# For a Keller map, invertible, injective and proper are the same

- kind: theorem
- status: proved (evidence tier V3)
- authors: Hadamard and Caccioppoli (global inverse function theorem); Keller (1939); Ax (1968) and Grothendieck (1966); Cynk–Rusek (1991); Rudin (1995)
- in one line: Keller F on ℂⁿ: automorphism ⇔ injective ⇔ proper ⇔ birational

**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).[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](/wiki/Statement:keller-birational.md).

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](/wiki/Statement:jc-n.md). Where properness fails is the [non-properness set](/wiki/Statement:jelonek-set.md).

Neither hypothesis can be dropped casually. [Pinchuk's real maps](/wiki/Statement:pinchuk-real.md) 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.

## Relations

- uses [keller-birational](/wiki/Statement:keller-birational.md) — (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.
- related to [jc-n](/wiki/Statement:jc-n.md) — 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.

## References

1. 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.

## Relations to this page

- used by [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.md) — Weighted scaling then makes it proper, and a proper Keller map is an automorphism.
- used by [jelonek-set](/wiki/Statement:jelonek-set.md) — If $S_F$ is empty, the map is proper, and a proper Keller map is an automorphism.

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