Henselian schemes in positive characteristic

TL;DR

This paper explores Henselian schemes in positive characteristic, establishing foundational properties and demonstrating their suitability as algebraic analogues of tubular neighborhoods, contrasting with issues in characteristic zero.

Contribution

The paper revisits the foundations of Henselian schemes, proving their well-behaved properties in positive characteristic and addressing issues present in characteristic zero.

Findings

Henselian schemes behave well in positive characteristic.

Pathological behaviors in characteristic zero are absent in positive characteristic.

Analogues of smooth and étale maps are well-behaved in the Henselian setting.

Abstract

The global analogue of a Henselian local ring is a Henselian pair: a ring A and an ideal I which satisfy a condition resembling Hensel's lemma regarding lifting coprime factorizations of polynomials over A/I to factorizations over A. The geometric counterpart is the notion of a Henselian scheme, which is an analogue of a tubular neighborhood in algebraic geometry. In this paper we revisit the foundations of the theory of Henselian schemes. The pathological behavior of quasi-coherent sheaves on Henselian schemes in characteristic 0 makes them poor models for an "algebraic tube" in characteristic 0. We show that such problems do not arise in positive characteristic, and establish good properties for analogues of smooth and \'etale maps in the general Henselian setting.