Difference between revisions
proper-etale-bijective
Status and infobox
- status:
—→ proved - tier:
—→ V3 - kind:
—→ theorem - statement:
—→ For a polynomial map $F:\mathbb C^n\to\mathbb C^n$ with $\det JF\in\mathbb C^{\times}$, these are equivalent: (i) $F$ is a polynomial automorphism; (ii) $F$ is injective; (iii) $F$ is proper; (iv) $\mathbb C(F)=\mathbb C(x)$. - short:
—→ Keller F on ℂⁿ: automorphism ⇔ injective ⇔ proper ⇔ birational
Relations
- + uses keller-birational: (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.
- + related to jc-n: So the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper.
Source +21 −0
+---
+title: For a Keller map, invertible, injective and proper are the same
+short: Keller F on ℂⁿ: automorphism ⇔ injective ⇔ proper ⇔ birational
+kind: theorem
+status: proved
+tier: V3
+authors: Hadamard and Caccioppoli (global inverse function theorem); Keller (1939); Ax (1968) and Grothendieck (1966); Cynk–Rusek (1991); Rudin (1995)
+statement: For a polynomial map $F:\mathbb C^n\to\mathbb C^n$ with $\det JF\in\mathbb C^{\times}$, these are equivalent: (i) $F$ is a polynomial automorphism; (ii) $F$ is injective; (iii) $F$ is proper; (iv) $\mathbb C(F)=\mathbb C(x)$.
+---
+This is the closed-universe theorem: a Keller map that loses nothing to infinity is invertible.
+- **(iii) ⇒ (ii).** A proper local biholomorphism of ℂⁿ is a covering map. Since ℂⁿ is simply connected, the covering has one sheet (the Hadamard–Caccioppoli global inverse function theorem).
+- **(ii) ⇒ (i).** An injective polynomial self-map of ℂⁿ is bijective (Ax 1968; Grothendieck, EGA IV) and has a polynomial inverse (Cynk–Rusek 1991; Rudin 1995).<ref>S. Cynk, K. Rusek, Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math. 56 (1991) 29–35; [Ax–Grothendieck theorem](https://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem), Wikipedia.</ref>
+- **(ii) ⇒ (iv).** An injective étale map has one-point generic fibers, so it is birational.
+- **(iv) ⇒ (i).** This is [[uses::Statement:keller-birational|Keller's birational theorem, which turns a birational Keller map into an automorphism]].
+
+So [[related::Statement:jc-n|the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper]]. Where properness fails is the [[Statement:jelonek-set|non-properness set]].
+
+Neither hypothesis can be dropped casually. [[Statement:pinchuk-real|Pinchuk's real maps]] are not proper. The entire map $(e^x,ye^{-x})$ has Jacobian 1 and is periodic in $x$ with period $2\pi i$, so it is not injective.
+
+<references />
+
Revision 277 as rendered
This is the closed-universe theorem: a Keller map that loses nothing to infinity is invertible.
- (iii) ⇒ (ii). A proper local biholomorphism of ℂⁿ is a covering map. Since ℂⁿ is simply connected, the covering has one sheet (the Hadamard–Caccioppoli global inverse function theorem).
- (ii) ⇒ (i). An injective polynomial self-map of ℂⁿ is bijective (Ax 1968; Grothendieck, EGA IV) and has a polynomial inverse (Cynk–Rusek 1991; Rudin 1995).[1]
- (ii) ⇒ (iv). An injective étale map has one-point generic fibers, so it is birational.
- (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.
So the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper. Where properness fails is the non-properness set.
Neither hypothesis can be dropped casually. Pinchuk's real maps are not proper. The entire map has Jacobian 1 and is periodic in with period , so it is not injective.
- ↑ S. Cynk, K. Rusek, Injective endomorphisms of algebraic and analytic sets, Ann. Polon. Math. 56 (1991) 29–35; Ax–Grothendieck theorem, Wikipedia.