Equivalence of germs (of mappings and sets) over k vs that over K
Pith reviewed 2026-05-22 21:10 UTC · model grok-4.3
The pith
If two map-germs or scheme-germs are equivalent over a faithfully flat ring extension K, then they are already equivalent over the base ring k.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For any faithfully flat ring extension k to K, two germs of mappings (or of sets) are right-equivalent, left-right-equivalent or contact-equivalent over K only if they are already equivalent over k. The same holds for scheme-germs via the reduction that contact equivalence of maps is equivalent to ambient isomorphism of their zero sets. The statements are proved for formal, analytic and Nash germs over arbitrary base rings k.
What carries the argument
Faithfully flat ring extension k to K, which is used to descend equivalences under the classical right, left-right and contact groups of singularity theory.
If this is right
- If a family of maps is trivial over K then it is already trivial over k.
- The same descent holds for scheme-germs via the zero-set reduction.
- The statements apply to deformations and unfoldings over arbitrary base rings.
- The result holds for arbitrary field extensions in any characteristic.
Where Pith is reading between the lines
- The descent may let one reduce questions about equivalences after algebraic closure or completion back to the original base.
- It supplies a tool for checking triviality of unfoldings without first extending the base ring.
- One could test the result by constructing explicit low-dimensional examples where the extension is finite flat but not split.
Load-bearing premise
That contact equivalence of maps is equivalent to ambient isomorphism of their zero sets, even when the target is singular.
What would settle it
An explicit pair of map-germs over a ring k that become equivalent after a faithfully flat base change to K but remain inequivalent over k.
read the original abstract
Consider real-analytic mapping-germs, (R^n,o)-> (R^m,o). They can be equivalent (by coordinate changes) complex-analytically, but not real-analytically. However, if the transformation of complex-equivalence is identity modulo higher order terms, then it implies the real-equivalence. On the other hand, starting from complex-analytic map-germs (C^n,o)->(C^m,o), and taking any field extension, C to K, one has: if two maps are equivalent over K, then they are equivalent over C. These (quite useful) properties seem to be not well known. We prove slightly stronger properties in a more general form: * for Maps(X,Y) where X,Y are (formal/analytic/Nash) scheme-germs, with arbitrary singularities, over a base ring k; * for the classical groups of (right/left-right/contact) equivalence of Singularity Theory; * for faithfully-flat extensions of rings k -> K. In particular, for arbitrary extension of fields, in any characteristic. The case ``k is a ring" is important for the study of deformations/unfoldings. E.g. it implies the statement for fields: if a family of maps {f_t} is trivial over K, then it is also trivial over k. Similar statements for scheme-germs (``isomorphism over K vs isomorphism over k") follow by the standard reduction ``Two maps are contact equivalent iff their zero sets are ambient isomorphic". This study involves the contact equivalence of maps with singular targets, which seems to be not well-established. We write down the relevant part of this theory.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for faithfully flat ring extensions k → K (including arbitrary field extensions in any characteristic), if two map-germs or scheme-germs are equivalent over K under right, left-right or contact equivalence, then they are equivalent over k. The setting covers formal/analytic/Nash germs with arbitrary singularities; the case of base rings k is emphasized for its implications on deformations. Scheme-germ statements are obtained from the map-germ statements via the reduction that two maps are contact-equivalent if and only if their zero sets are ambient-isomorphic. The paper develops the auxiliary theory of contact equivalence for maps whose targets may be singular.
Significance. If the claims hold, the results supply useful descent properties for equivalences in singularity theory that extend classical real/complex-analytic observations to arbitrary base rings and characteristics. The explicit treatment of contact equivalence for singular targets and the deformation-theoretic consequences (triviality over K implies triviality over k) are potentially valuable for unfolding theory. The manuscript supplies the relevant definitions and arguments rather than relying on an external reference for the singular-target case.
major comments (2)
- [section developing contact equivalence for singular targets] The scheme-germ statements rest on the claim that contact equivalence of maps is equivalent to ambient isomorphism of their zero sets (abstract, final paragraph). Because the paper simultaneously states that contact equivalence for maps to singular targets 'seems to be not well-established' and therefore develops the relevant theory, the manuscript must verify that the developed definitions of the contact group action and ambient isomorphism yield a true if-and-only-if correspondence without hidden smoothness hypotheses on the target. If only one direction is established, or if extra conditions appear when the target is singular or non-reduced, the scheme-germ claims do not follow from the map-germ claims.
- [proofs of the main theorems for map-germs] The central statements are implications 'equivalence over K implies equivalence over k' for faithfully flat k → K. The manuscript should make explicit where faithful flatness is used (e.g., in lifting coordinate changes or in the Nakayama-type arguments that descend the equivalence) and confirm that the same arguments apply when the target is singular; otherwise the reduction to the map-germ case may require additional verification.
minor comments (2)
- [abstract] The abstract refers to 'the standard reduction' without a forward reference to the precise statement or section where the if-and-only-if is proved for singular targets.
- [notation section] Notation for the contact group action on maps to singular targets should be introduced once and used consistently; currently the abstract and the body appear to employ slightly different conventions for the same objects.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points that will improve the clarity of the manuscript. We address each major comment below and will revise accordingly.
read point-by-point responses
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Referee: [section developing contact equivalence for singular targets] The scheme-germ statements rest on the claim that contact equivalence of maps is equivalent to ambient isomorphism of their zero sets (abstract, final paragraph). Because the paper simultaneously states that contact equivalence for maps to singular targets 'seems to be not well-established' and therefore develops the relevant theory, the manuscript must verify that the developed definitions of the contact group action and ambient isomorphism yield a true if-and-only-if correspondence without hidden smoothness hypotheses on the target. If only one direction is established, or if extra conditions appear when the target is singular or non-reduced, the scheme-germ claims do not follow from the map-germ claims.
Authors: We agree that an explicit verification of the if-and-only-if is necessary once the theory is developed for singular targets. In the relevant section we define the contact group action via the standard pullback construction on the ideal sheaf and define ambient isomorphism via the induced action on the zero-scheme; the equivalence then holds by construction for arbitrary (possibly singular or non-reduced) targets, with no smoothness hypotheses. We will add a short proposition (or remark) making this if-and-only-if explicit and confirming that both directions are obtained directly from the definitions. revision: yes
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Referee: [proofs of the main theorems for map-germs] The central statements are implications 'equivalence over K implies equivalence over k' for faithfully flat k → K. The manuscript should make explicit where faithful flatness is used (e.g., in lifting coordinate changes or in the Nakayama-type arguments that descend the equivalence) and confirm that the same arguments apply when the target is singular; otherwise the reduction to the map-germ case may require additional verification.
Authors: We will revise the proofs of the main theorems (Theorems 3.1 and 4.2 and their corollaries) to include explicit remarks indicating the precise uses of faithful flatness: (i) to descend the existence of coordinate changes via the faithful-flatness criterion for isomorphisms of modules, and (ii) in the Nakayama-type lifting arguments that descend infinitesimal equivalences. Because the statements and proofs are formulated entirely in the language of modules over the local rings of the (possibly singular) targets, the same arguments apply verbatim when the target is singular; no additional verification is required for the reduction step. revision: yes
Circularity Check
No circularity: derivation relies on algebraic properties of faithfully flat extensions and developed theory for singular targets
full rationale
The paper proves that equivalences (right/left-right/contact) of map-germs and scheme-germs over k are equivalent to those over K for faithfully flat k → K. The scheme-germ claims follow from the map-germ claims via the stated reduction to ambient isomorphism of zero sets; the paper explicitly develops the contact equivalence theory for maps with singular targets to make this reduction rigorous without hidden smoothness assumptions. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The central implications are established from the definition of faithfully flat morphisms and the developed group actions, making the derivation self-contained.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Faithfully flat ring extensions preserve and reflect the relevant equivalence relations for map-germs.
- domain assumption Two maps are contact equivalent if and only if their zero sets are ambient isomorphic.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
for faithfully-flat extensions of rings k→K ... for arbitrary extension of fields, in any characteristic
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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