Relative A 1-Contractibility of Koras-Russell Prototypes and Exotic Motivic Spheres

The Koras-Russell threefolds are smooth affine contractible threefolds exhibiting exotic behavior in the algebro-geometric context. Our goal in this note is to extend its A1-contractibility from a field to a general base scheme. As a consequence, we also give a general strategy to extend the A1-contractibility of Koras-Russell prototypes in higher dimensions over a general base scheme. A major consequence is that we establish the existence of exotic motivic spheres in all dimensions at least 4 over infinite perfect fields.

Motivation for Extension and Exotic Varieties

The paper addresses the problem of characterizing the affine n-space A n k among smooth affine A1-contractible n-schemes over a field k. The Koras-Russell threefolds K are cited as a counterexample to the uniqueness of A3 k over fields of characteristic zero. The authors aim to extend the A1-contractibility of K from a field of characteristic zero to arbitrary fields. This extension leads to results over the integers SpecZ and Noetherian base schemes S.

Extension of A1-Contractibility

The authors establish that for any perfect field k, the canonical morphism f: K → Spec k is an A1-weak equivalence in Spck. This implies that K is an A1-contractible scheme that is not isomorphic to A3 Z. Furthermore, for any Noetherian scheme S with perfect residue fields, the canonical morphism K → S is an A1-weak equivalence in SpcS of relative dimension 3. The generalized Koras-Russell varieties Xm(n, ψ) are shown to be A1-contractible over any perfect field k.

Existence of Exotic Motivic Spheres

The study focuses on the existence of exotic motivic spheres, defined as smooth n-dimensional schemes that are A1-homotopic but not isomorphic to A n k 0. The generalized Koras-Russell varieties Xm(n, ψ) pave the way for producing counterexamples to the question of whether two smooth schemes X≃A1 A n S 0 are A1-homotopic but not isomorphic as S-schemes. Specifically, Theorem 6.8 states that for any infinite perfect field k, the strictly quasi-affine variety Xm 0 is an exotic motivic sphere.

Motivic Homotopy Theory Tools

The paper utilizes concepts from motivic homotopy theory, including the A1-homotopy category H(S), and the functor Lmot which is the left adjoint of the inclusion into H(S). The base change properties are crucial, with Proposition 3.10 detailing how f∗ and f⋆ induce Quillen adjunctions between model categories. The Milnor-Witt K-theory is introduced as a tool to describe the endomorphism ring of the motivic spheres A n 0.

Construction and Conclusion

The authors construct an infinite collection of pairwise non-isomorphic, stably isomorphic, A1-contractible smooth affine schemes over arbitrary base schemes in relative dimension ≥ 3. This collection provides a moduli of arbitrary positive dimension of pairwise non-isomorphic, stably isomorphic, A1-contractible smooth affine schemes over arbitrary base schemes in relative dimension ≥ 3. The final result is that for every m ≥ 0, the strictly quasi-affine variety Xm 0 is A1-homotopic to A m+3 k 0 but is not isomorphic to A m+3 k 0 as a k-scheme. The question of whether these varieties are necessarily isomorphic as S-schemes remains open in relative dimension 2.

Exotic Motivic Spheres

The paper concludes by showing that the prototypes studied in Section 5 provide counterexamples to Question 6.3 in every dimension ≥ 4. The existence of exotic motivic spheres in relative dimensions ≥ 3 over an appropriate base scheme S is anticipated based on the failure of analogous gluing lemmas for motivic spheres.

Appendix A

Appendix A provides lemmas concerning model categories, including the very weak five lemma and a lemma on the functor F taking trivial cofibrations between cofibrant objects to weak equivalences. Lemma A.3 states that for any affine S-scheme X, the canonical morphism f: X → S is an affine morphism which is locally of finite presentation. The bibliography lists numerous foundational works in algebraic geometry and motivic homotopy theory. The paper includes results on Milnor-Witt K-theory, including the residue homomorphism and the motivic Brouwer degree. The work builds upon established results such as those by Morel and Voevodsky regarding A1-homotopy theory. The bibliography also includes references to the Stacks Project and foundational texts like Lurie's Higher Topos Theory. The paper is supported by various collaborations and funding sources. The work is presented at the Department of Mathematics Federiques Enriques, University of Milan, Italy & IMB UMR5584, Université de Bourgogne-Europe, Dijon, France. The email address for the author is krishna.madhavan@unimi.it & krishna.madhavan@ube.fr. The URL is https://krishmv.github.io/. The paper is published on arXiv as 2510.21594v1 and was uploaded on 24 Oct 2025. The paper is in the math.AG category and was submitted on arXiv as 2510.21594v1. The paper is dated 24 Oct 2025 and the arXiv ID is 2510.21594v1. The paper is in the math.AG category. The paper was uploaded on 24 Oct 2025.

How the training signal is built

The construction of the family of smooth affine varieties Xm(n, ψ) involves a coordinate ring Rm(n, ψ) defined by x = Qm i=0 xi and x n = Qm i=0 x ni i z + y r + t s + x0 psi(x). The smoothness of Xm is a consequence of the Jacobian criterion owing to the assumption that psi(0)!= 0. The proof for A1-contractibility relies on showing that the inclusion maps i and j are A1-weak equivalences in Spck. This is achieved by relating the quotient space to Th(NPmHm) and Th(NWmXm).

Where the gains come from

The main gain comes from proving that the composite map i ◦ g: A n 0 → A n 0 is an A1-weak equivalence in Spck. This is achieved by showing that the induced map on the Milnor-Witt K-theory sheaves i∗: H1(K L, KMW n) → H1(A n 0, KMW n) is an isomorphism. This isomorphism is established by showing that the induced map on the Milnor-Witt K-theory sheaves i∗: H1(K L, KMW n) → H1(A n 0, KMW n) is an isomorphism. This implies that j∗ is an isomorphism.

The main objective of this section is to prove the existence of exotic motivic spheres in higher dimensions. The results produce a moduli of arbitrary positive dimension of pairwise non-isomorphic, stably isomorphic, A1-contractible smooth affine schemes over arbitrary base schemes in relative dimension ≥ 3. This demonstrates that the generalized Koras-Russell varieties pave the way for producing counterexamples to this question in higher dimensions. The study aims to close this problem once and for all.

Summary of Key Findings

The paper establishes that the family of Koras-Russell threefolds K is relatively A1-contractible in H(Z), meaning K is an A1-contractible scheme in H(Z) that is not isomorphic to A3 Z. The results extend this to arbitrary perfect fields, showing that all fibers of the morphism f: K → S are A1-contractible over the corresponding (residue) fields. The key finding is that for every m ≥ 0, the strictly quasi-affine variety Xm 0 is A1-homotopic to A m+3 k 0 but is not isomorphic to A m+3 k 0 as a k-scheme. This result closes the question 6.3 in all dimensions ≥ 4 over fields using the generalized Koras-Russell varieties. The paper demonstrates that there are exotic motivic spheres in relative dimensions ≥ 3 over an appropriate base scheme S.

The existence of exotic motivic spheres in higher dimensions is shown by the prototypes studied in Section 5. The results produce a moduli of arbitrary positive dimension of pairwise non-isomorphic, stably isomorphic, A1-contractible smooth affine schemes over arbitrary base schemes in relative dimension ≥ 3. This demonstrates that the generalized Koras-Russell varieties pave the way for producing counterexamples to this question in higher dimensions. The study aims to close this problem once and for all.

The paper demonstrates that there are exotic motivic spheres in relative dimensions ≥ 3 over an appropriate base scheme S. The results produce a moduli of arbitrary positive dimension of pairwise non-isomorphic, stably isomorphic, A1-contractible smooth affine schemes over arbitrary base schemes in relative dimension ≥ 3. This demonstrates that the generalized Koras-Russell varieties pave the way for producing counterexamples to this question in higher dimensions. The study aims to close this problem once and for all.

1. Improved system can determine if a smooth affine variety is A1-contractible over arbitrary base schemes by checking if it is an A1-weak equivalence in SpcS or an isomorphism in H(S).

2. Improved system can construct examples of exotic motivic spheres in all dimensions at least 4 over infinite perfect fields using the generalized Koras-Russell varieties as prototypes.

3. Improved system can test whether a smooth scheme X over a base scheme S is A1-homotopic to AnS without being isomorphic to AnS without needing an explicit isomorphism proof, leveraging the result that the prototypes studied in Section 5 provide counterexamples to the Question 6.3 in every dimension at least 4.

4. Improved system can classify smooth affine varieties over arbitrary base schemes by checking if the morphism f: X -> S is an A1-weak equivalence in SpcS, which is equivalent to checking if for every point s of S with residue field kappa(s), the scheme theoretic fiber of f over s is a smooth A1-contractible kappa(s)-scheme.

5. Improved system can verify relative A1-weak equivalences between smooth schemes using the two out of three axiom for weak equivalence in a model category when appropriate, as stated in Corollary 3.14.

The Koras-Russell threefolds are a certain family of smooth, affine contractible threefolds exhibiting "exotic" behavior in the algebro-geometric context. Our goal in this note is to extend its A 1-contractibility from a field to a general base scheme. As a consequence, we also give a general strategy to extend the A 1-contractibility of Koras-Russell prototypes in higher dimensions over a general base scheme. As a major consequence, we establish the existence of "exotic" motivic spheres in all dimensions at least 4 over infinite perfect fields.