(x)$ is invertible in R, then (6.7) shows that so is x, with inverse $x^{1} = N_J(x)^{1}x^{\sharp }$ . Conversely, assume x is invertible in J. Then $y := (x^{1})^2$ satisfies $U_xy = 1_J$ , and (6.6) yields $1_J = N_J(U_xy)U_xy = N_J(x)^2N_J(y)1_J$ , hence
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
Without the assumption that J is finitely generated projective as an Rmodule, Lemma 6.5 would be false [Reference Petersson and RacinePeR85, Theorem 10].
Example 6.6. We endow the Rmodule $M := \operatorname {\mathrm {Her}}_3(C,\Gamma )$ from Definition 5.2 with a cubic norm Rstructure $\mathbf {M}= (M,1_{\mathbf {M}},\sharp ,N_{\mathbf {M}})$ , where $1_{\mathbf {M}}$ is the 3by3 identity matrix. An element $x \in \operatorname {\mathrm {Her}}_3(C, \Gamma )$ may be written as
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
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
with indices modulo (mod) 3, and the norm $N_{\mathbf {M}}$ by
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 Uoperator 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
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
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
for $i \in 1, 2, 3$ taken mod 3 and $a, b \in C$ .
Note that, for the Jordan algebra $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ , if we multiply $\Gamma $ by an element of $R^{\times }$ or any entry in $\Gamma $ by the square of an element of $R^{\times }$ , we obtain an algebra isomorphic to the original. Therefore, replacing $\Gamma $ by ${\left \langle { (\det \Gamma )^{1} \gamma _1, (\det \Gamma )^{1} \gamma _2, (\det \Gamma ) \gamma _3}\right \rangle }$ does not change the isomorphism class of $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ , and we may assume that $\gamma _1 \gamma _2 \gamma _3 = 1$ .
Example 6.8. When studying the Jordan Ralgebras $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ in the special case $R = \mathbb {R}$ , the preceding paragraph shows that it is sufficient to consider two choices for $\Gamma $ , namely, ${\left \langle {1, s, s}\right \rangle }$ for $s = \pm 1$ . We compute $T_{\operatorname {\mathrm {Her}}_3(C, \Gamma )}$ for each choice of C and $\Gamma $ . Regular symmetric bilinear forms over $\mathbb {R}$ are classified by their dimension and signature (an integer), so it suffices to specify the signature. If $C = \mathbb {R}$ , $\mathbb {C}$ , $\mathbb {H}$ , or $\mathbb {O}$ , the signature of $n_C$ is $2^r$ for $r = 0$ , 1, 2, 3, respectively. By (6.12), $T_J$ has signature $3(1 + 2^r)$ for $J = \operatorname {\mathrm {Her}}_3(C)$ and $3  2^r$ for $J = \operatorname {\mathrm {Her}}_3(C, {\left \langle {1, 1, 1}\right \rangle })$ . For $C$ the split composition algebra of rank $2^r$ for $r = 1$ , 2, or 3 and any $\Gamma$ , the signature of $n_C$ is 0 and the signature of $T_{\operatorname {\mathrm {Her}}_3(C, \Gamma)}$ is 3.
Remark 6.9. Alternatively, one could define the Jordan algebra structure on $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for an arbitrary ring R without referring to cubic norm structures as follows. Writing out the formulas for the Uoperator from Definition 5.2 in case $R = \mathbb {Q}$ , one finds that the formulas do not involve any denominators other than $\gamma _i$ terms and therefore make sense for any R regardless of whether 2 is invertible. This makes $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ a paraquadratic algebra. Because it is a Jordan algebra in case $R = \mathbb {Q}$ as in Definition 5.2, we conclude that $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is a Jordan algebra with no hypothesis on R by extension of identities [Reference BourbakiBouA2, Section IV.2.3, Theorem 2]. This alternative definition gives the same objects but is much harder to work with.
7 Albert algebras are Freudenthal algebras are Jordan algebras
Definition 7.1. A split Freudenthal Ralgebra is a Jordan algebra $\operatorname {\mathrm {Her}}_3(C)$ as in Definition 6.7 for some split composition Ralgebra C. Because split composition algebras are determined up to isomorphism by their rank function, so are split Freudenthal algebras.
A paraquadratic Ralgebra J is a Freudenthal algebra if $J \otimes S$ is a split Freudenthal Salgebra for some faithfully flat $S \in {R\text {}\textbf {alg}}$ . It is immediate that every Freudenthal algebra is a Jordan algebra. Since every split Freudenthal Ralgebra is finitely generated projective as an Rmodule for every R, the same is true for every Freudenthal Ralgebra J, and by the same reasoning, we see that the identity element $1_J$ is unimodular. Because the rank of a composition algebra takes values in $\{ 1, 2, 4, 8 \}$ , the rank of a Freudenthal algebra takes values in $\{ 6, 9, 15, 27 \}$ .
We are now prepared to define the objects named in the title of this paper.
Definition 7.2. An Albert Ralgebra is a Freudenthal Ralgebra of rank 27.
We continue to prove results about Freudenthal algebras, rather than merely Albert algebras. The extra generality comes at a low cost.
Proposition 7.3. For every composition Ralgebra C and every $\Gamma \in \operatorname {\mathrm {GL}}_3(R)$ , $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ is a Freudenthal algebra.
Proof. Replacing R with $R_e$ as in (4.4), we may assume that C has constant rank. There is a faithfully flat $S \in {R\text {}\textbf {alg}}$ such that $C \otimes S$ is a split composition algebra.
Consider $T := S[t_1, t_2, t_3] / (t_1^2  \gamma _1, t_2^2  \gamma _2, t_3^2  \gamma _3)$ . It is a free Smodule, so faithfully flat. Then $\operatorname {\mathrm {Her}}_3(C, \Gamma ) \otimes T$ is isomorphic to $\operatorname {\mathrm {Her}}_3(C \otimes T)$ as Jordan algebras, and the latter is a split Freudenthal algebra.
A Freudenthal algebra is said to be reduced if it is isomorphic to $\operatorname {\mathrm {Her}}_3(C, \Gamma )$ for some C and $\Gamma $ .
Example 7.4. Let J be a Freudenthal Ralgebra. 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
for each i, so $\alpha _i^2 = 1$ and $c_{i+2} = 0$ for all i. Then
Since $1_C$ is unimodular, $\alpha _{i+1} \alpha _{i+2} = 1$ for all i, proving the claim for this J.
For general J, let $S \in {R\text {}\textbf {alg}}$ be faithfully flat such that $J \otimes S$ is split. Then $x \in J$ maps to an element of $R1_J \otimes S \subseteq J \otimes S$ and so belongs to $R1_J \subseteq J$ . Since $U_{\zeta 1_J} = \zeta ^2 \operatorname {\mathrm {Id}}_J$ for $\zeta \in R$ , the claim follows.
The following result is well known when R is a field or perhaps a local ring (see, for example [Reference PeterssonPe19, Proposition 20]). We impose no hypothesis on R.
Proposition 7.5. Suppose C is a split composition Ralgebra of constant rank at least 2, that is, C is $R \times R$ , $\operatorname {\mathrm {Mat}}_2(R)$ , or $\operatorname {\mathrm {Zor}}(R)$ . Then $\operatorname {\mathrm {Her}}_3(C, \Gamma ) \cong \operatorname {\mathrm {Her}}_3(C)$ for all $\Gamma $ .
Proof. Define $\gamma _i$ via $\Gamma = {\left \langle {\gamma _1, \gamma _2, \gamma _3}\right \rangle }$ . We may assume $\gamma _1\gamma _2\gamma _3 = 1$ . Since $n_C$ is universal, there are invertible $p, q \in C$ such that $\gamma _2 = n_C(q^{1})$ and $\gamma _3 = n_C(p^{1})$ , so $\gamma _1 = n_C(pq)$ . We define $C^{(p,q)}$ to be a notnecessarilyassociative Ralgebra with the same underlying Rmodule structure and with multiplication $\cdot _{(p,q)}$ defined by
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
It is evidently an isomorphism of Rmodules, 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 Rlinear map
is an isomorphism of Ralgebras. 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
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 (twosided) 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 Ralgebra 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 Ralgebra C: Every onesided ideal in C is a twosided 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 paraquadratic Ralgebra J is the kernel of a homomorphism, that is, an Rsubmodule I such that
where we have written $U_I J$ for the Rspan 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 Rsubmodule I is an outer ideal if
Here are some observations about outer ideals:

1. Every ideal is an outer ideal.

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

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 Rmodule.

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 Rlinear $\lambda \!: J \to R$ such that $\lambda (1_J) = 1$ . Then $\alpha = \lambda (\alpha 1_J) = \sum \alpha _i \lambda (y_i)$ is in $\mathfrak {a}$ .
Theorem 8.2. Let J be a Freudenthal Ralgebra. Every outer ideal of J is an ideal. The maps $I \mapsto I \cap R1_J$ and are bijections between the set of outer ideals of J and the set of ideals of R.
Proof. It suffices to show that the stated maps are bijections, because then observation (3) implies that every outer ideal is of the form $\mathfrak {a} J$ and therefore an ideal. In view of (5) (noting that $1_J$ is unimodular), it suffices to verify that $(I \cap R1_J)J = I$ for every outer ideal I. First suppose that $J = \operatorname {\mathrm {Her}}_3(C)$ for some composition Ralgebra 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
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
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 Ralgebra J. There is a faithfully flat $S \in {R\text {}\textbf {alg}}$ , such that $J \otimes S$ is a split Freudenthal algebra. We have
where the first equality is because S is flat and the second is by the previous case, since $I \otimes S$ is an outer ideal. It follows that $I = (I \cap R1_J)J$ as desired.
Remark 8.3. In the proof above, the inclusion $(I \cap R1_J) J \subseteq I$ could instead have been argued as follows. Define $\operatorname {\mathrm {Sq}}(J)$ as the Rsubmodule of J generated by $x^2$ for $x \in J$ . Since $1_J$ is unimodular, one finds that $(I \cap R1_J) \operatorname {\mathrm {Sq}}(J) \subseteq I$ . Then, one argues that $\operatorname {\mathrm {Sq}}(J) = J$ for a split Freudenthal algebra, and that $\operatorname {\mathrm {Sq}}(J \otimes S) = \operatorname {\mathrm {Sq}}(J) \otimes S$ for all flat $S \in {R\text {}\textbf {alg}}$ .
Corollary 8.4. Let $\phi \colon J \to A$ be a homomorphism of paraquadratic R algebras. If J is a Freudenthal algebra and $1_A$ is unimodular in A, then $\phi $ is injective.
In particular, the corollary applies to every homomorphism between Freudenthal Ralgebras.
Proof. The kernel of $\phi $ is an ideal of J and therefore $\mathfrak {a} J$ for some ideal $\mathfrak {a}$ of R. For $\alpha \in \mathfrak {a}$ , we have $0 = \phi (\alpha 1_J) = \alpha \phi (1_J) = \alpha 1_A$ , so $\alpha = 0$ because $1_{A}$ is unimodular.
Because a Freudenthal algebra J is a projective Rmodule of rank $\ge 3$ , the corollary says that there is no homomorphism of paraquadratic Ralgebras $J \to R^+$ . This might be stated as J is not augmented or J has no counit.
A paraquadratic algebra J is said to be simple if the underlying module is not the zero module and if every ideal in J equals 0 or J. Theorem 8.2 immediately gives:
Corollary 8.5. If J is a Freudenthal Ralgebra and R is a field, then J is simple. $\Box $
Remark 8.6. There is also the notion of an inner ideal in a Jordan algebra (see [Reference McCrimmonMcC71b, Theorem 8] for a description of them for $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(R))$ ). The inner ideals are related to the projective homogeneous varieties associated with the group of isometries described in Section 15 and “outer automorphisms” relating these varieties (see [Reference RacineRac] and [Reference Carr and GaribaldiCarrG]).
9 Groups of type $\mathsf {F}_4$ and $\mathsf {C}_3$
In the following, for a Jordan Ralgebra J, we write $\operatorname {\mathrm {Aut}}(J)$ for the ordinary group of Rlinear automorphisms of J and $\mathbf {Aut}(J)$ for the functor from ${R\text {}\textbf {alg}}$ to groups such that $S \mapsto \operatorname {\mathrm {Aut}}(J \otimes S)$ . Recall that for every simple root datum, there is a unique simple group scheme over $\mathbb {Z}$ called a Chevalley group [Reference Demazure and GrothendieckDemG, Corollary XXIII.5.4], and every split simple algebraic group over a field is obtained from a unique Chevalley group by base change [Reference MilneMilne, Section 23g].
Lemma 9.1. Let J be a Freudenthal algebra of rank 15 or 27 over a ring R. Then $\mathbf {Aut}(J)$ is a semisimple Rgroup scheme that is adjoint (i.e., its center is the trivial group scheme). Its root system has type $\mathsf {C}_3$ if J has rank 15 and type $\mathsf {F}_4$ if J has rank 27. If J is the split Freudenthal algebra, then the group scheme $\mathbf {Aut}(J)$ is obtained from the Chevalley group over $\mathbb {Z}$ by base change.
Proof. First suppose that $R = \mathbb {Z}$ and J is split. If J has rank 15, then the proof of 14.19 in [Reference SpringerSp73] shows that the automorphisms of $J \otimes F$ for every field F are exactly the automorphisms of the algebra $\operatorname {\mathrm {Mat}}_6(F)$ with the split symplectic involution, which is the split adjoint group $\operatorname {\mathrm {PGSp}}_6$ of type $\mathsf {C}_3$ . For J of rank 27, $\mathbf {Aut}(J) \times F$ is split of type $\mathsf {F}_4$ by [Reference JacobsonJ71, Section 6] (written for Lie algebras), [Reference FreudenthalFr85, Satz 4.11] (written for $\mathbb {R}$ ), [Reference Springer and VeldkampSpV, Theorem 7.2.1] (if $\operatorname {\mathrm {char}} F \ne 2, 3$ ), or [Reference SpringerSp73, 14.24] in general.
Note that $\mathbf {Aut}(J) \times F$ is connected and smooth as a group scheme over F, and $\mathbf {Aut}(J)$ is finitely presented (because $\mathbb {Z}$ is noetherian and J is a finitely generated module), so it follows by [Reference Gan and YuGanY, Proposition 6.1] or [Reference Alsaody and GilleAlsG, Lemma B.1] that $\mathbf {Aut}(J)$ is smooth as a scheme over the Dedekind domain $\mathbb {Z}$ . In summary, $\mathbf {Aut}(J)$ is semisimple and adjoint of the specified type. Moreover, because $\mathbf {Aut}(J) \times \mathbb {Q}$ is split, $\mathbf {Aut}(J)$ is a Chevalley group [Reference ConradConrad, Theorem 1.4].
In the case of general R and J, let $S \in {R\text {}\textbf {alg}}$ be faithfully flat such that $J \otimes S$ is split. Then $\mathbf {Aut}(J) \times S$ is semisimple adjoint of the specified type. Certainly, $\mathbf {Aut}(J)$ is also smooth. Moreover, for each $\mathfrak {p} \in \operatorname {\mathrm {Spec}} R$ , there is a $\mathfrak {q} \in \operatorname {\mathrm {Spec}} S$ such that $\mathfrak {q} \cap R = \mathfrak {p}$ . Then the field of fractions $R(\mathfrak {p})$ of $R/\mathfrak {p}$ embeds in the field $S(\mathfrak {q})$ , so the algebraic closure $\overline {R(\mathfrak {p})}$ includes in the algebraic closure $\overline {S(\mathfrak {q})}$ . Because $\mathbf {Aut}(J) \times \overline {S(\mathfrak {q})}$ is adjoint semisimple of the specified type and this property is unchanged by replacing one algebraically closed field by a smaller one, the same holds over $\overline {R(\mathfrak {p})}$ . Since this holds for every $\mathfrak {p}$ , the claim is verified.
Remark 9.2. In case R is a field, the automorphism group of the split Freudenthal algebra of rank 6 or 9 can be deduced in a similar manner, referring to 14.17 and 14.16 in [Reference SpringerSp73]. The automorphism group of the split Freudenthal algebra of rank 9 is $\operatorname {\mathrm {PGL}}_3 \rtimes \mathbb {Z}/2$ . The automorphism group of the split Freudenthal algebra of rank 6 is the special orthogonal group of the quadratic form $x^2 + y^2 + z^2$ , that is, the group commonly denoted $\mathrm {SO}(3)$ . In particular, it is not smooth when R is a field of characteristic 2, and indeed, one can give examples of Freudenthal algebras of rank 6 over a field F of characteristic 2 that are not split by any separable field extension of F. (From this, it follows that such an algebra is not split by any étale Falgebra R.)
Suppose J, $J_0$ are Jordan Ralgebras 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 Ralgebra. 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 Ralgebra of rank $r = 15$ or 27. In the diagram
all arrows are bijections that are functorial in R.
Proof. The top arrow has the claimed codomain by Lemma 9.1.
Every Freudenthal algebra is split by some faithfully flat Ralgebra by definition, so the discussion preceding the theorem of the statement yields the lower left arrow.
As another consequence of Lemma 9.1, the conjugation map $\mathbf {Aut}(J_0) \to \mathbf {Aut}(\mathbf {Aut}(J_0))$ is an isomorphism that we use to identify $H^1(R, \mathbf {Aut}(J_0)) \xrightarrow {\sim } H^1(R, \mathbf {Aut}(\mathbf {Aut}(J_0)))$ . Every semisimple group scheme is split by some faithfully flat Ralgebra (even an étale cover) [Reference Demazure and GrothendieckDemG, Corollary XXIV.4.1.6], and the remarks before the statement of the theorem now define the lower right arrow.
The facts that the diagram commutes and that the diagonal arrows are bijective are general features of the machinery of descent as in [Reference WaterhouseWa, Theorem 17.6] or see [Reference Calmès and FaselCalF, 2.2.4.5].
The machinery of descent shows a more refined statement, where each of the boxes in the theorem are replaced by groupoids and the arrows are equivalences of groupoids. In that statement, the bottom box is replaced by the groupoid of $\mathbf {Aut}(J_0)$ torsors and the diagonal arrows are provided by [Reference GiraudGir, p. 151, Theorem III.2.5.1].
In the theorem, the set $H^1(R, \mathbf {Aut}(J_0))$ is naturally a pointed set and the bijections are actually of pointed sets, where the distinguished elements are $J_0$ in the upper left and $\mathbf {Aut}(J_0)$ in the upper right.
In case R is a field of characteristic different from 2, 3 and $r = 27$ , the theorem goes back to [Reference HijikataHij]. Or see [Reference Knus, Merkurjev, Rost and TignolKnMRT, 26.18]. The statement of the theorem is similar to various statements over a field that can be found in [Reference SerreSerre, Section III.1].
Corollary 9.4. For each Freudenthal Ralgebra J of rank 15 or 27, there is an étale cover $S \in {R\text {}\textbf {alg}}$ such that $J \otimes S$ is a split Freudenthal algebra.
Proof. Let $J_0$ be the split Freudenthal Ralgebra of the same rank as J. The image $\mathbf {Iso}(J, J_0)$ of J in $H^1(R, \mathbf {Aut}(J_0))$ is an $\mathbf {Aut}(J_0)$ torsor. Since $\mathbf {Aut}(J_0)$ is smooth (Lemma 9.1), there is an étale cover of R that trivializes $\mathbf {Iso}(J, J_0)$ .
Note that exactly the same kind of argument gives analogues of Lemma 9.1 and Theorem 9.3 for composition algebras, where $r = 4$ or 8, and the group is of type $\mathsf {A}_1$ or $\mathsf {G}_2$ , respectively.
10 Generic minimal polynomial of a Freudenthal algebra
Polynomials with polynomiallaw coefficients
Let J be a Jordan Ralgebra, $\mathscr {P}(J, R)$ the Ralgebra 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
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
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 Ralgebra 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 Ralgebras. An Rlinear map $\phi \!: J \to J'$ is an isomorphism of paraquadratic 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
Since $\phi (1_J) = 1_{J'}$ , the definition of the bilinear forms $T_J$ and $T_{J'}$ in (6.1) give:
for all $x, y$ . Therefore, on the one hand we have
On the other hand, we have
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 ith coordinate and define $\phi \colon \operatorname {\mathrm {Her}}_3(R \times R) \to \operatorname {\mathrm {Mat}}_3(R)^+$ by sending
This map is obviously Rlinear and surjective and sends the identity to the identity. One checks directly that $\phi $ preserves norms, that is, that $\det (\phi (x))$ equals $N(x)$ according to (6.11). Because $\operatorname {\mathrm {Her}}_3(R \times R)$ and $\operatorname {\mathrm {Mat}}_3(R)^+$ are both cubic Jordan algebras with regular trace bilinear forms, the proof of the “if” direction of Lemma 10.2 shows that $\varphi $ is an isomorphism of Jordan algebras.
11 Basic classification results for Albert algebras
In the case where R is a field, such as the real numbers, a finite field, a local field, or a global field, one can find in many places in the literature classifications of Albert algebras proved using techniques involving algebras as in [Reference Springer and VeldkampSpV, Section 5.8]. For such an R, groups of type $\mathsf {F}_4$ can be classified using techniques from algebraic groups, such as in [Reference Platonov and RapinchukPlRap, Chapter 6] or [Reference GilleGil19]. The two approaches are equivalent by Theorem 9.3.
Example 11.1 (Albert algebras over $\mathbb {R}$ )
Up to isomorphism, there are three Albert $\mathbb {R}$ algebras, namely, the split one $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(\mathbb {R}))$ , $\operatorname {\mathrm {Her}}_3(\mathbb {O}, {\left \langle {1, 1, 1}\right \rangle })$ , and $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ . Rather than proving this in the language of Jordan algebras as in [Reference Albert and JacobsonAlbJ, Theorem 10], one may leverage Theorem 9.3 as follows. The three algebras are pairwise nonisomorphic because their trace forms are (Example 6.8). At the same time, a computation in the Weyl group of $\mathsf {F}_4$ as in [Reference SerreSerre, Section III.4.5], [Reference Borovoi and EvenorBorE, 14.1], or [Reference Adams and TaïbiAdT, Table 3] shows that $H^1(\mathbb {R}, \mathbf {Aut}(\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(\mathbb {R}))))$ has three elements. That is, there are exactly three isomorphism classes of simple affine group schemes over $\mathbb {R}$ of type $\mathsf {F}_4$ , so we have found all of them.
Example 11.2 (Albert algebras over global fields)
Let A be an Albert Falgebra 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
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 Falgebras. 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 Falgebras. This result, for number fields and with a proof in the language of octonion algebras, dates back to [Reference ZornZ, p. 400].
In the special case where F has a unique real embedding (e.g., $F = \mathbb {Q}$ ), the two isomorphism classes of octonion algebras are $\operatorname {\mathrm {Zor}}(F)$ and $\mathcal {O} \otimes F$ , and the three isomorphism classes of Albert algebras are $\operatorname {\mathrm {Her}}_3(\operatorname {\mathrm {Zor}}(F))$ and $\operatorname {\mathrm {Her}}_3(\mathcal {O} \otimes F, {\left \langle {1, s, s}\right \rangle })$ for $s = \pm 1$ .
Below, we will focus our attention on classification results in the case where R is not a field. We translate known results about cohomology of affine group schemes into the language of Albert algebras.
Proposition 11.3. If R is: (1) a complete discrete valuation ring whose residue field is finite or (2) a finite ring, then every Freudenthal Ralgebra of rank 15 or 27 and every quaternion or octonion Ralgebra is split.
Proof. In view of Theorem 9.3 and its analogue for composition algebras, it suffices to prove that $H^1(R, \mathbf {G}) = 0$ for $\mathbf {G}$ a simple Rgroup scheme of type $\mathsf {F}_4$ or $\mathsf {C}_3$ obtained by base change from a Chevalley group over $\mathbb {Z}$ . In case (1), this is [Reference ConradConrad, Proposition 3.10]. In case (2), we apply the following lemma.
Lemma 11.4. If R is a finite ring and $\mathbf {G}$ is a smooth connected Rgroup scheme, then $H^1(R, \mathbf {G}) = 0$ .
Proof. If R is not connected, then it is a finite product $R = \prod R_i$ , where each ring $R_i$ is finite, so $H^1(R, \mathbf {G}) = \prod H^1(R_i, \mathbf {G} \times R_i)$ . Therefore, it suffices to assume that R is connected.
Suppose $\mathbf {X}$ is a $\mathbf {G}$ torsor. Our aim is to show that $\mathbf {X}$ is the trivial torsor, that is, $\mathbf {X}(R)$ is nonempty. Put $\mathfrak {a}$ for the nil radical $\operatorname {\mathrm {Nil}}(R)$ of R. Because R is finite, there is some minimal $m \ge 1$ such that $\mathfrak {a}^m = 0$ . We proceed by induction on m. If $m = 1$ , then R is reduced and connected, so it is a finite field and $H^1(R, \mathbf {G}) = 0$ by Lang’s theorem. For the case $m \ge 2$ , put $I := \mathfrak {a}^{m1}$ . The ring $R/I$ has $\operatorname {\mathrm {Nil}}(R/I)^{m1} = (\operatorname {\mathrm {Nil}}(R)/I)^{m1} = 0$ , so by induction, $\mathbf {X}(R/I)$ is nonempty. On the other hand, $I^2 = \mathfrak {a}^{2m2} = \mathfrak {a}^m \cdot \mathfrak {a}^{m2} = 0$ and $\mathbf {X}$ is smooth, so the natural map $\mathbf {X}(R) \to \mathbf {X}(R/I)$ is surjective.
Example 11.5. Suppose R is a Dedekind domain, and write F for its field of fractions. For $\mathbf {G}$ a Chevalley group of type $\mathsf {G}_2$ , $\mathsf {F}_4$ , or $\mathsf {E}_8$ , the map $H^1(R, \mathbf {G}) \to H^1(F, \mathbf {G})$ has zero kernel [Reference HarderHa67, Satz 3.3]. Consequently, if A is an Albert or octonion Ralgebra and $A \otimes F$ is split, then the Ralgebra A is split.
In particular, if F is a global field with no real embeddings, then every Albert or octonion Falgebra is split, so every Albert or octonion Ralgebra is split.
In the case where F is a number field with a real embedding, we provide the following partial result, which relies on Example 11.1.
Proposition 11.6. Suppose F is a number field and R is a localization of its ring of integers at finitely many primes. If A is an Albert (respectively, octonion) Falgebra such that $A \otimes \mathbb {R}$ is not isomorphic to $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ (respectively, $\mathbb {O}$ ) for every embedding $F \hookrightarrow \mathbb {R}$ , then there is an Albert (respectively, octonion) Ralgebra B such that $B \otimes F \cong A$ and B is uniquely determined up to Risomorphism.
Proof. Write $\mathbf {G}$ for the automorphism group of the split Albert (respectively, octonion) Falgebra. Write $H^1_{\mathrm {ind}}(R, \mathbf {G}) \subseteq H^1(R, \mathbf {G})$ for the isomorphism classes of Ralgebras B such that $B \otimes F_v$ is not $\operatorname {\mathrm {Her}}_3(\mathbb {O})$ (respectively, $\mathbb {O}$ ), that is, such that $\mathbf {Aut}(B) \times F_v$ is not compact, for all real places v of F. Since $\mathbf {G}$ is simply connected, Strong Approximation gives that the natural map $H^1_{\mathrm {ind}}(R, \mathbf {G}) \to H^1_{\mathrm {ind}}(F, \mathbf {G})$ is an isomorphism [Reference HarderHa67, Satz 4.2.4], which is what is claimed.
12 The number of generators of an Albert algebra
The goal of this section is to prove Proposition 12.1, which is inspired by the work of FirstReichstein [Reference First and ReichsteinFiR] generalizing the ForsterSwan theorem. Let J be a paraquadratic Ralgebra. By a (paraquadratic) subalgebra of J, we mean a submodule $J' \subseteq J$ containing $1_J$ and closed under the Uoperator, that is, such that $U_x y \in J'$ for all $x,y \in J'$ . For any subset $S \subseteq J$ , the smallest subalgebra of J containing S is called the subalgebra generated by S; if this subalgebra is all of J, we say that S generates J.
Proposition 12.1. For every noetherian ring R, every Albert Ralgebra can be generated in the sense of the preceding paragraph by $3 + \dim \operatorname {\mathrm {Max}} R$ elements.
In the statement, $\operatorname {\mathrm {Max}} R$ is the topological space whose points are the maximal ideals of R, endowed with the subspace topology inherited from $\operatorname {\mathrm {Spec}} R$ . It is evident that $\dim \operatorname {\mathrm {Max}} R \le \dim \operatorname {\mathrm {Spec}} R$ , also known as the Krull dimension. Beyond this inequality, the two numbers may be quite different (e.g., for a local ring, $\dim \operatorname {\mathrm {Max}} R = 0$ and $\dim \operatorname {\mathrm {Spec}} R$ can be any number).
If 2 is invertible in R, the bound in the proposition is Corollary 4.2c in [Reference First and ReichsteinFiR]. The contribution here is to remove the hypothesis on 2. In the special case where R contains an infinite field of characteristic $\ne 2$ , a different (and possibly smaller) upper bound is given in [Reference First, Reichstein and WilliamsFiRW, Section 13].
The results in [Reference First and ReichsteinFiR] reduce the proof of the proposition to the case where R is a field. For a field of characteristic not $2$ , a proof may be read off from [Reference McCrimmonMcC04, p. 112]; we give a characteristicfree 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 (finitedimensional) 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
by defining
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 Falgebra which we denote by $J(A, \mu )$ .
Example 12.2. (a): Let E be a cubic étale Falgebra. 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 Falgebra 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 Falgebra 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,