Hasse-Schmidt derivations and locally trivial deformations in positive characteristic

TL;DR

The paper investigates lifting problems for Hasse–Schmidt derivations and locally trivial deformations in positive characteristic. It develops an obstruction-theoretic framework, reducing general obstructions to a $p$-power decomposition and proving a decomposition theorem that streamlines extension problems. It then applies this theory to locally trivial deformations, describing fibers of lifting spaces via non-abelian Čech cohomology and obstruction maps. Finally, it constructs an explicit singular curve whose locally trivial deformation functor does not satisfy Schlessinger’s (H1), illustrating that positive characteristic can preclude standard pro-representability phenomena, while also highlighting tame aspects in affine or certain geometric settings.

Abstract

In this paper, we first study the lifting problem of Hasse-Schmidt derivations and then apply the results to the theory of locally trivial deformations of algebraic schemes in positive characteristic. As an application, we construct an algebraic curve whose locally trivial deformation functor does not satisfy Schlessinger's condition (H1).

Theorems & Definitions (122)

• Theorem 1.1: = Theorem \ref{['counterexample theorem']}
• Theorem 1.2: = Corollary \ref{['affine cor']}
• Theorem 1.3: = Theorem \ref{['decomposition thm']}
• Theorem 1.4: = Theorem \ref{['lt main thm']}
• Definition 2.1: = Mac
• Definition 2.2: = Mac
• Lemma 2.3
• proof
• Lemma 2.4
• proof