Smooth scheme morphisms: a fresh view

TL;DR

This paper offers a simplified and direct approach to understanding relations between formal and standard smoothness in scheme morphisms, providing elementary proofs for complex properties like local-to-global formal smoothness.

Contribution

It introduces a new perspective that simplifies the relations between formal and standard smoothness, avoiding complex machinery and making deep properties more accessible.

Findings

Simplified relations between formal and standard smoothness.

Elementary proof of local-to-global formal smoothness.

Clarification of smoothness concepts in scheme morphisms.

Abstract

Relations between some kinds of formal and standard smoothness, for morphisms of schemes, are clarified in surprisingly simple and direct ways, bypassing much of the customarily employed machinery. Even the deep local-to-global property of formal smoothness has a fairly elementary proof, under mild additional hypotheses.