                                                                      22 November 2005

Generically algebraic Jordan algebras over commutative rings

Ottmar Loos

Institut für Mathematik University of Innsbruck, A-6020 Innsbruck, Austria

Abstract. The theory of the generic minimum polynomial, norm and trace is developed for quadratic Jordan algebras which are finitely generated and projective modules over an arbitrary commutative base ring, using scheme-theoretic methods. We recover, with new proofs, most of the classical theory over fields, and also obtain a number of results which are new even in the classical setting.

Introduction

The theory of the generic minimum polynomial, norm and trace for linear and quadratic Jordan algebras of finite dimension over a field is well-known [14, 16, 26]. In this paper, we extend the theory to quadratic Jordan algebras which are finitely generated and projective modules over an arbitrary commutative base ring. With an appropriate definition of generically algebraic algebra, we are able to recover most of the results of the classical theory. Our methods involve scheme theory in an essential way. They yield new proofs of classical results as well as new ones even in the classical setting. The theory developed here applies to associative or alternative algebras by considering the associated Jordan algebra with quadratic operators U_{x}y = xyx. The results seem to be new even in this case.

Let k be an arbitrary commutative ring with unity, let J be a quadratic Jordan algebra which is finitely generated and projective as a k-module, and denote by A the algebra of polynomial laws on J. For a polynomial f(t) in the indeterminate t with coefficients in A and an element x in a base ring extension J ⊗_{k} R of J, we denote by f(t; x) ∈ R[t] the polynomial obtained by evaluating the coefficients of f at x. We say J is generically algebraic (see 2.2) if there exists a locally monic polynomial m(t) ∈ A[t] such that

(i) for all x in all base ring extensions, substitution of x for t in m(t; x) and tm(t; x) yields zero,

(ii) for every prime ideal 𝔭 of k, the base change of m(t) from k to the quotient field κ(𝔭) of k/𝔭 is the classical generic minimum polynomial of J ⊗_{k} κ(𝔭).

These conditions are natural for the following reasons: (i) just says that every x satisfies its own generic minimum polynomial m(t; x). (Note that the condition m(x; x) = 0 is sufficient in (i) provided J = B^{+} is the Jordan algebra associated to an associative or alternative algebra B). Condition (ii) is forced upon us if we wish the definition to be invariant under base change and consistent with that over fields. A polynomial m(t) satisfying (i) and (ii) is unique (Prop. 2.7); it is called the generic minimum polynomial of J.

The question then arises which finitely generated and projective Jordan algebras are generically algebraic in this sense. It turns out that there always exist polynomials satisfying (i), for instance, det(t^{2}Id − U_{x}). The well-known constructions of Jordan algebras from quadratic forms with base point and from cubic norm structures yield examples of polynomials of degree 2 and 3 satisfying (i) [22, 30, 20]. Condition (ii) is much more restrictive, and there are many examples of finitely generated and projective Jordan algebras which are not generically algebraic. On the other hand, the obvious analogues over rings of the simple finite-dimensional algebras over algebraically closed fields are all generically algebraic in this sense.

We now give a more detailed account of the contents. After a preliminary section collecting facts on locally monic polynomials, schemes and pure submodules, section 1 deals with algebraic elements. Let J be a quadratic Jordan algebra which is finitely generated and projective as a k-module. Ignoring for the moment the difficulties arising from the lack of power-associativity of quadratic Jordan algebras, we define an element a ∈ J to be algebraic if the subalgebra k[a] generated by a is a direct summand of J. (The actual definition in 1.4 is more involved and makes sense even without assumptions on the k-module structure of J). This is a much more stringent condition than that a be integral, i.e., that it satisfy some monic polynomial. Algebraic elements have well-defined minimum polynomials which behave well under base change. The functor of algebraic elements is a finitely presented quasi-affine k-scheme (Prop. 1.14).

In section 2, we develop the general theory of generically algebraic algebras, establish results on ascent and descent, and prove the uniqueness and the basic properties of the generic minimum polynomial (Prop. 2.7, Th. 2.11). An important tool is the fact that the primitive elements, i.e., the algebraic elements of highest degree, form an open dense subscheme (Lemma 2.6).

In Theorem 3.1 of section 3 we show that generically algebraic Jordan algebras are stable under isotopy and compute the generic minimum polynomial of an isotope. For algebras over fields, this is due to N. Jacobson [12] in the linear case, and to K. McCrimmon [26] in the quadratic case. For central separable algebras over rings containing \frac{1}{2}, it was proved by R. Bix [2] by a case-by-case verification. McCrimmon’s proof made use of the composition law N(U_{x}y) = N(x)^{2}N(y) for the generic norm. Our proof is actually a simplification of McCrimmon’s and yields the composition law as a corollary. Let us point out here that the classical proof of the composition law relies on the factoriality of the polynomial ring in several variables over a field and therefore does not carry over to base rings. We then prove the symmetry property of the coefficients of the generic minimum polynomial of an isotope and the fact that these coefficients are polynomial laws on J × J (Theorem 3.5). We finally derive in 3.11 explicit formulas for these coefficients which are new even for algebras over fields.

It is a curious phenomenon of quadratic Jordan algebras that they may contain elements which are not power-associative in the sense that the subalgebra generated by such an element is not special. As mentioned above, this brings complications in the definition of algebraic element. In section 4, we study this phenomenon in more detail and show in particular that algebraic elements of degree ⩽3 are automatically power-associative (Prop. 4.5). As a consequence, generically algebraic Jordan algebras of degree ⩽3 over fields are strictly power-associative (Cor. 4.4). This gives a partial answer to a question raised by K. McCrimmon in [23].

Finally, in section 5 we prove that a module isomorphism between generically algebraic Jordan algebras of degree 3 which preserves squares, traces and unit elements, is already an algebra isomorphism (Th. 5.1). As a consequence, the automorphism group of the exceptional Jordan algebra in characteristic 2 is isomorphic, by restriction, to the automorphism group of the 2-Lie algebra of its space of trace zero elements, and a similar result holds for derivations (Cor. 5.4).

Acknowledgements. I wish to thank H. P. Petersson who has read an earlier version of this paper and offered many valuable comments.

This paper was prepared in part while the author was a guest of the Department of Algebra, Geometry and Topology of the University of Málaga, and the Department of Mathematics of the University of Oviedo. The hospitality of these institutions is gratefully acknowledged.

0. Preliminaries

0.1. Notations and conventions. Throughout, k denotes an arbitrary commutative ring. Spec(k) is the set of prime ideals of k, with the Zariski topology, and for a ring homomorphism ϱ: k → k′, Spec(ϱ): Spec(k′) → Spec(k) is the continuous map 𝔮 ↦ ϱ^{−1}(𝔮). The quotient field of k/𝔭 (𝔭 ∈ Spec(k)) is written κ(𝔭). We denote by k-alg the category of commutative associative k-algebras. Unsubscripted tensor products are understood over k. For R ∈ k-alg and X a k-module, we often abbreviate X_{R} = X ⊗ R; for x ∈ X we put x_{R} := x ⊗ 1_{R} ∈ X_{R}, and for a homomorphism h: X → Y of k-modules, we denote by h_{R}: X_{R} → Y_{R} the R-linear extension of h; i.e., h_{R}(x ⊗ r) = h(x) ⊗ r, for all x ∈ X, r ∈ R. Thus we have h_{R}(x_{R}) = h(x)_{R}.

By a Jordan algebra J over k we always mean a unital quadratic Jordan algebra, unless otherwise specified. The unit element is written 1_{J} or simply 1, the set of invertible elements is J^{×}. For an alternative (or associative) algebra B, the associated Jordan algebra with quadratic operators U_{x}y = xyx is denoted B^{+}.

0.2. Vanishing sets. Let M be a k-module and x ∈ M. With x we associate the ideals

𝔞 := Ann(x) = {λ ∈ k : λx = 0} and 𝔟 := ⟨x, M^{∗}⟩ = {⟨x, β⟩ : β ∈ M^{∗}}

of k, where M^{∗} is the dual module and ⟨ , ⟩: M × M^{∗} → k is the canonical pairing. Clearly, 𝔞 · 𝔟 = 0.

Now let M be finitely generated and projective. In geometric terms, M may then be considered as a vector bundle over the affine scheme S = Spec(k) and x as a section of this bundle. To carry this a bit further, let x(𝔭) := x⊗1_{κ(𝔭)} ∈ M(𝔭) := M ⊗ κ(𝔭), for all 𝔭 ∈ S. Thus M(𝔭) is the fibre of M and x(𝔭) is the value of the section x at the point 𝔭. We claim that

x(𝔭) = 0 ⟺ 𝔟 ⊂ 𝔭.  (1)

Indeed, M(𝔭) being a vector space over κ(𝔭), we have x(𝔭) = 0 if and only if ⟨x(𝔭), α⟩ = 0, for all α ∈ (M(𝔭))^{∗}. Since M is finitely generated and projective, the canonical homomorphism M^{∗} ⊗ R → (M ⊗ R)^{∗} is an isomorphism, for all R ∈ k-alg [4, II, §4.2, Prop. 2(ii)]. Thus

x(𝔭) = 0 ⟺ ⟨x, β⟩ ⊗ 1_{κ(𝔭)} = 0 for all β ∈ M^{∗}
⟺ ⟨x, β⟩ ∈ 𝔭 for all β ∈ M^{∗}
⟺ 𝔟 ⊂ 𝔭.

We may express (1) by

x^{−1}(0) = V (𝔟),  (2)

where V (𝔟) = {𝔭 ∈ S : 𝔭 ⊃ 𝔟} is the vanishing set of 𝔟, cf. [3, II, §4.3].

0.3. Unimodular elements. An element x of a k-module M is said to be unimodular if k ·x is a free k-module of rank 1 and a direct summand of M, equivalently, if there exists an element β ∈ M^{∗} such that ⟨x, β⟩ = 1, i.e., 𝔟 = k. Thus 0.2.1 yields, in case M is finitely generated and projective,

x is unimodular ⟺ x(𝔭) ≠ 0 for all 𝔭 ∈ S.  (1)

Let J be a Jordan algebra over k, finitely generated and projective as a k-module, with unit element 1_{J}. We claim that

J is a faithful k-module ⟺ 1_{J} is unimodular.  (2)

Indeed, J = 0 if and only if 1_{J} = 0, and the unit element is compatible with base change: 1_{J_{R}} = (1_{J})_{R}, for all R ∈ k-alg. By general facts on finitely generated and projective modules, J is faithful if and only if J(𝔭) ≠ 0, for all 𝔭 ∈ S. Now (2) follows from (1).

0.4. Lemma. Let M be a finitely generated and projective k-module and x ∈ M. Then the following conditions are equivalent:

(i) x^{−1}(0) is open in S = Spec(k),

(ii) there exists an idempotent ε ∈ k such that 𝔟 = k · ε, (iii) k · x is a direct summand of M.

If these conditions hold, ε is uniquely determined and is called the support idempotent of x. The annihilator of x is then 𝔞 = k · (1 − ε).

Proof. (i) ⟹ (ii) follows from well-known facts about the correspondence between open and closed subsets of S and idempotents of k [3, II, §4.3, Prop. 15].

(ii) ⟹ (iii): Choose β ∈ M^{∗} with ⟨x, β⟩ = ε. Then we have x = εx: Indeed, ⟨(1 − ε)x, M^{∗}⟩ = (1 − ε)𝔟 = (1 − ε)ε · k = 0. Since the canonical map M → M^{∗∗} is injective, it follows that (1 − ε)x = 0. Now one checks easily that π(y) := ⟨y, β⟩x defines a projection of M with image k · x.

(iii) ⟹ (i): Since M is finitely generated and projective so is k · x. Hence the rank function of k · x is continuous on S, which implies that x^{−1}(0) = {𝔭 ∈ S : rk_{𝔭}(k · x) = 0} is open.

Uniqueness of ε follows from the fact that it is the unit element of 𝔟. Finally, 0 = 𝔞𝔟 = ε𝔞 implies 𝔞 ⊂ k·(1−ε), and the reverse inclusion follows from (1−ε)x = 0.

0.5. Lemma. Let J be a not necessarily unital Jordan algebra over k which is finitely generated and projective as a k-module, and let e ∈ J be an idempotent of J. Then e satisfies the equivalent conditions of 0.4.

Proof. Consider the Peirce decomposition J = J_{2}(e) ⊕ J_{1}(e) ⊕ J_{0}(e) of J with respect to e. Then the J_{i}(e), being direct summands of J, are finitely generated and projective. Moreover, Peirce decomposition is compatible with base change: For all R ∈ k-alg, we have (J_{R})_{i}(e) = J_{i}(e) ⊗ R. Finally, e = 0 if and only if J_{2}(e) = 0. Hence

e(𝔭) = 0 ⟺ J_{2}(e) ⊗ κ(𝔭) = 0 ⟺ rk_{𝔭} J_{2}(e) = 0.

Since the rank function of J_{2}(e) is continuous on S, it follows that e^{−1}(0) is open.

0.6. Lemma. Let X be a k-module, let x_{1}, …, x_{n} ∈ X and let N be the k-span of x_{1}, …, x_{n}. Then the following conditions are equivalent:

(i) N is free with basis x_{1}, …, x_{n} and a direct summand of X, (ii) y := x_{1} ∧ ⋯ ∧ x_{n} is unimodular in ⋀^{n} X.

If X is finitely generated and projective these conditions are equivalent to

(iii) x_{1}(𝔭), …, x_{n}(𝔭) are linearly independent over κ(𝔭), for all 𝔭 ∈ Spec(k).

If these conditions hold, we have

x ∈ N ⟺ y ∧ x = 0,  (1)

for all x ∈ X.

Proof. (i) ⟹ (ii): Let X = N ⊕P and define linear forms α_{i} ∈ X^{∗} by ⟨x_{i}, α_{j}⟩ = δ_{ij} and ⟨α_{i}, P ⟩ = 0. Let β ∈ ( ⋀^{n} X)^{∗} be the image of α_{1} ∧ ⋯ ∧ α_{n} under the canonical homomorphism ⋀^{n}(X^{∗}) → ( ⋀^{n} X)^{∗}. Then ⟨β, y⟩ = det(⟨α_{i}, x_{j}⟩) = 1.

(ii) ⟹ (i): Define α_{i} ∈ X^{∗} by

⟨x, α_{i}⟩ := ⟨x_{1} ∧ ⋯ ∧ x_{i−1} ∧ x ∧ x_{i+1} ∧ ⋯ ∧ x_{n}, β⟩.

Then ⟨x_{i}, α_{j}⟩ = δ_{ij}. This clearly implies that N is free with basis x_{1}, …, x_{n}. Furthermore, X = N ⊕ P where P = ⋂_{i=1}^{n} Ker(α_{i}), so N is a direct summand of X.

(ii) ⟺ (iii): Since exterior powers commute with base change, we have y(𝔭) = x_{1}(𝔭) ∧ ⋯ ∧ x_{n}(𝔭), so the assertion follows from 0.3.1. Finally, (1) follows from [4, III, §7.9, Prop. 13].

0.7. k-functors and schemes. Following [6], we will consider schemes over k as special functors from k-alg to the category of sets, also called k-functors. Morphisms between functors are natural transformations. To fix notations, we give some examples.

(a) Let A ∈ k-alg. The affine scheme defined by A is the k-functor Spec(A) given by

Spec(A)(R) = Hom_{k-alg}(A, R).

Note the following special cases: For A = {0} we obtain the “empty functor”, mapping R to ∅ if R ≠ {0} and to a one-point set (consisting of the unique homomorphism {0} → {0}) if R = {0}.

For A = k, Spec(k) is the “one-point” functor because Hom_{k-alg}(k, R) consists of the unique homomorphism k → R making R a k-algebra.

For A = k^{I} where I is a finite set, we obtain a functor denoted by I_{k} and called the constant k-functor defined by I, although “locally constant” would be more apt. It can be described as follows: For each R ∈ k-alg, I_{k}(R) is the set of complete families of orthogonal idempotents (ε_{i})_{i∈I} in R. By the well-known relation between idempotents and open and closed subsets of Spec(R), I_{k}(R) can also be considered as the set of continuous (= locally constant) maps Spec(R) → I, where I has the discrete topology.

(b) A k-scheme X is a local k-functor for which there exist open affine subschemes U_{i} covering X [6, I, §1, 3.11]. Refer to [6, I, §1, 3.6, §2, 4.1] for the notion of an open (closed) subfunctor. “Covering” means that X(K) = ⋃ U_{i}(K) for all fields K ∈ k-alg. The union need not be disjoint, and when R is not a field, X(R) may be strictly bigger than the union of the U_{i}(R).

(c) The example k^{I} of (a) generalizes in the obvious way to the case of an arbitrary set I, where now an element of I_{k}(R) is a family (ε_{i}) as before with only finitely many ε_{i} ≠ 0. This corresponds to the fact that Spec(R) is quasicompact, and hence a continuous map to the discrete space I can take only finitely many values. If I is infinite, I_{k} is still a scheme but no longer affine.

(d) Let X_{i} (i ∈ I) be a family of k-functors with the property that X_{i}({0}) is a one-point set. We define ∐_{i∈I} X_{i} to be the functor X given as follows: The elements of X(R) are the pairs (ε, x) where ε = (ε_{i})_{i∈I} ∈ I_{k}(R), and x = (x_{i})_{i∈I} ∈ ∏_{i∈I} X_{i}(ε_{i}R). If the X_{i} are local, this means that X is the local functor associated to the functor R ↦ ˙⋃_{i∈I} X_{i}(R). There is a unique morphism deg: ∐_{i∈I} X_{i} → I_{k} such that X_{i} = deg^{−1}({i}_{k}), given by (ε, x) ↦ ε.

0.8. Locally monic polynomials. Let t be an indeterminate. A polynomial f(t) ∈ k[t] is called locally monic if it satisfies the following equivalent conditions:

(i) For all 𝔭 ∈ S = Spec(k), the localizations f(t)_{𝔭} ∈ k_{𝔭}[t] are monic, (ii) there exists a finite subset D of ℕ and a family (ε_{d})_{d∈D} of orthogonal idempotents of k with sum 1 such that f(t) = ∑_{d∈D} ε_{d}f_{d}(t) where f_{d}(t) is monic of degree d.

If these conditions hold, the function deg f: S → ℕ, 𝔭 ↦ deg f(t)_{𝔭}, is locally constant and called the degree of f(t). It is given by

(deg f)(𝔭) = d ⟺ ε_{d} ≡ 1 mod 𝔭.  (1)

(i) ⟹ (ii): Write f(t) = ∑_{j=0}^{n} λ_{j}t^{j} where λ_{j} ∈ k and let S_{d} ⊂ S be the set of 𝔭 such that deg f(t)_{𝔭} = d, for d ∈ D := {0, …, n}. Then

S_{d} = {𝔭 ∈ S : (λ_{d})_{𝔭} = 1 and (λ_{j})_{𝔭} = 0 for j = d + 1, …, n}

which shows that the S_{d} are open in S, being finite intersections of open sets. Here we use the following general fact: If M is any k-module and x ∈ M then {𝔭 ∈ S : x_{𝔭} = 0} is the complement of the support of the monogenous k-module k · x and hence open in S, cf. [3, II, §4.4, Prop. 17]. By our assumption (i), S = ˙⋃_{d∈D} S_{d}. Hence the (S_{d})_{d∈D} form a decomposition of S into open and closed subsets, corresponding to a family (ε_{d})_{d∈D} of orthogonal idempotents with sum 1_{k} via

𝔭 ∈ S_{d} ⟺ ε_{d} ≡ 1 mod 𝔭.  (2)

Let f_{d}(t) = ε_{d}f(t). Then f(t) = ∑_{d∈D} ε_{d}f_{d}(t) and f_{d}(t)_{𝔭} = \{ \begin{matrix}0 & if 𝔭 ∉ S_{d} \\ f(t)_{𝔭} & if 𝔭 ∈ S_{d} \end{matrix} \}.

In particular, f_{d}(t)_{𝔭} is monic of degree d for all 𝔭 ∈ S_{d}. By standard facts on localization [3, II, §3.3, Cor. 2 of Th. 1], this implies that f_{d}(t) is monic of degree d. Moreover, deg f is constant equal to d on S_{d} and hence locally constant on S, and (1) follows from (2).

(ii) ⟹ (i): Let S_{d} := Spec(kε_{d}). Then the S_{d} define a decomposition of S into open and closed subsets, and for 𝔭 ∈ S_{d} and d ≠ d′ we have f_{d′}(t)_{𝔭} = 0. Hence f(t)_{𝔭} = f_{d}(t)_{𝔭} is monic for all 𝔭 ∈ S.

Remark. The set D can be chosen as any finite subset of ℕ containing the range of the function deg f. In particular, by reducing D to the range of deg f, all ε_{d} are nonzero and, correspondingly, the S_{d} non-empty. However, it would be inconvenient to require this condition, see, e.g., 1.8. To be consistent, we then must consider the unique element of the polynomial ring over the zero ring as monic.

Locally monic polynomials behave well under base change: Let R ∈ k-alg and ϱ: k → R the homomorphism making R a k-algebra. Then f(t)_{R} ∈ R[t], obtained by applying ϱ to the coefficients of f(t), is locally monic and the degree functions are related by

(deg f_{R})(𝔮) = (deg f)(ϱ^{−1}(𝔮)),  (3)

i.e., the diagram

Spec(R)  \overline{}^{deg f_{R}} ℕ
[figure: Spec(ϱ); deg f]
Spec(k)

(4)

commutes. This follows easily from (ii) above because the (ϱ(ε_{d}))_{d∈D} form a complete system of orthogonal idempotents in R.

We also note that the property of being locally monic descends from faithfully flat base extensions: If R ∈ k-alg is faithfully flat over k and f(t) ∈ k[t] has f(t)_{R} locally monic then f(t) is locally monic. Indeed, for every 𝔭 ∈ Spec(k) there exists 𝔮 ∈ Spec(R) such that ϱ^{−1}(𝔮) = 𝔭, and R_{𝔮} is faithfully flat over k_{𝔭}. Since the localization of f(t)_{R} at 𝔮 is monic and may be identified with the base extension of f(t)_{𝔭} from k_{𝔭} to R_{𝔮}, it follows by faithfully flat descent that f(t)_{𝔭} is monic.

0.9. The copolynomial. Let δ: Spec(k) → ℕ be a locally constant function, corresponding to a family of orthogonal idempotents (ε_{d})_{d∈D} with sum 1 in k via δ(𝔭) = d ⟺ 𝔭 ∈ Spec(kε_{d}). For example, δ = deg f could be the degree function of a locally monic polynomial. For R ∈ k-alg and r ∈ R, we define

r^{δ} := ∑_{d∈D} ε_{d}r^{d}  and  δ · r := ∑_{d∈D} ε_{d}dr.  (1)

Now let f(t) be locally monic and let f(t) = ∑_{d∈D} ε_{d}f_{d}(t) as in (ii) of 0.8. The copolynomial of f(t) is defined by

f̌(t) := t^{deg f} · f(t^{−1}) = ∑_{d∈D} ε_{d}t^{d}f_{d}(t^{−1}).  (2)

Clearly, f̌(0) = ∑_{d∈D} ε_{d} = 1, so the copolynomial is comonic. We define the coefficients c_{i} of f in descending order by the following ascending expansion of f̌(t):

f̌(t) = ∑_{i∈ℕ}(−1)^{i}c_{i}t^{i}.  (3)

Of course, c_{0} = 1 and c_{i} = 0 for i > max deg f. We can reconstruct f(t) from f̌(t) and deg f by the formula

f(t) = t^{deg f} · f̌(t^{−1}),  (4)

which yields the expansion

f(t) = ∑_{i∈ℕ}(−1)^{i}c_{i}t^{(deg f)−i} = ∑_{d∈D} ε_{d} ∑_{j=0}^{d}(−1)^{j}c_{j}t^{d−j}.  (5)

This also shows why it is more convenient to define the coefficients of f in the roundabout manner using f̌ rather than f. We finally remark that

f(0) ∈ k^{×} ⟹ f(0)^{−1}f̌(t) is locally monic of degree deg(f).  (6)

Indeed, this is easily reduced to the case where deg(f) = d is constant. Then f(t) = ∑_{i=0}^{d}(−1)^{i}c_{i}t^{d−i} and hence f(0)^{−1}f̌(t) = ∑_{i=0}^{d}(−1)^{d−i}c_{i}c_{d}^{−1}t^{i}.

0.10. Example: Characteristic polynomials. Let M be a finitely generated and projective k-module and let g ∈ End(M). We refer to [1] for the notion of the determinant of g. The characteristic polynomial of g is defined by

χ_{g}(t) := det(tId_{M} − g).  (1)

This is locally monic, has degree deg χ_{g} = rk M, and the associated copolynomial is

χ̌_{g}(t) = det(Id_{M} − tg) = ∑_{i∈ℕ}(−1)^{i}c_{i}(g)t^{i},  (2)

where the coefficients are given by c_{i}(g) = trace ⋀^{i} g. (Note that in [1], λ_{t}(g) := χ̌_{g}(−t) is called the characteristic polynomial of g.)

The following lemma is probably well known but we give a proof for lack of a convenient reference.

0.11. Lemma. The map associating with a polynomial f ∈ k[t] the quotient E = k[t]/(f) induces a bijection between the set of locally monic polynomials in k[t] and the set of quotient algebras of k[t] which are finitely generated and projective as k-modules. The inverse map is given by E ↦ χ_{L(z)}(t), where z := can(t) ∈ E and L(z) is multiplication by z in the algebra E. The degree of f equals the rank of E (as functions on Spec(k)), and f is monic of degree d if and only if E is free with basis 1, z, …, z^{d−1}.

Proof. (a) Let f(t) be locally monic. After decomposing 1_{k} = ∑_{d∈D} ε_{d} as in 0.8, we may assume that f(t) is monic of (constant) degree d. Then it follows from the usual division algorithm in k[t] that E is free of rank d with basis z^{0}, …, z^{d−1}. Now a standard computation shows that det(tId_{E} − L(z)) = f(t).

(b) Conversely, let 𝔑 be an ideal in k[t], let E = k[t]/𝔑 be finitely generated and projective and let f(t) := χ_{L(z)}(t). Then f(t) is locally monic of degree rk E, and by the Cayley-Hamilton Theorem [1, Th. 1.4], 0 = f(L(z)) · 1_{E} = can(f(t)) where can: k[t] → E is the canonical map. Thus f(t) ∈ 𝔑, so there exists a surjective homomorphism u: E′ := k[t]/(f(t)) → E. By what we proved in (a), E′ is finitely generated and projective, and rk E′ = deg f = rk E. By standard facts on finitely generated and projective modules [3, II, §3.3, Th. 1 and Cor. 5 of Th. 1], u is an isomorphism.

0.12. Schemes defined by modules. With any k-module M, we associate functors M_{a} and M_{u} defined by

M_{a}(R) = M ⊗ R, M_{u}(R) = {x ∈ M_{a}(R) : x unimodular}.

If M = k, then k_{u} is just the functor of units, i.e., k_{u}(R) = R^{×}, so k_{u} = G_{m}, the multiplicative group (over k). Now suppose M is finitely generated and projective. Then M_{a} ≅ Spec(A) is an affine, smooth and finitely presented k-scheme with connected fibres over k, where A = 𝒪(M), the symmetric algebra over the dual module M^{∗} of M [6, II, §1, 2.1]. Moreover, A is faithfully flat over k, so, denoting by ι: k → A the canonical homomorphism,

Spec(ι): Spec(A) → Spec(k) is open and surjective,  (1)

cf. [10, Th. 2.4.6]. Also, M_{u} is open in M_{a}, quasi-affine (but in general no longer affine) and also smooth, finitely presented and with connected fibres. Indeed, choose a generating set α_{1}, …, α_{n} for M^{∗}. Then M_{u} is the union of the open subschemes of M_{a} defined by α_{1}, …, α_{n}, i.e., x ∈ M_{u}(R) if and only if the ideal of R generated by α_{1}(x), …, α_{n}(x) is all of R.

Note that A = 𝒪(M) is compatible with base ring extension: For all R ∈ k-alg there is a canonical isomorphism A_{R} ≅ 𝒪(M_{R}) which will be treated as an identification. This comes from the canonical isomorphism (M_{R})^{∗} ≅ (M^{∗})_{R} (because M is finitely generated and projective) and the fact that the symmetric algebra commutes with base ring extension.

We may identify A with the algebra of k-valued polynomial laws on M in the sense of [31]. Thus if g ∈ A, then for all x ∈ M_{R} we have g(x) ∈ R, and for all kalgebra homomorphisms ϱ: R → S, we have ϱ(g(x)) = g((Id_{M} ⊗ϱ)(x)). Usually, this will be simply written as g(x)_{S} = g(x_{S}). Also, A = ⨁_{n∈ℕ} A_{n} is a graded algebra, where A_{n} is the n-th symmetric power of M^{∗}, corresponding to the homogeneous polynomial laws of degree n. In particular, A_{0} = k, A_{1} = M^{∗} and A_{2} is naturally identified with the quadratic forms on M. We claim that

an idempotent ε of A belongs to A_{0} = k.  (2)

Indeed, decompose ε = ε_{0} + ε_{+} where ε_{0} ∈ A_{0} and ε_{+} ∈ ∑_{i⩾1} A_{i}. Assume ε_{+} ≠ 0, and let n ⩾ 1 be maximal with the property that ε_{+} ∈ ⨁_{i⩾n} A_{i}. Then ε^{2} = ε_{0}^{2} +2ε_{0}ε_{+} +ε_{+}^{2} = ε_{0} +ε_{+} yields ε_{0}^{2} = ε_{0} and ε_{+} = 2ε_{0}ε_{+} +ε_{+}^{2}, or (1−2ε_{0})ε_{+} = ε_{+}^{2}. Now (1 − 2ε_{0})^{2} = 1 and hence ε_{+} = (1 − 2ε_{0})ε_{+}^{2} ∈ ⨁_{i⩾2n} A_{i}, contradiction.

0.13. Locally monic polynomials, continued. We keep the notations and conventions introduced in 0.12. Consider a polynomial f(t) = ∑_{i⩾0} g_{i} t^{i} ∈ A[t] and an element x ∈ M_{R}. By evaluating the coefficients of f at x we obtain a polynomial in R[t], written

f(t; x) := ∑_{i⩾0} g_{i}(x) t^{i} ∈ R[t].  (1)

Conversely, specifying an element in A[t] amounts to specifying a polynomial f(t; x) ∈ R[t] for all x ∈ M_{R} and all R ∈ k-alg, varying functorially with R, i.e., such that f(t; x)_{S} = f(t; x_{S}) for all R-algebras S. Note also that, for f(t) ∈ A[t] and any R ∈ k-alg,

f_{R} := f(t)_{R} := f(t) ⊗_{k} 1_{R} = ∑_{i⩾0}(g_{i})_{R} t^{i} ∈ A_{R}[t]  (2)

is a polynomial with coefficients in A_{R}.

Now let f(t) ∈ A[t] be locally monic, with degree function deg f: Spec(A) → ℕ, and let f(t) = ∑_{d∈D} ε_{d}f_{d}(t) as in 0.8, where (ε_{d})_{d∈D} is a complete system of orthogonal idempotents of A and f_{d}(t) is monic of degree d. By 0.12.2, the ε_{d} belong to k. Let R ∈ k-alg and let ϱ: k → R be the homomorphism making R a k-algebra. Then the ϱ(ε_{d}) = ε_{d} ⊗ 1_{R} form a family of orthogonal idempotents with sum 1_{R}, and f(t)_{R} = ∑_{d∈D}(ε_{d} ⊗1_{R})·f_{d}(t)⊗1_{R} shows that f(t)_{R} is locally monic.

Let 𝔓 ∈ Spec(A) and 𝔭: = k ∩ 𝔓 = Spec(ι)(𝔓) ∈ Spec(k). Then

(deg f)(𝔓) = d ⟺ ε_{d} ≡ 1 mod 𝔓  (by 0.8.1)
⟺ ε_{d} ≡ 1 mod 𝔭  (by 0.12.2).

Thus there exists a unique locally constant function \overline{deg} f: Spec(k) → ℕ making the diagram

Spec(A)  \overline{}^{deg f} ℕ
[figure: Spec(ι); \overline{deg} f]
Spec(k)

(3)

commutative. It can also be described as follows: A_{κ(𝔭)} is isomorphic to the polynomial algebra in n = rk_{𝔭} M variables over κ(𝔭), in particular, it is an integral domain. Hence the locally monic polynomial f(t)_{κ(𝔭)} ∈ A_{κ(𝔭)}[t] is actually monic, and we have

( \overline{deg} f)(𝔭) = deg f(t)_{κ(𝔭)},  (4)

for all 𝔭 ∈ Spec(k). Indeed, letting d be the unique element of D with ε_{d} ≡ 1 mod 𝔭 (which exists because 𝔭 is a prime ideal), we see that f(t)_{κ(𝔭)} = f_{d}(t)_{κ(𝔭)} has degree d, as required.

Let again R ∈ k-alg and let x ∈ M_{R}. Then f(t; x) = ∑_{d∈D} ε_{d}f_{d}(t; x) = ∑_{d∈D} ϱ(ε_{d})f_{d}(t; x). Furthermore, for 𝔮 ∈ Spec(R), ϱ(ε_{d}) ≡ 1 mod 𝔮 if and only if ε_{d} ≡ 1 mod ϱ^{−1}(𝔮). Hence f(t; x) is locally monic of degree

deg f(t; x) = (\overline{deg} f) ∘ Spec(ϱ).  (5)

In particular, for R = k (and hence ϱ = Id) we see that \overline{deg} f = deg f(t; x) for any x ∈ M. Applying this to f_{R} ∈ A_{R}[t] (cf. (2)) and noting that f_{R}(t; x) = f(t; x) for all x ∈ M_{R} yields

\overline{deg} f_{R} = (\overline{deg} f) ∘ Spec(ϱ).  (6)

We finally remark that passing to the copolynomial commutes with evaluating the coefficients, i.e., we have the formula

f̌(t; x) = f(t; x)ˇ .  (7)

Indeed, by the definition of the copolynomial in 0.9, f(t; x)ˇ = ∑ t^{d}ϱ(ε_{d})f_{d}(t^{−1}; x), and f̌(t) = ∑ t^{d}ε_{d}f_{d}(t^{−1}). Evaluating this at x yields f̌(t; x) = ∑ t^{d}ε_{d}f_{d}(t^{−1}; x). But since R is a k-algebra, ε_{d}r = ϱ(ε_{d})r for all r ∈ R, whence (7).

0.14. Density. A subfunctor U of a k-functor X is called dense if, for all open subfunctors V ⊂ X and all closed Z ⊃ U ∩ V we have Z = V, and this property remains valid in all scalar extensions. If X is a scheme then this notion agrees with “universally schematically dense” in the sense of [11, 11.10].

Let X be a smooth separated finitely presented k-scheme with connected nonempty fibres, and let U be an open subscheme of X. We refer to [7, Exp. XVIII, 1.7] and [11, 11.10.10] for the equivalence of the following conditions:

(i) U is dense in X, (ii) there exists a faithfully flat and finitely presented R ∈ k-alg such that

U(R) ≠ ∅, (iii) U(K) ≠ ∅ for all algebraically closed fields K ∈ k-alg.

Note that the assumptions on X are in particular satisfied when X = M_{a} is the k-scheme defined by a finitely generated and projective k-module M.

For example, let J be a Jordan algebra over k which is finitely generated and projective as a k-module, let J = J_{a} be the affine scheme defined by J and U = J^{×} the subfunctor of invertible elements, defined by U(R) = J_{R}^{×} for all R ∈ k-alg. Then U is open, being the inverse image of k_{u} under the morphism x ↦ det U_{x} for all x in all base extensions of J, and it is dense because 1_{J} ∈ U(k).

Density will be used mostly for the following type of argument: Suppose U ⊂ X is dense, that Y is a separated k-functor, and that f, g: X → Y are morphisms which agree on U. Then f = g. This is immediate from the definition applied to V = X and Z the subfunctor of X where f and g agree.

0.15. Pure submodules. Recall [18, §4J] that a submodule P of a k-module X is called pure if for every k-module N, the map u ⊗ Id_{N}: P ⊗ N → X ⊗ N induced from the inclusion u: P ⊂ X is injective. For example, this is so if P is a direct summand or if X/P is flat. We collect some facts on pure submodules.

(a) P is pure if and only if the map P ⊗R → X⊗R is injective, for all R ∈ k-alg.

(b) If P is pure in X then P_{R} (canonically identified with Im(u_{R})) is a pure submodule of X_{R}, for all R ∈ k-alg.

(c) Conversely, if R ∈ k-alg is faithfully flat and P_{R} is pure in X_{R} then P is pure in X.

(d) Suppose X is projective and P ⊂ X is pure and finitely generated. Then P is a direct summand of X (and hence both P and X/P are projective.)

(e) Suppose k is a principal ideal domain. Then P is pure in X if and only if P ∩ λX = λP, for all λ ∈ k.

Proof. (a) The stated condition is obviously necessary. To see that it is sufficient, let N be an arbitrary k-module, and let R := k⊕N be the split null extension. Then the injectivity of P ⊗R → X ⊗R implies that also P ⊗N → X ⊗N is injective.

(b) See [3, I, Ex. 24(e) of §2] or [18, 4.84(f)].

(c) Let N be a k-module. Since R is faithfully flat, the map P ⊗ N → X ⊗ N is injective provided the map (P ⊗ N) ⊗ R → (X ⊗ N) ⊗ R is injective. Now (P ⊗ N) ⊗ R ≅ (P ⊗_{k} R) ⊗_{R} (N ⊗_{k} R) by [4, II, §5.1, Prop. 3], and similarly for X in place of P. Since P_{R} is pure in X_{R}, the map (P_{R}) ⊗ (N_{R}) → X_{R} ⊗ N_{R} is injective, whence our assertion.

(d) See [18, p. 164, Ex. 42(a)].

(e) See [3, I, Ex. 24(a) of §2] or [18, Cor. 4.93].

1. Algebraic elements

1.1. In this section, J always denotes a unital quadratic Jordan algebra over an arbitrary commutative ring k of scalars. Denote by k[t] the polynomial ring in the indeterminate t and let a ∈ J. It is well known that there is a unique homomorphism j_{a}: k[t]^{+} → J of Jordan algebras sending t to a. The image of j_{a} is the subalgebra k[a] of J generated by (1 and) a. The kernel 𝔎(a) of j_{a} is a Jordan, but in general not an associative ideal of k[t]. In case k is a field, 𝔎(a) contains a unique largest associative ideal [16, Section 1] which is of course principal. If k is arbitrary, we proceed in a somewhat different manner as follows.

Write the elements of J^{2} = J × J as formal column vectors ( \begin{matrix}x \\ y \end{matrix} ), x, y ∈ J,

and correspondingly the endomorphisms of J^{2} as 2 × 2-matrices with entries from End J. For a ∈ J and i ∈ ℕ, define a^{[i]} ∈ J^{2} by

a^{[i]} := ( \begin{matrix}a^{i} \\ a^{i+1} \end{matrix} ) .  (1)

Let h_{a}: k[t] → J^{2} be the unique k-linear map with h_{a}(t^{i}) = a^{[i]}, and denote its kernel by 𝔑(a) and its image by M(a). Thus

h_{a}(f(t)) = ( \begin{matrix}j_{a}(f(t)) \\ j_{a}(tf(t)) \end{matrix} ) ,  (2)

whence j_{a} = pr_{1} ∘ h_{a} and 𝔑(a) ⊂ 𝔎(a). Also, by definition, we have an exact sequence

[figure: 0; 𝔑(a); i_{a}; k[t]; h′_{a}; M(a); 0]  (3)

of k-modules, where i_{a} is inclusion and h′_{a} is just h_{a} but with codomain M(a) = Im(h_{a}). If ψ: J → J̃ is a homomorphism of unital Jordan algebras, one checks immediately that

𝔎(a) ⊂ 𝔎(ψ(a)) and 𝔑(a) ⊂ 𝔑(ψ(a)).  (4)

1.2. Lemma and Definition. (a) Let θ_{a} := ( \begin{matrix}0 & Id \\ U_{a} & 0 \end{matrix} ) ∈ End(J^{2}). Then

θ_{a}^{2} = ( \begin{matrix}U_{a} & 0 \\ 0 & U_{a} \end{matrix} ) ,  (1)
θ_{a} · a^{[i]} = a^{[i+1]},  (2)
h_{a}(f(t)g(t)) = f(θ_{a}) · h_{a}(g(t)),  (3)

for all i ∈ ℕ and f(t), g(t) ∈ k[t]. Hence, regarding J^{2} as a k[t]-module by letting t act via θ_{a}, h_{a} is a homomorphism of k[t]-modules and M(a) is the k[t]-submodule generated by a^{[0]}.

(b) 𝔑(a) is the largest ideal of k[t] contained in 𝔎(a). We define

E(a) := k[t]/𝔑(a),  (4)

regarded as a commutative associative monogenous k-algebra, generated by the element z := can(t), and denote by π: E(a)^{+} → J the homomorphism of Jordan algebras induced from j_{a}. Then for all b ∈ Ker(π) = 𝔎(a)/𝔑(a), b^{2} = 2b = 0.

(c) Let M_{0}(a) = {w ∈ M(a) : pr_{1}(w) = 0}. Then M_{0}(a) ≅ Ker(π) as kmodules, and pr_{2}(M_{0}(a)) ⊂ J consists of absolute zero divisors.

Remark. Of course, h_{a} induces a canonical isomorphism

γ_{a}: E(a) \overline{}^{≅} M(a) ,  γ_{a}(z^{n}) = a^{[n]},  (5)

of k-modules and even of k[t]-modules. Nevertheless, it is useful to distinguish E(a) and M(a), in particular, since the algebra structure of E(a) does not correspond to a multiplication on M(a) induced in a natural way from the Jordan algebra structure on J^{2}.

Proof. (a) (1) and (2) are immediate from the definitions and (3) follows easily from (2).

(b) It is clear from (3) that 𝔑(a) is an ideal of k[t] and 𝔑(a) ⊂ 𝔎(a) by 1.1.2. Now suppose I is an ideal of k[t] contained in 𝔎(a), and let f(t) ∈ I. Then also tf(t) ∈ I, and hence h_{a}(f(t)) = 0 by 1.1.2, showing f(t) ∈ 𝔑(a). If g(t) ∈ 𝔎(a) then, because 𝔎(a) is a Jordan ideal, g(t)^{2} and tg(t)^{2} = U_{g(t)} · t belong to 𝔎(a), whence g(t)^{2} ∈ Ker(h_{a}) = 𝔑(a) by 1.1.2. Similarly, 2g(t) = 1 ∘ g(t) and 2tg(t) = t ∘ g(t) belong to 𝔎(a), showing 2g(t) ∈ 𝔑(a).

(c) The first statement follows easily from pr_{1} ∘ h_{a} = j_{a}. An element w = ( \begin{matrix}0 \\ y \end{matrix} )

belongs to M_{0}(a) if and only if there exists f(t) ∈ k[t] such that j_{a}(f(t)) = 0, i.e., f(a) = 0, and y = j_{a}(tf(t). By [15, Cor. 3.3.3], U_{y} = U_{a}U_{f(a)} = 0.

1.3. Lemma. Let a ∈ J and R ∈ k-alg. We use the notations of 1.2 and 0.1.

(a) Identify k[t] ⊗ R with R[t] in the canonical way. Then

(h_{a})_{R} = h_{a_{R}}: R[t] → J_{R}^{2}.  (1)

(b) The image of (i_{a})_{R} is contained in 𝔑(a_{R}) and hence induces a homomorphism φ: 𝔑(a) ⊗ R → 𝔑(a_{R}), for which Ker(φ) = Ker ((i_{a})_{R}). The image of the map u_{R}: M(a) ⊗ R → J_{R}^{2} induced from the inclusion u: M(a) ⊂ J^{2} is M(a_{R}), whence a surjective homomorphism u′_{R}: M(a) ⊗ R → M(a_{R}) of R-modules. The diagram

[figure: Ker(φ); 0; Ker(u_{R})]
[figure]
[figure: 𝔑(a) ⊗ R; (i_{a})_{R}; R[t]; (h′_{a})_{R}; M(a) ⊗ R; 0]
[figure: φ; u′_{R}]
[figure: 0; 𝔑(a_{R}); i_{a_{R}}; R[t]; h′_{a_{R}}; M(a_{R}); 0]
[figure]
[figure: Coker(φ); 0; 0]

(2)

is commutative with exact rows and columns, and there is a canonical isomorphism

∂: Ker(u_{R}) ⟶^{≅} Coker(φ).  (3)

(c) The unique R-module homomorphism η: E(a) ⊗ R → E(a_{R}) making the diagram

[figure: E(a) ⊗ R; η; E(a_{R}); (γ_{a})_{R}; ≅; ≅; γ_{a_{R}}; M(a) ⊗ R; u′_{R}; M(a_{R})]  (4)

commutative, is a homomorphism of R-algebras.

(d) If R is flat over k then Ker(u_{R}) = 0 and u′_{R} is an isomorphism.

Proof. (a) The map x ↦ x_{R} from J to J_{R} is a homomorphism of Jordan algebras over k. Hence (a^{n})_{R} = (a_{R})^{n} which implies

(h_{a})_{R}(t^{n}) = h_{a}(t^{n})_{R} = (a^{[n]})_{R} = (a_{R})^{[n]} = h_{a_{R}}(t^{n}),  (5)

from which (1) follows.

(b) Let f(t) ∈ 𝔑(a) and r ∈ R. Then by (1),

h_{a_{R}}(f(t) ⊗ r) = (h_{a})_{R}(f(t) ⊗ r) = h_{a}(f(t)) ⊗ r = 0.

Since the elements of the form f(t) ⊗ r span 𝔑(a) ⊗ R, this proves the existence of φ. Next, let f(t) ⊗ r ∈ k[t] ⊗ R = R[t]. Then

u_{R}(h′_{a}(f(t)) ⊗ r) = h_{a}(f(t)) ⊗ r = (h_{a})_{R}(f(t) ⊗ r) = h_{a_{R}}(f(t) ⊗ r)

(by (1)) shows that the image of u_{R} equals the image of h_{a_{R}} which is, by definition, M(a_{R}). This establishes the existence of u′_{R}.

It is easy to check that (2) is commutative. The exactness of the second row follows by tensoring 1.1.3 with R, while the exactness of the third row is just 1.1.3, but for a_{R} instead of a. Now (3) follows from the Snake Lemma [3, I, §1.4].

(c) Let z = can(t) ∈ E(a) and w = can′(t) ∈ E(a_{R}) (where can′: R[t] → E(a_{R}) is the canonical map) be the generators of E(a) and E(a_{R}), respectively. We show that η(z^{n} ⊗ 1_{R}) = w^{n} for all n ∈ ℕ, from which the homomorphism property of η follows easily. Now γ_{a_{R}}(w^{n}) = (a_{R})^{[n]} while u_{R}((γ_{a})_{R}(z^{n} ⊗ 1_{R})) = u_{R}(γ_{a}(z^{n}) ⊗ 1_{R}) = u_{R}(a^{[n]} ⊗ 1_{R}) = (a^{[n]})_{R}, and these two are equal by (5).

(d) Obvious.

1.4. Definition. Let again J be a unital Jordan algebra over k. We will use the expression “a satisfies a polynomial f(t)” for the fact that f(t) ∈ 𝔑(a), i.e., that j_{a}(f(t)) = j_{a}(tf(t)) = 0. An element a ∈ J is called integral if it satisfies some monic polynomial. Just as in the case of associative algebras, we have:

a is integral ⟺ M(a) is finitely generated as a k-module.  (1)

Indeed, the implication from left to right follows easily from 1.2.2 and induction. For the reverse, note that θ_{a} induces, by 1.2.2, an endomorphism

ζ_{a}: M(a) → M(a).

Then M(a) is a faithful k[ζ_{a}]-module. Hence by [3, V, §1.1, Lemma 1], there exists a monic f(t) ∈ k[t] satisfying f(ζ_{a}) = 0. It follows that 0 = f(ζ_{a})·a^{[0]} = f(θ_{a})·a^{[0]} = h_{a}(f(t)) (by 1.2.3), so a satisfies f(t).

In general, integral elements do not have well-defined minimum polynomials, i.e., the ideal 𝔑(a) need not be generated by a (locally) monic polynomial. For

example, let k = ℤ/4ℤ, J = Mat_{2}(k)^{+} and a = ( \begin{matrix}0 & 2 \\ 0 & 0 \end{matrix} ). Then 𝔑(a) is generated

by 2t and t^{2} and E(a) ≅ k ⊕ (k/2k) is not projective as a k-module. Therefore, we introduce the following notion. An element a ∈ J is called pre-algebraic if E(a) is a finitely generated and projective k-module. By Lemma 0.11, this is equivalent to 𝔑(a) being the principal ideal generated by a unique locally monic polynomial, called the minimum polynomial of a and denoted by μ_{a}(t). Because E(a) ≅ M(a) by 1.2.5, we have

a is pre-algebraic ⟺ M(a) is a finitely generated projective k-module. (2)

Obviously, by (1) and (2),

a pre-algebraic ⟹ a integral.  (3)

We note that, for a pre-algebraic,

μ_{a}(t) = det(tId_{M(a)} − ζ_{a}),  μ̌_{a}(t) = det(Id_{M(a)} − tζ_{a}).  (4)

Indeed, we have the commutative diagram

E(a) \overline{}^{L(z)} E(a)
[figure: γ_{a}; ≅; ≅; γ_{a}]
M(a)  \overline{}_{ζ_{a}} M(a)

where γ_{a} is the isomorphism 1.2.5. Hence the assertion follows from 0.10.2, Lemma 0.11 and the fact that the determinant is compatible with isomorphisms [1, Prop. 1.3(iv)].

The degree of a pre-algebraic a is defined to be the degree of μ_{a}(t), i.e., the locally constant function

deg a: 𝔭 ↦ deg μ_{a}(t)_{𝔭} = rk_{𝔭} E(a) = rk_{𝔭} M(a)  (5)

on Spec(k). If a has constant degree d, i.e., if μ_{a}(t) is monic of degree d, then by Lemma 0.11, E(a) is free as a k-module with basis 1, …, z^{d−1}. Hence 1.2.5 implies

a is pre-algebraic of degree d ⟺ a^{[0]}, …, a^{[d−1]} is a basis of M(a). (6)

Note that for a pre-algebraic a ∈ J, the exact sequence 1.1.3 splits because M(a) is in particular projective. It therefore remains exact upon tensoring with any R ∈ k-alg, so we have, with the notations of Lemma 1.3(b):

a pre-algebraic ⟹ φ: 𝔑(a) ⊗ R → 𝔑(a_{R}) injective.  (7)

Hence, for a pre-algebraic a, we may and often will identify 𝔑(a) ⊗ R with its image in 𝔑(a_{R}). However, even when also a_{R} is pre-algebraic, φ is not necessarily surjective, and hence μ_{a}(t)_{R} may be different from μ_{a_{R}}(t). For example, let k = ℤ and J = B^{+} where B is the associative commutative ℤ-algebra, free of rank 4, with basis 1, a, b, ab and multiplication table a^{2} = 2b, b^{2} = 0. Then 𝔎(a) = 𝔑(a) = (t^{4}) so μ_{a}(t) = t^{4}. But for R = ℤ/2ℤ, J_{R} is the tensor product of the algebra of dual numbers over R with itself, and the minimum polynomial of a_{R} is t^{2}. This leads to the following definition:

An element a ∈ J is called algebraic if a_{R} is pre-algebraic and satisfies

μ_{a_{R}}(t) = μ_{a}(t)_{R},  (8)

equivalently,

𝔑(a_{R}) = 𝔑(a) ⊗ R,  (9)

for all R ∈ k-alg. The equivalence of (8) and (9) follows from the fact that 𝔑(a) and 𝔑(a_{R}) are, respectively, the principal ideals generated by μ_{a}(t) and μ_{a_{R}}(t). Of course, if k is a field, the three notions just introduced all coincide with the usual definition of an algebraic element. They hold for all a ∈ J provided J is finite-dimensional over k.

1.5. Proposition. Let a ∈ J and R ∈ k-alg and denote by ϱ: k → R the ring homomorphism making R a k-algebra.

(a) a is algebraic if and only if M(a) is a finitely generated and projective k-module and a pure submodule of J^{2}.

(b) If J is finitely generated and projective as a k-module, then a is algebraic if and only if M(a) is a direct summand of J^{2}.

(c) Let a be algebraic. Then a_{R} is algebraic as well, and the homomorphisms φ, u′_{R} and η of Lemma 1.3 are isomorphisms. The degree functions of a and a_{R} are related by

(deg a_{R})(𝔮) = (deg a)(ϱ^{−1}(𝔮))  (1)

for all 𝔮 ∈ Spec(R).

(d) Conversely, let R be faithfully flat over k and assume that a_{R} is algebraic. Then a is algebraic.

Proof. (a) If a is algebraic then it is in particular pre-algebraic, so M(a) is finitely generated and projective by 1.4.2. Because of 1.4.9, the map φ of 1.3(b) is an isomorphism. Hence by 1.3.3, u_{R}: M(a) ⊗ R → J_{R}^{2} is injective for all R ∈ k-alg, so M(a) is pure by 0.15(a). Conversely, suppose M(a) is finitely generated and projective and pure, and let R ∈ k-alg. Then u_{R}: M(a) ⊗ R → J_{R}^{2} is injective, so u′_{R}: M(a) ⊗ R → M(a_{R}) is an isomorphism. Finitely generated projective modules remain so under base change. Hence M(a_{R}) is finitely generated and projective, so a_{R} is pre-algebraic by 1.4.2. Since in particular a itself is pre-algebraic, φ is injective. By 1.3.3, φ is surjective as well, so 1.4.9 holds.

(b) Direct summands of finitely generated and projective modules are themselves finitely generated and projective and of course pure, and pure submodules of projective modules are direct summands, by 0.15(d).

(c) From the definition of an algebraic element it is evident that a_{R} is algebraic along with a. The fact that φ, u′_{R} and η are isomorphisms follows from the proof of (a) and 1.3.4. Formula (1) follows from the definition of the degree function in 1.4.5 and 0.8.3.

(d) We use the characterization given in (a). By Lemma 1.3(d), u′_{R}: M(a)⊗R → M(a_{R}) is an isomorphism. Since M(a_{R}) is finitely generated and projective, the same is true of M(a) by [3, I, §3.6, Prop. 12]. Purity of M(a) follows from that of M(a_{R}) by 0.15(c).

1.6. Power-associativity. An element a ∈ J is called power-associative if the kernel 𝔎(a) = Ker(j_{a}) of the evaluation map j_{a}: k[t] → J (cf. 1.1) is an ideal of the associative algebra k[t] (and not just a Jordan ideal). By Lemma 1.2, equivalent conditions are that 𝔑(a) = 𝔎(a), or that pr_{1}: M(a) → k[a] or π: E(a) → J be injective. We will say an element a ∈ J is strictly power-associative if a_{R} is powerassociative for all R ∈ k-alg. Any one of the following conditions is sufficient to guarantee that every element of J is power-associative:

(a) J has no 2-torsion,

(b) J contains no absolute zero divisors, and any one of the following conditions implies that every element of J is strictly power-associative:

(c) 2 ∈ k^{×},

(d) J = B^{+} is the Jordan algebra associated with an associative or alternative algebra B.

Indeed, (a) and (b) follow from Lemma 1.2, and the rest is clear.

Let v: k[a] ⊂ J be the inclusion, let R ∈ k-alg and denote by v′_{R}: k[a] ⊗ R → R[a_{R}] the induced map. Then we have a commutative diagram of surjective maps

[figure: M(a) ⊗ R; pr_{1}⊗Id_{R}; k[a] ⊗ R; u′_{R}; v′_{R}; M(a_{R}); pr_{1}; R[a_{R}]]  (1)

where u′_{R} is as in Lemma 1.3. By 0.15(a), k[a] is pure if and only if, for all R ∈ k-alg, v′_{R} is injective, while M(a) is pure if and only if all u′_{R} are injective. Also, all top maps are bijective if and only if a is power-associative, and all bottom maps are bijective if and only if a is strictly power-associative. Thus we see that

a is power-associative and k[a] is pure ⟺

a is strictly power-associative and M(a) is pure.  (2)

If k is a field, the purity conditions are automatically satisfied, so power-associat-ivity of an element a implies strict power-associativity of a, and by (c), only the case where the characteristic is 2 is of interest. But over rings there may exist power-associative elements which are not strictly power-associative.

Example. Let A be the associative commutative algebra over ℤ obtained from ℚ[t]/(t^{4}) by restricting scalars to ℤ. Let a := can(t) ∈ A and b := a^{2}/2 and define J = ℤ1 ⊕ ℤa ⊕ ℤb ⊕ ℤa^{3} ⊂ A. Using the relations a^{2} = 2b, a ∘ b = a^{3}, b^{2} = 0, one shows easily that J is a Jordan subalgebra of A; it is not an associative subalgebra of A because ab ∉ J. Obviously, J is free of rank 4 as a ℤ-module. Since J has no 2-torsion, every element of J is power-associative. But the element a is not strictly power-associative, because for R = ℤ/2ℤ we have a_{R}^{2} = 0 ≠ a_{R}^{3}. We also see that k[a] is not pure, because 2b ∈ k[a] but b ∉ k[a] (cf. 0.15(e)). On the other hand, M(a) is a direct summand of J^{2} (hence pure), a complementary submodule being

spanned by ( \begin{matrix}b \\ 0 \end{matrix} ), ( \begin{matrix}0 \\ 1 \end{matrix} ), ( \begin{matrix}0 \\ a \end{matrix} ), ( \begin{matrix}0 \\ b \end{matrix} ). In particular, a is algebraic. The algebra

J is also an example of a special but not strictly special Jordan algebra: It loses speciality after the base change from ℤ to ℤ/2ℤ.

Remark. K. McCrimmon [23] has defined a Jordan algebra J over a field K to be power-associative if every element of J is power-associative. As remarked above, every element of J is then even strictly power-associative. But it is not clear if all elements of all field extensions J ⊗_{K} L are power-associative, even when K is infinite.

In the presence of (strict) power-associativity, the conditions that an element a be pre-algebraic or algebraic can be reformulated as follows:

1.7. Corollary. (a) Let a ∈ J be power-associative. Then a is pre-algebraic if and only if k[a] is finitely generated and projective.

(b) Let a ∈ J be strictly power-associative. Then a is algebraic if and only if k[a] is finitely generated and projective and a pure submodule of J.

(c) Let a ∈ J be strictly power-associative and let J be finitely generated and projective. Then a is algebraic if and only if k[a] is a direct summand of J.

Proof. (a) Immediate from 1.4.2 because pr_{1}: M(a) → k[a] is now an isomorphism of k-modules.

(b) By strict power-associativity, the horizontal arrows in 1.6.1 are isomorphisms. Now the assertion follows from Prop. 1.5(a).

(c) This is immediate from (b) and Prop. 1.5(b).

Example. The reader may find it instructive to use (c) for showing that the element a ∈ J = B^{+} of 1.4 is not algebraic. Here k[a] = ℤ1 ⊕ ℤa ⊕ ℤ2b ⊕ ℤ2ab is not a direct summand of J, because J/k[a] ≅ ℤ^{2} ⊕ (ℤ/2ℤ)^{2} is not projective as a ℤ-module.

1.8. Example: Idempotents. We claim that an idempotent e = e^{2} ∈ J is strictly power-associative. Since e_{R} remains an idempotent in J_{R} for all R ∈ k-alg, it suffices to show e power-associative. Now e^{2} = e implies e^{3} = U_{e}e = U_{e}e^{2} =

e^{4} = (e^{2})^{2} = e, and then e^{i} = e for all i ⩾ 1. Hence e^{[i]} = ( \begin{matrix}e \\ e \end{matrix} ) for all

i ⩾ 1, so M(e) = k ( \begin{matrix}1_{J} \\ e \end{matrix} ) + k ( \begin{matrix}e \\ e \end{matrix} ). If w = λ ( \begin{matrix}1_{J} \\ e \end{matrix} ) + μ ( \begin{matrix}e \\ e \end{matrix} ) ∈ M(e) and

pr_{1}(w) = λ1_{J} + μe = 0, then applying U_{e} to this equation yields λe + μe = 0 and therefore w = 0. Hence pr_{1}: M(e) → J is injective, as desired.

Now let us assume that J is finitely generated and projective as a k-module. We show that e is algebraic and compute its minimum polynomial. By 1.7(c), e will be algebraic if and only if k[e] is a direct summand of J. Consider the Peirce decomposition J = J_{2} ⊕ J_{1} ⊕ J_{0} of J with respect to e. Then 1_{J} = e ⊕ (1_{J} − e) ∈ J_{2} ⊕ J_{0}, and k[e] = k · 1_{J} + k · e = k · e ⊕ k · (1_{J} − e), so it suffices to show that k · e and k · (1 − e) are direct summands of J_{2} and J_{0}, respectively. This follows from Lemma 0.5 since e and 1 − e are idempotents (in fact, the unit elements) of J_{2} and J_{0}. Using the notations of 0.2, we define subsets of S = Spec(k) by

𝔭 ∈ S_{0} ⟺ 1_{J}(𝔭) = 0 ⟺ J(𝔭) = {0},
𝔭 ∈ S′_{1} ⟺ e(𝔭) = 0 ≠ 1_{J}(𝔭),
𝔭 ∈ S′′_{1} ⟺ e(𝔭) = 1_{J}(𝔭) ≠ 0,
𝔭 ∈ S_{2} ⟺ 0 ≠ e(𝔭) ≠ 1_{J}(𝔭).

These sets are open and closed, disjoint, and their union is S. Indeed, let γ, δ and ε be the support idempotents of 1_{J}, 1_{J} − e and e, respectively (cf. 0.4 and 0.5). Then these subsets correspond to the following orthogonal idempotents with sum 1_{k}: ε_{0} = 1 − γ, ε′_{1} = δ(1 − ε), ε′′_{1} = ε(1 − δ), and ε_{2} = εδ. The minimum polynomial of e satisfies

μ_{e}(t)_{𝔭} = \{ \begin{matrix}1 & if 𝔭 ∈ S_{0} \\ t & if 𝔭 ∈ S′_{1} \\ t − 1 & if 𝔭 ∈ S′′_{1} \\ t^{2} − t & if 𝔭 ∈ S_{2} \end{matrix} \} .

Put S_{1} = S′_{1} ∪ S′′_{1} and ε_{1} = ε′_{1} + ε′′_{1}. Then the degree function of e takes the value i on S_{i}, and the minimum polynomial of e and its copolynomial are given by

μ_{e}(t) = ε_{0} + ε_{1}(t − ε) + ε_{2}(t^{2} − t),  μ̌_{e}(t) = 1 − εt,  (1)

where of course some of the ε_{i} may be zero. We note the special cases e = 1_{J} and e = 0:

μ_{1_{J}} (t) = γ(t − 1) + (1 − γ),  μ_{0}(t) = γt + (1 − γ).  (2)

When J is faithful as a k-module, i.e., when 1_{J} is unimodular or γ = 1 (cf. 0.3.2), these formulas simplify to μ_{1_{J}} (t) = t − 1 and μ_{0}(t) = t.

1.9. Lemma. (a) Let λ ∈ k^{×} be a unit. Then an element a ∈ J is (pre-)algebraic if and only if λa is so, and in this case,

μ_{λa}(t) = λ^{deg a}μ_{a}(λ^{−1}t),  μ̌_{λa}(t) = μ̌_{a}(λt).  (1)

(b) Let a, b ∈ J^{×} be invertible and consider the isotopes J^{(b)} and J^{(a)}. Then a is (pre-)algebraic in J^{(b)} if and only if b is (pre-)algebraic in J^{(a)}, and then the respective minimum polynomials are related by

μ_{a}^{(b)}(t) = μ_{b}^{(a)}(t).  (2)

Proof. (a) Put σ_{λ} = ( \begin{matrix}Id & 0 \\ 0 & λId \end{matrix} ) ∈ GL(J^{2}). From 1.1.1, we have (λa)^{[i]} =

λ^{i}σ_{λ}(a^{[i]}) and hence σ_{λ}(M(a)) = M(λa). Thus M(λa) is finitely generated and projective (and pure) if and only if M(a) is so. In view of 1.4.2 and Prop. 1.5(a), this proves the statement about λa being (pre-)algebraic. The diagram

M(a)  \overline{}^{λζ_{a}} M(a)
[figure: σ_{λ}; ≅; ≅; σ_{λ}]
M(λa)  \overline{}_{ζ_{λa}} M(λa)

is easily seen to commute. Hence by 1.4.4, μ̌_{λa}(t) = det(Id_{M(λa)} − tζ_{λa}) = det(Id_{M(a)} − λtζ_{a}) = μ̌_{a}(λt), and then the first formula of (1) follows by 0.9.4.

(b) We indicate quantities computed in J^{(v)} by a superscript (v); in particular, the ith power of an element x in the isotope J^{(v)} is x^{(i,v)}. Induction shows

U_{a}b^{(i,a)} = a^{(i+1,b)} and U_{b}^{−1}b^{(i+1,a)} = a^{(i,b)}

for all i ∈ ℕ, equivalently,

( \begin{matrix}0 & U_{b}^{−1} \\ U_{a} & 0 \end{matrix} ) ( \begin{matrix}b^{(i,a)} \\ b^{(i+1,a)} \end{matrix} ) = ( \begin{matrix}a^{(i,b)} \\ a^{(i+1,b)} \end{matrix} ) .  (3)

Now ϕ := ( \begin{matrix}0 & U_{b}^{−1} \\ U_{a} & 0 \end{matrix} ) ∈ GL(J^{2}), and (3) says that ϕ induces an isomorphism

ϕ: M^{(a)}(b) → M^{(b)}(a) of k-modules. One easily checks that the diagram

M^{(a)}(b) \overline{}^{ζ_{b}^{(a)}} M^{(a)}(b)
[figure: ϕ; ≅; ≅; ϕ]
M^{(b)}(a) \overline{}_{ζ_{a}^{(b)}} M^{(b)}(a)

is commutative. Now similar arguments as before complete the proof.

1.10. Invertibility. Let a ∈ J be invertible. Then we define a^{[i]} for all i ∈ ℤ by 1.1.1 and extend h_{a} to a map h_{a}: k[t, t^{−1}] → J^{2}. It is immediately checked that θ_{a} ∈ GL(J^{2}) with inverse

θ_{a}^{−1} = ( \begin{matrix}0 & U_{a}^{−1} \\ Id & 0 \end{matrix} ) ,  (1)

and that 1.2.2 holds for all i ∈ ℤ. Thus J^{2} becomes a module over the Laurent polynomial ring k[t, t^{−1}] by letting again t act via θ_{a}, and formula 1.2.3 then holds for all f(t), g(t) ∈ k[t, t^{−1}]. The analogue of the usual formula (a^{−1})^{i} = (a^{i})^{−1} is

(a^{−1})^{[i]} = ω(a^{[−i−1]}) (i ∈ ℤ),  (2)

where ω = ( \begin{matrix}0 & Id \\ Id & 0 \end{matrix} ) ∈ GL(J^{2}) is the switch of factors. This is easily verified. We

finally note that, when a is invertible, M(a) is not necessarily a k[t, t^{−1}]-submodule of J^{2}. Indeed, 1.2.2 and the fact that M(a) is spanned by all a^{[i]}, i⩾0, implies that

θ_{a}^{−1} stabilizes M(a) ⟺ θ_{a}^{−1}(a^{[0]}) = a^{[−1]} ∈ M(a).  (3)

1.11. Lemma. For any polynomial f(t) ∈ k[t] define f^{♯}(t) by

tf^{♯}(t) = f(t) − f(0).  (1)

(a) If a ∈ J satisfies a polynomial f(t) with invertible constant term then a is invertible in J, with inverse

a^{−1} = −\frac{f^{♯}(a)}{f(0)}.  (2)

Moreover, θ_{a}^{−1} ∈ End(J^{2}) stabilizes M(a), and M(a^{−1}) ⊂ ω(M(a)) where ω ∈ GL(J) is as in 1.10. If in addition a is integral, then

M(a^{−1}) = ω(M(a)).  (3)

(b) Suppose a ∈ J is algebraic. Then a ∈ J^{×} if and only if μ_{a}(0) ∈ k^{×}. In this case, a^{−1} is algebraic as well, of the same degree as a, formula (3) holds, and the minimum polynomial of a^{−1} is

μ_{a^{−1}}(t) = \frac{μ̌_{a}(t)}{μ_{a}(0)}.  (4)

Proof. (a) For fields, (2) is proved in [16, Sec. 1, Lemma 1], and the proof is identical in case of a base ring. For the convenience of the reader, we repeat the argument. Squaring (1) gives t^{2}f^{♯}(t)^{2} = f(t)^{2} − 2f(0)f(t) + f(0)^{2} and applying j_{a} to this yields U_{a}f^{♯}(a)^{2} = f(0)^{2} · 1_{J}, or U_{a}b^{2} = 1_{J} where b := −f^{♯}(a)/f(0). Hence 1_{J} is in the range of U_{a} so a is invertible. Now multiply (1) with t^{2} to obtain t^{2}f^{♯}(t) = tf(t) − tf(0). As tf(t) ∈ 𝔑(a), applying j_{a} yields U_{a}f^{♯}(a) = −f(0)a or U_{a}b = a and therefore b = U_{a}^{−1}a = a^{−1}.

We show that θ_{a}^{−1} stabilizes M(a) by verifying the condition of 1.10.3. Apply h_{a} to (1) and use 1.2.3. Then θ_{a} · h_{a}(f^{♯}(t)) = −f(0)a^{[0]}, and hence θ_{a}^{−1} · a^{[0]} = −f(0)^{−1}h_{a}(f^{♯}(t)) ∈ M(a), as desired. Thus all a^{[i]} for i < 0 lie in M(a). From 1.10.2, it follows that M(a^{−1}) = ω(h_{a}( ∑_{j⩾1} k · t^{−j})) ⊂ ω(M(a)).

Now assume a is integral, so it satisfies a monic polynomial, say a^{[n]} +λ_{1}a^{[n−1]} + ⋯ + λ_{n}a^{[0]} = 0. Applying θ_{a}^{−n−1} to this and using 1.2.2 yields ∑_{i=0}^{n} λ_{i}a^{[−i−1]} = 0, and by 1.10.2, we obtain ∑_{i=0}^{n} λ_{i}(a^{−1})^{[i]} = 0, so a^{−1} satisfies the polynomial 1 + λ_{1}t + ⋯ + λ_{n}t^{n} with constant term 1. Thus we may switch the roles of a and a^{−1} and have M(a) ⊂ ω(M(a^{−1})). Now it follows from ω^{2} = Id that (3) holds.

(b) For the first statement, and by what we proved in (a), it remains to show that a ∈ J^{×} implies μ_{a}(0) ∈ k^{×}. Assume this is not the case. Then there exists a maximal ideal 𝔪 ⊂ k such that μ_{a}(0) ∈ 𝔪. Let K := k/𝔪. By 1.4.8, we have μ_{a_{K}} (0) = μ_{a}(0) ⊗ 1_{K} = 0, whence a_{K} is not invertible in J_{K} by [16, Sec. 1, Lemma 1]. On the other hand, invertible elements remain so under base change, contradiction.

Now let a be algebraic and invertible. Then a is in particular integral and satisfies μ_{a}(t) whose constant term is invertible. Hence part (a) shows that (3) holds. Since ω ∈ GL(J^{2}), it follows from (3) that M(a^{−1}) is finitely generated and projective and pure in J^{2}, so a^{−1} is algebraic by Prop. 1.5(a). From 1.10.2 one sees that the diagram

M(a) \overline{}^{(ζ_{a})^{−1}} M(a)
[figure: ω; ≅; ≅; ω]
M(a^{−1}) \overline{}_{ζ_{a^{−1}}} M(a^{−1})

commutes. Hence μ_{a^{−1}}(t) = det(tId_{M(a^{−1})} − ζ_{a^{−1}}) = det(tId_{M(a)} − (ζ_{a})^{−1}) = det(−ζ_{a})^{−1} · det(Id − tζ_{a}) = μ_{a}(0)^{−1} · μ̌_{a}(t). Finally, deg a = deg a^{−1} follows from (4) and 0.9.6.

1.12. Lemma. Let J be a finitely generated and projective Jordan algebra over k and a ∈ J. Then a is algebraic of degree d if and only if

a^{[0]} ∧ ⋯ ∧ a^{[d−1]} ∈ ⋀^{d} J^{2} is unimodular, and  (1)
a^{[0]} ∧ ⋯ ∧ a^{[d]} = 0.  (2)

Proof. “Only if”: As remarked in 1.4, M(a) is free with basis a^{[0]}, …, a^{[d−1]}. By 1.5(b), it is a direct summand of J^{2}, and obviously a^{[d]} ∈ M(a). Thus (1) and (2) follow from Lemma 0.6.

“If”: From (1) and (2) and Lemma 0.6, it follows that the span, say N, of a^{[0]}, …, a^{[d−1]} is a free direct summand of J^{2} with basis a^{[0]}, …, a^{[d−1]}, and that a^{[d]} ∈ N. Now 1.2.2 implies that N contains all a^{[i]} and hence that N = M(a). By 1.5(b), a is algebraic and obviously of degree d.

1.13. Definition. For a unital Jordan algebra J over k and d ∈ ℕ, we define the following subsets of J:

J_{alg} := {a ∈ J : a is algebraic}, J_{alg,d} := {a ∈ J_{alg} : deg a = d}.

Now let J = J_{a} be the functor from k-alg to the category of sets defined by J as in 0.12, i.e., J(R) = J ⊗ R, for all R ∈ k-alg. We put

J_{alg}(R) := (J_{R})_{alg}, J_{alg,d}(R) := (J_{R})_{alg,d},

for all R ∈ k-alg and denote by ℕ_{k} the constant functor determined by ℕ as in 0.7(c).

1.14. Proposition. (a) J_{alg} and J_{alg,d} are hard subsheaves of J in the sense of [6, III, §1, 3.3], in particular, they are local functors. The degree function a ↦ deg a associated to an algebraic element a defines a morphism deg: J_{alg} → ℕ_{k} such that J_{alg,d} = deg^{−1}({d}_{k}), for all d ∈ ℕ. The J_{alg,d} are open subfunctors of J_{alg} which cover J_{alg}.

(b) Suppose J is finitely generated and projective as a k-module and let P^{(n)} := ⋀^{n} J^{2}. Then ⋀^{n}(J_{R}^{2}) ≅ (P^{(n)})_{R} for all R ∈ k-alg, and hence there are morphisms p_{n}: J → P_{a}^{(n)} given by x ↦ x^{[0]} ∧ ⋯ ∧ x^{[n−1]}, for all x ∈ J_{R}, R ∈ k-alg. Let P_{u}^{(n)} be the open subscheme of unimodular elements of P_{a}^{(n)}, cf. 0.12. Then J_{alg} and the J_{alg,d} are finitely presented quasi-affine k-schemes, given by

J_{alg,d} = p_{d}^{−1}(P_{u}^{(d)}) ∩ p_{d+1}^{−1}(0) and J_{alg} = ∐_{d=0}^{r} J_{alg,d},

where r = max rk J^{2}.

Proof. (a) Let X = J_{alg} or X = J_{alg,d}. We first show that X is a subfunctor of J, i.e., that, for all homomorphisms ϱ: R → S of k-algebras, J(ϱ)(X(R)) ⊂ X(S). Since ϱ: R → S makes S an R-algebra, a_{S} ∈ X(S) follows from Prop. 1.5(c), (applied to J_{R} and S instead of J and R). To say that X is a hard sheaf just means that it commutes with finite direct products (which is obvious), and that for all R ∈ k-alg and all faithfully flat R-algebras S, the sequence of sets

X(R) \overline{}X(S) [figure] X(S ⊗_{R} S)

is exact, where the arrows are induced from R → S and the two embeddings S → S ⊗_{R} S into the first and second factor. After changing the base ring from k to R, this is precisely the statement of Prop. 1.5(d).

It is easy to check that there is a morphism (i.e., a natural transformation of functors) deg: J_{alg} → ℕ_{k} such that deg_{R}: J_{alg}(R) → ℕ_{k}(R) is given by

(deg_{R}(a))(𝔭) = (deg a)(𝔭),

for all R ∈ k-alg, a ∈ J_{alg}(R), 𝔭 ∈ Spec(R). Clearly a ∈ J_{alg,d}(R) if and only if the function deg_{R}(a): Spec(R) → ℕ is constant equal to d. This proves that deg^{−1}({d}_{k}) = J_{alg,d}. Finally, if K is a field then Spec(K) is a one-point set, so the function deg_{K}(a) is constant, for every a ∈ J_{alg}(K) (this is just saying that an algebraic element over a field has a well-defined (constant) degree). It follows that J_{alg}(K) is the disjoint union of the J_{alg,d}(K), proving the last assertion.

(b) Let r = max{rk_{𝔭}(J^{2}) : 𝔭 ∈ Spec(k)}. Then clearly J_{alg,d} = ∅ for d > r. Hence, to prove the statement for J_{alg}, it suffices to show that the J_{alg,d} are finitely presented quasi-affine schemes. Since J is finitely generated and projective so are the exterior powers P^{(n)}, and hence they define affine finitely presented schemes P_{a}^{(n)} and open quasi-affine finitely presented subschemes P_{u}^{(n)} of unimodular elements. Also, exterior powers commute with base change, so the natural homomorphism ⋀^{n}(J_{R}^{2}) → P_{R}^{(n)} is an isomorphism. This proves the existence of p_{n}. Now Lemma 1.12 (which of course holds in all base ring extensions as well) shows that J_{alg,d} = Z ∩ U where Z = p_{d+1}^{−1}(0) and U = p_{d}^{−1}(P_{u}^{(d)}) are, respectively, closed and open finitely presented subschemes of J [6, I, §3]. This completes the proof.

Remark. Let in particular k be a field and J finite-dimensional. Then every element of J is algebraic, so (b) shows J is the disjoint union of the J_{alg,d}. Over a ring, the union is still disjoint, but there may be non-algebraic elements.

2. Generically algebraic algebras

2.1. The function deg J. In this section, J always denotes a unital quadratic Jordan algebra over an arbitrary ring k which is finitely generated and projective as a k-module, and J = J_{a} the affine k-scheme defined by J as in 0.12. For every prime ideal 𝔭 ∈ Spec(k), J(𝔭) = J ⊗ κ(𝔭) is finite-dimensional and therefore generically algebraic over κ(𝔭) [16, Sec. 2, Th. 2]. In particular, it has a well-defined generic minimum polynomial whose degree is, by definition, the degree deg J(𝔭) of J(𝔭). We thus have a function deg J: Spec(k) → ℕ given by

(deg J)(𝔭) := deg J(𝔭).

This function is lower semicontinuous (where ℕ has the discrete topology). Indeed, let p_{d}: J → P_{a}^{(d)} be as in 1.14 and let Y = p_{d}^{−1}(P_{u}^{(d)}). Then Y is open in J. Let deg J(𝔭) = d and let K be an algebraic closure of κ(𝔭). Then it follows from [16, Section 3] that there exists x ∈ J_{K} such that x^{[0]}, …, x^{[d−1]} are linearly independent, i.e., Y(K) ≠ ∅. Let |Y| be the open subset of Spec(A) underlying the open subscheme Y, cf. [6, I, §1, No. 4]. Then by 0.12.1, U := p(|Y|) ⊂ Spec(k) is an open neighbourhood of 𝔭, and for all 𝔮 ∈ U the fibre of Y over 𝔮 is not empty. Thus there exists y ∈ J ⊗ L where L is an algebraic closure of κ(𝔮), for which y ∈ Y(L), i.e., such that y^{[0]}, …, y^{[d−1]} are linearly independent, and hence the degree of J(𝔮) is at least d.

The degree of J behaves as expected under base change, namely,

deg J_{R} = (deg J) ∘ Spec(ϱ) (R ∈ k-alg),  (1)

where ϱ: k → R is the homomorphism making R a k-algebra. Indeed, if 𝔮 ∈ Spec(R) and 𝔭 = ϱ^{−1}(𝔮) ∈ Spec(k), then κ(𝔮) is, via ϱ, an extension field of κ(𝔭), so by the invariance of the generic minimum polynomial of a Jordan algebra over a field under field extension, deg J(𝔭) = deg J(𝔭) ⊗ κ(𝔮) = deg J_{R}(𝔮). — For later use, we note:

a ∈ J algebraic ⟹ deg a ⩽ deg J.  (2)

Indeed, for every 𝔭 ∈ Spec(k), the degree of a at 𝔭 is the degree of μ_{a}(t)_{κ(𝔭)} by 1.4.5, and by 1.4.8, this is the same as μ_{a(𝔭)}(t), which divides the generic minimum polynomial of J(𝔭).

2.2. Definition. Let J be as above, let A = 𝒪(J) be the affine algebra of J (cf. 0.12), and let f(t) ∈ A[t] be locally monic. Using the notations of 0.13, we say J satisfies f(t) if, for all R ∈ k-alg, every x ∈ J(R) satisfies the polynomial f(t; x), i.e., f(t; x) ∈ 𝔑(x). Obviously, this condition is stable under base change, i.e., if J satisfies f(t) then J_{R} satisfies f(t)_{R}, for all R ∈ k-alg. In particular, J(𝔭) satisfies f(t)_{κ(𝔭)} for all 𝔭 ∈ Spec(k), so that f(t)_{κ(𝔭)} is a multiple of the generic minimum polynomial of J(𝔭). Thus by 0.13.4 and the definition of the degree function of J, we have:

J satisfies f(t) ⟹ deg J ⩽ \overline{deg} f(t).  (1)

Note that such f always exist. For example, f(t; x) = det(t^{2}Id − U_{x}) is locally monic, and by the Cayley-Hamilton Theorem, f(U_{x}; x) = 0 in End(J_{R}) for all R ∈ k-alg. Hence 0 = f(U_{x}; x) · 1_{J} = f(U_{x}; x) · x, cf. [16, Section 1, Lemma 2]. However, it is in general not true that J satisfies a locally monic f for which equality holds in (1). This leads to the following definition:

J is called generically algebraic if there exists a locally monic polynomial m(t) ∈ A[t] such that

(i) J satisfies m(t), i.e., for all R ∈ k-alg and all x ∈ J(R), x satisfies the polynomial m(t; x),

(ii) deg J = \overline{deg} m(t); equivalently, that for all prime ideals 𝔭 ∈ Spec(k), m(t) ⊗_{k} 1_{κ(𝔭)} is the generic minimum polynomial of J(𝔭).

An associative or alternative algebra B will be called generically algebraic if the associated Jordan algebra B^{+} has this property. — It is useful to note that (ii) is equivalent to the condition

(ii)′ m(t)_{K} is the generic minimum polynomial of J_{K}, for all fields K ∈ k-alg.

This follows easily from the fact that the kernel of the canonical map k → K is a prime ideal 𝔭 of k, so K is an extension field of κ(𝔭), and the well-known invariance of the generic minimum polynomial of a finite-dimensional Jordan algebra under base field extensions.

A polynomial m(t) satisfying (i) and (ii) will be called a generic minimum polynomial for J. Actually, we will show below in 2.7(a) that m(t) is uniquely determined by (i) and (ii).

2.3. Remarks. (a) The property of being generically algebraic is stable under base change (and descends from faithfully flat base extensions, see 2.7(b)): If J is generically algebraic over k and m(t) is a generic minimum polynomial for J then J_{R} is generically algebraic over R and m(t)_{R} is a generic minimum polynomial for J_{R}, for all R ∈ k-alg.

Indeed, by general facts, J_{R} is finitely generated and projective over R, and by 0.13, m(t)_{R} is locally monic. As noted before, J_{R} satisfies m(t)_{R}. Hence condition (i) holds for m(t)_{R}, and condition (ii) follows from 0.13.6 and 2.1.1.

(b) By condition (ii) and 0.13.3, the degree function deg J of a generically algebraic J is locally constant on Spec(k). Thus we can decompose k = ∏_{d∈D} k_{d} and correspondingly Spec(k) = ∐_{d∈D} Spec(k_{d}) such that deg J is constant equal to d on Spec(k_{d}); equivalently, that J ⊗ k_{d} is generically algebraic of constant degree d over k_{d}. In proofs, this fact will often be used by employing a phrase like “after decomposing the base ring, we may assume that J has constant degree equal to d.” There are, however, natural examples of generically algebraic Jordan algebras whose degree functions are not constant, see 2.4(d).

(c) In the literature, the terminology “generically algebraic Jordan algebra of degree d”, in particular for d = 2 and d = 3, has been used in an informal fashion for Jordan algebras over rings which possess a monic polynomial m(t) of degree d satisfying condition (i), e.g., in [29, 30, 20]. Since condition (ii) does in general not hold for these examples, they are not necessarily generically algebraic in the present sense.

2.4. Examples. (a) If k is a field, every finite-dimensional Jordan algebra over k is generically algebraic in the present sense, and its generic minimum polynomial is the usual one.

(b) Let either 𝒞 be a composition algebra of rank ⩾2 over k as defined in [28, 1.4] or let 𝒞 = k. (Note that k itself is not a composition algebra in the sense of [28] unless 2 is a unit in k). Consider the Jordan algebra J = H_{d}(𝒞, k) of hermitian d × d-matrices over 𝒞 with diagonal entries in k. Then it is easily seen from [22, pp. 501–503] that J is generically algebraic of degree d, provided d ⩽ 3. If 𝒞 is associative, then H_{d}(𝒞, k) is generically algebraic of degree d for all d ⩾ 1.

(c) We leave it to the reader to show that J is generically algebraic of degree 0 if and only if J = {0}, and of degree 1 if and only if J ≅ k. The generically algebraic algebras of degree 2 are precisely the Jordan algebras associated with quadratic forms with base point on finitely generated and projective k-modules of rank ⩾2, see 3.7. The generically algebraic Jordan algebras of degree 3 are all obtained by the “general cubic construction”, see 3.9.

(d) Let M be a finitely generated and projective k-module. Then J := End(M)^{+} is generically algebraic, with generic minimum polynomial given by the characteristic polynomial

m(t; x) = det(tId − x)

where det is the determinant of an endomorphism of a finitely generated and projective module, see [1]. The degree of J is given by deg(J) = rk(M) and therefore is in general not constant. More generally, Azumaya algebras over k are generically algebraic [17, III, §1], as are central separable Jordan algebras over rings containing \frac{1}{2} [2, §1].

(e) Let J′ and J′′ be generically algebraic with generic minimum polynomials m′ and m′′. Then it is easily seen that J = J′ × J′′ is generically algebraic with generic minimum polynomial m(t; (x′, x′′)) = m′(t; x′) · m′′(t; x′′) and that deg J = deg J′ + deg J′′.

(f) By 2.3(b), the degree function of a generically algebraic Jordan algebra J is locally constant on Spec(k). This necessary condition gives easy examples of finitely generated and projective Jordan algebras which are not generically algebraic. For example, let J = B^{+} where B is as in 1.4. Then

(deg J)(𝔭) = \{ \begin{matrix}2 & if 𝔭 = (2) \\ 4 & if 𝔭 ≠ (2) \end{matrix} \}

is not constant on the connected space Spec(ℤ). On the other hand, there are examples of finitely generated and projective Jordan algebras of constant degree which are not generically algebraic, see 3.7.

2.5. Primitive elements. Let J be finitely generated and projective as a kmodule. An element a ∈ J is called primitive if a is algebraic and deg a = deg J. We define

J_{prim} = {a ∈ J : a primitive} and J_{prim}(R) := (J_{R})_{prim} ⊂ J(R),

for all R ∈ k-alg. Then J_{prim} is a subfunctor of J (actually, of J_{alg}): Indeed, let R → S be a homomorphism of k-algebras, and let a ∈ J_{prim}(R). Then a_{S} ∈ J(S) = (J_{R})_{S} is algebraic by Prop. 1.5(c), and from 1.5.1 and 2.1.1, it follows that deg a_{S} = deg J_{S}.

In case J is generically algebraic and m(t) is a generic minimum polynomial of J, we have

a is primitive ⟺ a is algebraic and μ_{a}(t) = m(t; a).  (1)

Since deg a = deg μ_{a}(t) by definition, this is immediate from (ii) of 2.2 and the fact that μ_{a}(t) divides m(t; a). Also,

a primitive ⟹ μ̌_{a}(t) = m̌(t; a)  (2)

which follows immediately from 0.13.7.

2.6. Lemma. Let J be generically algebraic. Then J_{prim} is an open dense finitely presented subscheme of J, given as follows: Decompose k = ∏_{d∈D} k_{d} such that J_{d} := J ⊗ k_{d} has constant degree d, cf. 2.2(b). Then, in the notation of 1.13 and of 0.7(d),

J_{prim} = ∐_{d∈D}(J_{d})_{alg,d}.

In particular, J_{prim} = J_{alg,d} if J has constant degree d.

Proof. Since J = ∐_{d∈D} J_{d} where J_{d} = (J_{d})_{a} is the affine scheme defined by J_{d}, we may assume that deg J = d is constant. Then by definition, a ∈ J_{prim}(R) if and only if a is algebraic and has degree d, so J_{prim} = J_{alg,d}.

Now we show that J_{alg,d} is open and dense in J. Consider the morphisms p_{n}: J → P_{a}^{(n)} of 1.14(b). By (i) of 2.2, x^{[d]} is a linear combination of x^{[0]}, …, x^{[d−1]} in all base ring extensions, which shows that p_{d+1} = 0. Hence by Prop. 1.14(b), J_{alg,d} is the inverse image of the open subscheme P_{u}^{(d)} of P_{a}^{(d)} under p_{d} and therefore open in J. To prove that it is dense, it suffices by 0.14(iii) to show that J_{alg,d}(K) ≠ ∅, for all algebraically closed fields K ∈ k-alg. This follows from well-known results on Jordan algebras over fields, see [16, Section 3].

Remark. Although J_{prim} is dense in J, this does not imply that J itself contains any primitive elements. For example, let k = 𝔽_{2} and J = k^{3}, which by 2.4(e) is generically algebraic of degree 3. Then a^{n} = a for all n ⩾ 1 and all a ∈ J, so M(a) has dimension ⩽2, while for a to be primitive, we must have dim M(a) = 3. On the other hand, by 0.14(ii), there always exists a faithfully flat R ∈ k-alg such that J_{prim}(R) ≠ ∅.

2.7. Proposition. (a) The generic minimum polynomial m(t) of a generically algebraic Jordan algebra J is uniquely determined. The coefficients m_{i} of m(t), defined as in 0.9.4 by the expansion

m̌(t; x) = ∑_{i⩾0}(−1)^{i}m_{i}(x)t^{i},  (1)

are homogeneous of degree i. In particular, m_{0} = 1 and m_{1} is a linear form on J, called the generic trace of J. The generic norm, defined by

N(x) := m(0; −x) = (−1)^{deg J}m(0; x),  (2)

is locally homogeneous of degree deg J, i.e., it satisfies

N(rx) = r^{deg J}N(x),  (3)

for all r ∈ R, x ∈ J_{R}, R ∈ k-alg; see 0.9.1 for the notation r^{deg J}. The m_{i} and hence also m(t) and N are invariant under isomorphisms in the obvious sense and under derivations in the sense that

∂_{Δ(x)}m_{i}|_{x} = 0,  (4)

for all Δ ∈ Der(J).

(b) (Descent) Let J be a Jordan algebra over k, let R ∈ k-alg be faithfully flat over k and suppose that J̃ := J_{R} is generically algebraic over R. Then J is generically algebraic over k.

Remarks. (i) Despite appearances, the sum in (1) is finite, because m_{i} = 0 for i > max deg J. The derivative ∂_{v}f|_{x} of a polynomial law f at x in direction v is defined by f(x + εv) = f(x) + ε∂_{v}f|_{x} ∈ J ⊗ k(ε), where k(ε) = k[t]/(t^{2}) is the algebra of dual numbers.

(ii) In [30, Remark after 2.6], examples of isomorphisms are given which do not preserve norms. This is no contradiction to (a) because the “norms” in question are the cubic forms of a cubic norm structure. The algebras in these examples are not generically algebraic of degree 3 and hence have generic norms different from the cubic forms of the norm structure, see also the remark made in 3.9.

Proof. (a) Suppose m(t) and m′(t) are generic minimum polynomials for J. Then by 2.5.1, m(t; a) = μ_{a}(t) = m′(t; a) for all a ∈ J_{prim}(R) and all R ∈ k-alg, so m(t) = m′(t) follows from density of J_{prim}.

By [31, I, §8], m_{i} is homogeneous of degree i if and only if

m_{i}(rx) = r^{i}m_{i}(x),  (5)

for all r ∈ R, x ∈ J_{R}, R ∈ k-alg. Since k_{u} is dense in k_{a} and J_{prim} is dense in J, it suffices to show that this holds for all r ∈ R^{×} and x ∈ J_{prim}(R). Then by Lemma 1.9, rx is algebraic and we have μ̌_{rx}(t) = μ̌_{x}(rt). Hence, using 0.13.7,

m̌(t; rx) = μ̌_{rx}(t) = μ̌_{x}(rt) = m̌(rt; x).

Now (5) follows by comparing coefficients at powers of t in (1). In particular, m_{1} is homogeneous of degree 1 and thus a linear form on J [31, I, §11]. For (3), we may, after decomposing the base ring, assume that J has constant degree d. Then N = m_{d} is homogeneous of degree d.

The invariance under isomorphisms is clear from the uniqueness. In particular, the m_{i} are invariant under all automorphisms. If Δ ∈ Der(J) then Φ := Id + εΔ ∈ Aut(J_{R}) where R = k(ε). As remarked in 2.2(a), J_{R} is generically algebraic with generic minimum polynomial m(t)_{R}. Hence m_{i}(Φ(x)) = m_{i}(x), which after expansion yields (4).

(b) Since J̃ is in particular finitely generated and projective so is J by [3, I, §3.6, Prop. 12]. Hence, deg J is well-defined.

Let Ã = 𝒪(J̃) = A ⊗ R where A = 𝒪(J), and let m̃(t) ∈ Ã[t] be the generic minimum polynomial of J̃. We show that m̃(t) is “defined over k”, i.e., of the form m(t)_{R}, obtained by base change from a (unique) polynomial m(t) ∈ A[t]. The algebra S := R ⊗_{k} R can be considered as an R-algebra in two ways by means of the embeddings ϱ_{1}, ϱ_{2}: R → S into the first and second factor. We denote these R-algebra structures by S_{1} and S_{2}, respectively. By general facts on faithfully flat descent [17, Ch. III], m̃(t) is defined over k if and only if m̃(t)_{S_{1}} = m̃(t)_{S_{2}}. By 2.3(a) (applied to J̃ over the base ring R), J̃_{S_{i}} is generically algebraic over S with generic minimum polynomial m̃(t)_{S_{i}}, for i = 1, 2. Since J̃ = J ⊗_{k} R is defined over k, we have J̃_{S_{i}} = (J ⊗_{k} R)⊗_{R} S_{i} = J ⊗_{k} S_{i} = J ⊗_{k} S (because S_{i} = S as a k-algebra by restriction of scalars from R to k). Hence m̃(t)_{S_{1}} = m̃(t)_{S_{2}} follows from the uniqueness of the generic minimum polynomial of J ⊗_{k} S shown in (a). This proves the existence of m(t) ∈ A[t] with m(t)_{R} = m̃(t).

Now it is easy to see that m(t) meets the requirements of Definition 2.2. First, Ã is faithfully flat over A and m̃(t) can be considered as obtained from m(t) by base change from A to Ã. Hence m(t) is locally monic, as noted in 0.8. From the fact that J̃ satisfies m(t)_{R} and that R is faithfully flat, it follows easily that J satisfies m(t). Finally, let ϱ: k → R be the homomorphism making R a k-algebra. Then (\overline{deg} m) ∘ Spec(ϱ) = \overline{deg} m̃ (by 0.13.6) = deg J̃ (by condition (ii) of 2.2, because J̃ is generically algebraic over R) = (deg J) ∘ Spec(ϱ) (by 2.1.1). Now Spec(ϱ) is surjective because R is faithfully flat over k, so we conclude that condition (ii) of 2.2 holds for m(t).

2.8. Lemma. Let E ∈ k-alg be a commutative associative k-algebra which is finitely generated and projective as a k-module. Then the following conditions are equivalent:

(i) E is generically algebraic of degree deg E = rk E, (ii) E becomes monogenous after a faithfully flat base change.

If these conditions hold then the generic minimum polynomial of E is m(t; x) = det(tId − L(x)) and the primitive elements of E are precisely the generators of E as a k-algebra.

Remark. Algebras satisfying these conditions play an important role in the theory of the norm functor [8] where the property (ii) is called “locally simple”, see also [9].

Proof. (i) ⟹ (ii): Let E = E_{a} be the k-scheme defined by E. By Lemma 2.6, E_{prim} is open and dense in E, so there exists a faithfully flat (and even finitely presented) R ∈ k-alg such that E_{R} contains a primitive element z. Since z is strictly power-associative, R[z] ≅ M(z) and R[z] ⊂ E_{R} is a direct summand as an R-module by Cor. 1.7(c). It follows that rk R[z] = rk M(z) = deg(z) = deg E_{R} (because z is primitive) = rk E_{R} (by 2.1.1). Thus R[z] = E_{R} is monogenous, and we also see that primitive elements are generators.

(ii) ⟹ (i): By the descent property proved in Prop. 2.7(b) and the behaviour of the degree under base change (2.1.1), we may assume that E is monogenous, say, E = k[z]. Clearly, the indicated polynomial m(t) is locally monic, and from the Cayley-Hamilton Theorem it follows that E satisfies m(t). Hence deg E ⩽ \overline{deg} m(t) by 2.2.1. By Lemma 0.11, E ≅ k[t]/(m(t; z)). Since E is an associative algebra, z is strictly power-associative in the sense of 1.6. Because E = k[z], it follows from Cor. 1.7(c) that z is algebraic, and obviously μ_{z}(t) = m(t; z). Hence \overline{deg} m(t) = deg m(t; z) = (by 0.13.5 in the special case R = k) = deg μ_{z}(t) = deg z (by definition of the degree of an algebraic element in 1.4.5) ⩽ deg E (by 2.2.1). Thus deg E = \overline{deg} m(t), so E is generically algebraic with generic minimum polynomial m(t). As deg μ_{z}(t) = rk E by Lemma 0.11, we have deg E = rk E. We also see that z is primitive because deg z = deg E.

2.9. Lemma. Let J be generically algebraic with generic minimum polynomial m(t), and let a ∈ J be primitive. Let E := E(a) and π: E^{+} → J be as in 1.2(b). Then E is monogenous and finitely generated and projective, with rk E = deg J. For all y ∈ E_{R} and R ∈ k-alg,

m(t; π(y)) = det(tId − L(y)),  m̌(t; π(y)) = det(Id − tL(y)),  (1)

where L(y) is left multiplication with y in E.

Proof. E is finitely generated and projective by 1.2.5 and 1.4.2, and it is generically algebraic with generic minimum polynomial

m^{E}(t; y) = det(tId_{E} − L(y))  (2)

by Lemma 2.8. The affine scheme E := E_{a} defined by E is smooth, finitely presented and with connected fibres. By Lemma 2.6, E_{prim} is open in E and J_{prim} is open in J. Hence U: = E_{prim} ∩ π^{−1}(J_{prim}) is open in E. Moreover, z = can(t) ∈ E is primitive by Lemma 2.8, and π(z) = a ∈ J is primitive by assumption, whence z ∈ U(k). Thus U is dense in E by 0.14. We claim that

μ_{y}(t) = μ_{π(y)}(t) for all y ∈ U(R), R ∈ k-alg.  (3)

Indeed, y and b := π(y) are algebraic elements of E_{R} and J_{R}, respectively, so they have well-defined locally monic minimum polynomials, generating, respectively, the ideals 𝔑(y) and 𝔑(b) of R[t]. Moreover, 𝔑(y) ⊂ 𝔑(b) by 1.1.4, so μ_{y}(t) is a multiple of μ_{b}(t). Hence it suffices to show that both polynomials have the same degree, i.e., that deg b = deg y. Since y and b are primitive, deg y = deg E_{R} = rk E_{R} (by 2.8) and deg b = deg J_{R}. Also, since a is primitive, deg J = deg a = rk E, cf. 1.4.5, and therefore also deg J_{R} = rk E_{R}. This implies deg y = deg b, so we have (3). It follows that det(tId_{E} − L(y)) = m^{E}(t; y) (by (2)) = μ_{y}(t) (because y is primitive) = μ_{π(y)}(t) (by (3)) = m(t; π(y)) (because π(y) is primitive). Hence the first formula of (1) holds for all y ∈ U(R) and all R ∈ k-alg. Since U is dense in E, it holds for all y ∈ E(R). The second formula is clear from 0.10.1 and 0.10.2.

2.10. The adjoint. Let J be generically algebraic with generic minimum polynomial m(t). We define the polynomial m^{♯}(t) ∈ A[t] as in 1.11.1 by

tm^{♯}(t; x) = m(t; x) − m(0; x),  (1)

and the adjoint of an arbitrary x ∈ J_{R} (R ∈ k-alg) by

x^{♯} = m^{♯}(−x; −x).  (2)

From the homogeneity of the coefficients m_{i} of m(t) it follows that m(t; x) is locally homogeneous of degree deg J in (t; x), i.e., m(λt; λx) = λ^{deg J}m(t; x). Hence x ↦ x^{♯} is a locally homogeneous polynomial law of degree (deg J) − 1. If deg J = d is constant, we have

m^{♯}(t; x) = ∑_{i=1}^{d}(−1)^{i−1}m_{i−1}(x)t^{d−i},  x^{♯} = ∑_{i=1}^{d} m_{i−1}(x)(−x)^{d−i}.  (3)

We also note the formulas

U_{x}x^{♯} = N(x)x,  U_{x}(x^{♯})^{2} = U_{x^{♯}}x^{2} = N(x)^{2}1_{J},  (4)

which follow by applying j_{−x} to the equations t^{2}m^{♯}(t; −x) = tm(t; −x) − tN(x) and [tm^{♯}(t; −x)]^{2} = [m(t; −x) − N(x)]^{2}.

2.11. Theorem. Let J be a generically algebraic Jordan algebra over k. We use the notations introduced in 2.7 and 2.10.

(a) The generic minimum polynomial and copolynomial can be recovered from the generic norm by

m(t; x) = N(t1_{J} − x),  m̌(t; x) = N(1_{J} − tx).  (1)

In particular, m̌(t; 0) = N(1_{J}) = 1, m̌(t; 1_{J}) = (1 − t)^{deg J} and

m_{i}(1_{J}) = \binom{deg J}{i}  (i ∈ ℕ).  (2)

(b) For all x ∈ J, the generic norm is multiplicative on k[x] in the sense that

N((fg)(x)) = N(f(x))N(g(x))  (3)

for all f, g ∈ k[t]. In particular,

N(x^{j}) = N(x)^{j}  (j ∈ ℕ).  (4)

(c) The element x ∈ J is invertible if and only if N(x) ∈ k^{×}. In this case,

x^{−1} = \frac{x^{♯}}{N(x)},  (5)
m(t; −x^{−1}) = \frac{m̌(t; −x)}{N(x)},  (6)
N(x^{−1}) = \frac{1}{N(x)},  (7)
N(x^{♯}) = N(x)^{(deg J)−1},  (8)
x^{♯♯} = N(x)^{(deg J)−2}x.  (9)

(d) x ∈ J is nilpotent if and only the m_{i}(x) ∈ k are nilpotent, for all i ⩾ 1.

(e) The derivative of m_{i+1} in the direction of 1_{J} is

∂_{1_{J}} m_{i+1}|_{x} = ((deg J) − i)m_{i}(x).  (10)

Proof. (a) By density of J_{prim} (Lemma 2.6), it suffices to prove this for all x ∈ J_{prim}(R) and all R ∈ k-alg, and after changing the base from k to R to simplify notation, we may assume x = a ∈ J is primitive. As in Lemma 2.9, let a = π(z) and put y := t1_{E} − z ∈ E_{k[t]}. Then by 2.9.1,

N(t1_{J} − a) = N(π(y)) = m(0; −π(y)) = det L(y) = det(tId − L(z)) = m(t; a).

The second formula of (1) is proved similarly. By specializing t → 0 and because the copolynomial is comonic (0.9) we see 1 = m̌(0; x) = N(1_{J}), and putting x = 1_{J} yields m̌(t; 1_{J}) = N(((1 − t)1_{J}) = (1 − t)^{deg J} (by 2.7.3) = ∑_{i⩾0}(−1)^{i}m_{i}(1)t^{i} (by 2.7.1). This proves (2).

(b) By the density argument employed above, it suffices to prove this for x = a ∈ J primitive. Then the asserted formulas follow at once from the fact that N(π(y)) = det L(y) (by 2.9.1) and the multiplicativity of the determinant.

(c) If N(x) = m(0; −x) is invertible, then x is invertible, and formula (5) holds by Lemma 1.11(a). Conversely, let x ∈ J^{×} but assume N(x) ∉ k^{×}. Then there exists a maximal ideal 𝔪 ⊂ k with N(x) ∈ 𝔪. Let K = k/𝔪. Then 0 = N(x)_{K} = N(x_{K}), and hence x_{K} is not invertible in J_{K}, by [16, Theorem 2(iii)]. This contradicts the fact that invertible elements remain so under base change.

We prove (6). By density, we may assume x primitive. Then also x^{−1} is primitive because deg x = deg x^{−1} by 1.11(b). Now it follows from 1.11.4 and 2.5.2 that

m(t; x^{−1}) = \frac{m̌(t; x)}{m(0; x)},

and replacing x by −x in this formula yields (6). By specializing t → 0 we have (7).

Next, (8) and (9) are polynomial identities, so by density we may assume x invertible. Then N(x)^{−1} = N(x^{−1}) = N(x^{♯}/N(x)) = N(x^{♯})/N(x)^{deg J} yields the first formula. For the second, one argues similarly, using the fact that (x^{−1})^{−1} = x and that x^{♯} is homogeneous of degree (deg J) − 1.

(d) After decomposing the base ring, we may assume deg J = d constant. Then x^{d} = ∑_{i=1}^{d}(−1)^{i−1}m_{i}(x)x^{d−i}. Hence, if all m_{i}(x) are nilpotent, it follows from the multinomial expansion that x is nilpotent. Conversely, if x is nilpotent then so is x(𝔭), for all prime ideals 𝔭 ∈ Spec(k). By [16, Theorem 2(vi)], 0 = m_{i}(x(𝔭)) = m_{i}(x) ⊗ 1_{κ(𝔭)}, so m_{i}(x) belongs to the intersection of all prime ideals of k and therefore is nilpotent.

(e) Let R = k(ε) be the algebra of dual numbers. Then by (a), m̌(t; x + ε1_{J}) = N(1_{J} − t(x + ε1_{J})) = N((1 − tε)1_{J} − tx). Now 1 − tε is invertible with inverse (1 − tε)^{−1} = 1 + tε. Hence by 2.7.3,

m̌(t; x + ε1_{J}) = (1 − tε)^{deg J}N(1 − t(1 + tε)x) = (1 − tε)^{deg J} m̌(t + t^{2}ε; x).

Expanding both sides with formula 2.7.1 and comparing coefficients at εt^{i+1} yields (10).

2.12. Generic elements. For finite-dimensional algebras Jordan algebras over fields, the generic minimum polynomial is simply the minimum polynomial of the generic element, see [5, 14]. Does a similar statement hold over rings? It would be tempting to be able to say “J is generically algebraic if and only if the generic element of J is algebraic”. Unfortunately, this is not the case. Let us first recall the notion of generic element for finitely generated and projective modules [19, §18].

Let J be finitely generated and projective and A = 𝒪(J). The generic element x of J is the element x ∈ J ⊗ J^{∗} ⊂ J ⊗ A corresponding to Id_{J} under the canonical isomorphism J ⊗ J^{∗} ≅ End J. The name “generic element” is justified by the fact that x can be specialized to any x ∈ J_{R}, R ∈ k-alg, in the following sense: Evaluation of an element g ∈ A at x defines a homomorphism e_{x}: A → R, e_{x}(g) = g(x), and hence a map Id_{J} ⊗ e_{x}: J_{A} → J_{R}, which maps x to x. We thus have g = g(x) ∈ A, for all g ∈ A. This implies that, for any f(t) ∈ A[t], we have f(t; x) = f(t), and that J satisfies a locally monic f(t) ∈ A[t] (in the sense of 2.2) if and only if the generic element x satisfies f(t).

2.13. Proposition. (a) Let J be generically algebraic with generic minimum polynomial m(t). Then the generic element x of J is pre-algebraic and its minimum polynomial is m(t), but x is not an algebraic element of J_{A}, unless J = {0}.

(b) There are examples of finitely generated and projective Jordan algebras whose generic element is pre-algebraic but which are not generically algebraic.

Proof. (a) After decomposing the base ring, we may assume deg J = d constant. Since every element x ∈ J_{R} in every base ring extension R of k satisfies m(t; x), this is in particular so for x, whence m(t) ∈ 𝔑(x). To prove μ_{x}(t) = m(t) it suffices to show that x^{[0]}, …, x^{[d−1]} are free over A in J_{A}^{2}. Thus assume a relation ∑_{i=0}^{d−1} g_{i}(x)x^{[i]} = 0 in J_{A}^{2}. By specializing x to any a ∈ J_{prim}(R), R ∈ k-alg, it follows that ∑_{i=0}^{d−1} g_{i}(a)a^{[i]} = 0. Since a is algebraic of degree d, the powers a^{[0]}, …, a^{[d−1]} are linearly independent over R, so that all g_{i}(a) = 0. Thus the g_{i} vanish on the open and dense subscheme J_{prim} of J and are therefore zero. It follows that x is pre-algebraic of degree d, so its minimum polynomial is m(t).

Now assume that x is algebraic. Then, for every A-algebra R, the degree of x_{R} is still d. On the other hand, x can be specialized to 0. Hence the element 0 ∈ J is algebraic of degree d. Since 0 is in particular an idempotent, the computation in 1.8 shows μ_{0}(t) = γt + (1 − γ) where γ is the support idempotent of 1_{J}. It follows that either d = 0 and then γ = 0 hence J = 0, or d = 1 whence d = 1 and J = k. But the second case is impossible, because then x = t ∈ A = k[t], and the A-submodule of A generated by t is tA which is not a direct summand of A (as an A-module).

(b) Let J = B^{+} be the algebra of 2.4(f). Then it is easily seen that the generic element is pre-algebraic, but J is not generically algebraic because its degree function is not locally constant on Spec(ℤ).

3. The generic minimum polynomial of an isotope

3.1. Theorem. Let J be a generically algebraic Jordan algebra over k, with generic minimum polynomial m(t) and generic norm N. Let v ∈ J^{×} and let J^{(v)} be the visotope of J, with unit element 1^{(v)} = v^{−1} and U-operators U_{x}^{(v)} = U_{x}U_{v}. Then also J^{(v)} is generically algebraic, deg J^{(v)} = deg J, with generic minimum polynomial, copolynomial and generic norm given by

m^{(v)}(t; x) = N(v)N(tv^{−1} − x),  (1)
m̌^{(v)}(t; x) = N(v)N(v^{−1} − tx),  (2)
N^{(v)}(x) = N(v)N(x).  (3)

Proof. We follow the idea of the proof of [26, Th. 1] but avoid the use of the composition law for the generic norm. Clearly J^{(v)} is finitely generated and projective because J = J^{(v)} as k-modules. For the remainder of the proof, we may assume, after decomposing the base ring, that deg J = d is constant, and it is no restriction to assume d > 0, for else J = {0} by 2.4(c). Then N is homogeneous of degree d. Define polynomial laws N_{i,d−i}, bihomogeneous of degree (i, d − i), on J × J by the expansion

N(sx + ty) = ∑_{i=0}^{d} s^{i}t^{d−i}N_{i,d−i}(x, y),  (4)

where s, t are indeterminates and x, y ∈ J_{R}, R ∈ k-alg, cf. [31, II, §1]. Then N_{i,d−i}(x, y) = N_{d−i,i}(y, x) and N_{0,d}(x, y) = N(y). Define f(t) ∈ A[t] (where A = 𝒪(J)) by

f(t; x) = N(v)N(tv^{−1} − x) = ∑_{i=0}^{d}(−1)^{i}N(v)N_{i,d−i}(x, v^{−1})t^{d−i}.  (5)

This is obtained from (4) by s → −1 and y = v^{−1}. Then the coefficient of t^{d} in f(t) is N(v)N_{0,d}(x, v^{−1}) = N(v)N(v^{−1}) = 1 (by 2.11.7), so f(t) is monic of degree d.

Now we show that J^{(v)} satisfies f(t) in the sense of 2.2, i.e., that f(t; x) and tf(t; x) vanish upon substitution of x for t, for all x ∈ J_{R}^{(v)} and R ∈ k-alg. Since everything is compatible with base change, we may extend k to R and then write again k for R (for simpler notation), so it suffices to prove this for all x ∈ J^{(v)}. Consider the k-algebra S = k(λ) = k[s]/(s^{d+2}) which is free as a k-module with basis 1, λ, …, λ^{d+1} and satisfies λ^{d+2} = 0. Since λ is nilpotent, 1^{(v)} − λx ∈ J_{S}^{(v)} is invertible. We compute its inverse in two ways. First, again by nilpotence of λ, the inverse is given by the geometric series:

(1^{(v)} − λx)^{(−1,v)} = ∑_{j=0}^{d+1} λ^{j}x^{(j,v)},  (6)

where x^{(j,v)} is the jth power of x in J^{(v)}. On the other hand, an element a is invertible in J^{(v)} if and only if it is invertible in J, and then

a^{(−1,v)} = (U_{a}^{(v)})^{−1}a = (U_{a}U_{v})^{−1}a = U_{v}^{−1}U_{a}^{−1}a = U_{v}^{−1}a^{−1}.  (7)

Since J is generically algebraic, a^{−1} = a^{♯}/N(a) by 2.11.5. All this remains true in any base ring extension. In particular, let a = 1^{(v)} − λx = v^{−1} − λx. Substituting a^{♯}/N(a) for a^{−1} in (6) and multiplying the resulting equation with N(a) yields

U_{v}^{−1}(v^{−1} − λx)^{♯} = N(v^{−1} − λx) · ∑_{j=0}^{d+1} λ^{j}x^{(j,v)}.  (8)

Now x ↦ x^{♯} is a homogeneous polynomial law of degree d − 1 by 2.10. Hence the coefficients of λ^{d} and λ^{d+1} on the left hand side of (8) vanish, so they must vanish on the right hand side as well. From (4) we have the expansion N(v^{−1} − λx) = ∑_{i=0}^{d}(−λ)^{i}N_{i,d−i}(x, v^{−1}). By collecting terms at λ^{d} and λ^{d+1} on the right hand side, we obtain the relations

∑_{i=0}^{d}(−1)^{i}N_{i,d−i}(x, v^{−1})x^{(d−i,v)} = ∑_{i=0}^{d}(−1)^{i}N_{i,d−i}(x, v^{−1})x^{(d+1−i,v)} = 0.

After multiplying with N(v), this says precisely that x satisfies f(t; x) in the isotope J^{(v)}.

We have verified condition (i) of 2.2 for J^{(v)}, so it remains to show that J^{(v)} has degree d, i.e., that J^{(v)}(𝔭) = J^{(v)} ⊗ 1_{κ(𝔭)} has degree d, for all 𝔭 ∈ Spec(k). Now deg J^{(v)}(𝔭) ⩽ d = deg J(𝔭) because because J^{(v)}(𝔭) satisfies the polynomial

f(t)_{κ(𝔭)} of degree d. But J(𝔭) = (J^{(v)}(𝔭))^{(v^{−2}(𝔭))} is an isotope of J^{(v)}(𝔭), and J^{(v)}(𝔭) is generically algebraic, being finite-dimensional over κ(𝔭). Hence the above argument, applied to J^{(v)}(𝔭) instead of J, yields deg J(𝔭) ⩽ deg J^{(v)}(𝔭). We have shown that J^{(v)} is generically algebraic with generic minimum polynomial f(t) = m^{(v)}(t). The formulas for m̌^{(v)}(t) and the generic norm are then an immediate consequence.

3.2. Corollary. (a) The generic norm of a generically algebraic Jordan algebra J permits Jordan composition:

N(U_{x}y) = N(x)^{2}N(y),  (1)

for all x, y ∈ J_{R}, R ∈ k-alg.

(b) Let B be a generically algebraic associative or alternative algebra over k and let N be its generic norm, i.e., the generic norm of the associated Jordan algebra B^{+}. Then N is multiplicative:

N(xy) = N(x)N(y),  (2)

for all x, y ∈ B_{R}, R ∈ k-alg.

Proof. (a) By density of J^{×} (see 0.14) we may assume y invertible. Then N(y)N(U_{x}y) = N^{(y)}(x^{(2,y)}) = N^{(y)}(x)^{2} (by 2.11.4) = N(y)^{2}N(x)^{2} (by 3.1.3).

(b) Again by density, we may assume x invertible. Then left multiplication L_{x} with x in B is an isomorphism L_{x}: (B^{(x)})^{+} → B^{+} of Jordan algebras [24, Prop. 4],

and (B^{(x)})^{+} = (B^{+})^{(x)} by [24, (19)]. Since the generic norm is invariant under isomorphisms by Prop. 2.7(a), it follows that N(xy) = N(L_{x}(y)) = N^{(x)}(y) = N(x)N(y).

The following lemma is the analogue over rings of [26, Th. 2].

3.3. Lemma. Let J be generically algebraic and let u, v ∈ J^{×}. Then the generic minimum polynomials of u ∈ J^{(v)} and v ∈ J^{(u)} are related by

m^{(v)}(t; u) = m^{(u)}(t; v),  (1)
m̌^{(v)}(t; u) = m̌^{(u)}(t; v),  (2)

and the generic norm has the symmetry property

N(v)N(v^{−1} − u) = N(u)N(u^{−1} − v).  (3)

Proof. Let us first assume that u is primitive, hence in particular algebraic, in J^{(v)}. By Lemma 1.9(b), v is then algebraic in J^{(u)}, and μ_{v}^{(u)}(t) = μ_{u}^{(v)}(t) = m^{(v)}(t; u) (by 2.5.1). Hence deg μ_{v}^{(u)}(t) = deg m^{(v)}(t; u) = deg J = deg J^{(u)} by Th. 3.1. This shows that v is primitive in J^{(u)}, and hence (1) follows again from 2.5.1. Furthermore, (2) and (3) are immediate consequences.

In the general case, consider the subfunctor U of J^{×} × J^{×} defined by

( \begin{matrix}x \\ y \end{matrix} ) ∈ U(R) ⟺ x is primitive in (J_{R})^{(y)},

for all R ∈ k-alg. Note that (x, y) ∈ U(R) if and only if (y, x) ∈ U(R) and that (1) – (3) hold on U(R), by what we proved above. Hence, it suffices to show that U is an open and dense subscheme of J^{×} × J^{×}. After decomposing the base ring, we may assume deg J = d constant. Then also all isotopes of all J_{R} have degree d. Define a morphism p: J^{×} × J^{×} → P_{a}^{(d)} (cf. Prop. 1.14(b)) by

( \begin{matrix}x \\ y \end{matrix} ) ↦ ( \begin{matrix}x^{(0,y)} \\ x^{(1,y)} \end{matrix} ) ∧ ⋯ ∧ ( \begin{matrix}x^{(d−1,y)} \\ x^{(d,y)} \end{matrix} )

in all base ring extensions. Then U is the inverse image of P_{u}^{(d)} under p and therefore open. Furthermore, J^{×} × J^{×} is smooth, being open in J × J, and it has connected fibres: Indeed, for all algebraically closed fields K ∈ k-alg, J_{K}^{×} × J_{K}^{×} is the complement of the hypersurface N(x)N(y) = 0 in J_{K} × J_{K} and the latter is isomorphic to affine 2n-space over K where n = dim J_{K}. Thus by 0.14, it suffices to have U(K) ≠ ∅. Since J_{prim}(K) ≠ ∅ by density of J_{prim} and J_{prim} × {1_{J}} ⊂ U, we are done.

3.4. Suppose J is generically algebraic of constant degree d. In view of 3.1.1 and 3.1.5, the symmetry formulas 3.3.1 – 3.3.3 are equivalent to

m_{i}^{(y)}(x) = N(y)N_{i,d−i}(x, y^{−1}) = N(x)N_{i,d−i}(y, x^{−1}) = m_{i}^{(x)}(y).

Here the two left hand sides are defined for all x while the two right hand sides are defined for all y. This suggests the existence of polynomial laws F_{i} on J × J such that m_{i}^{(y)}(x) = F_{i}(x, y) for all x and all invertible y, and analogously with x and y interchanged. For i = 0 and i = d this is of course trivially true, with F_{0} = 1 and

F_{d}(x, y) = N(x)N(y).  (1)

For i = d − 1, we have N_{d−1,1}(z, v) linear in v, and hence, using 2.11.5, m_{d−1}^{(y)}(x) = N(y)N_{d−1,1}(x, y^{−1}) = N_{d−1,1}(x, y^{♯}), so

F_{d−1}(x, y) = N_{d−1,1}(x, y^{♯})  (2)

is such a polynomial law. We now show that such F_{i} exist for all i.

3.5. Theorem. Let J be generically algebraic. Then there exist unique polynomial laws F_{i} (i ∈ ℕ) on J × J such that

m_{i}^{(y)}(x) = F_{i}(x, y),  (1)

for all x ∈ J_{R}, y ∈ (J_{R})^{×}, R ∈ k-alg. The F_{i} are symmetric and bihomogeneous of bidegree (i, i). In particular, F_{1} is a symmetric bilinear form on J, called the bilinear trace. Explicitly, it is given by

F_{1}(x, y) = m_{1}(x)m_{1}(y) − m_{1,1}(x, y) = −∂_{x}∂_{y} log N|_{1_{J}}  (2)

where m_{1,1} denotes the bilinear form associated to the quadratic form m_{2}, and satisfies

F_{1}(x, y^{♯}) = ∂_{x}N|_{y} ,  (3)

for all x, y ∈ J_{R}, R ∈ k-alg.

Proof. After decomposing the base ring, we may assume deg J = d constant. Let R ∈ k-alg and consider the R-algebra R̃ = R[t](λ) = R[t, s]/(s^{d+1}). Let x, y ∈ J_{R} be arbitrary. Then 1 − λx ∈ J_{R} ⊗_{R} R̃ is invertible, with inverse given by the geometric series: (1 − λx)^{−1} = 1 + λx + ⋯ + λ^{d}x^{d}. Since R̃ is a free R-module with basis λ^{i}t^{j} (i = 0, …, d, j ∈ ℕ), we can write, using 3.1.2,

m̌^{(1−λx)}(t; y) = N(1 − λx)N((1 − λx)^{−1} − ty) = ∑_{i,j=0}^{d} λ^{i}t^{j}h_{ij}(x, y),  (4)

with uniquely determined h_{ij}(x, y) ∈ R. By “varying R”, one sees that (4) defines polynomial laws h_{ij} on J × J, and since the left hand side of (4) depends only on λx and ty, it is clear that h_{ij} is bihomogeneous of bidegree (i, j). Now assume y invertible. Then by 2.7.1 and 3.3.2,

∑_{i=0}^{d}(−1)^{i}m_{i}^{(y)}(1−λx)t^{i} = m̌^{(y)}(t; 1−λx) = m̌^{(1−λx)}(t; y) = ∑_{i,j=0}^{d} λ^{i}t^{j}h_{ij}(x, y). (5)

Since m_{i}^{(y)} is homogeneous of degree i we have m_{i}^{(y)}(1 − λx) = (−1)^{i}λ^{i}m_{i}^{(y)}(x) plus terms with lower powers of λ. Substituting this into (5) and comparing coefficients at λ^{i}t^{i} yields m_{i}^{(y)}(x) = h_{ii}(x, y) (and h_{ij} = 0 for i > j). Thus F_{i} := h_{ii} has the asserted property. Symmetry and homogeneity of the F_{i} follow from the corresponding properties of the m_{i}^{(y)}(x) = m_{i}^{(x)}(y) and the fact that J^{×} × J^{×} is dense in J × J. In particular, F_{1} is a bilinear form by [31, Prop. I.6].

We now determine F_{1} = h_{11} from (4) as the coefficient of λt. Computing modulo λ^{2} and t^{2}, we have (1 − λx)^{−1} ≡ 1 + λx, hence

m̌^{(1−λx)}(t; y) = N(1 − λx)N(1 + λx − ty) = m̌(1; λx)m̌(1; ty − λx) (by 2.11.1)
≡ {1 − m_{1}(λx)}{1 − m_{1}(ty − λx) + m_{2}(ty − λx)}  (by 2.7.1)
≡ 1 − tm_{1}(y) + tλ{(m_{1}(x)m_{1}(y) − m_{1,1}(x, y)}.

This proves (2). By density of J^{×}, it suffices to prove (3) for y invertible. Replace v by y^{−1} in 3.1.2 and multiply the result with N(y). Then, because N(y^{−1}) = N(y)^{−1} and y^{−1} = y^{♯}N(y)^{−1} by Th. 2.11(c), it follows that

N(y − tx) = N(y)m̌^{(y^{−1})}(t; x) ≡ N(y) − tF_{1}(x, y^{♯}) (mod t^{2}).

On the other hand, N(y − tx) ≡ N(y) − t∂_{x}N|_{y} (mod t^{2}), whence (3).

3.6. Corollary. We keep the assumptions and the notation of Th. 3.5. Let Str(J) be the structure group of J and let g ↦ g^{∗} be the antiautomorphism of Str(J) determined by U_{g(x)} = gU_{x}g^{∗}, for all x ∈ J.

(a) The F_{i} are invariant under the structure group in the sense that, for g ∈ Str(J),

F_{i}(g(x), y) = F_{i}(x, g^{∗}(y)),  (1)

for all x, y ∈ J_{R}, R ∈ k-alg.

(b) The generic norm is covariant under the structure group in the sense that

N(g(x)) = N(g(1_{J}))N(x)  (2)

for all g ∈ Str(J), x ∈ J_{R}, R ∈ k-alg. The map g ↦ N(g(1_{J})) is a character χ: Str(J) → k^{×} satisfying χ(g^{∗}) = χ(g) and χ(U_{x}) = N(x)^{2}.

Proof. (a) By density of J^{×}, it suffices to prove (1) for invertible y, and after changing the base ring from k to R, we may assume x, y ∈ J. Then g: J^{(g^{∗}(y))} → J^{(y)} is an isomorphism of Jordan algebras. Hence, by the invariance of the generic minimum polynomial under isomorphisms (Prop. 2.7(a)), F_{i}(g(x), y) = m_{i}^{(y)}(g(x)) = m_{i}^{(g^{∗}(y))}(x) = F_{i}(x, g^{∗}(y)).

(b) We may assume that J has constant degree d. Then by 3.4.1 and (1), N(g(x))N(y) = N(x)N(g^{∗}(y)). By putting x = y we see that N(g(x)) = N(g^{∗}(x)) and for y = 1 we obtain N(g(x)) = N(g^{∗}(1))N(x) = N(g(1))N(x). This easily implies that χ is a homomorphism. Finally, χ(U_{x}) = N(x)^{2} is clear from 3.2.1.

Remark. For a base field and i = 1, formula (1) is proved in [26, Th. 4]. An obvious rephrasing of (b) together with 3.1.3 shows that an isotopy between generically algebraic Jordan algebras is a strict norm similarity. The converse holds for central separable Jordan algebras over rings containing \frac{1}{2} [2, Th. 4.4] but fails for rings where 2 is not invertible, e.g., for the n×n symmetric matrices over k = K(ε) (dual numbers over a field K of characteristic 2), as discovered by Waterhouse [32].

3.7. Corollary. J is a generically algebraic Jordan algebra of degree 2 if and only if J = Jor(X, Q, 1) is the Jordan algebra defined by a unital quadratic form (X, Q, 1) as in [20, 1.5], where X is a finitely generated and projective k-module of rank ⩾2.

Proof. (a) Let J = Jor(X, Q, 1) with X finitely generated and projective of rank ⩾2. Thus

U_{x}y = B(x, ȳ)x − Q(x)ȳ,  (1)

where B is the bilinear form associated with Q and x̄ = T (x)1 − x, with T (x) = B(1, x). By [20, 1.5.2], J satisfies the polynomial m(t; x) = t^{2} − T (x)t + Q(x), and we have deg J = 2 because rk X ⩾ 2, cf. 2.4(c). Hence J is generically algebraic of degree 2.

(b) Conversely, let J be generically algebraic of degree 2, with generic minimum polynomial m(t; x) = t^{2} − m_{1}(x)t + m_{2}(x), and let X be the k-module underlying J. Because a Jordan algebra over a field is of degree 1 if and only if it is onedimensional, the condition deg J = 2 implies rk X ⩾ 2. Hence the unit element 1_{J} = 1 is a unimodular vector by 0.3.2. Next, Q := m_{2} is a quadratic form, and N(x) = m(0; −x) = Q(x), so Q(1) = 1 by Th. 2.11(a). Thus (X, Q, 1) is a unital quadratic form in the sense of [20]. It remains to show that J = Jor(X, Q, 1) is the associated Jordan algebra, i.e., that (1) holds.

Let B be the bilinear form associated with Q and put T (x) := B(1, x) as well as x̄ := T (x)1 − x. Now note that m_{1}(x) = ∂_{1}m_{2}|_{x} (by 2.11.10) = B(1, x) = T (x), and x^{♯} = m_{1}(x)1−x (by 2.10.3) = x̄. By density of J^{×}, it suffices to prove (1) for y invertible. Then U_{x}y = x^{(2,y)} = m_{1}^{(y)}(x)x − m_{2}^{(y)}(x)1^{(y)} = F_{1}(x, y)x − F_{2}(x, y)y^{−1}. Furthermore, F_{1}(x, y) = F_{1}(x, \overline{ȳ}) = ∂_{x}N|ȳ = B(x, ȳ) by 3.5.3, and y^{−1} = y^{♯}/N(y) (by 2.11.5) = ȳ/Q(y), as well as F_{2}(x, y) = Q(x)Q(y) by 3.4.1. Hence U_{x}y is given by (1).

3.8. Example. Not every Jordan algebra J which is finitely generated and projective as a k-module and whose degree is constant equal to 2 is generically algebraic of degree 2 and (hence the Jordan algebra of a unital quadratic form). For example, let J = k · 1 ⊕ k · a be free of rank 2 as a k-module. It can be shown that there exists a unique Jordan algebra structure on J such that

a^{2} = βa − α1, a^{3} = βa^{2} − (α + δ)a + γ1  (1)

if and only if the constants α, β, γ, δ in k satisfy the conditions

2γ = 2δ = βγ = βδ = γ^{2} = δ^{2} = γδ = 0.  (2)

Suppose these conditions are satisfied. Then J has constant degree equal to 2. Indeed, by (2), γ and δ are nilpotent, so γ_{K} = δ_{K} = 0 for all fields K ∈ k-alg. Thus J_{K} is just the commutative associative algebra K[t]/(t^{2} − β_{K}t + α_{K}1) considered as a Jordan algebra, hence of degree 2. We claim that J is generically algebraic if and only if γ = δ = 0. Indeed, assuming J to be generically algebraic, its generic minimum polynomial has the form m(t; x) = t^{2} − m_{1}(x)t + m_{2}(x)1. Now 0 = m(a; a) = a^{2} − m_{1}(a)a + m_{2}(a)1 shows, because 1 and a are a basis of J as a k-module, that β = m_{1}(a) and α = m_{2}(a). Moreover, 0 = j_{a}(tm(t; a)) = a^{3} − βa^{2} + αa implies, by (1), that γ = δ = 0. Conversely, if γ = δ = 0, then it is easy to see that J is the Jordan algebra associated with the unital quadratic form Q on J given by Q(λ1 + μa) = λ^{2} + λμβ + μ^{2}α, and is therefore generically algebraic of degree 2 by Cor. 3.7.

Since there are rings containing α, …, δ satisfying (2) with (γ, δ) ≠ (0, 0), for example, k = ℤ/4ℤ or the ring of dual numbers over a field of characteristic 2, this gives examples of Jordan algebras, free of rank 2 and of constant degree 2, which are not generically algebraic.

3.9. Corollary. (a) Let J be generically algebraic of degree 3, with generic norm N and adjoint map ♯. Then J is obtained by the general cubic construction [22, Th. 1] from the cubic norm structure (N, ♯, 1).

(b) Conversely, let (N, ♯, 1) be a cubic norm structure on a finitely generated and projective k-module X, let J be the Jordan algebra structure on X determined by (N, ♯, 1), and suppose that deg J = 3 (as a function on Spec(k)). Then J is generically algebraic of degree 3.

Remark. In (b), deg J is a well-defined function on Spec(k) by 2.1, because X is finitely generated and projective. Unlike the degree 2 case, a blanket assumption on the rank of X is not sufficient to guarantee that deg J = 3, cf. [30, 2.4 – 2.6].

Proof. (a) We first verify the conditions required for (N, ♯, 1): Condition (i) x^{♯♯} = N(x)x, holds by 2.11.9, and condition (ii) F_{1}(x, y^{♯}) = ∂_{x}N|_{y}, holds by 3.5.3. Condition (iii) says 1 × y = m_{1}(y)1 − y: By 2.10.3, x^{♯} = x^{2} − m_{1}(x)x + m_{2}(x)1. Linearization yields x × y = x ∘ y − m_{1}(x)y − m_{1}(y)x + ∂_{x}m_{2}|_{y}, and putting x = 1 results in

1 × y = 1 ∘ y − m_{1}(1)y − m_{1}(y)1 + ∂_{1}m_{2}|_{y}
= 2y − 3y − m_{1}(y)1 + 2m_{1}(y) (by 2.11.2 and 2.11.10)
= −y + m_{1}(y)1.

The remaining conditions (iv) 1^{♯} = 1 and (v) N(1) = 1 are clear from Th. 2.11. Now the same formal calculation as in [23, Th. 1] shows that U_{x}y = F_{1}(x, y)x − x^{♯} × y.

(b) It only remains to show that J satisfies the polynomial m(t; x) = t^{3} − T (x)t^{2} + S(x)t − N(x) (where we use the notations of [22]). Now x^{3} − T (x)x^{2} + S(x)x − N(x)1 = 0 is [22, (20)], and x^{4} = U_{x}x^{2} = U_{x}(x^{♯} + T (x)x − S(x)1) (by [22, (21)]) = N(x)x + T (x)x^{3} − S(x)x^{2} (by [22, (24)]), as desired.

3.10. Corollary. Let J be generically algebraic and x ∈ J. Then for all polynomials f(t), g(t) ∈ k[t],

F_{i}(f(x), g(x)) = m_{i}((fg)(x)).  (1)

In particular,

m_{i}(x^{l+n}) = F_{i}(x^{l}, x^{n}),  (2)

for all l, n ∈ ℕ.

Proof. By the standard density argument, we may assume x = a primitive. Then let E = E(a) = k[t]/(m(t; a)) and π: E^{+} → k[a] be as in Lemma 2.9, and let z = can(t) be the generator of E. The elements of E are of the form y = f(z) where f ∈ k[t]. If w = g(z) is a second element of E, then π(y) = f(a), π(w) = g(a), and π(yw) = (fg)(a). Hence, (1) is equivalent to

F_{i}(π(y), π(w)) = m_{i}(π(yw)),  (3)

for all y, w ∈ E. By 2.9.1, N(π(y)) = m(0; −π(y)) = det L(y), and m̌(t; π(y)) = det(Id − tL(y)), for all y ∈ E. Again by density, it suffices to prove (3) for all invertible w ∈ E. Then also π(w) is invertible in J with inverse π(w^{−1}), and we have

m̌^{(π(w))}(t; π(y)) = N(π(w))N(π(w^{−1}) − tπ(y)) (by 3.1.2)

= det L(w) det (L(w^{−1}) − tL(y))

= det(Id − tL(yw)) (because E is commutative and associative)

= m̌(t; π(yw)).

Expanding both sides with 2.7.1 and using 3.5.1 yields (3).

3.11. An explicit formula for the F_{i}. Let J be generically algebraic of constant degree d. It is natural to ask for an explicit formula, expressing the F_{i} of Th. 3.5 in terms of the coefficients m_{j} of the generic minimum polynomial and generalizing 3.5.2 in case i = 1. The proof of 3.5 shows that F_{i} = h_{ii} is the coefficient of λ^{i}t^{i} in 3.5.4. In order to free the index i for other purposes, we will use the letter p instead of i. In expanding the left hand side of 3.5.4, we are therefore only interested in terms containing λ^{p}t^{p}. Thus, it is no restriction to assume λ^{p+1} = 0.

Define polynomial laws g_{ip}(x, y) on J × J, bihomogeneous of bidegree (i, p), by the expansion

N((1 − λx)^{−1} − ty) = N(1 + λx + ⋯ + λ^{p}x^{p} − ty) ≡ (−t)^{p} ∑_{i=0}^{p} g_{ip}(x, y)λ^{i}, (1)

where ≡ means that both sides differ only by terms not involving t^{p}. Then

\begin{aligned}N(1 − λx)N((1 − λx)^{−1} − ty) & ≡ (−t)^{p}( ∑_{i=0}^{p}(−1)^{i}m_{i}(x)λ^{i})( ∑_{i=0}^{p} g_{ip}(x, y)λ^{i}) \\ & = λ^{p}t^{p}( ∑_{i=0}^{p} g_{ip}(x, y)(−1)^{i}m_{p−i}(x)) + ⋯ \end{aligned}

where the dots indicate terms not involving λ^{p}, and therefore

F_{p}(x, y) = ∑_{i=0}^{p} g_{ip}(x, y)(−1)^{i}m_{p−i}(x).  (2)

It remains to compute the g_{ip}. Since N(1 + z) = m̌(1; −z) = ∑_{j=0}^{d} m_{j}(z) by 2.7.1 and 2.11.1, we have

N(1 + λx + ⋯ + λ^{p}x^{p} − ty) ≡ ∑_{j=p}^{d} m_{j}(−ty + λx + ⋯ + λ^{p}x^{p}),  (3)

because m_{j} = 0 for j > d. Let t_{0}, …, t_{p} be indeterminates, and define the multihomogeneous polynomial laws m_{i_{0},...,i_{p}}(x_{0}, …, x_{p}) of multidegree (i_{0}, …, i_{p}) and total degree j = i_{0} + ⋯ + i_{p} by the expansion

m_{j}(t_{0}x_{0} + ⋯ + t_{p}x_{p}) = ∑_{i_{0}+···+i_{p}=j} m_{i_{0},...,i_{p}}(x_{0}, …, x_{p})t_{0}^{i_{0}} ⋯ t_{p}^{i_{p}},

cf. [31, Chap. II]. Note that the i_{ν} are ⩾0, and if i_{ν} = 0 then m_{i_{0},...,i_{p}} is independent of x_{ν}. Then we obtain from (3), putting x̃ = λx + ⋯ + λ^{p}x^{p} for short,

N(1 + x̃ − ty) ≡ (−t)^{p} ∑_{l=0}^{d−p} ∑_{i_{1}+···+i_{p}=l} λ^{1i_{1}+2i_{2}+···+pi_{p}} m_{p,i_{1},...,i_{p}}(y, x, x^{2}, …, x^{p}).

(4)

Now collect the terms involving λ^{i} to obtain g_{ip}:

g_{ip}(x, y) = ∑_{l=0}^{d−p} ∑_{{i_{1}+···+i_{p}=l \atop 1i_{1}+2i_{2}+···+pi_{p}=i}} m_{p,i_{1},...,i_{p}}(y, x, x^{2}, …, x^{p}).  (5)

The sum over l in (5) actually runs only from 0 to min(p, d − p) because l = i_{1} + ⋯ + i_{p} ⩽ 1i_{1} + ⋯ + pi_{p} = i and 0 ⩽ i ⩽ p by (1). Then formulas (2) and (5) together constitute the desired explicit expression for F_{p}. We note the following special cases:

F_{2}(x, y) = m_{2}(x)m_{2}(y) + m_{2,1}(y, x^{2} − m_{1}(x)x) + m_{2,2}(x, y),  (6)
F_{3}(x, y) = m_{3}(x)m_{3}(y) − m_{3,1}(y, x^{3} − m_{1}(x)x^{2} + m_{2}(x)x)
+ m_{1}(x)m_{3,2}(y, x) − m_{3,1,1}(y, x, x^{2}) − m_{3,3}(x, y).  (7)

Finally note that, because the F_{i}(x, y) are symmetric in x and y, so must be the right hand side of these formulas, which yields identities between the polarizations of the m_{i}.

Remark. An expansion similar to the right hand side of (4), but in a different context, occurs in [27, (1.13)] (personal communication by K. McCrimmon).

3.12. The exponential trace formula. Let J be generically algebraic. Switching x and y in Formula 3.5.3 we have F_{1}(x^{♯}, y) = ∂_{y}N|_{x}. For invertible x, this is equivalent to

\frac{∂_{y}N|_{x}}{N(x)} = ∂_{y} log N|_{x} = F_{1}(x^{−1}, y)  (1)

because x^{−1} = x^{♯}/N(x). Let k[[t]] be the algebra of formal power series and put J̃ := J ⊗ k[[t]]. Also let x ∈ J be arbitrary. Then 1 − tx ∈ J̃ is invertible in J̃, with inverse given by (1 − tx)^{−1} = ∑_{i=0}^{∞} x^{i}t^{i}. Indeed, the formal sum on the right makes sense in J[[t]] (by which we mean the direct product of countably many copies J · t^{i} (i ∈ ℕ) of J with the obvious operations making it a Jordan algebra) and has the right formal properties, so it suffices to show that J̃ = J[[t]]. Since J is finitely generated and projective there exist finitely many v_{j} ∈ J and α_{j} ∈ J^{∗} such that Id_{J} = ∑_{j=1}^{l} v_{j} ⊗ α_{j}. Hence an element ∑_{i=0}^{∞} z_{i}t^{i} ∈ J[[t]] equals ∑_{j=1}^{l} φ_{j}(t)v_{j} where φ_{j}(t) = ∑_{i=0}^{∞} α_{j}(z_{i})t^{i} ∈ k[[t]].

We now compute the logarithmic derivative with respect to t of

N(1 − tx) = m̌(t; x) = ∑_{i=0}^{n}(−1)^{i}m_{i}(x)t^{i}

(by 2.7.1 and 2.11.1, with n = max deg J) in two ways. On the one hand,

\frac{d}{dt} log N(1 − tx) = N(1 − tx)^{−1} \frac{d}{dt}m̌(t; x)
= N(1 − tx)^{−1} · ∑_{i=1}^{n}(−1)^{i}i m_{i}(x)t^{i−1}.  (2)

On the other hand, by (1), the chain rule and 3.10.2,

\frac{d}{dt} log N(1 − tx) = F_{1}((1 − tx)^{−1}, −x)
= − ∑_{i=0}^{∞} F_{1}(x^{i}, x)t^{i} = − ∑_{i=0}^{∞} m_{1}(x^{i+1})t^{i}.  (3)

Combining (2) and (3), we have

∑_{i=1}^{n}(−1)^{i}i m_{i}(x)t^{i−1} = ( ∑_{i=0}^{n}(−1)^{i}m_{i}(x)t^{i})( − ∑_{i=0}^{∞} m_{1}(x^{i+1})t^{i}).

By comparing coefficients at powers of t, one sees recursively that i! m_{i}(x) is a polynomial with coefficients in ℤ in m_{1}(x), …, m_{1}(x^{i}). The first two terms are

2m_{2}(x) = m_{1}(x)^{2} − m_{1}(x^{2}),  (4)
6m_{3}(x) = m_{1}(x)^{3} − 3m_{1}(x)m_{1}(x^{2}) + 2m_{1}(x^{3}).  (5)

Explicit formulas for all i are known and involve the cycle indicator polynomials, see, e.g., [21, §4].

Now assume k is a ℚ-algebra. Then we can integrate and exponentiate formula (3) and obtain the exponential trace formula

N(1 − tx) = exp ( − ∑_{i=1}^{∞} m_{1}(x^{i})\frac{t^{i}}{i} ).  (6)

Remark. In [1, Th. 1.10], formula (3) is proved in a different way for the special case J = End(M) (cf. 2.4(d)), and is referred to as the exponential trace formula, although (6) seems to be more deserving of this name.

4. Generically power-associative algebras

4.1. Definition. Let J be a Jordan algebra over k. For every R ∈ k-alg we define

J_{pa}(R) := {a ∈ J_{R} : a strictly power-associative},

cf. 1.6. From the very definition of strict power-associativity it is evident that J_{pa} is a subfunctor of J. We claim that it is in fact a hard subsheaf of J. After a base change, this just means that it has the following descent property: If a ∈ J and a_{R} is strictly power-associative for some faithfully flat R ∈ k-alg, then a is strictly power-associative as well. Thus we must show that pr_{1}: M(a_{S}) → J_{S} is injective, for all S ∈ k-alg. Now the S-algebra R_{S} = R ⊗ S is in particular flat over S, so the canonical map M(a_{S}) ⊗_{S} R_{S} → M(a_{S} ⊗_{S} 1_{R_{S}}) is an isomorphism by Lemma 1.3(d). We can consider R_{S} ≅ S ⊗ R = S_{R} as an R-algebra, and then a_{S} ⊗_{S} 1_{R_{S}} = a ⊗_{k} 1_{R⊗S} = a_{R} ⊗_{R} 1_{S_{R}} are canonically identified. Also, since a_{R} is strictly power-associative, pr_{1}: M(a_{R} ⊗_{R} 1_{S_{R}}) → J_{R} ⊗_{R} S_{R} is injective. From the commutative diagram

M(a_{S}) ⊗_{S} R_{S} \overline{}^{pr_{1}⊗Id_{R_{S}}} J_{S} ⊗_{S} R_{S}
[figure: ≅; ≅]
M(a_{R} ⊗_{R} 1_{S_{R}})  \overline{}_{pr_{1}} J_{R} ⊗_{R} S_{R}

we see that M(a_{S}) ⊗_{S} R_{S} → J_{S} ⊗ R_{S} is injective. Since R_{S} is faithfully flat over S, it follows that M(a_{S}) → J_{S} is injective, as desired.

4.2. Lemma. Let J be a Jordan algebra over k and let a ∈ J be algebraic of degree d. Consider the following conditions:

(i) a is strictly power-associative, (ii) a^{0}, …, a^{d−1} are k-free and span a direct summand of J, (iii) a^{0} ∧ ⋯ ∧ a^{d−1} is unimodular in ⋀^{d} J.

Then (iii) ⟺ (ii) ⟹ (i). If J is finitely generated and projective as a k-module or if k is a field then all conditions are equivalent.

Proof. (iii) ⟺ (ii) follows from Lemma 0.6.

(ii) ⟹ (i): Since a is algebraic of degree d, M(a) is free with basis a^{[0]}, …, a^{[d−1]} by 1.4.6. Now pr_{1}(a^{[i]}) = a^{i} shows that pr_{1}: M(a) → k[a] is a k-module isomorphism. Hence a is power-associative and k[a] is pure, being a direct summand. By 1.6.2, a is strictly power-associative.

Now let J be finitely generated and projective, and suppose (i) holds. Then M(a) is pure by Prop. 1.5(a), so k[a] is pure by 1.6.2 and therefore a direct summand by 0.15(d). As pr_{1}: M(a) → k[a] is an isomorphism and a^{[0]}, …, a^{[d−1]} is a basis of M(a), we have (ii). Finally, if k is a field, k[a] is automatically a direct summand, so the same argument applies.

4.3. Lemma. Let J be a Jordan algebra over a field K and let a ∈ J be an algebraic element of degree d ⩽ 3. Then a^{0}, …, a^{d−1} are linearly independent.

Proof. There are the following cases:

d = 0: Then J = {0} and the empty set is linearly independent.

d = 1: Then J ≠ {0} and a^{0} = 1_{J} ≠ 0 is linearly independent.

d = 2: Assume that 1 and a are linearly dependent. Then a = λ1 for some λ ∈ K, hence, a^{2} = λ^{2}1 = λa. This shows that a satisfies the polynomial t − λ of degree 1, contradicting the fact that μ_{a}(t) has degree 2.

d = 3: Assume again, by way of contradiction, that 1, a, a^{2} are linearly dependent. We must have 1 and a linearly independent, else a = λ1 and μ_{a}(t) = t − λ as in the previous case. Thus a^{2} ∈ K · 1 ⊕ K · a, and since μ_{a}(t) has degree 3, a^{3} is a linear combination of 1, a, a^{2}, hence of 1 and a. It follows that K[a] = K · 1 ⊕ K · a, so there exist unique α, β, γ, δ ∈ K such that

a^{2} = α1 + βa,  a^{3} = αa + βa^{2} + γ1 + δa.  (1)

It suffices to show that γ = δ = 0, because then (1) says that a satisfies the degree 2 polynomial t^{2} − βt − α1, contradicting the fact that deg a = 3. If K has characteristic ≠ 2 then the first equation (1) implies a∘a^{2} = 2a^{3} = 2αa+2βa^{2}, and we are done. If K has characteristic 2, we compute powers of a as follows, always using 2 = 0 in K:

a^{4} = (a^{2})^{2} = (α1 + βa)^{2} = α^{2}1 + β^{2}a^{2}  (2)
= U_{a}a^{2} = U_{a}(α1 + βa) = αa^{2} + βa^{3}
= αa^{2} + αβa + β^{2}a^{2} + βγ1 + βδa
= α^{2}1 + 2αβa + β^{2}a^{2} + βγ1 + βδa.  (3)

From (2) and (3) we see by comparing coefficients at 1 and a and using (1), that βγ = βδ = 0. Thus we are done if β ≠ 0. Now assume β = 0 and compute fifth and sixth powers:

a^{5} = U_{a^{2}}a = U_{α1}a = α^{2}a
= U_{a}a^{3} = U_{a}(γ1 + (α + δ)a)
= γa^{2} + (α + δ)a^{3} = αγ1 + (α + δ)(γ1 + (α + δ)a).

Comparing coefficients at a shows α^{2} = (α + δ)^{2} = α^{2} + δ^{2}, whence δ = 0. Next,

\begin{aligned}a^{6} & = (a^{2})^{3} = α^{3}1 \\ & = (a^{3})^{2} = (γ1 + αa)^{2} = γ^{2}1 + α^{2}a^{2} = (γ^{2} + α^{3})1, \end{aligned}

which implies γ = 0 and completes the proof.

4.4. Corollary. Let J be a (possibly infinite-dimensional) Jordan algebra over a field K which is generically algebraic of degree ⩽3 in the sense of [16]. Then J is strictly power-associative in the sense of [23]; i.e., every element of every base field extension of J is power-associative.

This is obvious from Lemma 4.2 and Lemma 4.3. Note, however, that it is not clear whether all elements of J_{R}, R a commutative K-algebra, are power-associative as well.

4.5. Proposition. Let J be finitely generated and projective as a k-module.

(a) J_{pa} ∩ J_{alg} is an open finitely presented subscheme of J_{alg}.

(b) Algebraic elements of degree ⩽3 are strictly power-associative, that is, ∐_{d⩽3} J_{alg,d} ⊂ J_{pa}.

Proof. (a) By Prop. 1.14(b), it suffices to show that J_{pa} ∩ J_{alg,d} is open and finitely presented, for all d. Define a morphism from J_{alg,d} to (⋀^{d} J)_{a} by x ↦ x^{0} ∧ ⋯ ∧ x^{d−1} in all base extensions. Then Lemma 4.2(iii) says that J_{pa} ∩ J_{alg,d} is the inverse image of (⋀^{d} J)_{u}, whence the assertion.

(b) Let a ∈ J be algebraic of constant degree d ⩽ 3 and let 𝔭 ∈ Spec(k) be arbitrary. Then a(𝔭) ∈ J(𝔭) is algebraic of degree d as well, so Lemma 4.3 shows that a(𝔭)^{0}, …, a(𝔭)^{d−1} are linearly independent over κ(𝔭). Now Lemma 0.6 and Lemma 4.2 imply that a is strictly power-associative. Since the same argument works in all base ring extensions, it follows that J_{alg,d} ⊂ J_{pa} for 0 ⩽ d ⩽ 3. Now ∐_{d⩽3} J_{alg,d} ⊂ J_{pa} follows because J_{pa}, being a hard sheaf (cf. 4.1), is in particular a local functor.

4.6. Definition. Let J be a generically algebraic Jordan algebra over k. By Lemma 2.6, J_{prim} is an open dense subscheme of J, contained in J_{alg}, and by Prop. 4.5(a), J_{pa} ∩ J_{alg} is open in J_{alg}. Hence J_{pa} ∩ J_{prim} is open in J. We will say J is generically power-associative if J_{pa} ∩ J_{prim} is dense in J. By 0.14 and Lemma 4.2, an equivalent condition is: For all 𝔭 ∈ Spec(k), and letting K denote an algebraic closure of κ(𝔭), there exists an element x ∈ J_{K} such that the powers 1, x, …, x^{(deg J(𝔭))−1} are linearly independent over K.

4.7. Corollary. Let J be generically algebraic over k. Any one of the following conditions is sufficient for J to be generically power-associative:

(i) 2 ∈ k^{×}, (ii) J = B^{+} where B is associative or alternative, (iii) deg J ⩽ 3 (as a function on Spec(k)).

Indeed, (i) and (ii) follow from (c) and (d) of 1.6, and (iii) is clear from Prop. 4.5(b).

4.8. Example. A Jordan algebra of degree 4 need not be generically powerassociative, the simplest example being J = k · 1 ⊕ k · a ⊕ k · a^{3} where k is a ring with 2k = 0. But there are such examples even over ℤ. Let J be the algebra, free of rank 4 over ℤ, with basis 1, a, b, a^{3} introduced in 1.6. It is easily seen that J is generically algebraic of degree 4, with generic minimum polynomial m(t; x) = (t − φ(x))^{4} where φ is the linear form determined by φ(1) = 1 and φ(a) = φ(b) = φ(a^{3}) = 0. Then J is not generically power-associative, because for K a field of characteristic 2, the powers 1, x, x^{2}, x^{3} of any x ∈ J_{K} are linearly dependent. Indeed, let us put c = a^{3}. The products in J are determined by the following relations:

U_{a}1 = a^{2} = 2b, U_{a}a = a^{3} = a ∘ b = c, U_{a}b = U_{a}c = 0,
U_{b} = U_{c} = U_{a,c} = U_{b,c} = 0,
U_{a,b}1 = a ∘ b = c, U_{a,b}a = U_{a,b}b = U_{a,b}c = 0.

Now let R be an arbitrary commutative ring and denote the basis of J_{R} obtained by base change from ℤ to R again by 1, a, b, c. Let x = λ1 + αa + βb + γc ∈ J_{R}. Then the powers of x are

\begin{aligned}x^{2} & = λ^{2}1 + 2λαa + 2(λβ + α^{2})b + (2λγ + αβ)c, \\ x^{3} & = λ^{3}1 + 3λ^{2}αa + (3λ^{2}β + 6λα^{2})b + (3λ^{2}γ + 3λαβ + α^{3})c. \end{aligned}

A straightforward computation shows that the determinant of the coefficients of 1, x, x^{2}, x^{3} with respect to the basis 1, a, b, c is

det ( \begin{matrix}1 & λ & λ^{2} & λ^{3} \\ 0 & α & 2λα & 3λ^{2}α \\ 0 & β & 2λβ + 2α^{2} & 3λ^{2}β + 6λα^{2} \\ 0 & γ & 2λγ + αβ & 3λ^{2}γ + 3λαβ + α^{3} \end{matrix} ) = 2α^{6}.

Hence J_{R} contains a power-associative element of degree 4 if and only if 2 ∈ R^{×}.

5. Algebras of degree 3

5.1. Theorem. Let J and J′ be generically algebraic Jordan algebras of degree 3 over a ring k and let m_{1} and m′_{1} be the generic traces of J and J′, respectively. Also let f: J → J′ be an isomorphism of k-modules. Then the following conditions are equivalent:

(i) f is an isomorphism of Jordan algebras, (ii) f preserves unit elements, squares and traces; i.e., f(1_{J}) = 1_{J′}, f(x^{2}) =

f(x)^{2}, and m′_{1}(f(x)) = m_{1}(x), for all x ∈ J.

Remark. This theorem is of course only of interest in case 2 is not a unit in k, because otherwise the quadratic operators can be recovered from the squaring operation by the formula 2U_{x}y = x ∘ (x ∘ y) − x^{2} ∘ y.

Proof. (i) ⟹ (ii): The first two conditions are obvious, and the third follows from Prop. 2.7(a).

(ii) ⟹ (i): First note that the conditions on f are preserved under arbitrary base change. We will show that f preserves third powers in all base ring extensions. Then the assertion will follow by differentiation, see also [25, Th. 1].

Let U := J_{prim} = J_{alg,3} ⊂ J and U′ := J′_{prim} = J′_{alg,3} ⊂ J′. By Lemma 2.6 these are open dense subschemes of J and J′, respectively. Since f induces an isomorphism f: J → J′ of schemes, f^{−1}(U′) is dense in J, hence U ∩ f^{−1}(U′) is so as well. Thus it suffices to prove f(x^{3}) = f(x)^{3} for all x ∈ U(R) ∩ f^{−1}(U′(R)) and all R ∈ k-alg. Everything is invariant under base change, so we may replace R by k and assume x ∈ J_{prim} ∩ f^{−1}(J′_{prim}).

Let E = k[x] and E′ = k[y] where y := f(x). By Prop. 4.5(b), x and y are strictly power-associative, so by Lemma 4.2, E and E′ are free of rank 3 with bases 1 = 1_{J}, x, x^{2} and 1′ = 1_{J′}, y, y^{2}, respectively. Let T = m_{1}, S = m_{2} and N = m_{3} and denote the corresponding quantities for J′ by T′, S′ and N′. Since x satisfies its generic minimum polynomial m(t; x) = t^{3} − T (x)t^{2} + S(x)t − N(x), we have

x^{3} = T (x)x^{2} − S(x)x + N(x)1,  (1)
x^{4} = T (x)x^{3} − S(x)x^{2} + N(x)x.  (2)

Substitution of (1) in (2) results in

x^{4} = [T (x)^{2} − S(x)]x^{2} + [N(x) − S(x)T (x)]x + T (x)N(x)1.  (3)

Since f preserves squares and hence also fourth powers, f applied to (3) yields

y^{4} = [T (x)^{2} − S(x)]y^{2} + [N(x) − S(x)T (x)]y + T (x)N(x)1′.  (4)

On the other hand, the analogue of (3) holds in J′ for y, so we have

y^{4} = [T′(y)^{2} − S′(y)]y^{2} + [N′(y) − S′(y)T′(y)]y + T′(y)N′(y)1′.  (5)

Since f preserves traces, we have T (x) = T′(y). Hence it follows from (4) and (5) and the linear independence of 1′, y, y^{2} that S(x) = S′(y) and N(x) = N′(y). Now f applied to (1) shows

f(x^{3}) = T (x)y^{2} − S(x)y + N(x)1′ = T′(y)y^{2} − S′(y)y + N′(y)1′ = y^{3} = f(x)^{3}.

By our initial reduction, f(x^{3}) = f(x)^{3} holds now for all x ∈ J(R) and all R ∈ k-alg. In particular, let R = k(ε) (dual numbers) and let x, y ∈ J. Then (x + εy)^{3} = x^{3} + ε(x^{2} ∘ y + U_{x}y), and since f preserves squares and circle products, we also have f(U_{x}y) = U_{f(x)}f(y). This completes the proof.

5.2. The characteristic 2 case. Let k be a commutative ring with 2k = 0, and let J be a unital Jordan algebra over k. Then the k-module J becomes a 2-Lie algebra, denoted 𝔏(J), with Lie bracket [x, y] = x ∘ y and second power operation x^{[2]} = x^{2} [13, Ch. I, Th. 4].

From 1 ∘ x = 2x = 0 for all x ∈ J it follows that k · 1_{J} is contained in the centre Z(𝔏(J)) of 𝔏(J). For a Jordan matrix algebra as in 2.4(b), it is easily seen that in fact

Z(𝔏(H_{n}(𝒞, k))) = k · 1_{J}.  (1)

Now let J be generically algebraic of constant degree d and let T = m_{1} be the generic trace. Then by 3.12.4, the trace commutes with the squaring operation:

T (x^{2}) = T (x)^{2}, for all x ∈ J.  (2)

Linearization of (2) yields T (x∘y) = 2T (x)T (y) = 0, which together with (2) shows that

J_{0} = Ker T is a 2-ideal of 𝔏(J).  (3)

Assume in particular that d is odd. Then T (1_{J}) = d · 1_{k} (by 2.11.2) = 1_{k} , which implies

𝔏(J) = k · 1_{J} ⊕ J_{0}  (4)

is a direct sum of 2-Lie algebras.

Let 𝔤 and 𝔤′ be 2-Lie algebras over k. We denote the set of Lie algebra isomorphisms from 𝔤 to 𝔤′ by Isom_{Lie}(𝔤, 𝔤′), and the set of 2-Lie algebra isomorphism by Isom_{2-Lie}(𝔤, 𝔤′), and employ similar notations for automorphism groups. Clearly, Isom_{2-Lie}(𝔤, 𝔤′) ⊂ Isom_{Lie}(𝔤, 𝔤′). Furthermore,

Z(𝔤′) = 0 ⟹ Isom_{2-Lie}(𝔤, 𝔤′) = Isom_{Lie}(𝔤, 𝔤′).  (5)

Indeed, this follows easily from the formula ad(x^{[2]}) = (ad x)^{2}, valid in any 2-Lie algebra, and the fact that the adjoint representation of 𝔤′ is faithful.

5.3. Corollary. We keep the assumptions of Theorem 5.1 and assume furthermore that 2k = 0. Let 𝔏(J) = k · 1_{J} ⊕ J_{0} and 𝔏(J′) = k · 1_{J′} ⊕ J′_{0} be the decompositions as in 5.2.4.

(a) The restriction map res: f ↦ f_{0} = f|J_{0} is a bijection res: Isom(J, J′) → Isom_{2-Lie}(J_{0}, J′_{0}), and

res: Aut(J) → Aut_{2-Lie}(J_{0})  (1)

is an isomorphism of groups.

(b) Assume that in addition Z(𝔏(J)) = k · 1_{J} and Z(𝔏(J′)) = k · 1_{J′}; equivalently, that J_{0} and J′_{0} have trivial centres. Then the assertions of (a) hold for Isom_{Lie} and Aut_{Lie} instead of Isom_{2-Lie} and Aut_{2-Lie} as well.

Proof. (a) It is clear that an isomorphism f of Jordan algebras induces an isomorphism f_{0} = f|J_{0} of 2-Lie algebras. Conversely, let f_{0}: J_{0} → J′_{0} be an isomorphism of 2-Lie algebras and extend f_{0} to an k-module isomorphism f: J → J′ by f(1_{J}) = 1_{J′} and f|J_{0} = f_{0}. Since (λ1_{J} + x_{0})^{2} = λ^{2}1_{J} + 2λx_{0} + x_{0}^{2} = λ^{2}1_{J} + x_{0}^{2} for all x_{0} ∈ J_{0}, we see that f satisfies the conditions (ii) of Th. 5.1, and therefore is an isomorphism of Jordan algebras.

(b) From 5.2.4 and the assumption on the centres of 𝔏(J) and 𝔏(J′) it follows that J_{0} and J′_{0} have trivial centres as Lie algebras. Now the assertion is clear from 5.2.5.

Using the fact that Δ is a derivation if and only if Id + εΔ is an automorphism (where k(ε) is the ring of dual numbers), it is easy to formulate an infinitesimal version of Cor. 5.3 which we leave to the reader. In particular, 5.2.1 and Cor. 5.3(b) imply the following result:

5.4. Corollary. Let J = H_{3}(𝒞, k) be the Jordan algebra of 3 × 3 hermitian matrices over a composition algebra 𝒞 over k with scalar diagonal entries as in 2.4(b), and assume that 2k = 0. Then Aut_{2-Lie}(J_{0}) = Aut_{Lie}(J_{0}) and Der_{2-Lie}(J_{0}) = Der_{Lie}(J_{0}). The restriction maps Aut(J) → Aut_{Lie}(J_{0}) and Der(J) → Der_{Lie}(J_{0}) are isomorphisms of groups and of 2-Lie algebras, respectively.

References

[1] G. Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring, Ark. Mat. 11 (1973), 263–301.

[2] R. Bix, Separable Jordan algebras over commutative rings. III, J. Algebra 86 (1984), no. 1, 35–59.

[3] N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972, Translated from the French.

[4] , Elements of mathematics. Algebra, Part I: Chapters 1-3, Hermann, Paris, 1974, Translated from the French.

[5] H. Braun and M. Koecher, Jordan-Algebren, Grundlehren der mathematischen Wissenschaften, vol. 128, Springer-Verlag, Berlin, 1966.

[6] M. Demazure and P. Gabriel, Groupes algébriques, vol. 1, Masson, Paris, 1970.

[7] M. Demazure and A. Grothendieck, Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin, 1962/1964.

[8] D. Ferrand, Un foncteur norme, Bull. Soc. Math. France 126 (1998), no. 1, 1–49.

[9] , Monogenous algebras. Back to Kronecker, Preprint, 2004.

[10] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. (1965), no. 24, 231.

[11] , Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. (1966), no. 28, 255.

[12] N. Jacobson, Generic norm of an algebra, Osaka Math. J. 15 (1963), 25–50.

[13] , Lectures on quadratic Jordan algebras, Tata Institute of Fundamental Research, Bombay, 1969, Lectures on Mathematics, No. 45.

[14] , Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., 1969.

[15] , Structure theory of Jordan algebras, University of Arkansas Lecture Notes in Mathematics, vol. 5, University of Arkansas, Fayetteville, Ark., 1981.

[16] N. Jacobson and J. Katz, Generically algebraic quadratic Jordan algebras, Scripta Math. 29 (1973), 215–227.

[17] M. A. Knus, Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, vol. 294, Springer-Verlag, Berlin, 1991.

[18] T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999.

[19] O. Loos, Jordan pairs, Lecture Notes in Math., vol. 460, Springer-Verlag, 1975.

[20] , Tensor products and discriminants of unital quadratic forms over commutative rings, Mh. Math. 122 (1996), 45–98.

[21] , Trace and norm for reduced elements of Jordan pairs, Comm. Algebra 25 (1997), no. 9, 3011–3042.

[22] K. McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495–510.

[23] , The Freudenthal-Springer-Tits constructions revisited, Trans. Amer. Math. Soc. 148 (1970), 293–314.

[24] , Homotopes of alternative algebras, Math. Ann. 191 (1971), 253–262.

[25] , Quadratic Jordan algebras and cubing operations, Trans. Amer. Math. Soc. 153 (1971), 265–278.

[26] , The generic norm of an isotope of a Jordan algebra, Scripta Math. 29 (1973), no. 3-4, 229–236.

[27] , A characterization of the nondegenerate radical in quadratic Jordan triple systems, Algebras Groups Geom. 4 (1987), no. 2, 145–164.

[28] H. P. Petersson, Composition algebras over algebraic curves of genus zero, Trans. Amer. Math. Soc. 337 (1993), 473–491.

[29] H. P. Petersson and M. L. Racine, Radicals of Jordan algebras of degree 3, Radical theory, Proc. 1st Conf., Eger/Hung. 1982, Colloq. Math. Soc. János Bolyai 38, 1985, pp. 349–377.

[30] , Jordan algebras of degree 3 and the Tits process, J. of Algebra 98 (1986), 211–243.

[31] N. Roby, Lois polynomes et lois formelles en théorie des modules, Ann. Sci. École Norm. Sup. (3) 80 (1963), 213–348.

[32] W. C. Waterhouse, Symmetric determinants and Jordan norm similarities in characteristic 2, Proc. Amer. Math. Soc. 93 (1985), no. 4, 583–589.