Affine Pseudocoverings and Their Endomorphisms

Definitions and mathematical statement

Over the source's intended ground field $k$, let $\mathbb A^2=\operatorname{Spec}k[x,y]$ and let $X$ be a smooth affine surface. Suppose there is a factorization \[ \mathbb A^2\xrightarrow{\ \varphi_1\ }X \xrightarrow{\ \varphi_2\ }\mathbb A^2, \qquad \varphi=\varphi_2\circ\varphi_1, \] where $\varphi$ is an étale affine-plane endomorphism and both $\varphi_1,\varphi_2$ are étale. An étale morphism is locally of finite presentation, flat and unramified; between smooth surfaces it has relative dimension zero and an isomorphism on the appropriate tangent spaces. The ground field and characteristic are not specified in the extracted statement.

The source asks to determine what properties $X$ and the two factor maps must have in this affine pseudo-covering construction. This is a broad classification/constraint request: no invariant, proposed necessary-and-sufficient list or single yes/no conclusion is stated. If an equivalence of such factorizations is intended, it must specify whether isomorphisms $\alpha:X\to X'$ are required to satisfy $\alpha\varphi_1=\varphi'_1$ and $\varphi'_2\alpha=\varphi_2$, and whether changes of coordinates on the two affine planes are allowed.

Étaleness must not be silently strengthened to finiteness, properness, surjectivity or invertibility. For example, in the usual characteristic-zero polynomial setting, an étale endomorphism has nonzero constant Jacobian determinant, but calling it an automorphism would add a substantial unstated conclusion. The two factors are morphisms between different surfaces, even though the original prose loosely calls them endomorphisms.

The supplied literature assessment describes $X$ as an affine pseudo-plane and mentions special subclass classifications, including torus-action cases. The restored statement only assumes a smooth affine surface with the displayed étale factorization; no definition or additional fibration/topological conditions for "pseudo-plane" are supplied. Those subclass assumptions must not be imported from a reference title. The triage checked 21 August 2026 reports no general classification resolving the broad request, which is a dated supplied finding rather than an independent current literature determination.

Short English statement

Which smooth affine surfaces can occur between two étale maps factoring an étale endomorphism of the affine plane, and what constraints hold on the factors? The record specifies the factorization but no precise classification invariant. Étale does not mean finite or invertible, and pseudo-plane subclass results need additional hypotheses.

Sources

• Local UnsolvedMath registry, permanent ID 30000222, exact source code catalogue entry; Affine Pseudocoverings and Their Endomorphisms. Proposed year 2005, status open. The supplied corrected_verified statement restores the smooth affine surface, factorization and étaleness of both maps.
• Supplied report citation: Kommutative Algebra (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/19, pp. 1073--1126, https://doi.org/10.4171/owr/2005/19. The title year differs from DOI/proposed year; the verification's claimed direct inspection is metadata, not an inspection performed here.
• Supplied literature triage checked 2026-08-21: "original Oberwolfach Report 19/2005", https://doi.org/10.4171/owr/2005/19; "Masuda--Miyanishi, affine pseudo-planes with torus actions", https://doi.org/10.1007/S00031-005-1108-3. This dated OWR-derived assessment requests independent verification of time-sensitive status and cited claims.