Equivariant Ax Problem for Affine Varieties

Definitions and mathematical statement

The exact extraction, labelled the equivariant Ax problem in the original, asks whether the following implication holds. Let $X$ be a normal affine ${\mathbb{C}}$-variety with effective algebraic $G$-action, let $\varphi:X\to X$ be $G$-equivariant and let $\psi:X//G\to X//G$ be induced. Does $\psi$ being an automorphism force $\varphi$ to be an automorphism, without assuming $\varphi$ unramified?

If $A={\mathbb{C}}[X]$ and the invariant quotient is the intended affine variety $X//G=\operatorname{Spec}(A^G)$, equivariance makes $\varphi^*$ preserve $A^G$ and $\psi^*=\varphi^*|_{A^G}$. Thus the displayed target is to deduce bijectivity of $\varphi^*:A\to A$ from bijectivity on invariants. The group class and finite-generation assumptions for the quotient are not specified; reductive groups provide a standard setting but are not silently added.

The literal stated hypotheses are insufficient. Take $X=\mathbb A^1_{\mathbb{C}}$, $G=\mathbb G_m$ acting by scalar multiplication, and $\varphi(z)=0$. This is a normal affine variety with an effective algebraic action, and the constant map is equivariant because zero is fixed. Its invariant ring is ${\mathbb{C}}$, so $X//G$ is a point and $\psi$ is its identity automorphism, while $\varphi$ is not an automorphism. Equivalently $z\mapsto0$ is noninjective on ${\mathbb{C}}[z]$ but the identity on the invariants. The example even has reductive $G$.

Consequently the extracted unrestricted implication cannot be treated as a nontrivial open assertion without further source qualifications. Dominance, restrictions on orbit/quotient geometry or another map hypothesis might belong to the intended problem, but none is supplied and no repair is selected. The removed unramified hypothesis is not reintroduced to bypass this issue.

The literature triage checked 21 August 2026 reports no authoritative complete resolution found, but does not address this elementary formulation counterexample. Its open label is retained as metadata concerning the intended source direction, not as validation of the literal implication. This diagnosis does not claim to resolve a stronger omitted-hypothesis version.

Short English statement

The extracted implication is false under its literal hypotheses: scalar multiplication on $\mathbb A^1$ makes the quotient a point, while the equivariant constant-zero map is not invertible. The intended equivariant Ax problem therefore needs source qualifications, which are not silently supplied or replaced by unramifiedness.

Sources

• Local UnsolvedMath registry, permanent ID 30000662, exact source code catalogue entry; Equivariant Ax Problem for Affine Varieties. Proposed year 2007, status open. Corrected_verified editing only restores notation/removes the label; neither dominance nor orbit restrictions are stated, and omission of unramifiedness is explicit.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/01, pp. 5--82, https://doi.org/10.4171/owr/2007/01.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2007/01, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no precise additional-hypothesis formulation is supplied.