Grothendieck topologies and ideal closure operations

TL;DR

This paper explores how Grothendieck topologies on affine schemes induce and relate to various ideal closure operations, linking topological and algebraic structures through cohomology and global sections.

Contribution

It establishes a systematic connection between Grothendieck topologies and ideal closure operations, providing a framework to construct topologies from closure properties and vice versa.

Findings

Relates radical to surjective and constructible topologies

Connects integral closure with submersive, proper, and Voevodsky's h-topologies

Links Frobenius closure to Frobenius topology and plus closure to finite topology

Abstract

We relate closure operations for ideals and for submodules to non-flat Grothendieck topologies. We show how a Grothendieck topology on an affine scheme induces a closure operation in a natural way, and how to construct for a given closure operation fulfilling certain properties a Grothendieck topology which induces this operation. In this way we relate the radical to the surjective topology and the constructible topology, the integral closure to the submersive topology, to the proper topology and to Voevodsky's h-topology, the Frobenius closure to the Frobenius topology and the plus closure to the finite topology. The topologies which are induced by a Zariski filter yield the closure operations which are studied under the name of hereditary torsion theories. The Grothendieck topologies enrich the corresponding closure operation by providing cohomology theories, rings of global sections, concepts of exactness and of stalks.