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An obstruction to lifting schemes to spectral schemes

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Pith's one-line read A scheme over the integers can lift to the sphere only if it carries a compatible system of p-adic Frobenius lifts.

desk verdict Clean, usable scheme-level obstruction from the Tate Frobenius; the new content is the descent theory and the concrete non-liftability statements. read the letter →

arxiv 2607.11853 v1 pith:I4ZDWEEV submitted 2026-07-13 math.AG

Robert Szafarczyk This is my paper · ORCID
classification math.AG MSC 14Fxx55P4313D03
keywords spectralschemesMoorespectrahat-deltaringsTate-valuedFrobeniusliftingobstructiongroupnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper gives a uniform algebraic test that decides whether an ordinary scheme over the integers can arise by base change from a spectral scheme over the sphere spectrum. The test is the existence of a hat-delta structure: on every p-adic completion of every affine open one must be able to lift the Frobenius endomorphism in a way that is compatible with localization and gluing. The same condition is necessary for morphisms and for group-scheme structures. With it the author rules out lifts for every ring of integers of a number field, for the additive group and for GLn (n greater than 1), and for many closed subschemes of projective space, while recovering the known unique lifts of etale algebras and of the multiplicative group. The obstruction therefore turns a purely topological question about E-infinity multiplications into a calculation with ordinary rings and schemes.

What carries the argument

Hat-delta rings (and the schemes they define): an animated ring R together with a delta_p-structure on the p-completion of R for every prime p. The structure is produced by base-changing the Tate-valued Frobenius of a Moore ring and is shown to descend along the Zariski topology.

What would settle it

Exhibit a discrete ring or scheme that admits no hat-delta structure yet whose Moore spectrum still carries an E-infinity multiplication extending the given ring structure; or, conversely, produce a hat-delta structure that does not arise from any Moore lift.

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Extended reading notes

Core claim

If a discrete scheme X admits a Moore-scheme lift to Spec S, then the structure sheaf of X carries a unique functorial hat-delta structure: each affine open is equipped with a derived Frobenius lift on every p-completion, and these lifts are compatible under localization and gluing. The assignment of a lift to its hat-delta structure is a functor on Moore schemes.

Load-bearing premise

The comparison between ordinary Z-linear maps and animated-ring maps must remain an equivalence for the 2-truncated targets that appear after base change of the Tate construction; if that comparison fails, the obstruction is no longer well-defined in the category of animated rings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper constructs an algebraic obstruction to lifting discrete schemes over Z to connective spectral schemes over the sphere spectrum S. Extending Nikolaus’s observation for rings, it shows that any Moore-scheme lift XS of a discrete scheme X induces a functorial ˆδ-scheme structure on X (compatible δ p-structures on the p-completions of the rings of sections of all affine opens). The argument proceeds by base-changing the Tate-valued Frobenius of the Moore rings, promoting the resulting Z-linear Frobenius lifts to animated-ring maps via a comparison of mapping spaces (Appendix B), and gluing via the sheaf property of ˆδ-structures established in §2.2. The obstruction is then applied to show non-liftability of rings of integers of number fields, torsion rings, Ga and GLn (n≥2) as group schemes, and certain closed subschemes of Pn, while recovering uniqueness for étale algebras and Gm.

Significance. The result supplies a uniform, purely algebraic necessary condition that applies simultaneously to rings and schemes, recovering several classical non-existence statements (S/n, rings of integers, Ga) by short calculations and producing new ones (GLn, certain projective hypersurfaces). The development of animated ˆδ-rings, their descent, and the associated notion of ˆδ-scheme is carefully written and of independent interest for spectral algebraic geometry. Strengths include the explicit functoriality of the obstruction, the concrete computations in §4, and the isolation of the technical comparison of Z-linear versus animated maps in a self-contained appendix. The work is a solid contribution that organises and extends known obstructions in a form usable for further geometric applications.

minor comments (4)
  1. [§2.1, Remark 2.1.11] Remark 2.1.11 and Warning 3.2.4 note that the formalism is specific to connective animated rings and to the pair (S,Z). A short forward-looking sentence on the expected behaviour for non-connective derived rings (or for other base pairs) would help the reader gauge the scope.
  2. [Construction 2.1.27] In Construction 2.1.27 the hat-truncation ˆτ≤n is introduced without an immediate comparison to the categorical truncation in Ringan_ˆδ; a one-line pointer to Proposition 2.1.30 would clarify the relationship.
  3. [§4.5, Proposition 4.5.1] The proof of Proposition 4.5.1 (uniqueness of the coherent ˆδ-structure on A1) works with p-adic valuations of the polynomial f; a brief remark that the same argument applies after base change to any p-adically complete ring would make the statement slightly more general.
  4. [Proof of Theorem 3.2.1] A few typographical inconsistencies appear (e.g., “Forbenius” in the proof of Theorem 3.2.1, occasional missing spaces around ˆδ). These are easily corrected in copy-editing.

Circularity Check

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No significant circularity: the obstruction is extracted from the external Tate-valued Frobenius and does not redefine its target.

full rationale

The central claim (Theorem 3.2.3) asserts that a Moore-scheme lift XS of a discrete scheme X induces a functorial ˆδ-scheme structure on X. Construction 3.2.2 obtains a Z-linear Frobenius lift φ_Z by base-changing the Tate-valued Frobenius of the E∞-ring SR (an external construction of Nikolaus–Scholze) and identifying StCpR ⊗ Z ≃ R∧p via Appendix A. The subsequent promotion of φ_Z to an animated-ring map (via Corollary B.4) is automatic for the discrete targets R∧p and R/p that arise; the comparison Map_an o Map_Z is an equivalence already in the base case of Proposition B.2. The resulting δ p-structures glue by the sheaf property of Str_δ̂ (Proposition 2.2.4) and the fact that Forget creates the relevant limits (Proposition 2.1.12). None of these steps defines the obstruction in terms of the desired ˆδ-structure, fits a parameter to the examples, or imports a uniqueness theorem from overlapping authors that forces the claim. Self-citations are limited to standard black-box background (Lurie, Bhatt–Lurie, Nikolaus–Scholze). The derivation is therefore self-contained against its external topological input and exhibits no circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper works entirely inside the standard foundations of spectral algebraic geometry (Lurie’s Higher Algebra and Spectral Algebraic Geometry) and the theory of animated δ-rings (Bhatt–Lurie). The only new entities are the ˆδ-structure and the ˆδ-scheme, both defined by explicit universal properties from already-existing δp-structures on p-completions. No numerical parameters are fitted.

assumptions (4)
  • domain assumption Existence and basic properties of the Tate-valued Frobenius on E∞-rings (Nikolaus–Scholze)
    Used as a black box in Construction 3.2.2 to produce the candidate Frobenius lift after base change to Z.
  • standard math Animated rings and the cotangent-complex formalism of Lurie
    Background for all mapping-space comparisons and for the definition of δp-structures via derived Frobenius lifts.
  • domain assumption Moore spectra are unique up to equivalence and have Tor-amplitude in [0,1]
    Appendix A; needed to identify base change of Moore rings with ordinary rings and to control Tate constructions.
  • ad hoc to paper The comparison Map_an(R,S) → Map_Z(R,S) has contractible fibres when S is 2-truncated (Corollary B.4)
    Proved in Appendix B via the cotangent complex; load-bearing for promoting the Z-linear Frobenius lift to an animated one.
invented entities (2)
  • ˆδ-ring (animated ring equipped with a δp-structure on every p-completion)
    purpose: Package the family of derived Frobenius lifts into a single algebraic object that can be sheafified.
    Defined in 2.1.1 by an explicit pullback formula; no independent existence claim beyond the obstruction theory.
  • ˆδ-scheme
    purpose: Globalise the obstruction from affine to arbitrary schemes via descent.
    Defined in 2.2.8 as a ringed space locally isomorphic to Spec of a ˆδ-ring; existence is proved by sheafification of the affine data.

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Pith. "Pith review of An obstruction to lifting schemes to spectral schemes." pith.science (2026). https://pith.science/paper/I4ZDWEEV

@misc{pith2026260711853,
  author       = {Pith},
  title        = {Pith review of: An obstruction to lifting schemes to spectral schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I4ZDWEEV}},
  note         = {Machine review of arXiv:2607.11853}
}
abstract

We develop and study an obstruction for lifting schemes over the integers to spectral schemes over the sphere spectrum. This extends a result of Nikolaus for rings, which states that a necessary condition for liftability is existence of a $\hat\delta$-structure. We prove descent properties for $\hat\delta$-rings, define $\hat\delta$-schemes, and prove an analogous statement. We then apply it to concrete examples such as number rings, closed subschemes of $\mathbb P^n$, and various group schemes.

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