POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS

Z. REICHSTEIN AND N. VONESSEN

Abstract. We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_{n}-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.

Contents

1. Introduction 2 2. Preliminaries 4

2.1 Matrix Invariants 4 2.4 The ring of generic matrices and its trace ring 5 2.7 Central polynomials 6

3. Definition and first properties of n-varieties 7 4. Irreducible n-varieties 9 5. The Nullstellensatz for prime ideals 10 6. Regular maps of n-varieties 12 7. Rational maps of n-varieties 15 8. Generically free PGL_{n}-varieties 17 9. Brauer-Severi Varieties 20 References 23

Date: August 5, 2005. 2000 Mathematics Subject Classification. Primary: 16R30, 16R20; Secondary 14L30,

14A10.

Key words and phrases. Polynomial identity ring, central simple algebra, trace ring, coordinate ring, the Nullstellensatz.

Z. Reichstein was supported in part by an NSERC research grant. N. Vonessen gratefully acknowledges the support of the University of Montana and the

hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

1. Introduction

Polynomial identity rings (or PI-rings, for short) are often viewed as being “close to commutative”; they have large centers, and their structure (and in particular, their maximal spectra) have been successfully studied by geometric means (see the references at the end of the section). In this paper we revisit this subject from the point of view of classical affine algebraic geometry. We will show that the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, can be extended to the setting of PI-rings.

Before proceeding with the statements of our main results, we will briefly introduce the objects that will play the roles of varieties, defining ideals, coordinate rings, etc. Throughout this paper we will work over an algebraically closed base field k of characteristic zero. We also fix an integer n ≥ 1, which will be the PI-degree of most of the rings we will consider. We will write M_{n} for the matrix algebra M_{n}(k). The vector space of m-tuples of n × n-matrices will be denoted by (M_{n})^{m}; we will always assume that m ≥ 2. The group PGL_{n} acts on (M_{n})^{m} by simultaneous conjugation. The PGL_{n}-invariant dense open subset

U_{m,n} = {(a_{1}, …, a_{m}) ∈ (M_{n})^{m} | a_{1}, …, a_{m} generate M_{n} as k-algebra}

of (M_{n})^{m} will play the role of the affine space 𝔸^{m} in the sequel. (Note that U_{m,1} = 𝔸^{m}.) The role of affine algebraic varieties will be played by PGL_{n}-invariant closed subvarieties of U_{m,n}; for lack of a better term, we shall call such objects n-varieties; see Section 3. (Note that, in general, n-varieties are not affine in the usual sense.) The role of the polynomial ring k[x_{1}, …, x_{m}] will be played by the algebra G_{m,n} = k{X_{1}, …, X_{m}} of m generic n×n matrices, see 2.4. Elements of G_{m,n} may be thought of as PGL_{n}- equivariant maps (M_{n})^{m} ⟶ M_{n}; if n = 1 these are simply the polynomial maps k^{m} ⟶ k. Using these maps, we define, in a manner analogous to the commutative case, the associated ideal I(X) ⊂ G_{m,n}, the PI-coordinate ring k_{n}[X] = G_{m,n}/I(X), and the central simple algebra k_{n}(X) of rational functions on an irreducible n-variety X ⊂ U_{m,n}; see Definitions 3.1 and 7.1. We show that Hilbert’s Nullstellensatz continues to hold in this context; see Section 5. We also define the notions of a regular map (and, in particular, an isomorphism) X ⟶ Y and a rational map (and, in particular, a birational isomorphism) X −−→ Y between n-varieties X ⊂ U_{m,n} and Y ⊂ U_{l,n}; see Definitions 6.1 and 7.5.

In categorical language our main results can be summarized as follows. Let

Var_{n} be the category of irreducible n-varieties, with regular maps of nvarieties as morphisms (see Definition 6.1), and

PI_{n} be the category of finitely generated prime k-algebras of PI-degree n (here the morphisms are the usual k-algebra homomorphisms).

1.1. Theorem. The functor defined by

\begin{matrix}X & ↦ & k_{n}[X] \\ (f: X ⟶ Y) & ↦ & (f^{∗}: k_{n}[Y] ⟶ k_{n}[X]) \end{matrix}

is a contravariant equivalence of categories between Var_{n} and PI_{n}.

In particular, every finitely generated prime PI-algebra is the coordinate ring of a uniquely determined n-variety; see Theorem 6.4. For a proof of Theorem 1.1, see Section 6.

Every n-variety is, by definition, an algebraic variety with a generically free PGL_{n}-action. It turns out that, up to birational isomorphism, the converse holds as well; see Lemma 8.1. To summarize our results in the birational context, let

Bir_{n} be the category of irreducible generically free PGL_{n}-varieties, with dominant rational PGL_{n}-equivariant maps as morphisms, and

CS_{n} be the category of central simple algebras A of degree n, such that the center of A is a finitely generated field extension of k. Morphisms in CS_{n} are k-algebra homomorphisms (these are necessarily injective).

1.2. Theorem. The functor defined by

\begin{matrix}X & ↦ & k_{n}(X) \\ (g: X −−→ Y) & ↦ & (g^{∗}: k_{n}(Y) ↪ k_{n}(X)) \end{matrix}

is a contravariant equivalence of categories between Bir_{n} and CS_{n}.

Here for any PGL_{n}-variety X, k_{n}(X) denotes the k-algebra of PGL_{n}- equivariant rational maps X −−→ M_{n} (with addition and multiplication induced from M_{n}), see Definition 8.2. If X is an irreducible n-variety then k_{n}(X) is the total ring of fractions of k_{n}[X], as in the commutative case; see Definition 7.1 and Proposition 7.3. Note also that g^{∗}(α) stands for α ∘ g (again, as in the commutative case). For a proof of Theorem 1.2, see Section 8.

Note that for n = 1, Theorems 1.1 and 1.2 are classical results of affine algebraic geometry; cf., e.g., [13, Corollary 3.8 and Theorem 4.4].

It is well known that central simple algebras A/K of degree n are in a natural bijection with n − 1-dimensional Brauer-Severi varieties over K and (if K/k is a finitely generated field extension) with generically free PGL_{n}- varieties X/k such that k(X)^{PGL^{n}} ≃ K. Indeed, all three are parametrized by the Galois cohomology set H^{1}(K, PGL_{n}); for details, see Section 9. Theorem 1.2 may thus be viewed as a way of explicitly identifying generically free PGL_{n}-varieties X with central simple algebras A, without going through H^{1}(K, PGL_{n}). The fact that the map X ↦ k_{n}(X) is bijective was proved in [24, Proposition 8.6 and Lemma 9.1]; here we give a more conceptual proof and show that this map is, in fact, a contravariant functor. In Section 9 we show how to construct the Brauer-Severi variety of A directly from X.

Many of the main themes of this paper (such as the use of PGL_{n}-actions, generic matrices, trace rings, and affine geometry in the study of polynomial identity algebras) were first systematically explored in the pioneering work of Amitsur, Artin and Procesi [2, 4, 19, 20, 21] in the 1960s and 70s. Our approach here was influenced by these, as well as other papers in this area, such as [3, 5, 7, 23]. In particular, Proposition 5.3 is similar in spirit to Amitsur’s Nullstellensatz [1] (cf. Remark 5.6), and Theorem 1.1 to Procesi’s functorial description of algebras satisfying the nth Cayley-Hamilton identity [22, Theorem 2.6]. (For a more geometric statement of Procesi’s theorem, along the lines of our Theorem 1.1, see [15, Theorem 2.3].) We thank L. Small and the referee for bringing some of these connections to our attention.

2. Preliminaries

In this section we review some known results about matrix invariants and related PI-theory.

2.1. Matrix invariants. Consider the diagonal action of PGL_{n} on the space (M_{n})^{m} of m-tuples of n × n-matrices. We shall denote the ring of invariants for this action by C_{m,n} = k[(M_{n})^{m}]^{PGL^{n}}, and the affine variety Spec(C_{m,n}) by Q_{m,n}. It is known that C_{m,n} is generated as a k-algebra, by elements of the form (A_{1}, …, A_{m}) ↦ tr(M), where M is a monomial in A_{1}, …, A_{m} (see [21]); however, we shall not need this fact in the sequel. The inclusion C_{m,n} ↪ k[(M_{n})^{m}] of k-algebras induces the categorical quotient map

(2.2)  π: (M_{n})^{m} ⟶ Q_{m,n} .

We shall need the following facts about this map in the sequel. Recall from the introduction that we always assume m ≥ 2, the base field k is algebraically closed and of characteristic zero, and

U_{m,n} = {(a_{1}, …, a_{m}) ∈ (M_{n})^{m} | a_{1}, …, a_{m} generate M_{n} as k-algebra }.

2.3. Proposition. (a) If x ∈ U_{m,n} then π^{−1}(π(x)) is the PGL_{n}-orbit of x. (b) PGL_{n}-orbits in U_{m,n} are closed in (M_{n})^{m}.

(c) π maps closed PGL_{n}-invariant sets in (M_{n})^{m} to closed sets in Q_{m,n}.

(d) π(U_{m,n}) is Zariski open in Q_{m,n}.

(e) If Y is a closed irreducible subvariety of Q_{m,n} then π^{−1}(Y) ∩ U_{m,n} is irreducible in (M_{n})^{m}.

Proof. (a) is proved in [4, (12.6)].

(b) is an immediate consequence of (a).

(c) is a special case of [18, Corollary to Theorem 4.6].

(d) It is easy to see that U_{m,n} is Zariski open in (M_{n})^{m}. Let U_{m,n}^{c} be its complement in (M_{n})^{m}. By (c), π(U_{n,m}^{c}) is closed in Q_{m,n} and by (a), π(U_{m,n}) = Q_{m,n} \ π(U_{m,n}^{c}).

(e) Let V_{1}, …, V_{r} be the irreducible components of π^{−1}(Y) in (M_{n})^{m}. Since PGL_{n} is connected, each V_{i} is PGL_{n}-invariant. By part (c), π(V_{1}), …, π(V_{r}) are closed subvarieties of Q_{m,n} covering Y. Since Y is irreducible, we may assume, after possibly renumbering V_{1}, …, V_{r}, that Y = π(V_{1}). It suffices to show that

π^{−1}(Y) ∩ U_{m,n} = V_{1} ∩ U_{m,n};

since V_{1} ∩ U_{m,n} is irreducible, this will complete the proof of part (e). As V_{1} is an irreducible component of π^{−1}(Y), we clearly have

V_{1} ∩ U_{m,n} ⊂ π^{−1}(Y) ∩ U_{m,n} .

To prove the opposite inclusion, let y ∈ π^{−1}(Y) ∩ U_{m,n}. We want to show y ∈ V_{1}. Since π(y) ∈ Y = π(V_{1}), there is a point v ∈ V_{1} such that π(y) = π(v). That is, v lies in π^{−1}(π(y)), which, by part (a), is the PGL_{n}-orbit of y. In other words, y = g · v for some g ∈ PGL_{n}. Since V_{1} is PGL_{n}-invariant, this shows that y ∈ V_{1}, as claimed. □

2.4. The ring of generic matrices and its trace ring. Consider m generic matrices

X_{1} = (x_{ij}^{(1)})_{i,j=1,...,n} , …, X_{m} = (x_{ij}^{(m)})_{i,j=1,...,n} ,

where x_{ij}^{(h)} are mn^{2} independent variables over the base field k. The k-

subalgebra generated by X_{1}, …, X_{m} inside M_{n}(k[x_{ij}^{(h)}]) is called the algebra of m generic n × n-matrices and is denoted by G_{m,n}. If the values of n and m are clear from the context, we will simply refer to G_{m,n} as the algebra of generic matrices.

The trace ring of G_{m,n} is denoted by T_{m,n}; it is the k-algebra generated, inside M_{n}(k[x_{ij}^{(h)}]) by elements of G_{m,n} and their traces. Elements

of M_{n}(k[x_{ij}^{(h)}]) can be naturally viewed as regular (i.e., polynomial) maps

(M_{n})^{m} ⟶ M_{n}. (Note that k[x_{ij}^{(h)}] in the coordinate ring of (M_{n})^{m}.) Here PGL_{n} acts on both (M_{n})^{m} and M_{n} by simultaneous conjugation; Procesi [21, Section 1.2] noticed that T_{m,n} consists precisely of those maps (M_{n})^{m} ⟶ M_{n} that are equivariant with respect to this action. (In particular, the i-th generic matrix X_{i} is the projection to the i-th component.) In this way the invariant ring C_{m,n} = k[(M_{n})^{m}]^{PGL^{n}} which we considered in 2.1, is naturally identified with the center of T_{m,n} via f ↦ fI_{n×n}.

We now recall the following definitions.

2.5. Definition. (a) A prime PI-ring is said to have PI-degree n if its total ring of fractions is a central simple algebra of degree n.

(b) Given a ring R, Spec_{n}(R) is defined as the set of prime ideals J of R such that R/J has PI-degree n; cf. e.g., [19, p. 58] or [26, p. 75].

The following lemma shows that Spec_{n}(G_{m,n}) and Spec_{n}(T_{m,n}) are closely related.

2.6. Lemma. The assignment J ↦ J ∩ G_{m,n} defines a bijective correspondence between Spec_{n}(T_{m,n}) and Spec_{n}(G_{m,n}). In addition, for any prime ideal J ∈ Spec_{n}(T_{m,n}), we have the following:

(a) The natural projection ϕ: T_{m,n} ⟶ T_{m,n}/J is trace-preserving, and T_{m,n}/J is the trace ring of G_{m,n}/(J ∩ G_{m,n}).

(b) tr(p) ∈ J for every p ∈ J.

Proof. The first assertion and part (a) are special cases of results proved in [3, §2]. Part (b) follows from (a), since for any p ∈ J, ϕ(tr(p)) = tr(ϕ(p)) = tr(0) = 0. In other words, tr(p) ∈ Ker(ϕ) = J, as claimed. □

2.7. Central polynomials. We need to construct central polynomials with certain non-vanishing properties. We begin by recalling two well-known facts from the theory of rings satisfying polynomial identities.

2.8. Proposition. (a) Let k{x_{1}, …, x_{m}} be the free associative algebra. Consider the natural homomorphism k{x_{1}, …, x_{m}} ⟶ G_{m,n}, taking x_{i} to the i-th generic matrix X_{i}. The kernel of this homomorphism is precisely the ideal of polynomial identities of n × n-matrices in m variables.

(b) Since k is an infinite field, all prime k-algebras of the same PI-degree satisfy the same polynomial identities (with coefficients in k).

Proof. See [19, pp. 20-21] or [26, p. 16] for a proof of part (a) and [27, pp. 106-107] for a proof of part (b). □

For the convenience of the reader and lack of a suitable reference, we include the following definition.

2.9. Definition. An (m-variable) central polynomial for n×n matrices is an element p = p(x_{1}, …, x_{m}) ∈ k{x_{1}, …, x_{m}} satisfying one of the following equivalent conditions:

(a) p is a polynomial identity of M_{n−1}, and the evaluations of p in M_{n} are central (i.e., scalar matrices) but not identically zero.

(b) p is a polynomial identity for all prime k-algebras of PI-degree n−1, and the evaluations of p in every prime k-algebra of PI-degree n are central but not identically zero.

(c) p is a polynomial identity for all prime k-algebras of PI-degree n−1, and the canonical image of p in G_{m,n} is a nonzero central element.

(d) The constant coefficient of p is zero, and the canonical image of p in G_{m,n} is a nonzero central element.

That the evaluations of p in an algebra A are central is equivalent to saying that x_{m+1}p − px_{m+1} is a polynomial identity for A, where x_{m+1} is another free variable. Thus the equivalence of (a)—(c) easily follows from Proposition 2.8. The equivalence of (c) and (d) follows from [19, p. 172]. The existence of central polynomials for n × n-matrices was established independently by Formanek and Razmyslov; see [11]. Because of Proposition 2.8(a), one can think of m-variable central polynomials of n×n matrices as nonzero central elements of G_{m,n} (with zero constant coefficient).

The following lemma, establishing the existence of central polynomials with certain non-vanishing properties, will be repeatedly used in the sequel.

2.10. Lemma. Let A_{1}, …, A_{r} ∈ U_{m,n}. Then there exists a central polynomial s = s(X_{1}, …, X_{m}) ∈ G_{m,n} for n × n-matrices such that s(A_{i}) ≠ 0 for i = 1, …, r. In other words, each s(A_{i}) is a non-zero scalar matrix in M_{n}.

Proof. First note that if A_{i} and A_{j} are in the same PGL_{n}-orbit then s(A_{i}) = s(A_{j}). Hence, we may remove A_{j} from the set {A_{1}, …, A_{r}}. After repeating this process finitely many times, we may assume that no two of the points A_{1}, …, A_{r} lie in the same PGL_{n}-orbit.

By the above-mentioned theorem of Formanek and Razmyslov, there exists a central polynomial c = c(X_{1}, …, X_{N}) ∈ G_{N,n} for n × n-matrices. Choose b_{1}, …, b_{N} ∈ M_{n} such that c(b_{1}, …, b_{N}) ≠ 0. We now define s by modifying c as follows:

s(X_{1}, …, X_{m}) = c(p_{1}(X_{1}, …, X_{m}), …, p_{N}(X_{1}, …, X_{m})) ,

where the elements p_{j} = p_{j}(X_{1}, …, X_{m}) ∈ G_{m,n} will be chosen below so that for every j = 1, …, N,

(2.11)  p_{j}(A_{1}) = p_{j}(A_{2}) = ⋯ = p_{j}(A_{r}) = b_{j} ∈ M_{n} .

We first check that this polynomial has the desired properties. Being an evaluation of a central polynomial for n × n matrices, s is a central element in G_{m,n} and a polynomial identity for all prime k-algebras of PI-degree n−1. Moreover, s(A_{i}) = c(b_{1}, …, b_{N}) ≠ 0 for every i = 1, …, r. Thus s itself is a central polynomial for n × n matrices. Consequently, s(A_{i}) is a central element in M_{n}, i.e., a scalar matrix.

It remains to show that p_{1}, …, p_{N} ∈ G_{m,n} can be chosen so that (2.11) holds. Consider the representation ϕ_{i}: G_{m,n} ⟶ M_{n} given by p ⟼ p(A_{i}). Since each A_{i} lies in U_{m,n}, each ϕ_{i} is surjective. Moreover, by our assumption on A_{1}, …, A_{r}, no two of them are conjugate under PGL_{n}, i.e., no two of the representations ϕ_{i} are equivalent. The kernels of the ϕ_{i} are thus pairwise distinct by [4, Theorem (9.2)]. Hence the Chinese Remainder Theorem tells us that ϕ_{1} ⊕ … ⊕ ϕ_{r}: G_{m,n} ⟶ (M_{n})^{r} is surjective; p_{j} can now be chosen to be any preimage of (b_{j}, …, b_{j}) ∈ (M_{n})^{r}. This completes the proof of Lemma 2.10. □

3. Definition and first properties of n-varieties

3.1. Definition. (a) An n-variety X is a closed PGL_{n}-invariant subvariety of U_{m,n} for some m ≥ 2. In other words, X = \overline{X} ∩U_{m,n}, where \overline{X} is the Zariski closure of X in (M_{n})^{m}. Note that X is a generically free PGL_{n}-variety (in fact, for every x ∈ X, the stabilizer of x in PGL_{n} is trivial).

(b) Given a subset S ⊂ G_{m,n} (or S ⊂ T_{m,n}), we define its zero locus as

Z(S) = {a = (a_{1}, …, a_{m}) ∈ U_{m,n} | p(a) = 0, ∀p ∈ S} .

Of course, Z(S) = Z(J), where J is the 2-sided ideal of G_{m,n} (or T_{m,n}) generated by S. Conversely, given an n-variety X ⊂ U_{m,n} we define its ideal as

I(X) = {p ∈ G_{m,n} | p(a) = 0, ∀a ∈ X} .

Similarly we define the ideal of X in T_{m,n}, as

I_{T} (X) = {p ∈ T_{m,n} | p(a) = 0, ∀a ∈ X} .

Note that I(X) = I_{T} (X) ∩ G_{m,n}.

(c) The polynomial identity coordinate ring (or PI-coordinate ring) of an n-variety X is defined as G_{m,n}/I(X). We denote this ring by k_{n}[X].

3.2. Remark. Elements of k_{n}[X] may be viewed as PGL_{n}-equivariant morphisms X ⟶ M_{n}. The example below shows that not every PGL_{n}-equiva-riant morphism X ⟶ M_{n}(k) is of this form. On the other hand, if X is irreducible, we will later prove that every PGL_{n}-equivariant rational map X −−→ M_{n}(k) lies in the total ring of fractions of k_{n}[X]; see Proposition 7.3.

3.3. Example. Recall that U_{2,2} is the open subset of M_{2,2} defined by the inequality c(X_{1}, X_{2}) ≠ 0, where

c(X_{1}, X_{2}) = (2 tr(X_{1}^{2}) − tr(X_{1})^{2})(2 tr(X_{2}^{2}) − tr(X_{2})^{2})−
(2 tr(X_{1}X_{2}) − tr(X_{1}) det(X_{2}))^{2} ;

see, e.g., [12, p. 191]. Thus for X = U_{2,2}, the PGL_{n}-equivariant morphism f: X ⟶ M_{2} given by (X_{1}, X_{2}) ⟼ \frac{1}{c(X_{1},X_{2})}I_{2×2} is not in k_{2}[X] = G_{2,2} (and not even in T_{2,2}).

3.4. Remark. (a) Let J be an ideal of G_{m,n}. Then the points of Z(J) are in bijective correspondence with the surjective k-algebra homomorphisms ϕ: G_{m,n} ⟶ M_{n} such that J ⊂ Ker(ϕ) (or equivalently, with the surjective k-algebra homomorphisms G_{m,n}/J ⟶ M_{n}). Indeed, given a ∈ Z(J), we associate to it the homomorphism ϕ_{a} given by ϕ_{a}: p ↦ p(a). Conversely, a surjective homomorphism ϕ: G_{m,n} ⟶ M_{n} such that J ⊂ Ker(ϕ) gives rise to the point

a_{ϕ} = (ϕ(X_{1}), …, ϕ(X_{m})) ∈ Z(J) ,

where X_{i} ∈ G_{m,n} is the i-th generic matrix in G_{m,n}. One easily checks that the assignments a ↦ ϕ_{a} and ϕ ↦ a_{ϕ} are inverse to each other.

(b) The claim in part (a) is also true for an ideal J of T_{m,n}. That is, the points of Z(J) are in bijective correspondence with the surjective k-algebra homomorphisms ϕ: T_{m,n} ⟶ M_{n} such that J ⊂ Ker(ϕ) (or equivalently, with the surjective k-algebra homomorphisms T_{m,n}/J ⟶ M_{n}). The proof goes through without changes.

3.5. Lemma. Let a = (a_{1}, …, a_{m}) ∈ U_{m,n} and let J be an ideal of G_{m,n} (or of T_{mn}). Let

J(a) = {j(a) | j ∈ J} ⊂ M_{n} .

Then either J(a) = (0) (i.e, a ∈ Z(J)) or J(a) = M_{n}.

Proof. Since a_{1}, …, a_{m} generate M_{n}, ϕ_{a}(J) is a (2-sided) ideal of M_{n}. Since M_{n} is simple, the lemma follows. □

3.6. Lemma. (a) Z(J) = Z(J ∩ G_{m,n}) for every ideal J ⊂ T_{m,n}. (b) If X ⊂ U_{m,n} is an n-variety, then X = Z(I(X)) = Z(I_{T} (X)).

Proof. (a) Clearly, Z(J) ⊂ Z(J ∩ G_{m,n}). To prove the opposite inclusion, assume the contrary: there exists a y ∈ U_{m,n} such that p(y) = 0 for every p ∈ J ∩ G_{m,n} but f(y) ≠ 0 for some f ∈ J. By Lemma 2.10 there exists a central polynomial s ∈ G_{m,n} for n × n-matrices such that s(y) ≠ 0. By [28, Theorem 1], p = s^{i}f lies in G_{m,n} (and hence, in J ∩ G_{m,n}) for some i ≥ 0. Our choice of y now implies 0 = p(y) = s^{i}(y)f(y). Since s(y) is a non-zero element of k, we conclude that f(y) = 0, a contradiction.

(b) Clearly X ⊂ Z(I_{T} (X)) ⊂ Z(I(X)). Part (a) (with J = I_{T} (X)) tells us that Z(I(X)) = Z(I_{T} (X)). It thus remains to be shown that Z(I_{T} (X)) ⊂ X. Assume the contrary: there exists a z ∈ U_{m,n} such that p(z) = 0 for every p ∈ I_{T} (X) but z ∉ X. Since X = \overline{X} ∩ U_{m,n}, where \overline{X} is the closure of X in (M_{n})^{m}, we conclude that z ∉ \overline{X}. Let C = PGL_{n} · z be the orbit of z in (M_{n})^{m}. Since z ∈ U_{m,n}, C is closed in (M_{n})^{m}; see Proposition 2.3(b). Thus C and \overline{X} are disjoint closed PGL_{n}-invariant subsets of (M_{n})^{m}. By [16, Corollary 1.2], there exists a PGL_{n}-invariant regular function f: (M_{n})^{m} ⟶ k such that f ≡ 0 on \overline{X} but f ≢ 0 on C. The latter condition is equivalent to f(z) ≠ 0. Identifying elements of k with scalar matrices in M_{n}, we may view f as a central element of T_{m,n}. So f ∈ I_{T} (X) but f(z) ≠ 0, contradicting our assumption. □

4. Irreducible n-varieties

Of particular interest to us will be irreducible n-varieties. Here “irreducible” is understood with respect to the n-Zariski topology on U_{m,n}, where the closed subsets are the n-varieties. However, since PGL_{n} is a connected group, each irreducible component of X in the usual Zariski topology is PGL_{n}-invariant. Consequently, X is irreducible in the n-Zariski topology if and only if it is irreducible in the usual Zariski topology.

4.1. Lemma. Let ∅ ≠ X ⊂ U_{m,n} be an n-variety. The following are equivalent:

(a) X is irreducible.

(b) I_{T} (X) is a prime ideal of T_{m,n}.

(c) I(X) is a prime ideal of G_{m,n}.

(d) k_{n}[X] is a prime ring.

Proof. (a) ⇒ (b): Suppose that I_{T} (X) is not prime, i.e., there are ideals J_{1} and J_{2} such that J_{1} · J_{2} ⊂ I_{T} (X) but J_{1}, J_{2} ⊄ I_{T} (X). We claim that X is not irreducible. Indeed, by Lemma 3.5, X ⊂ Z(J_{1}) ∪ Z(J_{2}). In other words, we can write X = X_{1} ∪ X_{2}, as a union of two n-varieties, where X_{1} = Z(J_{1}) ∩ X and X_{2} = Z(J_{2}) ∩ X. It remains to be shown that X_{i} ≠ X for i = 1, 2. Indeed, if say, X_{1} = X then every element of J_{1} vanishes on all of X, so that J_{1} ⊂ I_{T} (X), contradicting our assumption.

(b) ⇒ (c): Clear, since I(X) = I_{T} (X) ∩ G_{m,n}.

(c) ⇔ (d): I(X) ⊂ G_{m,n} is, by definition, a prime ideal if and only if k_{n}[X] = G_{m,n}/I(X) is a prime ring.

(c) ⇒ (a): Assume I(X) is prime and X = X_{1} ∪ X_{2} is a union of two n-varieties in U_{m,n}. Our goal is to show that X = X_{1} or X = X_{2}. Indeed, I(X_{1}) · I(X_{2}) ⊂ I(X) implies I(X_{i}) ⊂ I(X) for i = 1 or 2. Taking the zero loci and using Lemma 3.6(b), we obtain

X_{i} = Z(I(X_{i})) ⊃ Z(I(X)) = X ,

as desired. □

4.2. Proposition. Let J ∈ Spec_{n}(T_{m,n}). Then (a) Z(J) = Z(J ∩ C_{m,n})

(b) Z(J) is irreducible.

Proof. (a) Clearly Z(J ∩ C_{m,n}) ⊃ Z(J). To prove the opposite inclusion, suppose a ∈ Z(J ∩ C_{m,n}) and consider the evaluation map ϕ_{a}: T_{m,n} ⟶ M_{n} given by ϕ_{a}(p) = p(a).

Recall that ϕ_{a} is trace-preserving (see, e.g., [3, Theorem 2.2]). Since tr(j) ∈ J ∩ C_{m,n} for every j ∈ J, we see that tr(j(a)) = 0 for every j ∈ J. By Lemma 3.5 this implies that a ∈ Z(J), as claimed.

(b) Consider the categorical quotient map π: (M_{n})^{m} ⟶ Q_{m,n} for the PGL_{n}-action on (M_{n})^{m}. Recall that C_{m,n} = k[Q_{m,n}] is the coordinate ring of Q_{m,n}. Note that elements of C_{m,n} may be viewed in two ways: as regular functions on Q_{m,n} or (after composing with π) as a PGL_{n}-invariant regular function on (M_{n})^{m}. Let Y ⊂ Q_{m,n} be the zero locus of J ∩ C_{m,n} in Q_{m,n}. Then by part (a),

Z(J) = Z(J ∩ C_{m,n}) = π^{−1}(Y) ∩ U_{m,n} .

Since J is a prime ideal of T_{m,n}, J ∩ C_{m,n} is a prime ideal of C_{m,n}; see, e.g., [19, Theorem II.6.5(1)]. Hence, Y is irreducible. Now by Proposition 2.3(e), we conclude that Z(J) = π^{−1}(Y) ∩ U_{m,n} is also irreducible, as claimed. □

4.3. Corollary. If J_{0} ∈ Spec_{n}(G_{m,n}), then Z(J_{0}) is irreducible.

Proof. By Lemma 2.6, J_{0} = J ∩ G_{m,n} for some J ∈ Spec_{n}(T_{m,n}). By Lemma 3.6(a), Z(J_{0}) = Z(J), and by Proposition 4.2(b), Z(J) is irreducible. □

5. The Nullstellensatz for prime ideals

5.1. Proposition. (Weak form of the Nullstellensatz) Let A denote the algebra G_{m,n} or T_{m,n}, and let J be a prime ideal of A. Then Z(J) ≠ ∅ if and only if A/J has PI-degree n.

Note that for n = 1, Proposition 5.1 reduces to the usual (commutative) weak Nullstellensatz (which is used in the proof of Proposition 5.1). Indeed, a prime ring of PI-degree 1 is simply a nonzero commutative domain; in this case G_{m,1}/J = k[x_{1}, …, x_{m}]/J has PI-degree 1 if and only if J ≠ k[x_{1}, …, x_{m}].

Proof. First assume that Z(J) ≠ ∅. Since A has PI-degree n, its quotient A/J clearly has PI-degree ≤ n. To show PIdeg(A/J) ≥ n, recall that a point a = (a_{1}, …, a_{m}) ∈ Z(J) gives rise to a surjective k-algebra homomorphism A/J ⟶ M_{n}; see Remark 3.4.

Conversely, assume that R = A/J is a k-algebra of PI-degree n. Note that R is a Jacobson ring (i.e., the intersection of its maximal ideals is zero), and that it is a Hilbert k-algebra (i.e., every simple homomorphic image is finite-dimensional over k and thus a matrix algebra over k), see [2, Corollary 1.2]. So if c is a nonzero evaluation in R of a central polynomial for n × n-matrices, there is some maximal ideal M of R not containing c. Then R/M ≃ M_{n}, and we are done in view of Remark 3.4. □

5.2. Corollary. For any irreducible n-variety X, T_{m,n}/I_{T} (X) is the trace ring of the prime k-algebra k_{n}[X].

Proof. By Lemma 4.1 and Proposition 5.1, I_{T} (X) is a prime ideal of T_{m,n} of PI-degree n. Consequently, I(X) = I_{T} (X) ∩ G_{m,n} is a prime ideal of G_{m,n}, and the desired conclusion follows from Lemma 2.6(a). □

5.3. Proposition. (Strong form of the Nullstellensatz)

(a) I(Z(J_{0})) = J_{0} for every J_{0} ∈ Spec_{n}(G_{m,n}).

(b) I_{T} (Z(J)) = J for every J ∈ Spec_{n}(T_{m,n}).

For n = 1 both parts reduce to the usual (commutative) strong form of the Nullstellensatz for prime ideals (which is used in the proof of Proposition 5.3).

Proof. We begin by reducing part (a) to part (b). Indeed, by Lemma 2.6, J_{0} = J ∩ G_{m,n} for some J ∈ Spec_{n}(T_{m,n}). Now

I(Z(J_{0})) =^{(1)} I(Z(J)) = I_{T} (Z(J)) ∩ G_{m,n} =^{(2)} J ∩ G_{m,n} = J_{0} ,

where (1) follows from Lemma 3.6(a) and (2) follows from part (b).

It thus remains to prove (b). Let X = Z(J). Then X ≠ ∅ (see Proposition 5.1), X is irreducible (see Proposition 4.2(b)) and hence I_{T} (X) is a prime ideal of T_{m,n} (see Lemma 4.1). Clearly J ⊂ I_{T} (X); our goal is to show that J = I_{T} (X). In fact, we only need to check that

(5.4)  J ∩ C_{m,n} = I_{T} (X) ∩ C_{m,n} .

Indeed, suppose (5.4) is established. Choose p ∈ I_{T} (X); we want to show that p ∈ J. For every q ∈ T_{m,n} we have pq ∈ I_{T} (X) and thus

tr(p · q) ∈ I_{T} (X) ∩ C_{m,n} = J ∩ C_{m,n} ,

see Lemma 2.6(b). Hence, if we denote the images of p and q in T_{m,n}/J by \overline{p} and \overline{q} respectively, Lemma 2.6(a) tells us that tr(\overline{p} · \overline{q}) = 0 in T_{m,n}/J for every \overline{q} ∈ T_{m,n}/J. Consequently, \overline{p} = 0, i.e., p ∈ J, as desired.

We now turn to proving (5.4). Consider the categorical quotient map π: (M_{n})^{m} ⟶ Q_{m,n} for the PGL_{n}-action on (M_{n})^{m}; here C_{m,n} = k[Q_{m,n}] is the coordinate ring of Q_{m,n}. Given an ideal H ⊂ C_{m,n}, denote its zero locus in Q_{m,n} by

Z_{0}(H) = {a ∈ Q_{m,n} | h(a) = 0 ∀ h ∈ H} .

Since Z(J∩C_{m,n}) = Z(J) = X in U_{m,n} (see Proposition 4.2(a)), we conclude that Z_{0}(J ∩ C_{m,n}) ∩ π(U_{m,n}) = π(X). On the other hand, since π(U_{m,n}) is Zariski open in Q_{m,n} (see Lemma 2.3(d)), and J ∩ C_{m,n} is a prime ideal of C_{m,n} (see [19, Theorem II.6.5(1)]), we have

(5.5)  Z_{0}(J ∩ C_{m,n}) = \overline{π(X)} ,

where \overline{π(X)} is the Zariski closure of π(X) in Q_{m,n}. Now suppose f ∈ I_{T} (X) ∩ C_{m,n}. Our goal is to show that f ∈ J ∩ C_{m,n}. Viewing f as an element of C_{m,n}, i.e., a regular function on Q_{m,n}, we see that f ≡ 0 on π(X) and hence, on \overline{π(X)}. Now applying the usual (commutative) Nullstellensatz to the prime ideal J ∩ C_{m,n} of C_{m,n}, we see that (5.5) implies f ∈ J ∩ C_{m,n}, as desired. □

5.6. Remark. In this paper, we consider zeros of ideals of G_{m,n} in U_{m,n}. In contrast, Amitsur’s Nullstellensatz [1] (see also [2]) deals with zeros in the larger space (M_{n})^{m}. Given an ideal J of G_{m,n}, denote by Z(J; (M_{n})^{m}) the set of zeroes of J in (M_{n})^{m}. Since Z(J; (M_{n})^{m}) ⊃ Z(J), it easily follows that

J ⊂ I(Z(J; (M_{n})^{m})) ⊂ I(Z(J)) .

One particular consequence of Amitsur’s Nullstellensatz is that the first inclusion is an equality if J is a prime ideal. Proposition 5.3 implies that both inclusions are equalities, provided J is a prime ideal of PI-degree n. Note that the second inclusion can be strict, e.g., if J is a prime ideal of PI-degree < n (since then Z(J) = ∅ by Proposition 5.1).

The following theorem summarizes many of our results so far.

5.7. Theorem. Let n ≥ 1 and m ≥ 2 be integers.

(a) Z( ) and I( ) are mutually inverse inclusion-reversing bijections between Spec_{n}(G_{m,n}) and the set of irreducible n-varieties X ⊂ U_{m,n}.

(b) Z( ) and I_{T} ( ) are mutually inverse inclusion-reversing bijections

between Spec_{n}(T_{m,n}) and the set of irreducible n-varieties X ⊂ U_{m,n}.

□

6. Regular maps of n-varieties

Recall that an element g of G_{m,n} may be viewed as a regular PGL_{n}- equivariant map g: (M_{n})^{m} ⟶ M_{n}. Now suppose X is an n-variety in U_{m,n}. Then, restricting g to X, we see that g|_{X} = g′|_{X} if and only if g′ − g ∈ I(X). Hence, elements of the PI-coordinate ring of X may be viewed as PGL_{n}-equivariant morphisms X ⟶ M_{n}. All of this is completely analogous to the commutative case, where n = 1, M_{1} = k, k_{1}[X] = k[X] is the usual coordinate ring of X ⊂ k^{m}, and elements of k[X] are the regular functions on X. It is thus natural to think of elements of k_{n}[X] as “regular functions” on X, even though we shall not use this terminology. (In algebraic geometry, functions are usually assumed to take values in the base field k, while elements of k_{n}[X] take values in M_{n}.) We also remark that not every PGL_{n}-equivariant morphism X ⟶ M_{n} of algebraic varieties (in the usual sense) is induced by elements of k_{n}[X], see Example 3.3.

6.1. Definition. Let X ⊂ U_{m,n} and Y ⊂ U_{l,n} be n-varieties.

(a) A map f: X ⟶ Y is called a regular map of n-varieties, if it is of the form f = (f_{1}, …, f_{l}) with each f_{i} ∈ k_{n}[X] (so that f sends a = (a_{1}, …, a_{m}) ∈ X ⊂ (M_{n})^{m} to (f_{1}(a), …, f_{l}(a)) ∈ Y ⊂ (M_{n})^{l}). Note that a morphism of n-varieties X ⟶ Y extends to a PGL_{n}-equivariant morphism (M_{n})^{m} ⟶ (M_{n})^{l}.

(b) The n-varieties X and Y are called isomorphic if there are mutually inverse regular maps X ⟶ Y and Y ⟶ X .

(c) A regular map f = (f_{1}, …, f_{l}): X ⟶ Y ⊂ U_{l,n} of n-varieties induces a k-algebra homomorphism f^{∗}: k_{n}[Y] ⟶ k_{n}[X] given by \overline{X_{i}} ⟶ f_{i} for i = 1, …, l, where \overline{X_{1}}, …, \overline{X_{l}} are the images of the generic matrices X_{1}, …, X_{l} ∈ G_{l,n} in k_{n}[Y] = G_{l,n}/I(Y). One easily verifies that for every g ∈ k_{n}[Y], f^{∗}(g) = g ∘ f: X ⟶ Y ⟶ M_{n}.

(d) Conversely, a k-algebra homomorphism α: k_{n}[Y] ⟶ k_{n}[X] induces a regular map α_{∗} = (f_{1}, …, f_{l}): X ⟶ Y of n-varieties, where f_{i} = α(\overline{X_{i}}). It is easy to check that for every g ∈ k_{n}[Y], α(g) = g ∘ α_{∗}: X ⟶ Y ⟶ M_{n}.

6.2. Remark. It is immediate from these definitions that (f^{∗})_{∗} = f for any regular map f: X ⟶ Y of n-varieties, and (α_{∗})^{∗} = α for any k-algebra homomorphism α: k_{n}[Y] ⟶ k_{n}[X]. Note also that (id_{X})^{∗} = id_{k_{n}[X]}, and (id_{k_{n}[X]})_{∗} = id_{X}.

6.3. Lemma. Let X ⊂ U_{m,n} and Y ⊂ U_{l,n} be n-varieties, and let k_{n}[X] and k_{n}[Y] be their respective PI-coordinate rings.

(a) If f: X ⟶ Y and g: Y ⟶ Z are regular maps of n-varieties, then (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

(b) If α: k_{n}[Y] ⟶ k_{n}[X] and β: k_{n}[Z] ⟶ k_{n}[Y] are k-algebra homomorphisms, then (α ∘ β)_{∗} = β_{∗} ∘ α_{∗}.

(c) X and Y are isomorphic as n-varieties if and only if k_{n}[X] and k_{n}[Y] are isomorphic as k-algebras.

Proof. Parts (a) and (b) follow directly from Definition 6.1. The proofs are exactly the same as in the commutative case (where n = 1); we leave them as an exercise for the reader.

To prove (c), suppose f: X ⟶ Y and g: Y ⟶ X are mutually inverse morphisms of n-varieties. Then by part (a), f^{∗}: k_{n}[Y] ⟶ k_{n}[X] and g^{∗}: k_{n}[X] ⟶ k_{n}[Y] are mutually inverse k-algebra homomorphisms, showing that k_{n}[X] and k_{n}[Y] are isomorphic.

Conversely, if α: k_{n}[Y] ⟶ k_{n}[X] and β: k_{n}[X] ⟶ k_{n}[Y] are mutually inverse homomorphisms of k-algebras then by part (b), α_{∗} and β_{∗} are mutually inverse morphisms between the n-varieties X and Y. □

6.4. Theorem. Let R be a finitely generated prime k-algebra of PI-degree n. Then R is isomorphic (as a k-algebra) to k_{n}[X] for some irreducible nvariety X. Moreover, X is uniquely determined by R, up to isomorphism of n-varieties.

Proof. By our assumptions on R there exists a surjective ring homomorphism φ: G_{m,n} ⟶ R. Then J_{0} = Ker(φ) lies in Spec_{n}(G_{m,n}). Set X = Z(J_{0}) ⊂ U_{m,n}. Then X is irreducible (see Corollary 4.3), and J_{0} = I(X) (see Proposition 5.3(a)). Hence R is isomorphic to G_{m,n}/I(X) = k_{n}[X], as claimed. The uniqueness of X follows from Lemma 6.3(c). □

We are now ready to prove Theorem 1.1. Recall that for n = 1, Theorem 1.1 reduces to [13, Corollary 3.8]. Both are proved by the same argument. Since the proof of [13, Corollary 3.8] is omitted in [13], we reproduce this argument here for the sake of completeness.

Proof of Theorem 1.1. By Lemma 6.3, the contravariant functor F in Theorem 1.1 is well-defined. It is full and faithful by Remark 6.2. Moreover, by Theorem 6.4, every object in PI_{n} is isomorphic to the image of an object in Var_{n}. Hence F is a covariant equivalence of categories between Var_{n} and the dual category of PI_{n}, see, e.g., [8, Theorem 7.6]. In other words, F is a contravariant equivalence of categories between Var_{n} and PI_{n}, as claimed. □

We conclude our discussion of regular maps of n-varieties with an observation which we will need in the next section.

6.5. Lemma. Let X be an irreducible n-variety, and c a central element of k_{n}[X] or of its trace ring T_{m,n}/I_{T} (X). Then the image of c in M_{n} consists of scalar matrices.

Proof. Denote by k_{n}(X) the common total ring of fractions of k_{n}[X] and T_{m,n}/I_{T} (X). By Lemma 2.10, there exists a central polynomial s ∈ G_{m,n} for n×n matrices which does not identically vanish on X. Then R = G_{m,n}[s^{−1}] is an Azumaya algebra. Consider the natural map ϕ: R ⟶ k_{n}(X). Then the center of ϕ(R) is ϕ(Center(R)); see, e.g., [10, Proposition 1.11]. Note that k_{n}(X) is a central localization of ϕ(R). Hence the central element c of k_{n}(X) is of the form c = ϕ(p)ϕ(q)^{−1} for central elements p, q ∈ G_{m,n} with q ≢ 0 on X. All images of p and q in M_{n} are central, i.e., scalar matrices. Thus c(x) is a scalar matrix for each x in the dense open subset of X on which q is nonzero. Consequently, c(x) is a scalar matrix for every x ∈ X. □

7. Rational maps of n-varieties

7.1. Definition. Let X be an irreducible n-variety. The total ring of fractions of the prime algebra k_{n}[X] will be called the central simple algebra of rational functions on X and denoted by k_{n}(X).

7.2. Remark. One can also define k_{n}(X) using the trace ring instead of the generic matrix ring. That is, k_{n}(X) is also the total ring of fractions of T_{m,n}/I_{T} (X). Indeed, by Corollary 5.2, T_{m,n}/I_{T} (X) is the trace ring of k_{n}[X], so the two have the same total ring of fractions.

Recall that k_{n}(X) is obtained from k_{n}[X] by inverting all non-zero central elements; see, e.g., [26, Theorem 1.7.9]. In other words, every f ∈ k_{n}(X) can be written as f = c^{−1}p, where p ∈ k_{n}[X] and c is a nonzero central element of k_{n}[X]. Recall from Lemma 6.5 that for each x ∈ X, c(x) is a scalar matrix in M_{n}, and thus invertible if it is nonzero. Viewing p and c as PGL_{n}-equivariant morphisms X ⟶ M_{n} (in the usual sense of commutative algebraic geometry), we see that f can be identified with a rational map c^{−1}p: X −−→ M_{n}. One easily checks that this map is independent of the choice of c and p, i.e., remains the same if we replace c and p by d and q, such that f = c^{−1}p = d^{−1}q. We will now see that every PGL_{n}-equivariant rational map X −−→ M_{n} is of this form.

7.3. Proposition. Let X ⊂ U_{m,n} be an irreducible n-variety. Then the natural inclusion k_{n}(X) ↪ RMaps_{PGL_{n}}(X, M_{n}) is an isomorphism.

Here RMaps_{PGL_{n}}(X, M_{n}) denotes the k-algebra of PGL_{n}-equivariant rational maps X −−→ M_{n}, with addition and multiplication induced from M_{n}. Recall that a regular analogue of Proposition 7.3 (with rational maps replaced by regular maps, and k_{n}(X) replaced by k_{n}[X]) is false; see Remark 3.2 and Example 3.3.

First proof (algebraic). Recall that k_{n}(X) is, by definition, a central simple algebra of PI-degree n. By [24, Lemma 8.5] (see also [24, Definition 7.3 and Lemma 9.1]), RMaps_{PGL_{n}}(X, M_{n}) is a central simple algebra of PIdegree n as well. It is thus enough to show that the centers of k_{n}(X) and RMaps_{PGL_{n}}(X, M_{n}) coincide.

Let \overline{X} be the closure of X in (M_{n})^{m}. By [24, Lemma 8.5], the center of RMaps_{PGL_{n}}(X, M_{n}) = RMaps_{PGL_{n}}(\overline{X}, M_{n}) is the field k(X)^{PGL^{n}} of PGL_{n}- invariant rational functions f: X −−→ k (or equivalently, the field k(\overline{X})^{PGL^{n}}). Here, as usual, we identify f with f · I_{n}: X −−→ M_{n}. It now suffices to show that

(7.4)  k(X)^{PGL^{n}} = Center(k_{n}(X)) .

Recall from Lemma 2.6(a) that the natural algebra homomorphism G_{m,n} ⟶ k_{n}(X) extends to a homomorphism T_{m,n} ⟶ k_{n}(X). So the center of k_{n}(X) contains all functions f|_{X}: X ⟶ k, as f ranges over the ring C_{m,n} = k[(M_{n})^{m}]^{PGL^{n}}. Since X ⊂ U_{m,n}, these functions separate the PGL_{n}- orbits in X; see Proposition 2.3(b). Equality (7.4) now follows a theorem of Rosenlicht; cf. [18, Lemma 2.1]. □

Alternative proof (geometric). By Remark 7.2, it suffices to show that for every PGL_{n}-equivariant rational map f: X −−→ M_{n} there exists an h ∈ k[M_{n}]^{PGL^{n}} = C_{m,n} such that h ≢ 0 on X and hf: x ↦ h(x)f(x) lifts to a regular map (M_{n})^{m} ⟶ M_{n} (and in particular, hf is a regular map X ⟶ M_{n}).

It is enough to show that the ideal I ⊂ k[(M_{n})^{m}] given by

I = {h ∈ k[(M_{n})^{m}] | hf lifts to a regular map (M_{n})^{m} ⟶ M_{n}}

contains a PGL_{n}-invariant element h such that h ≢ 0 on X. Indeed, regular PGL_{n}-equivariant morphisms (M_{n})^{m} ⟶ M_{n} are precisely elements of T_{m,n}; hence, hf: X ⟶ M_{n} would then lie in k_{n}(X), and so would f = h^{−1}(hf), thus proving the lemma.

Denote by Z the zero locus of I in (M_{n})^{m} (in the usual sense, not in the sense of Definition 3.1(b)). Then Z ∩ X is, by definition the indeterminacy locus of f; in particular, X ⊄ Z. Choose a ∈ X \ Z and let C = PGL_{n} · a be the orbit of a in X. Since a ∈ X ⊂ U_{m,n}, Proposition 2.3(b) tells us that C is closed in (M_{n})^{m}. In summary, C and Z are disjoint PGL_{n}-invariant Zariski closed subsets of (M_{n})^{m}. Since PGL_{n} is reductive, they can be separated by a regular invariant, i.e., there exists a 0 ≠ j ∈ k[(M_{n})^{m}]^{PGL^{n}} such that j(a) ≠ 0 but j ≡ 0 on Z; see, e.g., [16, Corollary 1.2]. By Hilbert’s Nullstellensatz, h = j^{r} lies in I for some r ≥ 1. This h has the desired properties: it is a PGL_{n}-invariant element of I which is not identically zero on X. □

7.5. Definition. Let X ⊂ U_{m,n} and Y ⊂ U_{l,n} be irreducible n-varieties.

(a) A rational map f: X −−→ Y is called a rational map of n-varieties if f = (f_{1}, …, f_{l}) where each f_{i} ∈ k_{n}(X). Equivalently (in view of Proposition 7.3), a rational map X −−→ Y of n-varieties is simply a PGL_{n}-equivariant rational map (in the usual sense).

(b) The n-varieties X and Y are called birationally isomorphic or birationally equivalent if there exist dominant rational maps of n-varieties f: X −−→ Y and g: Y −−→ X such that f ∘ g = id_{Y} and g ∘ f = id_{X} (as rational maps of varieties).

(c) A dominant rational map f = (f_{1}, …, f_{l}): X −−→ Y of n-varieties induces a k-algebra homomorphism (i.e., an embedding) f^{∗}: k_{n}(Y) ⟶ k_{n}(X) of central simple algebras defined by f^{∗}(\overline{X_{i}}) = f_{i}, where \overline{X_{i}} is the image of the generic matrix X_{i} ∈ G_{l,n} in k_{n}[Y] ⊂ k_{n}(Y). One easily verifies that for every g ∈ k_{n}(Y), f^{∗}(g) = g∘f, if one views g as a PGL_{n}-equivariant rational map Y −−→ M_{n}.

(d) Conversely, a k-algebra homomorphism (necessarily an embedding) of central simple algebras α: k_{n}(Y) ⟶ k_{n}(X) (over k) induces a dominant rational map f = α_{∗}: X −−→ Y of n-varieties. This map is given by f = (f_{1}, …, f_{l}) with f_{i} = α(\overline{X_{i}}) ∈ k_{n}(X), where \overline{X_{1}}, …, \overline{X_{l}} are the images of the generic matrices X_{1}, …, X_{l} ∈ G_{l,n}. It is easy to check that for every g ∈ k_{n}(Y), α(g) = g ∘ α_{∗}, if one views g as a PGL_{n}-equivariant rational map Y −−→ M_{n}.

7.6. Remark. Once again, the identities (f^{∗})_{∗} = f and (α_{∗})^{∗} = α follow directly from these definitions. Similarly, (id_{X})^{∗} = id_{k_{n}(X)} and (id_{k_{n}(X)})_{∗} = id_{X}.

We also have the following analogue of Lemma 6.3 for dominant rational maps. The proofs are again the same as in the commutative case (where n = 1); we leave them as an exercise for the reader.

7.7. Lemma. Let X ⊂ U_{m_{1},n}, Y ⊂ U_{m_{2},n} and Z ⊂ U_{m_{3},n} be irreducible n-varieties.

(a) If f: X −−→ Y and g: Y −−→ Z are dominant rational maps of nvarieties then (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

(b) If α: k_{n}(Y) ↪ k_{n}(X) and β: k_{n}(Z) ↪ k_{n}(Y) are homomorphisms (i.e., embeddings) of central simple algebras then (α ∘ β)_{∗} = β_{∗} ∘ α_{∗}.

(c) X and Y are birationally isomorphic as n-varieties if and only if the central simple algebras k_{n}(X) and k_{n}(Y) are isomorphic as kalgebras. □

We are now ready to prove the following birational analogue of Theorem 6.4.

7.8. Theorem. Let K/k be a finitely generated field extension and A be a central simple algebra of degree n with center K. Then A is isomorphic (as a k-algebra) to k_{n}(X) for some irreducible n-variety X. Moreover, X is uniquely determined by A, up to birational isomorphism of n-varieties.

Proof. Choose generators a_{1}, …, a_{N} ∈ K for the field extension K/k and a K-vector space basis b_{1}, …, b_{n^{2}} for A. Let R be the k-subalgebra of A generated by all a_{i} and b_{j}. By our construction R is a prime k-algebra of PI-degree n, with total ring of fraction A. By Theorem 6.4 there exists an n-variety X such that k_{n}[X] ≃ R and hence, k_{n}(X) ≃ A. This proves the existence of X. Uniqueness follows from Lemma 7.7(c). □

8. Generically free PGL_{n}-varieties

An irreducible n-variety is, by definition, an irreducible generically free PGL_{n}-variety. The following lemma says that up to birational isomorphism, the converse is true as well.

8.1. Lemma. Every irreducible generically free PGL_{n}-variety X is birationally isomorphic (as PGL_{n}-variety) to an irreducible n-variety in U_{m,n} for some m ≥ 2.

Proof. Choose a ∈ U_{2,n}. By [24, Proposition 7.1] there exists a PGL_{n}- equivariant rational map ϕ: X −−→ (M_{n})^{2} whose image contains a. Now choose PGL_{n}-invariant rational functions c_{1}, …, c_{r} ∈ k(X)^{PGL^{n}} on X which separate PGL_{n}-orbits in general position (this can be done by a theorem of Rosenlicht; cf. e.g., [18, Theorem 2.3]). We now set m = r + 2 and define f: X −−→ (M_{n})^{m} by

f(x) = (c_{1}(x)I_{n×n}, …, c_{r}(x)I_{n×n}, ϕ(x)) ∈ (M_{n})^{r} × (M_{n})^{2} = (M_{n})^{m} .

Let \overline{Y} be the Zariski closure of f(X) in (M_{n})^{m}, and Y = \overline{Y} ∩ U_{m,n}. By our choice of ϕ, Y ≠ ∅. It thus remains to be shown that f is a birational isomorphism between X and Y (or, equivalently, \overline{Y}). Since we are working over a base field k of characteristic zero, it is enough to show that X has a dense open subset S such that f(a) ≠ f(b) for every pair of distinct k-points a, b ∈ S.

Indeed, choose S ⊂ X so that (i) the generators c_{1}, …, c_{r} of k(X)^{PGL^{n}} separate PGL_{n}-orbits in S, (ii) f is well-defined in S and (iii) f(S) ⊂ U_{m,n}. Now let a, b ∈ S, and assume that f(a) = f(b). Then a and b must belong to the same PGL_{n}-orbit. Say b = h(a), for some h ∈ PGL_{n}. Then f(a) = f(b) = hf(a)h^{−1}. Since f(a) ∈ U_{m,n}, h = 1, so that a = b, as claimed. □

Lemma 8.1 suggests that in the birational setting the natural objects to consider are arbitrary generically free PGL_{n}-varieties, rather than nvarieties. The relationship between the two is analogous to the relationship between affine varieties and more general algebraic (say, quasi-projective) varieties in the usual setting of (commutative) algebraic geometry. In particular, in general one cannot assign a PI-coordinate ring k_{n}[X] to an irreducible generically free PGL_{n}-variety in a meaningful way. On the other hand, we can extend the definition of k_{n}(X) to this setting as follows.

8.2. Definition. Let X be an irreducible generically free PGL_{n}-variety. Then k_{n}(X) is the k-algebra of PGL_{n}-equivariant rational maps f: X −−→ M_{n}, with addition and multiplication induced from M_{n}.

Proposition 7.3 tells us that if X is an irreducible n-variety then this definition is consistent with Definition 7.1. In place of k_{n}(X) we will sometimes write RMaps_{PGL_{n}}(X, M_{n}).

8.3. Definition. A dominant rational map f: X −−→ Y of generically free PGL_{n}-varieties gives rise to a homomorphism (embedding) f^{∗}: k_{n}(Y) ⟶ k_{n}(X) given by f^{∗}(g) = g ∘ f for every g ∈ k_{n}(Y).

If X and Y are n-varieties, this definition of f^{∗} coincides with Definition 7.5(c). Note that (id_{X})^{∗} = id_{k_{n}(X)}, and that (g ∘ f)^{∗} = f^{∗} ∘ g^{∗} if g: Y −−→ Z is another PGL_{n}-equivariant dominant rational map. We will now show that Definition 7.5 and Remark 7.6 extend to this setting as well.

8.4. Proposition. Let X and Y be generically free irreducible PGL_{n}-vari-eties and

α: k_{n}(X) → k_{n}(Y)

be a k-algebra homomorphism. Then there is a unique PGL_{n}-equivariant, dominant rational map α_{∗}: Y −−→ X such that (α_{∗})^{∗} = α.

Proof. If X and Y are n-varieties, i.e., closed PGL_{n}-invariant subvarieties of U_{m,n} and U_{l,n} respectively (for some m, l ≥ 2) then α_{∗}: Y −−→ X is given by Definition 7.5(d), and uniqueness follows from Remark 7.6.

In general, Lemma 8.1 tells us that there are birational isomorphisms X −−→ X′ and Y −−→ Y′ where X′ and Y′ are n-varieties. The proposition is now a consequence of the following lemma. □

8.5. Lemma. Let f: X −−→ X′ and g: Y −−→ Y′ be birational isomorphisms of PGL_{n}-varieties. If Proposition 8.4 holds for X′ and Y′ then it holds for X and Y.

Proof. Note that by our assumption, the algebra homomorphism

β = (g^{∗})^{−1} ∘ α ∘ f^{∗}: k_{n}(X′) → k_{n}(Y′)

is induced by the PGL_{n}-equivariant, dominant rational map β_{∗}: Y′ → X′:

\xymatrix{k_{n}(X) \ar[r]^{α} & k_{n}(Y) & Y \ar@{-->}[d]_{g} & X \ar@{-->}[d]^{f} \\ k_{n}(X′) \ar[r]^{β} \ar[u]^{f^{∗}} & k_{n}(Y′) \ar[u]_{g^{∗}} & Y′ \ar@{-->}[r]^{β_{∗}} & X′}

Now one easily checks that the dominant rational map

α_{∗} := f^{−1} ∘ β_{∗} ∘ g: Y −−→ X

has the desired property: (α_{∗})^{∗} = α. This shows that α_{∗} exists. To prove uniqueness, let h: Y ⟶ X be another PGL_{n}-equivariant dominant rational map such that h^{∗} = α. Then (f ∘ h ∘ g^{−1})^{∗} = (g^{−1})^{∗} ∘ α ∘ f^{∗} = β. By uniqueness of β_{∗}, we have f ∘ h ∘ g^{−1} = β_{∗}, i.e., h = f^{−1} ∘ β_{∗} ∘ g = α_{∗}. This completes the proof of Lemma 8.5 and thus of Proposition 8.4. □

8.6. Corollary. Let X, Y and Z be generically free irreducible PGL_{n}- varieties.

(a) If f: X −−→ Y, g: Y −−→ Z are PGL_{n}-equivariant dominant rational maps then (g ∘ f)^{∗} = f^{∗} ∘ g^{∗}.

(b) If α: k_{n}(Y) ↪ k_{n}(X) and β: k_{n}(Z) ↪ k_{n}(Y) are homomorphisms (i.e., embeddings) of central simple algebras then (α ∘ β)_{∗} = β_{∗} ∘ α_{∗}.

(c) X and Y are birationally isomorphic as PGL_{n}-varieties if and only if k_{n}(X) and k_{n}(Y) are isomorphic as k-algebras.

Proof. (a) is immediate from Definition 8.3.

(b) Let f = (α ∘ β)_{∗} and g = β_{∗} ∘ α_{∗}. Part (a) tells us that f^{∗} = g^{∗}. The uniqueness assertion of Proposition 8.4 now implies f = g.

(c) follows from (a) and (b) and the identities (id_{X})^{∗} = id_{k_{n}(X)}, and (id_{k_{n}(X)})_{∗} = id_{X}. □

Proof of Theorem 1.2. We use the same argument as in the proof of Theorem 1.1. The contravariant functor F is well defined by Corollary 8.6. Since (f^{∗})_{∗} = f and (α_{∗})^{∗} = α (see Proposition 8.4), F is full and faithful. By Theorem 7.8, every object in CS_{n} is isomorphic to the image of an object in Bir_{n}. The desired conclusion now follows from [8, Theorem 7.6]. □

9. Brauer-Severi Varieties

Let K/k be a finitely generated field extension. Recall that the following sets are in a natural (i.e., functorial in K) bijective correspondence:

(1) the Galois cohomology set H^{1}(K, PGL_{n}),

(2) central simple algebras A of degree n with center K,

(3) Brauer-Severi varieties over K of dimension n − 1,

(4) PGL_{n}-torsors over Spec(K),

(5) pairs (X, ϕ), where is X is an irreducible generically free PGL_{n}- variety and ϕ_{X}: k(X)^{PGL^{n}} ⟶^{≃} K is an isomorphism of fields (over k). Two such pairs (X, ϕ) and (Y, ψ) are equivalent, if there is a PGL_{n}-equivariant birational isomorphism f: Y −−→ X which is compatible with ϕ and ψ, i.e., there is a commutative diagram

\xymatrix{k(X)^{PGL^{n}} \ar[dr]^{ϕ}_{≃} \ar[rr]^{f^{∗}} &  & k(Y)^{PGL^{n}} \ar@{-->}[dl]^{≃}_{ψ} \\  & K}

Bijective correspondences between (1), (2), (3) and (4) follow from the theory of descent; see [29, Sections I.5 and III.1], [30, Chapter X], [6, (1.4)] or [14, Sections 28, 29]. For a bijective correspondence between (1) and (5), see [17, (1.3)].

For notational simplicity we will talk of generically free PGL_{n}-varieties X instead of pairs (X, ϕ) in (5), and we will write k(X)^{PGL^{n}} = K instead

of k(X)^{PGL^{n}} ≃^{ϕ} K, keeping ϕ in the background.

Suppose we are given a generically free PGL_{n}-variety X (as in (5)). Then this variety defines a class α ∈ H^{1}(K, PGL_{n}) (as in (1)) and using this class we can recover all the other associated objects (2) - (4). In the previous section we saw that the central simple algebra A can be constructed directly from X, as RMaps_{PGL_{n}}(X, M_{n}). The goal of this section is to describe a way to pass directly from (5) to (3), without going through (1); this is done in Proposition 9.2 below.

In order to state Proposition 9.2, we introduce some notation. We will write points of the projective space ℙ^{n−1} = ℙ_{k}^{n−1} as rows a = (a_{1} : …: a_{n}).

The group PGL_{n} acts on ℙ^{n−1} by multiplication on the right:

g: (a_{1} : …: a_{n}) ↦ (a_{1} : …: a_{n})g^{−1} .

Choose (and fix) a = (a_{1} : …: a_{n}) ∈ ℙ^{n−1} and define the maximal parabolic subgroup H of PGL_{n} by

(9.1)  H = {h ∈ PGL_{n} | ah^{−1} = a} .

If a = (1 : 0 : ⋯: 0) then H ⊂ PGL_{n} consists of of n × n-matrices of the form

(\begin{matrix}∗ & 0 & … & 0 \\ ∗ & ∗ & … & ∗ \\ ⋮ & ⋮ & & ⋮ \\ ∗ & ∗ & … & ∗ \end{matrix})

9.2. Proposition. Let X be an irreducible generically free PGL_{n}-variety, A = k_{n}(X) and σ: X/H −−→ X/PGL_{n}. Then the Brauer-Severi variety BS(A) is the preimage of the generic point η of X/PGL_{n} under σ.

Before we proceed with the proof, three remarks are in order. First of all, by X/H we mean the rational quotient variety for the H-action on X. Recall that X/H is defined (up to birational isomorphism) by k(X/H) = k(X)^{H}, and the dominant rational map σ: X/H −−→ X/PGL_{n} by the inclusion of fields k(X)^{PGL^{n}} ↪ k(X)^{H}. Secondly, recall that k(X/PGL_{n}) = k(X)^{PGL^{n}} = K, so that η ≃ Spec(K), and σ^{−1}(η) is, indeed, a K-variety. Thirdly, while the construction of BS(A) in Proposition 9.2 does not use the Galois cohomology set H^{1}(K, PGL_{n}), our proof below does. In fact, our argument is based on showing that σ^{−1}(η) and BS(A) are Brauer-Severi varieties defined by the same class in H^{1}(K, PGL_{n}).

Proof. Let X_{0}/k be an algebraic variety with function field k(X_{0}) = K, i.e., a particular model for the rational quotient variety X/PGL_{n}. The inclusion

K ≃^{ϕ} k(X)^{PGL^{n}} ↪ X induces the rational quotient map π: X −−→ X_{0}. After replacing X_{0} by a Zariski dense open subset, we may assume π is regular; after passing to another (smaller) dense open subset, we may assume π: X ⟶ X_{0} is, in fact, a torsor; cf., e.g., [18, Section 2.5].

We now trivialize this torsor over some etale cover U_{i} → X_{0}. Then for each i, j the transition map f_{ij}: PGL_{n} × U_{ij} ⟶ PGL_{n} × U_{ij} is an automorphism of the trivial PGL_{n}-torsor PGL_{n} × U_{ij} on U_{ij}. It is easy to see that f_{ij} is given by the formula

(9.3)  f_{ij}(u, g) = (u, g · c_{ij}(u)) ,

for some morphism c_{ij}: U_{ij} ⟶ PGL_{n}. The morphisms c_{ij} satisfy a cocycle condition (for Cech cohomology) which expresses the fact that the transition maps f_{ij} are compatible on triple “overlaps” U_{hij}. The cocycle c = (c_{ij}) gives rise to a cohomology class \overline{c} ∈ H^{1}(X_{0}, PGL_{n}), which maps to α under the natural restriction morphism H^{1}(X_{0}, PGL_{n}) ⟶ H^{1}(K, PGL_{n}) from X_{0} to its generic point; cf. [9, Section 8]. (Recall that by our construction the function field of X_{0} is identified with K.)

Now define the quotient Z of X by the maximal parabolic subgroup H ⊂ PGL_{n} as follows. Over each U_{i} set Z_{i} = H\PGL_{n} × U_{i}. By descent we can “glue” the projection morphisms Z_{i} ⟶ U_{i} into a morphism Z ⟶ X_{0} by the transition maps

\overline{f_{ij}}(u, g) = (u, \overline{g} · c_{ij}(u)) .

Moreover, since over each U_{i} the map π: X ⟶ X_{0} factors as

π_{i}: PGL_{n} × U_{i} ⟶^{p_{i}} H\PGL_{n} × U_{i} ⟶^{q_{i}} U_{i} ,

the projection maps p_{i} and q_{i} also glue together, yielding

π: X ⟶^{p} Z ⟶^{q} X_{0} .

By our construction the fibers of p are exactly the H-orbits in X; hence, k(Z) = k(X)^{H}, cf., e.g., [18, 2.1]. In other words, p is a rational quotient map for the H-action on X and we can identify Z with the rational quotient variety X/H (up to birational equivalence). Under this identification q becomes σ.

Now recall that by the definition of H, the homogeneous space H\PGL_{n} is naturally isomorphic with ℙ^{n−1} via \overline{g} ↦ g · a. Since over each U_{i} the map q: Z ⟶ X_{0} looks like the projection H\PGL_{n} × U_{i} ⟶ U_{i}, Z is, by definition, a Brauer-Severi variety over X_{0}. Moreover, π: X ⟶ X_{0} (viewed as a torsor over X_{0}) and q: Z ⟶ X_{0} (viewed as a Brauer-Severi variety over X_{0}) are constructed by using the same cocycle (c_{ij}) and hence, the same cohomology class \overline{c} ∈ H^{1}(X_{0}, PGL_{n}). Restricting to the generic point of X_{0}, we see that the cohomology class of Z as a Brauer-Severi variety over K = k(X_{0}) is the image of \overline{c} under the restriction map H^{1}(X_{0}, PGL_{n}) ⟶ H^{1}(K, PGL_{n}), i.e., the class α ∈ H^{1}(K, PGL_{n}) we started out with. □

9.4. Remark. Note that the choice of the maximal parabolic subgroup H ⊂ PGL_{n} is important here. If we repeat the same construction with H replaced by H^{transpose} we will obtain the Brauer-Severi variety of the opposite algebra A^{op}.

The following corollary of Proposition 9.2 shows that X, viewed as an abstract variety (i.e., without the PGL_{n}-action), is closely related to BS(A).

9.5. Corollary. Let X be a generically free PGL_{n}-variety, A = k_{n}(X) be the associated central simple algebra of degree n, and K = k(X)^{PGL^{n}} be the center of A. Then

k(X) ≃ K(BS(A))(t_{1}, …, t_{n^{2}−n}) ≃ K(BS(M_{n}(A))) .

Here k(X) denotes the function field of X (as a variety over k), and K(BS(A)) and K(BS(M_{n}(A)) denote, respectively, the function fields of the Brauer-Severi varieties of A and M_{n}(A) (both are defined over K). The letters t_{1}, …, t_{n^{2}−1} denote n^{2}−n independent commuting variables, and the isomorphisms ≃ are field isomorphisms over k (they ignore the PGL_{n}-action on X).

Proof. The second isomorphism is due to Roquette [25, Theorem 4, p. 413]. To show that k(X) ≃ K(BS(A))(t_{1}, …, t_{n^{2}−n}), note that by Proposition 9.2, K(BS(A)) = K(X/H) = k(X)^{H}, where H is the parabolic subgroup of PGL_{n} defined in (9.1). Since dim(H) = n^{2} − n, it remains to show that the field extension k(X)/k(X)^{H} is rational.

Now recall that H is a special group (cf. [18, Section 2.6]); indeed, the Levi subgroup of H is isomorphic to GL_{n−1}. Consequently, X is birationally isomorphic to (X/H) × H (over k). Since k is assumed to be algebraically closed and of characteristic zero, every algebraic group over k is rational. In particular, H is birationally isomorphic to 𝔸^{n^{2}−n} and thus X is birationally isomorphic to (X/H) × 𝔸^{n^{2}−n}. In other words,

k(X) ≃ K(X/H)(t_{1}, …, t_{n^{2}−n}) ≃ K(BS(A))(t_{1}, …, t_{n^{2}−n}) ,

as claimed. □

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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

E-mail address: reichst@math.ubc.ca
URL: www.math.ubc.ca/^{∼}reichst

Department of Mathematical Sciences, University of Montana, Missoula, MT 59812-0864, USA

E-mail address: Nikolaus.Vonessen@umontana.edu
URL: www.math.umt.edu/vonessen