Dp-finite fields III: inflators and directories

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 ${\mathrm{Sub}}_K(K^n)$ 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 ${\mathrm{Sub}}_K(K^n)$ to ${\mathrm{Sub}}_k(k^n)$, satisfying certain compatibility across $n$. An $r$-inflator is a similar family of maps $\{{\mathrm{Sub}}_K(K^n)\to {\mathrm{Sub}}_k(k^{rn}){\}}_{n\in \mathbb{N}}$ 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.

Preliminary draft, comments welcome

References in corpus (2)

• On dp-minimal fields
• Dp-finite fields II: the canonical topology and its relation to henselianity

Cited by in corpus (1)

• Dp-finite fields V: topological fields of finite weight