The complexity of tilting
Abstract
Many explicit constructions in algebra are not first-order definable in classical finitary logic, but are, however, definable in infinitary logic. The aim of this paper is to show how infinitary logic can be used to lift model-theoretic results in arithmetic to a quantifier-sensitive form. We apply this idea to a deep result of Jahnke and Kartas on the model theory of perfectoid fields. Our main result is that a first-order sentence ϕ holds in the tilt k♭ of a perfectoid field k if and only if a certain sentence ϕnatural holds in a non-principal ultrapower of k, where ϕnatural does not depend on k and is only slightly more complicated than ϕ itself. In particular, tilting preserves the Lévy hierarchy in a precise sense. To this end, we prove that the Standard Decomposition of a valued field at a constant t is infinitarily interpretable by formulas of rank at most 1.
Related
- From τ-tilting to τ2-tilting via splitting tilting triples
- Free-Monodromic Mixed Tilting Sheaves on Flag Varieties
- Self-Orthogonal τ-Tilting Modules and Tilting Modules III: Finitistic Dimension
- Self-Orthogonal τ-Tilting Modules and Tilting Modules II: Annihilator Separation
- Tilting theory on continuous quivers (I) maximal rigid and tilting subcategories
BibTeX
@misc{gitin2026complexity,
title = {{The complexity of tilting}},
author = {Leo Gitin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41064},
eprint = {2610.41064},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41064},
}
RIS
TY - UNPB
TI - The complexity of tilting
AU - Leo Gitin
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.41064
UR - https://arxiv.org/abs/2610.41064
AB - Many explicit constructions in algebra are not first-order definable in classical finitary logic, but are, however, definable in infinitary logic. The aim of this paper is to show how infinitary logic can be used to lift model-theoretic results in arithmetic to a quantifier-sensitive form. We apply this idea to a deep result of Jahnke and Kartas on the model theory of perfectoid fields. Our main result is that a first-order sentence ϕ holds in the tilt k^♭ of a perfectoid field k if and only if a certain sentence ϕ^natural holds in a non-principal ultrapower of k, where ϕ^natural does not depend on k and is only slightly more complicated than ϕ itself. In particular, tilting preserves the Lévy hierarchy in a precise sense. To this end, we prove that the Standard Decomposition of a valued field at a constant t is infinitarily interpretable by formulas of rank at most 1.
ER -