Lifting zero-dimensional schemes and divided powers

TL;DR

This paper investigates divided power structures on finitely generated algebras over fields of positive characteristic, providing examples of schemes that cannot be lifted to rings of different characteristics and analyzing Frobenius neighborhoods.

Contribution

It introduces new examples of zero-dimensional Gorenstein schemes that do not lift to certain rings and examines lifting obstructions for Frobenius neighborhoods of hypersurface singularities.

Findings

Zero-dimensional Gorenstein schemes do not lift to certain noetherian rings.

Frobenius neighborhoods of hypersurface singularities lack liftings to mildly ramified rings.

Obstructions to lifting are characterized in specific algebraic contexts.

Abstract

We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.