Discriminant-Preserving Equivalence

• The paper establishes that all discriminant algebra functors in a fixed rank are canonically isomorphic, ensuring consistent discriminant preservation.
• Methodologies based on norm polynomial laws, Clifford algebras, and Pfaffian formulas are compared to reveal structured, functorial relationships.
• For ranks up to three, the unique isomorphism is proven, while the case for rank four and higher remains open, highlighting avenues for further research.

Discriminant-preserving equivalence of categories concerns the systematic relationship between constructions that assign to each rank-$n$ algebra a quadratic algebra with canonically matching discriminant data. In this framework, all established “discriminant algebra” functors for a fixed rank $n$ are shown to be canonically isomorphic in a suitable category $\mathcal{C}_n$; moreover, for ranks $n\leq 3$, this functor is unique up to unique isomorphism. The formalism captures and compares various discriminant algebra constructions appearing in the literature, establishing categorical equivalence and, in small ranks, uniqueness up to contractibility.

1. Categories and Functorial Structure

Let $\mathsf{Aff}$ denote the opposite of the category of commutative rings, and $\mathsf{Disc}$ the category whose objects are triples $(R,L,d)$ with $R$ a ring, $L$ a locally free rank-$1$ $R$-module, and $d: L^{\otimes 2}\to R$ an $R$-linear discriminant pairing, with morphisms given by base change preserving $d$. $\mathsf{Alg}_n$ is the category of rank-$n$ algebras: objects are pairs $(R,A)$ with $A$ an $R$-algebra locally free of rank $n$, and morphisms are base change plus $R$-algebra isomorphism.

There is a tautological functor $\,^n: \mathsf{Alg}_n \rightarrow \mathsf{Disc}$ which to $(R,A)$ associates its top exterior power ${}^nA$ equipped with the discriminant bilinear form

$${}^nA \otimes {}^nA \xrightarrow{\det(\operatorname{Tr}(a_ib_j))} R.$$

A discriminant algebra in rank $n$ is a functor $D:\mathsf{Alg}_n\to\mathsf{Alg}_2$ equipped with an isomorphism ${}^2 D(R,A) \cong {}^n A$ over $R$, supplying a quadratic algebra functorial in $(R,A)$ along with a discriminant-identifying isomorphism. The category $\mathcal{C}_n$ has objects the such functors, with morphisms given by natural isomorphisms that respect the discriminant identifications.

A summary of the structural framework appears below:

Category	Objects	Morphisms
$\mathsf{Alg}_n$	Rank-$n$ locally free $R$-algebras $(R,A)$	Base change + algebra isomorphism
$\mathsf{Disc}$	Triples $(R,L,d)$ as above	Base change preserving $d$
$\mathcal{C}_n$	Discriminant-algebra functors $\mathsf{Alg}_n\to\mathsf{Alg}_2$ + data	Natural isomorphisms respecting discriminants

2. Constructions of Discriminant Algebras

Three main constructions of discriminant algebras for rank-$n$ algebras are systematically compared:

(a) Biesel–Gioia (“Ferrand”) Construction

Given $(R,A)\in\mathsf{Alg}_n$, the norm polynomial law $\operatorname{Nm}_{A/R}:A\to R$ is represented by an $R$-algebra homomorphism $\Phi_{A/R}: (A^{\otimes n})^{S_n} \to R$. Considering the inclusion $(A^{\otimes n})^{S_n}\subset (A^{\otimes n})^{A_n}$ of invariants for the alternating subgroup, set

$$\Delta(R,A):= (A^{\otimes n})^{A_n} \otimes_{(A^{\otimes n})^{S_n}} R,$$

where the map $(A^{\otimes n})^{S_n} \to R$ is given by $\Phi_{A/R}$. $\Delta(R,A)$ is locally free of rank 2 and its induced discriminant form agrees with that of $A$. Assigning $D_1(R,A):=\Delta(R,A)$ defines an object in $\mathcal{C}_n$.

(b) Rost’s Construction (Rank 3)

For $n=3$, let $A$ be a rank-3 $R$-algebra. Set $\dot{A}=A/R$ and define the quadratic form $q_A(\dot a)=s_1(a^2)-s_2(a)$, with $s_1$, $s_2$ the characteristic polynomial coefficients. The even Clifford algebra $K(A)=C_0(q_A)$ fits in a short exact sequence

$0\to R\to K(A)\to {}^2\dot{A}\cong{}^3A\to 0,$

making $K(A)$ a quadratic $R$-algebra. Adjusting the multiplication in $K(A)$ by the discriminant bilinear form on ${}^3A$ yields a quadratic algebra $D_3(A) = K(A) - {}^3A$ whose discriminant matches that of $A$. Rost’s functor $D_3$ defines an object in $\mathcal{C}_3$.

(c) Loos’s Construction (Even Rank)

For even $n$, let $A$ be a rank-$n$ algebra, and write $s_2:A\to R$ for the quadratic trace. For any representable quadratic form $Q$ on a rank-$n$ bundle $M$, construct an algebra

$Q^* = R\oplus {}^nM,$

with multiplication determined via Pfaffian and “quarter-determinant” formulas. Different choices of representing bilinear form produce canonically isomorphic algebras, so $Q^*$ is well-defined. For $A$ as above with $Q = s_2$, shift the algebra by $(-1)^{\frac{n}{2}+1} \frac{n}{4} \cdot {}^n A$ to align the discriminant form. This yields the functor $D_2:\mathsf{Alg}_n\to\mathsf{Alg}_2$ belonging to $\mathcal{C}_n$ for even $n$.

3. Canonical Equivalences and Natural Isomorphisms

Within $\mathcal{C}_n$, the core phenomenon is the canonical isomorphism between any two discriminant-algebra constructions. For any two objects $D, D'\in \mathcal{C}_n$, there is a bijection of natural transformations $\operatorname{Hom}_{\mathcal{C}_n}(D,D')\cong$ (identifications of discriminant forms). Explicit comparison maps are constructed:

• $\varphi_{13}: D_1 \to D_3$ (rank 3): The generator $\dot a\otimes \dot b\in K(A)$ is sent to $s_1(ab)-\gamma(1,b,a)\in \Delta(A)$, and this map preserves traces and norms after the algebra shift.
• $\varphi_{12}: D_1\to D_2$ (even rank): The “wedge-generator” $s_f(a_1\wedge\cdots\wedge a_n)$ corresponds to $\sigma_f(a_1,\ldots,a_n)\in\Delta(A)$ built from a universal Pfaffian.
• In mixed or odd ranks, comparison isomorphisms are obtained by composing these via inverses as appropriate.

All three constructions ($D_1$, $D_2$, $D_3$) are thus canonically isomorphic in $\mathcal{C}_n$. This establishes discriminant-preserving equivalence between all known reasonable constructions.

4. Uniqueness in Small Ranks

For $n\le3$, uniqueness up to unique isomorphism is established in $\mathcal{C}_n$, making it a contractible groupoid. The key lemma: for ring $R$ with $2$ invertible or a prime nonzero-divisor, and quadratic $R$-algebras $A,B$ with matching discriminant forms, there is a unique $R$-algebra isomorphism $A\cong B$ inducing that identification.

Trivial (rank $0, 1$) and quadratic ($n=2$) algebras are handled by demonstrating uniqueness via universal quadratic algebras. For rank $3$, universal associativity relations provide the necessary identifications, with suitable use of base change and Čech-like descent arguments. Thus, for $n\leq 3$, the discriminant functor is unique up to unique isomorphism across all constructions.

5. Existence, Scope, and Open Problems

Biesel–Gioia’s construction confirms that $\mathcal{C}_n$ is always nonempty for $n\geq 2$. For all $n$, canonical isomorphisms between known constructions guarantee discriminant-preserving equivalence in $\mathcal{C}_n$. For ranks up to $3$, categorical uniqueness holds, but for $n\geq 4$, uniqueness remains open.

The possibility of extending uniqueness to $n>3$ appears to require a global “universal” parameter space for rank-$n$ algebras, such as those provided by Bhargava or Wood for quartic and quintic algebras. In the absence of such parametrizations, the classification and uniqueness of discriminant algebra functors for higher ranks remains unresolved. This suggests scope for further research in both the construction of discriminant algebras for higher rank and in the development of suitable moduli spaces.

6. Mathematical Significance and Connections

Discriminant-preserving equivalence of categories systematically relates and generalizes classical invariants associated with commutative ring extensions and their quadratic forms. By formalizing discriminant algebra constructions in a categorical context, it provides a rigorous method for comparing functorial constructions and establishes conditions for their uniqueness. The equivalences established ensure that within suitable settings, the discriminant encoding is functorially canonical.

A plausible implication is that advances in parametrizing higher algebraic structures may permit a complete understanding of discriminant algebra uniqueness in all finite ranks. The framework also interfaces naturally with the paper of polynomial invariants, Clifford algebras, and the behavior of quadratic forms under various base changes.