WEDDERBURN’S THEOREM FOR REGULAR LOCAL RINGS

Manuel Ojanguren

In [Pa] Ivan Panin proved the following theorem.

Theorem 1. Let R be a regular local ring, K its field of fractions and (V, Φ) a quadratic space over R. Suppose R contains a field of characteristic zero. If (V, Φ)⊗_{R}K is isotropic over K, then (V, Φ) is isotropic over R.

The proof rests on a series of lemmas which can be summarized in the following one.

Lemma 2. Let k be a field of characteristic zero, u a closed point of a smooth k-variety and R = O_{U,u} the local ring of U at u. Let further X be a projective R-scheme, smooth over R. Let K be the field of fractions of R and suppose that X has a K-point. Then, for every prime number p there exist an integral R-etale algebra S of degree prime to p and an S-point of X.

Proof. See [Pa], Lemma 3, Lemma 4 and proof of Theorem 1..

I want to show that the argument used for proving Theorem 1 also yields the following extension of Wedderburn’s theorem to a large class of regular local rings.

Theorem 3. Let R be a regular local ring, K its field of fractions and A an Azumaya algebra over R. Suppose R contains a field k of characteristic zero. If A ⊗_{R} K is isomorphic to M_{n}(D) where D is a central division algebra over K, then A is isomorphic to M_{n}(Δ) where Δ is a maximal (unramified) R-order of D. In other words, every class of the Brauer group of R is represented by a maximal order in a division K-algebra.

Proof. Let d^{2} be the dimension of D over K. It suffices to show that A contains a right ideal I such that A/I is free of rank (n^{2} − n)d^{2} over R. In fact, since any A-module is projective over A if and only if it is projective over R, the projection A → A/I splits, I is a direct factor of the right A-module A, and Δ := End_{A}(I) is an Azumaya algebra equivalent to A. Clearly Δ ⊗_{R} K = D and by Morita theory

A = End_{Δ}(Hom_{A}(I, A)) = M_{n}(Δ).

In order to find a right ideal I of the right rank we consider the set I of all such ideals or, more precisely, we consider the functor I that associates to any R-algebra S the set of such ideals in A ⊗_{R} S.

Typeset by AMS-TEX

Lemma 4. I is a smooth closed subscheme of the Grassmannian scheme G consisting of all the free R-submodules of A which are direct factors of A and have rank nd^{2}.

Proof. We denote by m the maximal ideal of R. To show that I is closed we first remark that A, as an R-module, is generated by the set A^{∗} of all invertible elements of A. In fact for any a ∈ A and any λ ∈ k the reduced norm of λ + a is a polynomial

P (λ) = λ^{nd} + c_{1}λ^{n−1} + ⋯ + c_{nd}

whose coefficients are in R and only depend on a. Choosing λ in k^{∗} such that P (λ) is not 0 in R/m insures that λ + a is invertible and allows to write a = (λ + a) − λ. So an R- submodule M of A is an ideal if aM = M for every unit a. In other words, we must show that the set of fixed points of G under the action of A^{∗} is closed. This is well-known.

The second point is the smoothness of I. This means that for any R-algebra S and any ideal I of S, any S/I-point of X can be lifted to an S/I^{2}-point. But points correspond to right ideals generated by an idempotent and it is well-known that idempotents can be lifted.

Note that it suffices to treat the case when A is of prime power order in the Brauer group Br(R) of R. In fact the class of A is a product of classes [A_{i}] of order p_{i}^{e_{i}} for some distinct primes p_{1}, …, p_{r}. If each of them is represented by an order Δ_{i} in D_{i} = Δ_{i} ⊗_{R} K then A is Brauer equivalent to Δ_{1} ⊗_{R} ⋯ ⊗_{R} Δ_{r} which is an order in D = D_{1} ⊗_{K} ⋯ ⊗_{K} D_{r} and we know that D is a division algebra.

We now assume that R is of geometric type, in other words R is the local ring of a closed point u of a smooth k-variety. The general case then follows from this special case by a standard application of Popescu’s theorem, saying that a regular ring containing a field is an inductive limit of smooth algebras.

Suppose now that A is of prime power exponent in Br(R) and that the degree of D is p^{e} for some prime number p. Since A ⊗_{R} K = M_{n}(D) the scheme I has a K-point and according to Lemma 2 it also has an S-point, where S is an integral etale algebra whose degree d is prime to p. This means that A⊗_{R} S = M_{n}(B) for some maximal order B in D⊗_{K} L, L being the field of fractions of S. Note that D⊗_{K} L remains a division algebra because the degree of L over K is prime to p. So the Brauer class [A]_{S} of A ⊗_{R} S in Br(S) is represented by a degree p^{e} algebra. In [Ga] (see also [AdJ], Proposition 2.6.1) Gabber proved that any class α ∈ Br(R) which is represented by a degree m algebra when extended to a finite faithfully flat R-algebra S of degree d can be represented by an R-algebra of degree dn. We can thus find an Azumaya algebra A_{1} of degree dp^{e} in the same class as A. On the other hand, we dispose of Ferrand’s [Fe] norm functor N_{S/R} from S-algebras to R-algebras. Applying it to B we find that N_{S/R}(B) = A_{2} is an Azumaya R-algebra equivalent to A^{⊗d} ([Fe], section 7.3), of degree p^{ed} ([Fe], Thorme 4.3.4). If the integer c is an inverse of d modulo p^{e}, the algebra A_{3} = A_{2}^{⊗c} is Brauer equivalent to A and its degree is p^{cde}. Recall now that DeMeyer [DM] proved that every class in Br(R) is represented by a unique “minimal” Azumaya algebra Δ with the property that every algebra in the same class is isomorphic to some matrix algebra over Δ. What is the degree m of this Δ in our case? We must have A_{1} ≃ M_{s_{1}}(Δ) and A_{3} ≃ M_{s_{3}}(Δ), hence s_{1}m = dp^{e} and s_{3}m = p^{cde}. Since d is prime to p, this implies that m divides p^{e} and extending the scalars to K shows that m = p^{e}. The theorem is proved.

Easy and well-known examples (the simplest one being the usual quaternion algebra extended to ℝ[x, y, z]/(x^{2} + y^{2} + z^{2})) show that we cannot replace regularity by, say, normality.

In a very interesting, recent article, Benjamin Antieau and Ben Williams [AB] show that Theorem 3 fails for nonlocal regular rings.

References

[AB] Benjamin Antieau and Ben Williams, Azumaya maximal orders do not always exist, preprint, http://arxiv.org/abs/1209.2216 (2012).

[AdJ] Michel Artin and Aise Johan de Jong, Stable orders over surfaces, preprint.

[DM] Franck DeMeyer, Projective modules over central separable algebras, Canad. J. Math. 21 (1969), 39–43.

[Fe] Daniel Ferrand, Un foncteur norme, Bull. Soc. Math. France 126 (1998), 1–49.

[Ga] Ofer Gabber, Some theorems on Azumaya algebras, The Brauer group, Lecture Notes in Math, vol. 844, Springer, Berlin-New York, 1981, pp. 129–209.

[Pa] Ivan Panin, Rationally isotropic quadratic spaces are locally isotropic, Invent. Math. 176 (2009), 397–403.