DEFORMATIONS OF REPRESENTATIONS

C. Procesi

This is a short presentation of the general theory of equivalence of representations and their deformations.

This theory has a formal aspect which should be recalled and goes back to the work of several people in the 60’s.

So this is an expository paper which aims at reviving that theory which seems to have some interest also for the Theory of Galois representations.

As far as I understand this theory, there have been 4 stages of development.

1) The roots are in the theory of maximal orders (Deuring) which in turn was motivated by complex multiplication on Abelian varieties (the theory of Riemann matrices) [D].

2) The identification of special maximal orders with good reduction everywhere (Azumaya algebras) [Az].

3) The ideas of Grothendieck on the Brauer group and the work of Auslander Goldman [A-G],[Gr].

4) The approach to study finite dimensional representations as some kind of non commutative affine geometry (PI-rings) [P].

At the end we will discuss Mazur’s theorem [M] on universal local deformations of Galois modules and give some complements to this result. The paper does not intend to be a shortcut to this theorem but rather an attempt to sketch with fairly complete proofs the main theorems connected with the parametrizations of finite dimensional representations.

Typeset by AMS-TEX

Contents

(1) n−dimensional representations.

(2) Deformations

(3) An entirely non commutative approach

(4) Remarks on Mazur’s Theorem

In this paper by ring we mean an associative ring with 1.

§1 n−dimensional representations.

1.1 Given a ring B one denotes by M_{n}(B) the full ring of n × n matrices over B. If f : B → C is a ring homomorphism we can construct M_{n}(f) : M_{n}(B) → M_{n}(C) the homomorphism induced on matrices. We will use systematically the following simple (well known) Lemma which we leave to the reader.

Lemma. If I is a 2-sided ideal in M_{n}(B) then I = M_{n}(J) for a (unique) 2-sided ideal J in B (and M_{n}(B)/I = M_{n}(B/J)).

Let R be a ring, by an n−dimensional representation over a commutative ring B one means a homomorphism

(1.1.1)  ϕ : R → M_{n}(B)

We shall indicate by R_{R}^{n}(B) the set of all these representations. It is a set valued functor on the category of rings since, given f : B → C, we can compose

ϕ : R → M_{n}(B) ⟶^{M_{n}(f)} M_{n}(C).

Lemma. The functor R_{R}^{n}(−) is representable.

Proof. We consider a set of generators a_{α} for the ring R which is thus presented as quotient of the free (non commutative) algebra ℤ < x_{α} >:

π : ℤ < x_{α} >→ R = ℤ < x_{α} > /K, K := Kerπ.

For each α choose a set of n^{2} variables x_{i,j}^{α}. Let A := ℤ[x_{i,j}^{α}] be the polynomial ring in all these variables.

Define the generic matrices ξ_{α} in M_{n}(A) by setting ξ_{α} to be the matrix which, in the i, j entry, has value x_{i,j}^{α}.

Let j : ℤ < x_{α} >→ M_{n}(A) be defined by j(x_{α}) = ξ_{α}. Let finally I be the 2-sided ideal in M_{n}(A) generated by the image j(K).

By the previous Lemma I = M_{n}(J) for some ideal J of A and thus we have a commutative diagram

(1.1.2)  \begin{matrix}ℤ < x_{α} > & ⟶^{j} & M_{n}(A) \\ ↓^{π} & & ↓^{M_{n}(p)} \\ R & ⟶_{j_{R}} & M_{n}(A/J) \end{matrix}

where p : A → A/J is the quotient map.

If f : R → M_{n}(B) is any map f(a_{α}) = (b_{i,j}^{α}) and thus the map x_{i,j}^{α} → b_{i,j}^{α} of A → B induces the required map \overline{f} : A/J → B for which

R ⟶^{j_{R}} M_{n}(A/J)
f  ↓^{M_{n}(\overline{f})}
M_{n}(B)

commutes. □

Definition. We shall denote by A_{n}(R) the universal ring A/J and shall refer to the mapping

(1.1.3)  R ⟶^{j_{R}} M_{n}(A_{n}(R))

as the universal n−dimensional representation.

One should remark that, if instead of working with rings we work with algebras over a ring F all the discussion carries over. Moreover if R is a finitely generated algebra so is the universal ring A_{n}(R).

Notice that the construction R → A_{n}(R) is clearly functorial in R and for every morphism f : R → S we have a commutative diagram

(1.1.4)  \begin{matrix}R & ⟶^{j_{R}} & M_{n}(A_{n}(R)) \\ ↓^{f} & & ↓^{M_{n}(A_{n}(f))} \\ S & ⟶^{j_{S}} & M_{n}(A_{n}(S)) \end{matrix}

In other words Theorem. The functor B → M_{n}(B) from the category C of commutative rings to the category N of non commutative rings has a right adjoint

hom_{N} (R, M_{n}(B)) = hom_{C}(A_{n}(R), B).

1.2 When one studies representations one has a natural equivalence given by changing the basis. In the arithmetic theory one needs a little care.

The fact is that one has to define the automorphism group of matrices as group scheme. This is rather simple, suppose B is a commutative ring and we want to study the

automorphism group of M_{n}(B) as B algebra.

If B is a field it is an easy and well known fact that every automorphism is inner, i.e. it is of the form X → AXA^{−1} for an invertible matrix A. Since scalar matrices act trivially this defines the projective linear group P GL(n, B) as automorphism group.

This statement is no more true in general so we want to define it as group scheme over ℤ and the remark is the following.

Let e_{ij} denote the standard matrix units.^{1}

An automorphism ϕ maps the matrix units in a new basis ϕ(e_{ij}) = ∑_{h,k} c_{i,j}^{h,k}e_{h,k} and the homomorphism conditions are some quadratic relations plus the fact that ϕ(1) = ∑_{i} ϕ(e_{i,i}) = 1.

To this we have to add that the elements ϕ(e_{ij}) should be a basis, i.e. that a certain determinant is invertible.^{2}

All together these algebraic restrictions on the elements c_{i,j}^{h,k} define a finitely generated ℤ algebra P_{n} which represents the functor that to a commutative ring B associates the group Aut_{n}(B) of automorphisms of M_{n}(B).

Since this is a group valued functor P_{n} has a natural structure of Hopf ring (which could be explicited) and makes its spectrum the integral form of the projective group.

Notice that this presents the group scheme as a closed subgroup of the group GL(n^{2}, −) of invertible n^{2} × n^{2} matrices.

By construction the homomorphisms of P_{n} to a commutative ring B classify the automorphisms of M_{n}(B) in particular the identity of P_{n} classifies the generic automorphism of M_{n}(P_{n}).

Given an n-dimensional representation of a ring R ⟶^{f} M_{n}(B) and an automorphism ϕ of M_{n}(B) the composition ϕ ∘ f is another representation, i.e. the affine group scheme P GL(n, −) acts on the affine scheme R_{R}^{n}(−).

1e_{ij} is the matrix with 1 in the ij position 0 elsewhere.

2we shall see that this condition is automatically satisfied.

One can of course write this in terms of the coaction on rings.

We may compare the automorphism group Aut_{n}(B) with the group of inner automorphisms GL(n, B)/B^{∗}. Given an automorphism ϕ consider the idempotents u_{i} = ϕ(e_{ii}), We have a decomposition B^{n} = ⊕_{i}u_{i}B^{n} and it follows easily that u_{i}B^{n} is a rank 1 projective module over B. The elements ϕ(e_{ij}) establish isomorphisms between these rank 1 projectives. It is then clear that ϕ is inner if and only if these projectives are free.

In particular if B is a local ring every automorphism of M_{n}(B) is inner.

Equivalence classes of rank 1 projectives form a group under tensor product called the Picard group of B denoted P ic(B). One can prove that the map Aut_{n}(B) → u_{i}B^{n} is a homomorphism to the Picard group and we have seen that its kernel is the group of inner automorphisms.

We notice a corollary of the previous analysis

Corollary. The characteristic polynomial of a matrix a ∈ M_{n}(B) is invariant under Aut_{n}(B).

Proof. By the localization principle it is enough to do it when B is local, in this case every automorphism is inner and the proof is the usual one. □

1.3 The next point is to try to parametrize equivalence classes of representations.

As we well know in geometric invariant theory it is not possible in general to parametrize in a nice way orbit sets, one has to restrict to some kind of stable or semistable points. In this note we shall confine ourselves to stable points which, in our case, will be absolutely irreducible representations, the interested reader can see some treatment of the more general case in [P2],[P3],[P4].

In order to motivate our next discussion we should make a short geometric digression. Suppose we look at the set of n−dimensional representations with values in an alge-

braically closed field \overline{k}. We may, after base change, assume that R is a \overline{k} algebra and that the n−dimensional representations are parametrized by the \overline{k} commutative algebra A_{n}(R) ⊗ \overline{k}.

In representation theory one has the usual notion of irreducibility and

Theorem-Wedderburn. 1) A subalgebra of M_{n}(\overline{k}) is irreducible if and only if it is the entire algebra of matrices.

A finite dimensional algebra R over a field F with no 2 sided ideals is isomorphic to M_{n}(D) where D is a finite dimensional division ring.

These theorems for finite fields are to be combined with another theorem of Wedderburn which implies that if D is a finite division ring then it is a field.

Thus the set of irreducible representations of R\overline{k} with values in M_{n}(\overline{k}) coincides with the set of surjective homomorphisms ϕ : R\overline{k} → M_{n}(\overline{k}) → 0.

Given n^{2} elements a_{i}, i = 1, …, n^{2} of R\overline{k} they map to a basis of M_{n}(\overline{k}) if and only if the discriminant det(tr(ϕ(a_{i}a_{j})) is non 0.

Then we see that the condition for a homomorphism to be surjective is open since the space of surjective homomorphisms is covered by the open sets in which n^{2} prescribed elements map to a basis.

The basic example is when R is the free algebra in say m variables x_{i}, then a representation is just an m−tuple of n × n matrices, i.e. a point in the mn^{2} dimensional affine space M_{n}(\overline{k})^{m} over which the projective linear group P GL(n, \overline{k}) acts by simultaneous conjugation.

The irreducible representations form the open set of m−tuples which generate M_{n}(\overline{k}) as a \overline{k} algebra. If n > 1 it is non empty as soon as m ≥ 2, for instance take a diagonal matrix with distinct entries and the matrix of a cyclic permutation.

The general examples are subvarieties of this one.

When a map is irreducible it is clear that, under the action of the projective linear group, it has trivial stabilizer. It is in fact a basic property, which we are going to illustrate next, that the set of n−dimensional irreducible representations is in fact the total space of a principal bundle under the projective group.

The nature of this bundle is to be locally trivial only in the ètale topology and not in the Zariski topology, the analysis of this fact leads to the theory of Azumaya algebras.

The idea is that the projective linear group is also the automorphism group of matrices and so there is a dictionary which transforms facts on principal P GL(n) bundles into matrix bundles. In turn, over an affine scheme the global sections of a matrix bundle form a non commutative algebra of very special type called an Azumaya algebra.

It turns out that, to some extent, Azumaya algebras are very close to commutative algebras and the right objects to study deformation theory.

1.4 If R is an A−algebra we denote by R^{o} its opposite, which acts on R by right multiplications. We also set

R^{e} := R ⊗ R^{o}

R is naturally an R^{e} module. Then one sets:

Definition. An algebra R over a commutative ring A is called an Azumaya algebra^{3} if: 1) R is a finitely generated projective A module.

2) The natural map j : R^{e} → End_{A}(R) given by j(a ⊗ b)r := arb is an isomorphism.

The simplest example of Azumaya algebra is the ring End_{A}(P) where P is a finitely generated projective module. When P = A^{n} is free then of course End_{A}(P) = M_{n}(A).

The conditions of the definition can be formulated in a way which is more explicit and which we will use in 3.5.

3in [A-G2] the theory is a special case of the theory of separable algebras

1) The fact that P is finitely generated projective is expressed by exhibiting a surjective map p : A^{n} → P with a splitting j : P → A^{n}.

This is achieved by choosing n elements m_{i} ∈ P (image of the basis of A^{n}) and n linear forms ϕ_{i} ∈ P^{∗} such that x = ∑_{i=1}^{n} < ϕ_{i}|x > m_{i}, ∀x ∈ P.

Such a set of m_{i}, ϕ_{j} is called a set of projective generators.

For every prime ideal \underline{p} of A the localization P\underline{p} := P ⊗_{A} A\underline{p} is free of some rank n(\underline{p}). The rank is a locally constant function on the spectrum.

Once we know that P is finitely generated projective to verify 2) it is enough to see that j : R^{e} → End_{A}(R) is surjective, since R^{e}, End_{A}(R) are both finitely generated projective of the same rank (the square of the rank of P) and one can reduce to the local case remarking that a surjective map between two free modules (over a commutative ring) is necessarily an isomorphism.

Finally one can remark that End(R) is generated as A module by the elementary endomorphisms m_{i} ⊗ ϕ_{j} : r → m_{i} < ϕ_{j}|r > .

End_{A}(P) is a typical example of locally trivial in the Zariski topology matrix bundle. This is the only example when A is an algebraically closed field.

Theorem. If K is an algebraically closed field and R an Azumaya algebra over K then R = M_{n}(K) for some n.

Proof. One can use several more or less complicated approaches. I try to be most elementary.

A 2-sided ideal is just an R ⊗_{K} R^{o} submodule. Since by assumption this is the entire ring of endomorphisms it follows that R has no, non 0, 2 sided ideals.

Let then J ⊂ R be a minimal left ideal so that R acts on J as endomorphisms and the kernel is a 2 sided ideal hence it is 0.

Since J is irreducible by Wedderburn’s theorem R = End_{K}(J). □

Before showing the existence of non trivial Azumaya algebras we develop some basic facts. Proposition. If R is an Azumaya algebra over a ring A and B is a commutative A algebra R_{B} := R ⊗_{A} B is an Azumaya algebra over B.

Conversely if B is faithfully flat^{4}over A and R_{B} is Azumaya so is R.

Proof. If R is A projective then R_{B} is B projective and clearly we have the maps

R_{B}^{e} ⟶^{≅} (R^{e})_{B} → (End_{A}(R))_{B} ⟶^{≅} End_{B}(R_{B})

which give the required isomorphism for R_{B} given the one for R_{A}.

Conversely if B is faithfully flat an isomorphism over B comes from one over A and similarly for the finitely generated projective condition. □

4recall that a map of rings A → B is flat if an exact sequence of A−modules 0 → M → N → P → 0 remains exact under base change 0 → M_{B} → N_{B} → P_{B} → 0, it is faithfully flat if furthermore M_{B} = 0 if and only if M = 0.

As a corollary we have that, given elements a_{i} ∈ A which generate the unit ideal consider the localizations A[1/a_{i}]. Since ⊕_{i}A[1/a_{i}] is faithfully flat over A then R is an Azumaya algebra over A if and only if the algebras R ⊗ A[1/a_{i}] are Azumaya. In other words the Azumaya property is a local property (in the Zariski and ètale topology of the base) one could now define a sheaf of Azumaya algebras over a scheme etc. but we will not need all these facts.

1.5 The fact that there are non trivial Azumaya algebra comes from

Proposition. If A is a field an Azumaya algebra R over A is a matrix algebra M_{k}(D) where D is a finite dimensional division algebra with center A.

Proof. Let K be an algebraic closure of A, by the previous discussion it follows that R ⊗_{A} K = M_{n}(K) for some n. It follow that A is the center of R and that R has no 2 sided ideals. So we can use again the theorem of Wedderburn. □

The idea of faithful flatness can be used in various ways, the simplest is to analyze locally the structure of Azumaya algebras, one can take either the formal or the local algebraic approach.

In the first approach we let A be a complete local ring with algebraically closed residue field K. We need a simple^{5}

Lemma. Given a ring R an ideal J such that J^{2} = 0 and an idempotent \overline{e} ∈ R/J we can lift it to an idempotent e ∈ R.

More generally we can lift any k orthogonal idempotents to orthogonal idempotents.

Proof. We first lift an idempotent to some e so that e^{2} − e = j ∈ J then e + (1 − 2e)j is the required lift (since j^{2} = 0, e^{2}j = ej).

We can easily reduce to 2 orthogonal idempotents. Having lifted one to e we work in the ring (1 − e)R(1 − e) where we lift the second idempotent etc.. □

Theorem. A ring R over a complete local ring A with maximal ideal \underline{m} and algebraically closed residue field K is an Azumaya algebra of rank n^{2} if and only if

1) it is a free A module and R/\underline{m}R = M_{n}(K).

2) In this case R is necessarily isomorphic to M_{n}(A).

3) More generally if A is complete local with residue field F, R is a rank n^{2} Azumaya algebra over A then R/\underline{m}R = M_{h}(D) where D is a k^{2} dimensional division algebra over its center F, n = hk and moreover R = M_{h}(S) where S is a rank k^{2} Azumaya algebra over A and S/\underline{m}S = D.

Proof. The conditions 1) are clearly necessary from 1.4; we prove then 2).

Let R/\underline{m}R = R ⊗_{A} K = M_{n}(K) by Theorem 1.4 and base change.

The idea is to lift a set of matrix units e_{i,j} modulo the various powers of the maximal ideal and, by Nakayama’s lemma obtain always a basis of matrix units over A/\underline{m}^{k} for all k, then pass to the limit by completeness.

5it is really a small part of Hensel’s lemma.

In fact it suffices to lift the orthogonal idempotents e_{ii} to orthogonal idempotents u_{ii} which add to 1, and this we can do by the previous lemma, induction and completeness.

Next remark that R = ⊕_{ij}u_{ii}Ru_{jj} = ⊕_{i}Ru_{ii} and Ru_{ii} is a free rank n module over A for every i (by reducing mod \underline{m}). Then we need to show that R maps isomorphically to End(Ru_{11}) this follows easily by the isomorphism R^{e} = End(R).

3) is similar to 2) and we leave it to the reader. □

In particular this result can also be used when A/\underline{m} is a finite field 𝔽, in which case D = 𝔽 by Wedderburn’s theorem.

If one wants to work algebraically one can use Hensel’s lemma more profoundly and prove the same statement when A is Hensel with algebraically closed residue field (cf. Raynaud).

Given a local ring A one can pass to such Henselianization by a limit of ètale maps and then one easily obtains. Corollary. If R is an Azumaya algebra over a commutative ring A there is an ètale covering B of A such that R_{B} is locally a matrix algebra (of locally constant rank).^{6}

1.6 If R is an Azumaya algebra over A and \underline{m} is a maximal ideal of A we can form the local ring A\underline{m} and its completion Â\underline{m}. By a standard argument (cf. [R]) we can further extend in a faithfully flat way the residue field so it becomes an algebraically closed field K and obtain a new complete local ring Ã\underline{m} with residue field K, then we know that R ⊗_{A} Ã\underline{m} = M_{n}(Ã\underline{m}) for some n and n^{2} is the rank of R as projective module in the point \underline{m}.

It is now clear that, when working with Azumaya algebras over a connected spectrum we have a fixed rank which will be some number n^{2}. In general we can decompose A = ⊕A_{i} and R = ⊕R_{i} so that R_{i} is Azumaya over A_{i} of constant rank n_{i}^{2}.

This proved the:

Theorem. If R is Azumaya of rank n^{2}, we have a faithfully flat extension of A (for instance B := ∏\underline{m} Ã\underline{m}) such that R ⊗_{A} B = M_{n}(B). In fact we can take B to be also ètale over A.

One can use ideas of faithfully flat descent to derive properties of R.

Let B be faithfully flat over A then a map f : M → N of A modules is injective, surjective if and only if the same property holds for f ⊗ 1_{B} : M_{B} → N_{B}. Similarly two maps f, g : M → N coincide if and only if they coincide after base extension to B.

The general method of descent is to consider the two maps i_{1}, i_{2} : B ⇉ B ⊗_{A} B, where i_{1}(b) := b ⊗ 1, i_{2}(b) = 1 ⊗ b and for any A module M one has the exact sequence

M → M ⊗_{A} B ⇉ M ⊗_{A} B ⊗_{A} B

6if the spectrum is not connected we cannot fix the rank of these matrices globally.

i.e. M is identified to the subset of M ⊗_{A} B where the two maps M ⊗_{A} B ⇉ M ⊗_{A} B ⊗_{A} B coincide. In our case we will not really need the full force of descent.

For instance we can view R as a torsor, the idea is this.

Consider for every commutative A−algebra C the group of C automorphisms of R_{C}, call this A_{R}(C).

The same proof as in 1.2 show that this group valued functor is representable by an affine group scheme closed subgroup of the linear group GL(R).^{7} This group scheme is closely related to the group scheme Aut_{n}(C) in fact they are isomorphic over any extension C of B for which R_{C} = M_{n}(C) splits.

Proposition.

(1) Let I be a two sided ideal of R and let J = R∩A then I = JR and R/I = R⊗_{A}A/J is Azumaya.

(2) A is the center of R.

(3) There is an A−linear map tr : R → A the reduced trace which coincides with the trace in any splitting. Thought as an endomorphism tr : R → A ⊂ R it is associated to a canonical element ξ = ∑_{j} r_{j} ⊗ s_{j} ∈ R^{e}, tr(r) = ∑_{j} r_{j}rs_{j}.

(4) R is a projective R^{e} module.

(5) The functor N → R ⊗_{A} N is an equivalence between the category of A modules and that of R^{e} modules. Given a R^{e} module M the A−module

M^{R} := {x ∈ M|(1 ⊗ r − r ⊗ 1)x = 0}.

equals hom_{R^{e}}(R, M) and gives the isomorphism M = R^{e} ⊗_{A} M^{R}.

(6) If R ⊂ U with U any A algebra then U = R ⊗_{A} R′ where R′ is the centralizer of R in U.

Proof. 1) To prove I = JR one extends scalars to B and it suffices to prove that I_{B} = J_{B}R_{B}, this is the first Lemma of 1.1.

2) Let B be faithfully flat over A and R_{B} = M_{n}(B) if Z is the center of R then A ⊂ Z and Z_{B} is contained in the center of R_{B} which is B.

3) One shows that tr : M_{n}(B) = R ⊗_{n} B → B maps R to A by verifying the faithfully flat descent criterion using that the two isomorphisms of R⊗_{A} (B ⊗_{A} B) with M_{n}(B ⊗_{A} B) are conjugate by an automorphism that leaves trace invariant.

4) Since R^{e} is identified to End(R) it is enough to show that a finitely generated projective module P over a commutative ring A is also projective over End(P).

The property is easily seen to be local on the spectrum of A and thus one can reduce to the case A local and P free of some rank k, then End(P) = M_{k}(A), P = A^{k} and P is clearly projective over the ring of matrices which splits as direct sum of the spaces of columns, each isomorphic to P, hence 4).

5) Let J be the left ideal in R^{e} which annihilates the element 1 ∈ R, we claim that J is generated by the elements {1 ⊗ r − r ⊗ 1|r ∈ R} in fact if ∑_{i} a_{i}b_{i} = (∑_{i} a_{i} ⊗ b_{i})1 = 0 we have ∑_{i} a_{i} ⊗ b_{i} = ∑_{i} a_{i} ⊗ 1(1 ⊗ b_{i} − b_{i} ⊗ 1).

7If R is a free module this is the group on n × n matrices otherwise it is a form of it.

From 3) since the trace is surjective we can split R = A⊕\overline{R} as direct sum decomposition of A modules.

Since R = R^{e}/J the identification hom_{R^{e}}(R, M) = M^{R} is by ϕ → ϕ(1).

One has to prove that the natural map R^{e} ⊗_{A} M^{R} = R^{e} ⊗_{A} hom_{R^{e}}(R, M) → M is an isomorphism it suffices to show it under faithfully flat extension for R = M_{n}(B) and we leave it to the reader.

6) is a special case of 5). □

1.7 We can now analyze the universal n−dimensional representation in the case of an Azumaya algebra R of rank n^{2} over A. For this we need a

Lemma. Assume we have an embedding R ⊂ M_{n}(B) for some commutative ring B then A ⊂ B and M_{n}(B) = R ⊗_{A} B.

Proof. If we prove that A ⊂ B then we have a surjective map R ⊗_{A} B → M_{n}(B).

Since R ⊗_{A} B is Azumaya over B by Proposition 1.4 and the ideals of an Azumaya algebra over B are generated by ideals in B by Proposition 1.6 (1) this implies the rest of the claim.

To show that A ⊂ B one can assume B local.

Since B is the center of M_{n}(B) it suffices to show that M_{n}(B) = RB.

If by contradiction M_{n}(B) ≠ RB we can pass to the residue field and apply Nakayama’s lemma to obtain that for the image \overline{R} of R in M_{n}(K) we have \overline{R}K ≠ M_{n}(K).

We extend K to an algebraically closed field \overline{K}. The algebra \overline{R}\overline{K} modulo its radical is thus a direct sum of matrix algebras of rank strictly less than n and we can find a homomorphism ϕ : R → M_{h}(\overline{K}) with h < n and ϕ(R)\overline{K} = M_{h}(\overline{K}).

By the previous analysis this implies that ϕ(A) ⊂ \overline{K} and R ⊗_{A} \overline{K} = M_{h}(\overline{K}) a contradiction to the hypothesis on the rank of R. □

We can apply this discussion to the universal ring R → M_{n}(A_{n}(R)) and deduce that R ⊗_{B} A_{n}(R) = M_{n}(A_{n}(R)).

If C is any commutative ring over B we clearly get A_{n}(R_{C}) = A_{n}(R)_{C}.

With this discussion in mind we start to study the theory of a.i.r. in the case R = M_{n}(B).

A homomorphism ϕ : M_{n}(B) → M_{n}(C) induces a morphism B → C and then an isomorphism M_{n}(B) ⊗ C → M_{n}(C).

Since M_{n}(B) ⊗ C = M_{n}(C) we see that hom(M_{n}(B), M_{n}(C)) = hom(B, C) × Aut_{n}(C) hence the universal ring of M_{n}(B) is (notations of 1.2) B ⊗ P_{n} and the universal map for M_{n}(B) is given by embedding M_{n}(B) → M_{n}(B) ⊗ P_{n} = M_{n}(B ⊗ P_{n}) and then applying the universal element of the group scheme P_{n} extended as automorphism of M_{n}(C ⊗ P_{n}).

We will also need the following Proposition. If ϕ : R → S is a ring homomorphism between two rank n^{2} Azumaya algebras over two rings B, C then ϕ maps B to C and induces an isomorphism S = R⊗_{B} C.

Proof. We reduce to the case of matrix algebras by faithfully flat descent. □

1.8 Although we will not need this I should recall that the main motivation to introduce Azumaya algebras is in the theory of the Brauer group over commutative rings or even schemes generalizing the classical theory of Brauer over fields.

One easily verifies that, if R, S are two Azumaya algebras over the same commutative ring A so is R ⊗_{A} S.

Next one defines an equivalence on Azumaya algebras given by R ≡ S if there exist two f.g. projective modules P_{i} such that R ⊗_{A} End(P_{1}) = S ⊗_{A} End(P_{2}) and next one sees that the equivalence classes of Azumaya algebras form a commutative group under tensor product, in which the class of End(P) is the identity element and R^{0} gives an inverse to R.

This group, denoted Br(A) is functorial in A and can be described as an ètale cohomology group.

§2 Deformations

2.1 We want to analyze now equivalence classes of representations. Since we want the global theory we give a suitable functorial definition.

Definition. Given a ring R an absolutely irreducible representation of degree n with values in a commutative ring B consists of two data:

1) A rank n^{2} Azumaya algebra S over B.

2) A morphism ϕ : R → S such that S = Bϕ(R).

An equivalence between 2 absolutely irreducible representations

ϕ_{1} : R → S_{1}, ϕ_{2} : R → S_{2}

of degree n with values in B is an isomorphism ψ : S_{1} → S_{2} of B algebras for which ϕ_{2} = ψ ∘ ϕ_{1}.

We will denote by AI_{R}^{n}(B) the set of equivalence classes of irreducible representations which we will refer to as a.i.r..

Since, if f : B → C is a morphism of commutative rings and S is an Azumaya algebra over B, also S ⊗_{B} C is Azumaya over C, it follows that AI_{R}^{n}(−) is a set valued functor on the category of commutative rings.

We plan to show that it is representable by an open set of an affine scheme which is quotient under the projective group of an open set of the scheme of n−dimensional representations.

We go back to the universal map R ⟶^{j_{R}} M_{n}(A_{n}(R)) and define T (R) to be the subring of A_{n}(R) generated by all the elements tr(a), a ∈ R where tr is the usual trace of matrices.

We call T (R) the universal commutative trace ring.

We also define S(R) to be the subring of M_{n}(A_{n}(R)) generated by R and T (R) it is clear that S(R) = RT (R) and that the trace map tr : M_{n}(A_{n}(R)) → A_{n}(R) maps S(R) into T (R).

We want to give some useful examples.

First of all by the definition 2.1 it follows that, an a.i.r. ϕ : R → S where S is Azumaya over B the center of R maps to the center of S which is B.

In particular if R = R_{1} ⊕ R_{2} is a direct sum decomposition the two central idempotents e_{i} unit elements of R_{i} map into B which is thus decomposed into a direct sum B_{1} ⊕ B_{2} as well as S = S_{1} ⊕ S_{2}.

Thus the theory for R can be reduced to the theory for the two R_{i}.

The second remark is when R is itself a semisimple algebra, hence a direct sum of matrix algebras over division algebras.

By the previous remarks the theory can be reduced to the case R = M_{h}(D) where D is a finite dimensional division algebra of dimension k^{2} over its center F.

We claim that there exist n^{2}−dimensional a.i.r. if and only if kh = n.

In fact since R is a simple ring by the various theorems proved it is clear that if there is an a.i.r for R it can be realized as one into matrices over an algebraically closed field K and F ⊂ K since F is the center.

Now it is well known that D ⊗_{F} K = M_{h}(K) and hence R ⊗_{F} K = M_{kh}(K) and the claim follows quite easily.

The converse is clear since then R is a rank n^{2} Azumaya algebra over F.

More generally start from any rank n^{2} Azumaya algebra over some A. To give an n−dimensional a.i.r. one needs to give an Azumaya algebra S over some ring B and a morphism f : R → S with f(R)B = S.

In fact we have seen by Proposition 1.6 that the second condition is automatically satisfied and that S = R ⊗_{A} B.

It is indeed clear that in this case the functor AI_{R}^{n}(B) = Hom(A, B) is just represented by A, one verifies immediately that T (R) = F, S(R) = R.

The description of the universal n−dimensional representation of 1.7 explains in what sense the scheme of n−dimensional representations of R is the total space of a principal bundle with base the scheme of equivalence classes of a.i.r.

2.2 Consider n^{2} elements \underline{r} := r_{i} ∈ R to this set of elements one can associate a discriminant

(2.1.1)  d := d(\underline{r}) := det(tr(r_{i}r_{j})) ∈ T (R).

It is defined in such a way that, inverting d(\underline{r}) := det(tr(r_{i}r_{j})) makes the elements r_{i} a basis of M_{n}(A_{n}(R)[1/d]), of course it is possible that A_{n}(R)[1/d] = 0.

Now consider the algebra S(R)[1/d] ⊂ M_{n}(A_{n}(R)[1/d]) which is an algebra over the ring T (R)[1/d] the fact that the discriminant d is inverted implies that Lemma. The elements r_{i} are a basis of S(R)[1/d] over T (R)[1/d] and

S(R)[1/d] ⊗_{T (R)[1/d]} A_{n}(R)[1/d] = M_{n}(A_{n}(R)[1/d]).

Proof. Given r ∈ S(R)[1/d] we can solve r = ∑_{i} x_{i}r_{i} by computing tr(rr_{j}) = ∑_{i} x_{i}tr(r_{i}r_{j}) and using Cramer’s rule since d is invertible.

By the same rule the r_{i} are also a basis of M_{n}(A_{n}(R)[1/d]) over A_{n}(R)[1/d]. □

Now we will prove that Theorem. S(R)[1/d] is an Azumaya algebra over T (R)[1/d]. Proof. Let us denote S := S(R)[1/d], U := M_{n}(A_{n}(R)[1/d]), i : S → U to simplify notations.

S is a free module of rank n^{2} over the elements r_{i}. We consider the diagram^{8}

S ⊗ S^{o} ⟶^{i_{S}} End(S)
↓  ↓
U ⊗ U^{o} ⟶^{i_{U}} End(U)

we claim that the elements r_{i} ⊗ r_{j} are a basis of End(S) over T (R)[1/d]. In fact we have tr(r_{i} ⊗ r_{j} ∘ r_{h} ⊗ r_{k}) = tr(r_{i}r_{h})tr(r_{j}r_{k}) thus the discriminant of these n^{4} elements is d^{2n} and is invertible and we have an isomorphism i_{S}. □

We have also the more general result which can be proved in the same way :

Corollary of proof. If U is a rank n^{2} Azumaya algebra over a ring A and S ⊂ U is a subring closed under trace and contains a basis of U then S is an Azumaya algebra over its center.

We now come to deformation theory

Let ϕ : R → S be an absolutely irreducible representation of degree n with values in a commutative ring B. We know that S is a rank n^{2} Azumaya algebra over B and S = Bϕ(R). We consider now the case in which B is local and so we can select n^{2} elements r_{i} ∈ R such that ϕ(r_{i}) are a basis of S over B.

Next we split S by a faithfully flat extension C and S ⊗_{B} C = M_{n}(C), we can therefore complete the diagram

R  ⟶^{ϕ} S
↓  ↓
M_{n}(A_{n}(R)) ⟶ M_{n}(C)

By the Proposition 1.6 3) we the reduced trace maps ϕ(R) to B and induces a map of T (R) → B, S(R) → S which is independent of the splitting C.

We can localyze it at d ≠ 0 and it follows that ϕ induces a homomorphism ϕ : T (R)[1/d] → B and S = S(R)[1/d] ⊗_{T (R)[1/d]} B. This implies that Theorem. T (R)[1/d] classifies the absolutely irreducible representations such that the elements r_{i} are a basis of S.

8the tensor products are over the appropriate rings.

The set of equivalence classes of absolutely irreducible n−dimensional representations is parametrized by the open set of Spec(T (R)) covered by the sets U_{d} := Spec(T (R)[1/d]) as d varies among the discriminants of n^{2}−tuples of elements of R.

2.3 Before continuing it is necessary to discuss the finiteness properties of this construction.

We may as well work over some base ring A (sometimes A = ℤ, the most general case).

Suppose thus that R is a finitely generated algebra over A,^{9} by m elements a_{1}, …, a_{m}. We have seen that A_{n}(R) is also finitely generated over A and we wish to show that

Proposition. 1) T := T (R)[1/d] is also finitely generated.

2) If the base ring A is Noetherian, the scheme of a.i.r. is covered by a finite number of open sets U_{d} := Spec(T (R)[1/d]).

Proof. 1) For this remark that the generators a_{i} of R can be expressed as linear combinations a_{i} = ∑_{j} α_{i}^{j}r_{j} of the r_{j} by the usual trick of solving the equations tr(a_{i}r_{h}) = ∑_{j} α_{i}^{j}tr(r_{j}r_{h}) with α_{i}^{j} ∈ T also the multiplication table r_{i}r_{j} = ∑_{k} β_{ij}^{k}r_{k} gives β_{ij}^{k} ∈ T and then it is clear by the definition of T that it is generated by the elements

tr(a_{i}r_{h}), tr(r_{j}r_{h}), tr(r_{i}r_{j}r_{h}), d^{−1}.

□

2) For this it is enough to show that there is a fixed bound N to the length of monomials in the elements a_{i} so that, in any given a.i.r. R → M_{n}(K) the monomials of degree ≤ N span M_{n}(K).

We take the universal map of the free algebra A < x_{1}, …, x_{m} >→ M_{n}(A[x_{hk}^{i}]) defining the generic matrices. For every n^{2} monomials in the x_{i} we compute the discriminant of the corresponding generic matrices and then consider the ideal of A[x_{hk}^{i}] generated by all these discriminants. Since A is Noetherian it is finitely generated.

Thus we can find a finite list of n^{2}−tuples of monomials such that, if in a given representation all their discriminant vanish then the representation is not a. irreducible. This clearly gives the claim. □

To summarize given an A−algebra R one can consider the functor that to a given B associates the set of a.i.r. of R into M_{n}(B) this is representable by an open set of an affine scheme R_{n}(R) and over it acts the projective linear group. We also can define the functor equivalence classes of a.i.r. of R which is also represented by an open set of an affine scheme IR_{n}(R), moreover the map R_{n}(R) → IR_{n}(R) is a principal fibration under the projective linear group which is locally trivial in the ètale topology. If R is of finite type over a noetherian ring A the two schemes are of finite type.

9finitely generated as algebra not necessarily as module

In particular one can apply this idea when one gives a fixed absolutely irreducible representation ρ : R → M_{n}(k) where k is a field and wants to study the fiber of the absolutely irreducible representations which can be specialized to ρ.

Thus we want to parametrize equivalence classes of a.i.r. R → S where S is Azumaya over B local with maximal ideal \underline{m} and S/\underline{m}S = M_{n}(k) so the diagram of maps

R ⟶ S
↓
M_{n}(k)

is commutative.

Again we can localize the problem to a neighborhood where n^{2} chosen elements of R are a basis or even to the intersection of all such neighborhoods and then we can pass to the local ring T (R)\underline{n} where \underline{n} is the kernel of the specialization map T (R) → k.

The same argument as before shows that the given functor of a.i.r. over ρ is represented by the map T (R)\underline{n} → k in the category whose objects are maps B ⟶^{p_{B}} k of local rings and morphisms commutative diagrams

\begin{matrix}B & ⟶^{f} & C \\ ↓^{p_{B}} & ↓^{p_{C}} \\ k & ⟶^{1_{k}} & k \end{matrix} .

Before we pass to group theory we want to understand the theory in the case of finitely generated algebras.

Lemma. If R is a finitely generated ℤ algebra and S is a finite dimensional central simple algebra quotient of R then S is the ring of matrices over a finite field.

Proof. It is enough to see that the center F of S must be also a finitely generated ℤ algebra since then it is a finite field and by Wedderburn’s theorem also S is matrices over F.

To show the finite generation of F select a basis of S over F arising from R and proceed as in the proof of the previous Proposition. □

2.4 Let G be a finite group and R = ℤ[G] its group algebra which is finitely generated and even finite over ℤ.

I claim that the theory is non trivial only for representations over finite fields, in fact if we work over ℚ the algebra R_{ℚ} is semisimple and we have the analysis of the example 2.1.

Let us come to the relevant case take an a.i.r. V for G over a field 𝔽_{r} of prime characteristic.

Let \underline{m} be the kernel in ℤ[G] of the representation and denote the image residue ring by \overline{R}. By definition it is a ring of matrices which spans the matrices over 𝔽_{r}.

We claim that it must be \overline{R} = M_{n}(𝔽_{q}), q = p^{h}, h ≤ r. The proof is by standard non commutative algebra and we give a quick sketch. \overline{R} must be semisimple else the nilpotent ideal generates a nilpotent ideal in M_{n}(𝔽_{q}) which is a contradiction, then it must be simple otherwise it contains a non trivial central idempotent and again we can split M_{n}(𝔽_{r}) in a sum of rings, finally a finite simple algebra is the algebra of matrices over a finite field 𝔽_{q} by Wedderburn’s Theorem. Then 𝔽_{q} ⊂ 𝔽_{r} since by assumption the center of \overline{R} is contained in the scalars.

We can thus assume that V is a vector space over 𝔽_{q} and that \overline{R} = M_{n}(𝔽_{q}) = End(V).

Since every maximal ideal of ℤ[G] gives as quotient a ring M_{m}(𝔽_{q}) for some q and some m we may refer to the set of maximal ideals for which m = n as the n−dimensional Spectrum.^{10}

It can be easily seem that m is bounded by the maximal dimension of an irreducible complex representation of G but we will not need this.

We want to study in this case the complete local ring A in the given point of the spectrum of T (R)[1/d]. We clearly have a map of ℤ_{q} the unramified extension of p−adic numbers with residue field 𝔽_{q} into A, let now \underline{m} be the maximal ideal of A by universality A/\underline{m} = 𝔽_{q} and we want to analyze \underline{m}/(p, \underline{m}^{2}) and claim

Proposition. \underline{m}/(p, \underline{m}^{2}) = H^{1}(G, End(V)).

Proof. By the theory A/(p) is the local ring of the deformations of representations over 𝔽_{q} and now \underline{m}/(p, \underline{m}^{2}) is the set of points of this space of deformations with values in the dual numbers 𝔽_{q}[ϵ], with ϵ^{2} = 0, next the set of such deformations are equivalence classes of representations of type ρ(g)(1+u(g)ϵ) with u(g) ∈ M_{n}(𝔽_{q}) finally the automorphism group of M_{n}(𝔽_{q}[ϵ]) which induce the identity mod ϵ is the group of inner automorphisms induced by 1 + ϵM_{n}(𝔽) and finally this induces the identification with the desired H^{1}(G, End(V)).

Now for an inverse limit of groups Ĝ = \underleftarrow{lim} G_{i} one has a corresponding inverse limit of

complete local rings A_{i} and the limit Â := \underleftarrow{lim} A_{i} is a complete local ring with maximal

ideal \underline{m}/(p, \underline{m}^{2}) = H^{1}(Ĝ, End(V)). For each i one can then select in a compatible way a representation lifting ρ:

\begin{matrix}G_{i} & ⟶^{ρ_{i}} & M_{n}(A_{i}) \\ ↓ & & ↓ \\ G & ⟶^{ρ} & M_{n}(𝔽_{q}) \end{matrix}.

Definition-Theorem Mazur. The limit map Ĝ → M_{n}(Â) is a universal continuous lifting of ρ. If H^{1}(Ĝ, End(V)) is finite dimensional over 𝔽_{q} then Â is Noetherian.

10more correctly maximal spectrum.

§3 An entirely non commutative approach

3.1 There is a remarkable Theorem, due to M.Artin in the equalcharacteristic case and extended by myself in general, characterizing Azumaya algebras of rank n^{2} which has a nice implication for the universal continuous deformation of Mazur which I want to point out.

We return for a moment at the beginning of this paper, recall that, starting from the free (non commutative) algebra ℤ < x_{α} >, taking a set of n^{2} variables x_{i,j}^{α} we consider A := ℤ[x_{i,j}^{α}] the polynomial ring in all these variables.

We defined the generic matrices ξ_{α} in M_{n}(A) by setting ξ_{α} to be the matrix which, in the i, j entry, has value x_{i,j}^{α}.

We finally proved that the map j : ℤ < x_{α} >→ M_{n}(A) defined by j(x_{α}) = ξ_{α} is the universal map from the free algebra to matrices over a commutative ring. Definition. The image of the map j is the ring ℤ < ξ_{α} > and it is called the ring of generic matrices. The kernel of the map j is called the set of polynomial identities of n × n−matrices in the variables x_{α}.

One should remark that the universal property implies that the ideal of polynomial identities is the set of non commutative polynomials which vanish whenever computed into n × n matrices over a commutative ring.

It was a nice discovery of Amitsur Levitski and others that this ideal is in fact very big in the free algebra and that the rings which satisfy the same polynomial identities as matrices are rather close to commutative rings, the exposition of this Theory is the content of [P] (but many progresses have occurred since then). Here we will discuss some of the basics of this Theory.

Perhaps the best known identity is the Amitsur Levitski identity or Standard identity^{11}

(3.1.1)  S_{2m}(x_{1}, x_{2}, …, x_{2m}) := ∑_{σ∈S_{2m}} ϵ(σ)x_{σ(1)}x_{σ(2)} … x_{σ(2m−1)}x_{σ(2m)}.

Theorem Amitsur Levitski. S_{2m}(x_{1}, x_{2}, …, x_{2m}) is a polynomial identity for M_{m}(B), B any commutative ring. It is the only identity for any M_{m}(B) in degree 2m and M_{m}(B) does not satisfy any multilinear identity of degree < 2m.

Proof. That M_{m}(B) does not satisfy any multilinear identity of degree k < 2m can be seen as follows. One takes such a proposed identity and up to reordering the variables may assume that it is f(x) := ax_{1}x_{2} … x_{k} + M where a ≠ 0 and M collects the other terms.

Then one observes that the sequence of 2m − 1 matrix units

e_{11}, e_{12}, e_{22}, e_{23}, e_{33}, …, e_{m−1,m}, e_{mm}

gives product 0 unless it is multiplied in the order in which it has been presented and so, when one evaluates f(x) in the first k of this list one obtains a times a non 0 matrix unit, thus a non 0 value.

11S_{k} denotes the symmetric group on k letters and ϵ(σ) the sign of a permutation.

To prove that the matrices satisfy S_{2m} is much more subtle. There is now a very quick (and slick) proof due to Rosset.

Since it is a formal statement it is enough to do it on integral matrices. We select 2m matrices A_{i} and want to compute S_{2m}(A_{1}, …, A_{2m}). The trick is to add 2m Grassman variables u_{i} and compute on X := ∑_{i=1}^{2m} u_{i}A_{i}.

We see immediately that X^{2m} = S_{2m}(A_{1}, …, A_{2m})u_{1} ∧ u_{2} ∧ ⋯ ∧ u_{2m} so we have to show that X^{2m} = (X^{2})^{m} = 0.

Now X^{2} is a m × m matrix with entries in the even part of the Grassmann algebra generated by the u_{i} hence a matrix with entries in a commutative ring of characteristic 0.

To show that it is (X^{2})^{m} = 0 is equivalent to see that the traces of the powers of X^{2} vanish.

Now we compute

tr(X^{2k}) = ∑_{i_{1}<i_{2}<...i_{2k}} tr(S_{2k}(A_{i_{1}}, …, A_{i_{2k}}))u_{i_{1}} ∧ u_{i_{2}} ∧ … u_{i_{2k}}

We claim that tr(S_{2k}(A_{1}, …, A_{2k})) = 0, for all A_{i} in fact

\begin{matrix}tr(S_{2k}(A_{1}, …, A_{2k})) = tr(∑_{i=1}^{2k}(−1)^{i}A_{i}S_{2k−1}(A_{1}, …, Ǎ_{i}, … A_{2k}) \\ = tr(∑_{i=1}^{2k}(−1)^{i}S_{2k−1}(A_{1}, …, Ǎ_{i}, … A_{2k})A_{i}) = −tr(S_{2k}(A_{1}, …, A_{2k})). \end{matrix}

The uniqueness in degree 2m will not be used and so I skip the proof. □

12 The first main consequence of this theorem is

Corollary. If a ring R satisfies S_{2m}(x_{1}, x_{2}, …, x_{2m}) then it does not have any a.i.r. of degree n > m.

Proof. Otherwise an image of R would span the space M_{n}(K) of matrices for some field K but, since S_{2m}(x_{1}, x_{2}, …, x_{2m}) is multilinear this would imply that also M_{n}(K) satisfies S_{2m} a contradiction. □

3.2 In fact there is a more general theorem due to Kaplansky in these lines.

Theorem Kaplansky. Suppose R satisfies a multilinear identity with coefficients ±1 of degree d^{13} and M is an irreducible module, D := hom_{R}(M, M) the centralizer of R, which by Schur Lemma is a division ring with center (a field) F, then.

12the multilinear condition is required to avoid discussing identities which may exist when B is finite but not when B is infinite.

13this can be relaxed

dim_{F} D = h^{2} < ∞, dim_{D} M = k, 2hk ≤ d

Proof. It is in two steps.

1) Reduction to D commutative. Let K be a maximal commutative subfield of D the K equals the centralizer of K in D and if RK denotes the K algebra of operators of M spanned by K, R we clearly have End_{RK}M = K. Since the identity we start from is multilinear RK satisfies the same identity. By the commutative case we can deduce that 2 dim_{K} M ≤ d.

Clearly dim_{K} M = dim_{K} D dim_{D} M hence dim_{K} D < ∞ and it is well known that in this case dim_{F} D = [dim_{K} D]^{2} and 1) is proved.

2) To prove the commutative case D = K one has to apply the Jacobson density theorem. If we had dim_{K} M > \frac{d}{2} we could find k > \frac{d}{2} linearly independent elements over K and a subring S of R mapping surjectively to the ring of all endomorphisms of the subspace generated by these elements. This is clearly a contradiction because if an identity is satisfied by a ring it is satisfied by its subrings and quotients of them. Thus we contradict the previous theorem.

As I have remarked multilinear identities play a special role in the theory. Basically what we have already used in the proof of the previous Theorem is the fact

Lemma. Let R ⊂ S be rings, assume that S is a B algebra and that S = RB is spanned by R over B then any multilinear identity of R is also an identity of S.

3.3 Before discussing Azumaya algebras it is necessary to focus our attention again on the ring of generic matrices ℤ < ξ_{α} > which has several interesting properties.

It is a rather non trivial object and its structure is not completely understood for n > 2. The first property is that

Theorem Amitsur. ℤ < ξ_{α} > is an integral domain.

Proof. One starts remarking that there exist always some division algebra D over ℚ of dimension n^{2} over the center for every n (for instance one can add to the function field ℚ(x) a non commuting variable y with xy = qyx and q a primitive n^{th} root of 1, quantum torus).

Select a basis u_{i}, i = 1, …, n^{2} of D over its center F and add to F for each α n^{2} variables x_{i}^{α}, i = 1, …, n^{2}. Let A := F [x_{i}^{α}] be the polynomial ring in all these variables.

Define the generic elements η_{α} in D ⊗_{F} A by setting η_{α} = ∑_{i=1}^{n^{2}} x_{i}^{α}u_{i}.

Let j : ℤ < x_{α} >→ D ⊗_{F} A be defined by j(x_{α}) = η_{α}.

Since under extension of F to an algebraically closed field \overline{F} we get D ⊗_{F} \overline{F} = M_{n}(\overline{F}) we easily see that the rig of generic elements ℤ < η_{α} > of D is isomorphic to the ring of generic matrices, but ℤ < η_{α} >⊂ D ⊗_{F} A which is clearly an integral domain. □

In Amitsur original approach the next task was to prove that the generic matrices are an Öre domain and to study its division algebra of quotients, now the theory is strongly simplified by the study of its center which is a rather remarkable object.

3.4 A very interesting fact is the existence of a large center of ℤ < ξ_{α} >. This was conjectured by Kaplansky and proved independently by Formanek and Razmyslov.

Razmyslov argues by constructing a kind of non commutative discriminant as follows. Start remarking that in n × n matrices over a field F the form tr(xy) is non degenerate

and, if e_{i}, i = 1, …, n^{2}, e_{j}^{∗}, j = 1, …, n^{2} are dual bases for this form

Lemma. We have the identity:

∑_{i=1}^{n^{2}} e_{i}ae_{i}^{∗} = tr(a)

for all matrices a.

Then he introduces the polynomial

(3.4.1) F (y_{1}, y_{2}, …, y_{n^{2}+1}, x_{1}, x_{2}, …, x_{n^{2}}) := ∑_{σ∈S_{n^{2}}} ϵ_{σ}y_{1}x_{σ_{1}}y_{2}x_{σ_{2}} … y_{n^{2}}x_{σ_{n^{2}}} y_{n^{2}+1}.

By the following procedure.

Given a non commutative polynomial f in several variables which is linear in a given variable x_{i} write it in the form f = ∑_{k} a_{k}x_{i}b_{k}, also by f_{x_{i}=z} we mean that we substitute for x_{i} the new variable z.

We then set

(3.4.2)  A_{i}(f) := ∑_{k} b_{k}x_{i}a_{k}

It is constructed in such a way that, if we add an extra variable z and evaluate in matrices we have the identity

tr(zf) = tr(x_{i}A_{i}(f)_{x_{i}=z}).

Thus we deduce in particular that tr(f) = tr(x_{i}A_{i}(f)_{x_{i}=1}).

From the identity A_{i}(f) := ∑_{k} b_{k}x_{i}a_{k} we have

tr(x_{j}A_{i}(f)_{x_{i}=1}) = ∑_{k} tr(x_{j}b_{k}a_{k}) = tr(∑_{k} a_{k}x_{j}b_{k}) = tr(f_{x_{i}=x_{j}}).

The alternating nature of F implies that tr(x_{j}A_{i}(F)_{x_{i}=1}) = 0, ∀j ≠ 0, tr(x_{i}A_{i}(F)_{x_{i}=1}) = tr(F).

Let thus A_{i}(F)_{x_{i}=1}) be denoted as h_{i}(x_{j}, y_{h}), j ≠ i we then have that, if x_{1}, …, x_{n^{2}} are linearly independent n × n matrices then the elements h_{i}(x_{j}, y_{h}) are a dual basis of the x_{i} up to the scalar factor tr(F).

Now define and compute

(3.4.3)  h := ∑_{i=1}^{n^{2}} x_{i}y_{0}h_{i}(x_{j}, y_{h}).

From the identities proved and the Lemma it follows that

(3.4.4)  h = tr(y_{0})tr(F)

Thus Theorem. The polynomial h(x, y) := ∑_{i=1}^{n^{2}} x_{i}y_{0}h_{i}(x_{j}, y_{h}) evaluated in n × n matrices takes value in the center and it does not vanish identically.

Proof. We have already seen in 3.4.4 that the values are scalars.

To show that it does not vanish identically we must compute tr(F) for suitable values. Substitute in F: for x_{i+n(j−1)} = e_{ij}, i, j = 1, …, n and for the y_{i} set y_{1} = e_{11}, y_{n^{2}+1} = e_{n1} and for the

other y_{i} we look at the value of x_{i−1} and of x_{i}.

If x_{i−1} = e_{rk}, x_{i} = e_{ℓs} we set y_{i} = e_{kℓ}.

Now one verifies that all the monomials appearing in F vanish under this substitution except for the monomial corresponding to the identity permutation which gives value e_{11} thus tr(F) = 1 under this evaluation. □

The fact that ∑_{σ∈S_{n^{2}}} ϵ_{σ}y_{1}x_{σ_{1}}y_{2}x_{σ_{2}} … y_{n^{2}}x_{σ_{n^{2}}} y_{n^{2}+1} is connected to a discriminant can be understood as follows, any polynomial function which is multilinear in m vectors of an m−dimensional vector space and alternating is necessarily a multiple f the determinant. If moreover the vector space has a non degenerate symmetric form the square of the determinant is the discriminant of the form.

3.5 Now we can state the main Theorem. A ring R is an Azumaya algebra of rank n^{2} over its center if and only if it satisfies all the polynomial identities of n × n matrices and moreover it does not have any a.i.r of degree < n.

Proof. The conditions are necessary since an Azumaya algebra of rank n^{2} over its center can be embedded into matrices and all its a.i.r have degree n.

The converse is of course very subtle since one has to exhibit enough elements in the center C of R to prove the Azumaya property. We shall follow Schelter’s proof.

The first remark is that from the multilinear polynomial of Razmyslov h(x_{1}, …, x_{n^{2}}, y) we can deduce the first identity

(3.5.1) h(x_{1}, …, x_{n^{2}}, y)x + ∑_{j=1}^{n^{2}}(−1)^{j+n^{2}+1}h(x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n^{2}}, x, y)x_{j} = 0

on n × n matrices, since this is an alternating polynomial in n^{2} + 1 variables.

Consider all possible evaluations of the central polynomial h in R.

The ideal generated by all these evaluations must be the entire R, otherwise it is contained in a maximal ideal \underline{m} and R/\underline{m} is a simple algebra of degree n where h does not vanish a contradiction.

Next since R satisfies the identities of n × n matrices h takes value in the center C of R and we have 1 = ∑_{i=1}^{k} h_{i}t_{i} where the elements h_{i} are evaluations of h in elements y_{j} → b_{j}^{(i)}, x_{j} → a_{j}^{(i)} and t_{i} ∈ R.

h_{i} := h(b_{j}^{(i)}, a_{j}^{(i)})

First of all we want to exhibit a set of projective generators for R as a C module.

We take as elements of R the kn^{2} elements a_{j}^{(i)}t_{i}.

Take as linear forms elements f_{j}^{(i)} which are defined as follows:

Set h_{j} := (−1)^{n^{2}+1}h(x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n^{2}}, x, y)x_{j}.

Evaluate h_{i} for y_{j} → b_{j}^{(i)}, x_{h} → a_{h}^{(i)}, h ≠ j thus x_{j} is not evaluated at all and we obtain a linear form f_{j}^{(i)} defined by

< f_{j}^{(i)}|r >:= h(y_{j} → b_{j}^{(i)}, x_{h} → a_{h}^{(i)}, h ≠ j, x_{j} → r).

We need to verify that, for every r ∈ R we have r = ∑_{j,i} < f_{j}^{(i)}|r > a_{j}^{(i)}t_{i}.

From 1 = ∑_{i=1}^{k} t_{i}h_{i} we have r = ∑_{i=1}^{k} h_{i}rt_{i} we use the identity given from 3.5.1 to conclude.

Now we need to prove that the mapping from R^{e} to End(R) is surjective. For this, always from the analysis in 1.3, it is enough to exhibit the elementary endomorphisms a_{j}^{(i)}t_{i} ⊗ f_{h}^{(k)} in the image.

For this remark that < f_{j}^{(i)}|r > is the evaluation of a non commutative polynomial hence there exist some (explicit) elements d_{j,s}^{(i)}, e_{j,s}^{(i)} for which

< f_{j}^{(i)}|r >= ∑_{s} d_{j,s}^{(i)}re_{j,s}^{(i)}.

Thus m ⊗ f_{j}^{(i)}(r) = ∑_{s} md_{j,s}^{(i)}re_{j,s}^{(i)} hence ∑_{s} md_{j,s}^{(i)} ⊗ e_{j,s}^{(i)} ∈ R^{(e)} maps to the given endomorphism. □

§4 Remarks on Mazur’s Theorem.

4.1 Let us see how to use this theorem. Given an algebra R and a maximal ideal \underline{m} of R with R/\underline{m} a finite dimensional central simple algebra of rank n^{2} over a field F we know by the theory of a.i.r. that we have an Azumaya algebra S over a local ring B with maximal ideal \underline{n} and the universal map R → S lifting R → R/\underline{m}. Let us complete B to B̂.

We claim that the resulting algebra S ⊗_{B} B̂ = \underleftarrow{lim} S/(S\underline{n}^{k}) can be obtained by a completion construction directly from R.

In fact let \overline{R} denote the quotient of R modulo the ideal J_{n} generated by all the evaluations in R of the polynomial identities of matrices.

J_{n} is in the kernel of every n−dimensional representation and in particular in the kernel of the map R → S and \underline{m} ⊃ J_{n}:

Theorem. S ⊗_{B} B̂ = \underleftarrow{lim} R/(\underline{m}^{k} + J_{n})

Proof. The ring R̂\underline{m} := R/(\underline{m}^{k} + J_{n}) satisfies all the identities of n × n matrices by construction, moreover it contains a unique maximal ideal which is nilpotent and thus it has a unique a.i.r. of degree n so by the main theorem it is an Azumaya algebra over its center. It is in particular a free module of rank n^{2}. When we pass to the limit we see that R̂\underline{m} is again a free module over its center C which is complete local and also one has the limit isomorphism R̂e ; \underline{m} = End_{C}(R̂\underline{m}) so R̂\underline{m} is Azumaya.

The composite homomorphism R → S → S/(S\underline{n}^{k}) factors through R → R/(\underline{m}^{k} +J_{n}) → S/(S\underline{n}^{k}) which is a map of Azumaya algebras and so induces at the limit a map R̂\underline{m} → Ŝ. Conversely by universality we have compatible maps S → R/(\underline{m}^{k} + J_{n}) and thus the required isomorphism. □

This Theorem has an interesting corollary: Corollary. Let R be a finite ℤ module and \underline{m} a maximal ideal such that R/\underline{m} = M_{n}(𝔽_{q}) the universal complete local algebra at \underline{m} is a finite module over the ring ℤ_{q} of q−adic numbers (the unramified extension of p− adic numbers ℤ_{p} with residue field 𝔽_{q}) and a quotient of R ⊗ ℤ_{q}. Proof. We lift the representation ρ to a representation of R ⊗ ℤ_{q} and then, for every n the finite ring R/(\underline{m}^{k} + J_{n}) is a matrix algebra over a local ring with residue field 𝔽_{q}, thus we can extend the quotient R → R/(\underline{m}^{k} + J_{n}) to a quotient R ⊗ ℤ_{q} → R/(\underline{m}^{k} + J_{n}) in a compatible way.

Passing to the limit we see that the map ρ lifts to a surjective map of R ⊗ ℤ_{q} to R̂. □ In particular we can apply this to groups and profinite groups. Start from ℤ[G]/\underline{m} = M_{n}(𝔽_{q}) then the universal continuous representation of G into

matrices over a complete local ring is obtained by taking the inverse limit of the finite algebras ℤ[G]/I where:

(1) I is an ideal contained in \underline{m} and containing a power \underline{m}^{k}.

(2) ℤ[G]/I satisfies the identities of n × n matrices or, equivalently under 1), it is a (finite) Azumaya algebra hence isomorphic to M_{n}(B) for a finite local ring B with residue field 𝔽_{q}.

For a profinite group G the limit is on the finite algebras ℤ[G_{i}]/I. We have thus as profinite algebra the algebra M_{n}(A), and the complete compact profinite local ring A with maximal ideal \underline{m} and residue field 𝔽_{q}.

Thus we have the continuous lifting G → GL(n, A) of the given representation.

In this case one has that the kernel of the map GL(n, A) → GL(n, 𝔽_{q}) is the group K := 1 + M_{n}(\underline{m}) which is a pro p−group.

The Noetherian property for A is equivalent to the property for K to be finitely generated topologically.

In turn this is equivalent to the fact that the quotient of K by the product of the commutator subgroup [K, K] times K^{p} :=< k^{p}| k ∈ K > is finite. In fact one can easily see that

K/[K, K] := 1 + M_{n}(\underline{m}/\underline{m}^{2}), K/[K, K]K^{p} = 1 + M_{n}(\underline{m}/[(p) + \underline{m}^{2}]).

Thus given elements x_{1}, …, x_{k} which map to a basis of a basis of \underline{m}/[(p) + \underline{m}^{2}] one has by Nakayama’s lemma and completeness that A is a quotient of the power series algebra ℤ_{q}[[x_{1}, …, x_{k}]].

We see the exact relation between topological generators for K or for A.

In the applications to Galois representations the finiteness is insured by a suitable choice of the Galois group K (finite ramification).

I wish to make a very short historic note (or maybe more a reminiscence) on the non commutative part of the subject.

One of the motivations for PI theory was to study the analogue of the restricted Burnside problem for algebras, this would require a history in itself, anyway there were at least 3 groups of people working on the topic, the Russian school (Kurosh, Sirsov, etc), the Israeli (Amitsur Levitzki) and the American school (Jacobson, Kaplansky, Herstein). Sometimes there were lack of communications.

Kaplansky’s theorem is the key for the solution of this problem from a theoretical viewpoint while Sirsov furnished the most effective and complete combinatorial solution.

The ring of generic matrices appears first of all in Amitsur’s work as the free algebra modulo the identities of matrices, the remark that it can be described by generic matrices maybe is trivial anyway appears first in my Ph.D. thesis and I vividly recall explaining it to Amitsur at tea time in the University of Chicago in 1965, his first reaction was that he did not believe me.

When M.Artin got interested in the topic he was able to use the powerful ideas of faithfully flat descent which were unknown to the specialists in non commutative algebra, also that was the time of revival of geometric invariant theory and the invariant interpretation has changed completely the point of view.

Finally the technical breakthrough of discovering central polynomials simplified drastically the proofs.

In any case a full understanding of polynomial identities and central polynomials of matrices is quite a hard if not impossible challenge.

I have not discussed at all the theory of degenerations from irreducible to reducible representations, the interested reader could look at [P2-3-4] for some insights.

REFERENCES

[A-L] S. Amitsur, J. Levitzki, Minimal identities for algebras, Proc. A.M.S. 1 (1950), 449–463.

[Ar] M. Artin, On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11 (1969), 532–563.

[A-G1] M. Auslander, O. Goldman, Maximal Orders, Trans. A.M.S. 97 (1960), 1–24.

[A-G2] M. Auslander, O. Goldman, The Brauer Group of a Commutative Ring, Trans. A.M.S. 97 (1960), 367–409.

[Az] G. Azumaya, On maximally central algebras, Nagoya Math. J. 2 (1950), 119–150.

26 C. PROCESI

[D] M. Deuring, Algebren, Erg. der Math., 4 (1935), 143 pp. [Gr ] A. Grothendieck, Le groupe de Brauer I,II,III, Dix Exposés sur la Cohomologie des Schemas, North Holland (1968), 46–188.

[M] B. Mazur, Deforming Galois representations, Galois groups over ℚ (Ihara, Ribet, Serre eds.), MSRI Publ. 16 (1989), 385–437.

[P] C. Procesi, Rings with polynomial identities, Marcel Dekker (1973), 190 pp..

[P2] C. Procesi, Finite dimensional representations of algebras, Israel Journal of Mathematics 19 (1974).

[P3] C. Procesi, Trace identities and standard diagrams. Ring theory. Lecture notes in Pure and Appl. Mathematics 51 M. Dekker (1979), pp. 191–218.

[P4] C. Procesi, A formal inverse to the Cayley Hamilton theorem. Journal of Algebra 107 1 (1987), pp. 63–74.

[R] M. Raynaud, Anneaux locaux henseliens, S.L.N. 169 (1970), 129 pp..

[Ra] J. Razmyslov, Trace identities of matrix algebras via a field of characteristic zero, Math. USSR Izvestia (translation). 8 (1974), 727–760.

[Sc] W. Schelter, Azumaya algebras and Artin’s theorem, J. of Algebra 46 (1977), no.1, 303–304.

Dip. Mat. Castelnuovo, Univ. di Roma La Sapienza. e-mail claudio@mat.uniroma1.it

home-page: http://mercurio.mat.uniroma1.it/ordinari/procesi/home.html