Dagger closure in regular rings containing a field

TL;DR

This paper establishes the triviality of dagger closure in regular rings with a field and explores its relation to solid and tight closures, providing new insights into closure operations in algebraic geometry.

Contribution

It proves dagger closure is trivial in regular domains with a field and relates it to solid and tight closures, advancing understanding of closure operations.

Findings

Dagger closure is trivial in regular domains containing a field.

Graded dagger closure is trivial in polynomial rings over a field.

Dagger closure is contained in solid closure and forcing algebras are parasolid.

Abstract

We prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field. We also prove that Heitmann's full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions. Furthermore, we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid.