Infinitesimal-Fiber Criteria for Isomorphisms of Scheme Germs

Definitions and mathematical statement

Let $X$ and $Y$ be Noetherian schemes, let $x\in X$ and $y\in Y$, and let $f:(X,x)\to(Y,y)$ be a morphism of scheme germs with $f(x)=y$. The supplied record denotes the scheme-theoretic infinitesimal fibers at order $m$ by $X_m(x)$ and $Y_m(y)$, and the induced morphism by \[ f_m:X_m(x)\longrightarrow Y_m(y). \] With this source notation retained, the question is whether \[ \left[\text{for every infinitesimal order }m,\quad f_m\text{ is an isomorphism}\right] \quad\Longrightarrow\quad f:(X,x)\xrightarrow{\sim}(Y,y). \] The implication is asked universally for morphisms of germs of Noetherian schemes satisfying the indicated hypothesis.

The supplied record does not define these infinitesimal fibers. One possible interpretation, suggested by the supplied reference labels concerning jets, is the fiber of a jet scheme over its order-zero evaluation map. Conditionally, if a common base field $k$ is specified, the jet scheme $J_m(X/k)$, when representable, represents the functor \[ T\longmapsto\operatorname{Hom}_k\!\left(T\times_{\operatorname{Spec}k}\operatorname{Spec}\left(k[t]/(t^{m+1})\right),X\right), \qquad m\in{\mathbb{Z}}_{\geq 0}. \] Evaluation at $t=0$ gives $\pi_m:J_m(X/k)\to X$, and the corresponding scheme-theoretic fiber would be \[ X_m(x)=J_m(X/k)\times_X\operatorname{Spec}\kappa(x), \] where $\kappa(x)$ is the residue field at $x$. This is an explicitly conditional interpretation: the record specifies neither a base field nor this construction nor the order-index convention.

The meaning of an isomorphism of scheme germs also requires clarification. Under the local-scheme convention, it means that the induced local homomorphism $f^\#: \mathcal O_{Y,y}\to\mathcal O_{X,x}$ is an isomorphism. A formulation requiring an isomorphism on neighborhoods must be distinguished from this convention. These unresolved choices prevent the displayed source question from being a fully specified mathematical assertion.

Short English statement

Does a morphism between germs of Noetherian schemes become an isomorphism whenever it induces isomorphisms on every infinitesimal fiber? The source leaves the fiber construction, its base, and the meaning of a scheme germ insufficiently specified; the jet interpretation above is conditional.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001347$, exact source problem code catalogue entry, titled "Infinitesimal-Fiber Criteria for Isomorphisms of Scheme Germs." No individual proposer is identified.
• Supplied report attribution: Singularities (2010), Oberwolfach Reports, pp. 2405--2470, DOI 10.4171/owr/2009/43. https://doi.org/10.4171/owr/2009/43. The record separately gives proposed year 2009.
• Supplied literature reference labeled "jet closures and local isomorphism": https://doi.org/10.1016/j.jalgebra.2018.01.008.
• Supplied literature reference labeled "jet closures preprint": https://arxiv.org/abs/1704.07494.
• Supplied literature reference labeled "arc closures and local isomorphism": https://arxiv.org/abs/1811.12577.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the local isomorphism problem is negative in general, positive in several geometric settings, and unsettled in its full Noetherian form. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment.