Differential forms in positive characteristic avoiding resolution of singularities

TL;DR

This paper explores sheaves of differential forms in positive characteristic, proposing new approaches that behave well on singular varieties and relate to resolution of singularities.

Contribution

It introduces two promising sheaf constructions for differential forms in positive characteristic and proves they are cdh-sheaves, aligning with K"ahler differentials on smooth varieties.

Findings

Both sheaf constructions are cdh-sheaves.

They agree with K"ahler differentials on smooth varieties.

They coincide on all varieties assuming weak resolution of singularities.

Abstract

This paper studies several notions of sheaves of differential forms that are better behaved on singular varieties than K\"ahler differentials. Our main focus lies on varieties that are defined over fields of positive characteristic. We identify two promising notions: the sheafification with respect to the cdh-topology, and right Kan extension from the subcategory of smooth varieties to the category of all varieties. Our main results are that both are cdh-sheaves and agree with K\"ahler differentials on smooth varieties. They agree on all varieties under weak resolution of singularities. A number of examples highlight the difficulties that arise with torsion forms and with alternative candiates.