Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras

2025/05/16 by Yuki Kato, Kato, Yuki
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Logic, programming, and type systems

paper · pdf · doi:10.48550/arxiv.2505.10809

Abstract

In this paper, we prove tilting equivalence for the finite almost derived algebraic cobordism spectrum dMGL^{a,\rm fin} of perfectoid algebras. More precisely, if V is an integral perfectoid valuation ring and A is an integral perfectoid V-algebra, then the tilting functor induces a weak equivalence dMGL^{a,\rm fin}(A) ≃ dMGL^{a,\rm fin}(A^\flat). This invariant is a finite syntomic, derived, and non-\mathbbA¹-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed characteristic bases. To prove the result, we first establish the corresponding finite non-unital statement and isolate a form of excisive approximation for pointed ∞-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with the framework of Heuts. We also define approximation functors along natural transformations and apply them to the comparison between periodic algebraic cobordism and homotopy K-theory, obtaining Bott periodicity and Gabber rigidity.

Citations

• Almost mathematics of pointed symmetric monoidal model categories by Smith ideal theory
• Finite syntomic topology and algebraic cobordism of non-unital algebras
• Bivariant derived algebraic cobordism

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