2022/12/16 by Sheela Devadas, Devadas, Sheela
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2212.08644
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 étale maps in the general Henselian setting.