PROJECTIVELY SIMPLE RINGS

Z. REICHSTEIN, D. ROGALSKI AND J. J. ZHANG

Abstract. We introduce the notion of a projectively simple ring, which is an infinite-dimensional ℕ-graded k-algebra A such that dim_{k} A/I < ∞ for every nonzero 2-sided ideal I ⊂ A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B = B(X, L, σ). Here X is a smooth projective variety, σ is an automorphism of X with no proper σ-invariant subvariety (we call such automorphisms wild), and L is a σ-ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if dim X ≤ 2. In the case where X is an abelian variety, we describe all wild automorphisms of X. Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then proj A is Cohen-Macaulay and Gorenstein.

Contents

0. Introduction 2 1. Elementary properties of projectively simple rings 5 2. Twisted homogeneous coordinate rings 9 3. Wild automorphisms and algebraic group actions 12 4. Wild automorphisms and numerical invariants 15 5. Wild automorphisms and the Albanese map 18 6. Wild automorphisms of algebraic surfaces 19 7. Wild automorphisms of abelian varieties 23 8. Unipotency on the Neron-Severi Group 27 9. Classification of low-dimensional cases 32 10. The Cohen-Macaulay property 33 11. The Gorenstein property 36 Acknowledgments 40 References 40

2000 Mathematics Subject Classification. 16W50, 14A22, 14J50, 14K05. Key words and phrases. Graded ring, simple ring, twisted homogeneous coordinate

ring, projective variety, algebraic surface, wild automorphism, abelian variety, dualizing complex.

Z. Reichstein was partially supported by an NSERC research grant. D. Rogalski was partially supported by NSF grant DMS-0202479. J. J. Zhang was partially supported by NSF grant DMS-0245420.

0. Introduction

Let k be a field. The present study grew out of the following question: What are the simple graded k-algebras A = ⨁_{n=0}^{∞} A_{n}? Technically, such an algebra cannot be simple, since it always has ideals of the form A_{≥n} for each n ≥ 0. Thus we are let to the following natural definition: we call A projectively simple if dim_{k} A = ∞ and every nonzero graded ideal of A has finite codimension in A. If the reference to k is clear from the context, we will sometimes refer to A as a projectively simple graded ring.

Although it is easy to see that graded prime algebras of GK-dimension 1 which satisfy a polynomial identity are projectively simple, it is not immediately obvious that there are any examples more interesting than these. In fact, one interesting example has already appeared prominently in the literature: If S is the Sklyanin algebra of dimension 3 (as defined in [SV, Example 7.3]), then S has a central element g such that the factor ring B = S/gS is a projectively simple domain of GK-dimension 2. In this case B may be constructed as a twisted homogeneous coordinate ring B = B(E, L, σ), where E is an elliptic curve and σ is a translation automorphism of E. Thus it is natural to look for other examples of projectively simple rings which also arise as twisted homogeneous coordinate rings; let us briefly review this construction.

Given X a projective scheme, L a line bundle on X, and σ an automorphism of X, set L_{n} = L ⊗ σ^{∗}L ⊗ ⋯ ⊗ (σ^{n−1})^{∗}L for each n ≥ 0. The twisted homogeneous coordinate ring associated to this data is the vector space B(X, L, σ) = ⨁_{n=0}^{∞} H^{0}(L_{n}), which has a natural graded ring structure. As is true even in the commutative case where σ = Id, the rings obtained this way are typically well behaved only with additional assumptions on the sheaf L. In particular, if L is σ-ample (see §2 below), then B = B(X, L, σ) is noetherian and its properties are closely related to the geometry of X. In this case one can describe the ideals of B geometrically and thus prove the following: Proposition 0.1. (Proposition 2.2) Let X be a projective scheme, σ an automorphism of X, and L a σ-ample line bundle. Then B = B(X, L, σ) is projectively simple if and only if σ satisfies the condition

σ(Y) ≠ Y for every nonempty reduced closed subscheme Y ⊊ X.

In the sequel we shall refer to σ satisfying this condition as a wild automorphism of X. If we seek explicit examples of projectively simple twisted homogeneous coordinate rings, then Proposition 0.1 inevitably leads us to the following purely geometric questions:

(i) Can we classify all pairs (X, σ), where X is a projective variety and σ is a wild automorphism of X?

(ii) for each (X, σ) as above, can we find σ-ample line bundles on X? In the case where X is an abelian variety, we obtain the following complete

answer to these questions. Recall that every automorphism σ of an abelian variety X can be written in the form σ = T_{b} · α, where T_{b} : x ↦ x + b is a translation by some b ∈ X and α is an automorphism of X preserving the group structure; see, for example, [La1, Theorem 4, p.24]. Theorem 0.2. (Theorem 7.2, Theorem 8.5) Let X be an abelian variety over an algebraically closed field k of characteristic zero.

(a) Suppose that T_{b} : X ⟶ X is the translation automorphism by b ∈ X, and α is an automorphism of X preserving the group structure. Then σ = T_{b} · α is a wild automorphism of X if and only if β = α − Id is nilpotent and b generates X/β(X).

(b) If σ is a wild automorphism of X, then any ample invertible sheaf on X is σ-ample.

Given an abelian variety X, we shall see in §7 below that it is easy to find many automorphisms σ satisfying the conditions in Theorem 0.2(a). Then as L varies over all ample sheaves on X, by Theorem 0.2(b) and Proposition 0.1 one gets many projectively simple twisted homogeneous coordinate rings B(X, L, σ).

We conjecture that there are no examples of wild automorphisms, other than those described in Theorem 0.2. More precisely, Conjecture 0.3. If an irreducible projective variety X admits a wild automorphism then X is an abelian variety.

In Sections 3 – 6 we prove a number of results in support in Conjecture 0.3. In particular, in Theorem 6.5 below we show that this conjecture is true if dim(X) ≤ 2.

Having constructed some prototypical examples, we would like to say more about the structure of general projectively simple rings. Because twisted homogeneous coordinate rings are quite special among all graded rings, one might expect that the examples we have constructed so far are also very special. Rather surprisingly, the following theorem states that under certain hypotheses which are natural in the theory of noncommutative projective geometry, these are essentially the only possible examples. Theorem 0.4. (Theorem 2.4) Let k be an algebraically closed field and let A be a projectively simple noetherian k-algebra. Suppose that A is strongly noetherian, generated in degree 1 and has a point module. Then up to a finite dimensional vector space, A is isomorphic to a projectively simple twisted homogeneous coordinate ring B(X, L, σ) for a smooth projective variety X with σ-ample line bundle L and wild automorphism σ.

It would be very interesting to know what kinds of projectively simple rings can appear if the various hypotheses of Theorem 0.4 are relaxed. In Example 2.5 below we show that there do exist projectively simple rings which are not strongly noetherian.

Our final class of results concerns the noncommutative projective scheme Proj A associated to a projectively simple ring A, and takes its inspiration from some known results about ungraded simple rings. It is shown in [YZ2] that a noetherian simple ring A has finite injective dimension provided that A admits a dualizing complex. This last condition is a natural one which holds for many important classes of rings such as the universal enveloping algebras of finite dimensional Lie algebras and factor rings of Artin-Schelter regular rings. Following the ideas in [YZ2], we show that some similar homological results are true in the graded setting.

Theorem 0.5. (Proposition 10.1, Theorem 11.9) Let A be a projectively simple noetherian connected graded algebra with a balanced dualizing complex. Then

(a) Proj A is classically Cohen-Macaulay, namely, the dualizing complex for Proj A is isomorphic to ω[n] for a graded A-bimodule object ω in Proj A.

(b) Proj A is Gorenstein, namely, the dualizing bimodule ω in part (a) is invertible. As a consequence, the structure sheaf A has finite injective dimension.

If A satisfies the hypotheses of Theorem 0.5 and also A = B(X, L, σ) for some σ-ample L, then the conclusions of Theorem 0.5 are immediate, since QGr A is equivalent to the category of quasi-coherent sheaves on X and X is smooth by Lemma 3.1(b) below. So Theorem 0.5 is aimed primarily at projectively simple rings that do not satisfy the conditions of Theorem 0.4.

To conclude this introduction, we would like to say a bit about our motivation for introducing and studying projectively simple rings. Let us describe a potential application to the theory of GK-dimension; see [KL] for an introduction to this subject. It is known that no algebra may have a non-integer value of GK-dimension between 0 and 2. Moreover, it is conjectured that there does not exist a noetherian connected graded domain with GK-dimension strictly between 2 and 3 [AS1, p. 2]. Note that if A is a Goldie prime ring and I is a nonzero ideal of A, then GK(A/I) ≤ GK(A)−1 [KL, Proposition 3.15]. Thus if A is a connected graded noetherian prime ring with 2 < GK(A) < 3, then either A has a height one prime P with GK(A/P) = 1, or else A is projectively simple. A proof that projectively simple rings have integer GK-dimension would eliminate one of these cases, thus making progress towards the proof of the full conjecture. The results of this paper do at least show that a projectively simple ring which satisfies the hypotheses of Theorem 0.4 must have integer GK-dimension, since this is true of twisted homogeneous coordinate rings.

Another application is to the classification of graded rings of low GKdimension. Artin and Stafford have classified semiprime graded rings of GKdimension 2 in terms of geometric data in [AS1], [AS2]. The classification of rings of GK-dimension 3, which correspond to noncommutative surfaces, is a subject of much current interest. Those rings of GK-dimension 3 which are also projectively simple represent a special subclass. If such a ring satisfies all of the hypotheses of Theorem 0.4 above, then it must be a twisted homogeneous coordinate ring in large degree. We classify all such twisted homogeneous coordinate rings in Proposition 9.2 below.

1. Elementary properties of projectively simple rings

Throughout this paper k is a commutative base field, and all rings will be k-algebras. In Section 2 we will assume that k is algebraically closed, and from Section 3 to Section 9 we will assume that k is algebraically closed and that char k = 0. An algebra A is ℕ-graded (or graded) if A = ⨁_{i≥0} A_{i} with 1 ∈ A_{0} and A_{i}A_{j} ⊆ A_{i+j} for all i, j ≥ 0. The graded algebra A is locally finite if dim_{k} A_{i} < ∞ for all i. All algebras A in this article will be graded and locally finite, except when we consider localizations of A and in other obvious situations. If A_{0} = k, then A is called connected graded. Let 𝔪 denote the graded ideal A_{≥1}.

We recall from the introduction the property which is the subject of this article:

Definition 1.1. A locally finite graded algebra A is called projectively simple if dim_{k} A = ∞ and dim_{k} A/I < ∞ for every nonzero graded ideal I of A.

It is useful to notice that since A is locally finite, the condition dim_{k} A/I < ∞ for a graded ideal I is equivalent to the condition I ⊃ A_{≥n} for some n.

Our first result summarizes some easy observations about projectively simple rings. The proofs are straightforward, and we omit them.

Lemma 1.2. Let A be a graded ring such that 𝔪 ≠ 0.

(a) If A is projectively simple, then A is a finitely generated k-algebra.

(b) If A is projectively simple, then A is prime. If A is also connected, then the only nonzero graded prime ideal of A is 𝔪.

(c) If A is PI and finitely generated, then A is projectively simple if and only if A is prime of GK-dimension 1.

(d) If A is projectively simple, then dim_{k} A/J < ∞ for every (not necessarily graded) nonzero ideal J of A.

Before pursuing further the abstract notion of a projectively simple ring, we should note that there do exist examples besides the trivial ones in Lemma 1.2(c) above. Let us mention here just one important example, which will turn out to be the model for all of our subsequent examples.

Example 1.3. Let E be an elliptic curve and let σ be an translation automorphism of E given by the rule x ↦ x+a where a is a point on E of infinite order. Let L be a very ample line bundle of E. Then the twisted homogeneous coordinate ring B(E, L, σ) (see §2 for the definition) is a projectively simple ring of GK-dimension 2 (see [AS1, 6.5(ii)] for a proof).

We can also give a non-constructive proof of the existence of projectively simple factor rings of quite general graded algebras.

Lemma 1.4. Let A be finitely generated, connected graded and infinite dimensional over k. Then there is a graded ideal J ⊂ A such that A/J is projectively simple.

Proof. Let Φ be the set of all graded ideals I ⊂ A such that dim_{k} A/I = ∞. Suppose that {I_{α}}_{α∈S} is a subset of Φ consisting of an ascending chain of (possibly uncountably many) ideals. We claim that the union

U = ⋃_{α∈S} I_{α}

is in Φ. Suppose to the contrary that U is not in Φ. Then U contains A_{≥n} for some n. Let A be generated by elements of degree no more than d and let V be the finite dimensional vector space ⨁_{i=0}^{d} A_{n+i}. Then V ⊂ U implies that V ⊂ I_{α_{0}} for some α_{0} since {I_{α}}_{α∈S} is an ascending chain. Hence I_{α_{0}} contains A_{≥n} since A is generated by elements of degree no more than d. This yields a contradiction, whence U is in Φ. By Zorn’s lemma, the set Φ has a maximal object, say J. Then A/J is projectively simple. □

Next we will show that projective simplicity is preserved under base field extensions. Lemma 1.5. Let k ⊂ L be an extension of fields, with k algebraically closed. Then A is a projectively simple k-algebra if and only if A⊗_{k}L is a projectively simple L-algebra.

Proof. If R is any commutative k-algebra and V is any vector space over k, we write V_{R} for V ⊗_{k} R. Suppose that A is not projectively simple, but rather contains a nonzero ideal I with dim_{k} A/I = ∞. Then I_{L} is a nonzero ideal of A_{L} and dim_{L} A_{L}/I_{L} = ∞, so A_{L} is not projectively simple.

Now assume that A_{L} is not projectively simple, and let us show that A is not projectively simple. By assumption A_{L} has some nonzero ideal I such that A_{L}/I is not finite-dimensional. We may assume that I = (g) is generated by one nonzero homogeneous element g ∈ A_{L}. Write g = ∑_{i=0}^{q} b_{i}x_{i} where 0 ≠ b_{i} ∈ L and x_{i} ∈ A. We may choose q as small as possible so that {x_{0}, …, x_{q}} are linearly independent over k. Replacing g by b_{0}^{−1}g we may assume that b_{0} = 1. Let R = k[b_{1}, …, b_{q}] ⊂ L be the commutative affine k-algebra generated by the {b_{i}}. Let J be the ideal of A_{R} generated by g. Then (A_{R}/J) ⊗_{R} L ≅ A_{L}/I. Necessarily the degree n part (A_{R}/J)_{n} is not R-torsion for all n in an infinite subset Φ ⊂ ℤ. Now let θ : R → k be any k-algebra homomorphism (some such exists by the Nullstellensatz since k is algebraically closed), and extend it to a map θ : A_{R} → A_{k} = A. Then letting k be an R-module via θ, we have

A′ = (A_{R}/J) ⊗_{R} k ≅ A/(θ(J)).

For each n ∈ Φ, A′_{n} ≠ 0 since (A_{R}/J)_{n} is not R-torsion. Hence A′ has infinite k-dimension. Now θ(g) = x_{0} + ∑_{i=1}^{q} θ(b_{i})x_{i} is nonzero in θ(J) since {x_{0}, ⋯, x_{q}} is linearly independent over k. Therefore A is not projectively simple. □

One might ask if Lemma 1.2(a) could be strengthened to say that a projectively simple ring is necessarily noetherian. Using Lemma 1.4 we can see that the answer is no.

Example 1.6. Let k be a countable field. By the Golod-Shafarevitch construction [He, Chapter 8], there is a finitely generated connected graded k-algebra A such that dim_{k} A = ∞ and every element in A_{≥1} is nilpotent. By Lemma 1.4, there is a graded ideal J such that B := A/J is projectively simple, whence prime. Clearly B_{≥1} is a nil ideal, in other words every element is nilpotent, but B_{≥1} is not nilpotent since B is prime. By [MR, 2.3.7], B is neither left nor right noetherian.

Combining this with Lemma 1.5 one sees that if L is a field containing the algebraic closure of its prime field, then there exists a non-noetherian projectively simple algebra over L.

Now we discuss some further definitions which will be useful in the remainder of this section, and in §10-11. Let A be a noetherian graded ring, and let M and N be graded right A-modules. The group of module homomorphisms preserving degree is denoted Hom_{A}(M, N). Let N(n) denote the nth degree shift of the module N. We write

\underline{Hom}_{A}(M, N) = ⨁_{n∈ℤ}Hom_{A}(M, N(n)),

which is the group of all module homomorphisms from M to N in case M is finitely generated.

We say that a graded right module M is torsion if for every x ∈ M there is an n such that xA_{≥n} = 0. A noetherian module is torsion if and only if it is finite dimensional over k. We will also use a different kind of torsion property. A right A-module M is called Goldie torsion if for every element x ∈ M, the right annihilator r. ann(x) is an essential right ideal of A. Let Kdim M denote the Krull dimension of M; if A is prime then Kdim M < Kdim A if and only if M is Goldie torsion. We call M torsionfree (respectively, Goldie torsionfree) if it does not contain a nonzero torsion (respectively, Goldie torsion) submodule.

Let A and B be graded rings. An (A, B)-bimodule is called noetherian if it is noetherian on both sides. Note that if M is a noetherian (A, B)- bimodule, then the largest torsion left submodule τ_{A}(M) of M and the largest torsion right submodule τ_{B}(M) of M coincide, both being equal to the largest finite-dimensional sub-bimodule of M. So we simply write τ(M) for this module and call it the torsion submodule of M. The bimodule M/τ(M) is torsionfree on both sides.

The following useful result shows how the projectively simple property gives strong information about the structure of graded bimodules. In particular, the two different notions of torsion defined above actually coincide for noetherian bimodules over projectively simple rings.

Lemma 1.7. Let A be a noetherian projectively simple ring.

(a) Let B be any graded ring. If M is a noetherian (B, A)-bimodule such that M_{A} is Goldie torsion, then M is finite dimensional, whence torsion over A.

(b) Let M and N be noetherian graded (A, A)-bimodules such that M_{A} and N_{A} are not torsion. Then dim_{k} \underline{Hom}_{A}(M_{A}, N_{A}) = ∞.

Proof. (a) Since _{B}M is noetherian, M = ∑_{i} Bx_{i} for a finite set of elements {x_{i}} ⊂ M. Thus

I := r. ann(M_{A}) = ⋂_{i} r. ann(x_{i}) ≠ 0,

where the final inequality follows from the fact that every r. ann(x_{i}) is essential. Thus M_{A} is finitely generated over a finite dimensional algebra A/I. Therefore M is finite dimensional.

(b) By part (a), M_{A} and N_{A} are not Goldie torsion. Since A is a prime Goldie ring, by replacing M_{A} by a factor module and N_{A} by a submodule, we may assume that both M and N are uniform Goldie torsionfree graded right A-modules, or homogeneous right ideals of A. Since A is prime, N_{≥n}M ≠ 0 for all n. Hence for each n there is a homogeneous element x ∈ N_{≥n} such that the left multiplication

l_{x} : m → xm for all m ∈ M

is a nonzero element in \underline{Hom}_{A}(M_{A}, N_{A}). Thus (b) follows. □

In the final result of this section, we note some circumstances under which the property of projective simplicity passes from one algebra to closely related algebras. Lemma 1.8. Let A be a noetherian graded prime ring.

(a) Suppose that B is a graded subring of A such that dim_{k} A/B < ∞. Then A is projectively simple if and only if B is.

(b) Suppose that B is a projectively simple graded subring of a graded Goldie prime ring A such that A_{B} is finitely generated. Then A is projectively simple.

(c) Suppose that A is projectively simple, and let B = A^{(n)} be the nth Veronese subring ⨁_{i=0}^{∞} A_{ni} for some n ≥ 2. If B is prime and A_{B} (or _{B}A) is finitely generated, then B is projectively simple.

Proof. (a) Suppose that A is not projectively simple. Let 0 ≠ I be an ideal of A with dim_{k} A/I = ∞. Then J = I ∩ B is a nonzero ideal of B with dim_{k} B/J = ∞ and B is not projectively simple.

Conversely, suppose that A is projectively simple and let 0 ≠ J be an ideal of B. Since A is projectively simple, dim_{k} A/(A_{≥n}JA_{≥n}) < ∞. Also A_{≥n}JA_{≥n} ⊆ BJB = J for some n, so dim_{k} B/J < ∞ and B is projectively simple.

(b) Let I be a nonzero graded ideal of A. We want to show that I ∩ B is nonzero. Since A is Goldie prime, I contains a homogeneous regular element of A and thus

Kdim(A/I)_{B} < Kdim A_{B} = Kdim B.

Hence the map B → A/I cannot be injective. Thus B ∩I ≠ 0 and B/(B ∩I) is finite dimensional because B is projectively simple. Now A/I is finitely generated over B/(B ∩ I), so it is also finite dimensional.

(c) We think of B as a subring of A which is zero except in degrees which are multiples of n. By [AZ1, 5.10(1)], B is noetherian. Let J be any nonzero right ideal of A. Then we claim that J ∩ B ≠ 0. Suppose this is not true; then if 0 ≠ x ∈ J is any homogeneous element, then x^{n} ∈ J ∩ B = 0. Thus J is a right nil ideal, and so J is a nilpotent ideal since A is noetherian [MR, 2.3.7]. Then since A is prime, J = 0, a contradiction.

Now let I be a nonzero graded ideal of B. Since B is prime, I contains a homogeneous regular element x. If J = r. ann_{A} x, then r. ann_{B} x = J ∩B = 0 since x is regular in B; then by the claim above, J = 0 and x is regular in A. Then A/IA is a noetherian (B, A)-bimodule which is Goldie torsion as a right A-module. By Lemma 1.7(a), dim_{k} A/IA < ∞. Since I = IA ∩ B, we conclude that dim_{k} B/I < ∞. □

The result of Lemma 1.8(c) is false without the prime hypothesis on B, as is clear from the following example. Example 1.9. Let A = k⟨x, y⟩/(x^{2}, y^{2}). Then A is a PI prime ring of GKdimension one, so A is projectively simple by Lemma 1.2(c). The Veronese subring A^{(2)} is isomorphic to k[u, v]/(uv) where u = xy and v = yx. Hence A^{(2)} is semiprime, but not prime, so it cannot be projectively simple.

2. Twisted homogeneous coordinate rings

Starting now we work towards the goal of producing some interesting explicit examples of projectively simple rings. For this we take Example 1.3 as our model; we expect to find other projectively simple rings by looking at the class of twisted homogeneous coordinate rings, which we define and study in this section. We will also prove Theorem 0.4, which will show that under certain hypotheses, twisted homogeneous coordinate rings really are the only examples of projectively simple rings.

Assume throughout this section that k is an algebraically closed field. Let X be a commutative projective scheme, σ an automorphism of X and L an invertible sheaf on X. For any sheaf F on X, we use the notation F^{σ} for the pullback σ^{∗}(F). Now set

L_{n} = L ⊗ L^{σ} ⊗ ⋯ ⊗ L^{σ^{n−1}}

for all n ≥ 0. The twisted homogeneous coordinate ring (or twisted ring for short) B = B(X, L, σ) is defined to be the graded vector space ⨁_{n=0}^{∞} H^{0}(L_{n}), with the multiplication rule

fg = f ⊗ g^{σ^{m}} for f ∈ B_{m}, g ∈ B_{n}.

For more details about this construction see [AV] and [Ke].

The sheaf L is called σ-ample if for any coherent sheaf F on X, H^{i}(F ⊗ L_{n}) = 0 for all i > 0 and n ≫ 0. This reduces to the usual notion of ampleness in the commutative case when σ is the identity. In case L is σ-ample, the ring B = B(X, L, σ) is noetherian and there are many nice relationships between the properties of B and the properties of X. For example, in this case there is an equivalence of categories QGr B ≃ Qch X, where Qch X is the category of quasi-coherent sheaves on X and Proj B = (QGr B, πB) is the noncommutative projective scheme associated to B (see §10).

We now give in the next proposition a geometric characterization of which rings B(X, L, σ) are projectively simple, which proves Proposition 0.1 from the introduction. The answer involves the following geometric notion.

Definition 2.1. Let σ be an automorphism of a projective scheme X. Then we call σ wild if σ(Y) ≠ Y for every nonempty reduced closed subscheme Y ⊊ X.

Proposition 2.2. Let B = B(X, L, σ), where L is σ-ample. Then B is projectively simple if and only if σ is a wild automorphism of X.

Proof. Call two right ideals J, J′ of B equivalent if J_{≥n} = J′_{≥n} for some n ≥ 0. By [AS1, 4.4], the mapping I ↦ ⨁_{n≥0} H^{0}(I ⊗ L_{n}) gives a bijective correspondence between ideal sheaves I of X which are σ-invariant (in other words I^{σ} ≅ I), and equivalence classes of two-sided ideals of B.

Note that if I is a σ-invariant ideal sheaf which defines a closed subscheme Y of X, then the ideal sheaf I′ defining the reduction Y_{red} of Y is also σinvariant. Moreover, if Y is already reduced, then its defining ideal sheaf I is σ-invariant if and only if σ(Y) = Y. Thus B is projectively simple if and only if the only reduced subschemes Y of X with σ(Y) = Y are ∅ and X. □

Next, we want to prove Theorem 0.4, which allows us to say that a projectively simple ring with certain other hypotheses must be a twisted homogeneous coordinate ring in large degree. Our proof will largely rely on results from the literature. For the reader’s convenience, we will summarize the main ideas involved, but without full details.

Let A be a connected finitely ℕ-graded k-algebra. An R-point module for A, where R is a commutative k-algebra, is a cyclic graded right A ⊗_{k} Rmodule M, generated in degree 0, such that M_{0} ≅ R and M_{n} is a locally free R-module of rank 1 for each n ≥ 0. The algebra A is called strongly noetherian if for every commutative noetherian k-algebra C, the ring A ⊗_{k} C is noetherian. If A is strongly noetherian, then by work of Artin and Zhang [AZ2] there exists a projective scheme X which parametrizes the point modules for A. More exactly, if we associate to each commutative k-algebra R the set of all isomorphism classes of R-point modules for A, then this rule defines a functor Rings → Sets and X represents this functor.

Now let A be strongly noetherian and generated in degree 1. For each base ring R one gets a map from the set of R-point modules to itself which is defined by the rule M ↦ M_{≥1}(1), and these maps induce an automorphism σ of the representing scheme X [KRS, Proposition 10.2]. Then using the same methods as in [ATV], there is a natural way to construct a ring homomorphism ϕ : A → B(X, L, σ) where L is a very ample invertible sheaf on X. In [RZ], the authors prove the following additional facts about such maps ϕ: Proposition 2.3. [RZ] Let A be a strongly noetherian connected graded ring generated in degree 1. Let X be the scheme parametrizing the point modules, and let σ be the automorphism of X induced by the map M ↦ M_{≥1}(1) on point modules. Then there is a graded ring homomorphism ϕ : A → B(X, L, σ) which is surjective in large degree, and such that L is σ-ample and very ample.

Given the preceding proposition, it is now easy to show the following structure result for projectively simple rings, which proves Theorem 0.4 from the introduction. Theorem 2.4. Let A be a connected ℕ-graded k-algebra over an algebraically closed field k. Assume that A is generated in degree 1, strongly noetherian over k, and that some k-point module exists for A. Then if A is projectively simple, there is an injective ring map ϕ : A → B = B(X, L, σ), where L is σ-ample, and dim_{k} B/A < ∞. The ring B is also projectively simple and σ is a wild automorphism of X.

Proof. Let ϕ be the ring homomorphism of Proposition 2.3 above. By that proposition we have im ϕ ⊃ B_{≥m} for all m ≫ 0. Since we assumed that A has a point module, the scheme X representing the point modules for A must be nonempty. Then since L is σ-ample and X is non-empty, the sheaf L_{n} has some nonzero global section for all n ≫ 0 [Ke, Proposition 2.3], and so B_{n} ≠ 0 for all n ≫ 0. Then im ϕ is not finite dimensional, so ker ϕ is an ideal of A with dim_{k} A/ ker ϕ = ∞. Since A is projectively simple, ker ϕ = 0 and thus also dim_{k} B/A < ∞. The ring A is also prime by Lemma 1.2(b), and B must be prime as well. Then B is projectively simple by Lemma 1.8(a). The automorphism σ is wild by Proposition 2.2. □

Because of the preceding result, we will focus the majority of our attention on twisted homogeneous coordinate rings in the remaining sections of the paper. We are very curious, though, what kinds of more general projectively simple rings may appear if the hypotheses of Theorem 2.4 are relaxed. We wonder how stringent the hypothesis that A has a point module is; do there exist, for example, any projectively simple rings which have no finitely generated modules of GK-dimension one?

We can offer an example of a projectively simple ring which is not strongly noetherian. The example is not too far away from the strongly noetherian case, since it has a closely related overring which is strongly noetherian and projectively simple. It also has a large supply of point modules. Again, it would be interesting to know if there are non-strongly noetherian examples of projectively simple rings which are significantly different from this one. Example 2.5. Let k be algebraically closed of characteristic zero. Let B = B(X, L, σ) where X is integral (so B is a domain), and L is σ-ample and very ample. We construct special subrings of B which are “Naive noncommutative blowups” as studied in [KRS]. Let c ∈ X be a smooth closed point, with corresponding ideal sheaf I. Setting L_{n} = L ⊗ L^{σ} ⊗ ⋯ ⊗ L^{σ^{n−1}} and I_{n} = II^{σ} … I^{σ^{n−1}} for each n ≥ 0, we define the ring

R = R(X, L, σ, c) = ⨁_{n≥0} H^{0}(I_{n} ⊗ L_{n}) ⊂ B(X, L, σ).

Now assume in addition that X is an abelian variety, and that the point c ∈ X generates X in the sense that the subgroup C = {nc|n ∈ ℤ} of X is a Zariski dense set. Let σ be the translation automorphism x ↦ x+c of X. In Theorem 7.2 below we will prove that σ is wild, and so by Proposition 2.2, B = B(X, L, σ) is projectively simple. As in [KRS, Example 11.5], it follows that the set C is critically dense, which means that every infinite subset of it is dense. Then R is a noetherian domain by [KRS, Theorem 4.1].

Now let 0 ≠ P be a homogeneous prime ideal of R. It follows from [KRS, Theorem 4.1 and Lemma 6.1] that the R-module (R/P) has a compatible B-structure in large degree. More exactly, there is some n ≥ 0 and a left B-module M such that _{R}M ≅ _{R}(R/P)_{≥n}. Now either P = R_{≥1}, or else P = ann_{R}(R/P)_{≥n} = (ann_{B} M) ∩ R. In either case, P is the intersection with R of an ideal Q of B. Since B is projectively simple, dim_{k} B/Q < ∞, and so dim_{k} R/P < ∞. Thus R is projectively simple. However, R is not strongly noetherian [KRS, Theorem 9.2].

3. Wild automorphisms and algebraic group actions

As we saw in the preceding section, to study projectively simple twisted homogeneous coordinate rings we need to better understand wild automorphisms. This will be the subject of sections 3 – 7. We begin with several simple observations. Lemma 3.1. Let X be a projective scheme.

(a) If σ is a wild automorphism of X, then X is reduced.

(b) If σ is a wild automorphism, then X is smooth.

(c) Assume that X is reduced with irreducible components X_{1}, X_{2}, … X_{m}. Then σ is a wild automorphism of X if and only if the permutation of the X_{i} induced by σ is a single m-cycle, and σ^{m} restricts to a wild automorphism of each X_{i}. In this case, X is a disjoint union of X_{1}, ⋯, X_{m}.

(d) If X is integral, then an automorphism σ of X is wild if and only if σ^{n} is wild for all n ≥ 1.

Proof. (a) X_{red} is a non-trivial subscheme of X preserved by σ; hence, X = X_{red}.

(b) By part (a), Y is reduced. Let Y be the singular locus of X. Then Y is closed, σ-invariant and Y ≠ X. Since σ is wild, Y is empty.

(c) The orbit of each component is preserved by σ; hence, there can only be one such orbit. Since the subscheme

Y = ⋃_{i≠j} X_{i} ∩ X_{j} ⊊ X

is σ-invariant, it has to be empty, i.e., X_{1}, …, X_{m} are disjoint. If σ^{m}(Z) = Z for some subscheme Z ⊊ X_{i}, then σ(Z′) = Z′ where Z′ = ⋃_{j=0}^{m−1} σ^{j}(Z) ⊊ X; thus Z = ∅ and so σ^{m}|_{X_{i}} is wild.

(d) If σ preserves a subscheme Y ⊂ X, then so does σ^{n}. Conversely, if σ^{n} preserves Y ⊊ X then σ preserves ⋃_{j=0}^{n−1} σ^{j}(Y) ⊊ X. □

By part (d) of the Lemma, it is clear that we lose nothing essential in our understanding of wild automorphisms by restricting to the case where X is integral, and we will usually do so.

thus

The rest of this section will be devoted to proving Conjecture 0.3 under the additional assumption that σ can be extended to a finite-dimensional family of automorphisms (or even of birational isomorphisms) of X. Before proceeding, let us make a few remarks about our terminology. In order to avoid arithmetic complications (in what is already a difficult geometric problem), we shall assume throughout that k is an algebraically closed field of characteristic zero. For convenience, we often use the language of varieties rather than schemes; for us, a variety is a (not necessarily irreducible) reduced separated scheme of finite type over k. For a variety X the notation p ∈ X always means that p is a closed point of X. By an algebraic group we mean a variety with a compatible group structure. Note that we do not assume that algebraic groups are linear, but by definition they must have finitely many components. If G is an algebraic group, then for each h ∈ X one has a translation automorphism T_{h} defined by g ↦ hg. An abelian variety is a complete irreducible variety X over k which is also an algebraic group. Proposition 3.2. Let X be an irreducible projective variety. Suppose that an algebraic group G ⊂ Aut(X) acts regularly on X such that σ ∈ G acts by a wild automorphism. Then X is an abelian variety and some power of σ is a translation automorphism after a group structure in X is chosen properly.

Proof. First we make some reductions. Let H be the closure in G of the subgroup {σ^{i}}_{i∈ℤ}. Then H is an algebraic group. Since H is a closure of an abelian group, H is abelian. Without loss of generality, we may assume from now on that G = H, so in particular G is abelian. Since G has finitely many components, σ^{n} must lie in the connected component G_{e} containing the identity element of G, for some n ≥ 1. Since X is irreducible, σ^{n} is also wild (Lemma 3.1(d)). Thus we may replace σ by σ^{n} and G by G_{e}, and so we may assume that G is irreducible.

Choose any x ∈ X, and let Y be the closure of Gx in X. Since σ ∈ G, σ(Gx) ⊂ Gx and hence σ(Y) ⊂ Y. Since σ is wild, Y = X. Now the rule g ↦ gx defines a morphism f : G → X. By Chevalley’s theorem [Hart, Exercise II.3.19], the image Gx of the map f must be constructible, and so it must contain an open subset U of its closure Y = X. Hence Gx = GU = ⋃_{g∈G} gU is dense and open in X. Since Gx is σ-stable, Z := X − Gx is a σ-stable closed subvariety. Since σ is wild, we conclude that Z is empty and thus X = Gx.

Next, let G_{0} be the stabilizer of x. Since G is abelian, G_{0} is the stabilizer of every point in Gx = X. Since G is a subgroup of Aut(X) and X is a variety over an algebraically closed field, automorphisms of X are determined by their actions on closed points and so G_{0} is trivial. Hence the morphism f : G → X is bijective.

Since our standing assumption is that the base field k has characteristic 0, it now follows from [Hum, Theorem 4.6] that f is birational. Let V be the largest open set of X such that f|_{f^{−1}(V)} : f^{−1}(V) → V is an isomorphism. Then V is σ-invariant, and since σ is wild, we see that V = X and f is an isomorphism. Therefore the group structure of G may be transferred to X via f, G must be projective, and both X and G are abelian varieties. The isomorphism f transforms the translation automorphism T_{σ} of G to the translation automorphism T_{f(σ)} = σ of X. Since we replaced σ by σ^{n} during the proof, we see that some power of the original σ is a translation automorphism of X as required. □

The previous theorem can be generalized to the case of birational actions. Let Bir(X) be the group of all birational maps over k from X to itself.

Corollary 3.3. Let X be an irreducible projective variety admitting a wild automorphism σ. Suppose that P ⊂ Bir(X) is an algebraic group that acts birationally on X such that σ ∈ P. Then X is an abelian variety and some power of σ is a translation after a group structure in X is chosen properly.

Proof. Let G = {τ | στ = τσ} be the centralizer of σ in P. Since P is an algebraic group, so is G. Clearly σ ∈ G. We want to show that G ⊂ Aut(X). For any τ ∈ G, let Y_{τ} be the indeterminacy locus of the rational map τ: X −−→ X. Since στ = τσ, Y_{τ} is σ-stable. Since σ is wild, Y_{τ} is empty, whence τ is regular. Similarly, τ^{−1} is regular. Therefore G ⊂ Aut(X). The assertion now follows from Theorem 3.2. □

In Remark 8.3 below, we will see that the conclusions of Theorem 3.2 and Corollary 3.3 may be strengthened to say that σ is, in fact, itself a translation automorphism.

4. Wild automorphisms and numerical invariants

In this section we will show that the existence of a wild automorphism imposes strong restrictions on two important numerical invariants of a projective variety, the Kodaira dimension and the Euler characteristic. We will continue to assume that the base field k is algebraically closed and of characteristic zero.

We begin by considering the Kodaira dimension. Suppose that X is a smooth projective variety, and let ω_{X} be the canonical sheaf. For a sheaf F on X we write Γ(X, F) for the global sections of F. The Kodaira dimension (or canonical dimension) of X, denoted by κ(X), is defined to be the transcendence degree of the canonical ring ⨁_{i≥0} Γ(X, ω_{X}^{⊗i}) minus 1. Similarly, the anti-canonical dimension κ̄(X) of X is the transcendence degree of the anti-canonical ring ⨁_{i≥0} Γ(X, ω_{X}^{⊗−i}) minus 1. We will see next that the existence of a wild automorphism limits the possible values of κ and κ̄.

Let L be an invertible sheaf on a projective variety X and let σ be an automorphism of X. A σ-linearization of L is an isomorphism

p: σ^{∗}(L) ⟶^{≃} L .

Informally speaking, we can think of p as a way to lift σ from X to L. If p is chosen, we shall say that L is σ-linearized and identify σ^{∗}(L) and L via p. Proposition 4.1. Suppose an irreducible projective variety X admits a wild automorphism σ. If L is a σ-linearized invertible sheaf on X then either

(a) L is isomorphic to O_{X}, or

(b) Γ(X, L) = 0 = Γ(X, L^{−1}).

Proof. Suppose that (b) fails, in other words that either L or L^{−1} has a nonzero global section. Note that a σ-linearization of L induces a σlinearization of L^{−1}. Thus after switching the roles of L and L^{−1} if necessary, we may assume that V := Γ(X, L) ≠ (0).

Since L is linearized, pullback by σ induces an automorphism σ̃ = p ∘ σ^{∗} of V. Since k is algebraically closed, V must have an eigenvector f for the action of σ̃. The vanishing set Z(f) ⊊ X of the global section f is then fixed by σ. Since σ is wild, Z(f) = ∅. This implies that L is generated by the global section f, i.e., the map i : O_{X} → L sending 1 to f is an isomorphism. □

Corollary 4.2. Suppose an irreducible projective variety X of dimension d ≥ 1 admits a wild automorphism. Then either (i) κ(X) = κ̄(X) = −1 or (ii) ω_{X}^{⊗n} ≅ O_{X} for some n ≥ 1. (In the latter case κ(X) = κ̄(X) = 0.) In particular, X cannot be a Fano variety or a variety of general type.

Proof. Apply Proposition 4.1 with L = ω_{X}^{⊗n}. If (a) holds for some n ≥ 1 then ω_{X}^{⊗n} ≅ O_{X}; if instead (b) holds for all n ≥ 1 then κ(X) = κ̄(X) = −1.

The last assertion is an immediate consequence of the first: if X is Fano then κ̄(X) = d ≥ 1 and if X is of general type then κ(X) = d ≥ 1. □

Corollary 4.3. Let X be an abelian variety, and suppose that Y ⊆ X is an irreducible subvariety of X such that Y admits a wild automorphism. Then Y is a translate of an abelian subvariety of X.

Proof. By Corollary 4.2, since Y has a wild automorphism it follows that κ(Y) ≤ 0. Then a theorem of Ueno, which is proved for the case k = ℂ in [U, 10.1 and 10.3], and for a general algebraically closed field in [Mo, 3.7], states that Y must be a translate of an abelian subvariety of X. □

In the rest of this section we work out what the existence of a wild automorphism tells us about the Euler characteristic (or the arithmetic genus) of a variety. As a consequence, we will see that rationally connected varieties have no wild automorphisms.

Given a regular map f: X ⟶ X, we define the algebraic Lefschetz number L(f, X) by the formula

L(f, X) = ∑_{q≥0}(−1)^{q}Trace (f^{∗}: H^{q}(X, O_{X}) ⟶ H^{q}(X, O_{X})) .

Note that the sum on the right is well-defined because H^{q}(X, O_{X}) is a finitedimensional k-vector space for every q ≥ 0 [Hart, III.5.2] and H^{q}(X, O_{X}) = (0) for q > dim(X) [Hart, III.2.7]. We also define the algebraic Euler characteristic χ(O_{X}) of X as

χ(O_{X}) = L(id, X) = ∑_{q≥0}(−1)^{q} dim H^{q}(X, O_{X}) .

Proposition 4.4. Let X be a smooth irreducible projective variety and let σ be an automorphism of X.

(a) If χ(O_{X}) ≠ 0, then some power of σ has a fixed point.

(b) If dim(X) ≥ 1 and σ is wild then the algebraic Euler characteristic χ(O_{X}) = 0. Equivalently, the arithmetic genus p_{a}(X) = (−1)^{dim(X)+1}.

Proof. (a) By the “holomorphic Lefschetz fixed point theorem”, if σ has no fixed points then L(σ, X) = 0. A proof of this result over k = ℂ can be found in [GH, Section III.4, p. 426] and over an arbitrary algebraically closed field k of characteristic zero in [TT]. (Note that the sum in the right hand side of the formula is empty if there are no fixed points.)

Now assume that σ^{n} has no fixed point for any n ≥ 1. Set W = ⨁_{q≥0} H^{q}(X, O_{X}) and d = dim_{k}(W). Then the automorphism σ^{∗} of W induced by σ satisfies an equation of the form

(σ^{∗})^{d} + c_{1}(σ^{∗})^{d−1} + ⋯ + c_{d} id_{W} = 0

for some c_{i} ∈ k. Since σ^{∗} is an automorphism of W, c_{d} ≠ 0. By the linearity of the trace, we have

L(σ^{d}, X) + c_{1}L(σ^{d−1}, X) + ⋯ + c_{d−1}L(σ, X) + c_{d}L(id_{X}, X) = 0.

Then since L(σ^{n}, X) = 0 for all n ≥ 1, L(id_{X}, X) = χ(O_{X}) = 0.

(b) By Lemma 3.1(d) no power of σ can have a fixed point. Hence, part (a) tells us that χ(O_{X}) = 0. The second assertion is simply a restatement of the first, since by definition, p_{a}(X) = (−1)^{dim(X)}(χ(O_{X}) − 1) [Hart, p. 230]. □

Remark 4.5. If X is an irreducible variety defined over k = ℂ which has a wild automorphism, then the same arguments as in Proposition 4.4 show that e(X) = 0, where e(X) is the usual (topological) Euler characteristic. Proposition 4.6. Suppose an irreducible projective variety X of dimension d ≥ 1 has a wild automorphism. Then

(a) X carries a non-vanishing regular differential m-form ω for some odd m ≥ 1. Moreover, ω can be chosen so that

(4.7)  σ^{∗}ω = c ω for some c ∈ k^{∗}.

(b) X is not rationally connected.

(c) X is not unirational.

Recall that an algebraic variety X is called rationally connected if there is a family of irreducible rational curves in X such that two points in general position can be connected by a curve from this family; see [Ko, Section IV.3]. X is called unirational if it admits a dominant rational map ℙ^{n} −−→ X for some n ≥ 1.

Proof. (a) Let h^{n,m} = h^{n,m}(X) = dim H^{n}(X, ⋀^{m} Ω_{X}) be the Hodge numbers of X. Here Ω_{X} is the sheaf of differential 1-forms on X and ⋀^{m} Ω_{X} is the sheaf of differential m-forms.

We claim that h^{0,m} > 0 for some odd m ≥ 1. Since h^{n,m} = h^{m,n} (see, for example, [D, pp. 54-55]), we have

p_{a}(X) = ∑_{i=0}^{d−1}(−1)^{i}h^{d−i,0} = ∑_{m=1}^{d}(−1)^{d−m}h^{0,m} .

By Proposition 4.4, p_{a}(X) = (−1)^{d+1}, i.e.,

(4.8)  ∑_{m=1}^{d}(−1)^{m+1}h^{0,m} = 1 .

If h^{0,m} = 0 for every odd m ≥ 1, then every term in the left hand side of (4.8) is non-positive, a contradiction.

Thus for some odd m we have V := H^{0}(X, ⋀^{m} Ω_{X}) ≠ (0). Now we can choose ω to be an eigenvector for the linear automorphism σ^{∗} of the finitedimensional k-vector space V. In other words, (4.7) holds. Finally, the vanishing locus Y of ω is a closed σ-invariant subvariety of X. Since σ is wild, Y = ∅, i.e., ω is a non-vanishing differential form, as claimed.

(b) follows from (a). Indeed, if X is rationally connected then h^{0,m}(X) = 0 for all m ≥ 0; see [Ko, Corollary IV.3.8].

(c) follows from (b) because a unirational variety is rationally connected; see [Ko, Example IV.3.2.6.2] or [LBR, Lemma 3.4.1]. □

5. Wild automorphisms and the Albanese map

Let X be an irreducible variety. Then associated to X is the Albanese variety Alb(X), which is an abelian variety, and the Albanese map f : X → Alb(X). These constructions have the following properties: Alb(X) is generated as an algebraic group by the image of f (in the sense of Definition 7.1 below), and given any other regular map g : X → Y where Y is an abelian variety, there is a regular map h : Alb(X) → Y such that hf = g. The dimension of Alb(X) is called the irregularity of X and is denoted by q(X) (or just q if the reference to X is clear from the context). Equivalently, q(X) = dim_{k} H^{1}(X, O_{X}); see for example [BoHu, Section I.6] or [U, Lemma 9.22].

We shall now see that if an irreducible variety has a wild automorphism then its Albanese map must have various special properties. Proposition 5.1. Let X be an irreducible projective variety of dimension d and irregularity q and let π: X ⟶ Alb(X) be the Albanese map. Suppose that σ is a wild automorphism of X. By the universal property of the Albanese map, σ induces an automorphism of Alb(X), which we will denote by \overline{σ}.

(a) π is surjective.

(b) \overline{σ} is a wild automorphism of Alb(X).

(c) π is smooth.

(d) The fiber X_{t} = π^{−1}(t) is a smooth irreducible variety of dimension d − q for every t ∈ Alb(X).

(e) q ≤ d and if q = d then X is an abelian variety.

Note that by [Mo, Theorem 8.1], part (a) holds for every variety X with κ(X) = 0. On the other hand, if κ(X) = −1 and we do not assume that X has a wild automorphism, then part (a) may fail; for example, the Albanese map for the ruled surface X ⟶ C is surjective if and only if C is a curve of genus ≤ 1; see [GH, p. 554].

Proof. (a) Let \overline{X} = π(X). We claim that

(5.2)  \overline{σ} restricts to a wild automorphism of \overline{X}.

Indeed, assume the contrary, say \overline{σ}(\overline{Y}) ⊂ \overline{Y} for ∅ ⊊ \overline{Y} ⊊ \overline{X}. Then setting Y = π^{−1}(\overline{Y}), we see that ∅ ⊊ σ(Y) ⊆ Y, a contradiction. Now by Corollary 4.3, \overline{X} is a translate of an abelian subvariety in Alb(X). By the definition of the Albanese map, this implies that \overline{X} = Alb(X); see [La1, II.3].

(b) This follows from part (a) and (5.2).

(c) Since char k = 0 and we know that X must be nonsingular (see Lemma 3.1(b)), by generic smoothness there exists a Zariski open subset U ⊆ Alb(X) such that π is smooth over U [Hart, Corollary III.10.7]. Then clearly π is smooth over the \overline{σ}-invariant open subset

W = ⋃_{i∈ℤ} \overline{σ}^{i}(U)

of Alb(X). Since \overline{σ} is wild, W = Alb(X).

(d) The fact that each X_{t} is smooth of dimension d − q is immediate from part (c). To show that X_{t} is irreducible, consider the Stein decomposition

π: X ⟶^{α} X′ ⟶^{β} Alb(X) ,

where α has connected fibers and β is finite. The variety X′ may be defined explicitly as Spec π_{∗}(O_{X}) [Hart, III.1.5]; in particular, X′ is irreducible, since it is covered by the spectra of domains. Our goal is to show that β is an isomorphism. If we can prove this, then each X_{t} is connected and nonsingular, so must be irreducible.

Since the automorphism \overline{σ} of Alb(X) acts also on the sheaf of graded algebras π_{∗}(O_{X}) over Alb(X), we get an induced automorphism σ̃ of X′. Since β is a finite surjective map and \overline{σ} is a wild automorphism of Alb(X), it follows that σ̃ is a wild automorphism of X′. Thus X′ is nonsingular by Lemma 3.1(b). Then as in the proof of part (c), since β is smooth over an open set and \overline{σ} is wild, β must be a smooth morphism of relative dimension 0, in other words an étale map. Now by a theorem of Serre and Lang [Mu, Section IV.18], X′ has a structure of an abelian variety such that β is a regular homomorphism. By the universal property of the Albanese map, α factors through π. In other words, β has an inverse, so β is an isomorphism as desired.

(e) The inequality q ≤ d is an immediate consequence of part (a). If q = d, then by (c) π is a smooth morphism of relative dimension 0, so étale. By [Mu, Section IV.18] again, X has the structure of an abelian variety. □

6. Wild automorphisms of algebraic surfaces

In this section we will prove Conjecture 0.3 in the case where dim(X) ≤ 2. Our proof relies on the classification of algebraic surfaces. We will start with the most difficult cases, where X is assumed to be a ruled surface or a hyperelliptic surface, and defer the rest of the proof until the end of this section.

Recall that an algebraic surface X is called ruled if it admits a surjective morphism

(6.1)  π: X → C

to a smooth curve C, and such that the fiber of π over each point of C is isomorphic to ℙ^{1}. Such a morphism always has a section s: C ⟶ X; for example, see [Ko, Corollary IV.6.6.2]. Lemma 6.2. A (smooth projective) ruled surface cannot have a wild automorphism.

Proof. Assume the contrary: σ is a wild automorphism of a ruled surface X. Let π : X ⟶ C be as in (6.1) and let g be the genus of C. Since p_{a}(X) = −g (see [Hart, Corollary V.2.5]), Proposition 4.4 tells us that g = 1, so that C is an elliptic curve.

By [Hart, Corollary V.2.5] the irregularity q(X) = g = 1, so the Albanese variety Alb(X) is an elliptic curve. We claim that C is, in fact, the Albanese variety and π is the Albanese map for X. Indeed, let α: X ⟶ Alb(X) be the Albanese map. By the universal property of the Albanese map, π factors through α, in other words

π: X ⟶^{α} Alb(X) ⟶^{β} C ,

where β is a covering map of elliptic curves. Since the fibers of π are connected (each fiber is isomorphic to ℙ^{1}), we conclude that β is one-to-one, i.e., is a bijective morphism between smooth curves Alb(X) and C. We conclude that β is an isomorphism; this proves the claim. (For a different proof of this claim over k = ℂ, see [GH, p. 554].)

Now let s: C ⟶ X be a section of π, with the property that C_{0} = s(C) ⊂ X has the minimal possible self-intersection number. Following [Hart, Section V.2], we denote this number by −e. The group Num(X) of divisors in X up to numerical equivalence is isomorphic to ℤ ⊕ ℤ, and is generated by C_{0} and F, where F is a fiber of π; the intersection form on Num(X) is given by C_{0} · F = 1, F^{2} = 0 and C_{0}^{2} = −e [Hart, Proposition V.2.3]. As we saw in Proposition 5.1, σ acts on the fibers of π; since these fibers are all algebraically equivalent, we have σ(F) ≡ F in Num(X). We claim that C_{0} and σ(C_{0}) are numerically equivalent, that is,

(6.3)  σ(C_{0}) ≡ C_{0} in Num(X).

Indeed, suppose that σ(C_{0}) ≡ aC_{0} + bF for some a, b ∈ ℤ. Then a = σ(C_{0}) · F = 1, and since C_{0}^{2} = σ(C_{0})^{2}, we see that b = 0, thus proving (6.3). We now consider three cases.

1. e > 0. In view of (6.3), C_{0}·σ(C_{0}) = C_{0}^{2} < 0; consequently, σ(C_{0}) = C_{0}, contradicting the fact that σ is wild.

2. e = 0. Since σ is wild (and hence, so is σ^{2}), we may assume that C_{0}, σ(C_{0}) and σ^{2}(C_{0}) are three distinct curves in X. Since C_{0}^{2} = 0, formula (6.3) tells us that C_{0}, σ(C_{0}) and σ^{2}(C_{0}) are mutually disjoint in X. We now appeal to the general fact that any ruled surface X ⟶ C with three mutually disjoint sections C_{1}, C_{2} and C_{3} is isomorphic to ℙ^{1} ×C (over C). Indeed, for such X there is a (unique) isomorphism ℙ^{1} × C ⟶ X (over C), sending {0} × C, {1} × C and {∞} × C to C_{0}, C_{1} and C_{2}. Thus we may assume that X = ℙ^{1} × C, where C is an elliptic curve. In this case the canonical divisor of X is K_{X} = −2({pt}×C) [Hart, Lemma V.2.10], so that κ̄(X) > 0, contradicting Corollary 4.2.

3. e < 0. By [Hart, Theorems V.2.12 and V.2.15] there is only one surface X in this category, with e = −1. (This surface is often denoted by P_{1}.) By [Mar, Theorem 3(4)], the automorphism group Aut(X) has a normal subgroup

Δ = {id, δ_{1}, δ_{2}, δ_{3}} ≃ ℤ/2ℤ × ℤ/2ℤ

such that \overline{δ_{1}} = \overline{δ_{2}} = \overline{δ_{3}} = id_{C} for i = 1, 2, 3. Here \overline{δ} is the automorphism of C induced by δ (as in Proposition 5.1(b)). Let

Y = X^{δ_{1}} ∪ X^{δ_{2}} ∪ X^{δ_{3}} ,

where X^{δ} denotes the fixed point set of δ. Since σ(X^{δ}) = X^{σδσ^{−1}}, we see that σ(Y) = Y. Clearly, Y ≠ X; it thus remains to show that X^{δ} ≠ ∅ for δ = δ_{1}, δ_{2}, δ_{3}. Indeed, for every p ∈ C we have an automorphism

δ_{| π^{−1}(p)} : ℙ^{1} ⟶ ℙ^{1},

where ℙ^{1} stands for π^{−1}(p). Since every automorphism of ℙ^{1} has at least one fixed point, we see that X^{δ} ≠ ∅, as claimed.

This completes the proof of Lemma 6.2. □

Next we consider hyperelliptic surfaces. Recall that a hyperelliptic surface X is a surface of the form (E × F)/G, where E and F are elliptic curves, and G is a finite subgroup of E acting on E × F via

g · (x, y) ↦ (x + e, α(g) · y)

for some e ∈ X and injective homomorphism α: G ↪ Aut(F). There are only seven possibilities for E, F, G, and α; see [BeMi, Table 1.1], [GH, p. 590] or [BoMu, p. 37]. Lemma 6.4. A hyperelliptic surface cannot have a wild automorphism.

Proof. Suppose that X is a hyperelliptic surface and σ is an automorphism of X. By [BeMi, Lemmas 1.2 and 2.1], σ can be lifted to an automorphism σ̃ ∈ Aut(E) × Aut(F) which normalizes G and such that the diagram

E × F  \underrightarrow{σ̃}  E × F
↓  ↓
X = (E × F)/G ⟶^{σ} X = (E × F)/G

commutes. (In fact, by [BeMi, Lemma 3.1] σ̃ centralizes G, but we shall not use this.) Consequently, σ descends to an automorphism σ_{0} of F/α(G) ≃ ℙ^{1} (cf. [BoMu, Theorem 4, pp. 35-36]) such that the extended diagram

E × F  \underrightarrow{σ̃}  E × F
↓  ↓
X = (E × F)/G ⟶^{σ} X = (E × F)/G
↓  ↓
ℙ^{1} ≃ F/α(G) ⟶^{σ_{0}} ℙ^{1} ≃ F/α(G)

commutes; here the vertical arrows are the natural projections. Now recall that every automorphism of ℙ^{1} has a fixed point. If x ∈ F/α(G) ≃ ℙ^{1} is a fixed point of σ_{0} then the preimage of x in X = (E × F)/G is a σ-invariant curve. This shows that σ cannot be wild.

The above argument relied on the results of [BeMi] which are stated under the assumption that k = ℂ. However, we observe that the the proofs of Lemmas 1.2 and 2.1 in [BeMi] remain valid over any algebraically closed field k of characteristic 0. □

Theorem 6.5. Let X be an irreducible projective variety of dimension ≤ 2 over an algebraically closed field k of characteristic 0. If X admits a wild automorphism then X is an abelian variety.

Proof. By Lemma 3.1(b), X is smooth. The case dim X = 0 is trivial. Suppose dim(X) = 1. Recall that the arithmetic genus p_{a}(X) and the geometric genus p_{g}(X) of a smooth curve X coincide; see [Hart, Proposition IV.1.1]. Hence, by Proposition 4.4, p_{g}(X) = 1, i.e., X is an elliptic curve or equivalently, a 1-dimensional abelian variety.

From now on we will assume that X is a smooth surface. By Proposition 4.4, p_{a}(X) = −1 and thus the irregularity is

q(X) = p_{g}(X) − p_{a}(X) = p_{g}(X) + 1 ;

see [Hart, Remark 7.12.3]. Here the geometric genus p_{g}(X) = dim H^{0}(X, ω) is either 0 or 1; see Proposition 4.1. If p_{g}(X) = 1 then q(X) = 2, and X is an abelian variety; see Proposition 5.1(e).

Now suppose p_{g}(X) = 0 and q(X) = 1. To complete the proof, it is enough to show that this case cannot occur. Let π: X ⟶ C the Albanese map. Since q(X) = 1, C is an elliptic curve. We claim that X is minimal, i.e., X contains no smooth rational curves D with D^{2} = −1. Indeed, if such a curve existed on X, we would have π(D) = {p}, where p is a point of C. In other words, D would be contained in a fiber of π. By Proposition 5.1, π is a smooth map with irreducible fibers. In particular, D = π^{−1}(p). But then D^{2} = 0 (because the fibers of π are disjoint and algebraically equivalent). This contradicts our assumption that D^{2} = −1, thus proving the claim.

We now appeal to the Castelnuovo-Enriques classification of algebraic surfaces. By Corollary 4.2, the Kodaira dimension κ(X) = −1 or 0. If κ(X) = −1 then X is either rational or ruled [Hart, Theorem V.6.1]. By Proposition 4.6, X cannot be rational and by Lemma 6.2, X cannot be ruled. Thus we may assume without loss of generality that κ(X) = 0. Here there are four possibilities: (1) a K3 surface, (2) an Enriques surface, (3) an abelian surface, or (4) a hyperelliptic surface; see [BoHu, p. 373] or [Hart, Theorem V.6.3]. Of these four types, only (4) has p_{g}(X) = 0 and q = 1. On the the hand, by Lemma 6.4, X cannot be hyperelliptic. This shows that p_{g}(X) = 0 and q(X) = 1 is impossible, and the proof of the theorem is complete. □

7. Wild automorphisms of abelian varieties

In this section we will classify wild automorphisms of abelian varieties. We will assume throughout that k is algebraically closed of characteristic zero. We first review some basic definitions and facts from the theory of abelian varieties; see [Mu] and [La1] for detailed treatments of this material. As the name suggests, the group law of an abelian variety must be commutative [La1, Theorem II.1.1], and so we will use additive notation.

Let X and Y be arbitrary abelian varieties throughout the following discussion. We write Hom(X, Y) for the homomorphisms from X to Y, which are by definition the regular morphisms preserving the group structure. We also write End(X) for Hom(X, X), the endomorphisms of X. The words automorphism and morphism are used for regular maps which are not necessarily assumed to be homomorphisms of groups; for example, for any a ∈ X one has the translation automorphism T_{a} : x ↦ x + a. In fact, every regular map σ : X → Y between abelian varieties is of the form σ = T_{b} · α, where α ∈ Hom(X, Y) and b ∈ Y; see [La1, Theorem 4, p.24]. We say that ψ ∈ Hom(X, Y) is an isogeny if ψ is surjective and has finite kernel. For example, for any abelian variety X and p ∈ ℤ the map p · Id_{X} : X → X defined by x ↦ px is an isogeny [La1, Corollary IV.2.1]. If there exists an isogeny from X to Y, then there also must exist an isogeny from Y to X [La1, p. 29], and X and Y are said to be isogeneous.

A fundamental result is the complete reducibility theorem of Poincaré: if Z is an abelian subvariety of X, then there exists another abelian subvariety Z′ ⊆ X such that Z + Z′ = X and Z ∩ Z′ is a finite group [La1, Theorem II.1.6]; in this case clearly X is isogeneous to Z × Z′. An abelian variety X is simple if 0 and X are the only closed irreducible subgroups of X. Abelian varieties satisfy unique decomposition into simples in the following sense: every abelian variety X is isogeneous to a product X_{1} ×X_{2} ×⋯×X_{n} where each X_{i} is a simple abelian variety, and the simples appearing in such a decomposition are unique up to isogeny and order of the factors [La1, page 30]. If X′ is a closed subgroup of an abelian variety X, then the factor group X/X′ has a natural structure of an abelian variety [La1, p. 3].

Next, we recall some results about the structure of the endomorphism ring End(X). Let X be isogeneous to the product ∏_{i=1}^{h} X_{i}^{n_{i}} where the X_{i} are simple abelian varieties, mutually non-isogeneous. By [La1, Corollary VII.1.2], End(X) is a torsionfree ℤ-module, and so there is an embedding of rings End(X) ⊂ End_{ℚ}(X) := End(X) ⊗_{ℤ} ℚ. Then by [La1, Theorem 7, p.30], each D_{i} = End_{ℚ}(X_{i}) is a finite-dimensional division algebra over ℚ, and

End_{ℚ}(X) ≅ ∏_{i=1}^{h} M_{n_{i}}(D_{i}).

In particular, if X and Y are isogeneous then End_{ℚ}(X) ≅ End_{ℚ}(Y). If R is a ring with identity, we call an element r ∈ R unipotent if r − 1 is nilpotent and quasi-unipotent if some power of r is unipotent. We shall apply these terms in particular to endomorphisms α ∈ End(X). (The reader should not confuse these definitions with the use of the same terms in the theory of twisted homogeneous coordinate rings in [Ke]; see the discussion in §8 below.) Finally, Definition 7.1. The algebraic subgroup of an abelian variety X generated by S ⊂ X is the Zariski closure of the (abstract) subgroup <S> of X generated by S. In particular, we say that S generates X if S is not contained in any proper closed subgroup of X. If S = {s_{1}, s_{2}, … }, then we will sometimes say that s_{1}, s_{2}, … generate X.

Now we present the main theorem of this section, which gives a characterization of wild automorphisms of abelian varieties, and proves Theorem 0.2(a). Theorem 7.2. Let σ = T_{b} · α be an automorphism of X, where α ∈ End(X) is an automorphism and b ∈ X. Let β = α − Id, and set S = {b, β(b), β^{2}(b), … } ⊂ X. Then the following are equivalent:

(a) σ is wild.

(b) α is unipotent and S generates X.

(c) α is unipotent and the image \overline{b} of b generates X/β(X).

Proof. Let Z be the algebraic subgroup of X generated by S.

(b) ⇔ (c). If S generates X then clearly \overline{b} generates X/β(X). For the converse, assume that \overline{b} generates X/β(X). Then Z + β(X) = X. Applying β on both sides, we obtain β(Z) + β^{2}(X) = β(X). Now

X = Z + β(X) = Z + β(Z) + β^{2}(X) = Z + β^{2}(X) ;

the last equality follows from the fact that β(Z) ⊆ Z. Continuing in this manner, we obtain

X = Z + β^{i}(X) for i = 1, 2, …

Since β is nilpotent, we conclude that Z = X, as desired.

(a) ⇒ (b): Suppose that α − Id is not nilpotent. Let Y_{s} be the connected component containing 0 of the closed subgroup ker(α − Id)^{s} of X. Then Y_{s} ⊂ Y_{s+1} for all s, and this sequence of irreducible closed sets must stabilize, so we may set Y = Y_{s} for s ≫ 0. Clearly α(Y) ⊂ Y. Hence α induces an endomorphism of the abelian variety X/Y, denoted by \overline{α}. By the definition of Y, \overline{α}−Id is an isogeny of X/Y. Let π be the quotient map X → X/Y and let \overline{σ} = T_{π(b)}·\overline{α}, which is an automorphism of X/Y. Since \overline{α}−Id is an isogeny, it is surjective, and so there exists some \overline{x} such that (\overline{α} − Id)(\overline{x}) = −π(b). Then \overline{x} ∈ X/Y is a fixed point of the automorphism \overline{σ}. Set W = π^{−1}(\overline{x}) ⊂ X. Then σ(W) = W, and W ≠ X since Y ≠ X. Hence σ is not wild, a contradiction.

Thus we have proved that α is unipotent. Since σ(Z) ⊂ Z, and σ is wild, we conclude also that Z = X.

(b, c) ⇒ (a): First we observe that for the purpose of this proof we may replace σ by σ^{n}, where n ≥ 1; Indeed, σ is wild if and only if σ^{n} is wild; see Lemma 3.1(d). Moreover, σ^{n} = T_{b′} · α^{n}, where

b′ = ( ∑_{i=0}^{n−1} α^{i})(b) = ( ∑_{i=1}^{n} \binom{n}{i}β^{i−1})(b).

If β = α − Id is nilpotent, then so is α^{n} − Id. If \overline{b} is the the image of b in X′ = X/β(X), then the image of b′ in X′ is \overline{b′} = n\overline{b}, so that \overline{b′} generates X/β(X) if and only if \overline{b} does.

Let W ⊂ X be a nonempty subvariety of minimal possible dimension such that σ(W) ⊂ W. Our goal is to show that W = X; this will imply that σ is wild. Replacing σ by σ^{n} if necessary, we may assume also that W is irreducible. Note that by the minimality of dim(W), σ restricts to a wild automorphism of W. By Corollary 4.3, W is a translate H + x = T_{x}(H) of some abelian subvariety H of X. Now set

σ′ = T_{−x}σT_{x} = T_{b+β(x)} · α.

Then σ′(H) = H, so that c = σ′(0) = b + β(x) ∈ H. Thus α(H) ⊂ H, from which it follows that β(H) ⊂ H. In particular, H contains S′ = \{c, β(c), β^{2}(c), … \}. Since the image of c in X/β(X) coincides with \overline{b}, the implication (c) ⇒ (b) proved above shows that S′ generates X. Since S′ ⊂ H, we conclude that H = X and hence W = X, as we wished to show. □

For the remainder of this section we will study the conditions for wildness given by Theorem 7.2 a bit further. Following part (c) of that theorem, we see that we would like to understand when a single element of an abelian variety will generate that variety. We analyze this situation next. The first two parts of the proposition below show that if the base field k is uncountable, then “most” points a ∈ X will generate X. Since an arbitrary X is isogeneous to some product of simple abelian varieties, the remaining parts may be combined to give more precise information about which points a ∈ X generate X. Proposition 7.3. Let X be an abelian variety and a ∈ X some point.

(a) The element a generates X if and only if f(a) ≠ 0 for every 0 ≠ f ∈ End(X).

(b) There is a countable set of closed subgroups {G_{α}} of X such that a generates X if and only if a ∉ ⋃ G_{α}.

(c) If f : X → Y is an isogeny, then a generates X if and only if f(a) generates Y.

(d) If X is simple, then a generates X if and only if a is a point of infinite order on X.

(e) Let X = X_{1} × X_{2} × ⋯ × X_{n} where the X_{i} are abelian varieties such that Hom(X_{i}, X_{j}) = 0 for all i ≠ j. Then a = (a_{1}, a_{2}, …, a_{n}) generates X if and only if a_{i} generates X_{i} for every i = 1, …, n.

(f) Let Y = X^{×n} and a = (a_{1}, …, a_{n}) ∈ X^{×n}. Then a generates Y if and only if the following condition holds: given endomorphisms θ_{i} ∈ End(X) with ∑_{i=1}^{n} θ_{i}(a_{i}) = 0, one must have θ_{i} = 0 for i = 1, …, n.

Proof. (a) If f(a) = 0 for some 0 ≠ f ∈ End(X) then the algebraic subgroup Z generated by a is contained in Ker(f) ≠ X, so that a does not generate X.

Conversely, suppose that a does not generate X, so that a is contained in a closed subgroup Z ⊊ X. Let Z_{0} be the connected component of Z containing the identity element 0; then pa ∈ Z_{0} for some integer p ≥ 1. By complete reducibility, there is a non-trivial abelian subvariety W ⊂ X such that X = Z_{0} + W and Z_{0} ∩ W is finite. Then we may choose some isogeny θ : X/Z_{0} → W, and thus we have a nonzero endomorphism

f : X ⟶^{p·Id_{X}} X ⟶ X/Z_{0} ⟶^{θ} W ↪ X

such that f(a) = 0.

(b) For every 0 ≠ α ∈ End(X), let G_{α} = ker α. By [Mu, Theorem IV.19.3], End(X) is countable, so part (a) implies the result.

(c) Suppose that a generates a closed subgroup A in X and that f(a) generates B in Y. Then clearly f(A) = B. Now it is easy to see that

A = X ⟺ dim(A) = dim(X) ⟺ dim(B) = dim(Y) ⟺ B = Y.

(d) Every nonzero endomorphism of a simple abelian variety is an isogeny, which has finite kernel. Conversely, if a is a point on X of finite order and G = {na}_{n∈ℤ}, then a is in the kernel of the isogeny X → X/G. Now apply part (a).

(e) This is an easy consequence of the fact that every endomorphism of X maps X_{i} to X_{i} for all i.

(f) Suppose that a does not generate Y. Then by part (a), there exists an 0 ≠ α ∈ End(Y) such that α(a) = 0. If Z ⊂ im α is a simple abelian subvariety, then Z is isogeneous to some simple abelian variety appearing in the decomposition of X into simples. It follows that Hom(im α, X) ≠ 0. Then picking a nonzero homomorphism f : im α → X, we see that θ := fα : Y → X is a nonzero homomorphism such that θ(a) = 0. But θ(a) = ∑_{i=1}^{n} θ_{i}(a_{i}) for some θ_{i} ∈ End(X), at least one of which is nonzero. The converse is proved by reversing this argument. □

Let us also make some remarks concerning unipotent automorphisms of abelian varieties. Of course, any abelian variety X has at least the unipotent automorphism σ = Id. In the next result we will identify those X for which there exist non-identity unipotent automorphisms, and show how to construct some of them. Suppose that X = Y^{×n} where Y is an abelian variety. Then for any integer matrix M ∈ M_{n}(ℤ) we may define an endomorphism α_{M} ∈ End(X), as follows. Write an arbitrary point in X as a column vector \overline{x} with entries from Y. Then let α_{M} be defined by the formula α_{M}(\overline{x}) = M\overline{x}, where the right hand side is “matrix multiplication” performed using the ℤ-module structure of Y.

Proposition 7.4. (a) Let X = Y^{×n} for some abelian variety Y. Then for M ∈ GL_{n}(ℤ), the automorphism α_{M} ∈ End(X) is (quasi)- unipotent if and only if M is a (quasi)-unipotent matrix in GL_{n}(ℤ).

(b) Let X be an abelian variety which is isogeneous to a product ∏_{i} X_{i}, where the X_{i} are simple abelian varieties. Then X has a unipotent automorphism Id ≠ α ∈ End(X) if and only if X_{i} and X_{j} are isogeneous for some i ≠ j.

Proof. (a) Since the mapping ψ : M → α_{M} is a ring homomorphism from M_{n}(ℤ) to End(X), it is clear that M unipotent implies α_{M} unipotent. Conversely, suppose that M is not unipotent, so M − Id is not nilpotent. One may easily check that ker ψ = 0, and thus α_{M−Id} = α_{M} −Id is not nilpotent in End(X), so α_{M} is not unipotent. Thus M is unipotent if and only if α_{M} is unipotent; the same statement for quasi-unipotence follows immediately.

(b) Suppose first that the X_{i} are pairwise non-isogeneous. Then End(X) ⊂ End_{ℚ}(X), where End_{ℚ}(X) ≅ ∏ End_{ℚ}(X_{i}) is a product of division rings, and hence every nilpotent element in this ring is zero. Now if α ∈ End(X) is a unipotent automorphism, then α − Id = 0 and so α = Id.

Conversely, if X_{i} and X_{j} are isogeneous for some i ≠ j, then the ring R := End_{ℚ}(X_{i} × X_{i}) ≅ End_{ℚ}(X_{i} × X_{j}) embeds in End_{ℚ}(X). Here R ≅ M_{2}(D), where D = End_{ℚ}(X_{i}) is a division ring finite over ℚ. By part (a), R contains many unipotent automorphisms of the form α_{M} for non-identity unipotent matrices M ∈ GL_{2}(ℤ). □

8. Unipotency on the Neron-Severi Group

As usual, we assume in this section that our base field k is algebraically closed of characteristic zero.

In the last section we saw how to construct many examples of wild automorphisms σ of an abelian variety X. Then by Proposition 2.2, we see that we obtain lots of examples of projectively simple twisted homogeneous coordinate rings B(X, L, σ), as long as we can find σ-ample invertible sheaves L on X. We shall show in this section that it will suffice to take L to be any ample invertible sheaf on X. To do this, we apply criteria of Keeler for σ-ampleness, which will require us to consider the induced action of automorphisms of X on the group of divisors modulo numerical equivalence.

Let us begin with a review of Keeler’s results from [Ke]. Let X be any projective scheme, and let Pic X be the Picard group of all invertible sheaves on X. Two invertible sheaves L and L′ on X are numerically equivalent if (L.C) = (L′.C) for all integral curves C ⊂ X. Define the group Num(X) to be Pic X modulo numerical equivalence; it is isomorphic to ℤ^{m} for some m. When X is an abelian variety, then Num(X) is isomorphic to the Neron-Severi group NS(X) [Hart, p.367]. Any morphism σ : X → X naturally induces via pullback an endomorphism of the group Num(X), or equivalently a matrix P_{σ} ∈ M_{m}(ℤ).

Recall that we defined the notions of unipotence and quasi-unipotence for elements of a ring in the last section. Now let X be an projective scheme and σ : X → X any automorphism. Then we will call σ Num-unipotent if the corresponding matrix P_{σ} ∈ M_{m}(ℤ) is unipotent and Num-quasi-unipotent if P_{σ} ∈ M_{m}(ℤ) is quasi-unipotent. (Note that in [Ke] σ is called simply unipotent or quasi-unipotent, respectively.) Keeler proved that if σ is Numquasi-unipotent then every ample invertible sheaf L on X is σ-ample, and if σ is not Num-quasi-unipotent then no σ-ample invertible sheaf exists [Ke, Theorem 1.2]. Thus to construct σ-ample sheaves on abelian varieties, we just need to understand which automorphisms of abelian varieties are Numquasi-unipotent.

Let X be an abelian variety. For every invertible sheaf L ∈ Pic X and every a ∈ X, let ϕ_{L}(a) = (T_{a})^{∗}(L) ⊗ L^{−1} ∈ Pic X. The Theorem of the Square [Mu, Corollary 4, p.59] states that

(T_{a+b})^{∗}(L) ⊗ L ≅ (T_{a})^{∗}(L) ⊗ (T_{b})^{∗}(L) for all a, b ∈ X.

It follows that ϕ_{L} is a group homomorphism from X to Pic(X). Now we can easily handle the case of translation automorphisms. Lemma 8.1. Let X be an abelian variety with a ∈ X and let T_{a} be the corresponding translation automorphism. Then the induced automorphism P_{T_{a}} of Num(X) is the identity. In particular, T_{a} is Num-unipotent.

Proof. The Picard variety of X, denoted by Pic^{0}(X), is the subgroup of Pic(X) consisting of all L such that (T_{a})^{∗}(L) ≅ L for all a ∈ X, in other words those L such that ϕ_{L} is identically 0. Then another application of the Theorem of the Square shows that for every L ∈ Pic(X) and a ∈ X, ϕ_{L}(a) = (T_{a})^{∗}(L) ⊗ L^{−1} ∈ Pic^{0}(X) (see [Mu, p. 74]). Since any divisor in Pic^{0}(X) is numerically equivalent to 0 [La1, Proposition IV.3.4], L and (T_{a})^{∗}(L) induce the same element of Num(X), and so it follows that P_{T_{a}} = Id as required. □

Since an arbitrary automorphism σ ∈ Aut(X) has the form σ = T_{b} · α for b ∈ X and α ∈ End(X), we see from the preceding result that σ and α induce the same map on Num(X). Thus from now on we may restrict our attention to automorphisms in End(X). We would like to say that if α ∈ End(X) is a quasi-unipotent automorphism, then P_{α} is a quasiunipotent automorphism of Num(X). This is not obvious, because the map P : End(X) → End(Num(X)) defined by α ↦ P_{α} is only a homomorphism of multiplicative semigroups, not a ring map. Nevertheless, with a little more work we will succeed in proving that P preserves quasi-unipotency in Proposition 8.5 below. The main step is to express unipotency purely in terms of the multiplicative structure, which will require the following technical bit of algebra.

Lemma 8.2. Let D_{1}, D_{2}, …, D_{h} be division algebras over ℚ. Fix some integers n_{i} > 0 and let M ∈ ∏_{i=1}^{h} M_{n_{i}}(D_{i}).

(a) If M is unipotent, then M^{p} is conjugate to M for all p > 0. If M is quasi-unipotent, then M^{p} is conjugate to M^{q} for some 0 < p < q.

(b) Assume that each D_{i} is a (commutative) field. Then the converses of both statements in part (a) hold.

(c) Let g : ∏_{i=1}^{h} M_{n_{i}}(D_{i}) → M_{m}(F) be a homomorphism of multiplicative semigroups, where F is a field. Then g preserves quasiunipotency (respectively, unipotency).

Proof. In both parts it is easy to reduce to the case where h = 1, and we do so, writing D = D_{1} and n = n_{1}.

(a) Let M ∈ M_{n}(D) be unipotent. It is an exercise in linear algebra that in this case M has a Jordan canonical form (this is of course not necessarily true for arbitrary matrices in M_{n}(D)). Also, M and M^{p} must have the same Jordan form for all p ≥ 0, so they are conjugate.

Suppose instead that M is quasi-unipotent. Then M^{s} is unipotent for some s > 0, and so M^{s} and M^{sp} are conjugate for all p ≥ 0.

(b) Assume now that D is commutative, and let M ∈ M_{n}(D). Let λ_{1}, ⋯, λ_{n} be the eigenvalues of M in some algebraic closure of D. Then M is quasi-unipotent if and only if all of the λ_{i} are roots of unity.

Suppose that M^{p} is conjugate to M^{q} for some 0 < p < q. Then the ordered set {λ_{1}^{p}, ⋯, λ_{n}^{p}} is a permutation of the ordered set {λ_{1}^{q}, ⋯, λ_{n}^{q}}. Let τ ∈ S_{n} be that permutation, namely, λ_{i}^{p} = λ_{τ(i)}^{q} for all i. If w = n!, then τ^{w} is the identity. Hence we have

λ_{i}^{p^{w}} = λ_{τ(i)}^{p^{w−1}q} = ⋯ = λ_{τ^{t}(i)}^{p^{w−t}q^{t}} = ⋯ = λ_{τ^{w}(i)}^{q^{w}} = λ_{i}^{q^{w}}.

Thus λ_{i}^{q^{w}−p^{w}} = 1 for all i and M is quasi-unipotent.

If M is conjugate to M^{p} for all p > 0, then M is quasi-unipotent by the last paragraph. Hence M^{p} is unipotent for some p. Since M is conjugate to M^{p}, M is unipotent.

(c) This is an immediate consequence of parts (a) and (b). □

Remark 8.3. Lemma 8.2(a) also allows us to improve Theorem 3.2 and Corollary 3.3 slightly. Suppose that X is an abelian variety, that σ ∈ Aut(X) is wild, and that some power of σ is a translation. Then σ = T_{b} · α with α ∈ End(X) and α^{n} = Id for some n ≥ 0. By Theorem 7.2, α is unipotent. Then by Lemma 8.2(a), α is conjugate to α^{n} = Id, so α = Id. Thus the conclusion of Theorem 3.2 and Corollary 3.3 may be improved to the statement that σ is a translation.

Now we return to our abelian variety X, and show how the above lemma may be applied to the map P : End(X) → End(Num(X)). First, we want to know the action of P on the “scalars”. Lemma 8.4. Let X be an abelian variety. Let α = n · Id_{X} ∈ End(X) where n ∈ ℤ. Then P_{α} = n^{2} · Id_{Num(X)}.

Proof. This is immediate from [La1, Proposition 2, p. 92]. □

Let End_{ℚ}(X) = End(X) ⊗_{ℤ} ℚ and Num_{ℚ}(X) = Num(X) ⊗_{ℤ} ℚ. By Lemma 8.4, if we define P_{n^{−1}Id_{X}} = n^{−2}Id_{Num(X)} for n ≥ 1 then P naturally extends to a semigroup homomorphism P : End_{ℚ}(X) → End_{ℚ}(Num(X)).

Let X be isogeneous to a product ∏_{i=1}^{h} X_{i}^{n_{i}} where the X_{i} are simple abelian varieties, mutually non-isogeneous. As we stated in §7, each D_{i} = End_{ℚ}(X_{i}) is a finite dimensional division algebra over ℚ, and End_{ℚ}(X) ≅ ∏_{i=1}^{h} M_{n_{i}}(D_{i}). Also, letting m be the rank of Num(X) we have End_{ℚ}(Num(X)) ≅ M_{m}(ℚ). We may now prove our main result on Num-quasi-unipotence for automorphisms of abelian varieties. This settles Theorem 0.2(b) from the introduction. Theorem 8.5. Let X be an abelian variety.

(a) The map P : End_{ℚ}(X) → End_{ℚ}(Num(X)) preserves (quasi)-unipotency.

(b) If α ∈ Aut(X) is (quasi)-unipotent and b ∈ X is any point, then σ = T_{b} · α ∈ Aut(X) is Num-(quasi)-unipotent.

(c) If σ is a wild automorphism of X, then σ is Num-unipotent and any ample invertible sheaf L on X is σ-ample.

Proof. (a) As we saw in the comments before the theorem, the map P is a semigroup homomorphism from ∏_{i=1}^{h} M_{n_{i}}(D_{i}) to M_{m}(ℚ). Thus the result is immediate from Lemma 8.2(c).

(b) Since P is a semigroup homomorphism, this follows from part (a) and Lemma 8.1.

(c) This follows from part (b), Theorem 7.2(b), and [Ke, Theorem 1.2]. □

We have finally shown all of the necessary pieces to demonstrate the existence of a large supply of projectively simple twisted homogeneous coordinate rings. Corollary 8.6. Let X be an abelian variety with automorphism σ = T_{b} · α, where α ∈ End(X) is unipotent and the image of b generates X/(α−Id)(X). Then if L is any ample invertible sheaf on X, then the ring B(X, L, σ) is projectively simple.

Proof. By Theorem 7.2, σ is a wild automorphism. By Theorem 8.5, the sheaf L is σ-ample. Then by Proposition 2.2, B(X, L, σ) is a projectively simple ring. □

In [AV, Example 5.18], Artin and Van den Bergh gave an example of automorphism σ of a surface Y, defined by “translation along a pencil of elliptic curves”, which has the property that P_{σ} is unipotent, but not the identity. There seem to be few other explicit examples in the literature of automorphisms of projective schemes which are Num-quasi-unipotent, but whose action on Num(X) is non-trivial. In part (c) of the next result we show that we can produce numerous such examples in the setting of abelian varieties. Lemma 8.7. Let X be an abelian variety.

(a) If f ∈ End(X) is an isogeny, then P_{f} is nonzero.

(b) If β ∈ End_{ℚ}(X) is nonzero, then P_{β} is nonzero.

(c) If α ∈ End(X) is unipotent and α ≠ Id, then P_{α} is unipotent and P_{α} ≠ Id.

Proof. (a) This follows from Lemma 8.4 and the fact that there is a g ∈ End(X) such that fg = n · Id_{X} [La1, p. 29].

(b) Replacing β by nβ for some n ≥ 1 we may assume that β ∈ End(X). If Y is the image of β, then there are endomorphisms f : Y → X and g : X → Y such that gβf ∈ End(Y) is an isogeny. Now the assertion follows from (a).

(c) By Theorem 8.5, P_{α} is unipotent, so we just need to show that P_{α} is not the identity. Let β = α − Id, so that β is nilpotent. Let p be the maximal integer such that β^{p} ≠ 0. Then

α^{n} = ∑_{i=0}^{p} \binom{n}{i}β^{i}

for all n ≥ 0. By [La1, Proposition 2, p. 92],

(8.8)  P_{α^{n}} = \frac{1}{2} ∑_{i=0}^{p} ∑_{j=0}^{p} \binom{n}{i}\binom{n}{j}D(β^{i}, β^{j})

for all n ≥ 0, where D(f, g) = P_{f+g} − P_{f} − P_{g} for all f, g ∈ End(X). For all n the right hand side of (8.8) is equal to Q(n), where Q is some fixed polynomial

Q(z) = ∑_{i=0}^{2p} h_{i}z^{i} ∈ End(Num(X))[z] = M_{m}(ℤ)[z].

Now suppose that P_{α} is the identity. Then P_{α^{n}} is the identity for all n ≥ 1, and thus Q(n) = Id for all n ≥ 1. Let |M| be notation for the norm of an arbitrary matrix M ∈ M_{m}(ℤ) in the Euclidean topology on ℝ^{m^{2}}. Now if h_{i} ≠ 0 for some i ≥ 1, then clearly lim_{n→∞} |Q(n)| = ∞, contradicting the fact that Q(n) = Id for all n ≥ 1. Thus h_{i} = 0 for all i ≥ 1.

In particular, the leading coefficient h_{2p} of Q is (p!)^{−2}D(β^{p}, β^{p}) = 0. But using Lemma 8.4,

D(β^{p}, β^{p}) = P_{2β^{p}} − 2P_{β^{p}} = 4P_{β^{p}} − 2P_{β^{p}} = 2P_{β^{p}},

so P_{β^{p}} = 0 (recall our standing assumption that char k = 0). This contradicts part (b). Thus P_{α} ≠ Id. □

In order to illustrate the results of this section, we conclude with a simple example where the action of automorphisms on Num(X) may be calculated explicitly.

Proposition 8.9. Let E be an elliptic curve without complex multiplication

(in other words End(E) ≅ ℤ), and let X = E × E. Let M = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈

GL_{2}(ℤ) be arbitrary and let α = α_{M} ∈ End(X) be the corresponding automorphism; explicitly, α(x, y) = (ax + by, cx + dy).

(a) The divisors C^{1} = 0 × E, C^{2} = E × 0, and C^{3} = {(x, x) | x ∈ E} form a basis for Num(X) ⊗_{ℤ} ℚ ≅ ℚ^{3}.

(b) With respect to the basis given in (a), P_{α} ∈ M_{3}(ℤ) is equal to

(\begin{matrix}a^{2} − ab & c^{2} − cd & (a + c)^{2} − (a + c)(b + d) \\ b^{2} − ab & d^{2} − cd & (b + d)^{2} − (a + c)(b + d) \\ ab & cd & (a + c)(b + d) \end{matrix}) .

(c) det P_{α} = (det M)^{3}.

(d) α is quasi-unipotent if and only if P_{α} is.

Proof. (a, b) These parts are the statements of [Ko, Exercise II.4.16.2] and [Ko, Exercise II.4.16.6]), respectively, and we omit the proofs.

(c) This is an easy computation using part (b).

(d) By part (b), all entries of P_{α} are polynomial functions of a, b, c, d. Hence the semigroup homomorphism P : GL_{2}(ℤ) → GL_{3}(ℤ) can be extended to a group homomorphism P : GL_{2}(ℂ) → GL_{3}(ℂ) with the same formula given in (b). Since P is a group homomorphism, the desired assertion holds if and only if it holds after conjugation.

By conjugation we may assume that α has one of the two forms (\begin{matrix}a & 0 \\ 0 & d \end{matrix})

or (\begin{matrix}a & 1 \\ 0 & a \end{matrix}). Then one may check using the formula in part (d) that the

eigenvalues of P_{α} are either {a^{2}, d^{2}, ad} or {a^{2}, a^{2}, a^{2}}, respectively. Hence P_{α} is quasi-unipotent if and only if M is a quasi-unipotent matrix, if and only if α_{M} is a quasi-unipotent, by Proposition 7.4(a). □

9. Classification of low-dimensional cases

In this section we tie together our previous results to classify projectively simple twisted homogeneous coordinate rings of small GK-dimension. Again we assume that k is an algebraically closed field of characteristic zero.

We will use the following result of Keeler about the GK-dimension of twisted homogeneous coordinate rings [Ke, Theorem 6.1]. Let B = B(X, L, σ) where L is σ-ample. Then GKdim B is an integer satisfying the inequalities

(9.1)  j + dim X + 1 ≤ GKdim B ≤ j(dim X − 1) + dim X + 1,

where j + 1 is the size of the largest Jordan block of the matrix P_{σ} ∈ End(Num(X)) giving the induced action of σ on Num X. It is a fact that j is always even. Proposition 9.2. Let B = B(X, L, σ), where L is σ-ample. Then B is a projectively simple ring with GKdim(B) = 2 or 3 if and only if (a) X is an abelian variety of dimension GKdim(B) − 1,

(b) σ is the the translation T_{b}, and

(c) b generates X.

Proof. Suppose first that B is projectively simple. Since GKdim B ≤ 3, by (9.1) we must have dim X ≤ 2, and since j in that equation is even, the only possibility is j = 0, which forces GKdim(B) = dim X + 1. Also, σ is wild by Proposition 2.2. By Theorem 6.5, X must be an abelian variety. Let σ = T_{b}·α for α ∈ End(X) and b ∈ X. Note that P_{σ} = P_{α} by Lemma 8.1. By Theorem 7.2, α is unipotent. If α is not the identity, then by Lemma 8.7(c), P_{σ} = P_{α} is unipotent and not the identity. In this case P_{σ} has a non-trivial Jordan block, and so j ≠ 0 in (9.1), a contradiction. We conclude that α = Id and σ = T_{b} is a translation. By Theorem 7.2 again, b generates X.

Conversely, suppose that X is an abelian variety with dim X = 1 or 2 and that σ = T_{b} where b generates X. By Theorem 7.2, σ is wild. Then B is projectively simple by Proposition 2.2. Moreover, P_{σ} = Id by Lemma 8.1, so j = 0 in (9.1) and GKdim(B) = dim X + 1. □

One can prove analogous results for other small values of dimension. We omit the similar proof of the following. Proposition 9.3. Let B = B(X, L, σ) where X is an abelian variety and L is σ-ample.

(a) Suppose that GKdim B = 4. Then B is projectively simple if and only if dim X = 3, σ = T_{b}, and b generates X.

(b) Suppose that GKdim B = 5. Then B is projectively simple if and only if either

(1) dim X = 4, σ = T_{b}, and b generates X, or

(2) dim X = 2, σ = T_{b}, ·α for some non-identity unipotent α ∈ End(X), and the image \overline{b} of b generates X/(α − Id)(X).

10. The Cohen-Macaulay property

Our goal in the these last two sections is to study some of the homological properties of projectively simple rings and their associated noncommutative schemes. In particular we will prove Theorem 0.5 from the introduction. In order to get interesting results, we take as our basic hypothesis in these two sections that all algebras have a balanced dualizing complex. This condition holds for many important classes of algebras.

The objects of interest and the methods we employ here are rather different from those of the preceding sections, and we must begin with a review of the definitions of noncommutative projective schemes [AZ1], Serre duality [YZ1], and dualizing complexes [Ye, VdB]. We assume that the reader is familiar with the basics of derived categories and follow the notations used in [YZ1]. For example, the nth complex shift of a complex X is denoted by X[n].

Let A be a right noetherian graded k-algebra. Let Gr A be the category of graded right A-modules and let gr A be the full subcategory of Gr A consisting of the noetherian objects. Recall from §1 that a graded A-module M is called torsion if for every m ∈ M there is some n ≥ 0 such that mA_{≥n} = 0. Let Tors A denote the full subcategory of Gr A consisting of all torsion objects and let tors A be the full subcategory of noetherian torsion objects, or equivalently modules of finite k-dimension. The noncommutative projective spectrum of A is defined to be Proj A := (QGr A, A, s) where QGr A is the quotient category Gr A/ Tors A, the object A is the image of A in QGr A, and s is the auto-equivalence of QGr A induced by the degree shift M → M(1) for M ∈ Gr A. If we want to work with noetherian objects only, then proj A := (qgr A, A, s) is also called the projective spectrum of A. The canonical functor from Gr A → QGr A (and from gr A to qgr A) is denoted by π. If M ∈ Gr A, we will use the corresponding calligraphic letter M for π(M) if there is no chance of confusion. For example, A = π(A).

Let X = Proj A. For N ∈ QGr A, the i-th cohomology group of N is defined to be

H^{i}(X, N) = Ext_{QGr A}^{i}(A, N)

for all i ≥ 0. Then the (right) cohomological dimension of X is defined to be

cd(X) = max{i | H^{i}(X, N) ≠ 0 for some N ∈ QGr A}.

Suppose that d = cd(X) is finite. An object ω ∈ qgr A is called a dualizing sheaf for X if there is a natural isomorphism

θ^{0} : H^{d}(X, N)^{∗} → Hom_{qgr A}(N , ω)

for all N ∈ qgr A. Here (−)^{∗} means the k-vector space dual. We say that X satisfies Serre duality if a dualizing sheaf ω exists. In this case we say that X is classically Cohen-Macaulay if θ^{0} can be extended to a sequence of natural isomorphisms

θ^{i} : H^{d−i}(X, N)^{∗} → Ext_{qgr A}^{i}(N , ω)

for all N and 0 ≤ i ≤ d. If A is commutative, then the concepts of dualizing sheaf and the Cohen-Macaulay property which we have defined here agree with the usual commutative notions.

The notion of a balanced dualizing complex for a noncommutative graded algebra was introduced by Yekutieli in [Ye]. Let D(Gr A) denote the derived category of graded right A-modules. Given a complex Y, we use the notation h^{i}(Y) for the ith cohomology of Y. Let A^{∘} be the opposite algebra of A, and let A^{e} = A ⊗_{k} A^{∘}. A dualizing complex for a noetherian graded ring A is a bounded complex R ∈ D(Gr A^{e}) such that

(a) R has finite injective dimension over A and A^{∘},

(b) The A-bimodule h^{i}(R) is noetherian on both sides for all i ∈ ℤ, and (c) The natural maps A → RHom_{A}(R, R) and A → RHom_{A^{∘}}(R, R) are

isomorphisms in the derived category D(Gr A^{e}).

Let Γ_{𝔪} : Gr A^{e} → Gr A^{e} be the torsion functor lim_{n→∞} \underline{Hom}_{A}(A/𝔪^{n}, −). Writing 𝔪^{∘} = (A^{∘})_{≥1}, the functor Γ_{𝔪^{∘}} is defined similarly. If A has a dualizing complex R, then R is called balanced if there are isomorphisms R Γ_{𝔪}(R) ≅ A^{∗} and R Γ_{𝔪^{∘}}(R) ≅ A^{∗} in D(Gr A^{e}).

Yekutieli proved that if A is noetherian with a balanced dualizing complex R, then ω = h^{−(d+1)}(R) is a dualizing sheaf for Proj A, where R = π(R) [YZ1, 4.2(4)]. In addition, X is classically Cohen-Macaulay if and only if ω[d + 1] is isomorphic to R in the derived category D(QGr A) [YZ1, 4.2(5)], or equivalently if and only if R has nonzero cohomology in only one term.

A powerful criterion of Van den Bergh [VdB, 6.3] says that A admits a balanced dualizing complex if and only if A satisfies the left and right χ condition and X = Proj A has finite left and right cohomological dimension. Here, A is said to satisfy the (right) χ condition if dim_{k} \underline{Ext}_{A}^{i}(A/𝔪, M) < ∞ for all M ∈ gr A and all i ≥ 0, and similarly on the left. It follows from Van den Bergh’s criterion that many important classes of rings admit a balanced dualizing complex, although there are examples of noetherian rings which do not because they fail the χ condition [SZ], [KRS].

We may now prove Theorem 0.5(a).

Proposition 10.1. Let A be a connected graded, noetherian, projectively simple ring admitting a balanced dualizing complex. Then the associated projective scheme Proj A is classically Cohen-Macaulay.

Proof. By [VdB, 6.3], A satisfies the χ condition and has finite cohomological dimension on both sides, and moreover the balanced dualizing complex over A is given by

R = R Γ_{𝔪}(A)^{∗}.

Let X = Proj A. Note that R^{i} Γ_{𝔪}(N) ≅ H^{i−1}(X, N) for all N ∈ Gr A and all i ≥ 2 [AZ1, 7.2(2)]. Thus if cd(Proj A) = d, then R^{n} Γ_{𝔪} = 0 for n > d+1, and so h^{n}(R) = 0 for all n < −d − 1. Let R = π(R). Since the dualizing sheaf ω = h^{−(d+1)}(R) is necessarily nonzero, M := h^{−(d+1)}(R) is not finite dimensional over k. Let j_{0} = −d − 1 and put j = max{i | h^{i}(R) ≠ 0}. We want to show that j_{0} = j. If this is the case, then ω ≅ R[−d − 1] and we are done by [YZ1, 4.2(5)].

By the definition of a dualizing complex, h^{n}(R) is noetherian on both sides for every n. Thus for every n > j, h^{n}(R) is finite dimensional over k because h^{n}(R) = 0, but N := h^{j}(R) is not finite dimensional since h^{j}(R) ≠ 0.

Let Y be the truncation τ^{≤j}R and Z the truncation τ^{≥(j+1)}R. Then we have a distinguished triangle in the derived category D(Gr A)

Y → R → Z → Y [1]

which induces a long exact sequence

⋯ → Ext_{A}^{i}(Z, R) → Ext_{A}^{i}(R, R) → Ext_{A}^{i}(Y, R) → · · · .

Since h^{n}(Z) is finite dimensional and h^{n}(R) is noetherian for all n, it follows from the χ condition and induction on the lengths of the bounded complexes Z and R that Ext_{A}^{i}(Z, R) is finite dimensional for all i. If we show that Ext_{A}^{i}(Y, R) is infinite dimensional for some i, then it will follow that Ext_{A}^{i}(R, R) is also infinite dimensional. Since Y is bounded above at j with h^{j}(Y) = h^{j}(R) = N and R is bounded below at j_{0} with h^{j_{0}}(R) = M, every nonzero map from N to M induces a nonzero element in Ext_{A}^{j_{0}−j}(Y, R). Thus there is an injection of vector spaces

\underline{Hom}_{A}(N, M) → Ext_{A}^{j_{0}−j}(Y, R).

Now since N and M are not torsion, by Lemma 1.7(b) \underline{Hom}_{A}(N, M) is infinite dimensional. Hence Ext_{A}^{j_{0}−j}(Y, R) is infinite dimensional, whence Ext_{A}^{j_{0}−j}(R, R) is infinite dimensional. But since R is a dualizing complex, by definition we we have Ext_{A}^{i}(R, R) = 0 for all i ≠ 0. Thus j = j_{0}. □

11. The Gorenstein property

In this section, we will complete the proof of Theorem 0.5 by showing that if A is a projectively simple ring with a balanced dualizing complex, then the dualizing sheaf ω must be invertible. This is equivalent to X = Proj A being Gorenstein in the commutative case. In preparation for this result, we will first study graded bimodules over two rings.

Throughout this section, let A and B be noetherian finitely graded prime k-algebras, with graded quotient rings Q and T respectively. Given two noetherian (A, B)-bimodules M and N, we say M and N are p.isomorphic (short for projectively isomorphic), and write M ≅_{p} N, if there exists an isomorphism of bimodules f : M_{≥n} → N_{≥n} for some n ≫ 0. The map f induces an isomorphism π(M) → π(N) in QGr B (and similarly in QGr A^{∘}). A map f : M → N is called a p.isomorphism if f_{≥n} : M_{≥n} → N_{≥n} is an isomorphism for some n ≫ 0.

If M is a graded right B-module and N is a graded left B-module, the graded tensor product of M and N over B is denoted by M \underline{⊗}_{B} N. As usual, if M is a graded (A, B)-bimodule then M is called invertible if there is a graded (B, A)-bimodule N with bimodule isomorphisms M \underline{⊗}_{B} N ≅ A and N \underline{⊗}_{A} M ≅ B. Analogously, M is called p.invertible if there exists such a graded (B, A)-bimodule N with M \underline{⊗}_{B} N ≅_{p} A and N \underline{⊗}_{A} M ≅_{p} B, and N is called the p.inverse.

Next, we want to define a different but related notion of invertibility. We call a noetherian (A, B)-bimodule M generically invertible if

(a) M is Goldie torsionfree on both sides,

(b) M is evenly localizable in the sense that

Q \underline{⊗}_{A} M ≅ Q \underline{⊗}_{A} M \underline{⊗}_{B} T ≅ M \underline{⊗}_{B} T, and

(c) Q \underline{⊗}_{A} M = M \underline{⊗}_{B} T is an invertible (Q, T)-bimodule.

Although it is useful to state all three conditions in the above definition, we should note that in general condition (b) is actually a consequence of condition (a), as follows. Lemma 11.1. If M is a noetherian (A, B)-bimodule which is Goldie torsionfree on both sides, then M is evenly localizable.

Proof. First we claim that the (A, B)-bimodule M′ = Q \underline{⊗}_{A} M is Goldie torsionfree on the right. This is because every finitely generated graded right B-submodule of Q \underline{⊗}_{A} M is contained in qA \underline{⊗}_{A} M ≅ M for some homogeneous q ∈ Q.

Thus right multiplication by any element b ∈ B must induce a left Qmodule injection ψ_{b} : M′ → M′. Since _{Q}M′ is of finite rank over the gradedsemisimple ring Q, the map ψ_{b} must be a bijection for all b and so Q \underline{⊗}_{A} M ≅ Q \underline{⊗}_{A} M \underline{⊗}_{B} T. A symmetric argument gives M \underline{⊗}_{B} T ≅ Q \underline{⊗}_{A} M \underline{⊗}_{B} T. □

The reader may check that an invertible bimodule is generically invertible. The following key proposition is a kind of converse statement for projectively simple rings. The proof will be given below after several more lemmas. Proposition 11.2. Let A and B be connected graded, noetherian, projectively simple rings admitting balanced dualizing complexes. Then every generically invertible noetherian (A, B)-bimodule is p.invertible.

Let M be an (A, B)-bimodule. Then we may define the (B, A)-bimodules ∨M = \underline{Hom}_{B}(M_{B}, B_{B}) and M^{∨} = \underline{Hom}_{A^{∘}}(_{A}M,_{A}A). Induced by evaluation, we get a B-bimodule homomorphism

(11.3)  ϕ_{M} : ^{∨}M \underline{⊗}_{A} M → B

and an A-bimodule homomorphism

(11.4)  ξ_{M} : M \underline{⊗}_{B} M^{∨} → A,

both of which are graded morphisms of degree 0. The bimodules M^{∨} and ∨M will be our candidates for the p.inverse of M.

We shall need the following technical result, which says loosely that “Ext commutes with localization”. The proof is an easy generalization to the graded setting of ([BL, Proposition 1.6]). Lemma 11.5. Let B be a graded noetherian prime ring. Let T be the graded quotient ring of B. Let M_{B} be a finitely generated graded B-module and let N be a graded B-bimodule such that T \underline{⊗}_{B} N ≅ N \underline{⊗}_{B} T. Then

T \underline{⊗}_{B} \underline{Ext}_{B}^{i}(M_{B},_{B}N_{B}) ≅ \underline{Ext}_{T}^{i} (M \underline{⊗}_{B} T, N \underline{⊗}_{B} T)

for all i.

We next show that to prove Proposition 11.2 it is enough to know that M^{∨} and ^{∨}M are noetherian.

Lemma 11.6. Let A and B be two prime noetherian projectively simple rings. Let M be a generically invertible noetherian (A, B)-bimodule. If M^{∨} and ^{∨}M are noetherian on both sides, then M is p.invertible with p.inverse M^{∨} ≅_{p} ^{∨}M.

Proof. We will show that the evaluation map

ϕ_{M} : ^{∨}M \underline{⊗}_{A} M → B

of (11.3) is a p.isomorphism. By the definition of generic invertibility, the (Q, T)-bimodule L = M \underline{⊗}_{B} T ≅ Q \underline{⊗}_{A} M is invertible. By Morita theory, the inverse of L must be given by the (T, Q)-bimodule L^{−1} = \underline{Hom}_{T} (L, T_{T}). By Lemma 11.5, we have

T \underline{⊗}_{B} ^{∨}M ≅ \underline{Hom}_{T} (L, T_{T}) = L^{−1}.

Hence^{∨}M is nonzero and the the map ϕ_{M} is nonzero. Since B is projectively simple, the map ϕ_{M} must be surjective in large degree.

Now consider the map T \underline{⊗}_{B} ϕ_{M}. This map must be an isomorphism, since it may be identified with the chain of isomorphisms

T \underline{⊗}_{B} ^{∨}M \underline{⊗}_{A} M ≅ L^{−1} \underline{⊗}_{A} M ≅ L^{−1} \underline{⊗}_{Q}(Q \underline{⊗}_{A} M) ≅ T.

It follows that the left B-module ker ϕ_{M} is Goldie torsion. Since by assumption ^{∨}M is a noetherian bimodule, ^{∨}M \underline{⊗}_{A} M and thus ker ϕ_{M} are also noetherian bimodules. Hence ker ϕ_{M} is finite dimensional by Lemma 1.7, and ϕ_{M} is a p.isomorphism as we wished. The proof that the map ξ_{M} of (11.4) is a p.isomorphism is analogous. Finally, that ^{∨}M ≅_{p} M^{∨} is a formal consequence of the fact that ϕ_{M} and ξ_{M} are both p.isomorphisms. □

Although it is easy to see that ^{∨}M is left noetherian over B, in general there is no reason that ^{∨}M should be right noetherian over A. The dualizing complex provides the extra information needed to prove such a fact. Lemma 11.7. Let A and B be two graded noetherian rings admitting balanced dualizing complexes R_{A} and R_{B} respectively. Let M be a noetherian graded (A, B)-bimodule. If N is a graded noetherian right B-module, then \underline{Ext}_{B}^{i}(M, N) is a noetherian right A-module for all i.

Proof. First, note that by [VdB, 5.1 and 4.8], (11.8)

RHom_{A^{∘}}(M, R_{A}) ≅ R Γ_{𝔪_{A^{∘}}}(M)^{∗} ≅ R Γ_{𝔪_{B}}(M)^{∗} ≅ RHom_{B}(M, R_{B}).

The functor D(−) = RHom_{B}(−, R_{B}) gives a duality from D(Gr B) to D(Gr B^{∘}) which restricts to a duality between the subcategories of complexes with finitely generated cohomology groups [Ye, 3.4]. Thus RHom_{B}(M, R_{B}) is a complex of graded left B-modules with noetherian cohomologies. Applying the same reasoning to the duality functor RHom_{A^{∘}}(−, R_{A}) and using (11.8), we see that the cohomology groups of RHom_{B}(M, R_{B}) are also noetherian right A-modules.

Now since D(−) is a duality we have

RHom_{B}(M, N) ≅ RHom_{B^{∘}}(D(N), D(M)).

As we showed above, D(M) has noetherian cohomologies on the right, and so Ext_{B^{∘}}^{i}(D(N), D(M)) also has noetherian right cohomologies. Thus

Ext_{B}^{i}(M, N) ≅ Ext_{B^{∘}}^{i}(D(N), D(M))

is noetherian as a right A-module. □

Proof of Proposition 11.2. By Lemma 11.6 it suffices to show that M^{∨} and ∨M are noetherian bimodules. We only show this for ^{∨}M; the proof for M^{∨} is symmetric. It is clear that ^{∨}M = Hom_{B}(M_{B}, B_{B}) is a noetherian left B-module. By Lemma 11.7, Hom_{B}(M, B) is also noetherian as a right A-module. □

Let A be a graded ring with balanced dualizing complex R, and suppose that X = Proj A is classically Cohen-Macaulay. Set M = h^{−(d+1)}(R)/τ where d = cd(Proj A) and τ is the torsion submodule of h^{−(d+1)}(R). When we say that the dualizing sheaf ω = π(M) is invertible, we mean that M is an p.invertible (A, A)-bimodule in the sense defined earlier. Now we are ready to prove Theorem 0.5(b). Theorem 11.9. Let A be a connected graded, noetherian, projectively simple ring with a balanced dualizing complex R. Let ω be the dualizing bimodule of Proj A. Then ω is invertible and A = π(A) has finite injective dimension in the category QGr A.

Proof. Let M = h^{−(d+1)}(R)/τ as in the comments before the proof. Then M is torsionfree on both sides, so Goldie torsionfree on both sides by Lemma 1.7(a).

We need to show that M is p.invertible. We already know that M is Goldie torsionfree on both sides. Let Q be the graded fraction ring of A. Then M is evenly localizable to Q by Lemma 11.1. Since we have already proven that Proj A is classically Cohen-Macaulay in Proposition 10.1, we know that for every i ≠ −d − 1, h^{i}(R) is finite dimensional. By [YZ3, 6.2(1)], the complex Q⊗_{A} R ⊗_{A} Q is a graded dualizing complex for the ring Q, and since M is evenly localizable this complex is just a shift of M ⊗_{A} Q. Since Q is graded semisimple artinian with graded global dimension zero, a dualizing bimodule for Q is a progenerator on both sides. Thus M ⊗_{A} Q is invertible over Q and hence by definition M is generically invertible. Then by Proposition 11.2, M is p.invertible.

By the definition of a dualizing complex R has finite injective dimension. This implies that π(R) and hence ω has finite injective dimension in QGr A. Since M is a p.invertible bimodule, the functor − ⊗_{A} M^{∨} induces an autoequivalence of QGr A and maps ω to A. Therefore A has finite injective dimension in QGr A. □

Acknowledgments

We thank Mike Artin, Lawrence Ein, Dennis Keeler, János Kollár, Sándor Kovács, and Paul Smith, for helpful comments.

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(Reichstein) Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

   E-mail address: reichst@math.ubc.ca

(Rogalski) Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA

   E-mail address: rogalski@math.mit.edu

(Zhang) Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA

   E-mail address: zhang@math.washington.edu