Submission date: 12 November 2019

Abstract:

We develop some tools for analyzing dp-finite fields, including a notion of an “inflator” which generalizes the notion of a valuation/specialization on a field. For any field K, let Sub_K(K^n) denote the lattice of K-linear subspaces of K^n. An ordinary valuation on K with residue field k induces order-preserving dimension-preserving specialization maps from Sub_K(K^n) to Sub_k(k^n), satisfying certain compatibility across n. An r-inflator is a similar family of maps {Sub_K(K^n) → Sub_k(k^{rn})}_{n in ℕ} scaling dimensions by r. We show that 1-inflators are equivalent to valuations, and that r-inflators naturally arise in fields of dp-rank r. This machinery was “behind the scenes” in &10 of [10]. We rework &10 of [10] using the machinery of r-inflators.

Mathematics Subject Classification: 03C45, 13A18

Keywords and phrases:

Full text arXiv 1911.04727: pdf, ps.