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Endomorphisms of Symbolic Algebraic Varieties J. Eur. Math. Soc. 1, 109–197 c Springer-Verlag & EMS 1999 M. Gromov Endomorphisms of symbolic algebraic varieties Received August 3, 1998 / final version received January 22, 1999 Abstract. The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investi- gated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. 1. Ax’ surjunctivity theorem 1.A. Strict embeddings and surjunctivity. A map between sets is called a strict embedding, denoted f : X ⊂ Y, if it is one-to-one but not onto. 6= Then, following Gottschalk, (see [Gott, 1972]) a map f : X → Y is called surjunctive if it is not a strict embedding. In other words f is surjunctive iff it is either surjective or non-injective. 1.B. Theorem [Ax]1. Every regular selfmapping of a complex algebraic variety X is surjunctive. In other words no X admits a strict embedding X ⊂ X. Or, put it yet 6= another way, “one-to-one” implies “onto” for every regular map f : X → X. If X = Cn this specializes to the following earlier result by Bialynicki- Barula and Rosenlicht, (see [BB-R, 1962]). 1.B0. Every complex polynomial self-mapping of Cn is surjunctive. Repeat, this signifies that no strict polynomial embedding Cn ⊂ Cn is 6= possible, i.e. every injective polynomial map Cn → Cn is surjective. M. Gromov: IHES, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France, e-mail:[email protected] Mathematics Subject Classification (1991): 14A10, 14E10, 05C25, 03C45 110 M. Gromov 2. Proalgebraic spaces and their endomorphisms A complex proalgebraic space means a projective limit X of a projective system of complex algebraic varieties Xi where i runs over a directed set I. The simplest and most important case is where I = N ={1,2,...}. Here our projective system can be given by a string of C-varieties Xi and connecting regular maps, also called projections, X1 ← X2 ← ... ← π2 π3 πi Xi ← .... Then the projective limit X = lim Xi consists of the sequences πi+1 ← xi ∈ Xi such that πi(xi ) = xi−1 for all i = 2, 3,.... For example, the complex Euclidean spaces Ci, i = 1, 2,..., form such a projective system i+1 i ∞ for the natural projections πi+1 : C → C and the limit, denoted C , consists of all infinite sequences (z1, z2,... ,zi ...)for zi ∈ C. Recall that a partially ordered set I is called directed,ifforeverytwo elements i1, i2 ∈ I there exists a third one, say i3 ∈ I, dominating both of them, i ≥ i , i . In other words, the positive cones I + ={i∈I,i≥i }and 3 1 2 i1 1 I + ={i∈I,i≥i}have a non-empty intersection. A subset in a directed i2 2 set I is called cofinal if it intersects every positive cone in I. A subset in I is called residual, if it contains some positive cone. A partially defined order preserving map between directed sets, say I ⊃ I 0 −→ J , is called directed α if its domain of definition I 0 is residual in I and its image is cofinite in J.If J0 ⊂ Jis residual then its pullback α−1(J0) ⊂ I 0 ⊂ I is, obviously, residual in I. Consequently, a composition of directed maps I ⊃ I 0 −→ J and α J ⊃ J0 −→ K is directed with the domain of definition I 00 = α−1(J0) ⊂ I. β A proregular map f = f∞ between two proalgebraic spaces X = X∞ = lim X and Y = Y∞ = lim Y is given, by definition, by a directed map ← i ← j between the underlying directed sets, I ⊃ I 0 → J, denoted i 7→ j = j(i) and by regular maps fi : Xi → Y j such that all diagrams commute −→ Xi1 Y j 1 π ↓↓π i 1 , i 2 j1,j2 −→ Xi2 Y j 2 π π where i1,i2 and j1, j2 are the connecting maps of our projective systems defined for all i1 ≥ j2 and j1 ≥ j2. (In the above case of I = N we have π = π ◦ π ◦···◦π ). i1,i2 i2 i2−1 i1 Clearly, one can compose proregular maps with the composition being proregular. 2.A. Example : polynomial mappings C∞ →C∞. A function p : C∞ →C is called a polynomial if it is a polynomial depending on finitely many , ,... , ∈C , ... complex variables, say zi1 zi2 zik among the infinity of z1 z2 making up C∞. In other words p is obtained by composing the projection Endomorphisms of symbolic algebraic varieties 111 of C∞ to some Ck with an ordinary complex polynomial Ck → C.Then ∞ apolynomial map f of C into itself, f : z = (z1, z2,...zi ...) 7→ 0 = ( 0 , 0 ,... 0 ...) 0 = ( ) z z1 z2 zi is defined by zi pi z for some polynomials pi on C∞. It is easy to see that these polynomial maps C∞ → C∞ are the same as proregular self mapping (endomorphisms) of C∞ thought of as a proalgebraic space. 2.B. Our central problem. Find simple conditions which ensure surjunc- tivity of a proregular endomorphism of a proalgebraic space. First, let us make clear that some conditions are unavoidable. 2.B0. Counterexamples. (a) The complement X = C\{0, 1, 2,...}is pro- algebraic being the decreasing intersection of quasi-affine varieties Xi = C\{0, 1 ...i}. And the (obviously proregular) map z 7→ z −1 is one-to-one but not onto. ∞ ∞ (b) The polynomial map f : C → C for f :{z1,z2,...}7→ {0,z1,z2,...}is also one-to-one but not onto. The above may look discouraging. Yet we shall see that surjunctivity is rather typical for maps commuting with a sufficiently large automorphism group (or pseudo-group) acting on X. Here is our basic example. 2.C. Endomorphisms of X∞ = X1. Let X be an arbitrary (finite dimen- sional !) complex algebraic variety and 1 be a countable set. Then the infinite Cartesian power X = X1, i.e. the space of all X-valued functions on 1, comes along with a natural proalgebraic structure. One can see this, for example, by enumerating 1, i.e. by bijecting N ={1,2,...}↔1and thus identifying X1 with { , ,... } ∞ N i 1 2 i X = X ={x ,x ,... ,x ,...} ∈ =lim X = X . 1 2 i xi X ← Or, more invariantly, one can use the directed set I of all finite subsets ⊂ 1 (I can not bring myself to denote them i ⊂ 1, I rather have ∈ I). Here one has the projective system {X,∈ I}with the projections X → X6 for all ⊃ 6 corresponding to restrictions of functions x = x(ω) ∈ X from to 6. One can see that the projective limit of X is isomorphic to the above X∞ =lim Xi (where, recall, the Cartesian powers Xi form ← a projective system with the projections X ← X2 ← ... ← Xi ← ...) in the category of proalgebraic spaces and proregular maps. In fact, take an exhaustion of 1 by finite subsets 1 ⊂ 2 ⊂ ... ⊂ i ⊂ ...,∪ i = 1. i Then the projective limit lim X i obviously equals the projective limit ← lim X over all since the subsets exhausting 1 are cofinal in the set of ← i 112 M. Gromov i Ni all finite ’s in 1. On the other hand the spaces X = X , Ni = card i, i i make a cofinal subsystem in {X } = , ,... andsolimX i =lim X . i 1 2 ← ← Now, suppose we have a group 0 acting on 1 and thus on X = X1 and let f : X → X be a proregular 0-equivariant map. 2.C0.Theorem.If 0 is a locally compact (e.g. discrete) amenable (e.g. solvable) group and the action of 0 on 1 has finitely many orbits (e.g. transitive), then f is surjunctive, i.e. one-to-one ⇒ onto for these maps f . Remark. In many examples the isotropy subgroups 0δ ⊂ 0, δ ∈ 1,are compact and then 1 admits a structure of a locally finite graph invariant under 0. Yet we do not have to make this assumption as the general case of the theorem trivially reduces to the one where 0δ are compact. We prove the theorem in 5.M00 exploiting Ax’ idea of reduction surjunc- tivity from the algebraic category to finite sets X. 2.C00 Question.Are0-equivariant proregulator selfmappings of X0 are surjunctive for all, not necessarily amenable, groups 0 acting on 1 with finitely many orbits ? One knows, this is true for initially subamenable (e.g. residually amen- able) groups (see 4.G) but the question remains open even in the more traditional framework of symbolic dynamics,i.e.forfinite sets X, where the problem was raised by Gottschalk in [Gott] and persued in [La]. 2.C000 Generalizations. Theorem 2.C0 extends to some endomorphisms of proalgebraic spaces where the global symmetry group is replaced by partial symmetries. The spaces we consider in this paper arise starting from X1 where 1 is a countable connected graph with bounded valency,i.e.havingat most d < ∞ edges at each vertex. Such a 1 has many partial symmetries γ i.e. graph isomorphisms γ : D ↔ D0 between finite subgraphs D and D0 in 1 and we require our f to be compatible with the transformations 0 X D ↔ X D induced by some of these γ . Then we consider a certain space 0 X◦ of “orbits” of these transformations X D ↔ X D wherewehaveanatural map f ◦ : X◦ → X◦.
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