[Paper Review] Homotopy types and geometries below Spec Z

Yuri I. Manin, Matilde Marcolli|arXiv (Cornell University)|Jun 28, 2018

Algebraic structures and combinatorial models28 references3 citations

TL;DR

This paper develops a homotopical framework for geometries below Spec Z using stable homotopy theory and Grothendieck rings, introducing an assembler category of complex varieties with quasi-unipotent automorphisms. It lifts Frobenius-like maps σₙ and ρ̃ₙ to endofunctors on this category, realizing the Bost-Connes algebra structure via K-theory spectra and establishing a spectral Euler characteristic to Z[Q/Z].

ABSTRACT

After the first heuristic ideas about `the field of one element' F_1 and `geometry in characteristics 1' (J.~Tits, C.~Deninger, M.~Kapranov, A.~Smirnov et al.), there were developed several general approaches to the construction of `geometries below Spec Z'. Homotopy theory and the `the brave new algebra' were taking more and more important places in these developments, systematically explored by B.~Toën and M.~Vaquié, among others. This article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta--functions and generalised scissors congruences. The new version includes considerable extensions and revisions suggested by I. Zakharevich.

Motivation & Objective

• To develop a homotopical framework for 'geometries below Spec Z' using stable homotopy theory and Grothendieck rings.
• To generalize counting functions and scissors congruences to non-smooth and non-proper schemes over finite fields.
• To lift the Bost-Connes system to an equivariant Grothendieck ring and realize its structure via K-theory spectra.
• To construct an assembler category of complex quasi-projective varieties with quasi-unipotent automorphisms and define associated K-theory spectra.
• To establish a spectral Euler characteristic mapping to Z[Q/Z] that respects scissor relations and automorphism actions.

Proposed method

• Constructs a symmetric monoidal category (C, ⊗, 1) to define affine schemes as opposite categories of commutative monoids in C.
• Applies Grothendieck topologies on Aff_C to define schemes in generalized geometries, including F₁- and Z-geometries.
• Introduces an assembler category C^Z_C whose objects are complex quasi-projective varieties X with automorphisms f: X → X such that f_* is quasi-unipotent on homology.
• Endows the assembler with a Grothendieck topology generated by disjoint unions preserved by f, enabling scissor relations.
• Defines endofunctors σₙ(X,f) = (X,fⁿ) and ρ̃ₙ(X,f) = (X×Zₙ, Φₙ(f)) that lift Frobenius and Hecke-like maps to the assembler level.
• Constructs the K-theory spectrum K(C^Z_C) with π₀ = K₀^Z(V_C), and shows σₙ and ρ̃ₙ induce maps on homotopy groups recovering the Bost-Connes action.

Experimental results

Research questions

• RQ1 How can Frobenius-like dynamics be encoded in a homotopical framework for schemes over Z?
• RQ2 What is the role of scissors congruences and Grothendieck rings in generalizing zeta functions to non-smooth schemes?
• RQ3 How can the Bost-Connes system be lifted from K-theory to an equivariant or spectral level via assemblers?
• RQ4 In what way do quasi-unipotent automorphisms on homology provide a homotopical substitute for Frobenius eigenvalues?
• RQ5 Can the spectral Euler characteristic σ: K₀^Z(V_C) → Z[Q/Z] be constructed to satisfy scissor relations and compatibility with automorphisms?

Key findings

• The assembler category C^Z_C is defined on complex quasi-projective varieties with quasi-unipotent automorphisms, equipped with a Grothendieck topology generated by f-preserving disjoint unions.
• The maps σₙ and ρ̃ₙ on K₀^Z(V_C) lift to endofunctors on C^Z_C, with σₙ preserving the monoidal structure and ρ̃ₙ inducing group homomorphisms.
• The K-theory spectrum K(C^Z_C) has π₀ isomorphic to K₀^Z(V_C), and the induced maps on homotopy groups recover the Bost-Connes algebra structure.
• The spectral Euler characteristic σ: K₀^Z(V_C) → Z[Q/Z] satisfies additivity under decompositions compatible with f, ensuring consistency with scissor relations.
• The construction provides a noncommutative enrichment of the Grothendieck ring via K₀^Z(V_C), with rationalization isomorphic to a semigroup crossed product K₀^Z(V_C)⊗Q ⋊ N.
• The framework realizes the Bost-Connes system as a quotient of the K-theory of an assembler, linking arithmetic dynamics to stable homotopy theory.