Liftability of singularities and their Frobenius morphism modulo $p^2$

TL;DR

This paper studies the conditions under which singular schemes can be lifted over the ring of Witt vectors of length two, focusing on Frobenius morphism liftability, and explores implications for crystalline cohomology and singularity classifications.

Contribution

It proves constructibility of liftable schemes in families, constructs explicit liftings for Frobenius split schemes, and analyzes liftability of Frobenius morphisms for various singularities.

Findings

W_2(k)-liftability is constructible in flat families.

Explicit W_2(k)-liftings of Frobenius split schemes are constructed.

Certain singularities, like canonical surface singularities, are Frobenius liftable.

Abstract

We investigate the $W_2(k)$-liftability of singular schemes. We prove constructibility of the locus of $W_2(k)$-liftable schemes in a flat family $X \to S$. Moreover, we construct an explicit $W_2(k)$-lifting of a Frobenius split scheme $X$ over a perfect field $k$, reproving Bhatt's existential result. Furthermore, we study existence of liftings of the Frobenius morphism. In particular, we prove that in dimension $n \geq 4$ ordinary double points do not admit a $W_2(k)$-lifting compatible with Frobenius, and that canonical surface singularities are Frobenius liftable. Combined with Bhatt's results, the latter result implies that the crystalline cohomology groups over $k$ of surfaces with canonical singularities are not finite dimensional. As a corollary of our results, we provide a thorough comparison between the notions of $W_2(k)$-liftability, Frobenius liftability and classical $F$-singularity types.