Tight closure and plus closure in dimension two

TL;DR

This paper proves that in a two-dimensional N-graded domain over an algebraically closed finite field, the tight closure and graded plus closure of a homogeneous ideal are the same, resolving a longstanding question.

Contribution

It establishes the equivalence of tight closure and plus closure in a specific algebraic setting, answering Hochster's question.

Findings

Tight closure equals plus closure in the given setting

The result applies to two-dimensional N-graded domains over algebraically closed finite fields

Provides a definitive answer to a longstanding open problem

Abstract

We prove that the tight closure and the graded plus closure of a homogeneous ideal coincide for a two-dimensional N-graded domain of finite type over the algebraic closure of a finite field. This answers in this case a ``tantalizing question'' of Hochster.