Injective endomorphisms of algebraic and analytic sets

We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in C and its holomorphic bijection which is not a biholomorphism. Introduction. The Zariski Main Theorem (see e.g. [4]) asserts that every injective morphism of normal algebraic varieties over an algebraically closed field of characteristic zero is an automorphism onto its open image. For arbitrary complex algebraic varieties regularity of the inverse of a regular biholomorphic bijection is ensured by the Serre Theorem [7, 9]. The simple example C 3 t → (t, t) ∈ {x − y = 0} shows that in general, bijective morphisms are not isomorphisms. From the example {xy−1 = 0} 3 (u, v)→ (u, 0) ∈ {y = 0} we see that an injective morphism of non-singular algebraic sets may not be surjective. Grothendieck proved in [5, Prop. 17.9. 6] a counterpart of the Zariski Main Theorem for the category of S-preschemes, where S is a fixed prescheme. His theorem says that every injective S-endomorphism of an Sprescheme of finite presentation is an automorphism. Note that his theorem is unapplicable to endomorphisms of algebraic sets. Our main result gives a geometric counterpart of the Grothendieck result; Theorem 2.2 says that every injective endomorphism of an affine algebraic variety over an algebraically closed field k of characteristic zero is an automorphism. This generalizes the well-known result of Bia lynicki-Birula and Rosenlicht [2] saying that every injective polynomial transformation of k is a polynomial automorphism. Note that surjectivity of an injective endomorphism of an affine algebraic variety was earlier proved by Ax [1]. In the proof Ax applied the so-called 1991 Mathematics Subject Classification: Primary: 14A10, 32C15; Secondary 14E99, 32C25. 30 S. Cynk and K. Rusek “transfer principle” and the metamathematical notion of an “elementary formula”. A topological proof of Ax’s Theorem was given by Borel in his unpublished paper [3]. One can also consider Theorem 2.2 as an analogue of the Clements– Osgood Theorem on analyticity of the inverse of a holomorphic bijection of a complex manifold. In Section 3 we show that there is no counterpart of Theorem 2.2 for the category of analytic sets. In order to obtain a suitable counterexample we construct an irreducible one-dimensional complex analytic space and its holomorphic but not biholomorphic bijection and next we prove that this analytic space is biholomorphic with an analytic curve in C. 1. Dominating regular mappings with finite fibres. Let F : V → W be a regular dominating mapping of irreducible affine algebraic varieties over k, i.e. F (V ) = W . Then the induced homomorphism F∗ : k[W ]→ k[V ] of rings of regular functions is injective, so it has a unique extension to a monomorphism of fields of rational functions, also denoted by F∗ : k(W )→ k(V ). If we assume additionally that dimV = dimW then the field k(V ) is a finite extension of the field F∗(k(W )); we put d(F ) = [k(V ) : F∗(k(W ))]. It is well known that d(F ) is equal to the number of points in the generic fibre of F (see e.g. [4]). In some cases we have more precise information about the number d(F ): Theorem 1.1. Let V and W be normal affine varieties and dimV = dimW . If F : V → W is a regular dominating mapping with finite fibres then d(F ) = max{#F−1(y) : y ∈W}. For the proof we need the following Lemma 1.2. Let V and W be irreducible affine algebraic varieties such that dimV = dimW and V is normal. Assume that F : V → W is a regular dominating mapping with finite fibres. Then, for every y ∈ W with F−1(y) = {x1, . , xm} ⊂ V , there exists a function f ∈ k[V ] which is integral over F∗(k[W ]) and f(xi) 6= f(xj) for i 6= j. P r o o f. By [4, Corollary 1, p. 136] we can find a normal affine algebraic variety W ′, a regular injective mapping F ′ : V → W ′ and a regular finite mapping G : W ′ →W such that F = G ◦ F ′. Then the points {F (xj) : j = 1, . ,m} ⊂ W ′ are pairwise different, so there exists g ∈ k[W ′] such that g(F (xi)) 6= g(F (xj)) for i 6= j. Since G is finite, the ring k[W ′] is integral over the ring G∗ (k[W ]). Therefore f = g ◦ F ′ = F ′ ∗(g) has all desired properties. P r o o f o f T h e o r e m 1.1. Take a y ∈ W and the function f ∈ k[V ] from Lemma 1.2 which is integral over the ring F∗(k[W ]) and separates the Injective endomorphisms 31 points in the fibre F−1(y) = {x1, . , xm} ⊂ V . In the extension of integral rings F∗(k[W ]) ⊂ k[V ] the ring F∗(k[W ]) is normal. One can easily verify that f ∈ k[V ] is algebraic over the field F∗(k(W )) and the coefficients of its minimal equation of degree d over F∗(k(W )) lie in F∗(k[W ]). Since all functions from F∗(k[W ]) are constant on the set F−1(y) whose elements are separated by f , we have m ≤ d ≤ d(F ). Hence d(F ) = max{#F−1(y) : y ∈W}. 2. Bijectivity implies biregularity. Assume that all varieties occurring in this section are defined over a field k (algebraically closed of characteristic zero). Lemma 2.1. Let V be an affine algebraic variety and let V = V1∪ . ∪Vs be its decomposition into irreducible components. Denote by πi : V̂i → Vi a normalization of Vi for i = 1, . , s. Then MV = {f : V → k : f ◦ πi ∈ k[V̂i] for i = 1, . , s} is a noetherian k[V ]-module (under the standard operations). P r o o f. Since k[V̂i] is a noetherian k[Vi]-module, it is also a noetherian k[V ]-module for every i = 1, . , s (under the multiplication rf = (r ◦ πi)f for r ∈ k[V ], f ∈ k[V̂i]. ) The mapping MV 3 f → (f ◦ π1, . . . , f ◦ πs) ∈ s ⊕