THE DISCRIMINANT ALGEBRA OF A CUBIC ALGEBRA

MARKUS ROST

Introduction

In this note we describe a construction of the discriminant algebra D(A) of a flat cubic algebra A over any ring R (all rings and algebras are unital and commutative). The algebra D(A) is a flat quadratic algebra. Its discriminant is the discriminant of A.

The discriminant algebra of a cubic binary form has been constructed by Manjul Bhargava. His construction should be the same as the one considered here via the correspondence (binary cubic forms) ↔ (cubic algebras).

There are many further related questions and constructions not considered here. For example: If A is etale, then A ⊗_{R} A = A ⊕ A ⊗_{R} D(A). What is the relation between A ⊗_{R} A and D(A) for arbitrary A?

1. The discriminant of a quadratic algebra

Let L be a flat quadratic algebra over R.

Consider the bilinear form

b: L × L → R, b(x, y) = trace_{L/R}(xy)

We define the discriminant of L as

disc(L) = det b ∈ (⋀^{2} L)^{−⊗2}

2. Verschiebung

Let H = L/R. Since ⋀^{2} L = H, the discriminant disc(L) can also be considered as an element of H^{⊗−2}.

Let

p: L → H

be the projection. For any element e ∈ H^{⊗−2} we define the quadratic algebra L_{e} by L_{e} = L as R-module and with new product

x ∗ y = xy + e(p(x) ⊗ p(y))

We call L_{e} the Verschiebung (something like “translational displacement”, if you really insist on a translation) of L with e.

For the discriminant one finds:

disc(L_{e}) = disc(L) + 4e

Date: March 4, 2002.

3. Cubic algebras

Now let A be a flat cubic algebra over R.

Let N_{A}: A → R be the norm map and write

N_{A}(x + t) = t^{3} + T_{A}(x)t^{2} + S_{A}(x)t + N_{A}(x)

with x ∈ A and t ∈ R where T_{A} ∈ A^{∨} is the trace and S_{A} ∈ S^{2}A^{∨} is a quadratic form on A.

Let M = A/R and let

π: A → M

be the projection.

3.1. The discriminant. Consider the bilinear form

b: A × A → R, b(x, y) = T_{A}(xy)

We define the discriminant of A as

disc(A) = det b ∈ (⋀^{3} A)^{−⊗2}

Since ⋀^{3} A = ⋀^{2} M, the discriminant disc(A) can also be considered as an element of (⋀^{2} M)^{⊗−2}.

3.2. The reduced trace form. Consider the expression

T_{A}(x^{2}) − S_{A}(x)

for x ∈ A. One finds that it is invariant under x ↦ x + a with a ∈ R. Hence there exists a quadratic form

q_{A}: M → R

with

q_{A}(π(x)) = T_{A}(x^{2}) − S_{A}(x)

for x ∈ A.

Let

K(A) = C_{0}(q_{A})

be the even Clifford algebra of q_{A}. Note that

K(A)/R = C_{0}(q_{A})/R = ⋀^{2} M

For the discriminants one finds

disc(q_{A}) = disc(K(A)) = −3 disc(A) ∈ (⋀^{2} M)^{⊗−2}

We call K(A) the Kummer discriminant algebra of A. This is suggested by

Lemma. The following are equivalent (at least for a local ring R):

(1) K(A) is etale and split, i.e., K(A) ≃ R ⊕ R.

(2) 1/3 ∈ R and A ≃ R[t]/(t^{3} − a) for some a ∈ R^{∗}.

The proof is left to the reader.

3.3. The discriminant algebra. We are now ready to define the discriminant algebra D(A) as the Verschiebung

D(A) = K(A)_{disc(A)}

Then indeed

disc(D(A)) = disc(K(A)) + 4 disc(A) = disc(A)

We conclude with

Lemma. The following are equivalent (at least for a local ring R):

(1) D(A) is etale and split, i.e., K(A) ≃ R ⊕ R.

(2) A is etale and cyclic.

The proof is left to the reader.

Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA

   E-mail address: rost@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/~rost