Local rigid cohomology of singular points

TL;DR

This paper proves that contact equivalent singular points on schemes over a field of characteristic p have isomorphic local rigid cohomology spaces, with compatibility to Frobenius structures, highlighting invariance under contact equivalence.

Contribution

It establishes an isomorphism between local rigid cohomology spaces of contact equivalent singular points, extending understanding of their invariance properties.

Findings

Rigid cohomology spaces are isomorphic for contact equivalent singular points.

The isomorphism respects Frobenius structures.

Provides a new invariance property for local rigid cohomology.

Abstract

We show that if two singular points $x' \in X'$ and $x \in X$ on schemes over a field $k$ of characteristic $p > 0$ are contact equivalent then the rigid cohomology spaces $H^{\bullet}_{rig, \{x\}}(X)$ and $H^{\bullet}_{rig, \{x'\}}(X')$ are isomorphic. The isomorphism that we construct is moreover compatible with the Frobenius structure on rigid cohomology.